AP Calc by Steven Budd

AP Calculus AB S. Budd Lamar High School pdf September 29, 2010

Mr. Budd, compiled September 29, 2010

Contents 1 Area and Slope 1.1 Using Graphs to Multiply: Definite Integrals . . . . . . . . 1.1.1 What is Calculus? . . . . . . . . . . . . . . . . . . . 1.1.2 What is a Definite Integral? . . . . . . . . . . . . . . 1.1.3 Approximating Area: Counting Squares . . . . . . . 1.1.4 What is â&#x20AC;&#x153;Signedâ&#x20AC;? Area . . . . . . . . . . . . . . . . 1.1.5 Using Known Shapes to Evaluate Definite Integrals . 1.1.6 Using Symmetry . . . . . . . . . . . . . . . . . . . . 1.2 Approximating Definite Integrals from Tables . . . . . . . . 1.2.1 Using Tables of Data . . . . . . . . . . . . . . . . . . 1.2.2 Rectangular Approximation Method (RAM) . . . . 1.2.3 Trapezoidal Approximation: Quasi-RAM . . . . . . 1.2.4 Streamlining Calculations for Equal Widths . . . . . 1.2.5 Finding a Range of Values . . . . . . . . . . . . . . . 1.2.6 Midpoint RAM . . . . . . . . . . . . . . . . . . . . . 1.2.7 Unequal Subdivisions . . . . . . . . . . . . . . . . . 1.3 Approximating Definite Integral from Formulas . . . . . . . 1.3.1 Definite Integrals from Known Shapes . . . . . . . . 1.3.2 Approximating Definite Integrals . . . . . . . . . . . 1.3.3 Using Symmetry . . . . . . . . . . . . . . . . . . . . 1.4 Slope and Rate of Change . . . . . . . . . . . . . . . . . . . 1.4.1 Instantaneous Rate of Change . . . . . . . . . . . . . 1.4.2 Definition and Notation . . . . . . . . . . . . . . . . 1.4.3 Approximating Derivatives from Tabular Data . . . 1.5 IROC as a limit . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Approximating Rate of Change from a Formula . . . 1.5.2 Kinematics: Displacement, Velocity, Acceleration . . 1.6 Slope and Area: Pulling It Together . . . . . . . . . . . . .

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1 3 3 4 5 6 6 8 13 13 17 18 19 20 21 23 31 31 32 34 37 37 38 39 47 47 48 53

2 Limits 2.1 Introduction to Limits . . . . . . . . . . 2.1.1 Graphic Introduction to Limits . 2.1.2 Step Discontinuities & One-Sided 2.1.3 Limits from a Table . . . . . . .

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55 57 57 60 61

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CONTENTS

2.2

2.3

2.4

2.1.4 Limits from an Expression . . . . . . . 2.1.5 Substitution and Properties of Limits Limits at Cancelable Discontinuities . . . . . 2.2.1 Limits at Cancelable Discontinuities . 2.2.2 De-rationalizing with Conjugates . . . 2.2.3 Derivative at a Point . . . . . . . . . . 2.2.4 De-denominatorizing with LCDs . . . Limit Definition of Derivative as a Function . 2.3.1 Derivative as a Function . . . . . . . . 2.3.2 Tangent Lines . . . . . . . . . . . . . . Basic Calculus of Polynomials . . . . . . . . . 2.4.1 Notation . . . . . . . . . . . . . . . . . 2.4.2 Basic Properties of Derivatives . . . . 2.4.3 Power Rule . . . . . . . . . . . . . . . 2.4.4 Higher Order Derivatives . . . . . . . 2.4.5 Kinematics . . . . . . . . . . . . . . .

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62 63 67 67 68 69 71 75 75 77 81 81 82 83 84 84

3 Basic Differentiation 3.1 Antidifferentiation of Polynomials . . . . 3.1.1 Notation of Antiderivatives . . . 3.1.2 Anti-Power Rule . . . . . . . . . 3.1.3 Kinematics . . . . . . . . . . . . 3.1.4 General vs. Particular Solutions 3.2 Product and Quotient Rules . . . . . . . 3.2.1 Product Rule . . . . . . . . . . . 3.2.2 Quotient Rule . . . . . . . . . . . 3.3 Chain Rule . . . . . . . . . . . . . . . . 3.3.1 Chain Rule . . . . . . . . . . . . 3.4 Tangent Lines . . . . . . . . . . . . . . . 3.4.1 Tangent Lines . . . . . . . . . . . 3.4.2 Horizontal Tangents . . . . . . . 3.4.3 Vertical Tangents . . . . . . . . . 3.4.4 Normal Lines . . . . . . . . . . . 3.4.5 Tangent Line Approximations . . 3.4.6 Introduction to Slope Fields . . .

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89 91 91 91 93 94 99 99 102 107 107 113 113 114 115 116 116 118

4 Curve Sketching 4.1 Relating Graphs of f and f 0 4.1.1 Relative Extrema . . 4.1.2 First Derivative Test 4.2 Second Derivative Sketching 4.2.1 Concavity . . . . . . 4.2.2 Points of Inflection .

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123 125 125 126 135 135 137

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CONTENTS

5 Trigonometrics 5.1 Differentiation of Trigonometric Functions . . . . . . . 5.1.1 Special Limits . . . . . . . . . . . . . . . . . . . 5.1.2 Trigonometric Derivatives . . . . . . . . . . . . 5.2 Implicit Differentiation . . . . . . . . . . . . . . . . . . 5.2.1 Implicit Differentiation . . . . . . . . . . . . . . 5.3 Inverse Functions . . . . . . . . . . . . . . . . . . . . . 5.3.1 Inverse Functions . . . . . . . . . . . . . . . . . 5.3.2 Differentiating Inverse Functions . . . . . . . . 5.4 Related Rates (Triangles) . . . . . . . . . . . . . . . . 5.4.1 Introduction to Related Rates . . . . . . . . . . 5.4.2 Related Rates w/ Triangles . . . . . . . . . . . 5.5 Antidifferentiating Trig . . . . . . . . . . . . . . . . . 5.5.1 Antidifferentiating to Inverse Functions . . . . 5.5.2 Antidifferentiation of Trigonometric Functions

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141 143 143 144 149 149 153 153 155 157 157 158 161 161 162

6 Exponentials 6.1 Antidifferentiation by Simplification . . . . . . . . . 6.1.1 u-Simplification . . . . . . . . . . . . . . . . . 6.1.2 Simplification with Trigonometrics Inside . . 6.1.3 Simplification with Trigonometrics Outside . 6.2 The Happy Function . . . . . . . . . . . . . . . . . . 6.2.1 Differentiating the Exponential Function . . . 6.2.2 Antidifferentiating the Exponential Function 6.2.3 Skippable u-Simplification . . . . . . . . . . . 6.3 Inverse of the Happy Function . . . . . . . . . . . . . 6.3.1 Inverse of the Exponential Function . . . . . 6.3.2 Implicit Differentiation with ln . . . . . . . . 6.3.3 Antidifferentiating Reciprocals . . . . . . . . 6.3.4 Antidifferentiating Fractions . . . . . . . . . . 6.4 Separable Differential Equations . . . . . . . . . . . 6.4.1 Separable Differential Equations . . . . . . . 6.4.2 Separable Differential Equations with Logs . 6.5 Exponential Growth and Decay . . . . . . . . . . . . 6.5.1 Proportional Growth . . . . . . . . . . . . . . 6.5.2 Other Applications of Differential Equations .

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165 167 167 168 169 173 173 175 176 179 179 181 182 184 187 187 188 191 191 192

7 Existence Theorems 7.1 Quasi-Limits: One-Sided and Infinite . . . . . . 7.1.1 Step Discontinuities & One-Sided Limits 7.1.2 One-Sided Derivatives . . . . . . . . . . 7.1.3 Infinite Limits . . . . . . . . . . . . . . 7.2 Limits at Infinity and Horizontal Asymptotes . 7.2.1 Limits at Infinity . . . . . . . . . . . . . 7.2.2 Horizontal Asymptotes . . . . . . . . . . 7.3 Continuity and Differentiability . . . . . . . . .

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195 197 197 198 200 205 205 207 209

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Mr. Budd, compiled September 29, 2010

CONTENTS 7.4

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213 213 213 214 219 219 219 227 227 227 228 229 231

8 Integral Theorems 8.1 MVT for Integrals . . . . . . . . . . . . . . . . . . . . 8.1.1 Substitution of Variables . . . . . . . . . . . . . 8.1.2 Properties of Definite Integrals . . . . . . . . . 8.1.3 Average Value . . . . . . . . . . . . . . . . . . 8.1.4 Mean Value Theorem for Integrals . . . . . . . 8.2 Accumulation Functions . . . . . . . . . . . . . . . . . 8.2.1 Accumulation Functions . . . . . . . . . . . . . 8.2.2 Fundamental Theorem of Calculus, part II . . . 8.2.3 Curve Sketching with Accumulation Functions 8.3 Quick, Cheap Antiderivatives . . . . . . . . . . . . . . 8.3.1 Creating Quick, Cheap Antiderivatives . . . . .

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235 237 237 238 238 241 247 247 247 250 257 257

9 Area and Volume 9.1 More Definite Integrals . . . . . 9.1.1 Definite Integral . . . . 9.1.2 Area: Slicing dx . . . . 9.1.3 Total Distance . . . . . 9.1.4 Other Applications . . . 9.2 Area . . . . . . . . . . . . . . . 9.2.1 High and Low y Switch 9.2.2 Area: Slicing dy . . . . 9.2.3 Total Distance . . . . . 9.3 Volume . . . . . . . . . . . . . 9.3.1 Volumes of Rotation . . 9.4 Volume: Slicing with Washers . 9.4.1 Slicing with Washers . . 9.5 Non-Circular Slicing . . . . . . 9.6 Area and Volume . . . . . . . . 9.7 Related Rates with Volume . . 9.7.1 Volume problems . . . .

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261 263 263 263 264 266 271 271 272 272 277 277 281 281 285 289 295 295

7.5

7.6

Some Basic Calculus Theorems . . . . . . . 7.4.1 Intermediate Value Theorem . . . . 7.4.2 Extreme Value Theorem . . . . . . . 7.4.3 Rolleâ&#x20AC;&#x2122;s Theorem . . . . . . . . . . . Mean Value Theorem . . . . . . . . . . . . . 7.5.1 Average Rate of Change . . . . . . . 7.5.2 Mean Value Theorem . . . . . . . . Riemann Sums . . . . . . . . . . . . . . . . 7.6.1 Sigma Notation . . . . . . . . . . . . 7.6.2 Riemann Sums . . . . . . . . . . . . 7.6.3 Evaluating Definite Integrals Exactly 7.6.4 Proof . . . . . . . . . . . . . . . . . 7.6.5 Evaluating Definite Integrals . . . .

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Mr. Budd, compiled September 29, 2010

CONTENTS

vii

10 Extrema and Optimization 10.1 Absolute Extrema . . . . . . . . . . . . . . . 10.1.1 Absolute Extrema . . . . . . . . . . . 10.1.2 Absolute Extrema from the Derivative 10.1.3 Optimization . . . . . . . . . . . . . . 10.2 First Derivative . . . . . . . . . . . . . . . . . 10.2.1 First Derivative Test . . . . . . . . . . 10.3 Second Derivative . . . . . . . . . . . . . . . . 10.3.1 Concavity . . . . . . . . . . . . . . . . 10.3.2 Second Derivative Test . . . . . . . . .

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299 301 301 302 304 309 309 311 311 313

11 Review 11.1 Separable Differential Equations . . . . . . . . . . . 11.1.1 Separable Differential Equations . . . . . . . 11.1.2 Separable Differential Equations with Logs . 11.2 Exponential Growth and Decay . . . . . . . . . . . . 11.2.1 Proportional Growth . . . . . . . . . . . . . . 11.2.2 Other Applications of Differential Equations . 11.3 Related Rates . . . . . . . . . . . . . . . . . . . . . . 11.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Integral as Accumulator . . . . . . . . . . . . . . . . 11.6 Particle Motion . . . . . . . . . . . . . . . . . . . . . 11.7 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Extrema and Optimization . . . . . . . . . . . . . . 11.9 Implicit Differentiation . . . . . . . . . . . . . . . . . 11.10Differential Equations Again . . . . . . . . . . . . . . 11.11Related Rates Again . . . . . . . . . . . . . . . . . . 11.12Area and Volume . . . . . . . . . . . . . . . . . . . . 11.13Tangent Lines . . . . . . . . . . . . . . . . . . . . . . 11.14Miscellany . . . . . . . . . . . . . . . . . . . . . . . . 11.15More Miscellany . . . . . . . . . . . . . . . . . . . .

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319 321 321 323 327 327 328 331 337 345 351 357 363 367 371 377 381 387 393 405

12 Makeup 12.1 MU: 12.2 MU: 12.3 MU: 12.4 MU: 12.5 MU: 12.6 MU: 12.7 MU: 12.8 MU: 12.9 MU: 12.10MU:

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407 408 409 410 417 419 423 424 428 429 435

Differential Equations . . . Related Rates . . . . . . . . Graphs . . . . . . . . . . . . Integral as Accumulator . . Linear Motion . . . . . . . . Data . . . . . . . . . . . . . Extrema and Optimization Implicit Differentiation . . . Area and Volume . . . . . . Tangent Lines . . . . . . . .

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Mr. Budd, compiled September 29, 2010

viii

CONTENTS

Mr. Budd, compiled September 29, 2010

Unit 1

Introduction to Calculus: Area and Slope 1. The Definite Integral as Area 2. Approximating Definite Integral by Riemann Slicing 3. Rate of Change 4. Approximating Rate of Change from Graph, Table, or Equation 5. Slope and Area: Pulling It Together

Advanced Placement Concept of the derivative. • Derivative presented geometrically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. Interpretations and properties of definite integrals. • Computation of Riemann sums using left, right, and midpoint evaluation points. • Basic properties of definite integrals (Examples include additivity and linearity.) 1

AP Unit 1 (Area and Slope)

Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals by functions represented algebraically, geometrically, and by tables of values. International Baccalaureate (MM 8.6) The estimation of the numerical value of a definite integral using the trapezium rule. Included: an appreciation of the effect of doubling the number of sub-intervals.

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 1: Using Graphs to Multiply: Definite Integrals

1.1

Using Graphs to Multiply: Definite Integrals

Advanced Placement Interpretations and properties of definite integrals. • Basic properties of definite integrals (Examples include additivity and linearity.) Numerical approximations to definite integrals. Textbook We won’t be following the book too closely the first couple weeks, so that there is limited correspondence to a section in the book. The closest section in content would be §4.3 Area or §4.4 The Definite Integral. [16] Resources §5.1 Areas and Integrals in Ostebee and Zorn [17]. §1-3 One Type of Integral of a Function in Foerster [10]. Explorations 1-3a:“Introduction to Definite Integrals” and 1-4a:“Definite Integrals by Trapezoidal Rule” in [9].

1.1.1

What is Calculus?

Ostebee and Zorn describe the focus of calculus as follows: “The tangent-line problem and the area problem are the two main geometric problems of calculus.” [17] The tangent-line problem is an issue of slope, so that our main concerns in calculus are slope and area. Our interest in slope and area is not purely geometric, however. Slope is our codeword for rate of change, which can be rate of people entering AstroWorld, or the rate at which oil leaves a gash in an oil tanker. Likewise, area can be a whole range of things from people who have entered Super Happy Fun Land in a six-hour time period to the amount of oil that has bled out of a shipwrecked tanker. Area, as we shall see, can even be used to represent the volume of an object. Foerster, whose materials we will see much of this year, describes calculus as consisting of four things: limits, derivatives, integrals, and integrals [10]. These things probably have no meaning for you, and you are probably wondering why integrals is listed twice. Understanding these things is what we will seek to do over the next nine months. As a brief introduction, I will tell you that derivatives are related to slope, and one of the integrals (definite integrals) is related to area. Both derivatives and definite integrals are limits, and the other type of integral (indefinite) is also related to both derivatives and indefinite integrals. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

1.1.2

What is a Definite Integral?

Example 1.1.1 Begin with Exploration 1-3: Introduction to Definite Integrals.

Figure 1.1: Velocity, v(t), as a function of the number of seconds, t, since you started slowing. [9]

Distance is Velocity times Time. Looking at the graph in Figure 1.1, we can see that after 30 seconds, the velocity is basically constant at 60 feet per second. If we were to find the distance traveled between 30 and 50 seconds, we would multiply 20 seconds by 60 feet per second, yielding 1200 feet. Notice that the same product (20 seconds Ă&#x2014; 60 feet per second) is represented by the area under the curve of v(t) from 30 seconds to 50 seconds. The shape of this region is a rectangle, and the formula for finding the area of a rectangle is A = l Ă&#x2014; w. If you look at the region under the curve of v(t) from 30 seconds to 50 seconds, you should notice that it is a rectangle with width of 20 seconds and height of 60 feet per second. The area of this rectangle is 1200 feet, or 20 seconds Ă&#x2014; 60 feet per second. The distance traveled from a starting time to a stopping time is the area under the velocity curve between the two times. This area is called the definite integral. This is true for the simple case between 30 and 50 seconds, but it is also true for the less simple case between 0 and 20 seconds. To find the distance traveled between 0 and 20 seconds, we would need to multiply the time difference (20 seconds) by the velocity, except this is not so straightforward because the velocity is changing with time. So instead of finding the product by multiplying two numbers, I need a different approach. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 1: Using Graphs to Multiply: Definite Integrals

When we multiplied two constants, we were doing the same arithmetic as finding the area of a simple rectangle of constant height. In the less straightforward case of multiplying a varying v(t) by t, the distance is still the area under the curve of v(t) from the starting time to the ending time. But since we don’t have a simple formula (because we don’t have a simple shape) we need a different approach to finding the area. Definition 1.1 (Definite Integral). The definite integral of the function f from Rb x = a to x = b [written a f (x) dx] gives a way to find the product of (b − a) and f (x), even if f (x) is not a constant. [10] This definition of the definite integral tells us why we need the definite integral. It doesn’t tell us how to get it. Here’s a geometric explanation from a different author: Definition 1.2 (The Integral as Signed Area). Let f be a function defined for a ≤ x ≤ b. Z b f (x) dx a

denotes the signed area bounded by x = a, x = b, y = f (x), and the x-axis. [17]

1.1.3

Approximating Area: Counting Squares

Rb If f (x) is always positive, and a is less than b, then a f (x) dx is the area under f between a and b. We will talk more about the qualifier “signed” in a bit. One way to approximate an area is the “Counting Squares” approach.[10] This approach is fairly basic; you superimpose a grid on your graph and lightly shade the area you are approximating (or imagine the shading in your head). You count the number of whole squares that are shaded. Sometimes I like to count all the whole squares in vertical strips, and this approach will be helpful to us in the future. Then you count the partial squares, rounding the shaded portion of each square to the nearest tenth, 0.1. After you have added the total number of whole and partial squares, you multiply the number of squares by the area of each square, being conscious of your units. The number of squares multiplied by the area of each square is the area of the shape. Looking at the example Figure 1.1, we can see that the area under the curve from 30 to 50 seconds gives 24 squares. At 50 feet per square (10 feet per second times 5 seconds), that gives a displacement of 1200 feet. How many squares are between 0 and 20 seconds? Therefore what is the change in your position in the first twenty seconds? Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

1.1.4

What is “Signed” Area

To find the definite integral of f (x) from x = a to x = b, you are basically looking for the signed area under the curve of f between a and b. What do we mean by signed area? If the area is above the x-axis, it is counted as positive; if the area is below the x-axis it is counted as negative (i.e., a negative amount above the axis). Think about why this is important. What’s happening when the velocity is negative? If the velocity were negative, how should you count the area/ distance? [Another way to make the signed area negative is if b is less than a, so that taking you from a to b means that you go right to left on the graph, i.e., the change in x, ∆x ≈ dx, is negative.] Figure 1.2: [10]

Example 1.1.2 The graph in Figure 1.2 shows v(t) centimeters per second as a function of t seconds after an object starts moving. At what time does the object change direction? How far is the object from its starting point when t = 9 sec? What is the total distance traveled by the object? [adapted from [10]] [Ans: 5 sec, 7.1 cm] Note the distinction between displacement and total distance.

1.1.5

Using Known Shapes to Evaluate Definite Integrals

Example 1.1.3 Several areas are shown in Figure 1.3, labeled as integrals. Use familiar area formulas to evaluate each integral. [17] Ans: 6; k (b − a); 94 π + 32 Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 1: Using Graphs to Multiply: Definite Integrals

Figure 1.3: [17]

Example 1.1.4 (adapted from AB ’03) Let f be a function defined

Figure 1.4: From 2003 AP Calculus AB exam

on the closed interval −3 ≤ x ≤ 4. The graph of f 0 , a function that is related to f , but different from f , known as the derivative of f , consists of one line segment and a semicircle, as shown in Figure 1.4 Find (a)

(b)

−3

f 0 (x) dx

Ans: - 32 ; −8 + 2π Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

1.1.6

Using Symmetry

Example 1.1.5 (adapted from AB ’01) A car is traveling on a straight road with velocity 40 ft/sec at time t = 0. For 0 ≤ t ≤ 18 seconds, the car’s acceleration a(t), in ft/sec2 , is the piecewise linear function defined by the graph in Figure 1.5. Figure 1.5: A car’s acceleration

(a) How fast is the car going at time t = 0? (b) How much does the car’s velocity increase during the first second? During the first two seconds? (c) What is the car’s velocity at time t = 2? At time t = 6? (d) What change takes place to the car’s velocity at time t = 6? (e) At what time does the velocity of the car return to 40 ft/sec?

[Ans: 40 (ft/sec); 15, 30 (ft/sec); 70, 100 (ft/sec); v decreases; 12 s]

Problems big giant blue-green Calculus book p. 380: Writing Exercises # 1,2; # 41-44 1.A-1 The online supplement for AP Calculus AB during the academic year has been migrated to Lamar’s new Moodle site, so take the following steps to enroll. Go to http://moodle.houstonisd.org/lamarhs/ and follow the instructions for creating a new account. You will then need to search for AP Calculus AB to enroll. When asked for it, the enrollment key for this class will be area for Mr. Budd’s class, or thompson# for Mr. Thompson’s class, where # represents the period. This is a vital online supplement to what happens in class. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 1: Using Graphs to Multiply: Definite Integrals

Figure 1.6: [10]

1.A-2 In Figure 1.6, a car is slowing down from a speed of v = 60 ft/sec. Estimate the distance it goes from time t = 5 sec to t = 25 sec by finding the definite integral. [10] [Ans: about 680 feet] 1.A-3 In Figure 1.7, a car speeds up slowly from v = 55 mi/hr during a long trip. Figure 1.7: [10]

Estimate the distance it goes from time t = 0 hr to t = 4 hr by finding the definite integral. [10] [Ans: about 266 miles] 1.A-4 In the previous two problems, you found a distance using a definite integral. Suppose you use the formula d = vt, rearranged to v = dt . What velocities do you get in the previous two problems when you divide the distance by the change in time? What do you think this represents? 1.A-5 The rate at which people enter an amusement park on a given day is modeled by the function E of time t. E(t) is measured in people per hour and time t is measured in hours after midnight. When the parkR opens at 9 17 a.m., there are no people in the park. Explain the meaning of 9 E(t) dt. Is this equal to the number of people in the park? Why or why not? [Ans: The number of people who entered the park by 5 p.m.; no] 1.A-6 A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. If x represents the distance from one end of the blood Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) vessel, and B(x) is a function that represents the diameter at that point, 2 R 360 B(x) then using correct units explain the meaning of 0 π dx, and 2 2 R 275 B(x) dx. π 125 2

1.A-7 Let g be the function shown graphically in Figure 1.8. When asked to Figure 1.8: Graph of g [17]

R2 estimate 1 g(x) dx, a group of calculus students submitted the following answers: −4, 4, 45, and 450. Only one of these responses is reasonable; the others are “obviously” incorrect. Which is the reasonable one? [17] [Ans: 45] 1.A-8 The graph of a function f is shown in Figure 1.9. [Adapted from [17]] Figure 1.9: Graph of f [17]

R6 (a) Which of the following is the best estimate of 1 f (x) dx: −24, 9, 20, 38? Justify your answer. R8 (b) 6 f (x) dx ≈ 4. Does this approximation overestimate or underestimate the exact value of the integral? Justify your answer. R7 (c) Explain a quick way to tell that 12 ≤ 3 f (x) dx. [Ans: 20; underestimate] Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 1: Using Graphs to Multiply: Definite Integrals

Figure 1.10: Graph of f [17]

1.A-9 The graph of a function f (shown in Figure 1.10) consists of two straight lines and two one-quarter circles. Evaluate each of the following integrals. R2 (a) 0 f (x)dx R5 (b) 2 f (x)dx R5 (c) 0 f (x)dx R9 (d) 5 f (x)dx R4 (e) 4 f (x)dx R 15 (f) 0 f (x)dx R 15 (g) 0 |f (x)| dx Ans: 4,

9π 4 ,

−4π, 0, −8 −

7π 4 ,

16 +

25π 4

1.A-10 Suppose Mr. Budd is driving to the Utah Shakespearean Festival in Cedar City, UT. Once he gets on the road, he sets his cruise control for 55 mph. Let t be the number of hours since he started driving on cruise control. (a) How far has he gone during the first half hour on cruise control? the first hour? the first two hours? (b) Write an equation for the velocity, i.e., v(t) =(something). (c) Graph the velocity versus time. (d) Find the area under the curve of v(t) from t = 0 to t = 0.5. Also, find the area from t = 0 to t = 1 and also to t = 2. (e) What shape are these areas in? If I look at the area from t = 0 to t = tstop , what is the width of the figure (as an expression with tstop in it)? the height? the area (as an expression of tstop )? Call your expression for area A(tstop ). (f) Plot a graph of distance traveled versus time. Use the points (0.5, distance for 0.5), (1, distance for 1), and (2, distance for 2). Look for a pattern, and use your result for A(tstop ) to connect the dots. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) (g) On your graph of distance versus time, what is the slope at t = 0.5? at t = 1? at t = 2? Indicate units.

1.A-11 (from Explorations 1-3a [9]) As you drive on the highway you accelerate to 100 feet per second to pass a truck. After you have passed, you slow down to a more moderate speed. Table 1.1 shows your velocity, v(t), as a function of the number of seconds, t, since you started slowing. Table 1.1: Your velocity after passing a truck t (s) 0 5 10

v(t) ft/s 100 77.3755 67.5477

(a) How fast are you going at t = 0? How fast are you going at t = 5? Why is it not so straightforward to ask how fast you were going for 0 ≤ t ≤ 5? (b) If your speed is constantly decreasing, give an upper estimate of how far you traveled in the first 5 seconds. Give a lower estimate of your displacement in the first 5 seconds. (c) If you had to give one number for your distance in feet for the first 5 seconds, what might it be? Give a reason for how you obtained your answer. (d) How far did you travel for 5 ≤ t ≤ 10? For 0 ≤ t ≤ 10? [Ans: 100, 77.3755, ; 500, 386.878 (ft); 443.439 ft; 362.308, 805.747 ft] 1.A-12 Read the handout “How to Succeed in Calculus.” (a) Give examples of three things on the list that you already do. (b) Name one thing on the list that you will try to improve this year. Describe specifically what actions you will take this week. (c) Submit your answer in the appropriate place on the moodle site. 1.A-13 Start Exploration 1-4a: “Definite Integrals by Trapezoidal Rule”; do problems 1 through 3.

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

1.2

Approximating Definite Integrals from Tables

Advanced Placement Interpretations and properties of definite integrals. • Computation of Riemann sums using left, right, and midpoint evaluation points.

Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals by functions represented algebraically, geometrically, and by tables of values. Textbook §4.7 Numerical Integration [16] Resources §5.1 Areas and Integrals in Ostebee and Zorn [17]. §1-4 Definite Integrals by Trapezoids, from Equations and Data in Foerster [10]. Exploration 1-4a:“Definite Integrals by Trapezoidal Rule” in [9].

1.2.1

Using Tables of Data

Example 1.2.1 (from Explorations 1-3 [9]) As you drive on the highway you accelerate to 100 feet per second to pass a truck. After you have passed, you slow down to a more moderate speed. Table 1.2 shows your velocity, v(t), as a function of the number of seconds, t, since you started slowing. Table 1.2: Your velocity after passing a truck t (s) 0 5 10 15 20 25 30

v(t) (ft/s) 100 77.3755 67.5477 63.2786 61.4242 60.6187 60.2687

Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) (a) Whatâ&#x20AC;&#x2122;s the fastest you went in the first five seconds? The slowest? Give an upper and lower range for the displacement in the first 5 seconds. (b) How might one obtain a single best estimate for the change in position for the first 5 seconds? (c) On Figure 1.11, show that each estimate, upper and lower, is represented graphically by a rectangle with a width of 5 seconds. Graphically visualize why the upper-estimate rectangle includes too much area, and the lower-estimate rectangle does not include enough area. (d) If, instead of rectangles that are either too big or too small, suppose we represent the area with one trapezoid, with a width of 5 seconds. The formula for the area of a trapezoid is h1 + h2 AT = b 2 i.e., the base times the average of the two heights. For a trapezoid that best represents the area of the graph between t = 0 and t = 5, what are the two heights, and what is the area? (e) Estimate your change in position for each of the subintervals [5, 10], [10, 15], [15, 20], and for the overall interval [0, 20].

Figure 1.11: Velocity, v(t), as a function of the number of seconds, t, since you started slowing. [9]

Example 1.2.2 (adapted from Finney, et al. [8]) Try this in your mighty, mighty groups of four. A power plant generates electricity by Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

burning oil. Pollutants produced by the burning process are removed by scrubbers in the smokestacks. Over time the scrubbers become less efficient and eventually must be replaced when the amount of pollutants released exceeds government standards. Measurements taken at the end of each month determine the rate at which pollutants are released into the atmosphere as recorded in the Table 1.3.

Table 1.3: [8] Month Pollutant Release Rate (tons/day)

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

0.20

0.25

0.27

0.34

0.45

0.52

0.63

0.70

0.81

0.85

0.89

0.95

(a) What is an upper estimate for the total tonnage of pollutants released in the month of January? February? June? (b) Suppose you plotted the data on a graph of Pollution Rate (tons/day) vs. Time (day). Describe how the total tonnage released for each of those months represents a rectangle, one for each month. (c) What are the lower estimates for these months? (d) Give an upper estimate of the total tonnage of pollutants released from the beginning of January to the end of June. Assuming that new scrubbers allow only 0.05 ton/day released, what is a lower estimate? Why would this problem be easier if the scrubbers didnâ&#x20AC;&#x2122;t decline, and the pollution rate stayed at 0.05 tons/day? [Ans: 61.32, 47.04] (e) In the best case, approximately when will a total of 125 tons of pollutants have been released into the atmosphere? [Ans: Oct 27] (f) The upper and lower approximations give a range of reasonable values for the definite integral, but neither one of them is necessarily very reliable. Graphically, instead of having a rectangle at the highest possible y-value for each subinterval, or the lowest possible y-value, what might be a more reasonable approach. Numerically, rather than using the upper or lower approximations, what might be a more reasonable approach?

For this problem, the Upper Rectangular Approximation also happens to be a Right-Endpoint Rectangular Approximation. An Upper RAM will be the Right RAM so long as the function is always increasing. For this problem, the Lower Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Rectangular Approximation also happens to be the Left-Endpoint Rectangular Approximation. The Left RAM will happen to be the Lower RAM whenever the function is increasing. Were the function decreasing, the left RAM would be the upper approximation.

Terms In using a rectangular approximation to estimate

Rb a

f (x) dx:

• The interval starts at a and ends at b. For this problem, it is the beginning of January to the end of June. The interval width is b − a, e.g., 181 or 182 days depending on leap-hood. • The interval is divided into subintervals. For the pollution problem, the subintervals are the months. The subinterval widths would be 31 days, 28 (or 29) days, etc. • We will pretend that Riemann sum is German for RAM. With a rectangular approximation method, the actual area for each subinterval is replaced with the area of a rectangle. The rectangle will have the same width as the subinterval, and the height is determined by whichever Rectangular Approximation Method is chosen. There are countless types of Rectangular Approximation Methods, but five which you need to know: • Left endpoint Rectangular Approximation Method (RAM)[8] - the height of each approximating rectangle is the height of the left side of the corresponding subinterval (e.g., beginning of the month). • Right endpoint RAM - the height of each approximating rectangle is the height of the right side of the corresponding subinterval (e.g., end of the month). • Midpoint RAM - the height of each approximating rectangle is the height in the middle of the subinterval (e.g., the sixteenth of the month). • Upper RAM - the height of each approximating rectangle is the maximum height in the corresponding subinterval. • Lower RAM - the height of each approximating rectangle is the minimum height in the corresponding subinterval. In addition to the Rectangular Approximation Methods, there is also: Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

• Trapezoidal Approximation Method, in which the actual area of each subinterval is replaced by the area of a trapezoid. The base of the trapezoid is the same as the subinterval width, just as for rectangles. For each trapezoid, the two heights used are the two heights on the left and right of each subinterval.

Example 1.2.3 Why did I not ask for the Midpoint Approximation for the pollution problem? In what cases could I ask for a Midpoint RAM?

1.2.2

Rectangular Approximation Method (RAM)

There are many physical situations where we must multiply two quantities, one of which is not a set constant, but a continuously changing variable. In order to multiply two things, one of which is changing, we utilize the definite integral, which is nothing more than the signed area under a curve. Previously, we have estimated the area under a curve via the “counting squares” approach. Another way to estimate the area of a funky shape is to approximate the shape with a series of vertical rectangles, which, together, form a blocky or pixelized version of the original shape. Our approach here is to divide the shape or region into a number of funky strips that have three straight sides and a curved top (or bottom if below the x-axis) that follows the function whose definite integral we are finding, i.e., that we are integrating. Once you have divided the shape into a number of strips, the next thing to do is replace the strip with a rectangle of approximately the same size. The idea is that we are approximating the area of the strip, which we don’t know (since we don’t have a formula to find the area of a funky strip) with the area of a rectangle, for which we do have a formula. The area of a rectangle is base times height. • The width of each rectangle is the subinterval width, which is usually determined to some extent either by the way the problem is asked, or by the data itself. • The constant height of each approximating rectangle is based on the varying height of the funkily-shaped strip. There are several different rules for assigning a height to each rectangle. Here are two, but we will discuss others a little later: Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) • Left endpoint Rectangular Approximation Method (RAM)[8] - the height of each approximating rectangle is the height of the left side of the corresponding strip. • Right endpoint RAM - the height of each approximating rectangle is the height of the right side of the corresponding strip.

Figure 1.12: Left endpoint and right endpoint rules for the Rectangular Approximation Method [8]

The left endpoint rectangular approximation is designated by Ln , where n is the number of rectangles (or strips or slices or subintervals). The right endpoint rectangular approximation is designated by Rn .

1.2.3

Trapezoidal Approximation: Quasi-RAM

What is usually better than taking a constant height at the right endpoint, or a constant height at the left endpoint is joining the left and right endpoints of each funky strip with a line segment, creating a trapezoid instead of a rectangle. The Trapezoidal Approximation Method finds the funky area by adding up the areas of multiple replacement trapezoids, just like the Rectangular Approximation Method added the areas of multiple rectangles. Example 1.2.4 (adapted from AB ’98) A table of values for the velocity v(t), in ft/sec, of a car traveling on a straight road, at 5 second intervals of time t, for 0 ≤ t ≤ 50, is shown in Table 1.4. Table 1.4: Velocity of a car traveling on a straight road t (seconds) 0 5 10 15 20 25 30 35 40 45 v(t) (ft/sec) 0 12 20 30 55 70 78 81 75 60

50 72

R 50 (a) Approximate 0 v(t) dt with a left and right rectangular and trapezoidal approximations, each with five subintervals. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

(b) Draw rectangles or trapezoids on the graphs in Figure 1.13 to demonstrate each of the three methods of estimation. From the graphs, which seems to be the most accurate? (c) Find L10 , R10 and T10 . (d) Using correct units, explain the meaning of

R 50 0

v(t) dt.

(e) How could you estimate the average velocity of the car?

Figure 1.13: Draw the appropriate rectangles or trapezoids for L5 , R5 , and T5

1.2.4

Streamlining Calculations for Equal Widths

If all of the subinterval widths are the same, the calculations for rectangular and trapezoidal approximations can be simplified.

Left RAM

f (x) dx ≈ Ln = ∆x (y0 + y1 + y2 + · · · + yn−1 ) a

where ∆x =

b−a n .

Right RAM

f (x) dx ≈ Rn = ∆x (y1 + y2 + · · · + yn−1 + yn ) a

Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Trapezoidal Approximation

f (x) dx ≈ Tn = ∆x a

+ y1 + y2 + · · · + yn−1 +

yn ∆x = (y0 + 2y1 + 2y2 + · · · + 2yn−1 + yn ) 2 2

Example 1.2.5 Refer again to Table 1.2. (a) What is the width of each subinterval? (b) Use the above formulas to find T4 , R4 , and L4 to approximate R 20 v(t) dt on your calculator in one input step, using values 0 from the table.

Example 1.2.6 R 40 Refer once again to the data in Table 1.4 on page 18. Estimate 5 v(t) dt. What does this represent?

1.2.5

Finding a Range of Values

There may be a situation in which you want to find a range of values for the definite integral, i.e., what is the best case scenario, and what is the worst case scenario. Here are two more rules for approximating the height of each funky shape. • Upper RAM - the height of each approximating rectangle is the maximum height in the corresponding strip. • Lower RAM - the height of each approximating rectangle is the minimum height in the corresponding strip. The upper Riemann sum is designated Un , where n is the number of subintervals. The lower sum is designated Ln . That’s right, the lower and left sums have the same designation. You will have to tell which is which from context. If you’re asked to find L8 and U8 , you should find a lower and upper Riemann sum, not a left and upper. The importance of these rules are in giving a range of values. The upper RAM is always an overestimate, whereas the lower RAM always underestimates the actual integral. If I is the actual value of the definite integral, then Ln ≤ I ≤ Un Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

Example 1.2.7 Refer once again to the data in Table 1.4 on page 18. Find lower and upper rectangular approximations, using five subintervals. Then find U10 and L10 . Whatâ&#x20AC;&#x2122;s happening to the range as you increase the number of subintervals? Draw rectangles on Figure 1.14 to demonstrate these approximation methods. Figure 1.14: Draw the appropriate rectangles or trapezoids for U5 and L5 .

Key Questions 1. When is a lower sum always the same as a left sum? 2. What happens to the range between the upper and lower rectangular approximations as the number of subintervals increases?

1.2.6

Midpoint RAM

Recall that we have discussed the left and right rectangular approximation methods. A third, similar, method is the midpoint approximation, which typically gets confused with the trapezoidal approximation. â&#x20AC;˘ Midpoint RAM - the height of each approximating rectangle is the height of the strip in the middle. If you were to take the funky strip and fold it so that the right side and the left side touch, the creased side would be the midline, and the length of that folded side would be used as the height of the approximating rectangle. This would be done for each funky strip.

Example 1.2.8 Refer yet again to the data in Table 1.4 on page 18. Estimate the area under the graph by using the Midpoint Rectangular Approximation with n = 5. Draw rectangles on the graphs in Figure 1.16 to show that you understand the midpoint approximation. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Figure 1.15: Left endpoint, right endpoint, and midpoint rules for the Rectangular Approximation Method[8]

Figure 1.16: Draw the appropriate rectangles for M5

Key Questions 1. What is the difference between Mn and Tn ?

Accuracy Upper and Lower are obviously the worst methods in terms of accuracy, as they give us extreme values. Right and Left cannot be considered much more reliable than upper and lower, and frequently give upper and lower. (Why?) Trapezoidal approximations can be considered more accurate than right, left, upper, or lower approximations. But what about the midpoint rule? If you consider a trapezoidal and a midpoint approximation with the same number of subintervals, then generally the midpoint approximation is about twice as accurate. However, think about using a table of data. If I have nine equally spaced data points, i.e., eight subintervals, then I can use all eight data points to calculate T8 . I would not, however, be able to find M8 . (Why not?) The best I could do would be M4 . For midpoint and trapezoidal approximations, Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

doubling the number of subintervals generally quadruples your accuracy. So that: • Tn is accurate • Mn is roughly twice as accurate as Tn • T2n is roughly four times as accurate as Tn and therefore roughly twice as accurate as Mn It is worth noting that if Mn overestimates the actual integral, then Tn underestimates, and vice versa. Since the midpoint approximation is about twice as accurate as the trapezoidal approximation, we can make a super approximation that is a weighted average of the midpoint and trapezoid, with the midpoint being weighted twice as much as the trapezoid: S2n =

2Mn + Tn 2+1

Why does the number of subintervals double?

1.2.7

Unequal Subdivisions

Recall that the data you have may be such that it is not evenly spaced. In these cases, you must calculate each rectangle or trapezoid separately, and add them at the end. Example 1.2.9 (adapted from AB ’03) The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by function R of time t. A table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes is shown. (a) Approximate the amount of fuel consumed in the first 30 minutes using left and right endpoint and trapezoidal methods. Indicate units. (b) Approximate the amount of fuel consumed in the time interval 30 ≤ t ≤ 40 minutes, using left and right endpoint and trapezoidal methods. Indicate units. (c) If we know that R(t) is what we call strictly increasing, i.e., R always increases and never decreases, then what would be a range for the amount of fuel consumed by the plane in the first 90 minutes? Why is it important that we know that R is strictly increasing? Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Table 1.5: Rate of fuel consumption of a plane t (minutes) 0 30 40 50 70 90

R(t) (gallons per minute) 20 30 40 55 65 70

(d) Draw rectangles on Figure 1.17 to demonstrate that you understand the upper and lower approximation methods. R 90 (e) Approximate the value of 0 R(t) dt using the five subintervals indicated by the data in the table. What is the most appropriate method: left, right, trap, upper, lower, or mid? Draw appropriate polygons on Figure 1.17 to demonstrate which method you used. Rb (f) For 0 < b â&#x2030;¤ 90 minutes, explain the meaning of 0 R(t) dt in terms of fuel consumption for the plane. R 90 1 (g) What do you think is the physical meaning of 90 R(t) dt? 0 R 1 b Of b 0 R(t) dt

Figure 1.17: from 2003 AP Calculus AB exam

Problems 1.B-1 Refer to the graph which is repeated in Figure 1.18. (a) Find a range of values for the area under the graph using upper and lower rectangular approximation methods, with n = 4. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

Figure 1.18: [20] Draw the appropriate rectangles or trapezoids for M4 , T4 and R4

(b) Approximate the area under the graph using the Left, Right, and Midpoint Rectangular Approximation Methods and the Trapezoidal Approximation Method, with n = 4. Are these values within your upper/lower range? (c) Draw appropriate rectangles or trapezoids to demonstrate your understanding of M4 , T4 and R4 . (d) Approximate the area under the graph using the Left, Right, and Midpoint Rectangular Approximation Methods and the Trapezoidal Approximation Method, with n = 8. (e) What happens to the discrepancies between the various methods as you increased the number of subintervals? (f) How might you make all four methods of approximation get closer and closer to the same number? 1.B-2 Refer to Figure 1.19. By counting squares, find an approximation for the definite integral of f (x) from x = 2 to x = 14. Find an estimate of the definite integral of f (x) from x = 2 to x = 14, using rectangular approximation method with: (a) 3 subintervals and a midpoint method for finding the height of the rectangle. (b) 3 subintervals and a left-point method for finding the height of the rectangle. (c) 3 subintervals and a right-point method for finding the height of the rectangle. (d) 6 subintervals and a left-point method for finding the height of the rectangle. (e) 6 subintervals and a right-point method for finding the height of the rectangle. (f) 6 subintervals and an upper-point method for finding the height of the rectangle. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Figure 1.19: [10]

(g) 6 subintervals and a lower-point method for finding the height of the rectangle. [Ans: 308, 252, 356, 280, 332, 346, 266] 1.B-3 (adapted from AB â&#x20AC;&#x2122;04) A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v is a function of t. Selected values of v(t) are shown in Table 1.6. Table 1.6: Test plane velocities t (minutes) 0 5 10 15 20 25 v(t) (miles per minute) 7.0 9.2 9.5 7.0 4.5 2.4

30 2.4

35 4.3

40 7.3

(a) Use a midpoint Riemann sum and values from the table, what is the R 40 most number of subintervals that could be used to approximate v(t) dt? 0 (b) Find M4 . (c) Find T4 . Does T4 use the same data points as M4 ? (d) Find the super-approximation S8 =

2M4 +T4 . 3

1.B-4 Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in Table 1.7. [8] (a) Give an lower and upper estimate of the total quantity of oil that has escaped after 5 hours. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

Time (h) Leakage (gal/h)

0 50

1 70

Table 1.7: [8] 2 3 4 97 136 190

5 265

6 369

7 516

8 720

(b) Repeat (a) for the quantity of oil that has escaped after 8 hours. (c) The tanker continues to leak 720 gal/h after the first 8 hours. If the tanker originally contained 25,000 gal of oil, approximately how many more hours will elapse in the worst case before all the oil has leaked? in the best case? [Ans: 543–758 gal, 543 gal; 1693–2363 gal; 31.4 more hours, 32.4 hours] 1.B-5 (adapted from Acorn book) Table 1.8 gives the values for the rate (in gal/sec) at which water flowed into Lake Lamar, with readings taken at specific times. Table 1.8: Water Flow into Lake Lamar Time (sec) 0 10 25 37 46 60 Rate (gal/sec) 500 400 350 280 200 180

(a) Give a range of values for the total amount of water that flowed into the lake during the time period 0 ≤ t ≤ 60. (b) Find a trapezoidal approximation to the amount of water that flowed into the lake during that time period. (c) Does your trapezoidal approximation fall within the range you gave? [Ans: 16930–20520 gal, 18725 gal, yes] 1.B-6 An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in ft/sec2 and recorded every second after the drop for 5 sec, as shown in Table 1.9. Table 1.9: Acceleration of a falling object [8] t 0 1 2 3 4 5 a(t) 32.00 19.41 11.77 7.14 4.33 2.63

(a) Use L5 to find an upper estimate for the speed when t = 5. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Figure 1.20: [20]

(b) Use R5 to find a lower estimate for the speed when t = 5. (c) Use upper estimates for the speed during the first second, second second, and third second to find an upper estimate for the distance fallen when t = 3. [Ans: 74.65 ft/sec; 45.28 ft/sec; 146.59 ft] R4

1.B-7 Let I = 0 f (x) dx, where f is the function whose graph is shown in Figure 1.20. [20] (a) Use the graph to find L2 , R2 , and M2 . (b) Are these underestimates or overestimates of I? (c) Use the graph to find T2 . How does it compare with I? (d) For any value of n, list the numbers Ln , Rn , Mn , Tn , and I in increasing order. [Ans: 6, 12, 9.6; L2 : u, R2 : o, M2 : o; 9 < I; Ln < Tn < I < Mn < R] 1.B-8 As the fish and game warden of your Buddville, you are responsible for stocking the town pond with fish before the fishing season. The average depth of the pond is 20 feet. Using a scaled map, you measure the distances across the pond at 200-foot intervals, as shown in the diagram in Figure 1.21. [8] (a) Use the Trapezoidal Rule to estimate the volume of the pond. (b) You plan to start the season with one fish per 1000 cubic feet. You intend to have at least 25% of the opening dayâ&#x20AC;&#x2122;s fish population left at the end of the season. What is the maximum number of licenses the town can sell if the average seasonal catch is 20 fish per license? [Ans: 26.36 million cubic feet; 988]

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 2: Approximating Definite Integrals from Tables

Figure 1.21: [8]

Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 3: Approximating Definite Integral from Formulas

1.3

Approximating Definite Integral from Formulas

Advanced Placement Interpretations and properties of definite integrals. • Computation of Riemann sums using left, right, and midpoint evaluation points. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals by functions represented algebraically, geometrically, and by tables of values. Textbook §4.3 Area, §4.4 The Definite Integral, and §4.7 Numerical Integration [16] Resources §5.9 Approximate Integration in Stewart [20]. §5.1 Estimating with Finite Sums and §5.5 Trapezoidal Rule in Finney, et al. [8]. §1-4 Definite Integrals by Trapezoids, from Equations and Data by Foerster [10]. Exploration 1-4: “Definite Integrals by Trapezoidal Rule” in [9].

1.3.1

Definite Integrals from Known Shapes

We have looked at calculating definite integrals using graphs and using tables. Many times, however, instead of having a graph or a table, we have a formula or expression which we are integrating. Sometimes, we can find these definite integrals by looking at a graph of the expression. Example 1.3.1 Find: Z 50 (a) 60 dt 30 Z 0 (b) (−x − 2) dx −3 4 q

Z (c)

4 − (x − 2) − 2 dx 2

Ans: 1200, - 32 , −8 + 2π Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) Z

Example 1.3.2 Find

16 − (x + 3) dx.

−3

1.3.2

Approximating Definite Integrals

Unfortunately, very often, when we graph the integrated expression, we are not given a graph with nice shapes, for which we have area formulas. Although we may not be able to calculate the exact area under these curves (yet), we still can use the other techniques which we’ve already learned.

Graphing Example 1.3.3 Let f (x) = 1−x2 . Estimate a value for the integral R2 I1 = 0 f (x) dx. To graph use an xstep of 1 and a ystep of 1. [17] Example 1.3.4 Let g(x) = x3 . Estimate an xstep of 0.5 and a ystep of 0.5. [17]

R1 0

g(x) dx. To graph use

Riemann slicing Remember that when we had data points (or graphs), we approximated definite integrals by dividing the overall interval into subintervals. A definite integral was approximated for each subinterval, using area formulas for rectangles or trapezoids, and then the individual areas were added together. For tabular data, the subintervals were usually predetermined by what data was available. Graphically, this meant dividing the funky shape into several funky strips, each of which was replaced with a rectangle or trapezoid of similar area. The area of all the funky strips were added together, to get the area of the funky shape. For graphs, the number of subintervals was limited by our resolution to distinguish small changes in height or width. If we are given an expression to integrate, our approach will be similar. We divide the overall shape into smaller strips, and then replace the smaller strips with rectangles or trapezoids, whose areas we then add together. The advantage of using formulas is that we don’t have restrictions on which or how many subintervals to use. The number of strips can be a few, if we are going to calculate he areas by hand, or infinitely many, in a theoretically ideal case. If you are looking at an interval from t = 0 min to t = 8 min, you might naturally pick 8 strips, each of width 1 min. You might also pick a factor of 8, such as 2 Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 3: Approximating Definite Integral from Formulas

or 4, to give subinterval widths of 4 and 2 minutes, respectively. Alternatively, you might choose a multiple of 8, such as 16 (subinterval widths of 30 seconds). It is usually best to make all the strips of equal width. When we divide the shape into several strips we determined the width or base of each rectangle. total interval width wslice = number of subintervals or ∆x =

b−a n

When the areas of all the rectangles are added together, the sum of the areas is called a Riemann sum. The Riemann sum is an approximation to the definite integral. It is named after a German guy who apparently invented rectangles. P Definition 1.3 (Riemann sum). A sum of the form f (x)∆x where each term of the sum represents the area of a rectangle of altitude f (x) and base ∆x. A Riemann sum gives an approximate value for a definite integral.[10]

Example 1.3.5 Return to Exploration 1-4. (You should have already completed problems 1 through 3 for homework.) (a) Use your graphing calculator to make a table of values for the equation v(t) = t3 − 21t2 + 100t + 110 for the even values of t from t = 0 to t = 8. (b) Do problem 4 on Exploration 1-4 (c) Find a way to determine the answer for problem 4 on your graphing calculator in one line of input that doesn’t require you to copy data from a table.

Example 1.3.6 Using a program, calculate rectangular and/or R 20 2x dx trapezoidal approximations to the integral 0 60 + 40 (0.92) using 4, 8, and 16 subintervals. R2 Example 1.3.7 Without a program, find M3 for 0 1 − x2 dx. Check with a program, and then use more and more subintervals to find the actual value.

16 Ans: − 27 ; − 23 Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) R1 Example 1.3.8 Without a program, find T4 for 0 x3 dx. Check with a program, and then use more and more subintervals to find the actual value. Ans:

17 1 64 ; 4

Example R π 1.3.9 In your mighty, mighty groups of four: Approximate 0 2 sin2 x dx using: (a) M2 (b) T2 (c) M3 (d) T3 (e) R4

1.3.3

Using Symmetry

Example 1.3.10 Let f (x) = 1 − x2 . Find (or estimate) values for R2 R2 the integrals I1 = 0 f (x) dx and I2 = −2 f (x) dx. [17] Ans: − 32 , − 43 Note that f (x) is an even function. Example 1.3.11 Let g(x) = x3 . Find or estimate R1 g(x) dx. [17] −1

R1 0

g(x) dx and

[Ans: 0.25, 0] Note that g(x) is an odd function. (Why?)

Problems big giant blue-green Calculus book p. 368: #33; p. 380 #41-44; p. 413, #5 1.C-1 Quickly draw a graph of the appropriate functions, then calculate each definite integral. [17] Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 3: Approximating Definite Integral from Formulas (a)

(b)

(c)

−3 −3 −3

(x + 2) dx |x + 2| dx (|x| + 2) dx [Ans: 12, 13, 21]

Z 1q 2 1.C-2 Evaluate 1 − (x − 1) dx. [Hint: Sketch a graph of the integrand, 0 q 2 i.e., 1 − (x − 1) .] [17] Ans: π4 Z

6−

1.C-3 Evaluate

4 − (x − 3)

[Ans: 12 − π]

dx exactly. [17]

Z 1.C-4 Evaluate 0

4 − (x − 1) dx exactly.

√

Ans:

3 2

4π 3

1.C-5 Go to http://math.furman.edu/~dcs/java/NumericalIntegration.html Z 2 2 and estimate dt. 2 −1 1 + 4t (a) Start with a left-hand rule, using four subintervals. While doubling the number of subdivisions, watch what happens to the error (i.e., the difference between the estimate and the actual value). The “successive error ratio” that is reported if you use the “Double” button gives the ratio of the new error to the error from before, i.e., the one with half as many subintervals. What value does the successive error ratio approach as you continue to double the number of subintervals, while using the left-hand rule? [Ans: 0.5] (b) Now using the trapezoidal rule, what value does the successive error ratio approach as you continue to double the number of subintervals, i.e., as n → ∞? [Ans: 0.25] (c) Now using the midpoint rule, what value does the successive error ratio approach as you continue to double the number of subintervals? [Ans: 0.25] (d) For midpoint or trapezoidal methods, if you tripled the number of subintervals, what would you expect to happen to the error? [Ans: one-ninth of what it was before] (e) For a large number of subintervals, compare the absolute value of the error for the midpoint approximation and for the trapezoidal approximation. Which one is bigger? By roughly what percentage? (Make sure your approximations use the same number of subintervals) [Ans: error for trapezoidal is roughly double that of midpoint] Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

1.C-6 Go to http://math.hws.edu/javamath/config_applets/RiemannSums.html and set f (x) = ex +1. Let xmin be 0 and xmax be 1. Set ymin and ymax so that you can see the graph. As you increase the number of subintervals, what does the sum appear to be approaching? Do you recognize this number? [Ans: 2.718 = e] 1.C-7 Play around with http://www.plu.edu/~heathdj/java/calc2/Riemann.html Rπ 1.C-8 Estimate 0 sin x dx using trapezoids with 2 subintervals, 3 subintervals, and 4 subintervals. Give exact and decimal answers to three places after the decimal. Do your trapezoidal approximations over- or underestimate this definite integral? What’s happening to the values as you increase the number of subintervals? Make a√conjecture as to what the exact answeri h √ might be. Ans: π2 = 1.571, π 3 3 = 1.814, π4 1 + 2 = 1.896; under;

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 4: Slope and Rate of Change

1.4

Slope and Rate of Change

Advanced Placement Concept of the derivative. • Derivative presented geometrically and numerically. • Derivative interpreted as an instantaneous rate of change. Textbook §1.1 A Brief Preview of Calculus and §2.1 Tangent Lines and Velocity [16] Resources §1-1 The Concept of Instantaneous Rate in Foerster [10]. Exploration 1-1: “Instantaneous Rate of Change of a Function” in [9].

1.4.1

Instantaneous Rate of Change

Recall: average velocity and slope.

Example 1.4.1 (a) A car driving due east away from Houston is 20 miles from the city limits at 1 p.m. and 130 miles from the city limits at 3 p.m. What is the car’s average velocity between 1 p.m. and 3 p.m. (b) A ball thrown up into the air has an average velocity, between 3 seconds and 5 seconds after it was thrown, of −14 feet per second. If, 5 seconds after it was thrown, the ball was 20 feet above the ground, how high was it 3 seconds after it was thrown? (c) A ball thrown straight up into the air has a height above the ground of s(t) = −16t2 +96t feet, t seconds after it was thrown. Find the average velocity of the ball during the time period between 1 and 3 seconds after it was thrown.

Slope = Rate of Change 1. Average Rate of Change. This is the Algebra I version of slope. The slope of a secant line between two points. It is rise over run; change in y over change in x. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) 2. Instantaneous Rate of Change. This is the Calculus version of slope. It is the slope of the tangent line at one point. In a sense, it is an oxymoron, because there is no change in an instant.

1.4.2

Definition and Notation

Definitions and Notation The derivative is another way of saying instantaneous rate of change. It is denoted by a ‘prime’ after the function, i.e., the derivative of f (x) is written f 0 (x). Definition 1.4 (Derivative). The derivative of a function at a particular value of the independent variable is the instantaneous rate of change of the dependent variable with respect to the independent variable.[10] We’ve already noted that the rate of change is essentially the slope, so that the instantaneous rate of change is the slope at a point.

Notation The derivative of f is denoted by f 0 . f 0 (3) is the slope of the curve of f at the point where x = 3. The second derivative is the derivative of the derivative, and is denoted by f 00 .

Approximating the derivative given a graph Example 1.4.2 Graph s(t) = −16t2 + 96t. (a) Draw the line tangent to the graph of s(t) at t = 1. (b) Estimate s0 (1) by finding the slope of your tangent line. (c) What is the physical meaning of s0 (1), the rate of change of height, with respect to time, at t = 1? (d) Write an equation of the line tangent to the graph of s(t) at s = 1, and use it to approximate s(1.1). Compare this approximation to the actual value of s(1.1). (e) Estimate s0 (3). Why is s0 easy to find at t = 3? Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 4: Slope and Rate of Change

39 [Ans: ; 128; v(3); ; 0]

Example 1.4.3 Graph f (x) = x2 + x. (a) Draw the line tangent to the graph of f (x) = x2 + x at x = 2. (b) Estimate the slope of this line, i.e., f 0 (2) (c) Write the equation of this line. Plug 2.5 into the formula for your tangent line, and compare it to the actual value of 2.5. When do you suppose the tangent line approximation is a good approximation?

[Ans: ; 5; y = 5 (x − 2) + 6, 8.5, 8.75]

1.4.3

Approximating Derivatives from Tabular Data

Rate of Change = Difference Quotient • Rate from ratio is a quotient. • Change is the difference. Difference Quotients • Forward: an interval to the right of the point of interest. This interval starts at the point at which you are estimating the derivative (i.e., instantaneous rate of change), and ends at some very slightly higher x-value. • Backward: an interval to the left of the point of interest. This interval starts at some point very slightly lower x-value than where you are estimating the derivative, and ends at the point of interest. • Symmetric: an interval to the left and right of the point of interest. This interval starts with a very slightly lower x-value than where you are approximating the derivative, and ends at a very slightly higher x-value. Note that any difference quotient can be a forward, backward, or symmetric difference quotient. The point of interest helps decide which it is. Example 1.4.4 Use the table of values to answer the following questions [19]. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) x y

−1.5 2.027

−1 0.632

−0.5 −0.357

0 −1

0.5 −1.399

1 −1.718

(a) What is the best approximation of f 0 (1), the derivative at 1? Is this difference quotient forward, backwards, or symmetric? (b) What is the best estimate of f 0 (−1.5)? Name the type of interval. (c) Find the best estimate of f 0 (−1.25). Name the type of interval. (d) What is the best estimate of f 0 (−1)? Name the type of interval. (e) Extension Find the best estimate of f 00 (−1), i.e., the second derivative at −1, i.e., the rate of change of the rate of change.

[Ans: −0.638, b; −2.79, f ; −2.79, s; −2.384, s; 1.624] Key Questions 1. Before doing any calculations, how can you determine whether the derivative should be positive or negative? Example 1.4.5 Suppose that f is a function for which f 0 (2) exists. Use the values of f given in the table to estimate f 0 (1.9), f 0 (2), and f 0 (2.02). Name the type of difference quotient used. [17] x f (x)

1.9 6.6

1.97 6.905

2.0 7

2.02 7.059

2.2 7.5

[Ans: 4.357 f, 2.95 f, 2.95 b] Note that difference quotients might be used for graphs and expressions as well.

Problems big giant blue-green Calculus book p. 155, WE #3; #1-7 odd, 37 1.D-1 The position, s(t) (measured in inches), at any time, t (measured in seconds), of an object is described in Figure 1.22. Use the graph to determine: (a) s(0) (b) s(1) Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 4: Slope and Rate of Change

Figure 1.22: Displacement, s(t) [15]

[Ans: 1, 0, 0, Y, N]

1.D-2 The graph of a position function in Figure 1.23 represents the distance in miles that a person drives during a twelve minute drive to school. Make

Figure 1.23: Displacement, s(t) [15]

a sketch of the corresponding velocity function. [15] Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) 



         Ans:        

                 

1.D-3 The graph of a function f is shown in Figure 1.24. Rank the values of f 0 (−3), f 0 (−2), f (0), and f 0 (4) in increasing order. [17] [Ans: f 0 (4), f 0 (−3), f 0 (−2), f 0 (0)] Figure 1.24: Graph of f [17]

1.D-4 Suppose that f (x) = x3 − 5x2 + x − 1 and that g(x) = x3 − 5x2 + x + 4. Explain why f 0 (x) = g 0 (x) for every x. [Hint: How are the graphs of f and g related?] [17] 1.D-5 The graph of the derivative of a function f appears in Figure 1.25. [17] (a) Suppose that f (1) = 5. Find an equation of the line tangent to the graph of f at (1, 5). (b) Suppose that f (−3) = −6. Find an equation of the line tangent to the graph of f at (−3, −6). [Ans: y − 5 = 2 (x − 1); y = −6] 1.D-6 (adapted from AB ’06) The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function f , shown in Figure 1.26. (a) Find f 0 (22). Indicate units of measure. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 4: Slope and Rate of Change

Figure 1.25: Graph of f 0 [17]

Figure 1.26: Graph of f , Burning Calories

(b) For the time interval 0 ≤ t ≤ 24, at approximately what time t does f appear to be increasing at its greatest rate? Why? (c) Find the total number of calories burned over the time interval 6 ≤ t ≤ 18 minutes. (d) What do you suppose is the meaning of

1 (18 − 6) min

f (t) dt 6

[Ans: −3 cal/min/min; t = 2; 132; ]

1.D-7 The graph shows how the price of a certain stock varied over a recent trading day. [17] Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

(a) For each time interval below, find the total change in price and the average rate of change of price. (Be sure to indicate units used to measure these quantities.) i. ii. iii. iv.

8:00 to 11:00 9:00 to 1:00 9:30 to 2:00 11:00 to 1:00 Ans: 1, 0, 53 , −1 $/hr

(b) Estimate the instantaneous rate of change of the stock’s price at each of the following times. (Be sure to indicate units with your answers.) i. ii. iii. iv.

9:15 a.m. 10:30 a.m. 12:15 p.m. 1:45 p.m. [Ans: −4, 4, −2, 8 $/hr]

1.D-8 Go to http://math.hws.edu/javamath/basic_applets/SecantTangentApplet. html (a) For f (x), put in the function x2 + x, and hit ‘New Function’. (b) Put Tangent at x=2 (c) Change the window so that you can see the parabola, along with the red dot, the red tangent line, and the green secant line. (d) The green secant line is anchored at the point x = 2. You control the placement of the other point. Drag the green circle along the parabola, and notice how the slope of the secant line changes. (e) Pay careful attention to what happens to the green secant line as you drag the green dot closer and closer to the red dot. Make a table of the ‘Secant at x=’ values with the ‘Secant Slope =’ values. What do you notice? If the ‘Secant at x=’ value could be 2, what would the ‘Secant Slope =’ value be? Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 4: Slope and Rate of Change

1.D-9 (adapted from AB ’06) Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 80 seconds, as shown. t (seconds) v(t) (feet per second)

0 5

10 14

20 22

30 29

40 35

50 40

60 44

70 47

80 49

(a) Find the average rate of change of the velocity of Rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of measure. (b) Approximate the instantaneous rate of change of the velocity of Rocket A at t = 0. Repeat for t = 80 seconds, t = 20, and t = 55 seconds. Indicate units of measure. R 70 (c) Using correct units, explain the meaning of 10 v(t) dt in terms of the rocket’s flight. Use a midpoint R 70Riemann sum with 3 subintervals of equal length to approximate 10 v(t) dt. Z 70 1 v(t) dt? (d) What do you suppose is the meaning of (70 − 10) sec 10 Ans:

11 20

ft/s2 ; 0.2, 0.75, 0.4 ft/s2 ; , 2020 ft;

1.D-10 A differentiable function f has values shown. Estimate f 0 (1.5). [14] x f (x)

1.0 8

1.2 10

1.4 14

1.6 22 [Ans: 40]

1.D-11 Suppose that f is a function for which f 0 (2) exists. Use the value of f given below to estimate f 0 (1.99), f 0 (2), f 0 (2.01), and f 0 (2.1). Explain how you obtained your estimates. [17] x f (x)

1.9 25.34

1.99 33.97

1.999 34.896

2.0 35

2.001 35.104

2.01 36.05

2.1 46.18

[Ans: 102.889 fdq, 104 (sdq), 105.111 (bdq), 112.556 (bdq)]

Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 5: IROC as a limit

1.5

Instantaneous Rate of Change from a Limiting Process

Advanced Placement Concept of the derivative. • Derivative presented geometrically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. Textbook §1.1 A Brief Preview of Calculus and §2.1 Tangent Lines and Velocity [16] Resources §1-2 Rate of Change by Equation, Graph, or Table in Foerster [10]. §2.5 Average and Instantaneous Rates: Defining the Derivative in Ostebee and Zorn [17].

1.5.1

Approximating Rate of Change from a Formula

Graphs Example 1.5.1 In your mighty, mighty groups of four: Do Exploration 1.5: “Instantaneous Rate of Change”

Tables Example 1.5.2 Refer to f (x) = x2 + x. (a) Write a formula for the forward difference quotient used to estimate f 0 (2) using an interval of [2, 2 + h]. (b) Put your formula into the calculator, using a variable step-size. Evaluate the forward difference quotient for h = 0.1, h = 0.01, and h = 0.001. Continue using smaller step sizes (h) until there is no change in the thousandths place of your slope. (c) What would be the appropriate interval for a backward difference quotient of width h to approximate f 0 (2)? (d) As you did with the forward difference quotient, use smaller and smaller h’s until you see no change in the thousandths place of your slope. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Preview of a formal definition of derivative using formulas Example 1.5.3 Refer to f (x) = x2 + x. (a) Write an expression for the average rate of change of f (x) from 2 to 2.1. (b) Write an expression for the average rate of change of f (x) from 2 to 2.01. (c) Write an formula for the average rate of change of f (x) from 2 to x. If we want an instantaneous rate of change, we want x to be as close as possible to what? (d) Make a table of values for the AROC from 2 to x for different values of x that get closer and closer to 2 from both sides. (e) Rewrite your formula, using the expression for f (x) and the value for f (2). (f) Factor the numerator in your above expression. Cancel. When would canceling not be allowed? What happens to the remaining expression as x gets closer and closer to 2? (g) Write an expression for the difference quotient for f (x) from 2 to 2 + h. If we want this difference quotient to represent the f 0 (2), what do we want h to approach? (h) Multiply the numerator out and combine like terms. Cancel. When would canceling not be allowed? What happens to the remaining expression as h get closer and closer to 0? (2) in the above example. Try to see Look at the work for simplifying f (2+h)â&#x2C6;&#x2019;f h that you can replace the 2 with x and you could still do the problem.

1.5.2

Kinematics: Displacement, Velocity, Acceleration

Velocity is the instantaneous rate of change of displacement, i.e., velocity is the derivative of displacement. v(t) = d0 (t) or v(t) = x0 (t). Speed is the magnitude of velocity. In one dimension, speed is the absolute value of velocity. speed = |v(t)| Acceleration is the instantaneous rate of change of velocity, i.e., acceleration is the derivative of velocity. a(t) = v 0 (t) Acceleration is the derivative of the derivative of displacement, i.e., acceleration is the second derivative of displacement. a(t) = d00 (t) Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 5: IROC as a limit

Problems big giant blue-green Calculus book p. 156: # 11, (31 or 33), 37, 39 1.E-1 Visit http://www.slu.edu/classes/maymk/SecantTangent/SecantTangent. html and use the following settings: • Let f (x) be x2 − x. • Let X0 be 3. This is the reference point at which we will be estimating the derivative. • Let dX be 1 for now. • Don’t worry about guessing f 0 (x) for now. • Make sure the first scroll-down menu is on ‘Right Secant’. • Change the second scroll-down menu to ‘X1 click’. • Make sure the two boxes are unchecked. • Change your window as appropriate. (a) Try to make some sense out of all the information that you’re being given, e.g., what are the meanings of X0, dX, and X1? Can you identify the backward, symmetric, and forward difference quotients? What do you suppose a negative dX means? (b) Use your mouse pointer or the ‘dX In’ and ‘dX Out’ buttons to move the second point on the secant line closer to and farther away from the reference point at x = 3. What’s happening to the line? What’s happening to the difference between the three slopes (left, balanced, and right)? (c) Write down the left, balanced, and right slope for X0= 3.0, and dX= 0.1, 0.01, and 0.001. (d) Play around: use some different functions, different points, etc. Go nuts! (e) Can you find some other, similar, websites? Anything better that we should know about? 1.E-2 Refer to g(x) = x2 − x. (a) Write an expression for the difference quotient for g(x) from 3 to 3.1. Find the value. (b) Write an expression for the difference quotient for g(x) from 3 to 3.01. Find the value. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) (c) Write a formula for the difference quotient for g(x) from 3 to x, using the expression for g(x) and the value for g(3). (d) Factor the numerator in your above expression. Cancel. When would canceling not be allowed? What happens to the remaining expression as x gets closer and closer to 3? h i g(3.01)−g(3) g(x)−g(3) x2 −x−6 Ans: g(3.1)−g(3) = 5.1, = 5.01, , , x + 2, goes to 5 3.1−3 3.01−3 x−3 x−3 (e) Write a formula for the difference quotient for g(x) from 3 to 3 + h. (f) Multiply the numerator out and combine like terms. Cancel. When would canceling not be allowed? What happens to the remaining expression as h get closer and closer to 0? ((3+h)2 −(3+h))−6 5h+h2 = h ; goes to 5 Ans: (3+h)−3

1.E-3 Refer to q(x) = x2 . (a) Write a formula for the difference quotient for q(x) from 1 to x. Factor the numerator in your difference quotient and cancel. What happens to the remaining expression as x gets closer and closer to 1? h i x1 −1 Ans: x−1 = x + 1, goes to 2 (b) Repeat part (a), replacing 1 with 2, then 3, and then −1. Look for a pattern, and guess a formula for q 0 (x). √ 1.E-4 Let r(x) = x. [17] (a) Graph r(x) on your calculator. Zoom in to the point (1, 1) repeatedly, until the graph looks like a straight line. (b) Use that point, i.e., (1, 1), and one other nearby point on the ‘line’ to find the slope of the ‘line’, which is an estimate of r0 (1). [Ans: 0.5] (c) Use to estimate r0 (1/4), r0 (9/4), r0 (4), r0 (25/4), and r0 (9). zooming 1 1 1 1 Ans: 1; 3 ; 4 ; 5 ; 6 (d) Use these results to sketch a graph of r0 (x) over the interval [−1, 2]. i h 1 (e) Use your results and your graph to guess a formula for r0 (x). Ans: 2√ x 1.E-5 Let f (x) = ln x. [17] (a) Use zooming to estimate f 0 (1/5), f 0 (1/2), f 0 (1), f 0 (2), and f 0 (5). [Ans: 5, 2, 1, 0.5, 0.2] (b) Use your results to sketch a graph of f 0 over the interval [0, 5]. (c) Use your results to guess a formula for f 0 (x). Ans:

1 x

1.E-6 Repeat the zooming process for a function of your choosing. Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 5: IROC as a limit

1.E-7 (AB ’01) The temperature, in degrees Celsius (◦ C), of the water in a pond is a differentiable function W of time t. The table shows the water temperature as recorded every 3 days over a 15-day period. t (days) 0 3 6 9 12 15

W (t) (◦ C) 20 31 28 24 22 21

(a) Use data from the table to find an approximation for W 0 (12). Show the computations that lead to your answer. Indicate units of measure. (b) A student proposes the function P , given by P (t) = 20 + 10te(−t/3) , as a model for the temperature of the water in the pond at time t, where t is measured in days and P (t) is measured in degrees Celsius. Estimate P 0 (12). Using appropriate units, explain the meaning of your answer in terms of water temperature. h Ans:

21−24◦ C 15−9 days

◦

= − 12 C/day; −0.549◦ C/day: water in pond decreasing at a rate of 0.549◦ C/day

Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope)

Mr. Budd, compiled September 29, 2010

AP Unit 1, Day 6: Slope and Area: Pulling It Together

1.6

Slope and Area: Pulling It Together

Do the following problems neatly on graph paper. 1.F-1 Suppose Mr. Budd is driving to the Utah Shakespearean Festival in Cedar City, UT. Once he gets on the road, he sets his cruise control for 55 mph. Let t be the number of hours since he started driving on cruise control. (a) How far has he gone during the first half hour on cruise control? the first hour? the first two hours? (b) Write an equation for the velocity, i.e., v(t) =(something). (c) Graph the velocity versus time. (d) Find the area under the curve of v(t) from t = 0 to t = 0.5. Also, find the area from t = 0 to t = 1 and also to t = 2. (e) What shape are these areas in? If I look at the area from t = 0 to t = tstop , what is the width of the figure (as an expression with tstop in it)? the height? the area (as an expression of tstop )? Call your expression for area A(tstop ). (f) Plot a graph of distance traveled versus time. Use the points (0.5, distance for 0.5), (1, distance for 1), and (2, distance for 2). Look for a pattern, and use your result for A(tstop ) to connect the dots. (g) On your graph of distance versus time, what is the slope at t = 0.5? at t = 1? at t = 2? Indicate units. 1.F-2 Let f (t) = 2t (a) Find f (0), f (0.5), f (1), f (1.5), and f (2). (b) Graph f (t). R1 R 1.5 R0 R 0.5 (c) Using the graph of f (t), find 0 f (t) dt, 0 f (t) dt, 0 f (t) dt, 0 f (t) dt, R2 f (t) dt. Do you see a pattern? 0 (d) RThink of a generic area under f (t) that starts at 0 and ends at x, i.e., x f (t) dt. What shape is it in? Write an expression (with x in it) 0 for the height of the shape. What is the width of the shape? Write an expression in terms of x for the area from t = 0 to t = x. Call this expression A(x) (A for area). R R R 0 0.5 1 (e) On a separate graph, plot 0, 0 f (t) dt , 0, 0 f (t) dt , 1, 0 f (t) dt , R R 1.5 2 1.5, 0 f (t) dt , 2, 0 f (t) dt . Connect the dots using A(x). 1.F-3 Let G(t) = t2 . While doing this problem, keep the previous problem in the back of your mind. Mr. Budd, compiled September 29, 2010

AP Unit 1 (Area and Slope) (a) Write the expression for the Average Rate of Change of G(t) from 1 to 1 + h. If this slope is to be a better and better approximation of G0 (1), what should be happening to h? Is this a forward, backward, or symmetric difference quotient? (b) In your difference quotient to estimate G0 (1), replace G(1 + h) with 2 (1 + h) , and G(1) with the actual value of G(1). Multiply out and combine like terms. Factor and cancel. (c) If you let h â&#x2020;&#x2019; 0, what happens to 1 + h? What happens to your remaining expression for G0 (1)? (d) Repeat the above process to approximate G0 (2) and G0 (3).

1.F-4 Let f (x) = 3x2 . (a) Find f (1), f (2), and f (3). (b) Use the TRAP program with more and more subintervals to make R1 R2 conjectures for the following definite integrals: 0 f (x) dx, 0 f (x) dx, R3 and 0 f (x) dx. Do you see a pattern? 1.F-5 Let G(x) = x3 . While doing this problem, keep the previous problem in the back of your mind. (a) Use symmetric difference quotients with h = 0.1, h = 0.01, and h = 0.001 to approximate G0 (1). Continue using smaller hâ&#x20AC;&#x2122;s until there is no change in the hundredths place in your estimates of G0 (1). (b) Repeat for G0 (2) and G0 (3). What do you notice?

Mr. Budd, compiled September 29, 2010

Unit 2

Limits and the Definition of the Derivative 1. Limits for Continuous Functions and Removable Discontinuities 2. Limit Definition of the Derivative (at x = c form) 3. Limit Definition of the Derivative (h or ∆x form) 4. One-Sided Limits and Infinite Limits 5. Limits at Infinity

AP Unit 2 (Limits) • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

Continuity as a property of functions. • An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions. Concept of the derivative. • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. Derivative at a point. • Slope of a curve at a point. • Tangent line to a curve at a point. • Instantaneous rate of change as the limit of average rate of change. Derivative as a function.

Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 1: Introduction to Limits

2.1

Introduction to Limits

Advanced Placement Limits of functions (including one-sided limits). • An intuitive understanding of the limiting process. • Estimating limits from graphs or tables of data. Continuity as a property of functions. • An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions. Textbook §1.3 The Concept of Limit and §1.4 Computation of Limits [16] Resources §1-5 Limit of a Function in Foerster [10]. Exploration 1-5: “Introduction to Limits” in [9]. §2.4 Introduction to Limits in Varberg, et al. [21]

2.1.1

Graphic Introduction to Limits

Understanding limits is a fairly intuitive process, and usually the easiest way to understand is to study examples and counterexamples. The graph in Figure 2.1 is given by the following function:   x + 1 −2 < x < 0      x=0 2 f (x) = −x 0<x<2    0 x=2    x − 4 2 < x ≤ 4

Remember This? Recall the different types of discontinuities. One thing to notice is that this is one function, not several. It is what is called a piecewise function, because it is defined in terms of several pieces, rather than Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

Figure 2.1: A piecewise function [14]

one smooth, nice neat curve. One thing to realize is that functions don’t have to look pretty, they just have to pass the vertical line test. Make sure you understand what is going on with the function. Notice that f (1) = −1, f (0) = 2 and f (2) = 0. The domain includes 4, but not −2. The function is continuous over (−2, 0), which basically means that I can draw that portion of the graph without lifting my pencil. At x = 0, the function undergoes a step (or jump) discontinuity, because it goes from 1 to 2 all the way down to 0. The function is again continuous over (0, 2) and over (2, 4). At x = 2 there is a removable discontinuity. Removable discontinuities are recognized by a hole in an otherwise continuous portion of the graph. A function has a removable discontinuity at a point if changing that one point would make the function continuous. As x → −1, f (x) →? I want to look at the limits of f (x), but first lets go over some notation. When we write lim f (x) = −1, we mean that as x get closer and closer to 1, but not x→1

equal to 1, f (x) (or y) gets closer and closer to −1.

Simple Case: Continuity To find the limit of f (x) as x goes to −1, trace your right forefinger along the function to the right of x = −1, moving left towards the point (−1, 0). Trace your left forefinger along the function, moving right towards the same point (−1, 0). You will see that as the x value gets closer to −1, the y value gets closer to 0. Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 1: Introduction to Limits

Example 2.1.1 From the graph in Figure 2.1, find the limit of f (x) as x approaches −1. As x → −1, f (x) → 0. Notation lim f (x) = 0

x→−1

This is read: “the limit of f (x) as x approaches −1 is zero”.

The Important Case: Removable Discontinuity An important point should be made, however, and that is that if I’m looking at the limit as x approaches −1, I don’t actually care what happens at −1, only near −1. So let’s take a look at the limit of f (x) as x approaches 2 (Figure 2.1 still). If I let my right finger approach x = 2 along the function from the right, and let my left finger approach x = 2 from the left, I notice that as the x value gets closer and closer to 2, the f (x) value gets closer and closer to −2. Hence lim f (x) = −2

x→2

Notice that lim f (x) 6= f (2). In other words, the limit is not the same as the x→2

functional value. Remember: I don’t care what’s happening at 2 if I’m taking the limit near 2. Here are two authors’ informal definitions of limits: Definition 2.1 (Intuitive Meaning of Limit). To say that lim f (x) = L means x→c

that when x is near but different from c then f (x) is near L. [21] Definition 2.2 (Limit). L is the limit of f (x) as x approaches c if and only if L is the one number you can keep f (x) arbitrarily close to just by keeping x close enough to c, but not equal to c.[10] Remember This? Remember the difference between if and if and only if. The formal definition of limit is quite a bit more complicated. The trick is: how do you objectively, logically, and mathematically reason what close and near mean. Now You Quickly look at numbers 1-4, 7, and 8 in Foerster §1-5. Find lim f (x). x→c Why didn’t I ask you to find the limits for numbers 5,6 and 9, 10. What types of Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

discontinuities are represented in the problems that do have limits. For problems 5 and 6, what types of discontinuities are represented? For problems 9 & 10?

2.1.2

Step Discontinuities & One-Sided Limits

Notation 1. lim− h(x) means the limit as x approaches 2 from the left, i.e., the negative x→2

side of the axis. 2. lim+ h(x) means the limit as x approaches 2 from the right, i.e., the x→2

positive side of the axis.

Graphically Example 2.1.2 For the function f (x) graphed in Figure 2.1, find lim f (x).

x→0

Now let’s take a look at the limit as x goes to 0 of the function graphed in Figure 2.1. As x gets closer and closer to 0 from the right, my y values get closer and closer to 0. As x gets closer and closer to 0 from the left, however, f (x) or y is getting closer and closer to 1. Which is the limit? Neither. The limit does not exist as x approaches 0 because the one-sided limits don’t match. lim f (x) does not exist

x→0

because lim f (x) = 1 6= lim+ f (x) = 0.

x→0−

x→0

lim f (x) is the one-sided limit from the left, or negative side. The one-sided

x→0−

limit from the right, or positive side, is given by using a “+” superscript. Theorem 2.1. lim f (x) = L if and only if lim f (x) = L and lim f (x) = L. x→c

x→c−

x→c+

If we reexamine lim f (x), we see that it exists (and is equal to −2) because x→2

lim f (x) = lim f (x) = −2. So the overall limit exists because the two one-

x→2−

x→2+

sided limits match, and the value of that overall limit is the same as that of the one-sided limits. Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 1: Introduction to Limits

2.1.3

Limits from a Table

Continuity: The Simple Case Example 2.1.3 Use your calculator to make a table of values for f (x) = x2 + 2x + 4 for values of x near 2. Find lim f (x). Repeat x→2

for lim f (x). x→3

[Ans: 12; 19]

Removable Discontinuity: The Important Case Example 2.1.4 Make a table of values for g(x) =

x3 − 8 for values x−2

of x near 3. Find lim g(x). Repeat for lim . x→3

x→2

[Ans: 19; 12] Example 2.1.5 What is the difference between the graph of f (x) and the graph of g(x)? [Ans: g(x) has a hole at (2, 12), representing the removable discontinuity] Remember: in evaluating lim f (x), imagine yourself blind to what is happening x→c to f at x = c. From all the evidence near x = c, what is your best guess as to what f (c) is, or should be? Another way to think about finding limits is: what would f (c) have to be in order to make f continuous at c? That is the limit of f as x approaches c. In fact the definition of continuity is that the limit of f matches the value of f .

One-Sided Limits, Numerically Example 2.1.6 On your graphing calculator, make a table of x3 −8 h(x) = |x−2| near x = 2. Find (a) the limit of h(x) as x → 2 from the left, aka from below. (b) the limit of h(x) as x → 2 from the right, aka from above. (c) the limit of h(x) as x → 2. Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

2.1.4

Limits from an Expression

If a function is continuous at x = c, then, by definition lim f (x) = f (c). Therex→c fore, if I know ahead of time that a function is continuous, then I simply plug in the x value that I’m approaching. Polynomial functions and functions involving sine and cosine are continuous, so long as there aren’t any variable expressions in any denominators. Over the next few class periods, we will learn several different methods for dealing with discontinuities.

I. Continuous Functions A. The Simple Case 1. Be on the look out for: a. on a graph: graph you can draw without lifting your pencil b. on a table: no singular points that stick out or are undefined c. with an expression: adding or multiplying polynomials, sine, cosine 2. How we deal with it: plug it in, plug it in II. Discontinuities A. Removable Discontinuities These are the only types of discontinuities for which limits exist. Limits only exist where a function is continuous, or where there is a removable discontinuity. 1. Be on the look out for: a. on a graph: holes b. on a table: a single point that sticks out c. with an expression: • piecewise functions • Rational Functions that give you 00 . 00 is called an indeterminate form, and could be many things, but are typical candidates for removable discontinuities. Be careful, though, indeterminate forms may be vertical asymptotes, or even step discontinuities. 2. How we deal with it: a. Use Algebra to cancel factors, after • factoring • rationalizing with conjugates • de-denominatorizing with LCDs Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 1: Introduction to Limits

b. Comparing to known limits, such as the grand-daddy of all known sin θ limits, lim θ→0 θ B. Step Discontinuities 1. Be on the look out for: a. on a graph: y-values that all of the sudden jump to a different value. b. on a table: y-values that all of the sudden jump to a different value c. with an expression: • piecewise functions |x| • variations on lim x→0 x 2. How we deal with it: one-sided limits C. Vertical Asymptotes 1. Be on the look out for: a. on a graph: Curves that go up or down off the graph. b. on a table: Numbers that get hugely positive or hugely negative as x approaces a specific value. nonzero c. with an expression: rational expressions where you have 0 2. How we deal with it: a. one-sided limits b. infinity, ∞ D. Infinitesimal Oscillations This is a relatively minor and rare type of discontinuity. Be on the look out for: 1 on a graph: oscillations with periods that get smaller and smaller, eventually getting infinitesimally small near a specific x-value. 2 with an expression: variations on lim sin x1 x→0

2.1.5

Substitution and Properties of Limits

These are the theorems that allow me to simply plug in values if I know that my function is continuous. Theorem 2.2 (Main Limit Theorem). Let n be a positive integer, k be a constant, and f and g be functions that have limits at c. [21] Then 1. lim k = k x→c

2. lim x = c x→c

Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) 3. lim kf (x) = k lim f (x) x→c

x→c

4. lim [f (x) + g(x)] = lim f (x) + lim g(x) x→c

x→c

5. lim [f (x) − g(x)] = lim f (x) − lim g(x) x→c

x→c

6. lim [f (x) · g(x)] = lim f (x) · lim g(x) x→c

x→c

lim f (x) f (x) = x→c , provided lim g(x) 6= 0 lim g(x) x→c x→c g(x)

7. lim

x→c

h in n 8. lim [f (x)] = lim f (x) x→c

9. lim

x→c

p n

f (x) =

q n

lim f (x), provided lim f (x) > 0 when n is even.

x→c

An example of the first limit property is that lim 7 = 7. Think about why x→−3

this makes sense numerically and graphically. The second limit property might be exemplified by lim x = π. x→π

The third property can be seen in action by lim 2x = 2 lim x, which can then x→π x→π be simplified with the second property to 2π.

Example 2.1.7

As an exercise to familiarize yourself with the x3 − 4 properties, evaluate lim , naming the property used at each x→2 2x step. Theorem 2.3 (Limits of Trigonometric Functions). For every real number c in the function’s domain [21] 1. lim sin t = sin c x→c

2. lim cos t = cos c x→c

3. lim tan t = tan c x→c

4. lim cot t = cot c x→c

5. lim sec t = sec c x→c

6. lim sec t = sec c x→c

Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 1: Introduction to Limits

Example 2.1.8 Find lim sin x

x→π

[Ans: 0] Theorem 2.4 (Limit of a Composite Function). If f is continuous at b and lim g(x) = b, then lim f (g(x)) = f (b). In other words, [20] x→a

x→a

lim f (g(x)) = f

x→a

lim g(x)

x→a

Example 2.1.9 Find π limπ sin 2θ + θ→ 3 3

[Ans: 0]

Problems big giant blue-green Calculus book p. 85: # 1,2, 15, 17, 21, 23, 25, 29, 31 2.A-1 Go to http://www.calculus-help.com/funstuff/phobe.html Watch: (a) Chapter 1, Lesson 1: What is a Limit? (b) Chapter 1, Lesson 2: When Does a Limit Exist? (c) Chapter 1, Lesson 3: How do you evaluate a limit? (d) Chapter 2, Lesson 1: The Difference Quotient. What differences in terminology do you notice? What did the tutorial help to clarify? 2.A-2 For the function f graphed in Figure 2.2, find the indicated limit or function value, or state that it does not exist. [21] (a) lim f (x) x→−3

(b) f (−3) (c) f (−1) (d) lim f (x) x→−1

Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

Figure 2.2: [21]

(e) f (1) (f) lim f (x) x→1

(g) lim− f (x) x→1

(h) lim+ f (x) x→1

[Ans: 2; 1; d.n.e.; 2.5; 2; d.n.e.; 2; 1] Figure 2.3: [21]

2.A-3 Follow the directions of problem 2 for the function graphed in Figure 2.3. [21] [Ans: d.n.e.; 1; 1; 2; 1; d.n.e.; 1; d.n.e.]

Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 2: Limits at Cancelable Discontinuities

2.2

Limits at Cancelable Discontinuities

Advanced Placement Limits of functions (including one-sided limits). • An intuitive understanding of the limiting process. • Calculating limits using algebra. • Estimating limits from graphs or tables of data. Derivative at a point. • Slope of a curve at a point. • Tangent line to a curve at a point. • Instantaneous rate of change as the limit of average rate of change. Textbook §1.3 Computation of Limits and §2.1 Tangent Lines and Velocity [16] Resources §1-5 Limit of a Function in Foerster [10]. Exploration 1-5: “Introduction to Limits” in [9]. §2.4 Introduction to Limits in Varberg, et al. [21]

2.2.1

Limits at Cancelable Discontinuities

Theorem 2.5 (Functions that Agree at All But One Point). Let c be a real number and let f (x) = g(x) for all x 6= c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f (x) also exists and lim f (x) = lim g(x).

x→c

In order to find the lim f (x), the value of f (c) is irrelevant. Only the values x→c infinitesimally close to c matter. x3 −8 x→2 x−2

Example 2.2.1 Find lim

Example 2.2.2 Find lim

x→3

by comparing it to lim x2 + 2x + 4. x→2

3−x x2 − 9 Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

Why are limits important in calculus? Recall that f 0 (2), the derivative of f (x) at x = 2, may be considered two ways, both of which are average rates of change over an interval that always includes x = 2, but where the interval gets smaller and smaller: • the average rate of change of f form 2 to 2 + h, where h is getting smaller f (2 + h) − f (2) and smaller, i.e., as h → 0; (2 + h) − 2 • the average rate of change of f from 2 to x, where x is getting closer and f (x) − f (2) closer to 2, i.e., as x → 2. x−2 f (2 + h) − f (2) is a forward (2 + h) − 2 difference quotient, but if h < 0, i.e., h is negative, then you have a backward difference quotient. Also recall that for the limit as h → 0, you would include both positive and negative values of h. Remember that if h > 0, i.e., h is positive, then

Example 2.2.3 For f (x) = x3 , find lim

∆x→0

f (2 + ∆x) − f (2) (2 + ∆x) − 2

Example 2.2.4 In your mighty, mighty groups of four: Pick a quadratic function f (x). Find (a) f (2) (b) lim f (x) x→2

f (2 + h) − f (2) ; (2 + h) − 2

(d) lim

f (x) − f (2) . x−2

h→0

x→2

Talk about what each of these things means, in terms of the graph of f (x + h) − f (x) f . Put your answers on the board. When you finish, find lim h→0 h

2.2.2

De-rationalizing with Conjugates

Sometimes we have to squeeze the factors out of the expression. One technique for doing that when we have radicals is to use conjugates. Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 2: Limits at Cancelable Discontinuities

Example 2.2.5 [20] Find lim √

t→0

t2 t2 + 9 − 3

Example 2.2.6 [16] Find √ 4− x lim x→16 x − 16 Example 2.2.7 In your mighty, √ mighty groups of four: pick a linear function mx + b. Let f (x) = mx + b. Make sure that f (2) exists. (a) f (2) (b) lim f (x) x→2

f (x) − f (2) . x−2 f (2 + h) − f (2) (d) lim ; h→0 (2 + h) − 2 (c) lim

x→2

Talk about what each of these things means, in terms of the graph of f . When you finish, put your answers on the board. When you f (x + h) − f (x) finish, find lim h→0 h Example 2.2.8 Find √ lim

∆x→0

x + ∆x − ∆x

Note: compare this problem to lim

∆x→0

√

f (x+∆x)−f (x) , ∆x

which is the for-

mula for the instantaneous rate of change, √ or... derivative. This x, which means that problem is finding the derivative for f (x) = √ f (x + ∆x) = x + ∆x.

2.2.3

Derivative at a Point

Definition 2.3 (Derivative (at x = c form)). [16] The derivative of f (x), with respect to x, at the point x = c is given by

f 0 (c) = lim

x→c

f (x) − f (c) x−c Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

e.g., f 0 (2) = lim

x→2

f (x) − f (2) x−2

Meaning: The instantaneous rate of change of f with respect to x at x = c. Graphically: The slope of the line tangent to the graph of f at the point x = c. Example 2.2.9 Let f (x) = x4 . Find f 0 (2). In your mighty, mighty groups of four: Find (a) f 0 (−1); (b) f 0 (3); (c) f 0 (−2). What is the pattern? Example 2.2.10 The graph of the function f shown in Figure 2.4 consists of a semicircle and three line segments. Find the following Figure 2.4: Graph of f

limits of difference quotients: f (x) − f (−2) x→−2 x+2 f (x) − f (2.3) lim x→2.3 x − 2.3 f (x) − f (−3) lim x+3 x→−3− f (x) − f (−3) lim x+3 x→−3+ f (x) − f (−3) lim x→−3 x+3

(a) lim (b) (c) (d) (e)

Ans: − 13 [Ans: −1] [Ans: 2] Ans: − 13 [Ans: d.n.e.]

Example 2.2.11 Each of the following limits are derivatives. Tell for which function, and for what point Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 2: Limits at Cancelable Discontinuities

x4 − 81 x→−3 x + 3 √ t−5 (b) lim t→25 t − 25 (a) lim

x2 + x − 6 x→2 x−2

Using conjugates to fetch factors Example √ of the line tangent to the graph √ 2.2.12 Write an equation of y = 2x + 4 at the point −1, 2

2.2.4

Ans: y −

√

√1 2

(x + 1)

De-denominatorizing with LCDs

Example 2.2.13 Find 1 1 − x + 1 4 lim x→3 x−3 Example 2.2.14 In your mighty, mighty groups of four: Let f (x) = 1 . Pick a value of c 6= − 13 3x + 1 (a) f (c) (b) lim f (x) x→c

f (x) − f (c) . x→c x−c f (c + h) − f (c) (d) lim ; h→0 (c + h) − c (c) lim

Talk about what each of these things means, in terms of the graph of f . When you finish, put your answers on the board. After you put your answers on the board: How would you find the point at f (x + h) − f (x) which the derivative is − 43 ? Also, find lim . h→0 h Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) Example 2.2.15 Find 1 1 − x + h x lim h→0 h Note: compare this problem to lim

h→0

f (x+h)−f (x) , h

which is the formula

for the instantaneous rate of change, or... derivative. This problem is finding the derivative for f (x) = x1 , which means that f (x + h) = 1 x+h .

Problems big giant blue-green Calculus book p. 95: Writing Exercises # 2; #3, 11, 19 p. 156: # 23 x2 − a2 is x→a x4 − a4

2.B-1 (adapted from ?) If a 6= 0, then lim

 2 x − 64   x 6= 8 x−8 2.B-2 f (x) =   k x=8 What value of k will make f continuous at x = 8?

Ans:

1 2a2

[Ans: 16]

2.B-3 [16] Evaluate analytically, then check your answer by making a table on your handy-dandy calculator. x2 + x − 6 x→−3 x2 − 9 2 (x + h) − 2 (x + h) + 1 − x2 − 2x + 1 (b) lim h→0 h (a) lim

2.B-4 Evaluate lim √ v→3

3−v √ . v− 3

[Ans: 2x − 2] √ Ans: −2 3

2.B-5 [16] Evaluate analytically, then check your answer by making a table on your handy-dandy calculator, if possible. √ 4− x (a) lim Ans: − 18 x→16 x − 16 √ x+5−3 (b) lim Ans: 16 x→4 x−4 p √ h i 3 (x + h) − 3x (c) lim Ans: 2√33x h→0 h Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 2: Limits at Cancelable Discontinuities

1 1 − 2 − h 2. 2.B-6 Find lim h→0 h

Ans:

1 4

2 2 − 2.B-7 Find lim 2x − 1 3 . This limit happens to be the derivative of what x→2 x−2 i h 2 at x = 2 function, at what point? Ans: − 94 ; 2x−1 2.B-8 Find lim

x→3

x−3 2 2 − 5x − 8 5

49 Ans: − 10

1 1 − 3 (x + ∆x) + 1 3x + 1 2.B-9 Find lim ∆x→0 ∆x √ √ x x− 8 2.B-10 lim is the derivative of what function for what value of x? x→2 x−2 Evaluate the limit, and check your answer by making a table on your h √ i √ calculator. Ans: f 0 (2) for f (x) = x x; 3 2 2 2.B-11 Use the limit definition of the derivative to write an equation of the line tangent to f (x) = x2 at the point (−4, f (−4)). [Ans: y − 16 = −8 (x + 4)] 2.B-12 Each of the following limits is a derivative. For each limit, • state for what function it is a derivative, and at what x-value; • approximate the limit numerically, using a table on your calculator; • confirm your answer using algebraic techniques, without using a calculator. x2 − 1 x→−1 x + 1 x3 − 8 (b) lim x→2 x − 2 1 1 − 2 + x 2 (c) lim x→0 x √ √ x+2− 2 (d) lim x→0 x (a) lim

Ans: x2 at −1 Ans: x3 at 2

h Ans:

1 2+x

at 0

√ Ans: x + 2 at 0

Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 3: Limit Definition of Derivative as a Function

2.3

Limit Definition of Derivative as a Function

Advanced Placement Limits of functions. • An intuitive understanding of the limiting process. • Calculating limits using algebra. • Estimating limits from graphs or tables of data. Derivative at a point. • Slope of a curve at a point. • Tangent line to a curve at a point. • Instantaneous rate of change as the limit of average rate of change. Derivative as a function. Textbook §2.1 Tangent Lines and Velocity and §2.2 The Derivative [16] Resources §3-2 Difference Quotients and One Definition of Derivative in Foerster [10]. Exploration 3-2: “Exact Value of a Derivative” and Exploration 3-3 “Numerical Derivative by Grapher” in [9].

2.3.1

Derivative as a Function

1 Example 2.3.1 For f (x) = , find f 0 (−1), f 0 (0), f 0 (1), f 0 (2), 3x + 1 etc. Instead of using the formula for the derivative at a point formula several times, we will find an expression for f 0 (x), and then simply plug in −1, 0, 1, and 2. Definition 2.4 (Derivative (∆x or h form)). [16] The derivative function of f (x) is given by

f 0 (x) = lim

∆x→0

∆y f (x + ∆x) − f (x) f (x + h) − f (x) = lim = lim h→0 ∆x ∆x→0 ∆x h Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

This gives me a formula for finding the derivative at any generic point, x, instead of only at a specific point like 2 or −1. Note that this formula can be adapted to find the derivative at a specific point by replacing x with a specific value, like 2 or −1.

Example 2.3.2 Find the point on the graph of y = 5 3 the tangent line is parallel to y = − x + . 4 4

1 where 3x + 1

First, let’s graph the function on our calculator, with the line − 43 x + 54 . See if you can spot the point where the tangent line is parallel to the line that we’ve graphed. As it turns out, there are two points that will work for this problem. √ Example 2.3.3 Find the point on the graph of y = 7x + 4 where the tangent line is perpendicular to the line 10x + 7y = 5, i.e., where the normal line is parallel to 10x + 7y = 5.

Example 2.3.4 (adapted from [2]) A function g is defined for all real numbers and has the following property: g(a + b) − g(a) = 6a2 b + 6ab2 + 2b3 − 3b. Find g 0 (x).

Ans: 6x2 − 3

Example 2.3.5 (BC89) Let f be a function that is everywhere differentiable and that has the following properties. (i) f (x + h) =

f (x) + f (h) for all real numbers h and x. f (−x) + f (−h)

(ii) f (x) > 0 for all real numbers x. (iii) f 0 (0) = −1. (a) Find the value of f (0). 1 (b) Show that f (−x) = for all real numbers x. f (x)

[Ans: 1]

(c) Using part (b), show that f (x + h) = f (x)f (h) for all real numbers h and x. (d) Use the definition of the derivative to find f 0 (x) in terms of f (x). [Ans: f 0 (x) = −f (x)] Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 3: Limit Definition of Derivative as a Function

Notation dy dy The derivative of a function f (x) might be written as f 0 (x) or . is called dx dx Liebniz notation, and is technically the derivative of y. Other notations include d f (x). Dx f (x) and dx

Example 2.3.6 Find

2.3.2

1 dy when y = dx x

Tangent Lines

Example 2.3.7 (adapted from BC97) Refer to the graph in Figure 2.5. The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown. Think about what the graph is telling you. An equation of a line tangent to the graph of f is 3x â&#x2C6;&#x2019; y = â&#x2C6;&#x2019;1. Figure 2.5:

(a) What is the x-coordinate of the point of tangency? (b) What is the y-coordinate of the point of tangency?

[Ans: (1, 4)] Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) Example 2.3.8 (adapted from [2]) If p(x) = (x + 2) (x + k) and if the line tangent to the graph of p at the point (4, p(4)) is perpendicular to the line 2x + 4y + 5 = 0, then k =

[Ans: −8] Example 2.3.9 (adapted from [2]) If the line 3x − y + 5 = 0 is tangent in the second quadrant to the curve y = x3 + k, then k =

[Ans: 3]

Problems 2.C-1 Find g 0 (x) if g(a + b) − g(a) is defined as follows: Ans: 3x2

(a) 3a2 b + 3ab2 + b3 (b) 2ab + b2 − 2b

[Ans: 2x − 2]

2.C-2 Suppose E(x) is a function for which E(x + h) − E(x) = E(x) (E(h) − 1) E(h) − 1 and lim = 1. Show that E 0 (x) = E(x), i.e., show that, if these h→0 h criteria are met for a function, then that function is its own derivative. 2.C-3 Use the limit definition of the derivative to find: (a) f 0 (x) if f (x) = 12 − x2 ; (b) (c) (d)

dy dx d dx dy dx 0

if y = 14 x3 ; 3x2 − 4x + 1 ; √ if y = x;

(e) f (x) if f (x) = 2x2 + x + k, where k is a constant; d 1 (f) dx x h i 1 12 Ans: −2x; 43 x2 ; 6x − 4; 2√ ; 4x + 1; − x x 2.C-4 The following limit is the derivative of what function: q √ 2 3 2 (x + ∆x) − 1 − 3 2x2 − 1 lim ∆x→0 ∆x √ Ans: 3 2x2 − 1 Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 3: Limit Definition of Derivative as a Function

2.C-5 (BC90) Let f (x) = 12 − x2 for x ≥ 0 and f (x) ≥ 0. The line tangent to the graph of f at the point (k, f (k)) intercepts the x-axis at x = 4. What is the value of k? [Ans: k = 2] 1 2.C-6 (adapted from AB97) At what point on the graph of y = x3 is the 4 tangent line parallel to the line 3x − 4y = 7? Ans: 1, 41 2.C-7 (adapted from [2]) Let f (x) = 3x3 −4x+1. An equation of the line tangent to y = f (x) at x = 2 is [Ans: y = 32x − 47] √ 2.C-8 (adapted from [2]) Find the point on the graph of y = x between (4, 2) and (9, 3) at which the normal to the graph is perpendicular line to the 5 through (4, 2) and (9, 3). Ans: 25 , 4 2 2.C-9 (adapted from [3]) If the graph of the parabola y = 2x2 + x + k is tangent to the line 3x + y = 3, then k = [Ans: 5]

Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 4: Basic Calculus of Polynomials

2.4

Basic Calculus of Polynomials

Advanced Placement Applications of derivatives • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and accleeration. Computation of derivatives • Knowledge of derivatives of basic functions, including power. • Basic rules for the derivative of sums of functions. Textbook §2.3 Computation of Derivatives: The Power Rule [16] Resources Explorations 3–4a: “Algebraic Derivative of a Power Function”, 3–5a: “Velocity and Acceleration Reading”, and Exploration 3–5b: “ Deriving Velocity and Acceleration Data from Displacement Data” in [9].

2.4.1

Notation

Notation dy dy . is called The derivative of a function f (x) might be written as f 0 (x) or dx dx Liebniz notation, and is technically the derivative of y. Other notations include d Dx f (x) and f (x). dx Example 2.4.1 Find the f 0 (x) for f (x) = x4 Plug in 2, −1, and 3. Compare your answers to the answers you got by using the at x = c form of the derivative for each point. Example 2.4.2 Find the derivative of x3 Example 2.4.3 Find the derivative of x3 + 1 Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) Example 2.4.4 Find f 0 (x) for f (x) =

Example 2.4.5 Find

2.4.2

â&#x2C6;&#x161;

x and for f (x) =

â&#x2C6;&#x161;

x3 + 1

dy 1 1 when y = and when y = 3 dx x x +1

Basic Properties of Derivatives

Derivative of a Constant If f (x) = k, then f 0 (x) = 0, i.e.,

d k=0 dx

Derivative of a Line If f (x) = mx + b, then f 0 (x) = m, i.e.,

d (mx + b) = m dx

Derivative of a Sum (or Difference)

If f (x) = g(x) + h(x), then f 0 (x) = g 0 (x) + h0 (x), i.e.,

d du dv (u + v) = + . dx dx dx

Derivative of a Scalar Multiple d ky = If f (x) = k g(x), where k is some constant, then f 0 (x) = k g 0 (x), i.e., dx dy k . dx Theorem 2.6. Linearity of Differentiation d [a f (x) + b g(x)] = a f 0 (x) + b g 0 (x) dx Differentiation will be intuitive as long as you are adding, or multiplying by a constant. When things get more complicated than that, things will get more complicated than that. Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 4: Basic Calculus of Polynomials

2.4.3

Power Rule Power Rule d n x = nxn−1 dx

To differentiate a power function y = axb : • Multiply the coefficient by the old exponent; • Lower the exponent by one.

Example 2.4.6 (adapted from AB97) If f (x) = x3 + x − f 0 (−1) =

1 , then x

[Ans: 5] √ Example 2.4.7 (adapted from [2]) For f (x) = x, find the point on the graph, between (1, 1) and (4, 2), where the slope of the tangent line is equal to the slope of the line between (1, 1) and (4, 2).

9 3 Ans: , 4 2 √ Example 2.4.8 (adapted from AB97) Let f (x) = x 3 x. If the rate of change of f at x = c is thrice its rate of change at x = 1, then c =

[Ans: 27]

Example 2.4.9 (AB86) Let f be the function defined by f (x) = 7 − 15x + 9x2 − x3 for all real numbers x. Write an equation of the line tangent to the graph of f at x = 2.

[Ans: y − 5 = 9 (x − 2)] Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) Example 2.4.10 (adapted from AB98) Write an equation (in slopeintercept form) for the line tangent to the graph of f (x) = x4 − x2 at the point where f 0 (x) = 1 [Ans: y = x − 1.055] √ 3 Example 2.4.11 Use the power rule to find lim

h→0

8+h−2 h 1 Ans: 12

2.4.4

Higher Order Derivatives

f 00 (x) means the derivative of f 0 (x), or the second derivative of f (x). f 000 (x) would be the third derivative of f (x). Example 2.4.12 Find f 000 (x) if f (x) = x3 + 1. [Ans: 6] d dy In Liebniz notation, the second derivative is . To abbreviate this, dx dx 2 d y d d is multiplied: . In reality, is very much not we pretend like the 2 dx dx dx being multiplied, but it is a useful notation. (Think of the dx2 on the bottom d6 y 2 as (dx) .) The sixth derivative would be . dx6

Example 2.4.13 Find

d2 2t3 − 6t2 + 5 dt2 [Ans: 12t − 12]

2.4.5

Kinematics

Velocity is the derivative of displacement: v(t) = d0 (t) Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 4: Basic Calculus of Polynomials

Acceleration is the derivative of the velocity: a(t) = v 0 (t)

Acceleration is the derivative of the derivative of displacement. Acceleration is the second derivative of displacement: a(t) = v 0 (t) = s00 (t)

Also: speed is the magnitude of velocity. In one dimension, this means that speed is the absolute value of velocity. speed(t) = |v(t)|

Example 2.4.14 A particle moves along the x-axis so that its position at time t, where t is in seconds, is given by d(t) = 2t3 − 6t2 + 5, where d(t) is given in feet. What is the velocity when the acceleration is zero?

[Ans: −6 feet/sec]

Problems Power Rule 2.D-1 (AB89) Let f be the function given by f (x) = x3 − 7x + 6. (a) Find the zeros of f . [Ans: 1, 2, −3] (b) Write an equation of the line tangent to the graph of f at x = −1. [Ans: y − 12 = −4 (x + 1)] x4 − 625 is the derivative of what function, at what point? Use the x→5 x − 5 power rule to evaluate the limit, without going through all that factor and cancel shtuff. [Ans: 500] √ 4+h−2 2.D-3 As with the problem before, lim is a derivative at a point. h→0 h Evaluate the limit, by identifying at which point and for which function this is the derivative, then use the power rule instead of squeezing out an h using conjugates. Ans: 14 2.D-2 lim

Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits)

2.D-4 [10] Misconception Problem Mae Danerror needs to find f 0 (3), where f (x) = x4 . She substitutes 3 for x, gets f (3) = 81, differentiates 81, and gets zero for the answer. Explain why she also gets zero for her grade. x3 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; 3x + 5, plot the graphs of f and f 0 on the 3 same screen. Show that each place where the f 0 graph crosses the x-axis corresponds to a high or low point on the f graph.

2.D-5 [10] For f (x) =

2.D-6 [15] The graphs of a function f and its derivative f 0 are given on the same coordinate axes in Figure 2.6. Label the graphs as f or f 0 and state the Figure 2.6: [15]

reasons for your choice. 2.D-7 [15] The graphs of a function f and its derivative f 0 are given on the same coordinate axes in Figure 2.7. Label the graphs as f or f 0 and state the Figure 2.7: [15]

reasons for your choice. Mr. Budd, compiled September 29, 2010

AP Unit 2, Day 4: Basic Calculus of Polynomials

2.D-8 (adapted from [2])

d 2x3 ln e = dx

Ans: 6x2

2.D-9 [17] When an oil tank is drained for cleaning, there are V (t) = 100, 000 − 4000t + 40t2 gallons of oil left in the tank t minutes after the drain valve is opened. (a) At what average rate does oil drain from the tank during the first 20 minutes? [Ans: 3200 gal/min] (b) At what rate does oil drain out of the tank 20 minutes after the drain valve is opened? [Ans: 2400 gal/min] (c) Explain what V 00 (t) says about the rate at which oil is draining from the tank. [Ans: rate of change is increasing, i.e., rate is getting less negative, i.e., rate of drainage is decreasing] 2.D-10 Find the derivative of x3 + x2 − 2, x3 + x2 − 1, x3 + x2 , and x3 + x2 + 58.7 2 Ans: 3x + 2x 2.D-11 Work backwards: find the functions, f (x), for the following derivatives: (a) f 0 (x) = 6x5 (b) f 0 (x) = 13x12 (c) f 0 (x) = 10x (d) f 0 (x) = 3x2 + 2x (e) f 0 (x) = m 0 if f 0 is the derivative Make up6 a word: of f , then f is the (blank) of f . 13 2 3 2 Ans: x ; x ; 5x ; x + x ; mx + b;

Kinematics 2.D-12 (MM99(2)) A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by h = 2 + 20t − 5t2 , t ≥ 0. (a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released.) [Ans: 2 m] (b) Show that the height of the ball after one second is 17 metres. (c) At a later time the ball is again at a height of 17 metres.

(d)

i. Write down an equation that t must satisfy when the ball is at a height of 17 metres. ii. Solve the equation algebraically. [Ans: t = 1 s, t = 3 s] dh . i. Find dt Mr. Budd, compiled September 29, 2010

AP Unit 2 (Limits) ii. Find the initial velocity of the ball (that is, velocity at the instant when it is released). [Ans: v(0) = 20 m/s] iii. Find when the ball reaches its maximum height. [Ans: t = 2 s] iv. Find the maximum height of the ball. [Ans: 22 m]

2.D-13 [10] Find equations for the velocity, v, and the acceleration, a, of a moving object if y = 5t4 −3t2.4 +7t is its displacement. Ans: v = 20t3 − 7.2t1.4 + 7, a = 60t2 − 10.08t0.4 2.D-14 [10] Car Problem Calvin’s car runs out of gas as it is going up a hill. The car rolls to a stop, then starts rolling backward. As it rolls, its displacement, d(t) feet, from the bottom of the hill at t seconds since Calvin’s car ran out of gas is given by d(t) = 99 + 30t − t2 . (a) Plot graphs of d and d0 on the same screen. Use a window large enough to include the point where the d graph crosses the positive t-axis. Sketch the result.

(b) For what range of times is the velocity positive? How do you interpret this answer in terms of Calvin’s motion up the hill? [Ans: Velocity is positive for 0 ≤ t < 15. C

(c) At what time did Calvin’s car stop rolling up and start rolling back? How far was it from the bottom of the hill at this time? [Ans: At 15 seconds, his car stopped. d (d) If Calvin doesn’t put on the brakes, when will he be back down at the bottom of the hill? [Ans: t = 33 sec] (e) How far was Calvin from the bottom of the hill when the car ran out of gas? [Ans: 99 feet from the bottom]

Mr. Budd, compiled September 29, 2010

Unit 3

Basic Differentiation 1. Finding Derivatives Using Limits 2. Power Rule - Derivatives 3. Antiderivatives of Polynomials 4. Product and Quotient Rules 5. Chain Rule 6. Tangent Lines AB

I. Derivatives Concept of the derivative. • Derivative presented graphically, numerically, and analytically. • Derivative interpreted as an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. Derivative at a point. • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. • Instantaneous rate of change as the limit of average rate of change. 89

AP Unit 3 (Basic Differentiation) • Approximate rate of change from graphs and tables of values. Derivative as a function. • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 . Applications of derivatives. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. Computation of derivatives. • Knowledge of derivatives of basic functions, including power functions. • Basic rules for the derivative of sums, products, and quotients of functions. • Chain rule.

II. Integrals Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions.

Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 1: Antidifferentiation of Polynomials

3.1

Antidifferentiation of Polynomials

Advanced Placement Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. Textbook §4.1 Antiderivatives [16] Resources Exploration 5–2b: “A Motion Antiderivative Problem” in [9].

3.1.1

Notation of Antiderivatives

We need a notation for antiderivative that is as confusing as humanly possible, so we will use a notation already available to us for a completely unrelated concept: the definite integral. The antiderivative of f 0 (x) is written as Z f 0 (x) dx Notice that the function, f 0 (x), that you are antidifferentiating is surrounded R by two things. On the right side is the integration sign, . The integral sign is used because antiderivatives are also known as indefinite integrals. On the right, is dx, which tells us what the variable is. How will you be able to tell the difference between a definite integral, and an antiderivative?

3.1.2

Anti-Power Rule

Revisit problem 11 on page 87. To anti-differentiate, we need to do the opposite of differentiation, in the opposite order. Mr. Budd, compiled September 29, 2010

AP Unit 3 (Basic Differentiation)

Theorem 3.1. Anti-Power Rule Z

xn dx =

xn+1 +C n+1

To antidifferentiate a power function y = axb : • Raise the exponent by one; • Divide by the new exponent; • Add an arbitrary constant C (the antiderivative of 0). The antiderivative of a scalar multiple is the scalar multiple of the antiderivative. Z Z k f (x) dx = k f (x) dx As long as you are multiplying a function by a constant, you can pull the constant out. For your own safety, never try to pull a variable out of the antiderivative: 63% of the time when you pull a variable out of the antiderivative, your pencil will explode. If it doesn’t happen the first time, know that you got lucky, and you are tempting the fates by trying it again. The antiderivative of a sum is the sum of the antiderivatives: Z Z Z (f (x) + g(x)) dx = f (x) dx + g(x) dx Theorem 3.2. Linearity of Antidifferentiation Z Z Z [a f (x) + b g(x)] dx = a f (x) dx + b g(x) dx

Antidifferentiation is intuitive, so long as you are adding(/subtracting) functions, or multiplying by a constant. It will not be intuitive for multiplication or division.

Example 3.1.1 (adapted from AB93)

x3 + x

dx =

x7 2x5 x3 Ans: + + +c 7 5 3 Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 1: Antidifferentiation of Polynomials

Notice that we cannot get the correct answer by squaring the antiderivative of 4 3 x x2 4 + 2 2 4 2 x3 + x, i.e., the answer is not x4 + x2 + C, or even + C. All 3 n we know how to antidifferentiate right now is power functions (ax ) and sums of power functions. x3 − 2 g(x + h) − g(x) , = h→0 h x3

Example 3.1.2 (adapted from [3]) If lim then g(x) could be equal to (A) −2x−4 (B) 6x−4 x3 + 1 x2 (D) x + x2 (C)

(E) 1 − 2x−3 x3 + 1 Ans: x2 Notice that we cannot find the antiderivative of a quotient by taking the quotient x4 − 2x +C of the antiderivatives: i.e., the answer is not 4 x4 4

Example 3.1.3 (BC86) For all real numbers x and y, let f be a function such that f (x + y) = f (x) + f (y) + 2xy and such that f (h) lim = 7. h→0 h (a) Find f (0). Justify your answer.

[Ans: 0] 0

(b) Use the definition of the derivative to find f (x). [Ans: 7 + 2x] (c) Find f (x). Ans: 7x + x2

3.1.3

Kinematics

dx Velocity is the derivative of displacement (v(t) = ), so that displacement is dt the antiderivative of velocity: Z x(t) = v(t) dt Mr. Budd, compiled September 29, 2010

AP Unit 3 (Basic Differentiation)

Z x(t) =

x0 (t) dt dx dt dt

dv d2 x Acceleration is the derivative of the velocity (a(t) = = 2 ), so that velocity dt dt is the antiderivative of acceleration: Z v(t) = a(t) dt or

Z v(t) =

v 0 (t) dt dv dt dt

Example 3.1.4 (adapted from AB93) The acceleration of a particle moving along the x-axis at time t is given by a(t) = 6t − 4. If the velocity is 18 when the t = 3 and the position is 11 when t = 1, then the position x(t) =

Ans: t3 − 2t2 + 3t + 9

Example 3.1.5 Exploration 3-9: “Displacement and Acceleration from Velocity”

3.1.4

General vs. Particular Solutions

t1.6 In antidifferentiating v(t) = 50+6t0.6 [9], the displacement is d(t) = 50t+6 + 1.6 C. Notice the +C, which is adding some arbitrary constant, the antiderivative of +0. This d(t) with the +C is a family of parallel curves. There are an infinite number of curves in this family, corresponding to the infinite number of possibilities of C. Note that these curves are considered parallel because their slopes are the same for every x value. If your antiderivative includes +C, it is called a general solution. Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 1: Antidifferentiation of Polynomials

If you want one function d(t) instead of a family of functions, what you want is a particular solution, instead of the general solution. In order to narrow the general solution to a particular solution, you need to incorporate some more information into your answer. This is done by solving for C, after plugging in some known point (t, d(t)) into the formula for d(t). For example, if v(t) = 50 + 6t0.6 , then we can antidifferentiate to get the general solution for the displacement, d(t) = 50t + 3.75t1.6 + C. If we want a particular solution, we will have to use some more information. In this particular example, for instance, we may know that the displacement at time 0 is 100. Then we solve 1.6 100 = 50 (0)+3.75 (0) +C, to find that C = 100. Thus, our particular solution would be d(t) = 50t + 3.75t1.6 + 100. If the point used is (0, d(0)), this is said to be an initial value. Antidifferentiation problems that ask you to find a particular solution using an initial value are called IVP’s, or Initial Value Problems.

Problems 3.A-1 [10] Find a function whose derivative is given. That is, write the general equation for the antiderivative. (a) f 0 (x) = 7x6 (b) f 0 (x) = x5 (c) f 0 (x) = x−9 7

(d) f 0 (x) = 36x 2 1 1 9 Ans: x7 + C, x6 + C, − x−8 + C, 8x 2 + C 6 8 3.A-2 [10] Find the particular function f (x) that has the given function f 0 (x) for its derivative and contains the given point. (a) f 0 (x) = x4 and f (1) = 10 (b) f 0 (x) = x2 − 8x + 3 and f (−2) = 13 113 Ans: 51 x5 + 9.8, 13 x3 − 4x2 + 3x + 3 3.A-3 [10] Derivative and Antiderivative Problem Let g 0 (x) = 0.6x. (a) Find the general equation for the antiderivative, g(x). Ans: g(x) = 0.3x2 + C (b) Find the particular equation for g(x) in each case. i. g(0) = 0

Ans: 0.3x2 Mr. Budd, compiled September 29, 2010

AP Unit 3 (Basic Differentiation) Ans: 0.3x2 + 3 Ans: 0.3x2 + 5

ii. g(0) = 3 iii. g(0) = 5

(c) Plot the graph of g 0 (x) and the three graphs for g(x) on the same screen, then sketch the results. Why are the three graphs of g(x) called a family of functions? 3.A-4 [2] If functions f and g are defined so that f 0 (x) = g 0 (x) for all real numbers x with f (1) = 2 and g(1) = 3, then the graph of f and the graph of g (A) intersect exactly once; (B) intersect no more than once; (C) do not intersect; (D) could intersect more than once; (E) have a common tangent at each point of tangency. [Ans: C] Z

3.A-5 (adapted from AB93) x2 + 2 4 x5 + x3 + 4x + C Ans: 5 3 Z 2 3.A-6 3 (ice) d (ice)

dx =

[Ans: iceberg]

d2 y is the second derivative, what do you suppose would be the meaning dx2 −1 d y of ? dx−1

3.A-7 If

3.A-8 (BC86) For all real numbers x and y, let f be a function such that f (x + f (h) y) = f (x) + f (y) + 2xy and such that lim = 7. h→0 h (a) Find f (0). Justify your answer. (b) Use the definition of the derivative to find f 0 (x). (c) Find f (x). 3.A-9 [10] Displacement Problem Ann Archer shoots an arrow into the air. Let d(t) be its displacement above the ground at time t seconds after she shoots it. From physics she knows that the velocity is given by d0 (t) = 70 − 9.8t. (a) Write the general equation for d(t). Ans: 70t − 4.9t2 + C Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 1: Antidifferentiation of Polynomials

(b) Write the particular equation for d(t), using the fact that Ann is standing on a platform that puts the bow 6 m above the ground when she shoots the arrow. Ans: 70t − 4.9t2 + 6 (c) How far is the arrow above the ground when t = 5? When t = 6? When t = 9? How do you explain the relationship among the three answers? [Ans: 233.5 m, 249.6 m, 239.1 m] (d) When is the arrow at its highest? How high is it at that time? [Ans: 256 m, at about t = 7.1 sec] 3.A-10 (adapted from [2]) At t = 0, a particle starts at the origin with a velocity of 6 feet per second and moves along the x-axis in such a way that at time t its acceleration is 24t2 feet per second per second. Through how many feet does the particle move during the first 2 seconds? [Ans: 44 feet] 3.A-11 (adapted from [2]) The acceleration, a(t), of a body moving in a straight line is given in terms of time t by a(t) = 4 − 6t. If the velocity of the body is 20 at t = 0 and if s(t) is the distance of the body from the origin at time t, what is s(2) − s(1)? [Ans: 19]

Mr. Budd, compiled September 29, 2010

AP Unit 3 (Basic Differentiation)

Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 2: Product and Quotient Rules

3.2

Product and Quotient Rules

Advanced Placement Concept of the derivative • Derivative presented graphically, numerically, and analytically. Applications of derivatives • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. Computation of derivatives • Basic rules for the derivative of products and quotients of functions. Textbook §2.4 The Product and Quotient Rules [16] Resources Exploration 4-2: “Derivative of a Product” in [9]. §3.3 The Product and Quotient Rules in [20].

3.2.1

Product Rule

Example f (x) = x7 + 1, g(x) = x5 − 4, and p(x) = 3.2.1 Consider 7 5 12 x + 1 x − 4 = x − 4x7 + x5 − 4. (a) Write an equation of the line tangent to f (x) at x = 1. Write it in the form y = k + m (x − 1), i.e., treat (x − 1) as a single entity, which you will leave alone. (b) Write an equation of the line tangent to g(x) at x = 1. Write it in the form y = a0 + a1 (x − 1). (c) Write an equation of the line tangent to p(x) at x = 1. As before, treat (x − 1) as a single entity. (d) Multiply the tangent lines at x = 1 for f (x) and g(x), keeping (x − 1) as a single variable unto itself: don’t worry about distributing it. The only like terms you need to combine are the (x − 1) terms. Your answer should look like y = a0 + 2 a1 (x − 1) + a2 (x − 1) . What do you see? Mr. Budd, compiled September 29, 2010

100

AP Unit 3 (Basic Differentiation) Example 3.2.2 d 11 d 7 d 4 x 6= x · x dx dx dx (b) How could we squeeze the real derivative out of x4 and x7 . Think about: how can we get the correct exponents, and the correct coefficients. (a) Show that

Product Rule du dv d (u · v) = v+u dx dx dx

If h(x) = f (x)g(x), then h0 (x) = f 0 (x)g(x) + f (x)g 0 (x)

Derivative of First · Second + First · Derivative of Second.

Note that the order can be rearranged: Derivative of First · Second + Derivative of Second · First. First · Derivative of Second + Derivative of First · Second. First · Derivative of Second + Second · Derivative of First. Do whichever way you remember best.

Example 3.2.3 Use linear approximations of f (x) and g(x) at x = c to demonstrate the product rule.

Example 3.2.4 Use the limit definition of the derivative to show that, if p(x) = f (x)g(x), then p0 (x) = f 0 (x)g(x) + f (x)g 0 (x).

Example 3.2.5 Suppose that f is the function shown in Figure 3.1 and that h(x) = x2 f (x) [17]. Evaluate h0 (2) and h0 (4)

[Ans: −16, 64] Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 2: Product and Quotient Rules

101

Figure 3.1: Graph of f [17]

2 Example 3.2.6 If f (x) = x3 + 1 , find f 0 (x)

Example 3.2.7 Find

3 i d h 3 x +1 . dx

2 This can be done two ways, one using the derivative of x3 + 1 , and splitting 3 2 x3 + 1 into x3 + 1 x3 + 1 . Another way is to use a variation of the product rule for differentiating the product of three functions. For the product of three functions, u, v, and w: du dv dw d (u · v · w) = ·v·w+u· ·w+u·v· dx dx dx dx For e(x) = f (x)g(x)h(x), e0 (x) = f 0 (x)g(x)h(x) + f (x)g 0 (x)h(x) + f (x)g(x)h0 (x) Examine this formula and see if you can notice how the pattern for two functions has been expanded for three functions. Once you get your answer, simplify and factor it. In the back of your mind, be looking for a method to go straight to the answer very quickly. Example 3.2.8 (adapted from [2]) If p(x) = (x + 2) (x + k) and if the line tangent to the graph of p at the point (4, p(4)) is perpendicular to the line 2x + 4y + 5 = 0, then k = Mr. Budd, compiled September 29, 2010

102

AP Unit 3 (Basic Differentiation) [Ans: −8] Example 3.2.9 Use the product rule to prove the power rule (for n = 1, 2, . . . using mathematical induction.

3.2.2

Quotient Rule Quotient Rule dv du d u v dx − u dx = dx v v2

If h(x) =

(v 6= 0)

f (x) , then g(x) h0 (x) =

g(x)f 0 (x) − f (x)g 0 (x) 2

[g(x)]

LoDeHi − HiDeLo LoLo

Because of the subtraction, it is important that you keep things in this order (unlike the product rule, where order is not really important). Example 3.2.10 Use the limit definition of the derivative to prove the quotient rule. Example 3.2.11 Use linear approximations of f (x) and g(x) at x = c to demonstrate the product rule. 2 x3 + 1 dy Example 3.2.12 If y = , find . 3 x +1 dx Notice that this problem has a quick and simple answer, if you cancel x3 + 1, and take the derivative of that to be 3x2 . Confirm that the quotient rule works.

Example 3.2.13 Find

d dx

1 . x3 + 1 Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 2: Product and Quotient Rules

103

Figure 3.2:

2 3 Compare your answer with the derivative you get for x3 + 1 and x3 + 1 and be looking for the pattern, while thinking of the derivatives of x2 , x3 , and 1 x. Example 3.2.14 from 2000 AP Calculus AB-2. Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. The graph in Figure 3.2, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of Runner B is given by the function v defined by v(t) = 24t . Find the acceleration of Runner B at time t = 2 seconds. 2t + 3 Indicate units of measure. 10 = 3.333 meters/sec2 3

72 72

0 Runner B: a(2) = v (2) = = = 1.469 meters/sec2 2

49 (2t + 3)

Runner A: acceleration =

t=2

Tangent Lines When writing the equation of a line, use the point-slope form: y − y1 = m (x − x1 ) When talking about m, the slope of the tangent line, you should immediately be thinking about the derivative. m = f 0 (x1 ) y − y1 = y 0 (x1 ) (x − x1 ) Mr. Budd, compiled September 29, 2010

104

AP Unit 3 (Basic Differentiation) Example 3.2.15 (adapted from AB93) An equation of the line 3x − 2 at the point (−1, −5) is tangent to the graph of y = 2x + 3 [Ans: y + 5 = 13 (x + 1)]

Problems 3.B-1 (adapted from AB98) Let f and g be differentiable functions with the following properties: (a) g(x) < 0 for all x (b) f (1) = π − 1 If h(x) = f (x)g(x) and h0 (x) = f (x)g 0 (x), then f (x) =

[Ans: π − 1]

3.B-2 [20] (a) If F (x) = f (x)g(x), where f and g have derivatives of all orders, show that F 00 = f 00 g + 2f 0 g 0 + f g 00 (b) Find a similar formulas for F 000 . 3.B-3 Use the Product Rule to show that d d 2 [f (x)] = [f (x)f (x)] = 2f (x)f 0 (x) dx dx 3.B-4 [20] (a) Use the Product Rule twice to prove that if f , g, and h are differentiable, then d (f gh) = f 0 gh + f g 0 h + f gh0 dx (b) Taking f = g = h in part 4a, show that d 3 2 [f (x)] = 3 [f (x)] f 0 (x) dx 3.B-5 [20] In this exercise we estimate the rate at which the total personal income is rising in the Miami-Ft. Lauderdale metropolitan area. In July, 1993, the population of this area was 3,354,000 and the population was increasing at roughly 45,000 people per year. The average annual income was $21,107 per capita, and this average was increasing at about $1900 per year (well above the national average of about $660 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in Miami-Ft. Lauderdale in July, 1993. [Ans: $7.322 billion per year] Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 2: Product and Quotient Rules

105

3.B-6 [17] Suppose that f (1) = 2 and f 0 is the function shown in Figure 3.3. Let m(x) = x3 f (x). Figure 3.3: Graph of f 0 [17]

(a) Evaluate m0 (1).

[Ans: 10]

(b) Show that m is increasing at 2. (c) Estimate m00 (1).

[Ans: 34] 2x − 5 . x2 − 4

3.B-7 (AB85) Let f be the function given by f (x) =

[Ans: {x|x ∈ R ∩ x 6= ±2}]

(a) Find the domain of f .

(b) Write an equation for each vertical and each horizontal asymptote for the graph of f . [Ans: y = 0, x = 2, x = −2] h i 0 (c) Find f (x). Ans: −2(x−4)(x−1) (x2 −4)2 (d) Write an equation for the line tangent to the graph of f at 5the point (0, f (0)). Ans: y − 4 = − 21 x 3.B-8 Differentiable functions f and g have the values shown in the table. [14] x 0 1 2 3

f0 1 2 3 4

f 2 3 5 10

g 5 3 1 0

(a) If A(x) = f (x) + 2g(x), then A0 (3) = 0

(b) If B(x) = f (x) · g(x), then B (2) = f (x) (c) If K(x) = , then K 0 (0) = g(x)

g0 −4 −3 −2 −1 [Ans: 2] [Ans: −7] 13 Ans: 25 Mr. Budd, compiled September 29, 2010

106

AP Unit 3 (Basic Differentiation)

(d) If D(x) =

1 , then D0 (1) = g(x)

3.B-9 (adapted from [2]) If g(x) =

Ans:

x+2 , then g 0 (−2) = x−2

3.B-10 (adapted from [2]) Consider the function f (x) = 0.75. The value of a is 3.B-11 (adapted from [2]) If y =

4 dy , then = 2 3+x dx

1 3

Ans: − 41

6x for which f 0 (0) = a + x3 [Ans: 8] i h −8x Ans: (3+x 2 )2

1 is called a witch of Maria Agnesi. Find 1 + x2 1 an equation of the tangent line to this curve at the point −1, . On 2 your calculator, graph the curve and the tangent line on the same screen. Ans: y = 21 x + 1

3.B-12 [20] The curve y =

3.B-13 [20] If f and g are the functions whose graphs are shown in Figure 3.4, let f (x) u(x) = f (x)g(x) and v(x) = . g(x) Figure 3.4:

(a) Find u0 (1). 0

(b) Find v (5).

[Ans: 0] Ans: − 32

3.B-14 [17] Suppose that f is the function shown in Figure 3.1. Let m(x) = f (x) . Evaluate m0 (0). [Ans: −4] x2 + 1

Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 3: Chain Rule

3.3

107

Chain Rule

Advanced Placement Computation of derivatives • Chain rule Derivative at a point. • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. Textbook §2.5 The Chain Rule [16] Resources §3-7 Derivatives of Composite Functions- The Chain Rule in Foerster [10]. Exploration 3-7:“Rubber-Band Chain Rule Problem” in [9]. §3-5 The Chain Rule in Stewart [20]. §3.6 New Derivatives from Old: The Chain Rule in Ostebee & Zorn [17].

3.3.1

Chain Rule

2 d 3 Example 3.3.1 Find x +1 , dx d √ 3 x +1 dx

3 d 3 x +1 , dx

d dx

1 , x3 + 1

These are some of the results we found last class: Chain Rule dy dy du = · dx du dx For h(x) = f (g(x)), h0 (x) = f 0 (g(x)) · g 0 (x) Derivative of the Outside · Derivative of the Inside Mr. Budd, compiled September 29, 2010

108

AP Unit 3 (Basic Differentiation) Power Chain Rule d n du u = nun−1 dx dx Example 3.3.2 Use the product rule, together with the chain rule, to prove the quotient rule, by finding d 1 f (x) · dx g(x)

Example 3.3.3 Given that f 0 (x) = find F 0 (x) if F (x) = f (g(x)) [19].

√ x and g(x) = 3x − 1, +1

Ans:

1 2x

Example 3.3.4 Let h(x) = f (g(x)). Use the information about f and g given in the table below to fill in the missing information about h and h0 [17]. x 1 2 3 4

f (x) 1 2 4 3

f 0 (x) 2 1 3 4

g 0 (x) 3 4 2 1

g(x) 4 3 1 2

h(x)

h0 (x)

Example 3.3.5 The function F is defined by F (x) = G (x − G(x)) where the graph of the function G is shown in Figure 3.5. Find F 0 (7) [2]

Ans: − 32

Product and Quotient Rules did not vanish Example 3.3.6 [18] Differentiate y = (2x + 1)

Ans: 2 (2x + 1)

x2 − x + 1

x3 − x + 1

17x3 + 6x2 − 9x + 3

Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 3: Chain Rule

109

Figure 3.5: from Best & Lux [2]

Example 3.3.7 [18] Find the derivative of the function 9 t−2 g(t) = 2t + 1 h Ans:

45(t−2)8 (2t+1)10

Problems 3.C-1 [14] Differentiable functions f and g have the values shown in the table. x 0 1 2 3 (a) If H(x) =

f 2 3 5 10

f0 1 2 3 4

g 5 3 1 0

g0 −4 −3 −2 −1 h

f (x), then H 0 (3) =

(b) If M (x) = f (g(x)), then M 0 (1) =

Ans:

√2 10

[Ans: −12]

[Ans: 6]

3.C-2 [3] Let the function f be differentiable on the interval [0, 2.5] and define g by g(x) = f (f (x)). Use the table to estimate g 0 (1). x f (x)

0.0 1.7

0.5 1.8

1.0 2.0

1.5 2.4

2.0 3.1

2.5 4.4 [Ans: 1.2]

Mr. Budd, compiled September 29, 2010

110

AP Unit 3 (Basic Differentiation)

3.C-3 [20] If f and g are the functions whose graphs are shown in Figure 3.6, let u(x) = f (g(x)), v(x) = g(f (x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not, explain why. Figure 3.6: [20]

(a) u0 (1)

Ans:

3 4

(b) v 0 (1)

[Ans: nonexistent, g 0 (2) DNE]

[Ans: −2]

3.C-4 [17] Suppose that f is the function shown in Figure 3.7 and that g(x) = f (x2 ). Figure 3.7: Graph of f [17]

(a) For which values of x is g 0 (x) = 0?

Ans:

√

2, 0

(b) Is g increasing or decreasing at −1?

[Ans: increasing] √ √ (c) Is g positive or negative over the interval 2, 5 ? [Ans: positive] 0

3.C-5 [3] The graphs of functions f and g are shown in Figure 3.8. If h(x) = f (g(x)), which of the following statements are true about the function h? I. h(2) = 5. II. h is increasing at x = 4. Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 3: Chain Rule

111

Figure 3.8: from Best & Lux [3]

III. The graph of h has a horizontal tangent at x = 1. [Ans: II and III only] 3.C-6 (BC98) If f and g are twice differentiable and if h(x) = f (g(x)), theni h 2 00 00 h (x) = Ans: f (g(x)) [g 0 (x)] + f 0 (g(x))g 00 (x) 3.C-7 [2] A particle moves along the x-axis so that at time t, t >= 0, its position 3 is given by x(t) = (t + 1) (t − 3) . Findha formula for the velocity, v(t), andi the acceleration, a(t), of the particle.

Ans: 4t (t − 3) ; 12 (t − 3) (t − 1) 5

3.C-8 (adapted from AB93) Find the derivative of f (x) h = (2x − 3) (7x + 2) . i 4 3 0 At how many different values of x will f (x) be 0? Ans: 2 (2x − 3) (7x + 2) (63x − 32); Three √ 9x − 2 3.C-9 (adapted from AB97) If f (x) = x 6x − 2, then f 0 (x) = Ans: √ 6x − 2 3.C-10 [17] Suppose that f (1) = 2, that f 0 is the function shown in Figure 3.9, and that k(x) = f (x3 ). Evaluate k 0 (−1). [Ans: 12] Figure 3.9: Graph of f 0 [17]

Mr. Budd, compiled September 29, 2010

112

AP Unit 3 (Basic Differentiation)

3.C-11 [2] Suppose that g is a function with the following two properties: g(−x) = g(x) for all x and g 0 (a) exists. [I.e., g is an even, differentiable function.] Find g 0 (−x). [Ans: −g 0 (x)] 3.C-12 [20] Use the table to estimate the value of h0 (0.5), where h(x) = f (g(x)). [Ans: −17.4] x f (x) g(x)

0 12.6 0.58

0.1 14.8 0.40

0.2 18.4 0.37

0.3 23.0 0.26

0.4 25.9 0.17

0.5 27.5 0.10

0.6 29.1 0.05

3.C-13 [20] If f is the function whose graph is shown, let h(x) = f (f (x)) and g(x) = f (x2 ). Use the graph of f to estimate the value of each derivative.

Figure 3.10: [20]

(a) h0 (2)

[Ans: 0.64]

(b) g (2) Z 3.C-14 Find f 0 (g(x)) g 0 (x) dx Z 3.C-15 Find

13 x3 + 1

3x2 dx

[Ans: 9] [Ans: f (g(x)) + C] h i 13 Ans: x3 + 1 +C

Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 4: Tangent Lines

3.4

113

Tangent Lines

Advanced Placement Derivative at a point. • Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents. • Tangent line to a curve at a point and local linear approximation. Textbook §3.1 Linear Approximations and Newton’s Method: “Linear Approximations” [16]

3.4.1

Tangent Lines

Lines with known y-intercept: y = mx + b Otherwise, in writing equations for lines, it is best to use the point-slope formula, in which you know a point and a slope: y − y1 = m (x − x1 ) If we are talking about writing the equation of a tangent line, remember that m, the slope of the tangent line is the same as the derivative.

Example 3.4.1 (adapted from AB93) An equation of the line tan3x − 2 at the point (−1, −5) is gent to the graph of y = 2x + 3

[Ans: y + 5 = 13 (x + 1)] Note: if you are simply asked to write an equation for the line, this would be an acceptable answer. If you’re asked to put the equation in standard form, it would be 13x − y = −8. If you’re asked to put the equation in slope-intercept form, the answer would be y = 13x + 8. You should be aware of each of these forms because you may need to spot them on a multiple choice exam. Mr. Budd, compiled September 29, 2010

114

AP Unit 3 (Basic Differentiation) Example 3.4.2 (adapted from AB98) Write an equation (in slopeintercept form) for the line tangent to the graph of f (x) = x4 − x2 at the point where f 0 (x) = 1

[Ans: y = x − 1.055]

Example 3.4.3 (adapted from [2]) If p(x) = (x + 2) (x + k) and if the line tangent to the graph of p at the point (4, p(4)) is perpendicular to the line 2x + 4y + 5 = 0, then k =

[Ans: −8]

Example 3.4.4 (adapted from [2]) If the line 3x − y + 5 = 0 is tangent in the second quadrant to the curve y = x3 + k, then k =

[Ans: 3]

3.4.2

Horizontal Tangents

f (x) has a horizontal tangent at (c, f (c)) if the slope of the tangent line is zero, i.e., f 0 (c) = 0. The equation of the tangent line will be y = f (c).

Example 3.4.5 The composite function h is defined by h(x) = f (g(x)), where f and g are functions whose graphs are shown in Figure 3.11. Find the number of horizontal tangent lines to the graph of h. [2]

[Ans: 6]

Example 3.4.6 (AB92) Let f be the function defined by f (x) = 3x5 − 5x3 + 2. Write the equation of each horizontal tangent line to the graph of f .

[Ans: y = 0, y = 2, y = 4] Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 4: Tangent Lines

115

Figure 3.11: from Best & Lux [2]

3.4.3

Vertical Tangents

Definition 3.1 (Vertical Tangent). f (x) has a vertical tangent at the point (c, f (c)) if the following conditions are met: 1. There is point of tangency, i.e., f (c) exists. 2. f 0 (c) is infinite. The equation of the line of tangency is x = c. Recall that the usual way to get an infinite answer is to have nonzero over zero. Note that it is not sufficient to state that the denominator is zero. Notice: both conditions must be met. Recall the definition of limit and realize that the derivative cannot exist unless the point exists. Frequently we may find f 0 to be nonzero over zero when we plug the number into the formula, but f is as well. It is not sufficient to find an infinite derivative; the point must exist as well. Also, we cannot use the point-form of a line if the slope does not exist. Simply write the equation as x = c. Example 3.4.7 Write theqequations of all the vertical (and hori2 3 zontal tangents) to f (x) = (x2 − 4) . √ Ans: x = 2, x = −2, (y = 3 16 Mr. Budd, compiled September 29, 2010

116

AP Unit 3 (Basic Differentiation) Example 3.4.8 Write the equations of all the vertical tangents to 1 f (x) = . x−2 [Ans: none] Example 3.4.9 (adapted from [2]) Which of the following is a function with a vertical tangent at x = 0? (A) (B) (C) (D)

f (x) = x2 √ f (x) = 3 x f (x) = x1 f (x) = |x|

√ [Ans: f (x) = 3 x] d √ d 1 Note that the derivative is infinite at x = 0 for x and for , but the dx dx x 1 1 function is not defined at x = 0. For f (x) = , there is a vertical asymptote x x at x = 0, which is different from a vertical tangent.

3.4.4

Normal Lines

Normal lines are perpendicular to tangent lines. The slope of the normal line will be the negative reciprocal of the slope of the tangent line. (Recall that the slope of the tangent line is the . . .) Example 3.4.10 The line that is normal to the curve y = x2 +2x−3 at (1, 0) intersects the curve at what other point? [19] 13 17 Ans: − , 4 16

3.4.5

Tangent Line Approximations

Tangent Lines to Approximate Values of f Example 3.4.11 You are stranded on a deserted island with your best friend, Wilson. He building some right triangles and wants √ to know the square root of 9.3. Without a calculator, estimate 9.3. Mr. Budd, compiled September 29, 2010

AP Unit 3, Day 4: Tangent Lines

Example 3.4.12 Let f (x) = graph of f at x = −

117 1 . Use the tangent line to the x+2

3 to find approximations for: 2

(a) f (−1.4) (b) f (−1.45) (c) f (−1.501)

[Ans: 1.6, 1.8, 2.004] Compare these tangent-line approximations to the actual functional values.

Example 3.4.13 (adapted from BC98) Let f be the function given by x2 − 2x + 3. The tangent line to the graph of f at x = 3 is used to approximate values of f (x). For what range of values is the error resulting from this tangent line approximation less than 0.1? Which of the following is the greatest value of x for which the error is less than 0.1? (A) 3.1 (B) 3.2 (C) 3.3 (D) 3.4 (E) 3.5

[Ans: 3.3] Note that the linear approximation has an error below 0.1 from 2.684 . . . to 3.316 . . .

Tangent Lines to Approximate Zeros of f Example 3.4.14 (adapted from AB97) Let f be a differentiable function such that f (2) = 1 and f 0 (2) = −4. If the tangent line to the graph of f at x = 2 is used to find an approximation to a zero of f , that approximation is

[Ans: 2.25] Mr. Budd, compiled September 29, 2010

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3.4.6

Introduction to Slope Fields

Slope fields give you a sense of the direction or current of the general solutions. To find the particular solution, start at the known point, and draw a curve to the left and right that is always parallel to the slopes as seen in the slope field. dy = −2x + 2 at twentydx 1 3 five points using x-coordinates of 0, 2 , 1, 2 , 2, and y-coordinates of −1, − 12 , 0, 12 , 1. Sketch an antiderivative within your slope field. Example 3.4.15 Draw a slope field for

Example 3.4.16 With a window of [0, 4.7] × [−3.1, 3.1], use the dy BIGSLOPE program to draw a slope field of = −4x + 8. Then dx R use the 5. Draw Solution option to graph (−4x + 8) dx for several values of C (−10, −8, −6, −4).

Problems 3.D-1 99 MM2 The function f is given by f (x) = (a)

2x + 1 x−3

x∈R

x 6= 3.

i. Show that y = 2 is an asymptote of the graph of y = f (x). ii. Find the vertical asymptote of the graph. [Ans: x = 3] iii. Write down the coordinates of the point P at which the asymptotes intersect. [Ans: (3, 2)]

(b) Find the points of the graph and the axes. of intersection Ans: − 21 , 0 , 0, − 13 (c) Hence sketch the graph of y = f (x), showing the asymptotes by dotted lines. −7 (d) Show that f 0 (x) = 2 and hence find the equation of the tan(x − 3) gent at the point S where x = 4. [Ans: y − 9 = −7 (x − 4)] (e) The tangent at the point T on the graph is parallel to the tangent at S. Find the coordinates of T . [Ans: (2, −5)] (f) Show that P is the midpoint of ST . 3.D-2 (adapted from BC97) Refer to the graph in Figure 3.12. The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown. The point (1, 4) is on the graph of y = f (x). An equation of the line tangent to the graph of f at (1, 4) is [Ans: y − 4 = 3 (x − 1)] Mr. Budd, compiled September 29, 2010

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Figure 3.12:

3.D-3 (BC90) Let f (x) = 12 − x2 for x ≥ 0 and f (x) ≥ 0. The line tangent to the graph of f at the point (k, f (k)) intercepts the x-axis at x = 4. What is the value of k? [Ans: k = 2] 1 3.D-4 (adapted from AB97) At what point on the graph of y = x3 is the 4 tangent line parallel to the line 3x − 4y = 7? Ans: 1, 41 3.D-5 (adapted from [2]) Let f (x) = 3x3 −4x+1. An equation of the line tangent to y = f (x) at x = 2 is [Ans: y = 32x − 47] √ 3.D-6 (adapted from [2]) Find the point on the graph of y = x between (4, 2) and (9, 3) at which the normal to the graph has the same slope line as the 5 through (4, 2) and (9, 3). Ans: 25 4 , 2 3.D-7 (adapted from [3]) If the graph of the parabola y = 2x2 + x + k is tangent to the line 3x + y = 3, then k = [Ans: 5] 8x at 1 + x3 the point (1, 4) forms a right triangle with the coordinate axes. The area of the triangle is [Ans: 9]

3.D-8 (adapted from [3]) A tangent line drawn to the graph of y =

3.D-9 (adapted from AB93) Find the derivative of f (x) = (x − 3) (x + 2) . At how many different values of x will the tangent line be horizontal? [Ans: Three] 3.D-10 (adapted from [2]) Which of the following is a function with a vertical tangent at x = 0? Mr. Budd, compiled September 29, 2010

120

AP Unit 3 (Basic Differentiation) (A) (B) (C) (D)

f (x) = x2 √ f (x) = 3 x f (x) = x1 f (x) = |x| [Ans: f (x) =

√ 3

3.D-11 (adapted from [2]) An equation of the normal to the graph of f (x) = x at (1, f (1)) is [Ans: x − 2y = −1] 3x − 2 3.D-12 (adapted from [3]) An equation of the line normal to the graph of f (x) = x at (3, 3) is x−2 [Ans: x − 2y + 3 = 0] √ 3.D-13 (AB88) Let f be the function given by f (x) = x4 − 16x2 .

(a) Find the domain of f . [Ans: Df : x ≥ 4, x ≤ −4, x = 0] (b) Describe the symmetry, if any, of the graph of f . [Ans: Graph of f is symmetric w.r.t. the y-a 3 (c) Find f 0 (x). Ans: √2x2 −16x 2 x (x −16)

(d) Find the slope of the line normal to the graph of f at x = 5 3 Ans: − 34 3.D-14 (adapted from AB97) No calculator! Let f be a differentiable function such that f (4) = 1.5 and f 0 (4) = 3. If the tangent line to the graph of f at x = 4 is used to find an approximation to a zero of f , that approximation is [Ans: 3.5] 3.D-15 (adapted from [3]) No √ calculator! Let f be a function with f (2) = 6 and derivative f 0 (x) = x3 + 1. Using a tangent line approximation to the graph of f at x = 2, estimate f (2.05) [Ans: 6.15] √ 3.D-16 (adapted from [3]) No calculator! The approximate value of y = x2 + 3 at x = 1.08, obtained from the tangent to the graph at x = 1, is [Ans: 2.04] 3.D-17 With a window of [0, 4.7] × [−3.1, 3.1], use the BIGSLOPE program to dy 1 1 draw a slope field of = √ , i.e., put √ into Y1 . After the slope dx √ x x field is drawn, then graph 2 x + C for C = −3, −2, −1 using the 5. Draw Solution option. 3.D-18 With a window of [0, 2.35] × [−3.1, 3.1], use the BIGSLOPE program to dy draw a slope field of = 4 (x − 1). Then draw solution to 2 x2 − 2x +C dx for C = −1, 1, 3. Mr. Budd, compiled September 29, 2010

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3.D-19 With a window of [−2.35, 2.35] × [−3.1, 3.1], use the BIGSLOPE program R 2 dy to draw a slope field of = x2 −1. Then draw solutions to x − 1 dx dx for C = −2, 0, 2. dy = 2x − 2 at twenty-five points using dx 1 3 x-coordinates of 0, 2 , 1, 2 , 2, and y-coordinates of −1, − 21 , 0, 12 , 1. Sketch an antiderivative within your slope field.

3.D-20 Draw by hand a slope field for

Basic Differentiation Review Problems 3.D-21 (AB97) If f (x) = −x3 + x +

1 , then f 0 (−1) = x

[Ans: −3]

3.D-22 (AB97) A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t) = 3t2 − 2t − 1. The position x(t) is 5 for t = 2. (a) Write a polynomial expression for the position of the particle at any time t ≥ 0. (b) For what values of t, 0 ≤ t ≤ 3, is the particle’s instantaneous velocity the same as its average velocity on the closed interval [0, 3] √ x−1 1 3.D-23 [2] What is lim ? Ans: x→1 x − 1 2 3/4 3.D-24 [2] If h(x) = x2 − 4 + 1, then the value of h0 (2) is [Ans: nonexistent] 3.D-25 [2] Which of the following is a function with a vertical tangent at x = 0? (A) f (x) = x3 √ (B) f (x) = 3 x 1 (C) f (x) = x [Ans: f (x) =

3.D-26 [2] The derivative of

√

1 x− √ is x3x

√ 3

x] 1 4 Ans: x−1/2 + x−7/3 2 3

3.D-27 (from [2]) A function f is defined for all real numbers and has the following property: f (a + b) − f (a) = 3a2 b + 2b2 . Find f 0 (x). Ans: 3x2 3.D-28 [2] If

d d d2 (f (x)) = g(x) and (g(x)) = f (3x), then f x2 is dx dx dx2

(A) 4x2 f (3x2 ) + 2g(x2 ) (B) f (3x2 ) Mr. Budd, compiled September 29, 2010

122

AP Unit 3 (Basic Differentiation) (C) f (x4 ) (D) 2xf (3x2 ) + 2g(x2 ) (E) 2xf (2x2 )

3.D-29 [2] If y =

dy 3 , then = 2 4+x dx

3 3.D-30 [2] lim

h→0

Ans: 4x2 f (3x2 ) + 2g(x2 ) " # −6x Ans: 2 (4 + x2 )

5 5 1 1 +h −3 2 2 = h

15 Ans: 16

3.D-31 [2] An object moves along the x-axis so that at time t, t > 0, its position is given by x(t) = t4 +t3 −30t2 +88t. At the instant when the acceleration becomes zero, the velocity of the object is [Ans: 12] 3.D-32 Using the limit definition of derivative, prove the Product Rule. 3.D-33 Use the Chain and Product Rule to prove the Quotient Rule.

Mr. Budd, compiled September 29, 2010

Unit 4

Curve Sketching 1. Curve Sketching 2. Second Derivative Sketching

Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 . Second derivatives. • Corresponding characteristics of the graphs of f , f 0 , and f 00 . • Relationship between the concavity of f and the sign of f 00 . • Points of inflection as places where the concavity changes.

123

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AP Unit 4, Day 1: Relating Graphs of f and f 0

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Relating Graphs of f and f 0

4.1

4.1.1

Relative Extrema

Definition 4.1 (Critical Point). A critical point is a point on the graph of f where f 0 is either 0 or undefined. Definition 4.2 (Critical Number). A critical number is the x-coordinate of the critical point. Note: the critical number is sometimes and confusingly referred to as the critical point. Judge from context whether the critical point really means the critical point or actually means the critical number. Example 4.1.1 How many critical points does the function f (x) = 4 5 (x + 2) x2 − 1 have? [2] Example 4.1.2 How many critical points does the function f (x) = q 1 2 3 2 (x − 4) + 1 have? How about f (x) = ? x−2 A point on the graph of f where f 0 is 0 has a horizontal tangent line. A point on the graph of f where f 0 is undefined is either a cusp or a vertical tangent. An extremum is either a maximum or a minimum. Definition 4.3 (Relative Extrema). If c is in the domain of a function f , then f (c) is a 1. relative maximum of f at c if and only if f (c) ≥ f (x) for all x in near c. Mr. Budd, compiled September 29, 2010

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AP Unit 4 (Curve Sketching)

2. relative minimum of f at c if and only if f (c) â&#x2030;¤ f (x) for all x in near c. Relative extrema are also called local extrema.

4.1.2

First Derivative Test

Critical points are important because they tell me where relative minima or maxima might occur. Just because the derivative is zero or undefined does not mean that I have a local extrema. But finding the critical points allows me to test only a few points for extremity, as opposed to an infinite number of points in a typical domain. To find out whether the critical points actually are relative extrema, we test them. There are two tests, the First Derivative Test, and the Second Derivative Test. The First Derivative Test checks the sign of the first derivative before and after the critical point. If the sign of the first derivative is changing, then the original function is changing direction, and there is a local extremum. In order to perform the First Derivative Test, I need to cut the number line at each critical number, so that I know how close I need to be when I check on either side of the critical number. Theorem 4.1 (First Derivative Test for Relative Extrema). (Notes stolen from [8]) At a critical point c: 1. If f 0 changes sign from positive to negative at c (f 0 > 0 for x < c and f 0 < 0 for x > c), then f has a relative maximum value at c. 2. If f 0 changes sign from negative to positive at c (f 0 < 0 for x < c and f 0 > 0 for x > c), then f has a relative minimum value at c. 3. If f 0 does not change sign at c (f 0 has the same sign on both sides of c), then f has no relative extreme value at c. Mr. Budd, compiled September 29, 2010

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Example 4.1.3 (AB98) The graph of f 0 , the derivative of f , is shown in Figure 4.1. Which of the following describes all relative extrema of f on the open interval (a, b)? (A) (B) (C) (D) (E)

One relative maximum and two relative minima. Two relative maxima and one relative minimum. Three relative maxima and one relative minimum. One relative maximum and three relative minima. Three relative maxima and two relative minima. [Ans: A]

Example 4.1.4 [2]The function f (x) = x4 − 18x2 has a relative minimum at x = [Ans: −3 and 3] Example 4.1.5 Determine at what values of x the function f (x) has a relative minimum for: • f 0 (x) = x2 x2 − 9 • f 0 (x) = x2 (x − 9) 2

• f 0 (x) = x (x − 3) (x + 3)

• f 0 (x) = x2 (x − 3) (x + 3) Example 4.1.6 If the derivative of f is given by f 0 (x) = sin (ln x), at which value of x, x ∈ [0, 1.07], does f have a relative maximum value? a relative minimum? [Ans: 0.0432, 1] Mr. Budd, compiled September 29, 2010

128

AP Unit 4 (Curve Sketching)

Figure 4.1: Graph of f 0 (x)

Connecting f and f 0 Figure 4.2: Graph of f

Example 4.1.7 (AB98) The graph of f is shown in Figure 4.2. Sketch the derivative of f .

Example 4.1.8 The graph of f 0 , the derivative of f , is shown in Figure 4.1. Sketch a graph of f .

Example 4.1.9 (AB96) Figure 4.4 shows the graph of f 0 , the derivative of a function f . The domain of f is the set of all real numbers such that â&#x2C6;&#x2019;3 < x < 5. Mr. Budd, compiled September 29, 2010

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Figure 4.3: Derivative of f from Figure 4.2

Figure 4.4: AB ’98 Note: This is the graph of the derivative of f , not the graph of f .

(a) For what values of x does f have a relative maximum? Why? (b) For what values of x does f have a relative minimum? Why? (c) On what intervals is the graph of f concave upward? Use f 0 to justify your answer. (d) Suppose that f (1) = 0. Draw a sketch that shows the general shape of the graph of the function f on the open interval 0 < x < 2. [Ans: x = −2,f 0 (x) changes from positive to negative at x = −2] [Ans: x = 4, f 0 (x) changes from negative to positive at x = 4] [Ans: (−1, 1) and (3, 5), f 0 is increasing on these intervals] [Ans: Figure 4.5]

Mr. Budd, compiled September 29, 2010

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AP Unit 4 (Curve Sketching)

Figure 4.5: AB â&#x20AC;&#x2122;96

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131

Problems 4.A-1 (AB00) Figure 4.6 shows the graph of f 0 , the derivative of the function f , or −7 ≤ x ≤ 7. The graph of f 0 has horizontal tangent lines at x = −3, x = 2, and x = 5, and a vertical tangent line at x = 3. Figure 4.6: AB ’00

(a) Find all values of x, for −7 < x < 7, at which f attains a relative minimum. Justify your answer. [Ans: x = −1, f 0 (x) changes from negative to positive at x = −1] (b) Find all values of x, for −7 < x < 7, at which f attains a relative maximum. Justify your answer. [Ans: x = −5, f 0 (x) changes from positive to negative at x = −5] (c) Find all values of x, for −7 < x < 7, at which f 00 (x) < 0. [Ans: on the intervals (−7, −3), (2, 3), and (3, 5),f 00 (x) exists and f 0 is decreasing ] 4.A-2 (AB98) The graphs of the derivatives of the functions f , g, and h are shown in Figure 4.7. Which of the functions f , g, or h have a relative maximum on the open interval a < x < b? [Ans: f only] Figure 4.7: AB98

4.A-3 (adapted from [3]) Which of the following are (is) true about a particle that starts at t = 0 and moves along a number line if its position at time 3 t is given by s(t) = (t − 1) (t − 5)? I. The particle is moving to the right for t > 4. II. The particle is at rest at t = 1 and t = 4. III. The particle changes direction at t = 1. Mr. Budd, compiled September 29, 2010

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4.A-4 (adapted from [3]) A particle starts at time t = 0 and moves along a number line so that its position, at time t ≥ 0, is given by x(t) = 3 (t − 3) (t − 7) . The particle is moving to the left for what values of t? [Ans: t < 4] 2

4.A-5 (adapted from [3]) If the derivative of the function f is f 0 (x) = −2 (x − 3) (x + 1) (x + 2) , then f has a local minimum at x = [Ans: −2] Figure 4.8: Problem 6: Graph of the derivative of f

4.A-6 (adapted from AB97) The graph of the derivative of f is shown in Figure 4.8. Graph the function f . [Ans: Figure 4.9] 4.A-7 (adapted from AB93) For what value of x does the function f (x) = 2 (x − 3) (x − 2) have a relative minimum? Ans: 38 4.A-8 (adapted from AB93) How many critical points does the function f (x) = 5 4 (x − 3) (x + 2) have? [Ans: Three] 4.A-9 (adapted from AB93) The function f given by f (x) = x3 − 12x + 24 is increasing for what values of x? [Ans: x < −2, x > 2]

Figure 4.9: Answer to problem 6: Graph of f

Mr. Budd, compiled September 29, 2010

AP Unit 4, Day 1: Relating Graphs of f and f 0

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Figure 4.10: Problem 10: Graph of h(x)

Figure 4.11: Answer to problem 10: Graph of h0 (x)

4.A-10 (BC98) The graph of y = h(x) is shown in Figure 4.10. Sketch the graph of y = h0 (x) [Ans: Figure 4.11] 4.A-11 [2] A particle moves along the x-axis so that at time t, t >= 0, its position 3 is given by x(t) = (t + 1) (t − 3) . For what values of t is the velocity of the particle increasing? [Ans: t < 1 or t > 3] x4 x5 − . The derivative of f attains it 2 10 maximum value at x = [Ans: 3] 2 x 2 4.A-13 The derivative of f is given by f 0 (x) = ex /5 cos . Find the x– 5 coordinates of all relative minima of f on the interval x ∈ [0, 7]. [Ans: 4.854]

4.A-12 [2] Consider the function f (x) =

4.A-14 (adapted from AB98) The first derivative of the function f is given by 3 cos2 x − . How many critical values does f have on the open f 0 (x) = x 20 interval (0, 10)? [Ans: Five] 4.A-15 (adapted from AB97) If the derivative of f is given by f 0 (x) = 2x2 − ex , at which value of x does f have a relative minimum value? [Ans: 1.488]

Mr. Budd, compiled September 29, 2010

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AP Unit 4, Day 2: Second Derivative Sketching

4.2

135

Second Derivative Sketching

4.2.1

Concavity

There are four basic curve types... (Under Construction) The graph of f is concave upward if f 0 is increasing, i.e., f 00 is positive. The graph of f is concave downward if f 0 is decreasing, i.e., f 00 is negative. If the function is concave down, then the curve is below the tangent line. If the function is concave up, then the curve is above the tangent line. If the function is concave up, then the tangent line underestimates the actual function; if the function is concave down, the tangent line overestimates the actual function. Figure 4.12:

Mr. Budd, compiled September 29, 2010

136

AP Unit 4 (Curve Sketching) Example 4.2.1 The graph of a twice-differentiable function f is shown in Figure 4.12. Which of the following is true? (A) f (1) < f 0 (1) < f 00 (1) (B) f (1) < f 00 (1) < f 0 (1) (C) f 0 (1) < f (1) < f 00 (1) (D) f 00 (1) < f (1) < f 0 (1) (E) f 00 (1) < f 0 (1) < f (1)

[Ans: D]

Second Derivative Test Recall the First Derivative Test. The First Derivative Test tells me that: • there is a local maximum if the first derivative changes from positive to negative; • there is a local maximum if the first derivative changes from negative to positive. Note that, if the first derivative is changing from positive to negative through zero, then it is decreasing. So, for a local maximum, if the second derivative exists, it is negative. This makes sense, because if the second derivative is negative, the function is concave down. If the function is concave down, it has a local maximum at the place where f 0 is zero. Similar arguments can be made for local minima. Theorem 4.2 (Second Derivative Test, Graphically). Based on f 0 : • If f 0 is decreasing through zero at x = c, then f (c) is a local maximum, since the graph of f is concave up with a horizontal tangent. • If f 0 is increasing through zero at x = c, then f (c) is a local minimum, since the graph of f is concave down with a horizontal tangent. Or, based on f 00 : • If f 0 (c) = 0 and if f 00 (c) < 0, then f (c) is a local maximum; • If f 0 (c) = 0 and if f 00 (c) > 0, then f (c) is a local maximum; Mr. Budd, compiled September 29, 2010

AP Unit 4, Day 2: Second Derivative Sketching

137

Figure 4.13: BC Acorn ’02

• If f 0 (c) = 0 and if f 00 (c) is 0 or undefined, then the Second Derivative Test fails to determine whether f (c) is a local extremum. Note: we only use the Second Derivative Test for f 0 (c) = 0. Why would we not use the Second Derivative Test on those critical points where f 0 (c) is undefined?

4.2.2

Points of Inflection

An inflection point is a point on the graph of f where the concavity changes. At inflection points on the graph of f , • f changes concavity; • f 0 changes direction, i.e., extrema of f 0 yield points of inflection of f ; • f 00 changes sign, i.e., if f 00 crosses the x-axis, then f has a point of inflection. For inflection points, the tangent line will overestimate the curve on one side of the point, and underestimate it on the other side. Be careful: for there to be an inflection point, it is not enough for the second derivative to be zero. The second derivative must change sign. This is similar to finding relative extrema: it is not enough for the first derivative to be 0, the first derivative must change sign for there to be a relative extremum.

Example 4.2.2 (BC Acorn ’02) Let f be a function whose domain is the open interval (1, 5). Figure 4.13 shows the graph of f 00 . Count the number of extrema of f 0 and points of inflection of the graph of f 0. Mr. Budd, compiled September 29, 2010

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AP Unit 4 (Curve Sketching) [Ans: one rel. min, two p.i.] 2

Example 4.2.3 (adapted from AB98) If f 00 (x) = x (x − 2) (x + 1) , then the graph of f has inflection points at what value(s) of x?

[Ans: 0 and 2 only]

Example 4.2.4 (adapted from BC98) If f is the function defined by f (x) = 3x5 +10x4 +10x3 −60x+7, what are all the x-coordinates of points of inflection for the graph of f ?

[Ans: 0]

Example 4.2.5 [3] Which of the following statements are true 3 about the function f if its derivative f 0 is defined by f 0 (x) = x (x − a) , a > 0. I. The graph of f is increasing at x = 2a. II. The function f has a local maximum at x = 0. III. The graph of f has an inflection point at x = a.

[Ans: I and II only]

Problems 4.B-1 (BC97) The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown in Figure 4.14. How many points of inflection does the graph of f have? [Ans: Six] 4.B-2 (adapted from AB98) What is the x-coordinate of the point of inflection 1 [Ans: 5] on the graph of y = x3 − 5x2 − 13? 3 2

4.B-3 (adapted from AB98) If f 00 (x) = x (x + 2) (x − 1) , then the graph of f has inflection points when x = [Ans: 0 and −2 only] 4.B-4 [3] Suppose a function f is defined so that it has derivatives f 0 (x) = x2 (1 − x) and f 00 (x) = x (2 − 3x). Over which interval is the graph of f both increasing and concave up? Ans: 0 < x < 32 Mr. Budd, compiled September 29, 2010

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Figure 4.14: BC ’97 The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown.

4.B-5 [3] Which of the following are true about the function f if its derivative is defined by 2 f 0 (x) = (x − 1) (4 − x) ? I. f is decreasing for all x < 4. II. f has a local maximum at x = 1. III. f is concave up for all 1 < x < 3. [Ans: III only] 2 4.B-6 (adapted from AB97) The graph of y = 3x4 − 8x3 − 24x + 162is concave down for what values of x? Ans: − 3 < x < 2

Mr. Budd, compiled September 29, 2010

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AP Unit 4 (Curve Sketching)

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Unit 5

Calculus of Trigonometric Functions 1. Implicit Differentiation and Related Rates 2. Differentiation of Sine, Cosine 3. Antidifferentiation of Trigonometrics 4. Derivatives of Inverse Functions Advanced Placement

I. Derivatives Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of derivatives. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions. Computation of derivatives. • Knowledge of derivatives of basic functions, including exponential, logarithmic, and trigonometric functions. 141

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AP Unit 5 (Trigonometrics)

II. Integrals Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables. Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth.

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 1: Differentiation of Trigonometric Functions

5.1

143

Differentiation of Trigonometric Functions

Advanced Placement Computation of derivatives. • Knowledge of derivatives of basic functions, including trigonometric functions.

5.1.1

Special Limits

Theorem 5.1 (Squeeze Theorem). Let f , g, and h be functions satisfying f (x) ≤ g(x) ≤ h(x) for all x near c, except possibly at c. If lim f (x) = x→c

lim h(x) = L, then lim g(x) = L.

x→c

Example 5.1.1 Show that lim x2 sin x→0

1 = 0. [20] x

Example 5.1.2 Find f 0 (0) for f (x) = sin x using the definition of the derivative. (a) Show that f 0 (0) can be written as lim

h→0

sin h . h

sin h = 1 for positive angles h along the unit circle. h h→0 (c) Find the limit of the average rate of change using a table.

(b) Prove lim+

(d) Use a symmetric difference quotient using ∆x = 0.001 on both sides of x = 0 to find f 0 (0). (e) Use nDeriv to show find f 0 (0). Example 5.1.3 Find g 0 (0) for g(x) = cos x using the definition of the derivative. cos h − 1 . h cos h − 1 sin h (b) Prove lim = 0 using lim = 1. h→0 h→0 h h (c) Find the limit of the average rate of change using a table. (a) Show that g 0 (0) can be written as lim

h→0

(d) Use a symmetric difference quotient using ∆x = 0.001 on both sides of x = 0 to find g 0 (0). (e) Use nDeriv to show find g 0 (0). Mr. Budd, compiled September 29, 2010

144

AP Unit 5 (Trigonometrics)

5.1.2

Trigonometric Derivatives

Derivative of Sine Example 5.1.4 Find the derivative of f (x) = sin x: (a) If f (x) = sin x, sketch f 0 (x). d sin x is. dx (c) Prove or disprove your conjecture:

(b) Make a conjecture about what

(a) Start with the definition of the derivative as the limit of an average rate of change. (b) Use the sum of angle formula for sine: sin (α + β) = sin α cos β+ cos α sin β. h as h approaches (c) Use the known limits for sinh h and 1−cos h 0. (d) Give graphic evidence that the derivative is the same as your conjecture by comparing the graphs of nDeriv(Y1 , X, X) and your proposed derivative using the happy bouncing ball.

d sin x = cos x dx Using the Chain Rule d sin (g(x)) = cos (g(x)) g 0 (x) dx Derivative of Cosine Example 5.1.5 Graph the derivative of cos x. Make and prove a conjecture for the derivative of cos x.

d cos x = − sin x dx Using the Chain Rule d cos (g(x)) = − sin (g(x)) g 0 (x) dx Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 1: Differentiation of Trigonometric Functions

145

Derivative of Tangent Example 5.1.6 Find

d tan x. dx

d tan x = sec2 x dx Using the Chain Rule: d tan (g(x)) = sec2 (g(x)) g 0 (x) dx

Derivative of Secant Example 5.1.7 Find

d sec x. dx

Use the Quotient Rule: d sec x = sec x tan x dx Using the Chain Rule: d sec (g(x)) = sec (g(x)) tan (g(x)) g 0 (x) dx

Example 5.1.8 (adapted from AB93) A particle moves along a line so that at time t, where 0 ≤ t ≤ π, its position is given by t2 s(t) = −3 cos t − + 10. What is the velocity when its acceleration 2 is zero?

[Ans: 1.597]

Example 5.1.9 Let f (x) = sin(2x). Find the intersection of the π π two lines tangent to the graph of f at x = and at x = . 6 3 Mr. Budd, compiled September 29, 2010

146

AP Unit 5 (Trigonometrics) h

Ans:

Example 5.1.10 (adapted from AB97)

π π 4 , 12

√

3 2

i ≈ (0.785, 1.128)

d cos3 (x2 ) = dx Ans: −6x cos2 (x2 ) sin(x2 )

Example 5.1.11 (BC97) The position of an object attached to a 1 1 spring is given by y(t) = cos(5t) − sin(5t), where t is time in 6 4 seconds. In the first 3 seconds, how many times is the velocity of the object equal to 0? [You may use a calculator.] [Ans: Five] Example 5.1.12 (adapted from [2]) Administrators at Massachusetts General Hospital believe that the hospital’s expenditures E(B), measured in dollars, are a function of how many beds B are in use with 2

E(B) = 7000 + (B + 1) . On the other hand, the number of beds B is a function of time, t, measured in days, and it is estimated that t B(t) = 25 sin + 50. 10 At what rate are the expenditures decreasing when t = 100? [Ans: $157/day]

Checking Antiderivatives Example 5.1.13 (adapted from AB97) Which of the following are antiderivatives of f (x) = sin x cos x? sin2 x 2 cos2 x II. F (x) = − 2 cos(2x) III. F (x) = − 4 I. F (x) = −

[Ans: II and III only] Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 1: Differentiation of Trigonometric Functions

147

Problems cos 5.A-1 (adapted from AB acorn ’02) What is lim

h→0

3π 3π + h − cos 2 2 ? h

[Ans: 1] 5.A-2 (adapted from AB acorn ’02) If f (x) = sin2 [Ans: −1]

π 4

− x , then f 0 (0) =

5.A-3 (adapted from AB98) Find equation of the line tangent to the graph of y = 2x + cos(πx) at the point (1, 1). Graph the function and its tangent line on your calculator. Find the equation of the line tangent to the graph of y = 2x + cos(πx) at the point (−1, −3). [Ans: y = 2x − 1] 2

5.A-4 (adapted from AB93) If f (x) = (x − 1) cos x, then f 0 (0) = π = 5.A-5 (adapted from AB98) If f (x) = tan(2x), then f 0 8

[Ans: −2]

5.A-6 (AB93) The fundamental period of 2 cos(3x) is

Ans:

[Ans: 4] 2π 3

5.A-7 (adapted from [2]) For x 6= 0, the slope of the tangent to y = x sin x equals zero whenever (A) tan x = −x 1 (B) tan x = x (C) tan x = x (D) sin x = x (E) cos x = x [Ans: tan x = −x] 5.A-8 (adapted from [2]) If y = 2 sin x cos x =, then y 0 =

[Ans: 2 cos(2x)]

2 2 5.A-9 (adapted h π π i from [2]) For f (x) = cos x and g(x) = −0.5x on the interval − , , the instantaneous rate of change of f is greater than the instan2 2 taneous rate of change of g for which value of x? [Note: you should use a calculator.]

(A) −1.5 (B) −1.2 (C) −0.8 (D) 0 (E) 0.9 Mr. Budd, compiled September 29, 2010

148

AP Unit 5 (Trigonometrics) [Ans: −0.8]

5.A-10 (adapted from [2]) lim

h→0

sin (x + h) − sin x h

[Ans: cos x]

5.A-11 (adapted from [2]) If g(x) = x2 − sin x, then lim

h→0

[Ans: 2x − cos x]

g(x + h) − g(x) = h

5.A-12 (adapted from [2]) At how many points on the interval −2π ≤ x ≤ 2π π does the tangent to the graph of the curve y = x sin x have slope ? 2 [Ans: Three] 5.A-13 (adapted from [2]) If y = cos3 (3x), then

dy = Ans: −9 cos2 (3x) sin(3x) dx

5.A-14 (from Stewart [20]) Let f (x) = sin x. (a) Find values of the constants a and b so that the linear function L(x) = a + bx has the properties L(0) = f (0) and L0 (0) = f 0 (0). (b) Find the value of the constant c for which the quadratic function Q(x) = L(x) + cx2 has the properties Q(0) = f (0), Q0 (0) = f 0 (0), and Q00 (0) = f 00 (0). (c) Find the value of the constant d for which the cubic function C(x) = Q(x) + dx3 has the properties C(0) = f (0), C 0 (0) = f 0 (0), C 00 (0) = f 00 (0), and C 000 (0) = f 000 (0). (d) Plot f , L, Q, and C on the same axes over the interval [−4, 4]. Ans: L(x) = x = Q(x); C(x) = x − 61 x3 5.A-15 Repeat the previous problem for f (x) = cos x Ans: L(x): Q(x) = 1 − 21 x2 = C(x)

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 2: Implicit Differentiation

5.2

149

Implicit Differentiation

Advanced Placement Computation of derivatives. • Chain rule and implicit differentiation. Textbook §2.8 Implicit Differentiation and Inverse Trigonometric Functions [16] Resources Exploration 4-8: “Implicit Relation Derivatives” in [9]

5.2.1

Implicit Differentiation

Example 5.2.1 For x = tan (f (x)), find f 0 (x) in terms of x only. d arctan x. Note that your answer is dx Example 5.2.2 Write an equation √ of the line tangent to the graph of x2 + y 2 = 16 at the point 2 3, 2 √ √ Ans: y − 2 = − 3 x − 2 3 Implicit Differentiation is Latin for “Remember the Freakin’ Chain Rule”. Remember: anytime you differentiate a y-term with respect to x, you must dy multiply by . dx d dy O(y) = O0 (y) · dx dx Example 5.2.3 Identify the variable, then Remember the Freakin’ Chain Rule: d 2 x dx d 2 (b) y dy d 2 (c) [f (x)] dx (a)

Mr. Budd, compiled September 29, 2010

150

AP Unit 5 (Trigonometrics) d 2 y dx d 2 (e) x dt

(d)

Example 5.2.4 Remember the Product Rule: d (xy) dx d (b) (−4xy) dx (a)

Example 5.2.5 For x2 + y 2 = 16: (a) For what coordinate pairs is the graph increasing? (b) For what coordinate pairs does the curve have a vertical tangent? (c) Remember the Quotient Rule: For what values of x or y is d2 y dx2 > 0 ? Example 5.2.6 Do Foerster’s Exploration 4-8 in your mighty, mighty groups of four. Example 5.2.7 Second Derivative Problems (adapted from BC98) p dy d2 y = 1 − y 2 , what is ? [Ans: −y] dx dx2 p dy d2 y (b) If = − 1 − y 2 , what is ? [Ans: −y] dx dx2 (c) Name two functions for which f 00 (x) = −f (x). Is it true for p p dy dy those functions that = 1 − y 2 and = − 1 − y2 ? dx dx (a) If

Example 5.2.8 Find

dy for y = cos (xy) dx

The derivative at a specific point You may plug in the values for x and y at any point after differentiating. If you dy only need a slope at a specific value, and don’t need a formula for , it may dx Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 2: Implicit Differentiation

151

behoove you to plug in values right after differentiation. It’s a lot easier to deal with 12, 6, and −24 than with 8y, 3x2 y, and 4xy. Note that sometimes, you must find one of the coordinate pairs from the other.

The explicit option Some equations can be solved explicitly. Just solve for y, and take the derivative per usual. Example 5.2.9 Solve x2 + y 2 = 16 explicitly for y, then find dy , and show that the answer is the same as we get using implicit dx differentiation.

Problems 5.B-1 (from [2]) Consider the curve x + xy + 2y 2 = 6. The slope of the line tangent to the curve at the point (2, 1) is Ans: − 31 5.B-2 (from [2]) If x2 + 2xy − 3y = 3, then the value of 5.B-3 (from [2]) If y 2 − 3x = 7, then

d2 y = dx2

5.B-4 (from [2]) If tan (x + y) = x, then

dy = dx

dy at x = 2 is [Ans: −2] dx h i Ans: − 4y93

Ans: − sin2 (x + y)

5.B-5 (from [2]) If y is a differentiable function of x, then the slope of the tangent 1 to the curve xy − 2y + 4y 2 = 6 at the point where y = 1 is Ans: − 10 5.B-6 (adapted from AB ’93) If 2x3 + 2xy + 4y 3 = 17, then in terms of x and y, h i dy 3x2 +y = Ans: − x+6y 2 dx 5.B-7 (adapted from AB ’97) If x2 + y 2 = 10, what is the value of point (1, −3)?

d2 y at the dx 2 10 Ans: 27

5.B-8 If x2 − y 2 = 25, for what coordinate pairs will the curve have vertical tangents? [Ans: (5, 0), (−5, 0)] 5.B-9 If y 2 − x2 = 25, for what coordinate pairs will the curve have vertical tangents? [Ans: none] Mr. Budd, compiled September 29, 2010

152

AP Unit 5 (Trigonometrics)

5.B-10 (adapted from AB ’98) If x2 +xy = −10, then when x = 2,

dy = Ans: dx

5.B-11 (adapted from BC ’97) If 2y = xy + x2 + 1, then when x = 1, [Ans: 4]

3 2

dy = dx

4 5.B-12 (adapted from BC ’98) The slope of the line tangent to the curve y 3+ 3 (xy + 1) = 0 at (2, −1) is Ans: 2

5.B-13 (adapted from AB Acorn ’00) What is the slope of the tangent to the 5 curve y 3 x2 + y 2 x = 6 at (2, 1)? Ans: − 16 5.B-14 (adapted from BC Acorn ’00) If x = y+cos (xy), what is

dy h ? Ans: dx

1+y sin(xy) 1−x sin(xy)

5.B-15 (HL 5/02) A curve has equation xy 3 + 2x2 y = 3. Find the equation of the tangent to this curve at the point (1, 1). [Ans: y − 1 = − (x − 1)] 5.B-16 (HL 5/03) A curve has equation x3 y 2 = 8. Find the equation of the normal to the curve at the point (2, 1). [Note: the normal sticks out of the curve like hairs, and is perpendicular to the tangent.] Ans: y − 1 = 34 (x − 2)

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 3: Inverse Functions

5.3

153

Inverse Functions

Advanced Placement Applications of derivatives. • Use of implicit differentiation to find the derivative of an inverse function. Computation of derivatives. • Knowledge of derivative of basic functions, including inverse trigonometric functions. • Chain rule and implicit differentiation.

5.3.1

Inverse Functions

To get an inverse relation, switch the x’s and y’s, then solve for y. Example 5.3.1 (adapted from D&S) Let f (x) = x3 −9x2 +31x−39 and let g be the inverse of f . What is the value of g 0 (0)?

Ans:

1 4

If you like, you can use d −1 1 f (x) = 0 −1 dx f (f (x))

Example 5.3.2 (adapted from AB ’07) The functions f and g are differentiable for all real numbers, and g is strictly increasing. Table 5.1 gives values of the functions and their first derivatives at selected values of x. (a) If g −1 is the inverse function of g, write an equation for the line tangent to the graph of y = g −1 (x) at (a) x = 3 (b) x = 4 (c) x = 5 Mr. Budd, compiled September 29, 2010

154

AP Unit 5 (Trigonometrics)

Table 5.1: From AB 2007 Exam x 1 2 3 4

f (x) 6 9 10 −1

g(x) 4 2 −4 3

f 0 (x) 2 3 4 6

g 0 (x) 5 1 2 7

Ans: y − 2 = 1 (x − 3); y − 3 =

1 2

(x − 4);

(b) What is the problem with asking for an equation for the line tangent to the graph of y = f −1 (x) at x = 9?

[Ans: the inverse of f ain’t a function] In order for a function f to have an inverse function f −1 , i.e., an inverse relation that is also a function, then f −1 must pass the vertical line test, and f must pass a horizontal line test. One way to ensure that f passes a horizontal line test is if f is monotonically increasing or monotonically decreasing. The inverse relation won’t always be a function. Sometimes we must take selective pieces of the graph to make sure that the inverse is a function. • For example, with

√

x, we use only the top half of the parabola.

• For arcsin x, we use only a portion of the graph that will give a full spectrum of sines, i.e., from −1 to 1. We use that portion of the graph in the fourth and first where the cosine is always positive. The range h quadrants π πi of arcsin x is − , . 2 2 • For arccos x, we use a portion of the graph that gives a full spectrum of cosine values, so the domain is x ∈ [−1, 1]. The range is the angles in quadrants I and II where the sine is positive. • The domain of arctan x is all real numbers, but the range is limited to the π π open interval − , . 2 2 • If f (x) passes a horizontal line test, then its inverse relation passes a vertical line test, and is therefore a function.

Example 5.3.3 Find

(−3)

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 3: Inverse Functions

155

Example 5.3.4 √

(a) Find arcsin (b) Find arccos

3 2 √1 2

(e) Graph y = arcsin (sin x) Example 5.3.5 Use a right triangle to find cos arcsin 35

Example 5.3.6 Use a right triangle to find sec (arctan 2x)

Example 5.3.7 For x = cos y, find that your answer is

5.3.2

d dx

dy in terms of x only. Note dx

arccos x.

Differentiating Inverse Functions

Example 5.3.8 Find an inverse function, g(x) = f −1 (x), for f (x) = x3 , and find g 0 (x) two ways. Example 5.3.9 Find the derivatives of (a) arcsin x (b) arccos x (c) arctan x

1 d arcsin x = √ dx 1 − x2 d 1 arccos x = − √ dx 1 − x2 d 1 arctan x = dx 1 + x2

Example 5.3.10 Find

d arcsin x5 dx Mr. Budd, compiled September 29, 2010

156

AP Unit 5 (Trigonometrics)

Example 5.3.11 Find

d dx

√ x 1 − x2 + arcsin x √ Ans: 2 1 − x2

Problems 5.C-1 For x = tan f (x), find f 0 (x) in terms of x only. Use right triangles, withi h O x 1 Ans: 1+x 2 H = 1. dy in terms of x only. Remember that sin2 y + cos2 y = dx O x A x h1, or use right triangles, i with A = 1 . Repeat for x = cos y, so H = 1 . 1 1 Ans: √1−x ;− √1−x 2 2

5.C-2 For x = sin y, find

x 5.C-3 (adapted from [2]) An equation for a tangent to the graph of y = arctan 3 π π 1 at the point 3, is: Ans: y − 4 = 6 (x − 3) 4 √ 5.C-4 [2] If g(x) = 3 x − 1 and f is the inverse function of g, then f 0 (x) = Ans: 3x2 5.C-5 (adapted from [2])

d [arctan 2x] = dx

Ans:

2 1+4x2

5.C-6 (adapted from AB ’07) The functions f and g are differentiable for all real numbers, and g is strictly increasing. Table 5.1 gives values of the functions and their first derivatives at selected values of x. If g −1 is the inverse function of g, write an equation for the line graph tangent to the of y = g −1 (x) at x = 6. Ans: y − 4 = 17 (x − 6) 5.C-7 (adapted from D&S)

d 4x arcsin = dx 3

Ans:

4 9−16x2

5.C-8 (adapted from D&S) If f (x) = x− 4 , what is the derivative of the inverse of f (x)? Ans: − x45

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 4: Related Rates (Triangles)

5.4

157

Related Rates (Triangles)

5.4.1

Introduction to Related Rates

Example 5.4.1 (adapted from Acorn BC ’00) A point (x, y) is moving along a curve y = f (x). At the instant when the slope of 1 the curve is − , the x-coordinate of the point is increasing at the 5 rate of 3 units per second. The rate of change, in units per second, of the y-coordinate of the point is

Ans: − 35 √ Example 5.4.2 (BC ’98) When x = 8, the rate at which 3 x is 1 increasing is times the rate at which x is increasing. What is the k value of k?

[Ans: 12] Example 5.4.3 [2] If xy 2 = 20, and x is decreasing at the rate of 3 units per second, the rate at which y is changing when y = 2 is approximately

[Ans: 0.6 units/sec] Mr. Budd, compiled September 29, 2010

158

AP Unit 5 (Trigonometrics)

5.4.2

Related Rates w/ Triangles

Example 5.4.4 (adapted from AB ’93) The top of a 13-foot ladder is sliding down a vertical wall at a constant rate of 2 feet per minute. When the top of the ladder is 5 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

Ans:

5 6

feet per minute

Example 5.4.5 (AB 2002 Form B) Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 10 km/hr. Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in Figure 5.1. Figure 5.1: From 2002 AP Calculus AB Exam

(a) Find the distance, in kilometers, between Ship A and Ship B when x = 4 km and y = 3 km. (b) Find the rate of change, in km/hr, of the distance between the two ships when x = 4 km and y = 3 km. (c) Let θ be the angle shown in Figure 5.1. Find the rate of change of θ, in radians per hour, when x = 4 km and y = 3 km. Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 4: Related Rates (Triangles)

159

Problems 5.D-1 [2] One ship traveling west is W (t) nautical miles west of a lighthouse and Figure 5.2: From [2]

a second ship traveling south is S(t) nautical miles south of the lighthouse at time t (hours). The graphs of W and S are shown in Figure 5.2. At what approximate time is the distance between the ships increasing at t = 1? (nautical miles per hour = knots) [Ans: 4 knots] 5.D-2 (adapted from AB ’97) A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 80 meters per second. At how many meters per second is the train moving away from the observer 3 seconds after it passes through the intersection? [Ans: 76.8] 5.D-3 (from HL 11/03) An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At at given instant the observer notes that the angle θ is 1 1 π radians and is increasing at radians per second. Find the speed, in 3 60 kilometers per hour, at which the airplane is moving towards the observer. 1 Ans: 15 km/s = 240 km/hr Figure 5.3: Airplane flying towards an observer.

Mr. Budd, compiled September 29, 2010

160

AP Unit 5 (Trigonometrics)

5.D-4 Figure 5.4 shows an isosceles triangle ABC with AB = 10 cm and AC = BC. The vertex C is moving in a direction perpendicular to (AB) with Figure 5.4: Growing isosceles triangle.

speed 2 cm per second. Calculate the rate of of the angle CAB increase 1 at the moment the triangle is equilateral. Ans: 10 radians per second 5.D-5 (adapted from D&S) Two cars start at the same place and at the same time. One car travels east at a constant velocity of 40 miles per hour and a second car travels north at a constant velocity of 48 miles per hour. Approximately how fast is the distance between them changing after half an hour? Round your answer to the nearest mile per hour. [Ans: 62 mph] 5.D-6 (adapted from D&S) A missile rises vertically from a point on the ground 65, 000 feet from a radar station. If the missile is rising at the rate of 17, 500 feet per minute at the instant when it is 38, 000 feet high, what is the rate of change, in radians per minute, of the missileâ&#x20AC;&#x2122;s angle of elevation from the radar station at this instant? [Ans: .201 radians per minute]

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 5: Antidifferentiating Trig

5.5

161

Antidifferentiating Trig

Advanced Placement Techniques of antidifferentiation

• Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables.

Textbook §2.8 Implicit Differentiation and Inverse Trigonometric Functions [16]

5.5.1

Antidifferentiating to Inverse Functions Z

√

1 dx = arcsin x + C 1 − x2

Example 5.5.1 Antidifferentiate: Z

√

1 dx 1 − 9x2

Ans:

1 3

arcsin 3x + C

1 dx = arctan x + C 1 + x2

Example 5.5.2 Antidifferentiate: Z

√

1 dx x (1 + x)

[Ans: 2 arctan

√

x + C]

Mr. Budd, compiled September 29, 2010

162

AP Unit 5 (Trigonometrics)

5.5.2

Antidifferentiation of Trigonometric Functions

Antiderivative of Sine dy Example 5.5.3 Use a program to draw a slope field of = sin x. dx Conjecture the antiderivative of sine. Prove your conjecture using u-substitution.

Z sin x dx = â&#x2C6;&#x2019; cos x + C

u-substitution gives: Z

sin (g(x)) g 0 (x) dx = â&#x2C6;&#x2019; cos (g(x)) + C

Antiderivative of Cosine dy Example 5.5.4 Use a program to draw a slope field of = cos x. dx Conjecture the antiderivative of cosine.

Z cos x dx = sin x + C

u-substitution gives: Z

cos (g(x)) g 0 (x) dx = sin (g(x)) + C

Other Trigonometric Antiderivatives Z

sec2 x dx = tan x + C

Z sec x tan x dx = sec x + C

Mr. Budd, compiled September 29, 2010

AP Unit 5, Day 5: Antidifferentiating Trig

163

Example 5.5.5 (adapted from AB97) At time t ≥ 0, the acceleration of a particle moving on the x-axis is a(t) = t + sin t. At t = 0, the velocity of the particle is −3. For what value of t will the velocity of the particle be zero?

[Ans: 1.855] Be careful! C 6= −3.

Problems Z 5.E-1 Find

√

3x2 dx using the u-substitution u = x3 . Ans: arcsin x3 + C 6 1−x

5.E-2 (AB93) If the second derivative of f is given by f 00 (x) = 2x − cos x, which of the following could be f (x)? x3 + cos x − x + 1 3 x3 (B) − cos x − x + 1 3 (C) x3 + cos x − x + 1

(A)

(D) x2 − sin x + 1 (E) x2 + sin x + 1 h

Ans:

x3 3

+ cos x − x + 1

5.E-3 (adapted from [2]) A particle moves along the x-axis with velocity at time t given by: v(t) = t + 2 sin t. If the particle is at the origin when t = 0, its position at the time when v = 5 is x = [Ans: 17.277]

Mr. Budd, compiled September 29, 2010

164

AP Unit 5 (Trigonometrics)

Mr. Budd, compiled September 29, 2010

Unit 6

Calculus of Exponential Functions 1. Derivatives Involving e 2. Anti-derivatives Involving e 3. Separable Differential Equations

Advanced Placement

I. Derivatives Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Computation of derivatives. • Knowledge of derivatives of basic functions, including exponential and logarithmic, and trigonometric functions. II. Integrals Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables. 165

166

AP Unit 6 (Exponentials) Applications of antidifferentiation. â&#x20AC;˘ Finding specific antiderivatives using initial conditions, including applications to motion along a line. â&#x20AC;˘ Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth.

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 1: Antidifferentiation by Simplification

6.1

167

Antidifferentiation by Simplification

Advanced Placement Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables. Textbook §4.6 Integration by Substitution [16]

6.1.1

u-Simplification

Example 6.1.1 Z

6x2 x3 + 1 dx

h i 2 Ans: x3 + 1 + C

Example 6.1.2 Z

3x2 √ dx 2 x3 + 1

Ans:

√

x3 + 1 + C

Example 6.1.3 (adapted from AB93) Z 4x3 √ dx = x4 − 3 √ Ans: 2 x4 − 3 + C

Completing the Derivative of the Inside Example 6.1.4 Z

x2 x3 + 1

dx =

Mr. Budd, compiled September 29, 2010

168

AP Unit 6 (Exponentials) h

Ans:

1 9

x3 + 1

Example 6.1.5 x2

(x3 + 1)

h i Ans: − 3(x31+1)

Example 6.1.6 (adapted from [2]) Z 4 x x2 − 1 dx =

Ans:

1 15

x3 + 1

Example 6.1.7 [11] r2 − 1

(r3 − 3r + 3)

Example 6.1.8 [11] Z

6.1.2

1 x2

r 5

1 + 5 dx x

Simplification with Trigonometrics Inside

R Example 6.1.9 Solve cos x sin x dx two different ways, and then show that the solutions are really the same. Do the same for R sec2 x tan x dx.

Example 6.1.10 (adapted from Stewart [20])

sec4 x tan x dx

Ans:

sec4 x 4

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 1: Antidifferentiation by Simplification

169

Example 6.1.11 [18] Z

√

sin x dx 1 + cos x √ Ans: −2 1 + cos x + C

6.1.3

Simplification with Trigonometrics Outside

Example 6.1.12 (adapted from [20]) Find

x2 cos 2x3

Ans:

Z Example 6.1.13 (adapted from [18]) Calculate

sec

√

1 6

dx.

sin 2x3 + C

x tan √ x

√

[Ans: 2 sec

Z Example 6.1.14 Calculate

√

x + C]

x dx cos2 x2

Ans:

1 2

tan x2 + C

Motivation for finding the happy function Example 6.1.15 What happens when we try to find

I need a function whose derivative is

tan x dx?

1 u.

Suppose I had a function h(x) (the happy function), for which h0 (x) = h(x). Then suppose I had a function I(x) = h−1 (x), which was the inverse function of h(x). Remember, to get an inverse function, I switch the x’s and y’s Mr. Budd, compiled September 29, 2010

170

AP Unit 6 (Exponentials)

So: y = I(x) x = h(y) d d x= h(y) dx dx dy 1 = h0 (y) dx 1 dy = h0 (y) dx 1 dy = h(y) dx dy 1 = x dx

I = h−1 Differentiate both sides Chain Rule, inside is y

h0 = h h(y) = x from before

So, if I can find a happy function, h(x), such that h0 (x) = h(x), then the inverse of the happy function has a derivative x1 , and I can finally find the antiderivative of tan x. Now, if I only had a happy function...

Problems Z 6.A-1

x Z

6.A-2 Z 6.A-3

p 3

h i 4/3 Ans: − 38 1 − x2 +C

1 − x2 dx =

G0 (v(x)) v 0 (x) dx =

[Ans: G (v(x)) + C]

H 0 (ax + b) dx =

6.A-4 [11]

Ans:

sin7 (θ) cos (θ) dθ

6.A-5 (adapted from [2]) If Z 6.A-6 (adapted from [2]) Z 6.A-7 (adapted from [20]) Z 6.A-8 [18] Find

dy = cos3 x sin x, then y = dx 6 cos x sin2 x dx = tan3 θ sec2 θ dθ

cos4 x sin x dx.

1 a H(ax

Ans:

+ b) + C

sin8 θ 8

Ans: − 41 cos4 x + C Ans: 2 sin3 x + C h Ans:

tan4 θ 4 +C

Ans: − 51 cos5 x + C Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 1: Antidifferentiation by Simplification Z 6.A-9 [18] Find Z 6.A-10 [18] Find Z 6.A-11 [18] Find Z 6.A-12 [18] Find

√

1 + sin x cos x dx.

171

h Ans:

x sin3 x2 cos x2 dx. √

sec2 x dx. 1 + tan x

1 + tan2 x sec2 x dx.

R 6.A-13 (adapted from [2]) cos (3x + 2) dx = R 6.A-14 [20] Find x sin 1 − x2 dx √ R sin x √ 6.A-15 [18] Find dx x R 6.A-16 [18] Find x sec2 x2 dx. R 6.A-17 [18] Find cos (3x + 1) dx. R 6.A-18 [18] Find sin (3 − 2x) dx.

2 3

Ans:

sin4 x2 + C

(1 + sin x) 1 8

3/2

√ Ans: 2 1 + tan x + C

Ans: tan x + Ans: Ans:

1 3

tan3 x + C

1 3

sin (3x + 2) + C 1 2 +C 2 cos 1 − x

√ [Ans: −2 cos x + C] Ans: Ans: Ans:

1 2

tan x2 + C

1 3

sin (3x + 1) + C

1 2

cos (3 − 2x) + C

Mr. Budd, compiled September 29, 2010

172

AP Unit 6 (Exponentials)

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 2: The Happy Function

6.2

173

The Happy Function

Advanced Placement Computation of derivatives. • Knowledge of derivatives of basic functions, including exponential and logarithmic functions. Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables. Textbook §2.7 Derivatives of Exponential and Logarithmic Functions: “Derivatives of the Exponential Functions”; §4.1 Antiderivatives; §4.6 Integration by Substitution [16]

6.2.1

Differentiating the Exponential Function d x e = ex dx

Example 6.2.1 Find

d x b dx

x d x d b = eln b dx dx d x ln b e = dx x ln b =e ln b ln b x = e ln b = bx ln b Applying the Chain Rule: d g(x) e = eg(x) g 0 (x) dx Mr. Budd, compiled September 29, 2010

174

AP Unit 6 (Exponentials) Example 6.2.2 (adapted from AB acorn ’02) Two particles start at the origin and move along the x-axis. For 0 ≤ t ≤ 10, their respective position functions are given by x1 = cos t and x2 = e−2t − 1. For how many values of t do the particles have the same velocity? [Ans: Four] eh − 1 is h→0 3h

Example 6.2.3 (adapted from BC97) lim

Ans:

1 3

Example 6.2.4 (adapted from AB97) Let f be the function given 2 by f (x) = 2e4x . For what value of x is the slope of the line tangent to the graph of f at (x, f (x)) equal to 5? [Ans: 0.246] 3

Example 6.2.5 (adapted from BC93) If f (x) = etan h

, then f 0 (x) = 3

Ans: 3 tan2 x sec2 x etan

Example 6.2.6 For ef (x) = x, find f 0 (x) in terms of x only. Note d ln x. that your answer is dx u-Substitution with exponentials Example 6.2.7 If

e−x dy = 2 , then y = dx (3 + 2e−x ) h Ans:

Example 6.2.8 [18] Z

1 2(3+2e−x )

e−x 1 + cos e−x dx

[Ans: −e−x − sin (e−x ) + C] Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 2: The Happy Function

6.2.2

175

Antidifferentiating the Exponential Function

Since the derivative of ex is ex the antiderivative of ex is ex .

ex dx = ex + C

Example 6.2.9 Find the antiderivative of

2x dx.

x Change 2x to eln 2 to ex ln 2 . Let u = x ln 2. If the antiderivative of your inside function is just a constant number (such as ln 2), you can skip the whole process of u-substitution and just divide by that constant number. Z

bx dx =

bx +C ln b

Applying u-substitution to the general antiderivative of ex : Z

eg(x) g 0 (x) dx = eg(x) + C

Example 6.2.10 (BC93) A particle moves along the x-axis so that at any time t ≥ 0 the acceleration of the particle is a(t) = e−2t . If 5 17 at t = 0 the velocity of the particle is and its position is , then 2 4 its position at any time t > 0 is x(t) =

h Ans:

e−2t 4

+ 3t + 4

Example 6.2.11 (adapted from [2]) Z

ex − x2 dx = ex3

e−x Ans: x + +C 3

Mr. Budd, compiled September 29, 2010

176

AP Unit 6 (Exponentials)

6.2.3

Skippable u-Simplification

If the inside function is linear, i.e.

• u = mx + b; •

du = m or k; dx

• the derivative of the inside is constant;

Then all that u-substitution will accomplish is to divide by the derivative of the inside. Do not try this unless the derivative of the inside is a constant.

f 0 (mx + b) dx =

f (mx + b) +C m

Example 6.2.12 Prove the above equation.

Z Example 6.2.13 Find

(4 − 7x) dx. 2

Example 6.2.14 Find

Example 6.2.15 Find

Ans: − (4−7x) +C 56

cos (3x + 2) dx.

Ans:

sin(3x+2) 3

sec (πx) tan (πx) dx.

Ans:

1 π

sec (πx) + C

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 2: The Happy Function

177

Problems 6.B-1 (adapted from AB98) Let f be the function given by f (x) = 2e3x and let g be the function given by g(x) = 6x3 . At what value of x do the graphs of f and g have parallel tangent lines? [Ans: −0.344] 6.B-2 (adapted from AB98) If f (x) = cos (e−x ), then f 0 (x) = [Ans: e−x sin (e−x )] ex−5 , then f 0 (5) = Ans: 72 2 2 2 ex 2ex (x2 −1) 0 Ans: 6.B-4 (adapted from AB97) If f (x) = 2 , then f (x) = x3 x 3

6.B-3 (adapted from BC97) If f (x) = (x − 1) 2 +

6.B-5 (adapted from AB93)

d (3x ) = dx

[Ans: (3x ) ln 3]

√ 6.B-6 (adapted from [2]) The approximate value of y = 3 + ex at x = 0.04, obtained from the tangent to the graph at x = 0, is [Ans: 2.01] 6.B-7 (adapted from [2]) Two particles move along the x-axis and their positions at time 0 ≤ t ≤ 2π are given by x1 = sin 3t and x2 = e(t−3)/2 − 0.75. For how many values of t do the two particles have the same velocity? [Ans: Six] 6.B-8 (adapted from [2]) If y = ekx , then 6.B-9 [11]

d6 y = dx6

Ans: k 6 ekx Ans:

ex cos (πex ) dx

1 π

sin (πex ) + C

6.B-10 [18] Z

√

ex dx ex + 1 √ Ans: 2 ex + 1 + C

6.B-11 (from [2]) Find

dy for ey = xy. dx

6.B-12 (from [2]) If cos x = ey , 0 < x < [Ans: − tan x]

h Ans:

y xy−x

π dy , what is in terms of x? 2 dx

6.B-13 (from Stewart [20]) Let f (x) = ex . (a) Find the value of the constants a, b, c, and d cubic function C(x) = a + bx + cx2 + dx3 has the properties C(0) = f (0), C 0 (0) = f 0 (0), C 00 (0) = f 00 (0), and C 000 (0) = f 000 (0). (b) Plot f and C on the same axes over the interval [−4, 4]. Ans: C(x) = 1 + x + 21 x2 + 16 x3 Mr. Budd, compiled September 29, 2010

178

AP Unit 6 (Exponentials)

dy in terms of x only. dx R t 6.B-15 (adapted from AB97) π1 e π dt =

Ans:

6.B-14 For ey = x, find

1 x

Ans: e π + C

6.B-16 (adapted from AB97) etan x dx = cos2 x

[Ans: etan x + C] 6.B-17 (adapted from [2]) The acceleration of a particle at time t moving along the x-axis is given by: a = 9e3t . At the instant when t = 0, the particle is at the point x = 2 moving with velocity v = −2. The position of the 1 Ans: e − 32 particle at t = is 3 6.B-18 (adapted from [2]) Z

√ 3

e x √ dx 3 3 x2 h

Ans: e

√ 3 x

6.B-19 (adapted from [2]) The number of bacteria in a culture is growing at a rate of 1000e2t/3 per unit of time t. At t = 0, the number of bacteria present was 1500. Find the number present at t = 3. Ans: 1500e2 R 6.B-20 [11] ex cos (πex ) dx Ans: π1 sin (πex ) + C R 6.B-21 (adapted from [18]) sin x ecos x dx [Ans: −ecos x + C] 6.B-22 [18] Z

e1/x dx x2 Ans: −e1/x + C

Z 6.B-23 Find

√

ex dx using the u-substitution u = ex . [Ans: arcsin ex + C] 1 − e2x

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 3: Inverse of the Happy Function

6.3

179

Inverse of the Happy Function

Advanced Placement Computation of derivatives. • Knowledge of derivatives of basic functions, including exponential and logarithmic functions. Techniques of antidifferentiation. • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables. Textbook §2.7 Derivatives of Exponential and Logarithmic Functions: “Derivative of the Natural Logarithm”; §4.1 Antiderivatives; §4.6 Integration by Substitution [16]

6.3.1

Inverse of the Exponential Function y = ln x ey dy ey dx dy dx dy dx

=x =1 1 ey 1 = x =

d 1 ln x = dx x

x>0

Using the Chain Rule gives: d 1 0 ln (g(x)) = g (x) dx g(x) d g 0 (x) ln (g(x)) = dx g(x)

g(x) > 0

Mr. Budd, compiled September 29, 2010

180

AP Unit 6 (Exponentials) Example 6.3.1 Find the derivative: d ln (−x) dx

x<0

d 1 ln x = dx x

x>0

Combining

and

d 1 ln (−x) = dx x

x<0

1 d ln |x| = dx x

x 6= 0

into one statement gives

Example 6.3.2 (adapted from AB97) If f (x) = ln 3x2 − 1 , then f 0 (x) =

(A) 2 3x − 1

6x (B) 2 |3x − 1| 6 |x| (C) 3x2 − 1 6x (D) 3x2 − 1 1 (E) 2 3x − 1 h

Ans:

6x 3x2 −1

Example 6.3.3 Show that d d ln xa = a ln x dx dx Example 6.3.4 Show that d d (ln a + ln x) = ln (ax) dx dx Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 3: Inverse of the Happy Function

181

Example 6.3.5 Show that d d ln ex = x dx dx

Example 6.3.6 Show that d ln x d e = x dx dx 1 π Example 6.3.7 (adapted from [2]) If y = cos u, u = 2v − + − 1, v 2 dy and v = ln x, then the value of at x = e is dx

Ans: − 3e

Example 6.3.8 Show that y = ln x2 + e is a solution to the dy differential equation = 2xe−y . dx

6.3.2

Implicit Differentiation with ln

The derivative at a specific point Example 6.3.9 [3] The slope of the tangent to the graph of ln (x + y) = x2 at the point where x = 1 is

[Ans: 2e − 1] Note that sometimes, you must find one of the coordinate pairs from the other.

The explicit option Example 6.3.10 Solve ln (x + y) = x2 explicitly for y, then find dy when x = 1. dx Mr. Budd, compiled September 29, 2010

182

6.3.3

AP Unit 6 (Exponentials)

Antidifferentiating Reciprocals

Why does the Anti-Power Rule not work for

x−1 dx?

But, we recently found a function whose derivative is

1 x.

1 dx = ln |x| + C x

Why is the antiderivative ln |x| and not just ln x? 1 is x 6= 0, but the domain of ln x is only the real numbers x R greater than 0. So, x1 dx = ln x only for x > 0. The problem is that x in x1 can be anything but zero, but ln x is impossible for those x’s below zero. I didn’t have this problem before, when I was differentiating ln x, because the domain of ln x is just those numbers above 0, and my derivative, x1 can handle those numbers. The domain of

I need an antiderivative of

1 for x < 0, since ln x only works for x > 0. x

So let’s take a look at the real numbers less than 0, For x > 0, we’ve already d 1 seen that ln x = . Now let’s look at x < 0: dx x = ln (−x) dy 1 d = (−x) dx −x dx dy 1 = − (−1) dx x dy 1 = dx x

So what we see is that (for x < 0).

1 is the derivative of both ln x (for x > 0) and ln (−x) x

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 3: Inverse of the Happy Function

183

d 1 ln x = dx x

(x > 0)

1 d ln (−x) = dx x

(x < 0)

Combining these into one statement for all x 6= 0 gives me d 1 ln |x| = dx x d ln x = x1 , is basically the same The typical expression that I’m use to seeing, dx thing, only your restricting yourself to positive x’s, i.e., those numbers for which x = |x|

1 1 is the derivative of ln |x| (for all x 6= 0, the antiderivative of is ln |x| x x (plus some arbitrary constant).

Since

dx = ln |x| + C x

u-substitution gives: Z

g 0 (x) = ln |g(x)| + C g(x)

Example 6.3.11 Find Z tan x dx

[Ans: − ln |cos x| + C]

Example 6.3.12 Use the substitution u = sec x + tan x to find Z sec x dx

[Ans: ln |sec x + tan x| + C] Mr. Budd, compiled September 29, 2010

184

AP Unit 6 (Exponentials) Example 6.3.13 (adapted from [2]) Find a family of curves that intersect at right angles every curve of the family y = 0.5x2 + k for every real value of k?

[Ans: y = − ln |x| + C]

6.3.4

Antidifferentiating Fractions

If the order on top is the same or higher than that on bottom, use division (long or synthetic) to reduce the problem to a polynomial + some remainder over the original denominator. The polynomial is easy to antidifferentiate using the anti-power rule, and the remainder over the original denominator should be easier to antidifferentiate than the original fraction.

Example 6.3.14 (adapted from [2]) Z x−1 dx = x−2

[Ans: x + ln |x − 2| + C]

Example 6.3.15 If the velocity of Runner B, in meters per second, 24t , find an expression for the runner’s position is given by v(t) = 2t + 3 if her position at time t = 0 seconds is 0 meters.

[Ans: 12t − 18 ln |2t + 3| + 18 ln 3]

Problems 6.C-1 (BC97) lim

1 Ans: e

2 6.C-2 (adapted from BC93) If f (x) = ln e3x , then f 0 (x) =

[Ans: 6x]

ln(e + h) − 1 is h→0 h

6.C-3 (adapted from AB93) The slope of the line normal to the graph of y = π 3 ln (sec x) at x = is [Note: normal lines are perpendicular to tangent 4 lines.] Ans: − 13 Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 3: Inverse of the Happy Function

185

6.C-4 (adapted from [2]) If y = 2u + 3eu and u = 1 + ln x, find [Ans: 5e]

dy 1 when x = . dx e

t2 + 1 gives the position 6.C-5 (adapted from [2]) The formula x(t) = 2 ln t + 9 of an object moving along the x-axis during the time interval 1 ≤ t ≤ 5. At the instant when the acceleration of the object is zero, what is the velocity? Ans: 43 6.C-6 (adapted graph1 of √ from [2]) What is the slope of the line tangent to the Ans: 3e3 y = ln 3 x at e3 , 1 ? 6.C-7 [3] There is a point between P (1, 0) and Q (e, 1) on the graph of y = ln x such that the tangent to the graph at that point is parallel to the line through points P and Q. What is the x-coordinate of this point? [Ans: e − 1] 6.C-8 (adapted from AB acorn ’02) Which of the following are antiderivatives ln3 x of ? x ln4 x 4 ln4 x II. +6 4 3 ln x − ln3 x III. x2 I.

(A) I only (B) III only (C) I and II only (D) I and III only (E) II and III only [Ans: I and II only] 3

6.C-9 (adapted from AB98) Let F (x) be an antiderivative of then F (9) = 6.C-10 (adapted from [3]) If eg(x) = 3x − 1, then g 0 (x) =

(ln x) . If F (1) = 2, x [Ans: 7.827] h i 3 Ans: 3x−1

6.C-11 (MM spec) The function f is given by 2x 1 + x2 R By using an appropriate substitution, find f (x) dx. Ans: x − ln 1 + x2 + C f (x) = 1 −

Mr. Budd, compiled September 29, 2010

186

AP Unit 6 (Exponentials)

6.C-12 (adapted from [2]) Find Z

sec2 x dx tan x [Ans: ln |tan x| + C]

6.C-13 (adapted from [2]) A particle starts at (3, 0) when t = 0 and moves along the x-axis in such a way that at time t > 0 its velocity is given by v(t) = 1 . Determine the position of the particle at t = 5. [Ans: 3 + ln 6] 1+t 6.C-14 The antiderivative of

1 cabin

[Ans: Evan’s ark]

6.C-15 (adapted from [2]) Z

12x2 dx = 1 + x3

Ans: 4 ln 1 + x3 + C

6.C-16 (adapted from AB98) Z

x2 − 1 dx = x h

Ans:

x2 2

− ln |x| + C

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 4: Separable Differential Equations

6.4

187

Separable Differential Equations

Advanced Placement Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. Textbook §7.1 Modeling with Differential Equations; §7.2 Separable Differential Equations [16]

6.4.1

Separable Differential Equations

Definition 6.1. Differential Equation A differential equation is an equation that contains the derivative of a function. [10]

Example 6.4.1 The solution to the differential equation 2xe−y , where y(0) = 1, is

dy = dx

Ans: y = ln x2 + e Example 6.4.2 (adapted from AB acorn ’02) The solution to the dy x2 = 3 where y(3) = 2, is differential equation dx y h

Ans: y =

q 4

4 3 3x

− 20

Example 6.4.3 (adapted from [2]) If the graph of y = f (x) contains 2x sin x2 dy the point (0, 1) and if = , then f (x) = dx y Mr. Budd, compiled September 29, 2010

188

AP Unit 6 (Exponentials) h

6.4.2

Ans:

i 3 − cos (x2 )

Separable Differential Equations with Logs

Example 6.4.4 (AB97) Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t ≥ 0. After her parachute opens, dv her velocity satisfies the differential equation = −2v − 32, with dt initial condition v(0) = −50. (a) Use separation of variables to find an expression for v in terms of t, where t is measured in seconds. (b) Terminal velocity is defined as lim v(t). Find the terminal t→∞ velocity of the skydiver to the nearest foot per second. (c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed? Ans: v = −34e−2t − 16, −16, 1.070 Example 6.4.5 A turkey is cooking in the oven at 300 degrees Fahrenheit. It starts out at room temperature (70 degrees). After 1 hour, it is ? degrees. How long before it reaches 170 degrees, at which point it will be done. The rate of change in the temperature of the turkey is proportional to the difference between the temperatures of the environment and the turkey.

Problems 6.D-1 (adapted from AB93) If x = 3, y =

dy = 2y 2 and if y = 1 when x = 2, then when dx [Ans: −1]

6.D-2 (adapted from AB acorn ’02) The solution to the differential equation q h i x2 dy = 3 where y(3) = 2, is Ans: y = 4 43 x3 − 20 dx y dy −xy 2 = . Let dx 2 y = f (x) be the particular solution to this differential equation with the initial condition f (−1) = 2.

6.D-3 (AB05B) Consider the differential equation given by

(a) On the axes provided (Figure 6.1), sketch a slope field for the given differential equation at the twelve points indicated. Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 4: Separable Differential Equations

189

Figure 6.1: 2005B AB Exam

(b) Write an equation for the line tangent to the graph of f at x = −1. (c) Find the solution y = f (x) to the given differential equation with the initial condition f (−1) = 2. h

Ans: y − 2 = 2 (x + 1);y =

x2 +1

6.D-4 (adapted from [2]) The point (1, 4) lies on the graph of an equation y = dy √ f (x) for which = 6x2 y where x ≥ 0 and y ≥ 0. When x = 0 the dx value of y is [Ans: 1] 6.D-5 (adapted from [2]) If the graph of y = f (x) is defined for all x ≥ 0, dy √ contains the point (0, 4), has = 3 xy and f (x) > 0 for all x, then dx h 2 i f (x) = Ans: x3/2 + 2 6.D-6 (AB98) Let f be a function with f (1) = 4 such that for all points (x, y) 3x2 + 1 on the graph of f the slope is given by . 2y (a) Find the slope of the graph of f at the point where x = 1. Ans: 21 (b) Write an equation for the line tangent to the graph of f at x = 1 and use it to approximate f (1.2). Ans: y − 4 = 12 (x − 1), 4.1 dy 3x2 + 1 (c) Find f (x) by solving the separable differential equation = dx 2y √ with the initial condition f (1) = 4. Ans: x3 + x + 14 (d) Use your solution to find f (1.2).

[Ans: 4.114] Mr. Budd, compiled September 29, 2010

190

AP Unit 6 (Exponentials)

dy 6.D-7 (adapted from [2]) If = 2xy and if y = 3 when x = 0, then y = dx h i 2 Ans: 3ex ` dy 6.D-8 (adapted from [2]) A solution of the equation + 2xy = 0 that contains dx h i 2 the point 0, e2 is Ans: y = e2â&#x2C6;&#x2019;x

Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 5: Exponential Growth and Decay

6.5

191

Exponential Growth and Decay

Advanced Placement Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth. Textbook §7.1 Modeling with Differential Equations; §7.2 Separable Differential Equations [16] Resources §7.2 Exponential Growth and Decay and §7.3 Other Differential Equations for Real-World Applications in [10]

6.5.1

Proportional Growth

Example 6.5.1 The rate growth of the population of Escherichia coli is proportional to the number of E. coli. Find a general expression for the population as a function of time if the initial population is P0 .

Ans: P = P0 ekt

Example 6.5.2 (adapted from AB98) Population y grows according dy to the equation = ky, where k is a constant and t is measured in dt years. If the population doubles every 8 years, then the value of k is

[Ans: 0.087] Mr. Budd, compiled September 29, 2010

192

AP Unit 6 (Exponentials) Example 6.5.3 [10] Chemical Reaction Problem Calculus buddite (a rare substance) is converted chemically into Glamis thanus. Buddite reacts in such a way that the rate of change in the amount left unreacted is directly proportional to that amount. (a) Write a differential equation that expresses this relationship. Solve it to find an equation that expresses amount in terms of time. Use the initial conditions that theh amount is 50 mg when i t/20 = 50e−0.025541...t t = 0 min and 30 mg when t = 20 min. Ans: dB dt = kB, B = 50 (0.6) (b) Sketch the graph of amount versus time. (c) How much buddite remains an hour after the reaction starts? [Ans: 10.8 mg] (d) When will the amount of buddite equal 0.007 mg? [Ans: 5 hr 47 min]

6.5.2

Other Applications of Differential Equations

Example 6.5.4 Tin Can Leakage Problem [10] Suppose you fill a tall (topless) tin can with water, then punch a hole near the bottom with an ice pick. The water leaks quickly at first, then more slowly as the depth of the water increases. In engineering or physics, you will learn that the rate at which the water leaks out is directly proportional to the square root of its depth. Suppose that at time t = 0 min, the depth is 12 cm and dy dt is −3 cm/min. (a) Write a differential equation stating that the instantaneous rate of change of y with respect to t is directly proportional to the square root of y. Find the proportionality constant. (b) Solve the differential equation to find y as a function of t. Use the given information to find the particular solution. What kind of function is this? (c) Plot the graph of y as a function of t. Sketch the graph. Consider the domain of t in which the function gives reasonable answers. (d) Solve algebraically for the time at which the can becomes empty. Compare your answer with the time it would take at the initial rate of −3 cm/min. Ans: k = −3 12−1/2 ; y =

3 2 16 t

− 3t + 12; ;8 (twice as long)

Example 6.5.5 Dam Leakage Problem [10] A new dam is constructed across Scorpion Gulch. Engineers want to predict the amount Mr. Budd, compiled September 29, 2010

AP Unit 6, Day 5: Exponential Growth and Decay

193

of water in the lake behind the dam as a function of time. At t = 0 days the water starts flowing in at a fixed rate F ft3 /hr. Unfortunately, as the water level rises, some leaks out. The leakage rate, L, is directly proportional to the amount of water, W ft3 , present in the lake. Thus the instantaneous rate of change of W is equal to F − L. (a) What does L equal in terms of W ? Write a differential equation that expresses dW/dt in terms of F , W , and t. Ans: dW dt = F − kW (b) Solve for W in terms of t, using the W = 0 initial condition when t = 0. Ans: W = Fk 1 − e−0.04t (c) Water is known to be flowing in at F = 5000 ft3 /hr. Based on geological considerations, the proportionality constant in the leakage equation is assumed to be 0.04/hr. Write the equation for W , substituting these quantities. Ans: W = 125000 1 − e−0.004t (d) Predict the amount of water after 10 hr, 20 hr, and 30 hr. After these numbers of hours, how much water has flowed in and how much has leaked out? [Ans: L : 8790, 31166, 72649] (e) When will the lake have 100000 ft3 of water? [Ans: bit more than 40 hr] (f) Find the limit of W as t approaches infinity. State the real world meaning of this number. Ans: 125000 ft3 (g) Draw the graph of W versus t. Clearly show an asymptote. (h) The lake starts filling with water. The actual amount of water at time t = 10 is exactly 40000 ft3 . The flow rate is still 5000 ft3/hr, as predicted. Use this information to find a more precise value of the leakage constant k. [Ans: k = 0.0464...]

Problems 6.E-1 (AB98) If

dy = ky and k is a nonzero constant, then y could be dt

(A) 2ekty (B) 2ekt (C) ekt + 3 (D) kty + 5 (E)

1 2 1 ky + 2 2 Ans: 2ekt Mr. Budd, compiled September 29, 2010

194

AP Unit 6 (Exponentials)

6.E-2 (adapted from AB93) A puppy weighs 2.1 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 4 months old (to the nearest 0.1 pound)? [Ans: 5.8 pounds] 6.E-3 (adapted from [2]) If g 0 (x) = 3g(x) and g(−1) = 1, then g(x) is Ans: e3x+3 6.E-4 (adapted from [2]) The change in N , the number of bacteria in a culture dN = 3N . If N = 4, when t = 0, the dish at time t is given by: dt approximate value of t when N = 1614 is [Ans: 2] 6.E-5 (AB ’96) The rate of consumption of cola in the United States is given by S(t) = Cekt , where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980. The consumption rate doubles every 5 years and the consumption rate at the beginning of1 1980 was 6 billion gallons per year. Find C and k. Ans: C = 6, k = 5 ln 2 6.E-6 [10] You run over a nail. As the air leaks out of your tire, the rate of change of air pressure inside the tire is directly proportional to that pressure. (a) Write a differential equation that states this fact. Evaluate the proportionality constant if the pressure was 35 psi and decreasing at 0.28 psi/min at time zero. Ans: dP dt = −0.008P (b) Solve the differential equation subject to the initial condition implied in step (a). Ans: P = 35e−0.008t (c) Sketch the graph of the function. Show its behavior a long time after the tire is punctured. (d) What will be the pressure at 10 min after the tire was punctured? [Ans: about 32.3 psi] (e) The car is safe to drive as long as the tire pressure is 12 psi or greater. For how long after the puncture will the car be safe to drive? [Ans: about 134 min] 6.E-7 (AB93) Let P (t) represent the number of wolves in a population at time t years, when t ≥ 0. The population P (t) is increasing at a rate directly proportional to 800 − P (t), where the constant of proportionality is k. (a) If P (0) = 500, find P (t) in terms of t and k. Ans: P (t) = 800 − 300e−kt (b) If P (2) = 700, find k. Ans: k = ln23 ≈ 0.549 (c) Find lim P (t) t→∞

[Ans: 800]

Mr. Budd, compiled September 29, 2010

Unit 7

Existence Theorems 1. Continuity 2. Extreme and Intermediate Value Theorems 3. Differentiability and Rolle’s Theorem 4. Average Rate of Change and Mean Value Theorem 5. Riemann Sums: Evaluating Definite Integrals 6. Fundamental Theorem of Calculus 7. Area Advanced Placement

1. Derivatives Concept of the derivative. • Relationship between differentiability and continuity. Derivative as a function. • The Mean Value Theorem and its geometric consequences. 2. Integrals Fundamental Theorem of Calculus. • Use of the Fundamental Theorem to evaluate definite integrals. • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. 195

196

AP Unit 7 (Existence Theorems)

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 1: Quasi-Limits: One-Sided and Infinite

7.1

197

Quasi-Limits: One-Sided and Infinite

Advanced Placement Limits of functions (including one-sided limits). • An intuitive understanding of the limiting process. • Calculating limits using algebra. • Estimating limits from graphs or tables of data. Asymptotic and unbounded behavior. • Understanding asymptotes in terms of graphical behavior. • Describing asymptotic behavior in terms of limits involving infinity. Continuity as a property of functions. • An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions. Textbook §1.3 Computation of Limits and §1.5 Limits Involving Infinity; Asymptotes [16] Resources §2-5 Limits Involving Infinity in Foerster [10].

7.1.1

Step Discontinuities & One-Sided Limits

Piecewise Functions  3 x −8   − x<2    x−2   x=2 Example 7.1.1 For the function f (x) = 0.2     x3 − 8    x>2 x−2 Is f (x) continuous? Where do you expect to have problems? If Mr. Budd, compiled September 29, 2010

198

AP Unit 7 (Existence Theorems) you are looking at values near 2, but different from 2, what formula should you use? Determine: (a) f (2) (b) lim− f (x) x→2

(d) lim f (x) x→2

(e) lim f (x) x→0+

(f) lim f (x) x→3

Example 7.1.2 In your mighty, mighty groups of four: [18] Let ( 2x − 1, x ≤ 2 g(x) = x2 − x x > 2 Is g(x) continuous? Where do you expect to have problems? Determine: (a) g(2) (b) lim− g(x) x→2

(d) lim g(x) x→2

(e) lim+ g(x) x→0

(f) lim g(x) x→3

Absolute Value x3 − 8 analytically. Check your anx→2 |x − 2| swer with a table. Predict the behavior of the graph near x = 2.

Example 7.1.3 Examine lim

7.1.2

One-Sided Derivatives

Example 7.1.4 Find (a)

lim

∆x→0+

|2 + ∆x| − |2| ∆x Mr. Budd, compiled September 29, 2010

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199

|−3 + h| − |−3| h h→0 |0 + ∆x| − |0| (c) lim ∆x ∆x→0+

(b) lim+

In your mighty, mighty groups of four: Find (a) (b) (c) (d) (e) (f)

|2 + ∆x| − |2| ∆x |−3 + ∆x| − |−3| lim ∆x ∆x→0− |0 + h| − |0| lim h h→0− |2 + ♥| − |2| lim ♥→0 ♥ |−3 + ∆x| − |−3| lim ∆x→0 ∆x |0 + ∆x| − |0| lim ∆x→0 ∆x lim

∆x→0−

What do each one of these limits represent, in terms of a graph of d y = |x|? Use your answers to sketch a graph of |x|. dx Example 7.1.5 The graph of the function f shown in Figure 7.1 Figure 7.1: Graph of f

consists of a semicircle and three line segments. Find the following limits of difference quotients: f (x) − f (−2) x+2 f (2.3 + h) − f (2.3) (b) lim h→0 h f (x) − f (−3) (c) lim − x+3 x→−3 (a) lim

x→−2

Ans: − 31 [Ans: −1] [Ans: 2] Mr. Budd, compiled September 29, 2010

200

AP Unit 7 (Existence Theorems)

(d)

lim +

x→−3

f (x) − f (−3) x+3

(e) lim

f (x) − f (−3) x+3

(f) lim+

f (2 + h) − f (2) h

x→−3

h→0

Ans: − 13 [Ans: d.n.e.] [Ans: −1]

Example 7.1.6 In your mighty, mighty groups of four: Write a linear function, mx + b, such that if you plug 2 in, you get 7 out. For example, I will use 3x + 1, which works because 3 (2) + 1 = 7. Let √  √3x + 1, x < 2 R(x) = 7, x=2  √ mx + b, x > 2 Why is it important that m (2) + b is 7? (a) Find lim− x→2

R(x) − R(2) x−2

R(x) − R(2) , x−2 x→2 and put your answer in a table next to your function mx + b.

(b) On the board, claim your function mx+b. Find lim+ R(x) − R(2) x−2 R(2 + h) − R(2) (d) Find lim h h→0+ (c) Find lim

x→2

1 R(x) − R(c) (e) Pick a value c, c > − . Find lim x→c 3 x−c

7.1.3

Vertical Asymptotes: Infinite Limits

Nonexistent, Infinite Limits Example 7.1.7 How many vertical asymptotes does the function 2x2 − x − 6 f (x) = 2 have? What are they? Check your answers on x −x−2 your grapher.

Remember This? How do you distinguish between vertical asymptotes and removable discontinuities? Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 1: Quasi-Limits: One-Sided and Infinite

201

Example 7.1.8 Using a table or graph, examine

lim x→2 x − 2

Technically, as x goes to 2, x−2

does not go to one particular number (infinity

is not a number). Therefore, we say that lim x−2

does not exist, or that it is x→2

nonexistent. however, we like to treat infinity like a number, so we

Sometimes,

write lim x−2 = ∞, and we describe the limit as being infinite. It is true that x→2 the limit is both nonexistent and infinite.

Positive and Negative Zero On a case by case basis, you need to decide the sign (positive or negative) of any zeros in the denominator which yield infinite limits.

Example 7.1.9 Find

lim tan θ and

− θ→ π 2

lim tan θ. Confirm your

+ θ→ π 2

answer with a graph and with a table.

Derivatives at a Step Discontinuity Example 7.1.10 (adapted from [18]) Let ( 2x − 1, x ≤ 2 g(x) = x2 − x x > 2 Is g(x) continuous? Where do you expect to have problems? Determine: (a) g(2) (b) lim− g(x) x→2

(d) lim g(x) x→2

g(x) − g(2) x−2 g(x) − g(2) (f) lim x−2 x→2+

(e) lim

x→2−

Mr. Budd, compiled September 29, 2010

202

AP Unit 7 (Existence Theorems) g(x) − g(2) x→2 x−2 g(x) − g(3) (h) lim x→3 x−3 (g) lim

Problems ( 7.A-1 [18] Let f (x) =

x2 , x ≤ 1 5x x > 1

Determine: (a) f (2) (b) lim− f (x) x→2

(d) lim f (x) x→2

(e) lim+ f (x) x→1

(f) lim− f (x) x→1

(g) lim f (x) x→1

(h) lim+ f (x) x→0

(i) lim f (x) x→3

 2  x , 7.A-2 [18] Let g(x) = 7,   2x + 3, Find lim f (x)

x<3 x=3 x>3 [Ans: 9]

x→3

 x−4  √ 0<x≤4 x−2 7.A-3 For the function f (x) =   |4 − 2x| 4 < x ≤ 4.27 determine whether lim f (x) exists, and if so, what the limit is. [Ans: 4] x→4

 3−x  0<x≤4  1 7.A-4 For the function g(x) = x−2 − 1  √ x+5 4 < x ≤ 4.27 determine whether lim g(x) exists, and if so, what the limit is. x→3

[Ans: 1]

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 1: Quasi-Limits: One-Sided and Infinite

203

 4x − 16   0<x≤4 |x − 4| 7.A-5 For the function h(x) =  √ x + 12 4 < x ≤ 4.27 determine whether lim h(x) exists, and if so, what the limit is. x→4 Ans: d.n.e. b/c lim h(x) = −4 6= lim h(x) = 4 x→4−

x→4+

[Ans: −∞]

7.A-6 lim cot x is x→π −

7.A-7 f (x) =

(x − 1) 3x2 − 5x + 2

(a) Is f continuous at x = 1? (b) Give the equation(s) of all vertical asymptotes. Ans: no; x = 23 (x + 2) x2 − 1 . 7.A-8 Let W be the function given by W (x) = x2 − c (a) Will f be continuous if c = 1? Explain. (b) Will f be continuous if c = 4? Explain. (c) For what positive values of c is f continuous for all real numbers x? [Ans: no ; no; none] 7.A-9 Find lim− t→1

t , being as specific as possible. ln t

[Ans: −∞]

7.A-10 Do you know your asymptote from a hole in the graph?

Mr. Budd, compiled September 29, 2010

204

AP Unit 7 (Existence Theorems)

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 2: Limits at Infinity and Horizontal Asymptotes

7.2

205

Limits at Infinity and Horizontal Asymptotes

Advanced Placement Asymptotic and unbounded behavior. • Understanding asymptotes in terms of graphical behavior. • Describing asymptotic behavior in terms of limits involving infinity. • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) Textbook §1.5 Limits Involving Infinity; Asymptotes [16] Resources §2-5 Limits Involving Infinity in Foerster [10].

7.2.1

Limits at Infinity

If lim f (x) = 0 then lim x→c

x→c

1 is infinite. In other words, 0 in the denominator f (x)

wants an infinite limit. 1 = 0, i.e., ∞ in the denominator x→c f (x)

Similarly, if lim f (x) is infinite, then lim x→c

wants the limit to be zero. Note also that 0 in the numerator wants a zero limit, and ∞ in the numerator wants an infinite limit. 0 = indeterminate 0 0 =0 nonzero 0 =0 ±∞

nonzero = ±∞ 0 nonzero = answer nonzero nonzero =0 ±∞

±∞ = ±∞ 0 ±∞ = ±∞ nonzero ±∞ = inderterminate ±∞

As x → ∞ or x → −∞, f (x) behaves like its highest order term.

Example 7.2.1 Find the horizontal asymptotes of y =

2x2 − x − 6 x2 − x − 2

Mr. Budd, compiled September 29, 2010

206

AP Unit 7 (Existence Theorems) 2x2 − x − 6 x→−∞ x2 − x − 2 2x6 (b) lim x→−∞ x2 − x − 2 2x2 − x − 6x3 (c) lim x→∞ x2 − x − 2 (a)

lim

Confirm your answers on your handy, dandy calculator with a graph, and with a table.

Example 7.2.2 Find 12 + 7x − 5x3 x→∞ 3x2 − 12x + 9 12 + 7x − 5x2 (b) lim x→−∞ 3x2 − 12x + 9 12 + 7x − 5x2 (c) lim x→∞ 3x2 − 12x3 + 9 (a) lim

Do it the textbook way, and then using highest order terms. What happens if instead of taking the limit to ∞, you take x → −∞? In using the highest order term, I’m saying that 3x2 − 12x + 9 behaves as 3x2 for larger and larger values of x. Essentially, I’m ignoring the −12x. But if x → ∞, −12x should be extremely massive, not negligible. The difference is in the relative values. Think of how many molecules are in a drop of water. Conceptually, we will say an infinite amount. Add a drop of water to the Atlantic Ocean. Has the amount of water in the ocean really changed significantly? Even though you’ve added an infinite number of water molecules? So, now imagine that you have three oceans, take out 12 drops, and add nine molecules of water. You still have three oceans. Which is why we say that 3x2 − 12x + 9 behaves as 3x2 as x → ∞. Note that if the degree is higher on top, the limit at infinity is nonexistent (infinite). If the degree on top is lower, then the limit at infinity is zero. If the degree is the same, then the limit at infinity is the ratio of the coefficients. In determining degrees, ex >> · · · >> xn >> · · · >> x3 >> x2 >> x >> ln x >> sin x.

Example 7.2.3 Find 3x2 − 13x + 7 sin x x→∞ 5x2 + 4x − 9

(a) lim

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AP Unit 7, Day 2: Limits at Infinity and Horizontal Asymptotes

207

3x2 − 13ex + 7 sin x x→∞ 5x2 + 4 ln x − 9ex 3x2 − 13x183 + 7 sin x (c) lim x→∞ 5x2 + 4 ln x − 9ex

(b) lim

You could also see that the sin x becomes irrelevant by using a variation on the squeeze theorem. 1 2 h Example 7.2.4 [18] Find lim 1 h→0 1− h 1−

[Ans: does not exist]

7.2.2

Horizontal Asymptotes

There is a horizontal asymptote at y = L if (and only if) at least one of the following conditions is met: • lim f (x) = L x→∞

•

lim f (x) = L

x→−∞

Note that there are two possibilities for the horizontal asymptote. There might be two horizontal asymptotes, or one, or none. In typical examples of rational expressions, where we just have polynomials over polynomials, the limits at positive and negative infinity are the same. So, to make sure that you know to take the limits at both positive and negative infinity, test-makers devise problems where the limits are different, yielding two horizontal asymptotes instead of one. Typically, this is done with radicals, or with absolute values. Remember This? When is

√

9x2 6= 3x? When is

√

9x4 6= 3x2 ?

Example 7.2.5 Find all horizontal asymptotes of √

−7x + 12 9x2 + 3x + 12

Ans: y = 37 , y = − 73 Mr. Budd, compiled September 29, 2010

208

AP Unit 7 (Existence Theorems)

Problems big giant blue-green Calculus book p. 96: # 33; p. 118: #25, 27, 29, 39, 43, 57, 59

7.B-1 Find all the horizontal asymptotes of h(x) = √ 7.B-2 (a) What is lim

x→−∞

x−1 . Ans: y = 21 , y = − 12 2 −3 − x + 4x

x−1 ? Show your process. −3 − x + 4x2

1 (b) Make a single change to the problem so that the answer would be 4 Ans: y = 0; x2 − 1 in numerator 7.B-3 Let f (x) =

x−1 . −3 − x + 4x2

(a) Create a new function g(x) by making a single change to f (x) so that 1 lim g(x) = − . x→−∞ 2 (b) Show the analysis necessary to evaluate the limit at negative infinity of your new function. A table is not sufficient analysis. (c) Use your calculator to find

lim g(x). Does this confirm your pre-

x→−∞

vious answer? (d) Use your calculator to write a table to find lim g(x). x→+∞

1 h 7.B-4 [18] Find lim 1 h→0 1+ h a t+ t 7.B-5 [18] Find lim b t→0 t+ t

Ans:

1 2

1−

[Ans: −1]

Ans:

a b

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 3: Continuity and Differentiability

7.3

209

Continuity and Differentiability

Advanced Placement Continuity as a property of functions. • An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.) • Understanding continuity in terms of limits. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). Concept of the derivative. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. Definition 7.1 (Continuity at a point). A function f is continuous at x = c if and only if • lim f (x) exists x→c

• f (c) exists • lim f (x) = f (c) x→c

Ways for a function to be not continuous: • removable discontinuity • step discontinuity • vertical asymptote √ 4x + 9 − 2x + 5 , Example 7.3.1 (adapted from AB 1969) If x+2  f (−2) = k and if f is continuous at x = −2, then k =  

√

f (x) =

[Ans: 1] Mr. Budd, compiled September 29, 2010

for x 6= −2

210

AP Unit 7 (Existence Theorems)

Definition 7.2 (Differentiability at a point). A function f is differentiable at f (x) − f (c) x = c if and only if lim exists x→c x−c Ways for a function to be not continuous: • cusp • vertical tangent • discontinuity When checking differentiability, make sure • the y-values on the left and right match; • the slopes on the left and right match. Example 7.3.2 Determine whether f (x) = |x − 2| is differentiable at x = 2. Write f 0 (x). ( 2x Example 7.3.3 Let g(x) = 1 2 2x + x

x≤1 x>1

Determine whether the function g(x) is differentiable at x = 1. Write g 0 (x). Example 7.3.4 How could you change g(x) to make it differentiable? Differentiability Implies Continuity Differentiability is a higher standard than continuity. If f is differentiable, then f is continuous. If f is continuous, then f may or may not be differentiable. If f is not continuous, then f cannot be differentiable. Definition 7.3 (Continuity over an open interval). Definition 7.4 (Continuity over a closed interval). Definition 7.5 (Differentiability over an open interval). Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 3: Continuity and Differentiability

211

Problems 7.C-1 (adapted from Acorn ’05) Let f be defined as follows, where a 6= 0,  2  x − a2 for x 6= a f (x) = x−a  0 for x = a Describe each of the following as true or false: (a) lim f (x) exists x→a

(b) f (a) exists (c) f (x) is continuous at x = a 7.C-2 (adapted from Acorn ’05) Let f be defined as follows, where a 6= 0,  3  x − a3 for x 6= a f (x) = x−a  2 3a for x = a Describe each of the following as true or false: (a) lim f (x) exists x→a

(b) f (a) exists (c) f (x) is continuous at x = a 7.C-3 (adapted from √ √ [2]) The function f is continuous at x = 1. If f (x) =  x + 7 − 5x − 1 for x 6= 2 then k = Ans: − 23 x−1  k for x = 2  2  x + 4x + 3 7.C-4 (adapted from [2]) Given f (x) = x+1  k

for x 6= −1 for x = −1

Determine the value of k for which f is continuous for all real x. [Ans: 2] 7.C-5 (adapted ( from [2]) The function f is defined for all real numbers such that x2 + kx − 3 for x ≤ 1 f (x) = If f is both continuous and differentiable, 5x + b for x > 1 what are the values of k and b? What changes if you are only told that f needs to be differentiable? [Ans: k = 3, b = −4] 7.C-6 (adapted from AB ’97) Let f be a function such that lim

h→0

5. Why must f be continuous at x = 2?

f (2 + h) − f (2) = h

Mr. Budd, compiled September 29, 2010

212

AP Unit 7 (Existence Theorems)

7.C-7 (adapted from [2]) Let m and b be real numbers and let the function f be defined by ( 1 + 3bx + 2x2 for x ≤ 2 f (x) = mx + b for x > 2 Find m and b if f is differentiable at x = 2.

[Ans: m = −13, b = −7]

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 4: Some Basic Calculus Theorems

7.4

213

Some Basic Calculus Theorems

Advanced Placement Continuity as a property of functions. • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). Concept of the derivative. • Relationship between differentiability and continuity. Resources Workshop Calculus [12]

Notation ⇒

implies because therefore for all, for any there exists is a member of such that

∀ ∃ ∈ 3

7.4.1

Intermediate Value Theorem

Example 7.4.1 Connect the dots. Theorem 7.1 (Intermediate Value Theorem). If f is a function that is continuous over the closed interval [a, b], and k is a y-value between f (a) and f (b), then there exists an x-value, c, which is between a and b, such that f (c) = k. In other words, if f is a nice function (i.e., continuous function) on some interval, then it goes through every y-value between the two endpoints.

7.4.2

Extreme Value Theorem

Theorem 7.2 (Extreme Value Theorem). If f is a function that is continuous over the closed interval [a, b], then there exists Mr. Budd, compiled September 29, 2010

214

AP Unit 7 (Existence Theorems) • at least one x-value cM , where a ≤ cM ≤ b, such that f (cM ) ≥ f (x) for all x, a ≤ x ≤ b. • at least one x-value cm , where a ≤ cm ≤ b, such that f (cM ) ≤ f (x) for all x, a ≤ x ≤ b.

In other words, if f is nice on some interval, then there is a maximum value of f somewhere on that interval, and a minimum value somewhere on that interval.

7.4.3

Rolle’s Theorem

Theorem 7.3 (Rolle’s Theorem). If f is a function that is continuous over the closed interval [a, b], and differentiable over the open interval (a, b), and if f (a) = f (b) [= 0], then there exists at least one x-value c, where a < c < b, such that f 0 (c) = 0. If f is a nice, smooth function (i.e., a continuous and differentiable function) which happens to start and stop on a horizontal line, then somewhere in between f will have a horizontal tangent.

Problems 7.D-1 Memorize the conditions and conclusions of the Intermediate Value Theorem. 7.D-2 [12] Apply the Intermediate value Theorem for Continuous Functions to a specific example. Consider f (x) = 2x + 1, where −2 ≤ x ≤ 4. (a) Sketch the graph of f for −2 ≤ x ≤ 4. Label −2 and 4 on the horizontal axis and f (−2) and f (4) on the vertical axis. (b) In order to apply the theorem, f must be continuous over the given closed interval. Explain why f is continuous on the closed interval [−2, 4]. (c) The theorem claims that if y is any value between f (−2) and f (4), there exists an input value c between −2 and 4 such that f (c) = y. Apply the theorem for y = 6; that is, find a value for c between −2 and 4 such that f (c) = 6. Label c and f (c) on the graph you sketched in part 2a and indicate the relationship between c and f (c). (d) Apply the theorem for y = −2; that is, find a value for c between −2 and 4 such that f (c) = −2. Label c and f (c) on the graph you sketched in part 2a and indicate the relationship between c and f (c). Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 4: Some Basic Calculus Theorems

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7.D-3 [12] Explain why the Intermediate Value Theorem for Continuous Functions makes sense. Support your explanation with an appropriate diagram; that is, on a pair of axes: (a) Label a and b on the horizontal axis, where a < b. (b) Sketch the graph of a squiggly function f that is continuous over the closed interval [a, b]. (c) Label f (a) and f (b) on the vertical axis. (d) Pick a y-value between f (a) and f (b) and label it k (e) Show that the conclusion of the Intermediate Value Theorem for Continuous Functions holds; that is, show that there exists an x-value c between a and b such that f (c) = k. Label c on the horizontal axis. 7.D-4 [12] For a given value of y between f (a) and f (b), is it possible for there to be more than one choice of c between a and b such that f (c) = y? If so, draw a diagram supporting your conclusion. If not, explain why not. 7.D-5 [12] Apply the Intermediate Value Theorem to some real-life situations. For each of the following situations: i Model the scenario with a graph. ii Illustrate the conclusion on your graph. (a) Scenario: The light turns green, and you step on the gas pedal in your car. Fifteen seconds later, you level off your speed at 60 mph. Conclusion: According to the Intermediate Value Theorem, there exists a time in the 15-second time interval when you are going 28 mph. (b) Scenario: You walk back and forth in front of a motion detector for 20 seconds. You vary your velocity. Your minimum distance is 0.5 meters from the detector and your maximum distance is 8.5 meters. Conclusion: According to the Intermediate Value Theorem, there exists a time in the 20-second time interval when you are 5.25 meters from the detector . (c) Scenario: You have the flu. Over a three-day period, your temperature fluctuates between 99.8◦ and 103.2◦ . Conclusion: According to the Intermediate Value Theorem, there exists a time in the three-day period when your temperature is 100◦ . 7.D-6 [12] Use the Intermediate Value Theorem to show that the equation x5 − 3x4 − 2x3 − x + 1 = 0 has a solution between 0 and 1. 7.D-7 Sketch a continuous function f for which does not meet the conclusions of the Intermediate Value Theorem, (i.e., there is no c ∈ (a, b) 3 f (c) = k) because k is not between f (a) and f (b). Mr. Budd, compiled September 29, 2010

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AP Unit 7 (Existence Theorems)

7.D-8 Sketch a function f which does not meet the conclusions of the Intermediate Value Theorem because f is not continuous. 7.D-9 Sketch a function f which does meet the conclusions of the Intermediate Value Theorem (i.e., there is c ∈ (a, b) 3 f (c) = k) despite the fact that k is not between f (a) and f (b). 7.D-10 Sketch a function f which does meet the conclusions of the Intermediate Value Theorem (i.e., there is c ∈ (a, b) 3 f (c) = k) despite the fact that f is not continuous. 7.D-11 Memorize the conditions and conclusions of the Extreme Value Theorem. 7.D-12 [12] Apply the Max-Min Theorem (aka the Extreme Value Theorem) for Continuous Functions to a specific example. Consider f (x) = x2 − 1, where −2 ≤ x ≤ 3. (a) Sketch the graph of f for −2 ≤ x ≤ 3. Label −2 and 3 on the horizontal axis and f (−2) and f (3) on the vertical axis. (b) In order to apply the theorem, f must be continuous over the given closed interval. Explain why f is continuous on the closed interval [−2, 3]. (c) The theorem claims that f has an absolute minimum on [−2, 3] – that is, there exists a value cm between −2 and 3 such that f (cm ) ≤ f (x) for all x between −2 and 3. Find cm . Label cm and f (cm ) on the graph you sketched in part 12a. (d) The theorem claims that f has an absolute maximum on [−2, 3] – that is, there exists a value cM between −2 and 3 such that f (x) ≤ f (cM ) for all x between −2 and 3. Find cM . Label cM and f (cM ) on the graph you sketched in part 12a. 7.D-13 [12] Explain why the Max-Min Theorem for Continuous Functions makes sense. Support your explanation with an appropriate diagram. On a pair of axes: (a) Label a and b on the horizontal axis, where a < b. (b) Sketch the graph of a (squiggly) function f which is continuous over the closed interval [a, b]. (c) Show that there exists an input value cm between a and b such that f (m) ≤ f (x) for all x between a and b. Label cm and f (cm ) on your diagram. (d) Show that there exists an input value cM between a and b such that f (x) ≤ f (cM ) for all x between a and b. Label cM and f (cM ) on your diagram. Mr. Budd, compiled September 29, 2010

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7.D-14 [12] The Max-Min Theorem for Continuous Functions states that every function f which is continuous over a closed interval [a, b] takes on both an absolute minimum and an absolute maximum on [a, b]. If you want to determine where the absolute extrema exist, what values would you examine? In other words, which values in the interval [a, b] would be candidates for the absolute maximum and absolute minimum? 7.D-15 [12] Is it possible for there to be more than one choice of cm between a and b such that f (cm ) ≤ f (x) for all x between a and b? If so, draw a diagram supporting your conclusion. If not, explain why not. Similarly, is it possible for there to be more than one choice of cM between a and b such that f (x) ≤ f (cM ) for all x between a and b? If so, draw a diagram supporting your conclusion. If not, explain why not. 7.D-16 [12] Apply the Max-Min Theorem to some real-life situations. For each of the following situations: i Model the scenario with a graph. ii Label the absolute extrema for your model. (a) You work an 8-hour shift at a pretzel factory. At the start of your shift, your production rate is low, but it continues to increase as you settle into a routine. Two hours before the end of the shift, you start thinking about what you are going to do after work, and your production rate decreases until it’s time to quit. (b) You’re home alone watching a scary movie, on a dreary night. Each time a scary part comes on, your heart rate increases dramatically and then returns to normal when the scary part is over. The movie is 117 minutes long, and there are 7 scenes that frighten you. (c) You create a distance-versus-time graph using a motion detector. Starting 7.75 meters from the detector, you walk toward the detector for 6.5 seconds. You walk faster and faster for the first 4 seconds, and then slower and slower for the next 2.5 seconds. You stop at the half–meter mark. 7.D-17 Sketch a function that has no maximum because it has a vertical asymptote. 7.D-18 Sketch a function that has no minimum because it has a removable discontinuity. 7.D-19 Sketch a function that has a maximum despite the fact that it is not continuous over [a, b]. 7.D-20 Memorize the conditions and conclusions of Rolle’s Theorem. 7.D-21 Sketch a continuous and differentiable function which does not meet the conclusions of Rolle’s Theorem (i.e., the function does not have a horizontal tangent). Mr. Budd, compiled September 29, 2010

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AP Unit 7 (Existence Theorems)

7.D-22 Sketch a function which does not meet the conclusions of Rolleâ&#x20AC;&#x2122;s Theorem (i.e., the function does not have a horizontal tangent) even though it starts and stops on the same horizontal line. 7.D-23 Sketch a function that does not start and stop on the same horizontal line, but still has a horizontal tangent.

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 5: Mean Value Theorem

7.5

219

Average Rate of Change and the Mean Value Theorem

Advanced Placement Concept of the derivative. • Derivative interpreted as an instantaneous rate of change. • Relationship between differentiability and continuity. Derivative as a function. • The Mean Value Theorem and its geometric consequences. Textbook §2.9 The Mean Value Theorem [16] Resources Workshop Calculus

7.5.1

Average Rate of Change

7.5.2

Mean Value Theorem

Theorem 7.4 (Mean Value Theorem). If f (x) is a function that is continuous over [a, b] and differentiable over (a, b), then ∃ c ∈ (a, b) 3 f 0 (c) =

f (b) − f (a) b−a

A nice, smooth function has a spot where the tangent line is parallel to the secant line, i.e., where the instantaneous rate of change matches the average rate of change. Example 7.5.1 The function L(t) = 10 000 e−0.2 − e−0.2t gives the number of gallons that have leaked out of a tanker, where t the time in hours after noon. L(3) − L(1) . Explain the meaning of this value. [Ans: 1349.596] 3−1 (b) Find the average rate at which oil leaked out of the tanker from 3 p.m. to 9 p.m. Indicate units. [Ans: 639.188 gal/hr] (a) Find

Mr. Budd, compiled September 29, 2010

220

AP Unit 7 (Existence Theorems) (c) Find L0 (t). Using correct units, explain the meaning of L0 (t). (d) At what time between 1 p.m. and 3 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (e) At what time between 3 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? Ans: 2 000e−0.2t , rate of leakage; t = 1.96671 (1:58); t = 5.70352 (5:42) Example 7.5.2 (adapted slightly from AB ’02) Let f be a function that is differentiable for all real numbers. Table 7.1 gives the values Table 7.1: AB ’02 x f (x) f 0 (x)

−1.5 −1 −7

−1.0 −4 −5

−0.5 −6 −3

0 −7 0

0.5 −6 3

1.0 −4 5

1.5 −1 7

of f and its derivative f 0 for selected points x in the closed interval −1.5 ≤ x ≤ 1.5. The second derivative of f has the property that f 00 (x) > 0 for −1.5 ≤ x ≤ 1.5. (a) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and f 00 (c) = r. Give a reason for your answer. ( 2x2 − x − 7 for x < 0 (b) Let g be the function given by g(x) = 2x2 + x − 7 for x ≥ 0 The graph of g passes through each of the points (x, f (x)) given in the table above. Is it possible that f and g are the same function? Give a reason for your answer. (c) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to approximate the value of f (1.2). Is this approximation greater than or less than the actual value of f (1.2)? Give a reason for your answer. (d) Write an equation of the line tangent to the graph of f at the point where x = 1.5. Use this line to approximate the value of f (1.2). Which of these approximations do you suppose is more accurate, and why? [Ans: 6; no; −3 < f (1.2); −3.1 < −3 < f (1.2)] Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 5: Mean Value Theorem

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Example 7.5.3 (from Acorn ’02) Let f be a function such that f 00 (x) < 0 for all x in the closed interval [1, 2]. Selected values of f are shown in Table 7.2. Which of the following must be true about Table 7.2: Acorn ’02 # 18 x f (x)

1.1 4.18

1.2 4.38

1.3 4.56

1.4 4.73

f 0 (1.2)? (a) (b) (c) (d) (e)

f 0 (1.2) < 0 0 < f 0 (1.2) < 1.6 1.6 < f 0 (1.2) < 1.8 1.8 < f 0 (1.2) < 2.0 f 0 (1.2) > 2.0

Example 7.5.4 (adapted slightly from AB ’06B) A car travels on a straight track. During the time interval 0 ≤ t ≤ 60 seconds, the car’s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. Table 7.3 shows selected values of these functions. Table 7.3: AB ’06B t (sec) v(t) (ft/sec) a(t) (ft/sec2 )

−20

−30

−20

−14

−10

Z 60 (a) Using appropriate units, explain the meaning of v(t) dt in Z 60 30 terms of the car’s motion. Approximate v(t) dt using a 30

trapezoidal approximation with the three subintervals determined by the table. Z 60 (b) Using appropriate units, explain the meaning of |v(t)| dt Z 60 30 in terms of the car’s motion. Approximate |v(t)| dt using 30 Mr. Budd, compiled September 29, 2010

222

AP Unit 7 (Existence Theorems) a trapezoidal approximation with the three subintervals determined by the table. (c) For 0 < t < 60, must there be a time t when v(t) = −5? Justify your answer. (d) For 0 < t < 60, must there be a time t when a(t) = 0? Justify your answer. (e) For 0 < t < 60, must there be a time t when a0 (t) = 0? Justify your answer.

Example 7.5.5 (adapted slightly from AB ’03) A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. Table 7.4 gives the measurements of the diameter of the Table 7.4: AB ’04B Distance x (mm) Diameter B(x) (mm)

120

180

240

300

360

blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B(x) is a twice–differentiable function that represents the diameter at that point. 1 (a) Using correct units, explain the meaning of 360

360

B(x) dx 2 0 Z 360 1 B(x) dx in terms of the blood vessel. Approximate the value of 360 0 2 using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer. Is your answer reasonable? 2 Z 360 B(x) (b) Using correct units, explain the meaning of π dx 2 0 in terms of the blood vessel. (c) Explain why there must be at least one value x, for 0 < x < 360, such that B 00 (x) = 0.

[Ans: 14 mm; volume of the b.v. from x = 0 to x = 360; B 0 (c1 ) = B 0 (c2 ) = 0, . . . ]

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Problems 7.E-1 (from [2]) The graph of a function f whose domain is the interval [−4, 4] is shown in the Figure 7.2. Which of the following statements are true? Figure 7.2: Venture III IB #11

(a) The average rate of change of f over the interval from x = −2 to 1 x = 0 is 2 (b) The slope of the tangent line at the point where x = 2 is 0. Z 3 (c) The left–sum approximation of f (t) dt with four equal subdivi−1

sions is 4. [Ans: T, F, T] 7.E-2 (from [2]) The line x − 3y + 7 = 0 is tangent to the graph of y = f (x) at (2, f (2)) and is also parallel to the line through (1, f (1)) and (7, f (7)). If f is differentiable on the closed interval [1, 7] and f (1) = 2, then find (a) f (7) (b) f (2) [Ans: 4, 3] x , then there exists a number c in the interval 7.E-3 (AB ’93) If f (x) = sin 2 π 3π < x < that satisfies the conclusion of the Mean Value Theorem. 2 2 What is c? [Ans: π] 7.E-4 (AB ’97) Let f be the function given by f (x) = 3 cos x. As shown in Figure 7.3, the graph f crosses the y–axis at point P and the x–axis at point Q. (a) Write an equation for the line passing through points P and Q. (b) Write an equation for the line tangent to the graph at point Q. Mr. Budd, compiled September 29, 2010

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Figure 7.3: AB ’97 Section II, # 2

(c) Find the x–coordinate of the point on the graph of f , between points P and Q, at which the line tangent to the graph of f is parallel to the line P Q. Ans: y − 3 = − π6 (x − 0); y − 0 = −3 (x − π/2); 0.690 7.E-5 (AB ’89) Let f be the function given by f (x) = x3 − 7x + 6. (a) Find the zeros of f . (b) Write an equation of the line tangent to the graph of f at x = −1. (c) Find the number c that satisfies the conclusion of the Mean Value Theorem for f on the closed interval [1, 3] q i h Ans: 1, 2, −3; y − 12 = −4 (x + 1); 13 3 √ 7.E-6 (adapted from [2]) Find the point on the graph of y = 3 x between (0, 0) and (1, 1) at which the line tangent to the graph has the same h slope asi 1 the line through (0, 0) and (1, 1). Ans: 3√ 3 7.E-7 (AB ’04B) A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v is a differentiable function of t. Selected values of v(t) for 0 ≤ t ≤ 40 are shown in the Table 7.5. (a) Use a midpoint Riemann sum with four subintervals of equal length Z 40 and values from the table to approximate v(t) dt. Show the 0 Mr. Budd, compiled September 29, 2010

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Table 7.5: AB ’04B t (min) v(t) (mpm)

7.0

9.2

9.5

7.0

4.5

2.4

4.3

7.3

computations that Z 40lead to your answer. Using correct units, explain v(t) dt in terms of the plane’s flight. the meaning of 0

(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval 0 < t < 40? Justify your answer. [Ans: 229 miles flown during the forty minutes; Twice] 7.E-8 (AB ’05) A metal wire of length 8 centimeters (cm) is heated at one end. Table 7.6 gives selected values of the temperature T (x), in degrees Celsius Table 7.6: AB ’05 Distance x (cm) Temperature T (x) (◦ C)

100

(◦ C), of the wire x cm from the heated end. The function T is decreasing and twice differentiable. (a) Estimate T 0 (7). Show the work that leads to your answer. Indicate units of measure. (b) Are the data in the table consistent with the assertion that T 00 (x) > 0 for every x in the interval 0 < x < 8? Explain your answer. h i ◦ Ans: − 27 C/cm; No, T 0 (c1 ) = −5.75, T 0 (c2 ) = −8, . . . 7.E-9 The function L(t) = 10 000 e−0.2 − e−0.2t gives the number of gallons that have leaked out of a tanker, where t the time in hours after noon. L(5) − L(1) . Explain the meaning of this value. [Ans: 1127.128] 5−1 (b) Find the average rate at which oil leaked out of the tanker from 5 p.m. to 9 p.m. Indicate units. [Ans: 506.451 gal/hr] (a) Find

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226

AP Unit 7 (Existence Theorems) (c) Find the average rate at which oil leaked out of the tanker from 1 p.m. to 9 p.m. Indicate units. [Ans: 816.790 gal/hr] (d) Find L0 (t). Using correct units, explain the meaning of L0 (t). (e) At what time between 1 p.m. and 5 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (f) At what time between 5 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (g) At what time between 1 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? Ans: 2 000eâ&#x2C6;&#x2019;0.2t , rate of leakage; t = 2.86737 (2:52); t = 6.86737 (6:52); t = 4.477603 (4:29)

Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 6: Riemann Sums

7.6

227

Riemann Sums: Evaluating Definite Integrals

Advanced Placement Interpretations and properties of definite integrals. • Computation of Riemann sums using left, right, and midpoint evaluation points. • Definite integral as a limit of Riemann sums over equal subdivisions.

7.6.1

Sigma Notation

Example 7.6.1 Find

5 P

i=1

[Ans: 55] Example 7.6.2 Write in sigma notation: (a)

7.6.2

1 2 3 15 + + + ··· + 1+1 1+2 1+3 1 + 15

Ans:

P15

i i=1 1+i

Riemann Sums

Example 7.6.3 (adapted from Acorn ’05) Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. R9 (a) Using correct units, explain the meaning of 1 R(t) dt. R9 (b) Approximate 1 R(t) dt using (a) M4 , a midpoint Riemann sum with 4 equal subintervals; (b) T4 , a trapezoidal approximation with 4 equal subintervals; 2M4 + T + 4 (c) S8 = 3 (d) A Riemann sum, using the partition a = x0 = 1 < x1 = 3 < x2 = 9, and evaluation points c1 = 1.96671 and c2 = 5.70352 [Ans: 6490.959; 6621.211; 6534.376; 6534.319] Mr. Budd, compiled September 29, 2010

228

AP Unit 7 (Existence Theorems) Example 7.6.4 (AB ’97) The expression r r r r ! 1 1 2 3 50 + + + ... + 50 50 50 50 50 is a Riemann sum approximation for r R1 x (a) 0 dx 50 R1√ (b) 0 x dx r 1 R1 x (c) dx 50 0 50 1 R1√ x dx (d) 50 0 1 R 50 √ (e) x dx 50 0 h i R1√ Ans: 0 x dx

7.6.3

Evaluating Definite Integrals Exactly

Example 7.6.5 (a) I will give you a table of values for certain definite integrals. Find a pattern, in terms of a and b, for: Rb (a) a x dx Rb (b) a x2 dx Rb (b) Use geometric reasoning to justify the pattern for a x dx Rb (c) Pick a function, f (x), and deduce a pattern for a f (x) dx. (d) Deduce a pattern for: Rb (a) a ex dx Rb (b) a cos x dx Rb 1 (c) a dx x Rb 1 (d) a dx 1 + x2 Rb 2 (e) a sec x dx Example 7.6.6 Write the Mean Value Theorem, as it applies to G(x). Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 6: Riemann Sums

7.6.4

229

Proof

Given: Let f (x) be the derivative of some function G(x) over the interval [a, b]. G(x) is differentiable over [a, b] and over any subinterval inside [a, b]. Why? G(x) is continuous over [a, b] and any subinterval therein. Why? If G(x) is continuous and differentiable, what conclusions can we draw? Recall:

• We partition the interval [a, b] into n subintervals: a = x0 < x1 < x2 < · · · < xn−1 < xn

Pn • The Riemann sum is given by i=1 f (ci )4xi , where 4xi = xi −xi−1 , and ci ∈ [xi−1 , xi ]. ci represents the sample points, or evaluation points within each and every subinterval, where the height is taken for the rectangular approximation of the funky strip.

• There are multiple rules for how to choose the sample points, e.g., left, right, midpoint, Monte Carlo, etc.

The proof involves setting up a Riemann sum such that the sample points, ci , for each subinterval were chosen as theR points guaranteed by the Mean Value Theorem for the antiderivative G(x) = f (x) dx for each subinterval [xi−1 , xi ]. That is,

G0 (ci ) =

G(xi ) − G(xi−1 ) xi − xi−1

Recognizing that G0 (ci ) = f (ci ) and xi − xi−1 = 4xi , the conclusion of the MVT becomes

f (ci ) =

G(xi ) − G(xi−1 ) 4xi Mr. Budd, compiled September 29, 2010

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AP Unit 7 (Existence Theorems)

Replacing this in the formula for the Riemann sum, we have Rn

= =

n X i=1 n X i=1

n X

f (ci )4xi G(xi ) − G(xi−1 ) 4xi 4xi G(xi ) − G(xi−1 )

i=1

G(x1 ) − G(x0 )

+ G(x2 ) − G(x1 ) + G(x3 ) − G(x2 ) .. . + G(xn ) − G(xn−1 )

G(xn ) − G(x0 )

G(b) − G(a)

Note that if we choose our sample points this way, then the value we get for Rn , G(b) − G(a), is independent of n, the number of subintervals. So that we can have three, six, one hundred, one million, or even one subinterval, and we get the same value for the Riemann sum, which is therefore the same value for the definite integral. Z

f (x) dx

lim Rn

n→∞

lim G(b) − G(a)

n→∞

= G(b) − G(a)

Example 7.6.7 (adapted from BC ’97) The closed interval [a, b] is partitioned into n equal subintervals, each of width ∆x, by the numbers x0 , x1 , . . . , xn where a = x0 < x1 < x2 < · · · < xn−1 < n √ P 3 x ∆x? xn = b. What is lim i n→∞ i=1

3 4/3 Ans: b − a4/3 4 Mr. Budd, compiled September 29, 2010

AP Unit 7, Day 6: Riemann Sums

7.6.5

231

Evaluating Definite Integrals

Example 7.6.8 (adapted from AB ’97) (a) Find Z 2 i Z1 3 ii 1 Z b iii Z1 x iv

4x3 − 6x2 dx 4x3 − 6x2 dx 4x3 − 6x2 dx 4t3 − 6t2 dt

(b) What do you suppose is the meaning of Z 2 1 i 4x3 − 6x2 dx 2−1 1 Z 3 1 ii 4x3 − 6x2 dx 3−1 1 Z b 1 4x3 − 6x2 dx iii b−1 1 Z x (c) Let A(x) = 4t3 − 6t2 dt. Find A0 (x). 1

Ans: 1, 28, b4 − 2b3 + 1, ; ; 4x3 − 6x2

Example 7.6.9 (AB97) If

Rb a

f (x) dx = a+2b, then

Rb a

(f (x) + 5) dx =

[Ans: 7b − 4a] Example 7.6.10 (adapted from Acorn ’05) Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. R9 (a) Using correct units, explain the meaning of 1 R(t) dt. R9 (b) Approximate 1 R(t) dt using (a) M4 , a midpoint Riemann sum with 4 equal subintervals; (b) T4 , a trapezoidal approximation with 4 equal subintervals; 2M4 + T + 4 [Ans: 6490.959; 6621.211; 6534.376] (c) S8 = 3 Mr. Budd, compiled September 29, 2010

232

AP Unit 7 (Existence Theorems) (c) Find the exact amount of oil that leaked out between 1 p.m. and 9 p.m. Ans: 10 000 e−0.2 − e−1.8 = 6534.319 gallons R9 1 R(t) dt. Using correct units, explain the meaning (d) Find 9−1 1 of this value. You can think of this as (signed) area divided by width. [Ans: 816.790 gal/hr; average of the rate of leakage from 1 p.m. to 9 p.m.] (e) Let L(T ) be the amount of oil that has leaked out from 1 p.m. to time t = T . Write an integral expression for L(T ). Find a formulah for L(T ) that does not require an integral, then findi RT L0 (T ) Ans: 1 R(t) dt = 10 000 e−0.2 − e−0.2T ; 2 000e−0.2T

Problems In your big giant calculus book, do problems 1-19 odd on page 390. 7.F-1 (adapted from Acorn ’05) Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. R9 (a) Using correct units, explain the meaning of 1 R(t) dt. R9 (b) Approximate 1 R(t) dt using a Riemann sum, using the partition a = x0 = 1 < x1 = 5 < x2 = 9, and sample points c1 = 3 and c2 = 7. How were these sample points chosen? R9 (c) Approximate 1 R(t) dt using a Riemann sum, using the partition a = x0 = 1 < x1 = 5 < x2 = 9, and evaluation points c1 = 2.86737 and c2 = 6.86737. Refer to your homework on the Mean Value Theorem to determine how these sample points were chosen. R9 (d) Approximate 1 R(t) dt using a Riemann sum with only one rectangle, using the evaluation point c1 = 4.477603. What is the height of this rectangle? [Ans: 6363.269 gal; 6534.319 gal; 6534.319 gal, 816.790 gal/hr] 7.F-2 (from [2]) Use a right-hand Riemann sum with four equal subdivisions to Z 3 approximate the integral |2x − 3| dx [Ans: 8] −1

7.F-3 (from [2]) When

x3 − x + 1 dx is approximated by using the mid-

−1

points of three rectangles of equal width, then the approximation is nearest to [Ans: 22.9] Z 1 2 7.F-4 [2] sin (πx) dx = Ans: π 0 Mr. Budd, compiled September 29, 2010

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7.F-5 What is the area of the region above the x-axis, below the graph of 1 y = sec2 (πx), between the lines x = 0 and x = ? What do you 3 1 i of dividing this area by the width, 3 − 0 ? hsuppose√ is the significance Ans: π3 ; average height

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Unit 8

Integral Theorems 1. Average Value and Mean Value Theorem for Integrals 2. Accumulation Functions and Fundamental Theorem of Calculus part II 3. Curve Sketching with Accumulation Functions Advanced Placement Interpretations and properties of definite integrals. • Definite integral as a limit of Riemann sums over equal subdivisions. • Definite integral as the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: Z b f 0 (x) dx = f (b) − f (a) a

• Basic properties of definite integrals. (Examples include additivity and linearity.) Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, 235

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the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

Fundamental Theorem of Calculus. â&#x20AC;˘ Use of the Fundamental Theorem to evaluate definite integrals. â&#x20AC;˘ Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

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8.1

237

Average Value and Mean Value Theorem for Integrals

Advanced Placement Applications of Integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. The emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing the limit as a definite integral. To provide a common foundation, specific applications should include the average value of a function. Interpretations and properties of definite integrals. • Definite integral as a limit of Riemann sums over equal subdivisions. • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: Z

f 0 (x) dx = f (b) − f (a)

• Basic properties of definite integrals. (Examples include additivity and linearity.) Fundamental Theorem of Calculus • Use of Fundamental Theorem to evaluate definite integrals. Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables (including change of limits for definite integrals).

8.1.1

Substitution of Variables

Example 8.1.1 (adapted from Acorn ’05) The area √ of the region in the first quadrant between the graph of y = x 9 − x2 and the x-axis is Mr. Budd, compiled September 29, 2010

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AP Unit 8 (Integral Theorems) [Ans: 9]

Z Example 8.1.2 (AB ’97) 0

π 4

cos2 x

dx is

8.1.2

Properties of Definite Integrals

8.1.3

Average Value

Suppose we were to average the following numbers: 17, 7, 5, 11. We would add 17 + 7 + 5 + 11, and then divide by 4. Note that 4 is the same as 1 + 1 + 1 + 1. So, what we have is essentially 17 + 7 + 5 + 11 = 10 1+1+1+1 Also note that 17 is 7 more than the average, 7 is 3 less, 5 is 5 less, and 11 is 1 more. Overall there is a total of 8 above the average, and 8 below the average. In general, to find the average, we take n P

i=1 n P

i=1

. Now consider averaging a continuous function f (x) over an interval [a, b]. There are an infinite number of points to average, so how do we add an infinite number of numbers? Rb f (x) dx a Rb 1 dx a Definition 8.1 (Average Value). The average value of a function f over the interval [a, b] is given by Rb a

f (x) dx 1 = b−a b−a

f (x) dx a

The average value is basically the average height of the function over an interval. We get this by taking the “area” (the definite integral), and dividing by the width (b − a). Mr. Budd, compiled September 29, 2010

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Example 8.1.3 (adapted from AB97) What is the average value of cos x on the interval [−2, 6]?

Ans:

sin 6+sin 2 8

Example 8.1.4 Graphically estimate the average value of sec2 (πx) 1 over the interval 0, . Confirm your answer analytically. 3 Example 8.1.5 (adated from BC Acorn ’02) If f is a continuous R a+h function for all real x, then what is lim h1 a f (x) dx? h→0

[Ans: f (a)]

Substitution of Variables Example 8.1.6 (adapted from AB ’98) Graphically estimate the √ mean value of y = x2 x3 + 1 on the interval [−1, 2]. Confirm or reject analytically.

Average Rate of Change given the derivative If you are given f 0 (x), the derivative of f with respect to x, then the average rate of change of f is the same as the average value of f 0 . The average rate of change is the average value of the rate of change, i.e., the average value of the derivative. dy 1 Example 8.1.7 (adapted from BC93) If = , what is the dx x average rate of change of y with respect to x on the closed interval [1, e]?

Ans:

1 e−1

Average Value of f : Rb a

f (x) dx b−a Mr. Budd, compiled September 29, 2010

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AP Unit 8 (Integral Theorems)

Average Rate of Change of f : f (b) − f (a) = b−a

Rb a

f 0 (x) dx b−a

Connection between the average value of a function and the Fundamental Theorem of Calculus Recall the proof of the Fundamental Theorem of Calculus that shows that Rb 0 G (x) dx = G(b) − G(a). The proof involved setting up a Riemann sum a such that the sample points, ci , for each subinterval were chosen as Rthe points that satisfied the Mean Value Theorem for the antiderivative G(x) = G0 (x) dx for each subinterval [xi−1 , xi ]. That is, G0 (ci ) =

G(xi ) − G(xi−1 ) xi − xi−1

Recognizing that xi − xi−1 = 4xi , the MVT becomes G0 (ci ) =

G(xi ) − G(xi−1 ) 4xi

Replacing this in the formula for the Riemann sum, we have Rn

n X

G0 (ci )4xi

i=1

= =

n X G(xi ) − G(xi−1 ) i=1 n X

4xi

G(xi ) − G(xi−1 )

i=1

G(x1 ) − G(x0 ) + G(x2 ) − G(x1 ) + G(x3 ) − G(x2 ) .. . + G(xn ) − G(xn−1 )

G(xn ) − G(x0 )

G(b) − G(a)

Note that if we choose our sample points this way, then the value we get for Rn , G(b) − G(a), is independent of n, the number of subintervals. So that we Mr. Budd, compiled September 29, 2010

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can have three, six, one hundred, one million, or even one subinterval, and we get the same value for the Riemann sum, which is therefore the same value for the definite integral. Suppose that we choose one rectangle instead of three, or six, or one hundred. If we choose our one sample point as above, so that the Mean Value Theorem is satisfied for the antiderivative, then our one rectangle will have exactly the same area as the definite integral. Both shapes have the same “area” and width, but the rectangle has a constant height, and the definite integral has a (possibly) variable height. Since the one constant height gives the same signed area as the variable height, that one height behaves similarly to all the combined variable heights. This one constant height that mimics our changing heights is said to be the average or mean height. The proof of the Fundamental Theorem basically involves taking sample points such that the height of each rectangle is equal to the average value of the function over the corresponding subinterval. Finding the average value in some ways involves replacing the funky shape under f (x) with a rectangle of exactly the same “area” and width. The rectangle has a height equal to the average value of the function over the interval. When comparing the funky shape to the rectangle, parts sticking out above the rectangle exactly match the parts drooping below the rectangle in size.

8.1.4

Mean Value Theorem for Integrals

Theorem 8.1. Mean Value Theorem for Integrals If f (x) is continuous over the interval [a, b], then there exists a number c ∈ [a, b] such that Z b f (c) (b − a) = f (x) dx a

or f (c) =

1 b−a

f (x) dx a

Example 8.1.8 (BC93) If f is continuous on the closed interval Rb [a, b], then there exists c such that a < c < b and a f (x) dx = f (c) b−a f (b) − f (a) (B) b−a (C) f (b) − f (a)

(A)

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AP Unit 8 (Integral Theorems) (D) f 0 (c) (b − a) (E) f (c) (b − a)

[Ans: E]

Example 8.1.9 Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. Z 3 1 (a) Find R(t) dt. Explain the meaning of this value. [Ans: 1349.596] 3−1 1 (b) Find the average rate at which oil leaked out of the tanker from 3 p.m. to 9 p.m. Indicate units. [Ans: 639.188 gal/hr] (c) At what time between 1 p.m. and 3 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (d) At what time between 3 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? Ans: 2 000e−0.2t , rate of leakage; t = 1.96671 (1:58); t = 5.70352 (5:42)

Connection between the Mean Value Theorem for Integrals and the average value of a function Note that f (c) is the average (or mean) value of the function f (x) over the interval [a, b]. The Mean Value Theorem for Integrals is little more than applying the Mean Value Theorem to the antiderivative. If we look at the Mean Value Theorem for G(x), the antiderivative of G0 (x), over [a, b]: G(b) − G(a) G0 (c) = b−a This is similar to the approach we take in the proof of the Fundamental Theorem, except that we only have one subinterval. Now we learned (from the Rb Fundamental Theorem) that G(b) − G(a) is the same as a G0 (x) dx. Using these gives us: Rb 0 G (x) dx 0 G (c) = a , b−a Mr. Budd, compiled September 29, 2010

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which is the signed area divided by the width, so that G0 (c) is the average value of G0 (x) over the interval [a, b]. If we let f (x) = G0 (x), i.e., G(x) is the antiderivative of f (x), then Rb f (c) =

f (x) dx b−a

at some point c ∈ (a, b).

Problems 8.A-1 [2]

π/3 R π/4

√ Ans: ln 3

sec2 x dx = tan x

8.A-2 (Acorn ’05) If f is continuous for all x, which of the following integrals Rb necessarily have the same value as a f (x) dx? I

R b−a

R b+c

f (x + a) dx

f (x + c) dx R 2b III 12 2a f x2 dx a+c

Use u-simplification, as well as geometric arguments.

[Ans: I, III]

π is Ans: π4 4

8.A-4 ([2]) The average rate of change of the function f (x) = x2 − 2 |x + 2|

over the interval −3 < x < −1 is [Ans: −3] 8.A-3 ([2]) The average value of sec2 x over the interval 0 ≤ x ≤

1 8.A-5 ([2]) The average (mean) value of over the interval 1 ≤ x ≤ e is x h i 1 Ans: e−1 8.A-6 ([2]) The average value of f (x) = e2x + 1 on the interval 0 ≤ x ≤ [Ans: e]

1 is 2

8.A-7 (adapted from [2]) What is the average value of −2t3 + 6t2 + 4 over the interval −1 ≤ t ≤ 1? [Ans: 6] 8.A-8 (adapted from BC97) Let f be a twice differentiable function such that f (1) = −2 and f (4) = 7. Which of the following must be true for the function f on the interval 1 ≤ x ≤ 4? I. The average rate of change of f is 3. Mr. Budd, compiled September 29, 2010

244

AP Unit 8 (Integral Theorems) 5 . 3 III. The average value of f 0 is 3. II. The average value of f is

[Ans: I and III only] 8.A-9 Draw a function that has an average value of 0 over the closed interval [−π, π]. 8.A-10 (AB ’97) Let f be the function given by f (x) = x3 − 6x2 + p, where p is an arbitrary constant. (a) Write an expression for f 0 (x) and use it to find the relative maximum and minimum values of f in terms of p. Show the analysis that leads to your conclusion. [Ans: max: f (0) = p, min: f (m = 4) = p − 32] (b) For what values of the constant p does f have three distinct roots? [Ans: 0 < x < 32] (c) Find the value of p such that the average value of f over the closed interval [−1, 2] is 1. Ans: 23 4 8.A-11 (AB ’96) The rate of consumption of cola in the United States is given by S(t) = Cekt , where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980. (a) The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion gallons per year. Find C and k. Ans: C = 6, k = 15 ln 2 (b) Find the average rate of consumption of cola over the 10-year time period beginning January 1, 1983. Indicate units of measure. [Ans: 19.680 billion gallons/year] (c) Use the trapezoidal rule with four equal subdivisions to estimate R7 S(t) dt. [Ans: 27.668] 5 R7 (d) Using correct units, explain the meaning of 5 S(t) dt in terms of cola consumption.

[Ans: amount, in billions of gallons, of cola consumed in the two year period from 1/1/85 to 1/1/8 8.A-12 (AB) A particle moves along the x-axis so that its velocity at time t, 0 ≤ t ≤ 5, is given by v(t) = 3 (t − 1) (t − 3). At time t = 2, the position of the particle is x(2) = 0. Find the average velocity of the particle over the interval 0 ≤ t ≤ 5. [Ans: 4] x on 8.A-13 (AB ’88) Without a calculator, find the average value of y = 2 hx + 2 i √ √2 the interval 0, 6 . Ans: ln 6 8.A-14 Using a calculator √ only to graph the function, exactly find the average value of y = − 1 − x2 + 1 over the interval [−1, 1]. Ans: 4−π 4 Mr. Budd, compiled September 29, 2010

AP Unit 8, Day 1: MVT for Integrals 8.A-15 (AB ’85) Let f (x) = 14πx2 and g(x) = k 2 sin

245 πx 2k

for k > 0.

(a) Find the average value of f on [1, 4]. (b) For what value of k will the average value of g on [0, k] be equal to the average value of f on [1, 4]? [Ans: 98π; 7π] 8.A-16 [3] Which of the following statements is true? I. If the graph of a function is always concave up, then the left-hand Riemann sums with the same subdivisions over the same interval are always less than the right-hand sums. Rb II. If the function f is continuous on the interval [a, b] and a f (x) dx = 0, then f must have at least one zero between a and b. III. If f 0 (x) > 0 for all x in an interval, then the function f is concave up in that interval. [Ans: II only] 8.A-17 Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. Z 5 1 R(t) dt. Explain the meaning of this value. [Ans: 1127.128] (a) Find 5−1 1 (b) Find the average rate at which oil leaked out of the tanker from 5 p.m. to 9 p.m. Indicate units. [Ans: 506.451 gal/hr] (c) Find the average rate at which oil leaked out of the tanker from 1 p.m. to 9 p.m. Indicate units. [Ans: 816.790 gal/hr] (d) At what time between 1 p.m. and 5 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (e) At what time between 5 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (f) At what time between 1 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? [Ans: t = 2.86737 (2:52); t = 6.86737 (6:52); t = 4.477603 (4:29)] 8.A-18 Sometimes the Mean Value Theorem for Integrals is called the Average Value Theorem. Explain why, in that case, the Mean Value Theorem proper could be called the Average Value Theorem for Derivatives.

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AP Unit 8, Day 2: Accumulation Functions

8.2

247

Accumulation Functions

Advanced Placement Fundamental Theorem of Calculus. • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Textbook §4.4 The Fundamental Theorem of Calculus: “The Second Fundamental Theorem of Calculus” [16]

8.2.1

Accumulation Functions

Example 8.2.1 (AB98)

Rx 0

sin t dt =

[Ans: 1 − cos x]

Example 8.2.2 (AB97) Let f (x) =

Rx a

h(t) dt, where h has the

Figure 8.1:

graph shown in Figure 8.1. Sketch the graph of f .

[Ans: Figure 8.2]

8.2.2

Fundamental Theorem of Calculus, part II d dx

f (t) dt = f (x) a

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AP Unit 8 (Integral Theorems)

Figure 8.2:

As soon as you see an accumulation function, differentiate it. Stop immediately, do not continue reading the problem until you have taken the derivative of the accumulation function. It is very easy to do, and the chances are extremely high that you will need the derivative. You can start reading the problem again, realizing that you know the derivative.

Example 8.2.3 (AB98) If F (x) = F 0 (2)?

Rx√ 0

t3 + 1 dt, then what is

[Ans: 3]

Example 8.2.4 [20] The error function Z x 2 2 erf(x) = √ e−t dt π 0 is used in probability, statistics, and engineering. Show that the 2 function y = ex erf(x) satisfies the differential equation y 0 = 2xy + 2 √ . π Example 8.2.5 (BC93) If F and f are differentiable functions Rx such that F (x) = 0 f (t) dt, and if F (a) = −2 and F (b) = −2 where a < b, which of the following must be true? (A) f (x) = 0 for some x such that a < x < b. (B) f (x) > 0 for all x such that a < x < b. (C) f (x) < 0 for all x such that a < x < b. (D) F (x) ≤ 0 for all x such that a < x < b. Mr. Budd, compiled September 29, 2010

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(E) F (x) = 0 for some x such that a < x < b.

[Ans: A] Using the Chain Rule, d dx

v(x)

f (t) dt = f (v(x)) v 0 (x)

R x2 Example 8.2.6 (BC97) Let f√(x) = 0 sin t dt. At how many points in the closed interval [0, π] does the instantaneous rate of change of f equal the average rate of change of f on that interval?

[Ans: Two]

Example 8.2.7 [18] Let Z F (x) = 2x + 0

sin 2t dt 1 + t2

Determine (a) F (0) (b) F 0 (0) (c) F 00 (0)

[Ans: 0;2;2] To carry extend part II even further, d dx

v(x)

f (t) dt = f (v(x)) v 0 (x) − f (u(x)) u0 (x)

u(x)

Example 8.2.8 Let F (x) = answer by simplifying F (x).

R 3x 1 dt. Find F 0 (x) and explain your x/2 t

[Ans: 0; ln 6] Mr. Budd, compiled September 29, 2010

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Example 8.2.9 [20] Find the derivative of g(x) =

R 3x u2 − 1 du. 2x u2 + 1

2(4x2 −1) 3(9x2 −1) Ans: 9x2 +1 − 4x2 +1

8.2.3

Curve Sketching with Accumulation Functions

Example 8.2.10 (BC97) The graph of f is shown in Figure 8.3. If Figure 8.3:

g(x) =

Rx a

f (t) dt, for what value of x does g(x) have a maximum?

[Ans: c]

Example 8.2.11 (BC97) Refer to the graph in Figure 8.4. At Figure 8.4: The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown.

what value does the absolute minimum of f occur? The absolute maximum? Mr. Budd, compiled September 29, 2010

AP Unit 8, Day 2: Accumulation Functions

251 [Ans: 0]

Example R x 8.2.12 (AB Acorn ’02) If the function g is defined by g(x) = 0 sin(t2 ) dt on the closed interval −1 ≤ x ≤ 3, then g has a local minimum at x = [Ans: 2.507] Example 8.2.13 (AB 2002) The graph of function f shown in Figure 8.5: From AP Calculus AB 2002 Exam

Figure 8.5. Let g be the function given by g(x) = (a) (b) (c) (d)

f (t) dt. Find g(−1), g 0 (−1), and g 00 (−1). Ans: − 32 ; 0; 3 For what values of x in the open interval (−2, 2) is g increasing? Explain your reasoning. [Ans: −1 < x < 1] For what values of x in the open interval (−2, 2) is the graph of g concave down? Explain your reasoning. [Ans: 0 < x < 2] Sketch the graph of g on the closed interval [−2, 2]. 0

Problems 8.B-1 (AB ’97) Let f be the function given by f (x) =

√

x − 3.

(a) Sketch the graph of f and shade the region R enclosed by the graph of f , the x-axis, and the vertical line x = 6. (b) Find the area of the region R. (c) Rather than using the line x = 6, consider the line x = w, where w can be any number greater than 3. Let A(w) be the area of the region enclosed by the graph of f , the x-axis, and the vertical line x = w. Write an integral expression for A(w). Mr. Budd, compiled September 29, 2010

252

AP Unit 8 (Integral Theorems) (d) Find the rate of change of A with respect to w when w = 6. √ Rw√ √ Ans: ; 2 3; 3 x − 3 dx; 3

8.B-2 (AB ’93) What is

d Rx cos(2πu) du? dx 0

8.B-3 (adapted from [2]) Suppose F (x) =

[Ans: cos(2πx)] R x2 0

F 0 (−1) =

1 dt for all real x, then 2 + t3 Ans: − 32

5 8.B-4 (adapted from [2]) Consider the function F defined so that F (x) + = 2 Rx πt 0 cos dt. The value of F (2) + F (2) is [Ans: −3] 2 3 8.B-5 (adapted from [2]) If the function G is defined for all real numbers by R 3x √ G(x) = 0 cos t2 dt, then G0 ( π) = [Ans: −3] R 2 sin x √ 8.B-6 (adapted from [2]) Suppose F (x) = 0 9 + t3 dt for all real x, then 0 F (π) = [Ans: −6] π Rx 8.B-7 (adapted from [2]) If for all x > 0, G(x) = 1 cos ln t dt, then the 2 π 00 value of G (e) is Ans: − 2e Rx 8.B-8 [2] Which of the following are true about the function F (x) = 1 ln (2t − 1) dt? I. F (1) = 0 II. F 0 (1) = 0 III. F 00 (1) = 1 [Ans: I and II only] 8.B-9 (AB ’94) Let F (x) =

Rx 0

sin t

dt for 0 ≤ x ≤ 3.

(a) Use the trapezoidal rule with four equal subdivisions of the closed interval [0, 1] to approximate F (1). [Note: you may use a calculator, but don’t use a program.] (b) On what intervals is F increasing? (c) If the average rate of change of F on the closed interval [1, 3] is k, R3 find 1 sin t2 dt in terms of k. √ √ 2π, 3 ; 2k Ans: 0.316; (0, π), 8.B-10 [20] The sine integral function Z Si (x) = 0

sin t dt t

is important in electrical engineering. Using your calculator to help you: Mr. Budd, compiled September 29, 2010

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(a) Draw the graph of Si . (b) At what values of x does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation: Z x 0

sin t dt = 1 t

8.B-11 (AB 2002B) The graph of a differentiable function f on the closed interval Figure 8.6: From AP Calculus AB 2002 Exam

[−3, 15] is shown in Figure 8.6. R x The graph of f has a horizontal tangent line at x = 6. Let g(x) = 5 + 6 f (t) dt for −3 ≤ x ≤ 15. (a) Find g(6), g 0 (6), and g 00 (6).

[Ans: 5; 3; 6]

(b) On what intervals is g decreasing? Justify your answer. [Ans: [−3, 0] and [12, 15]] (c) On what intervals is the graph of g concave down? Justify your answer. [Ans: (6, 15)] R 15 (d) Find a trapezoidal approximation of −3 f (t) dt using six subintervals of length ∆t = 3. [Ans: 12] 8.B-12 [2] The graph of the function f is shown in Figure 8.7. If the function G Rx is defined by G(x) = −4 f (t) dt, for −4 ≤ x ≤ 4, which of the following statements about G are true? I. G is increasing on (1, 2) II. G is decreasing on (−4, −3) III. G(0) < 0 [Ans: II and III only] Rx 2 8.B-13 (BC Acorn ’02) Let g be the function given by g(x) = 1 100 t2 − 3t + 2 e−t dt. Which of the following statements about g must be true? Mr. Budd, compiled September 29, 2010

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AP Unit 8 (Integral Theorems)

Figure 8.7: From [2]

I. g is increasing on (1, 2). II. g is increasing on (2, 3). III. g(3) > 0. [Ans: II only] 8.B-14 (BC Acorn ’02) The graph of f in Figure 8.8 consists of four semicircles.

Figure 8.8: Graph of f

If g(x) =

Rx 0

f (t) dt, where is g(x) nonnegative?

8.B-15 (BC ’98) Let g(x) =

Rx a

[Ans: [−3, 3]]

f (t) dt, where a ≤ x ≤ b. Figure 8.9 shows the

Figure 8.9:

Mr. Budd, compiled September 29, 2010

AP Unit 8, Day 2: Accumulation Functions

255 

      graph of g on [a, b]. Sketch the graph of f on [a, b].  Ans:     

Mr. Budd, compiled September 29, 2010

             

256

AP Unit 8 (Integral Theorems)

Mr. Budd, compiled September 29, 2010

AP Unit 8, Day 3: Quick, Cheap Antiderivatives

8.3

257

Quick, Cheap Antiderivatives

Advanced Placement Fundamental Theorem of Calculus. • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

8.3.1

Creating Quick, Cheap Antiderivatives

Suppose I have an initial value problem, such as finding velocity given acceleration. Remember that velocity is the antiderivative of acceleration. In doing these problems, we antidifferentiate, and then we plug in a value so that we can find the value of the arbitrary constant C. The antiderivative of f (x), which we will call G(x), given the point G(c) = k can be written in one step as an accumulation function: Z x G(x) = f (t) dt + k c

Note that this function satisfies our two conditions, namely: • G0 (x) = f (x) by the Fundamental Theorem of Calculus part II Rc • G(c) = k since c f (t) dt = 0 Example 8.3.1 (BC93) If p is a polynomial R xof degree n, n > 0, what is the degree of the polynomial Q(x) = 0 p(t) dt [Ans: n + 1] Example 8.3.2 (AB97) At time t ≥ 0, the acceleration of a particle moving on the x-axis is a(t) = t + sin t. At t = 0, the velocity of the particle is −2. For what value of t will the velocity be zero? [Ans: 1.48] Example 8.3.3 (adapted from AB ’03) A particle moves along the x–axis so that at any time t > 0, its acceleration is given by a(t) = ln (1 + 2t ). If the velocity of the particle is 2 ln 3 − 1 at time t = 1, then the velocity of the particle at time t = 2 is Mr. Budd, compiled September 29, 2010

258

AP Unit 8 (Integral Theorems) [Ans: 2.544]

Example 8.3.4 (adapted from BC ’97) If f is the antiderivative of x3 such that f (1) = 0, then f (4) is what? 2 + x5

[Ans: 0.577]

Example 8.3.5 Let A(x) =

Rx √ −1

t3 + 1 dx.

(a) Find the average rate of change of A over [0, 2]. (b) Find the value of c guaranteed by the Mean Value Theorem for A(x) over [0, 2].

[Ans: 1.621; 1.176]

Example 8.3.6 The graph of f 0 , the derivative of the function f , Figure 8.10: From AP Calculus BC 2003 Exam

is shown in Figure 8.10. If f (0) = 0, which of the following must be true? (a) f (0) > f (1) (b) f (2) > f (3) (c) f (1) > f (3) Mr. Budd, compiled September 29, 2010

AP Unit 8, Day 3: Quick, Cheap Antiderivatives

259

Problems 8.C-1 (BC ’97) If f is the antiderivative of is what?

x2 such that f (1) = 0, then f (4) 1 + x5 [Ans: 0.376] √

8.C-2 (BC Acorn ’04-05) If the function f is defined by f (x) = x3 + 2 and g is an antiderivative of f such that g(3) = 5, then what is g(1)? [Ans: −1.585] 8.C-3 (adapted from BC ’03) A particle moves along the x–axis so that at any time t ≥ 0, its velocity is given by v(t) = cos 2 − t2 . The position of the particle is 3 at time t = 0. What is the position of the particle at the second time when its velocity is equal to zero? [Ans: 3.563] 8.C-4 (adapted from [2]) The rate at which ice is melting in a pond is given by √ dV = 1 + 2t , where V is the volume of ice in cubic feet, and t is the dt time in minutes. What amount of ice has melted in the first 4 minutes? Ans: 9.645 ft3 8.C-5 (adapted from [2]) Oil is leaking from a tanker at the rate of R(t) = 500e−0.2t gallons per hour, where t is measured in hours. The amount of oil that has leaked out, starting at the end of the second hour, until the end of the tenth hour is 8.C-6 [2] A particle moves along the x-axis with velocity at time t given by v(t) = t + 2 sin t. If the particle is at the origin when t = 0, its position at the time when v = 5 is x = [Ans: 17.277] 2

8.C-7 (adapted from AB ’98) Let F (x) be an antiderivative of 10, then what is F (9)? 0

8.C-8 (adapted from AB Acorn ’04-05) If f (x) = cos f (1) =

πex 2

(ln x) . If F (2) = x [Ans: 13.425]

and f (0) = 2, then [Ans: 1.351]

1 1 2 8.C-9 Let C(x) be the antiderivative of n(x) = √ e−x /2 such that C(0) = . 2 2π (a) Find C(2). (b) Find limx→∞ C(x) [Ans: 0.97725; 1] 8.C-10 (adapted from BC ’03) A particle starts at point A on the positive x-axis at time t = 0 and travels along the x-axis. The particle’s √ is a dif position x(t) dx πt π t+1 0 ferentiable function of t, where x (t) = = −9 cos sin . dt 6 2 Mr. Budd, compiled September 29, 2010

260

AP Unit 8 (Integral Theorems) At time t = 9, the particle reaches its final destination at point D on the positive x-axis. How far apart are points A and D, the initial and final positions, respectively, of the particle? [Ans: 39.255 apart]

8.C-11 (adapted slightly from AB ’07) The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 ≤ t ≤ 7, where t is measured in hours. In this model, rates are given as follows: √ (a) The rate at which water enters the tank is f (t) = 100t2 sin t gallons per hour for all seven hours. (b) The rate at which water leaves the tank is: 250 gallons per hour for the first three hours, and 2000 gallons per hour for the next four hours. At t = 0, the amount of water in the tank is 5000 gallons. (a) On the same graph, sketch the rate at which water enters the tank, together with the rate at which water leaves the tank. At what times are the two rates the same? Z 7 (b) Using correct units, explain the meaning of f (t) dt in terms of water in the tank.

(c) How many gallons of water are there in the tank at t = 0? t = 3? t = 7? [Ans: ; ; 5000, 5126.591, 4513.807] 8.C-12 (adapted from AB ’07B) A particle moves along the x–axis so that its velocity v at time t ≥ 0 is given by v(t) = sin t2 . The position of the particle at time t is x(t) and its position at time t = 0 is x(0) = 5. (a) Find the position of the particle at t = 0 and at t = 3. [Ans: 5, 5.774] (b) Find the first time at which the particle changes direction. Does it change from right to left, or left to right? Find the position of the particle at that time. [Ans: 1.772, 5.895]

Mr. Budd, compiled September 29, 2010

Unit 9

Area and Volume 1. Area 2. Slicing with Discs 3. Slicing with Washers 4. Non-circular Slicing 5. Volume-Oriented Related Rates Advanced Placement Interpretations and properties of definite integrals â&#x20AC;˘ Computation of Riemann sums using left, right, and midpoint evaluation points. â&#x20AC;˘ Definite integral as a limit of Riemann sums over equal subdivisions. â&#x20AC;˘ Basic properties of definite integrals. Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, 261

262

AP Unit 9 (Area and Volume)

and the distance traveled by a particle along a line. Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals. Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables (including change of limits for definite integrals) Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 1: More Definite Integrals

9.1

263

More Definite Integrals

Advanced Placement Applications of Integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. The emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing the limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the average value of a function, and the distance traveled by a particle along a line.

9.1.1

Definite Integral

The definite integral can be seen as several things: 1. An infinite sum. Area is an infinite sum of infinitesimally thin rectangles. 2. A product. Displacement is velocity times time. How is a product really a sum? 3. An accumulated change. If oil is leaking out of a tank at certain rate Rb R(t), then a R(t) dt represents how much oil has leaked out from t = a to t = b. This is the accumulation of all the oil that has leaked out at a rate R(t). Change in velocity is the accumulation of accleration over a certain time period.

9.1.2

Area: Slicing dx

The area between two curves is a sum of an infinite number of infinitesimally thin rectangles. If you slice the area into vertical strips, then the area of each infinitesimally thin rectangle is given by dA = (highy â&#x2C6;&#x2019; lowy) dx, and the area is given by Z b Area = (highy â&#x2C6;&#x2019; lowy) dx a

Example 9.1.1 (adapted from Ostebee & Zorn [17]) Find the area of the region R bounded by the curves x = 0, y = 2, and y = ex . Mr. Budd, compiled September 29, 2010

264

AP Unit 9 (Area and Volume) (a) Find the area of R, with and without a calculator. (b) If the line x = h divides the region R into two regions of equal area, what is the value of h?

[Ans: 0.386; 0.219 ; 1.683]

Example 9.1.2 (adapted from AB ’00) Let R be the region in the 2 first quadrant enclosed by the graphs of y = e−x , y = 1 − cos x, and the y-axis. (a) Find the area of R. (b) If the line x = k divides the region R into two regions of equal area, what is the value of k?

[Ans: 0.591; 0.310]

Example 9.1.3 Find the area of the region bounded by the graphs 1 of y = ex/2 , y = 2 , x = 2, and x = 3. Try this with, and without x a calculator.

Ans: 2 e3/2 − e −

1 6

= 3.360

π Example 9.1.4 (adapted from AB ’98) If 0 ≤ k ≤ and the area 2 π under the curve y = cos x from x = k to x = is 0.2, then k = 2 [Ans: 0.927]

9.1.3

Total Distance

Distance : Displacement :: Area : Definite Integral There is a difference between the total distance traveled and the displacement. When you go backward, distance is counted positively, but displacement is counted negatively. How is that similar to the relationship between area and the definite integral? Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 1: More Definite Integrals

265

Speed is the magnitude (absolute value of velocity), and total distance traveled is the accumulation (i.e., definite integral) of speed. Displacement is the accumulation of velocity. D=

Rb a

|v(t)| dt

Just like area, total distance traveled must be positive. In a graph of velocity, the definite integral yields the displacement, i.e., change in position. The area yields the total distance traveled. Example 9.1.5 (adapted from AB 1997) A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by the function whose graph is shown in Figure 9.1 Figure 9.1: Vertical velocity of a bug

(a) At value of t does the bug change direction? (b) What is the total distance the bug traveled from t = 0 to t = 8? (c) What is the net displacement of the bug between t = 0 and t = 8? (d) What is the total distance traveled downward by the bug? upward? (e) What is the bug’s velocity at t = 5? The acceleration at t = 5? (f) What other questions could we ask about the bug? Example 9.1.6 (AB ’03) A particle moves along the x-axis so that its velocity at time t is given by 2 t v(t) = − (t + 1) sin 2 At time t = 0, the particle is at position x = 1. Mr. Budd, compiled September 29, 2010

266

AP Unit 9 (Area and Volume) (a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not? (b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time t = 0 until time t = 3. (d) During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.

Ans: 1.588, yes;

√

2π; 4.334; 2.265

Example 9.1.7 (adapted slightly from AB ’83) A particle moves along the x-axis so that at time t its position is given by x(t) = t3 − 6t2 + 9t + 11. (a) What is the velocity of the particle at time t? (b) During what time intervals is the particle moving to the left? (c) What is the total distance traveled by the particle from t = 0 to t = 2? Do this two ways. R2 (d) What does 0 v(t) dt represent?

Ans: 3t2 − 12t + 9; 1 < t < 3; 6; x(2) − x(0)

9.1.4

Other Applications

Example 9.1.8 (adapted minimally from AB ’06) At an intersection in Thomasville, Oregon, cars turn left at the rate L(t) = √ t 2 60 t sin cars per hour over the time interval 0 ≤ t ≤ 9 hours. 3 (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 ≤ t ≤ 9 hours. (b) Traffic engineers will consider turn restrictions when L(t) ≥ 100 cars per hour. Find all values of t for which L(t) ≥ 100 and compute the average value of L over this time interval. Indicate units of measure. Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 1: More Definite Integrals

267

Example 9.1.9 (adapted minimally from AB ’05) The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4πt R(t) = 2 + 4 cos 25 A pumping station adds sand to the beach at a rate modeled by the function S, given by 12t S(t) = 1 + 3t Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for 0 ≤ t ≤ 6. At time t = 0, the beach contains 2000 cubic yards of sand. (a) Find the rate at which sand is being removed by the tide at time t = 4. Find the rate at which sand is being added by the pumping station at time t = 4. (b) How much sand will the tide removed from the beach during this 6–hour period? What is the average rate at which the tide is removing sand from the beach during this period? Indicate units of measure. (c) Write an expression for the amount of sand removed by the tide during the first t hours. (d) How much sand will the pumping station add to the beach during this 6–hour period? What is the average rate at which the pumping station is adding sand to the beach during this period? Indicate units of measure. (e) Write an expression for the amount of sand added by the pumping station during the first t hours. (f) Write an expression for Y (t), the total number of cubic yards of sand on the beach at time t. Find Y 0 (t). (g) Find the rate at which the total amount of sand on the beach is changing at time t = 4. (h) For 0 ≤ t ≤ 6, what times are the best candidates for amount of sand on the beach to be an (absolute) maximum? minimum? How would you choose amongst the candidates?

Problems π 9.A-1 (adapted from AB ’98) If 0 ≤ k ≤ and the area under the curve y = cos x 2 π from x = k to x = is 0.2, then k = [Ans: 0.927] 2 Mr. Budd, compiled September 29, 2010

268

AP Unit 9 (Area and Volume)

9.A-2 (AB ’96) Let R be the region in the first quadrant under the graph of 1 y = √ for 4 ≤ x ≤ 9. Ans: 2; 25 4 x 9.A-3 (AB ’02) Let f and g be the functions given by f (x) = ex and g(x) = ln x. Find the area of the region enclosed by the graphs of f and g between 1 x = and x = 1. [Ans: 1.223] 2 2

9.A-4 (AB acorn ’02) What is the average value of the function f (x) = e−x on the closed interval [−1, 1]? [Ans: 0.747] Rxp 9.A-5 [2] The average rate of change of the function f (x) = 0 1 + cos (t2 ) dt over the interval [1, 3] to three decimal places is: [Ans: 0.858] 9.A-6 (adapted The approximate average rate of change of the function R xfrom [2]) f (x) = 0 sin t2 dt over the interval [2, 5] is [Ans: −0.092] 9.A-7 (BC97) following, which is the greatest value of x such R x If 0 ≤ x ≤ 4, ofRthe x that 0 t2 − 2t dt ≥ 2 t dt? (A) 1.35 (B) 1.38 (C) 1.41 (D) 1.48 (E) 1.59 [Ans: B] 9.A-8 (adapted from AB ’04) Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t F (t) = 80 + 6 cos for 0 ≤ t ≤ 30. 2 where F (t) is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection during the middle twenty minutes, i.e., for 5 ≤ t ≤ 25? (b) What is the average traffic flow during the middle twenty minutes? Indicate units of measure. (c) What is the average rate of change of traffic flow during the middle twenty minutes? Indicate units of measure. [Ans: 1592 cars; 79.601 cars per minute; 0.540 cars per minute per minute] Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 1: More Definite Integrals

269

9.A-9 (AB ’05B) A water tank at Camp Newton holds 960 gallons of water at time t = 0. During the time interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate of √ t 2 W (t) = 76 t cos gallons per hour. 6 During the same time interval, water is removed from the tank at the rate t R(t) = 220 cos2 gallons per hour. 3 (a) At what rate is water being removed from the tank at t = 15? Indicate units of measure. (b) Is the amount of water in the tank increasing at t = 15? Why or why not? [Ans: Y;] (c) How much water has been pumped into the tank during the first fifteen hours? What is the average rate at which water is pumped into the tank during that time period? [Ans: 933.042 gal; 62.203 gph] (d) How much water has been removed from the tank during the first fifteen hours? What is the average rate at which water is removed from the tank during that time period? (e) To the nearest gallon, how much water is in the tank at t = 15? t = 18? [Ans: 333; ] (f) Write an expression for the gallons of water in the tank at time t. (g) Write an expression for the rate at hwhich the total gallons of wateri Rt in the tank is changing at time t. Ans: 960 + 0 W (z) − R(z) dz (h) Find the best candidates for the time, 0 ≤ t ≤ 18 at which the amount of water in the tank is an absolute minimum. [Ans: 2.487, 12.450] (i) Find the best candidates for the time, 0 ≤ t ≤ 18 at which the amount of water in the tank is an absolute maximum. [Ans: 0, 6.198, 18] (j) How would you choose between the best candidates to determine the actual time in which the water is at an absolute extremum? 9.A-10 (BC Acorn 2000) A particle moves along the x-axis so that at any time t ≥ 0 its velocity is given by v(t) = ln (t + 1) − 2t2 + 4t − 1. (a) What is the total distance traveled by the particle from t = 0 to t = 2? [Ans: 2.178] (b) What is the net displacement of the particle between t = 0 and t = 2? [Ans: 1.963] 9.A-11 (AB ’87) A particle moves along the x-axis so that its acceleration at any time t is given by a(t) = 6t − 18. At time t = 0 the velocity of the particle is v(0) = 24, and at time t = 1 its position is x(1) = 20. Mr. Budd, compiled September 29, 2010

270

AP Unit 9 (Area and Volume) (a) Write an expression for the velocity v(t) of the particle at any time t. (b) For what values of t is the particle at rest? (c) Write an expression for the position of the particle at any time t. (d) Find the total distance traveled by the particle from t = 1 to t = 3. Ans: 3t2 − 18t + 24; 2, 4; t3 − 9t2 + 24t + 4; 6

9.A-12 (AB ’93) A particle moves on the x-axis so that its position at any time t ≥ 0 is given by x(t) = 2te−t . Find the total distance traveled by the particle from t = 0 to t = 5. Try this using fnInt, then try this on your calculator, without calculating a definite integral. [Ans: 1.404] 9.A-13 (AB 1997) A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t) = 3t2 − 2t − 1. The position x(t) is 5 for t = 2. (a) Write a polynomial expression for the position of the particle at any time t ≥ 0. (b) For what values of t, 0 ≤ t ≤ 3, is the particle’s instantaneous velocity the same as its average velocity on the closed interval [0, 3]? (c) Find the total distance traveled by the particle from t = 0 until time t = 3. Try it with a calculator, then without a calculator. Ans: t3 − t2 − t + 3; t = 1.786; 17

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 2: Area

9.2

271

Area

Advanced Placement Applications of Integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. The emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing the limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Fundamental Theorem of Calculus • Use of Fundamental Theorem to evaluate definite integrals.

9.2.1

High and Low y Switch

At times, the graphs may intersect between the bounds of integration, and the high and low y functions might switch. In this case, you should divide the integral up. When you get to use the calculator, feel free to use abs When you must use the fundamental theorem instead of a calculator to evaluate the definite integral, the absolute value will be worthless. Why? In that case, you need to separate

Example 9.2.1 Find the area of the region bounded by the graphs 4 of y = x, y = , x = 1, and x = 4. Try this with x (a) without a calculator; (b) with a calculator, but without using absolute value; (c) with a calculator, using absolute value.

Ans:

9 2

+ 4 ln 4 − 8 ln 2 =

9 2

= 4.500

Example 9.2.2 (adapted from AB ’83) Do the following problem without a calculator, then with a calculator. Find the area bounded Mr. Budd, compiled September 29, 2010

272

AP Unit 9 (Area and Volume) by the curve f (x) = 3x2 − 12x + 9 and the x-axis, between the lines x = 0 and x = 2. [Ans: 6] Example 9.2.3 Find the area of the region bounded by the graphs 1 of y = 2 , y = x, and y = 2. x Ans:

9.2.2

7 2

√ −2 2

Area: Slicing dy

If you slice the area into vertical strips, then the area of each infinitesimally thin rectangle is given by dA = (rightx − leftx) dy, and the area is given by Z b Area = (rightx − leftx) dy a

Example 9.2.4 (adapted from Ostebee & Zorn [17]) Find the area of the region R bounded by the curves x = 0, y = 2, and y = ex . (a) Find the area of R, with and without a calculator. (b) If the line y = k divides the region R into two regions of equal area, what is the value of k? [Ans: 0.386; 0.219 ; 1.683] Example 9.2.5 Find the area of the region bounded by the graphs 1 of y = 2 , y = x, and y = 2. Try this two ways. x Ans:

9.2.3

7 2

√ −2 2

Total Distance

Remember that Distance : Displacement :: Area : Definite Integral Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 2: Area

273

Example 9.2.6 (AB ’02B) A particle moves along the x-axis so that its velocity v at any time t, for 0 ≤ t ≤ 16, is given by v(t) = e2 sin t − 1. At time t = 0, the particle is at the origin. (a) Sketch the graph of v(t) for 0 ≤ t ≤ 16. (b) During what intervals of time is the particle moving to the left? Give a reason for your answer. (c) Find the total distance traveled by the particle from t = 0 to t = 4. Do this using absolute value, and without using absolute value. (d) Is there any time t, 0 < t ≤ 16, at which the particle returns to the origin? Justify your answer.

[Ans: ; (π, 2π), (3π, 4π), (5π, 16]; 10.542, no]

Problems 9.B-1 (AB ’93) Set up a definite integral to find the area shaded regioni h of the Rb in Figure 9.2. Ans: a (d − f (x)) dx Figure 9.2: AP Calculus AB (1993)

9.B-2 (AB ’96) [NO CALCULATOR] Let R be the region in the first quadrant 1 under the graph of y = √ for 4 ≤ x ≤ 9. x (a) Find the area of R. (b) If the line x = k divides the region R into two regions of equal area, what is the value of k? Ans: 2; 25 4 Mr. Budd, compiled September 29, 2010

274

AP Unit 9 (Area and Volume)

9.B-3 (adapted from AB ’97) [NO CALCULATOR] The area of the region enclosed by the graph of y = x2 − 2x + 2 and the line y = 10 is [Ans: 36] 9.B-4 (adapted from AB ’98) [NO CALCULATOR] What is the area of the region between the graphs of y = x3 and y = −x from x = 0 to x = 3? 99 Ans: 4 9.B-5 (AB ’02B) Let R be the region bounded by the y–axis and the graphs of x3 and y = 4 − 2x. Find the area of R. [Ans: 3.215] y= 1 + x2 9.B-6 (adapted from BC Acorn) Find the area of the region R bounded by the curves x = 1, y = 1, and y = e3x . (a) Find the area of R, by slicing dx, then again by slicing dy. (b) If the vertical line x = h divides the region R into two regions of equal area, what is the value of h? (c) If the horizontal line y = k divides the region R into two regions of equal area, what is the value of k? [Ans: 5.362; 0.814 ; 5.065] x2 y2 9.B-7 Find the area enclosed by the ellipse + = 1. What is area in the 16 9 form kπ? What do you suppose is the area enclosed by any generic ellipse x2 y2 + 2 = 1? [Ans: 12π] 2 a b 9.B-8 (AB ’02) Let f and g be the functions given by f (x) = ex and g(x) = ln x. Find the area of the region enclosed by the graphs of f and g between 1 [Ans: 1.223] x = and x = 1. 2 √ 9.B-9 (AB ’03) Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. Find the area of R. [Ans: 0.443] 9.B-10 (adapted from AB ’03B) Let f be the function given by f (x) = 4x2 − x3 , and let ` be the line y = 18 − 3x, where ` is tangent to the graph of f . Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f , the line `, and the x-axis. (a) At what point is ` tangent to f (x)? (b) Find the area of R. (c) Find the area of S. Ans: (3, 0);

64 3

;7.917

9.B-11 (AB ’05) Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 2: Area

275

1 Let f and g be the functions given by f (x) = + sin (πx) and g(x) = 4−x . 4 Let R be the shaded region in the first quadrant enclosed by the y–axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Find the area of S. [Ans: 0.0648; 0.410] 9.B-12 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. Find the area of R. [Ans: 0.429] 9.B-13 (adapted from AB ’86) A particle moves along the x-axis so that at any 1 time t ≥ 1 its acceleration is given by a(t) = . At time t = 1, the velocity t of the particle is v(1) = −2 and its position is x(1) = 4. (a) Find the velocity at time t = 9. (b) What is the position at time t = 9? (c) What is the total distance traveled from t = 1 to t = 9? (d) How far did the particle travel backwards, starting from t = 1? [Ans: ln 9 − 2; −0.225; 4.553; 4.389] 9.B-14 (AB ’00) Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. Figure 9.3, which consists of two line segments, shows the Mr. Budd, compiled September 29, 2010

276

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Figure 9.3: from AP Calculus AB 2000 exam

velocity, in meters per second, of Runner A. The velocity, in meters per 24t . second, of Runner B is given by the function v defined by v(t) = 2t + 3 Find the total distance run by Runner A and the total distance run by Runner B over the time interval 0 ≤ t ≤ 10 seconds. Indicate units of measure. Then find Runner B’s distance without using a calculator. Recall what to do when antidifferentiating and the degree on top is the same or higher as on bottom. Ans: 85 m; 83.336 m; 120 − ln 23 3

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 3: Volume

9.3

277

Volume: Sweet, Sweet Loaves of Calculus

Advanced Placement Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Resources §8.2 Finding Volumes by Integration in Ostebee and Zorn [17]. §6.2 Volumes in Stewart [20]. §7.4 Volumes in Finney [8].

9.3.1

Volumes of Rotation

In general, b

Z V =

A(x) dx a

or Z V =

A(y) dy c

where A(x) or (A(y)) represents the cross-sectional area of the solid at a particular value of x (or y) 2

For volumes of rotation where cross sections are discs, then A(x) = π [r(x)] or 2 A(y) = π [r(y)] . Example 9.3.1 (adapted from AB 1997) Let R be the √ region bounded by the y-axis, the line y = 2, and the curve y = x. (a) Find the area of region R. (b) Find the volume of the solid generated when region R is rotated about the y-axis. Mr. Budd, compiled September 29, 2010

278

AP Unit 9 (Area and Volume)

Ans:

8 32π 3; 5

Things to keep in mind for volume of rotation problems: • If the axis of rotation is the x-axis, or parallel to the x-axis, slice dx. If the axis of rotation is the y-axis, or parallel to the y-axis, slice dy. Rb • V = a πr2 d • If slicing dy, the radius will be a high x minus a low x. If slicing dx, the radius will be a high y minus a low y. If the axis of rotation is either the x- or y-axis, one of these values will be zero. 1 Example 9.3.2 [3] A region in the plane is bounded by y = √ , x the x-axis, the line x = m, and the line x = 2m where m > 0. A solid is formed by revolving the region about the x-axis. The volume of this solid (A) is independent of m (B) increases as m increases (C) decreases as m decreases 1 (D) increases until m = , then decreases 2 (E) is none of the above

[Ans: A]

Example 9.3.3 Derive, from scratch, the formula for the volume of a sphere of radius r.

Example 9.3.4 Derive, from scratch, the formula for the volume of a cone with height h and base radius r.

Example 9.3.5 [17] The following table gives the circumference (in inches) of a pole at several heights (in feet). Height Circumference

0 16

10 14

20 10

30 5

40 3

50 2

60 1

Assuming that cross sections of the pole taken parallel to the ground are circles, estimate the volume of the pole using: Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 3: Volume

279

(a) T3 (b) M3 (c) T6 (d) S2·3 =

M3 + 2T3 , a weighted average of M3 and T3 . 3

Nonstandard axes of rotation Remember that each radius is a high y minus a low y (or high x minus low x).

Example 9.3.6 Let R be the region bounded by the graphs of y = ex/2 , y = 1, and x = ln 2. (a) Set up, but do not solve, a definite integral that could be used to find the area of R. (b) Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about the line y = 1. (c) Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about the line x = ln 2. (d) Preview: what changes if the axes of rotation are y = −1 or x = 0?

Problems 9.C-1 (adapted from AB ’03B) Let R be the region bounded by the graph of f (x) = 4x2 − x3 and the x-axis. Find the volume of the solid generated when R is revolved about the x-axis. [Ans: 490.208] 9.C-2 [20] A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log) are listed in the table. x (m) 0 1 2 3 4 5

A (m2 ) 0.68 0.65 0.64 0.61 0.58 0.59

x (m) 6 7 8 9 10

A (m2 ) 0.53 0.55 0.52 0.50 0.48

Estimate the volume of the log using: Mr. Budd, compiled September 29, 2010

280

AP Unit 9 (Area and Volume) Ans: 5.8 m3 (b) the Trapezoid Rule with n = 5. Ans: 5.7 m3 (c) the Trapezoid Rule with n = 10. Ans: 5.75 m3 M5 + 2T10 (d) S2·5 = , a weighted average of M5 and T10 . Ans: 5.767 m3 3 (a) the Midpoint Rule with n = 5.

9.C-3 [8] We wish to estimate the volume of a flower vase using only a calculator, a string, and a ruler. We measure the height of the vase to be 6 inches. We then use the string and the ruler to find the circumference of the vase (in inches) at half-inch intervals. (We list them from starting at the top Circumferences 5.4 10.8 4.5 11.6 4.4 11.6 left, moving down) 5.1 10.8 6.3 9.0 7.8 6.3 9.4 (a) Sketch the vase. (b) Find the areas of the cross sections that correspond to the given circumferences. (c) Express the volume of the vase as an integral with respect to y over the interval [0, 6]. (d) Approximate the integral using the Trapezoidal Rule with n = 12. h i R6 2 1 3 Ans: 2.3, 1.6, 1.5, . . .; 4π [C(y)] dy; 34.7 in 0 9.C-4 (AB 1993) [No calculator! ]The region enclosed by the x-axis, the line √ x = 3, and the curve y = x is rotated about the x-axis. What is the volume of the solid generated? Ans: 92 π

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 4: Volume: Slicing with Washers

9.4

281

Volume: Slicing with Washers

9.4.1

Slicing with Washers

For washers, h i h i 2 2 2 2 A(x) = π (R(x)) − π (r(x)) = π (R(x)) − (r(x)) or

h i h i 2 2 2 2 A(y) = π (R(y)) − π (r(y)) = π (R(y)) − (r(y))

Remember that each radius is [high y - low y]. Example 9.4.1 Let R be the region bounded by the graphs of y = ex/2 , y = 1, and x = ln 2. Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about (a) the y-axis; (b) the line y = −1; (c) the line x = −1; (d) the line x = 1; √ (e) the line y = 2 Example 9.4.2 ([3]) Let R be the region in the first quadrant bounded above by the graph of f (x) = 2 arctan x and below by the graph of y = x. What is the volume of the solid generated when R is rotated about the x-axis? Mr. Budd, compiled September 29, 2010

282

AP Unit 9 (Area and Volume) [Ans: 7.151] Example 9.4.3 (AB â&#x20AC;&#x2122;02) Let f and g be the functions given by f (x) = ex and g(x) = ln x. Find the volume of the solid generated 1 when the region enclosed by the graphs of f and g between x = 2 and x = 1 is revolved about the line y = 4.

[Ans: 23.609] Example 9.4.4 (AB â&#x20AC;&#x2122;97) Let f be the function given by f (x) = 3 cos x. As shown in Figure 9.4, the graph of f crosses the y-axis at point P and the x-axis at point Q. Figure 9.4: AB Exam 1997

(a) Write an equation for the line passing through points P and Q. (b) Write an equation for the line tangent to the graph of f at point Q. Show the analysis that leads to your equation. (c) Find the x-coordinate of the point on the graph of f , between points P and Q, at which the line tangent to the graph of f is parallel to line P Q. (d) Let R be the region in the first quadrant bounded by the graph of f and the line segment P Q. Write an integral expression for the volume of the solid generated by revolving the region R about the x-axis. Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 4: Volume: Slicing with Washers h

283

Ans: y − 3 = − π6 (x − 0); y − 0 = −3 (x − π/2); 0.690; π

R π/2 h 0

(3 cos x) − − π6 x + 3

Homework For Monday, February 22, 2010, put # 1 on the front and #2,3 on the back. For Tuesday, February 23, 2010, put #4 on the front and #5,6 on the back. Each is due before the final bell at 3:42 p.m. 9.D-1 (adapted from AB ’00) Let R be the region in the first quadrant enclosed 2 by the graphs of y = e−x , y = 1 − cos x, and the y-axis. (a) Find the volume of the solid generated when the region R is revolved about the x–axis. (b) Find the volume of the solid generated when the region R is revolved about the line y = −1. (c) Suppose R is revolved around the line y = k, where k > 1 so that the line is above the region. Find k if the volume of this solid is the same as the volume of the solid in the previous part, where R is revolved around the line y = −1. [Ans: 1.747; 5.460; 1.941] √ 9.D-2 (AB ’03) Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. Find the volume of the solid generated when R is revolved about the horizontal line y = 1. [Ans: 1.424] 9.D-3 (AB Acorn ’04-05) Let S be the region enclosed by the graphs of y = 2x and y = 2x2 for 0 ≤ x ≤ 1. Write a definite integral for the volsolid generated when S is revolved about the line y = 3. hume of the i R1 2 2 2 Ans: π 0 3 − 2x − (3 − 2x) dx 9.D-4 (adapted from AB ’02B) Let R be the region bounded by the y–axis and x3 the graphs of y = and y = 4 − 2x. Find the volume of the solid 1 + x2 generated when R is revolved about (a) the x–axis. (b) the line y = 4. [Ans: 31.885;48.906] 9.D-5 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. Find the volume generated when R is revolved about the x–axis. [Ans: 4.267] Mr. Budd, compiled September 29, 2010

2 i

i dx

284

AP Unit 9 (Area and Volume)

9.D-6 (AB ’05)

1 Let f and g be the functions given by f (x) = + sin (πx) and g(x) = 4−x . 4 Let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. Find the volume of the solid generated when S is revolved about the horizontal line y = −1. [Ans: 4.558]

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 5: Non-Circular Slicing

9.5

285

Flat-Bottomed Volume: Non-Circular Slicing

Advanced Placement Interpretations and properties of definite integrals • Computation of Riemann sums using left, right, and midpoint evaluation points. • Definite integral as a limit of Riemann sums over equal subdivisions. Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Figure 9.5: AB 1998

Example 9.5.1 (AB 1998) The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x + 2y = 8, as shown in the Figure 9.5. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid? [Ans: 16.755] Example 9.5.2 (AB 1997) √ The base of a solid S is the region enclosed by the graph of y = ln x, the line x = e, and the x-axis. Mr. Budd, compiled September 29, 2010

286

AP Unit 9 (Area and Volume) If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is

[Ans: 1]

Example 9.5.3 (BC 1997) The base of a solid is the region in the first quadrant enclosed by the graph of y = 2−x2 and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume of the solid is given by

Ans:

R2 0

(2 − y) dy

Example 9.5.4 Repeat for equilateral triangles, isosceles right triangles, etc.

Problems For Friday, February 26, 2010, put # 1 on the front and #2,3 on the back. For Monday, March 1, 2010, put #4,5 on the front and #6 on the back. Each is due before the final bell at 3:42 p.m. 9.E-1 (AP ’96) Let R be the region in the first quadrant under the graph of 1 y = √ for 4 ≤ x ≤ 9. Find the volume of the solid whose base is x the region R and whose cross-sections cut by planes perpendicular to the x-axis are squares. Ans: ln 49 9.E-2 (AB ’00) Let R be the region in the first quadrant enclosed by the graphs 2 of y = e−x , y = 1 − cos x, and the y-axis. The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a square. Find the volume of the solid. [Ans: 0.461] 9.E-3 (adapted from AB ’02B) Let R be the region bounded by the y–axis and x3 the graphs of y = and y = 4 − 2x. The region R is the base of 1 + x2 a solid. For this solid, each cross section perpendicular to the x–axis is a semicircle. Find the volume of this solid. [Ans: 3.533] 9.E-4 (AB ’03) Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 5: Non-Circular Slicing

287

√ Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. [Ans: 1.554] 9.E-5 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles with diameters extending from y = f (x) to y = g(x). Find the volume of this solid. [Ans: 0.078] 9.E-6 [20] A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. (a) Use the Midpoint Rule to estimate the volume of the liver. Ans: 1110 cm3 (b) Use the Trapezoidal Rule to estimate the volume of the liver. Ans: 1053 cm3

Mr. Budd, compiled September 29, 2010

288

AP Unit 9 (Area and Volume)

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 6: Area and Volume

9.6

289

Area and Volume

Problems For Tuesday, March 2, 2010, put #1 on the front and #2 on the back. For Thursday, March 4, 2010, put #3 on the front and #4 on the back. For Monday, March 8, 2010, put #5 on the front. For Tuesday, March 9, 2010, put #6 on the front. Each is due before the final bell at 3:42 p.m. 9.F-1 (2010–4) [NO CALCULATOR] Let √ R be the region in the first quadrant bounded by the graph of y = 2 x, the horizontal line y = 6, and the y–axis, as shown in the figure below.

(a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 7. (c) Region R is the base of a solid. For each y, where 0 ≤ y ≤ 6, the cross section of the solid taken perpendicular to the y–axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid.

Mr. Budd, compiled September 29, 2010

290

AP Unit 9 (Area and Volume)

9.F-2 (2010B–1) In the figure above, R is the shaded region in the first quadrant bounded by the graph of y = 4 ln (3 − x), the horizontal line y = 6, and the vertical line x = 2.

(a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y = 8. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a square. Find the volume of the solid.

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 6: Area and Volume

291

9.F-3 (2004B AB-1) Let R be the region enclosed by the graph of y = the vertical line x = 10, and the x–axis.

√

x − 1,

(a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y = 3. (c) Find the volume of the solid generated when R is revolved about the vertical line x = 10. 9.F-4 (2005B AB-1)

Let f and g be the functions given by f (x) = 1 + sin(2x) and g(x) = ex/2 . Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the x–axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles with diameters extending from y = f (x) to y = g(x). Find the volume of this solid. 9.F-5 (2008P AB–2) Mr. Budd, compiled September 29, 2010

292

AP Unit 9 (Area and Volume)

Let R and S in the figure above be defined as follows: R is the region in the first and second quadrants bounded by the graphs or y = 3 − x2 and y = 2x . S is the shaded region in the first quadrant bounded by the two graphs, the x–axis, and the y–axis. (a) Find the area of S. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = −1. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is an isosceles right triangle with one leg across the base of the solid. Write, but do not evaluate, an integral expression that gives the volume of the solid.

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 6: Area and Volume

293

9.F-6 (2007B AB–1)

Let R be the region bounded by the graph of y = e2x−x and the horizontal 2 line y = 2, and let S be the region bounded by the graph of y = e2x−x and the horizontal lines y = 1 and y = 2, as shown above. (a) Find the area of R. (b) Find the area of S. (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 1.

Mr. Budd, compiled September 29, 2010

294

AP Unit 9 (Area and Volume)

9.F-7 (1999 AB-2)

The shaded region, R, is bounded by the graph of y = x2 and the line y = 4, as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated by revolving R about the x–axis. (c) There exists a number k, k > 4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of k. 9.F-8 (2007 AB–1) Let R be the region in the first and second quadrants bounded 20 above by the graph of y = and below by the horizontal line y = 2. 1 + x2 (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the x–axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles. Find the volume of this solid.

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 7: Related Rates with Volume

9.7

295

Related Rates with Volume

Advanced Placement Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of derivatives. • Modeling rates of change, including related rates problems. Textbook §2.6 Related Rates [16] Resources §10-4 Related Rates in Foerster; Exploration 10-4

9.7.1

Volume problems

Example 9.7.1 (adapted from AB ’98) The radius of a circle is decreasing at a constant rate of 0.2 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second?

[Ans: −0.2C]

Example 9.7.2 (AB 2002) A container has the shape of an open right circular cone, as shown in Figure 9.6. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at the 3 cm/hr. constant rate of − 10 (The volume of a cone of height h and radius r is given by V = 1 2 πr h.) 3 (a) Find the volume V of water in the container when h = 5 cm. 3 Indicate units of measure. Ans: 125 12 π cm (b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate units of measure. 3 Ans: − 15 8 π cm /hr Mr. Budd, compiled September 29, 2010

296

AP Unit 9 (Area and Volume)

Figure 9.6: From 2002 AP Calculus AB Exam

(c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportion- 3 ality? Ans: â&#x2C6;&#x2019; 10 Example 9.7.3 [2] The function V whose graph is sketched in Figure 9.7 gives the volume of air, V (t), (measured in cubic inches) that a man has blown into a balloon after t seconds. 4 3 V = Ď&#x20AC;r 3 The rate at which the radius is changing after 6 seconds is approximately what? Figure 9.7: From [2]

Mr. Budd, compiled September 29, 2010

AP Unit 9, Day 7: Related Rates with Volume

297 [Ans: 0.1 in/sec]

Problems 9.G-1 (adapted from Acorn AB ’00) If r is positive and increasing, for what value of r is the rate of increase of r3 forty-eight times that of r? [Ans: 4] 9.G-2 [2] Let y = 2ecos x . Both x and y vary with time in such a way that y increases at the constant rate of 5 units per second. The rate at which x π is changing when x = is [Ans: −2.5 units/sec] 2 9.G-3 [2] When the area of an expanding square, in square units, is increasing three times as fast as its side is increasing, in linear units, the side is Ans: 32 9.G-4 [2] Water is flowing into a spherical tank with 6 foot radius at the constant rate of 30π cubic ft per hour. When the water is h feet deep, the volume of water in the tank is given by V =

πh2 (18 − h) 3

What is the rate at which the depth of the water in the tank is increasing at the moment when the water is 2 feet deep? [Ans: 1.5 ft per hr] 9.G-5 [2] The edge of a cube is increasing at the uniform rate of 0.2 inches per second. At the instant when the total surface area becomes 150 square inches, what is the rate of increase, in cubic inches per the second, of volume of the cube? Ans: 15 in3 /sec 9.G-6 (adapted from AB ’98) If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 6 inches per minute, what relationship must exist between b and h for the area A of the triangle to be decreasing? Ans: b > 21 h 9.G-7 [2] Sand is being dumped on a pile in such a way that it always forms a cone whose base radius is always 3 times its height. The function V whose graph is sketched in Figure 9.8 gives the volume of the conical sand pile, V (t), measured in cubic feet, after t minutes. At what approximate rate is the radius of the base changing after 6 minutes. [Ans: 0.22 ft/min]

Mr. Budd, compiled September 29, 2010

298

AP Unit 9 (Area and Volume)

Figure 9.8: From [2]

Mr. Budd, compiled September 29, 2010

Unit 10

Extrema and Optimization 1. Solving Inequalites 2. First Derivative Test 3. Points of Inflection 4. Second Derivative Test 5. Absolute Extrema 6. Curve Sketching 7. Optimization Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 . Second derivatives. • Corresponding characteristics of the graphs of f , f 0 , and f 00 . • Relationship between the concavity of f and the sign of f 00 . • Points of inflection as places where the concavity changes. 299

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Applications of derivatives. â&#x20AC;˘ Optimization, both absolute (global) and relative (local) extrema.

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AP Unit 10, Day 1: Absolute Extrema

10.1

301

Absolute Extrema

Advanced Placement Applications of derivatives. • Optimization, both absolute (global) and relative (local) extrema. Textbook §3.7 Optimization Problems and §3.10 Business and Economic Applications [16] Resources §8-3 Maxima and Minima in Plane and Solid Figures in [10]

10.1.1

Absolute Extrema

First of all know the difference between absolute extrema and relative extrema • absolute = global = higher (or lower) than everything • relative = local = higher (or lower) than everything nearby The absolute extremum (of a continuous function over a closed interval) will occur either at an endpoint or at a critical point. To find absolute, aka global, extrema of f over [a, b]. [16] 1. Find the critical numbers of f in (a, b). 2. Evaluate f at each critical number in (a, b). 3. Evaluate f at the endpoints, i.e., at a and at b. 4. The lowest of these values is the absolute minimum, the highest is the absolute maximum. 5. Answer the question that’s asked. Do you need an x-value, or a y-value? Example 10.1.1 Find the absolute minimum and the absolute maximum values of the following functions, over the following intervals: (a) f (x) = πx2 (12 − 3x) over [0, 3] 3 (b) f (x) = x 200 − x over [5, 146.25] 4 Mr. Budd, compiled September 29, 2010

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10.1.2

Absolute Extrema from the Derivative

Example 10.1.2 (adapted from AB ’02) Let H(t) be a function for 9 ≤ t ≤ 23 whose derivative is given by H 0 (t) =

15600 9890 − 2 − 24t + 160 t − 38t + 370

such that H(9) = 0. (a) For what values of t, t ∈ [9, 23], is H(t) increasing? Justify your answer. (b) Based on a graph of H 0 (t), why are the endpoints not good candidates for the absolute maximum? (c) What is the maximum value (i.e., the absolute maximum) of H(t)? At what value of t does the maximum occur? (d) Based on a graph of H 0 (t), why is the critical point not a good candidate for the absolute minimum? How would you choose from the remaining candidates? (e) At what time does the absolute minimum occur? What is the minimum value of H(t) over [9, 23]. (f) How many points of inflection will the graph of H(t) have? Justify your answer. Example 10.1.3 (adapted from AB ’02B) The number of gallons, P (t), of a pollutant in a lake changes at the rate P 0 (t) = √ 2πt 2 cos − 3e−0.2 t gallons per day, where t is measured in days. 24 There are 100 gallons of the pollutant in the lake at time t = 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant. (a) Find the critical values of P (t) over (0, 50). (b) Find all values of x, x ∈ [0, 50], for which P (t) is increasing. (c) Based on knowledge of the intervals for which P (t) is increasing or decreasing, which are the two best candidates for the absolute minimum? Explain your reasoning. (d) How would you decide which is smaller, P (20.489 . . .) or P (43.574 . . .)? (e) Is the lake safe when the number of gallons of pollutant is at its absolute minimum? Justify your answer. (f) Is the lake safe for both relative minima of P (t)? (g) Based on knowledge of the intervals for which P (t) is increasing or decreasing, which are the three best candidates for the absolute minimum? Explain your reasoning. Mr. Budd, compiled September 29, 2010

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(h) What is the maximum number of gallons of pollutant over [0, 50]? At what time does it occur?

[Ans: 20.490 = c, 27.904 = d, 43.574 = e; [c, d], [e, 50]] [Ans: lake is safe at t = e, but not at c] Example 10.1.4 (adapted from AB ’00) Figure 10.1 shows the Figure 10.1:

graph of f 0 , the derivative of the function f , for −7 ≤ x ≤ 7. The graph of f 0 has horizontal tangent lines at x = −3, x = 2, and x = 5, and a vertical tangent line at x = 3. (a) What are the critical values of f over (−7, 7)? (b) Of the endpoints and critical points, which are good candidates for the absolute maximum? For the absolute minimum? Explain your reasoning. (c) Without being able to exactly find f (−5) or f (7), explain how you can still use the fundamental theorem to decide which is larger. (d) Which is smaller, f (−7) of f (−1)? Explain your reasoning. (e) Find all values of x, x ∈ [−7, 7], for which f (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

has relative maxima; has relative minima; has absolute maxima; has absolute minima; is increasing; is decreasing; is concave down; is concave up; has a point of inflection. Mr. Budd, compiled September 29, 2010

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10.1.3

Optimization

Example 10.1.5 Old Mac-Donald had a farm. And on that farm, he had some cows, which fought incessantly. In order to separate the Cowpulets from the Montamoos, he built two identical, adjacent rectangular pens which share a side. Old Mac-Donald hires you to find out the dimensions of the pen which give you the greatest area if Old MacDonald has, say, 600 yards of fencing.

[Ans: 75 yards by 100 yards adjoining]

• Draw a (big) picture. • Label variables. • Write an expression for what you’re maximizing or minimizing. • Write an equation for the constraints. • Use the constraint to reduce your maximizing/ minimizing expression to one variable. • Use the physical reality of the problem to determine end values. • Differentiate to find critical values. • Plug end and critical values into your maximizing/ minimizing expression. • Answer the question that’s asked, not the question that you want to answer.

Technique: Analysis of Maximum-Minimum Problems [10]

1. Make a sketch if one isn’t already drawn. 2. Write an equation for the variable you are trying to maximize or minimize. 3. Get the equation in terms of one variable and specify a domain. 4. Find an approximate maximum or minimum by grapher. 5. Find the exact maximum or minimum by seeing where the derivative is zero or infinite. Check any endpoints of the domain. 6. Answer the question by writing what was asked for in the problem statement. Mr. Budd, compiled September 29, 2010

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College Board does not recognize maxima or minima calculated on the graphing calculator. Justifications require mathematical (i.e., noncalculator) reasoning.

Example 10.1.6 A 400 meter track is to be built around a field that consists of a rectangle with two semicircles at either end. (The base of each semicircle spans the entire width of the rectangle.) How should the track be built in order to maximize (a) the area of the rectangle; (b) the total enclosed area.

Ans: length 100 m, radius

100 π

m ; circle of radius

200 π

Example 10.1.7 [from Acorn ’02] Consider one arch of cos x above the x-axis. Draw a rectangle that lies on the x-axis so that its top two vertices lie on the curve y = cos x. Shade the area between y = cos x and the x-axis that is not in the rectangle. Find the minimum area of the shaded region.

[Ans: 0.878]

Example 10.1.8 [10] Barb Dwyer must build a rectangular corral along the river bank. Three sides of the corral will be fenced with a barbed wire. The river forms the fourth side of the corral (Figure 10.2). The total length of fencing available is 1000 feet. What is the maximum area the corral could have? How should the fence be built to enclose this maximum area? Justify your answers.

[Ans: 125,000 square feet, 250 feet ⊥ to river by 500 feet parallel] Example 10.1.9 [10] The part of the parabola y = 4 − x2 from x = 0 to x = 2 is rotated about the y-axis to form a surface. A cone is inscribed in the resulting paraboloid with its vertex at the origin and its base touching the parabola (Figure 10.3). At which radius and altitude does the maximum volume occur? What is the maximum volume? Justify your answer. √ Ans: x = 2, y = 2, V =

4π 3

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Figure 10.2: Figure 10.3: Example 10.1.10 (adapted from BC93) Consider all right circular cylinders for which the sum of the height and the circumference is 30Ď&#x20AC; centimeters. What is the radius of the one with maximum volume?

[Ans: 10 cm]

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Curve Sketching Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 . Second derivatives. • Corresponding characteristics of the graphs of f , f 0 , and f 00 . • Relationship between the concavity of f and the sign of f 00 . • Points of inflection as places where the concavity changes. Resources §4.3 Connecting f 0 and f 00 with the Graph of f in [8]

Mr. Budd, compiled September 29, 2010

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Mr. Budd, compiled September 29, 2010

AP Unit 10, Day 2: First Derivative

10.2

309

First Derivative

Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 .

10.2.1

First Derivative Test

Example 10.2.1 Find all local extrema of f (x) = the interval (0, 2π).

Example 10.2.2 Find all local extrema of g(x) =

1 x + sin x over 2

q 2 3 (x2 − 2x − 8) .

Example 10.2.3 Show that the maximum value of h(x) = bxe−bx , b > 0 is independent of b.

Example 10.2.4 (MM 2004) Consider the function f (x) = 2 + 1 . x−1 (a) Sketch f (x) (b) Write down the x-intercepts and y-intercepts of f (x). (c) Write down the equations of the asymptotes of f (x). (d) Find f 0 (x) (e) There are no maximum or minimum points on the graph of f (x). Use your equation for f 0 (x) to explain why.

Example 10.2.5 (MM 2003) Figure blah-blah-blah shows the graph of y = ex (cos x + sin x), −2 ≤ x ≤ 3. The graph has a maximum turning point at C (a, b). dy . dx (b) Find the exact value of a and b. (a) Find

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AP Unit 10 (Extrema and Optimization) Example 10.2.6 (adapted from MM 2002) Let the function f (x) = 2 , x 6= 1. 1 + x3 (a) Write down the equation of the vertical asymptote of the graph of f . (b) Write down the equation of the horizontal asymptote of the graph of f . (c) Show that f 0 (x) =

−6x2 (1 + x3 )

(d) Determine the x-coordinates for which f (a) (b) (c) (d)

is increasing; is decreasing; has a relative maximum; has a relative minimum.

Problems 10.B-1 For the function h(x) = xe−x (a) Find the critical values of h(x); (b) Find all x-coordinates for which h is increasing; (c) Find the x-coordinates of any relative maxima on the graph of h; (d) Find the x-coordinates of any relative minima on the graph of h. Justify all answers.

[Ans: 1; x < 1; x = 1; ∅]

10.B-2 Exactly find all critical values of f (x) = x2 ln x and classify each as a relative minimum, a relative maximum, or neither. Justify your answer. h i Ans: √1e , relative min

Mr. Budd, compiled September 29, 2010

AP Unit 10, Day 3: Second Derivative

10.3

311

Second Derivative

Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . Second derivatives. • Corresponding characteristics of the graphs of f , f 0 , and f 00 . • Relationship between the concavity of f and the sign of f 00 . • Points of inflection as places where the concavity changes. Textbook §3.4 Concavity and the Second Derivative Test: [16]

10.3.1

Concavity

Recall concavity If f is concave up, then: • f is like a cup; • f 0 is increasing (if f 0 exists); • f 00 is positive (if f 00 exists); • the curve will lie above the tangent line.

Example 10.3.1 (AB97) For what interval is the graph of y = x4 − 9x3 + 27x2 − 45x + 36 is concave up?

Ans: x <

3 2

or x > 3

Example 10.3.2 For x2 + y 2 = 16: Remember the Quotient Rule: For what values of x or y is the graph of x2 + y 2 = 16 concave up? Mr. Budd, compiled September 29, 2010

312

AP Unit 10 (Extrema and Optimization) Example 10.3.3 (MM 2003) Figure blah-blah-blah shows the graph of y = ex (cos x + sin x), −2 ≤ x ≤ 3. The graph has a maximum turning point at C (a, b) and a point of inflection at D. dy . dx (b) Find the exact value of a and b. √ π (c) Show that at D, y = 2e 4 . (a) Find

Example 10.3.4 (adapted from MM 2002) Let the function f (x) = 2 , x 6= 1. 1 + x3 (a) Write down the equation of the vertical asymptote of the graph of f . (b) Write down the equation of the horizontal asymptote of the graph of f . (c) Using the fact that f 0 (x) =

−6x2

(1 + x3 ) 3 12x 2x − 1 derivative f 00 (x) = . 3 (1 + x3 )

show that the second

(d) Find the x-coordinates of the points of inflection of the graph of f . (e) Use the trapezium rule with five subintervals to approximate R3 the integral f (x) dx. 1

(f) Given that

f (x) dx = 0.637599, use a diagram to explain why

your answer is greater than this.

Points of Inflection An inflection point is a point on the graph of f where the concavity changes. At inflection points on the graph of f , • f changes concavity; • f 0 changes direction (points of inflection of f occur at extrema of f 0 ); • f 00 changes sign (points of inflection of f occur when f 00 crosses the x-axis). Mr. Budd, compiled September 29, 2010

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Be careful: for there to be an inflection point, it is not enough for the second derivative to be zero. The second derivative must change sign. This is similar to finding relative extrema: it is not enough for the first derivative to be 0, the first derivative must change sign for there to be a relative extremum.

10.3.2

Second Derivative Test

Theorem 10.1 (Second Derivative Test for Local Extrema). Let c be a critical value of f such that f 0 (c) = 0 and f 00 (c) exists. 1. If f 00 (c) > 0, then f (c) is a relative minimum. 2. If f 00 (c) < 0, then f (c) is a relative maximum. 3. If f 00 (c) = 0, then the test fails. In such cases, you can use the First Derivative Test. Example 10.3.5 Find the critical values of y = 5x3 − 3x5 , and test each value to decide whether it corresponds to a relative maximum, a relative minimum, or neither. Use the second derivative test.

[Ans: −1 (rel min); 0 (neither); 1 (rel max)] For this problem, there is one point for which the second derivative test fails. Can you determine whether it is a local extremum merely from the information you’ve received from the second derivative test? b Example 10.3.6 Let f (x) = x + , where b is a positive number. x Use the second derivative test to find and distinguish any relative extrema. h

i √ √ Ans: − b (rel max); b (rel min)

Example 10.3.7 (BC97) Where are the relative extrema of the function f given by f (x) = 3x5 − 4x3 − 3x.

[Ans: rel max at x = −1, rel min at x = 1] Mr. Budd, compiled September 29, 2010

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AP Unit 10 (Extrema and Optimization) Example 10.3.8 [3] Which of the following are true about the function f if its derivative is defined by 2

f 0 (x) = (x − 1) (4 − x) I. f is decreasing for all x < 4. II. f has a local maximum at x = 1. III. f is concave up for all 1 < x < 3.

[Ans: III only]

Problems In textbook, §3.4: # 11, 17, 19; 27-35 odd; §5.1:#71; §5.4:#53, 57 10.C-1 [3] At x = 0, which of the following is true of the function f (x) = sin x+e−x (a) f is increasing (b) f is decreasing (c) f is discontinuous (d) f is concave up (e) f is concave down [Ans: concave up] 1 1 + 3 x2 x [Ans: −2]

10.C-2 (adapted from AB93) At what value of x does the graph of y = have a point of inflection?

3 10.C-3 (adapted from [2]) Consider the function f (x) = x2 − 5 for all real numbers x. At what x-coordinates are the inflection √ √ for the graph points of f ? Ans: − 5, −1, 1, 5 10.C-4 (adapted from [3]) The slope of the curve y = 32 x2 − e−x at its point of inflection is [Ans: 3 − ln 27] 10.C-5 (BC98) Let f be a function defined and continuous on the closed interval [a, b]. If f has a relative maximum at c and a < c < b, which of the following statements must be true? I. f 0 (c) exists. II. If f 0 (c) exists, then f 0 (c) = 0. III. If f 00 (c) exists, then f 00 (c) ≤ 0. Mr. Budd, compiled September 29, 2010

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315 [Ans: II and III only]

10.C-6 (AB99) Suppose that the function f has a continuous second derivative for all x, and that f (0) = 2, f 0 (0) = −3, and f 00 (0) = 0. Let g be a function whose derivative is given by g 0 (x) = e−2x (3f (x) + 2f 0 (x)) for all x. (a) Write an equation of the line tangent to the graph of f at the point where x = 0. [Ans: y − 2 = −3 (x − 0)] (b) Is there sufficient information to determine whether or not the graph of f has a point of inflection when x = 0? Explain your answer. [Ans: No] (c) Given that g(0) = 4, write an equation of the line tangent to the graph of g at the point where x = 0. [Ans: y = 4] (d) Show that g 00 (x) = e−2x (−6f (x) − f 0 (x) + 2f 00 (x)). Does g have a local maximum at x = 0? Justify your answer. [Ans: Yes; g 0 (x) = 0, g 00 (x) < 0]

Mr. Budd, compiled September 29, 2010

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Mr. Budd, compiled September 29, 2010

AP Unit 10, Day 3: Second Derivative

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Second Derivative Test Advanced Placement Second derivatives. • Relationship between the concavity of f and the sign of f 00 . Applications of derivatives. • Optimization, both absolute (global) and relative (local) extrema. Textbook §3.4 Concavity and the Second Derivative Test: “Concavity” and “The Second Derivative Test” [16]

Mr. Budd, compiled September 29, 2010

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Mr. Budd, compiled September 29, 2010

Unit 11

Review 1. Solving Inequalites 2. First Derivative Test 3. Points of Inflection 4. Second Derivative Test 5. Absolute Extrema 6. Curve Sketching 7. Optimization Advanced Placement Derivative as a function • Corresponding characteristics of graphs of f and f 0 . • Relationship between the increasing and decreasing behavior of f and the sign of f 0 . Second derivatives. • Corresponding characteristics of the graphs of f , f 0 , and f 00 . • Relationship between the concavity of f and the sign of f 00 . • Points of inflection as places where the concavity changes. 319

320

AP Unit 11 (Review)

Applications of derivatives. â&#x20AC;˘ Optimization, both absolute (global) and relative (local) extrema.

Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 1: Separable Differential Equations

11.1

321

Separable Differential Equations

Advanced Placement Derivative as a function.

• Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Applications of antidifferentiation.

• Finding specific antiderivatives using initial conditions, including applications to motion along a line.

• Solving separable differential equations and using them in modeling.

Textbook §7.1 Modeling with Differential Equations; §7.2 Separable Differential Equations [16]

11.1.1

Separable Differential Equations

Definition 11.1. Differential Equation A differential equation is an equation that contains the derivative of a function. [10]

Example 11.1.1 (2005B AB-6) [NO CALCULATOR] Consider dy −xy 2 the differential equation = . Let y = f (x) be the particdx 2 ular solution to this differential equation with the initial condition f (−1) = 2.

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. Mr. Budd, compiled September 29, 2010

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AP Unit 11 (Review)

(b) Write an equation for the line tangent to the graph of f at x = −1. (c) Find the solution y = f (x) to the given differential equation with the initial condition f (−1) = 2. Example 11.1.2 (2002 AB–5) Consider the differential equation dy 3−x = . dx y (a) Let y = f (x) be the particular solution to the given differential equation for 1 < x < 5 such that the line y = −2 is tangent to the graph of f . Find the x–coordinate of the point of tangency, and determine whether f has a local maximum, local minimum, or neither at this point. Justify your answer. (b) Let y = g(x) be the particular solution to the given differential equation for −2 < x < 8 with the initial condition g(6) = −4. Find y = g(x)

Example 11.1.3 The solution to the differential equation 2xe−y , where y(0) = 1, is

dy = dx

Ans: y = ln x2 + e Example 11.1.4 (adapted from AB acorn ’02) The solution to the dy x2 differential equation = 3 where y(3) = 2, is dx y h

Ans: y =

q 4

4 3 3x

− 20

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Example 11.1.5 (adapted from [2]) If the graph of y = f (x) con 2x sin x2 dy tains the point (0, −1) and if = , then f (x) = dx y

11.1.2

i p Ans: − 3 − 2 cos (x2 )

Separable Differential Equations with Logs

Example 11.1.6 (AB97) Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t ≥ 0. After her parachute opens, dv her velocity satisfies the differential equation = −2v − 32, with dt initial condition v(0) = −50.

(a) Use separation of variables to find an expression for v in terms of t, where t is measured in seconds. (b) Terminal velocity is defined as lim v(t). Find the terminal t→∞ velocity of the skydiver to the nearest foot per second. (c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?

Ans: v = −34e−2t − 16, −16, 1.070

Example 11.1.7 (2004 AB-5) Consider the differential equation dy = x2 (y − 1). dx

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. Mr. Budd, compiled September 29, 2010

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(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xyâ&#x20AC;&#x201C;plane. Describe all points in the xyâ&#x20AC;&#x201C;plane for which the slopes are positive. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = 3.

Example 11.1.8 A turkey is cooking in the oven at 300 degrees Fahrenheit. It starts out at room temperature (70 degrees). After 1 hour, it is ? degrees. How long before it reaches 170 degrees, at which point it will be done. The rate of change in the temperature of the turkey is proportional to the difference between the temperatures of the environment and the turkey.

Homework For Wednesday, March 10, put 107 on the front and 108 on the back. For Thursday, March 11, put 109 on the front and 110 on the back.

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dy 11.A-107 (2005 AB–6) [NO CALCULATOR] Consider the differential equation = dx 2x − . y (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (b) Let y = f (x) be the particular solution to the differential equation with the initial condition f (1) = −1. Write an equation for the line tangent to the graph of f at (1, −1) and use it to approximate f (1.1). (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (1) = −1. 11.A-108 (2004B AB-5) Consider the differential equation

dy = x4 (y − 2). dx

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy–plane. Describe all points in the xy–plane for which the slopes are negative. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = 0.

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dy 11.A-109 (2008–5) [NO CALCULATOR] Consider the differential equation = dx y−1 , where x 6= 0. x2 (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

(b) Find the particular solution y = f (x) to the differential equation with the initial condition f (2) = 0. (c) For the particular solution y = f (x) described in part (b), find lim f (x). x→∞

dy 11.A-110 (2006 AB-5) [NO CALCULATOR] Consider the differential equation = dx 1+y , where x 6= 0. x (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.

(b) Find the particular solution y = f (x) to the differential equation with the initial condition f (−1) = 1 and state its domain.

Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 2: Exponential Growth and Decay

11.2

327

Exponential Growth and Decay

Advanced Placement Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth. Textbook §7.1 Modeling with Differential Equations; §7.2 Separable Differential Equations [16] Resources §7.2 Exponential Growth and Decay and §7.3 Other Differential Equations for Real-World Applications in [10]

11.2.1

Proportional Growth

Example 11.2.1 The rate growth of the population of Escherichia coli is proportional to the number of E. coli. Find a general expression for the population as a function of time if the initial population is P0 .

Ans: P = P0 ekt

Example 11.2.2 (adapted from AB98) Population y grows accorddy = ky, where k is a constant and t is measured ing to the equation dt in years. If the population doubles every 8 years, then the value of k is

[Ans: 0.087] Mr. Budd, compiled September 29, 2010

328

AP Unit 11 (Review) Example 11.2.3 [10] Chemical Reaction Problem Calculus buddite (a rare substance) is converted chemically into Glamis thanus. Buddite reacts in such a way that the rate of change in the amount left unreacted is directly proportional to that amount. (a) Write a differential equation that expresses this relationship. Solve it to find an equation that expresses amount in terms of time. Use the initial conditions that theh amount is 50 mg when i t/20 = 50e−0.025541...t t = 0 min and 30 mg when t = 20 min. Ans: dB dt = kB, B = 50 (0.6) (b) Sketch the graph of amount versus time. (c) How much buddite remains an hour after the reaction starts? [Ans: 10.8 mg] (d) When will the amount of buddite equal 0.007 mg? [Ans: 5 hr 47 min]

11.2.2

Other Applications of Differential Equations

Example 11.2.4 Tin Can Leakage Problem [10] Suppose you fill a tall (topless) tin can with water, then punch a hole near the bottom with an ice pick. The water leaks quickly at first, then more slowly as the depth of the water increases. In engineering or physics, you will learn that the rate at which the water leaks out is directly proportional to the square root of its depth. Suppose that at time t = 0 min, the depth is 12 cm and dy dt is −3 cm/min. (a) Write a differential equation stating that the instantaneous rate of change of y with respect to t is directly proportional to the square root of y. Find the proportionality constant. (b) Solve the differential equation to find y as a function of t. Use the given information to find the particular solution. What kind of function is this? (c) Plot the graph of y as a function of t. Sketch the graph. Consider the domain of t in which the function gives reasonable answers. (d) Solve algebraically for the time at which the can becomes empty. Compare your answer with the time it would take at the initial rate of −3 cm/min. Ans: k = −3 12−1/2 ; y =

3 2 16 t

− 3t + 12; ;8 (twice as long)

Example 11.2.5 Dam Leakage Problem [10] A new dam is constructed across Scorpion Gulch. Engineers want to predict the amount Mr. Budd, compiled September 29, 2010

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Problems For Friday, March 12, put 111 on the front and 112 on the back. It would be a good idea to turn the following in before spring break: For Monday, March 22, put 113 on the front, and be on the lookout for a quiz on Moodle due Monday.

11.B-111 (a) (adapted from AB93) A puppy weighs 2.1 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 4 months old (to the nearest 0.1 pound)? (b) (adapted from [2]) If g 0 (x) = 3g(x) and g(−1) = 1, find g(x). Mr. Budd, compiled September 29, 2010

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11.B-112 (a) (adapted from [2]) The change in N , the number of bacteria in a dN culture dish at time t is given by: = 3N . If N = 4, when t = 0, dt the approximate value of t when N = 1614 is (b) (AB â&#x20AC;&#x2122;96) The rate of consumption of cola in the United States is given by S(t) = Cekt , where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980. The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion gallons per year. Find C and k. 11.B-113 [10] You run over a nail. As the air leaks out of your tire, the rate of change of air pressure inside the tire is directly proportional to that pressure. (a) Write a differential equation that states this fact. Evaluate the proportionality constant if the pressure was 35 psi and decreasing at 0.28 psi/min at time zero. (b) Solve the differential equation subject to the initial condition implied in step (a). (c) Sketch the graph of the function. Show its behavior a long time after the tire is punctured. (d) What will be the pressure at 10 min after the tire was punctured? (e) The car is safe to drive as long as the tire pressure is 12 psi or greater. For how long after the puncture will the car be safe to drive? 11.B-114 Be on the lookout for a quiz on moodle due Monday, March 22. (Hereafter, moodle quizzes will be due on Sunday evening, but I give you some leeway because of spring break.)

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AP Unit 11, Day 3: Related Rates

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Related Rates

Example 11.3.15 (2002 AB-5) [NO CALCULATOR]

A container has the shape of an open right circular cone, as shown in the figure above. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evapo3 rating so that its depth h is changing at the constant rate of â&#x2C6;&#x2019; 10 cm/hr. (Note: the volume of a cone of height h and radius r is given by 1 V = Ď&#x20AC;r2 h.) 3 (a) Find the volume V of water in the container when h = 5 cm. Indicate units of measure. (b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate units of measure. (c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

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AP Unit 11 (Review) Example 11.3.16 (1987 AB-5) [NO CALCULATOR]

The trough shown in the figure above is 5 feet long, and its vertical cross sections are inverted isosceles triangles with base 2 feet and height 3 feet. Water is being siphoned out of the trough at the rate of 2 cubic feet per minute. At any time t, let h be the depth and V be the volume of water in the trough. (a) Find the volume of water in the trough when it is full. (b) What is the rate of change in h at the instant when the trough 1 is full by volume? 4 (c) What is the rate of change in the area of the surface of the water (shaded in the figure) at the instant when the trough is 1 full by volume? 4

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Example 11.3.17 (2002B AB-6) [NO CALCULATOR]

Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 10 km/hr. Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in the figure above. (a) Find the distance, in kilometers, between Ship A and Ship B when x = 4 km and y = 3 km. (b) Find the rate of change, in km/hr, of the distance between the two ships when x = 4 km and y = 3 km. (c) Let Î¸ be the angle shown in the figure. Find the rate of change of Î¸, in radians per hour, when x = 4 km and y = 3 km.

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AP Unit 11 (Review) Example 11.3.18 (1985 AB-5) [NO CALCULATOR]

The balloon shown above is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 261π cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the balloon is 144π cubic centimeters and the radius of the cylinder is increasing at the rate of 2 centimeters per minute. (The volume of a cylinder with radius r and height h is πr2 h, and the volume of 4 a sphere with radius r is πr3 .) 3 (a) At this instant, what is the height of the cylinder? (b) At this instant, how fast is the height of the cylinder increasing?

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Homework 11.C-115 (1988 BC-3) [NO CALCULATOR]

The figure above represents an observer at point A watching balloon B as it rises from point C. The balloon is rising at a constant rate of 3 meters per second and the observer is 100 meters from point C. (a) Find the rate of change in x at the instant when y = 50. (b) Find the rate of change in the area of right triangle BCA at the instant when y = 50. (c) Find the rate of change in θ at the instant when y = 50. 11.C-116 (1995 AB-5)

As shown in the figure above, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base with area 400π square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of (h − 12) feet per minute. (The 1 volume V of a cone with radius r and height h is V = πr2 h.) 3 (a) Write an expression for the volume of water in the conical tank as a function of h. (b) At what rate is the volume of water in the conical tank changing when h = 3? Indicate units of measure. (c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h = 3? Indicate units of measure.

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11.C-117 (1992 AB-6) [NO CALCULATOR] At time t, t ≥ 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 and at t = 15, the radius is 2. (The volume 4 V of a sphere with radius r is V = πr3 .) 3 (a) Find the radius of the sphere as a function of t. (b) At what time t will the volume of the sphere be 27 times its volume at t = 0? 1 2 11.C-118 (1984 AB-5) [NO CALCULATOR] The volume of a cone V = πr h 3 is increasing at the rate of 28π cubic units per second. At the instant when the radius r of the cone is 3 units, its volume is 12π cubic units and 1 the radius is increasing at unit per second. 2 (a) At the instant when the radius of the cone is 3 units, what is the rate of change of the area of its base? (b) At the instant when the radius of the cone is 3 units, what is the rate of change of its height h? (c) At the instant when the radius of the cone is 3 units, what is the instantaneous rate of change of the area of its base with respect to its height h?

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AP Unit 11, Day 4: Graphs

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Graphs

Example 11.4.21 (2004 AB-5) [NO CALCULATOR]

The graph of the function f shown above consists of a semicircle and three line segments. Let g be the function given by g(x) = Z x

f (t) dt. −3

(a) Find g(0) and g 0 (0). (b) Addendum: Find g 00 (0) (c) Find all values of x in the open interval (−5, 4) at which g attains a relative maximum. Justify your answer. (d) Find the absolute minimum value of g on the closed interval [−5, 4]. Justify your answer. (e) Find all values of x in the open interval (−5, 4) at which the graph of g has a point of inflection. (f) Addendum: Find the average rate of change of g on the interval −3 ≤ x ≤ 0; on the interval −5 ≤ x ≤ 0. (g) Addendum: For how many values c, where −3 < c < 0, if any, is g 0 (c) equal to the average rate of change of g on the interval −3 ≤ x < 0? (h) Addendum: For how many values c, where −5 < c < 0, if any, is g 0 (c) equal to the average rate of change of g on the interval −5 ≤ x < 0? (i) Addendum: For how many values c, where −5 < c < 0, if any, is g 0 (c) equal to the average rate of change of g on the interval −5 ≤ x < 0? (j) Addendum: Write an equation of the line tangent to the graph of g at x = −3. Write an equation of the line tangent to the graph of g at x = 0. Do the tangent lines lie above or below the graph of g?

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AP Unit 11 (Review) Example 11.4.22 (1999 AB-5)

The graph of the function Z xf , consisting of three line segments, is given above. Let g(x) = f (t) dt. 1

(a) Compute g(4) and g(â&#x2C6;&#x2019;2). (b) Find the instantaneous rate of change of g, with respect to x, at x = 1. (c) Find the absolute minimum value of g on the closed interval [â&#x2C6;&#x2019;2, 4]. Justify your answer. (d) The second derivative of g is not defined at x = 1 and x = 2. How many of these values are x-coordinates of points of inflection of the graph of g? Justify your answer.

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Example 11.4.23 (2005 AB-4) [NO CALCULATOR] x f (x) f 0 (x) f 00 (x)

0 −1 4 −2

0<x<1 Negative Positive Negative

1 0 0 0

1<x<2 Positive Positive Positive

2 2 DNE DNE

2<x<3 Positive Negative Negative

3 0 −3 0

3<x<4 Negative Negative Positive

Let f be a function that is continuous on the interval [0, 4). The function f is twice differentiable except at x = 2. The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at x = 2. (a) For 0 < x < 4, find all values of x at which f has a relative extremum. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer. (b) On the axes provided, sketch the graph of a function that has all the characteristics of f .

Z (c) Let g be the function defined by g(x) =

f (t) dt on the open 1

interval (0, 4). For 0 < x < 4, find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer. (d) For the function g defined in part (c), find all values of x, for 0 < x < 4, at which the graph of g has a point of inflection. Justify your answer.

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AP Unit 11 (Review) Example 11.4.24 (2009 AB–6) [NO CALCULATOR]

( g(x) for − 4 ≤ x ≤ 0 The derivative of a function f is defined by f (x) = 5e−x/3 − 3 for 0 < x ≤ 4 The graph of the continuous function f 0 , shown in the figure above, 5 . The graph of g on has x–intercepts at x = −2 and x = 3 ln 3 −4 ≤ x ≤ 0 is a semicircle, and f (0) = 5. 0

(a) For −4 < x < 4, find all values of x at which the graph of f has a point of inflection. Justify your answer. (b) Find f (−4) and f (4). (c) For −4 ≤ x ≤ 4, find the value of x at which f has an absolute maximum. Justify your answer.

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Homework 11.D-121 [NO CALCULATOR]

Let f be a function defined on the closed interval −3 ≤ x ≤ 4 with f (0) = 3. The graph of f 0 , the derivative of f , consists of one line segment and a semicircle, as shown above. (a) On what intervals, if any, is f increasing? Justify your answer. (b) Find the x-coordinate of each point of inflection of the graph of f on the open interval −3 < x < 4. Justify your answer. (c) Find an equation for the line tangent to the graph of f at the point (0, 3). (d) Find f (−3) and f (4). Show the work that leads to your answers.

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AP Unit 11 (Review)

11.D-122 [NO CALCULATOR]

The graph of the function f shown above Z x consists of two line segments. Let g be the function given by g(x) = f (t) dt. 0 0

(a) Find g(−1), g (−1), and g (−1). (b) For what values of x in the open interval (−2, 2) is g increasing? Explain your reasoning. (c) For what values of x in the open interval (−2, 2) is the graph of g concave down? Explain your reasoning. (d) On the axes provided, sketch the graph of g on the closed interval [−2, 2]. 11.D-123 [NO CALCULATOR]

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The graph of the function f above consists of three line segments. Z x (a) Let g be the function given by g(x) = f (t) dt. For each of g(−1), −4

g 0 (−1), and g 00 (−1), find the value or state that it does not exist. (b) For the function g defined in part (a), find the x-coordinate of each point of inflection of the graph of g on the open interval −4 < x < 3. Explain your reasoning. Z 3 (c) Let h be the function given by h(x) = f (t) dt. Find all values of x

x in the closed interval −4 ≤ x ≤ 3 for which h(x) = 0. (d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning. 11.D-124 [NO CALCULATOR]

Let f be a function defined on the closed interval [0, 7]. The graph of f , consisting of fourZ line segments, is shown above. Let g be the function x

given by g(x) =

f (t) dt. 2

(a) Find g(3), g 0 (3), and g 00 (3). (b) Find the average rate of change of g on the interval 0 ≤ x ≤ 3. (c) For how many values c, where 0 < c < 3, is g 0 (c) equal to the average rate found in part (b)? Explain your reasoning. (d) Find the x-coordinate of each point of inflection of the graph of g on the interval 0 < x < 7. Justify your answer.

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AP Unit 11 (Review)

Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 5: Integral as Accumulator

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Integral as Accumulator

Example 11.5.25 (2007 AB–2)

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 ≤ t ≤ 7, where t is measured in hours. In this model, rates are given as follows: √ (i) The rate at which water enters the tank is f (t) = 100t2 sin t gallons per hour for 0 ≤ t ≤ 7. (ii) The rate at which water leaves the tank is ( 250 for 0 ≤ t < 3 g(t) = gallons per hour 2000 for 2 < t ≤ 7 The graphs of f and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure above At time t = 0, the amount of water in the tank is 5000 ga1lons. (a) How many gallons of water enter the tank during the time interval 0 ≤ t ≤ 7? Round your answer to the nearest gallon. (b) For 0 ≤ t ≤ 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer. (c) For 0 ≤ t ≤ 7, at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer. (d) Addendum: For how many times t, 0 ≤ t ≤ 7, is the instantaneous rate at which water enters the tank equal to the average rate at which water enters the tank over the time interval 0 ≤ t ≤ 7? Mr. Budd, compiled September 29, 2010

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AP Unit 11 (Review) (e) Addendum: For how many times t, 0 ≤ t ≤ 7, is the instantaneous rate at which water leaves the tank equal to the average rate at which water leaves the tank over the time interval 0 ≤ t ≤ 7? (f) Addendum: What is the average rate of change of the rate at which the water leaves the tank over the time interval 0 ≤ t ≤ 7?

Example 11.5.26 (2002 AB-2) The rate at which people enter an amusement park on a given day is modeled by the function E defined by 15600 E(t) = 2 (t − 24t + 160) The rate at which people leave the same amusement park on the same day is modeled by the function L defined by L(t) =

9890 (t2 − 38t + 370)

Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 ≤ t ≤ 23, the hours during which the park is open. At time t = 9, there are no people in the park. (a) How many people have entered the park by 5:00 p.m. (t = 17)? Round your answer to the nearest whole number. (b) The price of admission to the park is $15 until 5:00 p.m. (t = 17). After 5:00 p.m., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. Z t (c) Let H(t) = (E(x) − L(x)) dx for 9 ≤ t ≤ 23. The value of 9

H(17) to the nearest whole number is 3725. Find the value of H 0 (17), and explain the meaning of H(17) and H 0 (17) in the context of the amusement park. (d) At what time t, for 9 ≤ t ≤ 23, does the model predict that the number of people in the park is a maximum?

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Example 11.5.27 (2008P AB–1) The rate at which raw sewage πt2 enters a treatment tank is given by E(t) = 850 + 715 cos gal9 lons per hour for 0 ≤ t ≤ 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per hour. The treatment tank is empty at time t = 0. (a) How many gallons of sewage enter the treatment tank during the time interval 0 ≤ t ≤ 4? Round your answer to the nearest gallon. (b) For 0 ≤ t ≤ 4, at what time t is the amount of sewage in the treatment tank greatest? To the nearest gallon, what is the maximum amount of sewage in the tank? Justify your answers. (c) For 0 ≤ t ≤ 4, the cost of treating the raw sewage that enters the tank at time t is (0.15 − 0.02t) dollars per gallon. To the nearest dollar, what is the total cost of treating all the sewage that enters the tank during the time interval 0 ≤ t ≤ 4?

Example 11.5.28 (2005B AB-2) A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate √ t W (t) = 95 t sin2 gallons per hour. 6 During the same time interval, water is removed from the tank at the rate t 2 R(t) = 275 sin gallons per hour. 3 (a) Is the amount of water in the tank increasing at time t = 15? Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time t = 18? (c) At what time t, for 0 ≤ t ≤ 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion. (d) For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(t) until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value for k.

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AP Unit 11 (Review)

Homework 11.E-125 A tank contains 125 gallons of heating oil at time t = 0. During the time interval 0 ≤ t ≤ 12 hours, heating oil is pumped into the tank at the rate H(t) = 2 +

10 gallons per hour. (1 + ln (t + 1))

During the same time interval, heating oil is removed from the tank at the rate 2 t R(t) = 12 sin gallons per hour. 47 (a) How many gallons of heating oil are pumped into the tank during the time interval 0 ≤ t ≤ 12 hours? (b) Is the level of heating oil in the tank rising or falling at time t = 6 hours? Give a reason for your answer. (c) How many gallons of heating oil are in the tank at time t = 12 hours? (d) At what time t, for 0 ≤ t ≤ 12, is the volume of heating oil in the tank the least? Show the analysis that leads to your conclusion. 11.E-126 For 0 ≤ t ≤ 31, the rate of change of the number√of mosquitoes on Tropical Island at time t days is modeled by R(t) = 5 t cos 5t mosquitoes per day. There are 1000 mosqitoes on Tropical Island at time t = 0. (a) Show that the number of mosquitoes is increasing at time t = 6. (b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your anwer. (c) According to the model, how many mosquitoes will be on the island at time t = 31? (d) To the nearest whole number, what is the maximum number of mosquitoes for 0 ≤ t ≤ 31? Show the analysis that leads to your conclusion.

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11.E-127 The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4πt R(t) = 2 + 5 sin . 25 A pumping station adds sand to the beach at a rate modeled by the function S, given by 15t S(t) = . 1 + 3t Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for 0 ≤ t ≤ 6. At time t = 0, the beach contains 2500 cubic yards of sand. (a) How much sand will the tide remove from the beach during this 6– hour period? Indicate units of measure. (b) Write an expression for Y (t), the total number of cubic yards of sand on the beach at time t. (c) Find the rate at which the total amount of sand on the beach is changing at time t = 4. (d) For 0 ≤ t ≤ 6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answer. 11.E-128 [NO CALCULATOR] Water is pumped into an underground tank at a constant √ rate of 8 gallons per minute. Water leaks out of the tank at the rate of t + 1 gallons per minute, for 0 ≤ t ≤ 120 minutes. At time t = 0, the tank contains 30 gallons of water. (a) How many gallons of water leak out of the tank from time t = 0 to t = 3 minutes? (b) How many gallons of water are in the tank at time t = 3 minutes? (c) Write an expression for A(t), the total number of gallons of water in the tank at time t. (d) At what time t, for 0 ≤ t ≤ 120, is the amount of water in the tank a maximum? Justify your answer.

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Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 6: Particle Motion

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Particle Motion

Example 11.6.29 (2003 AB-2) A particle moves along the x-axis so that its velocity at time t is given by 2 t v(t) = − (t + 1) sin 2 At time t = 0, the particle is at position x = 1. (a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not? (b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time t = 0 until time t = 3. (d) During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer. (e) Addendum: For what values of t is the particle moving to the right? Justify your answer. (f) Addendum: Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the origin at time t = 2? Justify your answer.

Example 11.6.30 (2004 AB-3) A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 − tan−1 (et ). At time t = 0, the particle is at y = −1. (Note: tan−1 x = arctan x) (a) Find the acceleration of the particle at time t = 2. (b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your answer. (c) Find the time t ≥ 0 at which the particle reaches its highest point. Justify your answer. (d) Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the origin at time t = 2? Justify your answer.

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AP Unit 11 (Review) Example 11.6.31 (2006 AB-4) [NO CALCULATOR] t (seconds) v(t) (feet per second)

Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 80 seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of measure. Z 70 (b) Using correct units, explain the meaning of v(t) dt in terms 10

of the rocket’s flight. Use a midpoint Riemann Z 70 sum with 3 subintervals of equal length to approximate v(t) dt. 10

(c) Rocket B is launched upward with an acceleration of a(t) = 3 √ feet per second per second. At time t = 0 seconds, the t+1 initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your answer. Example 11.6.32 (2007B AB–2)

A particle moves along the x–axis so that its velocity v at time t ≥ 0 is given by v(t) = sin t2 . The graph of v is shown above for √ 0 ≤ t ≤ 5π. The position of the particle at time t is x(t) and its position at time t = 0 is x(0) = 5. Mr. Budd, compiled September 29, 2010

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(a) Find the acceleration of the particle at time t = 3. (b) Find the total distance traveled by the particle from time t = 0 to t = 3. (c) Find the position of the particle at time t = 3. √ (d) For 0 ≤ t ≤ 5π, find the time t at which the particle is farthest to the right. Explain your answer.

Homework 11.F-129 [NO CALCULATOR]

A particle moves along the x–axis so that its velocity at time t, for 0 ≤ t ≤ 6, is given by a differentiable function v whose graph is shown above. The velocity is 0 at t = 0, t = 3, and t = 5, and the graph has horizontal tangents at t = 1 and t = 4. The areas of the regions bounded by the t–axis and the graph of v on the intervals [0, 3], [3, 5], and [5, 6] are 8, 3, and 2, respectively. At time t = 0, the particle is at x = −2. (a) For 0 ≤ t ≤ 6, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer. (b) For how many values of t, where 0 ≤ t ≤ 6, is the particle at x = −8? Explain your reasoning. (c) On the interval 2 < t < 3, is the speed of the particle increasing or decreasing? Give a reason for your answer. (d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer. 11.F-130 [NO CALCULATOR] Mr. Budd, compiled September 29, 2010

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AP Unit 11 (Review) t (seconds) v(t) (meters per second)

−10

−8

−4

−7

The velocity of a particle moving along the x–axis is modeled by a differentiable function v, where the position x is measured in meters, and time t is measured in seconds. Selected values of v(t) are given in the table above. The particle is at position x = 7 meters when t = 0 seconds. (a) Estimate the acceleration of the particle at t = 36 seconds. Show the computations that lead to your answer. Indicate units of measure. Z 40 (b) Using correct units, explain the meaning of v(t) dt in the context 20

of this problem. Use a trapezoidal sum with the three Z 40 subintervals indicated by the data in the table to approximate v(t) dt 20

(c) For 0 ≤ t ≤ 40, must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not. (d) Suppose that the acceleration of the particle is positive for 0 < t < 8 seconds. Explain why the position of the particle at t = 8 seconds must be greater than x = 30 meters. 11.F-131 Caren rides her bicycle along a straight road from home to school, starting at home at time t = 0 minutes and arriving at school at time t = 12 minutes. During the time interval 0 ≤ t ≤ 12 minutes, her velocity v(t), in miles per minute, is modeled by the piecewise-linear function whose graph is shown below.

(a) Find the acceleration of Carens bicycle at time t = 7.5 minutes. Indicate units of measure. Z (b) Using correct units, explain the meaning of |v(t)| dt in terms of Z Carens trip. Find the value of |v(t)| dt. Mr. Budd, compiled September 29, 2010

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(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. (d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function w given by π π sin t , where w(t) is in miles per minute for 0 ≤ t ≤ w(t) = 15 12 12 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. 11.F-132 An object moves along the x-axis with initial position x(0) = 2. The π velocity of the object at time t ≥ 0 is given by v(t) = sin t . 3 (a) What is the acceleration of the object at time t = 4? (b) Consider the following two statements. Statement I: For 3 < t < 4.5, the velocity of the object is decreasing. Statement II: For 3 < t < 4.5, the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct. (c) What is the total distance traveled by the object over the time interval 0 ≤ t ≤ 4? (d) What is the position of the object at time t = 4?

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Example 11.7.33 (2007 AB–3) x 1 2 3 4

f 0 (x) 4 2 −4 3

f (x) 6 9 10 −1

g 0 (x) 5 1 2 7

g(x) 2 3 4 6

The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by h(x) = f (g(x)) − 6. (a) Explain why there must be a value r for 1 < r < 3 such that h(r) = −5. (b) Explain why there must be a value c for 1 < c < 3 such that h0 (c) = −5. Z g(x) (c) Let w be the function given by w(x) = f (t) dt. Find the 1

value of w0 (3).

(d) Addendum: Use either a trapezoidal approximation or a midpoint Riemann sum with three subdivisions of equal length to approximate w(3). (e) Addendum: Estimate w00 (3). Z

(f) Addendum: Find the exact value of

(3f 0 (x) − 2g 0 (x) + 3) dx.

(g) If g −1 is the inverse function of g, write an equation for the line tangent to the graph of y = g −1 (x) at x = 2. Example 11.7.34 (2009 AB–5) [NO CALCULATOR] x f (x)

2 1

3 4

5 −2

8 3

13 6

Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval 2 ≤ x ≤ 13. (a) Estimate f 0 (4). Show the work that leads to your answer. Z 13 (b) Evaluate (3 − 5f 0 (x)) dx. Show the work that leads to 2 your answer. Mr. Budd, compiled September 29, 2010

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AP Unit 11 (Review) (c) Use a left Riemann sum with subintervals indicated by the data Z 13 in the table to approximate f (x) dx. Show the work that 2

leads to your answer.

(d) Suppose f 0 (5) = 3 and f 00 (x) < 0 for all x in the closed interval 5 ≤ x ≤ 8. Use the line tangent to the graph of f at x = 5 to show that f (7) ≤ 4. Use the secant line for the graph of f on 4 5 ≤ x ≤ 8 to show that f (7) ≥ . 3 Example 11.7.35 (2003B AB–3) Distance x (mm)

120

180

240

300

360

Diameter B(x) (mm)

A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table above gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B(x) is a twice-differentiable function that represents the diameter at that point. (a) Write an integral expression in terms of B(x) that represents the average radius, in mm, of the blood vessel between x = 0 and x = 360. (b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer. 2 Z 275 B(x) (c) Using correct units, explain the meaning of π dx 2 125 in terms of the blood vessel. (d) Explain why there must be at least one value, x, for 0 < x < 360, such that B 00 (x) = 0.

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Example 11.7.36 (2005 AB–3) Distance x (cm) Temperature T (x) (◦ C)

100

A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T (x), in degrees Celsius (◦ C), of the wire x cm from the heated end. The function T is decreasing and twice differentiable. (a) Estimate T 0 (7). Show the work that leads to your answer. Indicate units of measure. (b) Write an integral expression in terms of T (x) for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure. Z 8 (c) Find T 0 (x) dx, and indicate units of measure. Explain the 0 Z 8 meaning of T 0 (x) dx in terms of the temperature of the wire.

(d) Are the data in the table consistent with the assertion that T 00 (x) > 0 for every x in the interval 0 < x < 8? Explain your answer.

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Homework 11.G-133 The graph of the velocity v(t), in ft/sec, of a car traveling on a straight road, for 0 ≤ t ≤ 50, is shown below. A table of values for v(t), at 5 second intervals of time t, is shown to the right of the graph.

(a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. (b) Find the average acceleration of the car, in ft/sec2 , over the interval 0 ≤ t ≤ 50. (c) Find one approximation for the acceleration of the car, in ft/sec2 , at t = 40. Show the computations you used to arrive at your answer. Z 50 (d) Approximate v(t) dt with a Riemann sum, using midpoints of 0

five subintervals of equal length. Using correct units, explain the meaning of this integral. 11.G-134 [NO CALCULATOR] t (sec) v(t) (ft/sec) a(t) (ft/sec2 )

−20

−30

−20

−14

−10

A car travels on a straight track. During the time interval 0 ≤ t ≤ 60 seconds, the car’s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. The table above shows selected values of these functions. Mr. Budd, compiled September 29, 2010

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Z 60 |v(t)| dt in terms (a) Using appropriate units, explain the meaning of 30 Z 60 |v(t)| dt using a trapezoidal of the car’s motion. Approximate 30

approximation with the three subintervals determined by the table. Z 30 a(t) dt in terms (b) Using appropriate units, explain the meaning of Z 30 0 a(t) dt. of the car’s motion. Find the exact value of 0

(c) For 0 < t < 60, must there be a time t when v(t) = −5? Justify your answer. (d) For 0 < t < 60, must there be a time t when a(t) = 0? Justify your answer. 11.G-135 Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 ≤ t ≤ 9. Values of L(t) at various times t are shown in the table below. t (hours) L(t) (people)

0 120

1 156

3 176

4 126

7 150

8 80

9 0

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. (t = 5.5). Show the computations that lead to your answer. Indicate units of measure. (b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale. (c) For 0 ≤ t ≤ 9, what is the fewest number of times at which L0 (t) must equal 0 ? Give a reason for your answer. (d) The rate at which tickets were sold for 0 ≤ t ≤ 9 is modeled by r(t) = 550te−t/2 tickets per hour. Based on the model, how many tickets were sold by 3 P.M. (t = 3), to the nearest whole number? 11.G-136 [NO CALCULATOR] Let f be a twice-differentiab1e function such that f (2) = 5 and f (5) = 2. Let g be the function given by g(x) = f (f (x)). (a) Explain why there must be a value c for 2 < c < 5 such that f 0 (c) = −1. (b) Show that g 0 (2) = g 0 (5). Use this result to explain why there must be a value k for 2 < k < 5 such that g 00 (k) = 0. (c) Show that if f 00 (x) = 0 for all x, then the graph of g does not have a point of inflection. (d) Let h(x) = f (x) − x. Exp1ain why there must be a value r for 2 < r < 5 such that h(r) = 0.

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Extrema and Optimization

Example 11.8.37 (1996 AB–4) This problem deals with functions defined by f (x) = x + b sin x, where b is a positive constant and −2π ≤ x ≤ 2π. (a) Sketch the graphs of two of these functions, y = x + sin x and y = x + 3 sin x, as indicated below. (b) Find the x-coordinate of all points, −2π ≤ x ≤ 2π, where the line y = x + b is tangent to the graph of f (x) = x + b sin x. (c) Are the points of tangency described in part (b) relative maximum points of f ? Why? (d) For all values of b > 0, show that all inflection points of the graph of f lie on the line y = x.

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AP Unit 11 (Review) Example 11.8.38 (2008B–4) [NO CALCULATOR] The functions Z 3x p f and g are given by f (x) = 4 + t2 dt and g(x) = f (sin x). 0 0

(a) Find f (x) and g (x). (b) Write an equation for the line tangent to the graph of y = g(x) at x = π. (c) Write, but do not evaluate, an integral expression that represents the maximum value of g on the interval 0 ≤ x ≤ π. Justify your answer.

Example 11.8.39 (1997 AB–4) Let f be the function given by f (x) = x3 − 6x2 + p, where p is an arbitrary constant. (a) Write an expression for f 0 (x) and use it to find the relative maximum and minimum values of f in terms of p. Show the analysis that leads to your conclusion. (b) For what values of the constant p does f have 3 distinct real roots? (c) Find the value of p such that the average value of f over the closed interval [−1, 2] is 1.

Example 11.8.40 (2008P AB–6) [NO CALCULATOR] Let g(x) = xe−x + be−x , where b is a positive constant. (a) Find lim g(x) x→∞

(b) For what positive value of b does g have an absolute maximum 2 at x = ? Justify your answer. 3 (c) Find all values of b, if any, for which the graph of g has a point of inflection on the interval 0 < x < ∞. Justify your answer.

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Homework 11.H-137 [NO CALCULATOR] A cubic polynomial function f is defined by f (x) = 4x3 + ax2 + bx + k where a, b, and k are constants. The function f has a local minimum at x = −1, and the graph of f has a point of inflection at x = −2. (a) Find the values of a and b. Z 1 f (x) dx = 32, what is the value of k? (b) If 0

11.H-138 Let f be the function given by f (x) = 2xe2x . (a) Find lim f (x) and lim f (x). x→−∞

x→∞

(b) Find the absolute minimum value of f . Justify that your answer is an absolute minimum. (c) What is the range of f ? (d) Consider the family of functions defined by y = bxebx , where b is a nonzero constant. Show that the absolute minimum value of bxebx is the same for all nonzero values of b.

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11.H-139 The figure below shows the graph of f 0 , the derivative of the function f , for −7 ≤ x ≤ 7. The graph of f 0 has horizontal tangent lines at x = −3, x = 2, and x = 5, and a vertical tangent at x = 3.

(a) Find all values of x, for −7 < x < 7, at which f attains a relative minimum. Justify your answer. (b) Find all values of x, for −7 < x < 7, at which f attains a relative maximum. Justify your answer. (c) Find all values of x, for −7 < x < 7, at which f 00 (x) < 0. (d) At what value of x, for −7 ≤ x ≤ 7, does f attain its absolute maximum? Justify your answer. √ 11.H-140 [NO CALCULATOR] Let f be a function defined by f (x) = k x − ln x for x > 0, where k is a positive constant. (a) Find f 0 (x) and f 00 (x). (b) For what value of the constant k does f have a critical point at x = 1? For this value of k, determine whether f has a relative minimum, relative maximum, or neither at x = 1. Justify your answer. (c) For a certain value of the constant k, the graph of f has a point of inflection on the x–axis. Find this value of k.

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AP Unit 11, Day 9: Implicit Differentiation

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Implicit Differentiation

Example 11.9.41 (2005B ABâ&#x20AC;&#x201C;5) [NO CALCULATOR] Consider the curve given by y 2 = 2 + xy. (a) Show that

y dy = . dx 2y â&#x2C6;&#x2019; x

d2 y . dx2 (c) Find all points (x, y) on the curve where the line tangent to the 1 curve has slope . 2 (d) Addendum: At these points, do the lines tangent to the graph lie above or below the graph? (b) Addendum: Find

(e) Show that there are no points (x, y) on the curve where the line tangent to the curve is horizontal. (f) Addendum: Show that there are no points (x, y) on the curve where the line tangent to the curve is vertical. (g) Let x and y be functions of time t that are related by the equation y 2 = 2 + xy. At time t = 5, the value of y is 3 and dy dx = 6. Find the value of at time t = 5. dt dt

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AP Unit 11 (Review) Example 11.9.42 (1995 AB–3) Consider the curve defined by −8x2 + 5xy + y 3 = −149. dy . dx (b) Write an equation for the line tangent to the curve at the point (4, −1). (a) Find

(c) There is a number k so that the point (4.2, k) is on the curve. Using the tangent line found in part (b), approximate the value of k. (d) Write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the curve. (e) Solve the equation in part (d) for the value of k. Example 11.9.43 (1998 AB–6) Consider the curve defined by 2y 3 + 6x2 y − 12x2 + 6y = 1. dy 4x − 2xy = 2 . dx x + y2 + 1 (b) Write an equation of each horizontal tangent line to the curve. (a) Show that

(c) The line through the origin with slope −1 is tangent to the curve at point P . Find the x- and y-coordinates of point P .

Example 11.9.44 (1992 AB–4) [NO CALCULATOR] Consider the curve defined by the equation y + cos y = x + 1 for 0 ≤ y ≤ 2π. dy in terms of y. dx (b) Write an equation for each vertical tangent to the curve. (a) Find

d2 y in terms of y. dx2

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Homework 11.I-141 [NO CALCULATOR] Consider the curve defined by x2 + xy + y 2 = 27. (a) Write an expression for the slope of the curve at any point (x, y). (b) Determine whether the lines tangent to the curve at the x–intercepts of the curve are parallel. Show the analysis that leads to your conclusion. (c) Find the points on the curve where the lines tangent to the curve are vertical. 11.I-142 [NO CALCULATOR] Consider the curve given by xy 2 − x3 y = 6. 3x2 y − y 2 dy = . dx 2xy − x3 (b) Find all points on the curve whose x–coordinate is 1, and write an equation for the tangent line at each of these points. (a) Show that

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11.I-143 [NO CALCULATOR] Consider the curve given by x2 + 4y 2 = 7 + 3xy. dy 3y − 2x = . dx 8y − 3x (b) Show that there is a point P with x–coordinate 3 at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P . (a) Show that

d2 y at the point P found in part (b). Does the (c) Find the value of dx2 curve have a local maximum, a local minimum, or neither at the point P ? Justify your answer. 11.I-144 [NO CALCULATOR] Consider the closed curve in the xy-plane given by x2 + 2x + y 4 + 4y = 5. (a) Show that

dy − (x + 1) = . dx 2 (y 3 + 1)

(b) Write an equation for the line tangent to the curve at the point (−2, 1). (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the x–axis? Explain your reasoning.

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AP Unit 11, Day 10: Differential Equations Again

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Differential Equations Again

Example 11.10.45 (2003 AB–5) [NO CALCULATOR]

A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. √ The volume V of coffee in the pot is changing at the rate of −5π h cubic inches per second. (The volume V of a cylinder with radius r and height h is V = πr2 h.) √ dh h (a) Show that =− . dt 5 (b) Given that √ h = 17 at time t = 0, solve the differential equation dh h =− for h as a function of t. dt 5 (c) At what time t is the coffeepot empty?

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AP Unit 11 (Review) Example 11.10.46 (2005B AB-6) [NO CALCULATOR] Consider dy −xy 2 the differential equation = . Let y = f (x) be the particdx 2 ular solution to this differential equation with the initial condition f (−1) = 2. (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(b) Write an equation for the line tangent to the graph of f at x = −1. (c) Find the solution y = f (x) to the given differential equation with the initial condition f (−1) = 2. Example 11.10.47 (2004 AB-6) [NO CALCULATOR] Consider dy the differential equation = x2 (y − 1). dx (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

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(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are positive. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = 3.

Example 11.10.48 (1993 AB-6) [NO CALCULATOR] Let P (t) represent the number of wolves in a population at time t years, when t ≥ 0. The population P (t) is increasing at a rate directly proportional to 800 − P (t), where the constant of proportionality is k. (a) If P (0) = 500, find P (t) in terms of t and k. (b) If P (2) = 700, find k. (c) Find lim P (t). t→∞

Homework 11.J-145 [NO CALCULATOR] Consider the differential equation

1 dy = x + y − 1. dx 2

(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

d2 y (b) Find , in terms of x and y. Describe the region in the xy–plane dx2 in which all solution curves to the differential equation are concave up. (c) Let y = f (x) be a particular solution to the differential equation with the initial condition f (0) = 1. Does f have a relative minimum, a relative maximum, or neither at x = 0? Justify your answer. (d) Find the values of the constants m and b, for which y = mx + b is a solution to the differential equation. Mr. Budd, compiled September 29, 2010

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11.J-146 [NO CALCULATOR] Consider the differential equation

dy x = , where dx y

y 6= 0. (a) The slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point (3, −1), and sketch the solution curve that passes through the point (1, 2). (Note: The points (3, −1) and (1, 2) are indicated in the figure.)

(b) Write an equation for the line tangent to the solution curve that passes through the point (1, 2). (c) Find the particular solution y = f (x) to the differential equation with the initial condition f (3) = −1, and state its domain. 11.J-147 [NO CALCULATOR] Consider the differential equation given by 2

dy = dx

x (y − 1) . (a) On the axes provided, sketch a slope field for the given differential equation at the eleven points indicated.

(b) Use the slope field for the given differential equation to explain why a solution could not have the graph shown below. Mr. Budd, compiled September 29, 2010

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(c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = −1. (d) Find the range of the solution found in part (c). 11.J-148 [NO CALCULATOR] Consider the differential equation

dy 2 = (y − 1) cos (πx). dx

(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

(b) There is a horizontal line with equation y = c that satisfies the differential equation. Find the value of c. (c) Find the particular solution y = f (x) to the differential equation with the initial condition f (1) = 0.

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AP Unit 11, Day 11: Related Rates Again

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Related Rates Again

Example 11.11.49 (1996 AB-5)

An oil storage tank has the shape shown above, obtained by revolv9 4 x from x = 0 to x = 5 about the y-axis, ing the curve y = 625 where x and y are measured in feet. Oil flows into the tank at the constant rate of 8 cubic feet per minute. (a) Find the volume of the tank. Indicate units of measure. (b) To the nearest minute, how long would it take to fill the tank if the tank was empty initially? (c) Let h be the depth, in feet, of oil in the tank. How fast is the depth of the oil in the tank increasing when h = 4? Indicate units of measure.

Example 11.11.50 (1999 AB-6)

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AP Unit 11 (Review) 1 In the figure above, line ` is tangent to the graph of y = 2 at point x 1 P , with coordinates w, 2 , where w > 0. Point Q has coordiw nates (w, 0). Line ` crosses the x-axis at point R, with coordinates (k, 0). (a) Edited Write an equation of the line tangent to the graph at 1 . 3, 9 (b) Edited Write of line `, which is tangent to the graph an equation 1 at point P w, 2 , for w > 0. Hence show that, for w > 0, w 3 k = w. 2 (c) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of k with respect to time? (d) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of the area of 4P QR with respect to time? Determine whether the area is increasing or decreasing at this instant. Example 11.11.51 (2008B–2) For time t ≥ 0 hours, let r(t) = −10t2 120 1 − e represent the speed, in kilometers per hour, at which a car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is modeled by g(x) = 0.05x 1 − e−x/2 . (a) How many kilometers does the car travel during the first 2 hours? (b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when t = 2 hours. Indicate units of measure. (c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per hour?

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Homework 11.K-149 At a certain height, a tree trunk has a circular cross section. The radius R(t) of that cross section grows at a rate modeled by the function dR 1 = 3 + sin t2 centimeters per year dt 16 for 0 ≤ t ≤ 3, where time t is measured in years. At time t = 0, the radius is 6 centimeters. The area of the cross section at time t is denoted by A(t). (a) Write an expression, involving an integral, for the radius R(t) for 0 ≤ t ≤ 3. Use your expression to find R(3). (b) Find the rate at which the cross-sectional area A(t) is increasing at time t = 3 years. Indicate units of measure. Z 3 (c) Evaluate A0 (t) dt. Using appropriate units, interpret the meaning 0

of that integral in terms of cross-sectional area. 11.K-150 [NO CALCULATOR] t (minutes) r0 (t) (feet per minute)

5.7

4.0

2.0

1.2

0.6

0.5

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 < t < 12, the graph of r is concave down. The table above gives selected values of the rate of change, r0 (t), of the radius of the ba1loon over the time interval 0 ≤ t ≤ 12. The radius of the balloon is 30 feet when t = 5. 4 (Note: The Volume of a sphere of radius r is given by V = πr3 .) 3 (a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer. (b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. (c) Use a right Riemann sum with the five subintervals indicated by the R 12 data in the table to approximate 0 r0 (t) dt. Using correct units, exR 12 plain the meaning of 0 r0 (t) dt in terms of the radius of the balloon. R 12 (d) Is your approximation in part (c) greater than or less than 0 r0 (t) dt? Give a reason for your answer.

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11.K-151 The wind chill is the temperature, in degrees Fahrenheit (◦ F), a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour (mph). If the air temperature is 32◦ F, then the wind chill is given by w(v) = 55.6 − 22.1v 0.16 and is valid for 5 ≤ v ≤ 60. (a) Find W 0 (20). Using correct units, explain the meaning of W 0 (20) in terms of the wind chill. (b) Find the average rate of change of W over the interval 5 ≤ v ≤ 60. Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 ≤ v ≤ 60. (c) Over the time interval 0 ≤ t ≤ 4 hours, the air temperature is a constant 32◦ F. At time t = 0, the wind velocity is v = 20 mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at t = 3 hours? Indicate units of measure. 11.K-152 Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume V of a right circular cylinder with radius r and height h is given by V = πr2 h.) (a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute? (b) A recovery device arrives on the scene and √ begins removing oil. The rate at which oil is removed is R(t) = 400 t cubic centimeters per minute, where t is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time t when the oil slick reaches its maximum volume. Justify your answer. (c) By the time the recovery device began removing oil, 60, 000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

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AP Unit 11, Day 12: Area and Volume

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Area and Volume

Example 11.12.53 (2006 AB-1)

Let R be the shaded region bounded by the graph of y = ln x and the line y = x − 2, as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = −3. (c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y–axis.

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AP Unit 11 (Review) Example 11.12.54 (1998 AB-1) Let R be the region bounded by √ the x–axis, the graph of y = x, and the line x = 4. (a) Find the area of the region R. (b) Find the value of h such that the vertical line x = h divides the region R into two regions of equal area. (c) Find the volume of the solid generated when R is revolved about the x-axis. (d) The vertical line x = k divides the region R into two regions such that when these two regions are revolved about the x–axis, they generate solids with equal volumes. Find the value of k.

Example 11.12.55 (2001 AB-1)

Let R and S be the regions in the first quadrant shown in the figure above. The region R is bounded by the x–axis and the graphs of y = 2 − x3 and y = tan x. The region S is bounded by the y–axis and the graphs of y = 2 − x3 and y = tan x. (a) Find the area of R. (b) Find the area of S. (c) Find the volume of the solid generated when S is revolved about the x–axis.

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Example 11.12.56 (2004 AB-2)

Let f and√g be the functions given by f (x) = 2x (1 − x) and g(x) = 3 (x − 1) x for 0 ≤ x ≤ 1. The graphs of f and g are shown in the figure above. (a) Find the area of the shaded region enclosed by the graphs of f and g. (b) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved about the horizontal line y = 2. (c) Let h be the function given by h(x) = kx (1 − x) for 0 ≤ x ≤ 1. For each k > 0, the region (not shown) enclosed by the graphs of h and g is the base of a solid with square cross sections perpendicular to the x–axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k.

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Homework 11.L-153 Let R be the region bounded by the graphs of y = sin(πx) and y = x3 −4x, as shown in the figure below.

(a) Find the area of R. (b) The horizontal line y = −2 splits the region R into two parts. Write, but do not evaluate, an integral expression for the area of the part of R that is below this horizontal line. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a square. Find the volume of this solid. (d) The region R models the surface of a small pond. At all points in R at a distance x from the y–axis, the depth of the water is given by h(x) = 3 − x. Find the volume of water in the pond. 11.L-154 [NO CALCULATOR] Let R be the region bounded by the graphs of y = √ x x and y = , as shown in the figure below. 2

(a) Find the area of R. (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Find the volume of this solid. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = 2.

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11.L-155 Let R be the region in the first quadrant bounded by the graphs of y = x and y = . 3

√

(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the vertical line x = −1. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y–axis are squares. Find the volume of this solid. 11.L-156 [NO CALCULATOR] Let R be the region in the first quadrant enclosed by the graphs of y = 2x and y = x2 , as shown in the figure below.

(a) Find the area of R. (b) The region R is the base of a solid. For this solid, at each x the cross π x . Find section perpendicular to the x–axis has area A(x) = sin 2 the volume of the solid. (c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y–axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.

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Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 13: Tangent Lines

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Tangent Lines

Example 11.13.57 (1999 AB-4) Suppose that the function f has a continuous second derivative for all x, and that f (0) = 2, f 0 (0) = −3, and f 00 (0) = 0. Let g be a function whose derivative is given by g 0 (x) = e−2x (3f (x) + 2f 0 (x)) for all x. (a) Write an equation of the line tangent to the graph of f at the point where x = 0. (b) Is there sufficient information to determine whether or not the graph of f has a point of inflection when x = 0? Explain your answer. (c) Given that g(0) = 4, write an equation of the line tangent to the graph of g at the point where x = 0. (d) Show that g 00 (x) = e−2x (−6f (x) − f 0 (x) + 2f 00 (x)). Does g have a local maximum at x = 0? Justify your answer.

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AP Unit 11 (Review) Example 11.13.58 (2002B AB–2) The number of gallons, P (t), of √ a pollutant in a lake changes at the rate P 0 (t) = 1 − 3e−0.2 t gallons per day, where t is measured in days. There are 50 gallons of the pollutant in the lake at time t = 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant. (a) Is the amount of pollutant increasing at time t = 9? Why or why not? (b) For what value of t will the number of gallons of pollutant be at its minimum? Justify your answer. (c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer. (d) An investigator uses the tangent line approximation to P (t) at t = 0 as a model for the amount of pollutant in the lake. At what time t does this model predict that the lake becomes safe?

Example 11.13.59 (1996 AB-6)

x2 Line ` is tangent to the graph of y = x − at the point Q, as 500 shown in the figure above. (a) Find the x-coordinate of point Q. (b) Write an equation for line `. x2 (c) Suppose the graph of y = x − shown in the figure, where x 500 and y are measured in feet, represents a hill. There is a 50-foot tree growing vertically at the top of the hill. Does a spotlight at point P directed along line ` shine on any part of the tree? Show the work that leads to your conclusion.

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Example 11.13.60 (2001 AB–4) [NO CALCULATOR] Let h be a function defined for all x 6= 0 such that h(4) = −3 and the derivative x2 − 2 of h is given by h0 (x) = for all x 6= 0. x (a) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of h at x = 4. (d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4 Why?

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Homework 11.M-157 (Wed, 21-Apr, front) [NO CALCULATOR] Let f be the function given by ln x 1 − ln x . f (x) = for all x > 0. The derivative of f is given by f 0 (x) = x x2 (a) Write an equation for the line tangent to the graph of f at x = e2 . (b) Find the x–coordinate of the critical point of f . Determine whether this point is a relative minimum, a relative maximum, or neither for the function f . Justify your answer. (c) The graph of the function f has exactly one point of inflection. Find the x–coordinate of this point. (d) Find lim+ f (x). x→0

11.M-158 (Wed, 21-Apr, back) [NO CALCULATOR] Let f be the function given by f (x) = (ln x) (sin x). The figure below shows Z the graph of f for 0 < x

x ≤ 2π. The function g is defined by g(x) =

f (t) dt for 0 < x ≤ 2π. 1

(a) Find g(1) and g 0 (1). (b) On what intervals, if any, is g increasing? Justify your answer. (c) For 0 < x ≤ 2π, find the value of x at which g has an absolute minimum. Justify your answer. (d) For 0 < x ≤ 2π, is there a value of x at which the graph of g is tangent to the x–axis? Explain why or why not. 11.M-159 (Thu, 22-Apr) No problems due today. Test 10 is on Thursday, Apr 22. There will not be retests; make-ups will not be taken from classwork and homework. 11.M-160 (Thu, 22-Apr) You may want to get a jump start on your homework that is due Friday, 23-Apr. Mr. Budd, compiled September 29, 2010

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11.M-161 (Fri, 23-Apr, front) [NO CALCULATOR] The figure below shows the graph of f 0 , the derivative of the function f , on the closed interval −1 ≤ x ≤ 5. The graph of f 0 has horizontal tangent lines at x = 1 and x = 3. The function f is twice differentiable with f (2) = 6.

(a) Find the x-coordinate of each of the points of inflection of the graph of f . Give a reason for your answer. (b) At what value of x does f attain its absolute minimum value on the closed interval −1 ≤ x ≤ 5? At what value of x does f attain its absolute maximum value on the closed interval −1 ≤ x ≤ 5? Show the analysis that leads to your answers. (c) Let g be the function defined by g(x) = xf (x). Find an equation for the line tangent to the graph of g at x = 2.

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11.M-162 (Fri, 23-Apr, back) The temperature, in degrees Celsius (◦ C), of the water in a pond is a differentiable function W of time t. The table below shows the water temperature as recorded every 3 days over a 15-day period. t (days) 0 3 6 9 12 15

W (t) (◦ C) 20 31 28 24 22 21

(a) Use data from the table to find an approximation for W 0 (12). Show the computations that lead to your answer. Indicate units of measure. (b) Approximate the average temperature, in degrees Celsius, of the water over the time interval 0 ≤ t ≤ 15 days by using a trapezoidal approximation with subintervals of length 4t = 3 days. (c) A student proposes that function P , given by P (t) = 20 + 10te(−t/3) , as a model for the temperature of the water in the pond at time t, where t is measured in days and P (t) is measured in degrees Celsius. Find P 0 (12). Using appropriate units, explain the meaning of your answers in terms of water temperature. (d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of P (t) over the time interval 0 ≤ t ≤ 15 days. 11.M-163 (Mon, 26-Apr) No problems due today. 11.M-164 (Mon, 26-Apr) Get ready for the big push over the next week.

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AP Unit 11, Day 14: Miscellany

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Miscellany

Example 11.14.65 (2009 AB–2) The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R(t) = 1380t2 − 675t3 for 0 ≤ t ≤ 2 hours; R(t) is measured in people per hour. No one is in the auditorium at time t = 0, when the doors open. The doors close and the concert begins at time t = 2. (a) How many people are in the auditorium when the concert begins? (b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. (c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by w0 (t) = (2 − t) R(t). Find w(2) − w(1), the total wait time for those who enter the auditorium after time t = 1. (d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

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AP Unit 11 (Review) Example 11.14.66 (1998 AB-5) The temperature outside a house during a 24-hour period is given by πt , 0 ≤ t ≤ 24, F (t) = 80 − 10 cos 12 where F (t) is measured in degrees Fahrenheit and t is measured in hours. (a) Sketch the graph of F on the grid below

(b) Find the average temperature, to the nearest degree Fahrenheit, between t = 6 and t = 14. (c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of t was the air conditioner cooling the house? (d) The cost of cooling the house accumulates at the rate of $0.05 per hour for each degree the outside temperature exceed 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24-hour period?

Example 11.14.67 (2009 AB–3) Mighty Cable Company manufactures cable that sells for $120 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is x meters from the beginning of √ the cable is 6 x dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the companys cost of producing the cable.) (a) Find Mightys profit on the sale of a 25–meter cable. Mr. Budd, compiled September 29, 2010

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(b) Using correct units, explain the meaning of

√ 6 x dx in the

context of this problem. (c) Write an expression, involving an integral, that represents Mightys profit on the sale of a cable that is k meters long. (d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer. Example 11.14.68 (2006 AB-2)

At an intersection in Thomasville, Oregon, cars turn left at the rate √ t L(t) = 60 t sin2 cars per hour over the time interval 0 ≤ t ≤ 18 3 hours. The graph of y = L(t) is shown above. (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 ≤ t ≤ 18 hours. (b) Traffic engineers will consider turn restrictions when L(t) ≥ 150 cars per hour. Find all values of t for which L(t) ≥ 150 and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200, 000. In every two–hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

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Homework 11.N-165 (Tue, 27-Apr, front) Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t F (t) = 82 + 4 sin for 0 ≤ t ≤ 30, 2 where F (t) is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection over the 30–minute period? (b) Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer. (c) What is the average value of the traffic flow over the time interval 10 ≤ t ≤ 15? Indicate units of measure. (d) What is the average rate of change of the traffic flow over the time interval 10 ≤ t ≤ 15? Indicate units of measure. 11.N-166 (Tue, 27-Apr, back) A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the√edge of the water was changing during the storm is modeled by f (t) = t + cos t − 3 meters per hour, t hours after the storm began. The edge of the water was 35 meters from the road when the storm began, and the storm lasted 5 hours. The derivative of 1 f (t) is f 0 (t) = √ − sin t. 2 t (a) What was the distance between the road and the edge of the water at the end of the storm? (b) Using correct units, interpret the value f 0 (4) = 1.007 in terms of the distance between the road and the edge of the water. (c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer. (d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of g(p) meters per day, where p is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water.

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11.N-167 (Wed, 28-Apr, front) The figure below is the graph of a function of x, which models the height of a skateboard ramp. The function meets the following requirements.

(i) At x = 0, the value of the function is 0, and the slope of the graph of the function is 0. (ii) At x = 4, the value of the function is 1, and the slope of the graph of the function is 1. (iii) Between x = 0 and x = 4, the function is increasing. (a) Let f (x) = ax2 , where a is a nonzero constant. Show that it is not possible to find a value for a so that f meets requirement (ii) above. x2 , where c is a nonzero constant. Find the value of 16 c so that g meets requirement (ii) above. Show the work that leads to your answer.

(b) Let g(x) = cx3 −

(c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above. xn , where k is a nonzero constant and n is a positive k integer. Find the values of k and n so that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.

(d) Let h(x) =

11.N-168 (Wed, 28-Apr, back) The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes, are shown below. Mr. Budd, compiled September 29, 2010

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(a) Use data from the table to find an approximation for R0 (45). Show the computations that lead to your answer. Indicate units of measure. (b) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R00 (45)? Explain your reasoning. Z 90 (c) Approximate the value of R(t) dt using a left Riemann sum with 0

the five subintervals indicated by the data in Zthe table. Is this nu90 merical approximation less than the value of R(t) dt? Explain 0

your reasoning.

Z (d) For 0 < b ≤ 90 minutes, explain the meaning of

R(t) dt in terms of Z 1 b R(t) dt fuel consumption for the plane. Explain the meaning of b 0 in terms of fuel consumption for the plane. Indicate units of measure in both answers. 0

11.N-169 (Thu, 29-Apr, front) [NO CALCULATOR] Let f be a function that is differentiable for all real numbers. The table below gives the values of f and its derivative f 0 for selected points x in the closed interval −1.5 ≤ x ≤ 1.5. The second derivative of f has the property that f 00 (x) > 0 for −1.5 ≤ x ≤ 1.5. −1.5 −1 −7

x f (x) f 0 (x) Z (a) Evaluate answer.

−1.0 −4 −5

−0.5 −6 −3

0 −7 0

0.5 −6 3

1.0 −4 5

1.5 −1 7

1.5

(3f 0 (x) + 4) dx. Show the work that leads to your

(b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to approximate the value of f (1.2). Is this Mr. Budd, compiled September 29, 2010

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approximation greater than or less than the actual value of f (1.2)? Give a reason for your answer. (c) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and f 00 (c) = r. Give a reason for your answer. ( 2x2 − x − 7 for x < 0 (d) Let g be the function given by g(x) = . The 2x2 + x − 7 for x ≥ 0 graph of g passes through each of the points (x, f (x)) given in the table above. Is it possible that f and g are the same function? Give a reason for your answer. 11.N-170 (Thu, 29-Apr, back) [NO CALCULATOR] The temperature, in degrees Celsius (◦ C), of an oven being heated is modeled by an increasing differentiable function H of time t, where t is measured in minutes. The table below gives the temperature as recorded every 4 minutes over a 16–minute period. t (minutes) H(t) (◦ C)

0 65

4 68

8 73

12 80

16 90

(a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is changing at time t = 10. Show the computations that lead to your answer. Indicate units of measure. (b) Write an integral expression in terms of H for the average temperature of the oven between time t = 0 and time t = 16. Estimate the average temperature of the oven using a left Riemann sum with four subintervals of equal length. Show the computations that lead to your answer. (c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature? Give a reason for your answer. (d) Are the data in the table consistent with or do they contradict the claim that the temperature of the oven is increasing at an increasing rate? Give a reason for your answer. 11.N-171 (Fri, 30-Apr, front) [NO CALCULATOR] The twice–differentiable function f is defined for all real numbers and satisfies the following conditions: f (0) = 2, f 0 (0) = −4, and f 00 (0) = 3. (a) The function g is given by g(x) = eax + f (x) for all real numbers, where a is a constant. Find g 0 (0) and g 00 (0) in terms of a. Show the work that leads to your answers. (b) The function h is given by h(x) = cos (kx) f (x) for all real numbers, where k is a constant. Find h0 (x) and write an equation for the line tangent to the graph of h at x = 0. Mr. Budd, compiled September 29, 2010

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11.N-172 (Fri, 30-Apr, back) The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table below shows the rate as measured every 3 hours for a 24-hour period.

t (hours) 0 3 6 9 12 15 18 21 24

R(t) (gallons per hour) 9.6 10.4 10.8 11.2 11.4 11.3 10.7 10.2 9.6

(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to Z 24 approximate R(t) dt. Using correct units, explain the meaning 0

of your answer in terms of water flow. (b) Is there some time t, 0 < t < 24, such that R0 (t) = 0? Justify your answer. 1 768 + 23t − t2 . 79 Use Q(t) to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

11.N-173 (Mon, 3-May, front) Let f be the function given by f (x) = 4x2 − x3 , and let ` be the line y = 18 − 3x, where ` is tangent to the graph of f . Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f , the line `, and the x-axis, as shown below.

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(a) Show that ` is tangent to the graph of y = f (x) at the point x = 3. (b) Find the area of S. (c) Find the volume of the solid generated when R is revolved about the x-axis. x3 x2 x 11.N-174 (Mon, 3-May, back) Let f be the function given by f (x) = − − + 4 3 2 3 cos x. Let R be the shaded region in the second quadrant bounded by the graph of f , and let S be the shaded region bounded by the graph of f and line `, the line tangent to the graph of f at x = 0, as shown below.

(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = −2. (c) Write, but do not evaluate, an integral expression that can be used to find the area of S. Mr. Budd, compiled September 29, 2010

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11.N-175 (Tue, 4-May, front of the provided sheet) [NO CALCULATOR] Consider dy 2x the differential equation =− . dx y (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(b) Let y = f (x) be the particular solution to the differential equation with the initial condition f (1) = −1. Write an equation for the line tangent to the graph of f at (1, −1) and use it to approximate f (1.1). (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (1) = −1.

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11.N-176 (Tue, 4-May, back of the provided sheet) We will talk more about this as the time approaches, but commit to the following: For Tuesday: (a) Turn in problems 175 and 176 on the provided sheet. Beyond that, restrict yourself to very limited, light reviewing. No cramming the night before (b) Take some time to reflect on how much you’ve learned this year. (c) Get at least eight hours sleep before your exam. This is not a suggestion. It is an assignment, part of your final assignment for this class. Some of you have already heard me say this, but I will repeat it anyway. When neurologists study sleep deprivation, with PET-scans and the like, they notice that one of the quickest things to go is your ability to do mathematics. And the more complicated the math, the faster it goes. So get a good night’s sleep. For Wednesday: (a) Eat a good breakfast. Your brain uses a lot of energy. The ancient Greeks used to think that the center of thinking was the stomach, because they noticed when they sat around thinking a lot, they got hungry. We may know better now, but the point is that your brain needs fuel to keep going, so you need to give it some good fuel that will last you through the end of your exam. Eat a good breakfast, with some protein; I’ll even go so far as to invoke the f-word: fiber. (b) No cramming just before the exam, but do wake your brain up. Go through your flash cards. Relax. The hard part of AP Calculus is the studying and the preparation. By this point, all that is finished. (c) When you get to the test site, please go out of your way to be sincerely nice to everyone: including Mr. Guinther, any other proctors, and your peers. I want people to to out of their way to tell me how nice you were. (d) During the exam, know that there will be some problems that you don’t immediately know how to do. Don’t panic. Do breathe. Don’t stare. Do write something. Jot down formulas you think might be appropriate. A lot of times you may start writing, having no real sense of how you’re going to get to the answer, but once you start writing, it may take only a couple steps to get to the answer. (e) After the exam, take some time to acknowledge what you’ve accomplished. Celebrate in a way that’s not going to interfere with your other upcoming exams.

Mr. Budd, compiled September 29, 2010

404

AP Unit 11 (Review)

Mr. Budd, compiled September 29, 2010

AP Unit 11, Day 15: More Miscellany

11.15

405

More Miscellany

Example 11.15.77 (2008B–3) Distance from the rivers edge (feet) Depth of the water (feet)

A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at √ Picnic Point, in feet per minute, is modeled by v(t) = 16 + 2 sin t + 10 for 0 ≤ t ≤ 120 minutes. (a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer. (b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from t = 0 to t = 120 minutes. πx (c) The scientist proposes the function f , given by f (x) = 8 sin , 24 as a model for the depth of the water, in feet, at Picnic Point x feet from the river’s edge. Find the area of the cross section of the river at Picnic Point based on this model. (d) Recall that the volumetric flow is the product of the crosssectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 2100 cubic feet per minute for a 20–minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval 40 ≤ t ≤ 60 minutes. Does this value indicate that the water must be diverted?

Example 11.15.78 (2006 AB–3) The graph of the function f shown below consists of six line segments. Let g be the function given by Z x

g(x) =

f (t) dt. 0 Mr. Budd, compiled September 29, 2010

406

AP Unit 11 (Review)

(a) Find g(4), g 0 (4), and g 00 (4). (b) Does g have a relative minimum, a relative maximum, or neither at x = 1? Justify your answer. (c) Suppose that f is defined for all real numbers x and is periodic with a period of length 5. The graph above shows two periods of f . Given that g(5) = 2, find g(10) and write an equation for the line tangent to the graph of g at x = 108.

Mr. Budd, compiled September 29, 2010

Unit 12

Makeup

407

408

AP Unit 12 (Makeup)

12.1

MU: Differential Equations

Replaces problems 107-110, 145-148, 175

Homework dy 12.A-107 (2010–6) [NO CALCULATOR] Solutions to the differential equation = dx 2 d y xy 3 also satisfy = y 3 1 + 3x2 y 2 . Let y = f (x) be a particular dx2 dy solution to the differential equation = xy 3 with f (1) = 2. dx (a) Write an equation for the line tangent to the graph of y = f (x) at x = 1. (b) Use the tangent line equation from part (a) to approximate f (1.1). Given that f (x) > 0 for 1 < x < 1.1, is the approximation for f (1.1) greater than or less than f (1.1)? Explain your reasoning. (c) Find the particular solution y = f (x) with initial condition f (1) = 2. dy 12.A-108 (2010B–5) [NO CALCULATOR] Consider the differential equation = dx x+1 y (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and for −1 < x < 1, sketch the solution curve that passes through the point (0, −1).

(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy–plane for which y 6= 0. Describe all dy = −1. points in the xy–plane, y 6= 0, for which dx (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = −2.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 2: MU: Related Rates

12.2

409

MU: Related Rates

Replaces problems 115-118, 149-152

Makeup 12.B-115 (2010B–3) The figure below shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet.

The pool contains 1000 cubic feet of water at time t = 0. During the time interval 0 ≤ t ≤ 12 hours, water is pumped into the pool at the rate P (t) cubic feet per hour. The table below gives values of P (t) for selected values of t. t P (t)

0 0

2 46

4 53

6 57

8 60

10 62

12 63

During the same time interval, water is leaking from the pool at the rate R(t) cubic feet per hour, where R(t) = 25e−0.05t . (Note: The volume V of a cylinder with radius r and height h is given by V = πr2 h.) (a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval 0 ≤ t ≤ 12 hours. Show the computations that lead to your answer. (b) Calculate the total amount of water that leaked out of the pool during the time interval 0 ≤ t ≤ 12 hours. (c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time t = 12 hours. Round your answer to the nearest cubic foot. (d) Find the rate at which the volume of water in the pool is increasing at time t = 8 hours. How fast is the water level in the pool rising at t = 8 hours? Indicate units of measure in both answers.

Mr. Budd, compiled September 29, 2010

410

AP Unit 12 (Makeup)

12.3

MU: Graphs

Replaces problems 121-124, 158, 160

Makeup 12.C-121 (2010–5) [NO CALCULATOR] The function g is defined and differentiable on the closed interval [−7, 5] and satisfies g(0) = 5. The graph of y = g 0 (x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure below.

(a) Find g(3) and g(−2). (b) Find the x–coordinate of each point of inflection of the graph of y = g(x) on the interval −7 < x < 5. Explain your reasoning. 1 (c) The function h is defined by h(x) = g(x) − x2 . Find the x– 2 coordinate of each critical point of h, where −7 < x < 5, and classify each critical point as the location of a relative minimum, relative maximum, or neither a minimum nor a maximum. Explain your reasoning.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 3: MU: Graphs

411

12.C-122 (2009B AB–3)

A continuous function f is defined on the closed interval −4 ≤ x ≤ 6. The graph of f consists of a line segment and a curve that is tangent to the x–axis at x = 3, as shown in the figure above. On the interval 0 < x < 6, the function f is twice differentiable, with f 00 (x) > 0. (a) Is f differentiable at x = 0? Use the definition of the derivative with one-sided limits to justify your answer. (b) For how many values of a, −4 ≤ a < 6, is the average rate of change of f on the interval [a, 6] equal to 0 ? Give a reason for your answer. (c) Is there a value of a, −4 ≤ a < 6, for which the Mean Value Theorem, applied to the interval [a, 6], guarantees a value c, a < c < 6, at which 1 f 0 (c) = ? Justify your answer. 3 Z x (d) The function g is defined by g(x) = f (t) dt for −4 ≤ x ≤ 6. 0

On what intervals contained in [−4, 6] is the graph of g concave up? Explain your reasoning.

Mr. Budd, compiled September 29, 2010

412

AP Unit 12 (Makeup)

12.C-123 (2009B AB–5) [NO CALCULATOR]

Let f be a twice-differentiable function defined on the interval −1.2 < x < 3.2 with f (1) = 2. The graph of f 0 , the derivative of f , is shown above. The graph of f 0 crosses the x–axis at x = −1 and x = 3 and has a horizontal tangent at x = 2. Let g be the function given by g(x) = ef (x) . (a) Write an equation for the line tangent to the graph of g at x = 1. (b) For −1.2 < x < 3.2, find all values of x at which g has a local maximum. Justify your answer. h i 2 (c) The second derivative of g is g 00 (x) = ef (x) (f 0 (x)) + f 00 (x) . Is g 00 (−1) positive, negative, or zero? Justify your answer. (d) Find the average rate of change of g 0 , the derivative of g, over the interval [1, 3].

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 3: MU: Graphs

413

12.C-124 (2006B AB–2)

Let f be the function defined for x ≥ 0 with f (0) = 5 and f 0 , the first derivative of f , given by f 0 (x) = e−x/4 sin x2 . The graph of y = f 0 (x) is shown above. (a) Use the graph of f 0 to determine whether the graph of f is concave up, concave down, or neither on the interval 1.7 < x < 1.9. Explain your reasoning. (b) On the interal 0 ≤ x ≤ 3, find the value of x at which f has an absolute maximum. Justify your answer. (c) Write an equation for the line tangent to the graph of f at x = 2. 12.C-125 (2008B–5) [NO CALCULATOR]

Let g be a continuous function with g(2) = 5. The graph of the piecewiselinear function g 0 , the derivative of g, is shown above for −3 ≤ x ≤ 7. (a) Find the x–coordinate of all points of inflection of the graph of y = g(x) for −3 < x < 7. Justify your answer. Mr. Budd, compiled September 29, 2010

414

AP Unit 12 (Makeup) (b) Find the absolute maximum value of g on the interval −3 ≤ x ≤ 7. Justify your answer. (c) Find the average rate of change of g(x) on the interval −3 ≤ x ≤ 7. (d) Find the average rate of change of g 0 (x) on the interval −3 ≤ x ≤ 7. Does the Mean Value Theorem applied on the interval −3 ≤ x ≤ 7 guarantee a value of c, for −3 < c < 7, such that g 00 (c) is equal to this average rate of change? Why or why not?

12.C-126 (2002B AB-4) [NO CALCULATOR]

The graph of a differentiable function f on the closed interval [−3, 15] is shown in the figure above. Z The graph of f has a horizontal tangent line x

f (t) dt for −3 ≤ x ≤ 15.

at x = 6. Let g(x) = 5 + 6

(a) Find g(6), g 0 (6), and g 00 (6). (b) On what intervals is g decreasing? Justify your answer. (c) On what intervals is the graph of g concave down? Justify your answer. Z 15 (d) Find a trapezoidal approximation of f (t) dt using six subintervals of length 4t = 3.

−3

12.C-127 (2000 AB–2) Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 3: MU: Graphs

415

Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. The graph above, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of 24t Runner B is given by the function v defined by v(t) = . 2t + 3 (a) Find the velocity of Runner A and the velocity of Runner B at time t = 2 seconds. Indicate units of measure. (b) Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units of measure. (c) Find the total distance run by Runner A and the total distance run by Runner B over the time interval 0 ≤ t ≤ 10 seconds. Indicate units of measure. 12.C-128 (1996 AB-1)

The figure above shows the graph of f 0 , the derivative of a function f . The domain of f is the set of all real numbers x such that −3 < x < 5. Mr. Budd, compiled September 29, 2010

416

AP Unit 12 (Makeup) (a) For what values of x does f have a relative maximum? Why? (b) For what values of x does f have a relative minimum? Why? (c) On what intervals is the graph of f concave upward? Use f 0 to justify your answer. (d) Suppose that f (1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f on the open interval 0 < x < 2.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 4: MU: Integral as Accumulator

12.4

417

MU: Integral as Accumulator

Replaces problems 125-128, 165, 166

Makeup 12.D-125 (2010–1) There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 a.m., snow accumulates on the driveway at a rate modeled by f (t) = 7tecos t cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 a.m. (t = 6). The rate g(t), in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by   for 0 ≤ t < 6 0 g(t) = 125 for 6 ≤ t < 7   108 for 7 ≤ t ≤ 9 (a) How many cubic feet of snow have accumulated on the driveway by 6 a.m.? (b) Find the rate of change of the volume of snow on the driveway at 8 a.m. (c) Let h(t) represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain 0 ≤ t ≤ 9. (d) How many cubic feet of snow are on the driveway at 9 a.m.?

Mr. Budd, compiled September 29, 2010

418

AP Unit 12 (Makeup)

12.D-126 (2010â&#x20AC;&#x201C;3) There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people move onto the ride at a rate of 800 people per hour. The graph below shows the rate, r(t), at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride begins operation.

(a) How many people arrive at the ride between t = 0 and t = 3 ? Show the computations that lead to your answer. (b) Is the number of people waiting in line to get on the ride increasing or decreasing between t = 2 and t = 3 ? Justify your answer. (c) At what time t is the line for the ride the longest? How many people are in line at that time? Justify your answers. (d) Write, but do not solve, an equation involving an integral expression of r whose solution gives the earliest time t at which there is no longer a line for the ride.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 5: MU: Linear Motion

12.5

419

MU: Linear Motion

Replaces problems 129-132

Makeup 12.E-129 (2010B–4) [NO CALCULATOR] A squirrel starts at building A at time t = 0 and travels along a straight, horizontal wire connected to building B. For 0 ≤ t ≤ 18, the squirrel’s velocity is modeled by the piecewise-linear function defined by the graph below.

(a) At what times in the interval 0 < t < 18, if any, does the squirrel change direction? Give a reason for your answer. (b) At what time in the interval 0 ≤ t ≤ 18 is the squirrel farthest from building A? How far from building A is the squirrel at that time? (c) Find the total distance the squirrel travels during the time interval 0 ≤ t ≤ 18. (d) Write expressions for the squirrel’s acceleration a(t), velocity v(t), and distance x(t) from building A that are valid for the time interval 7 < t < 10.

Mr. Budd, compiled September 29, 2010

420

AP Unit 12 (Makeup)

12.E-130 (2010B–6) [NO CALCULATOR] Two particles move along the x–axis. πFor 0 ≤ t ≤ 6, the position of particle P at time t is given by p(t) = 2 cos t , 4 while the position of particle R at time t is given by r(t) = t3 −6t2 +9t+3. (a) For 0 ≤ t ≤ 6, find all times t during which particle R is moving to the right. (b) For 0 ≤ t ≤ 6, find all times t during which the two particles travel in opposite directions. (c) Find the acceleration of particle P at time t = 3. Is particle P speeding up, slowing down, or doing neither at time t = 3? Explain your reasoning. (d) Write, but do not evaluate, an expression for the average distance between the two particles on the interval 1 ≤ t ≤ 3. 12.E-131 (1993 AB-2) [NO CALCULATOR] A particle moves on the x–axis so that its position at any time t ≥ 0 is given by x(t) = 2te−t . (a) Find the acceleration of the particle at t = 0. (b) Find the velocity of the particle when its acceleration is 0. (c) Find the total distance traveled by the particle from t = 0 to t = 5. 12.E-132 (2007 AB–4) [NO CALCULATOR] A particle moves along the x–axis with position at time t given by x(t) = e−t sin t for 0 ≤ t ≤ 2π. (a) Find the time at which the particle is farthest to the left. Justify your answer. (b) Find the value of the constant A for which x(t) satisfies the equation Ax00 (t) + x0 (t) + x(t) = 0 for 0 < t < 2π. 12.E-133 (2003B AB-4) [NO CALCULATOR] A particle moves along the x-axis with velocity at time t ≥ 0 given by v(t) = −1 + e1−t . (a) Find the acceleration of the particle at time t = 3. (b) Is the speed of the particle increasing at time t = 3? Give a reason for your answer. (c) Find all values of t at which the particle changes direction. Justify your answer. (d) Find the total distance traveled by the particle over the time interval 0 ≤ t ≤ 3.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 5: MU: Linear Motion

421

12.E-134 (2005 AB-5) [NO CALCULATOR]

A car is traveling on a straight road. For 0 ≤ t ≤ 24 seconds, the car’s velocity v(t), in meters per second, is modeled by the piecewise-linear function defined in the graph above. Z 24 Z 24 (a) Find v(t) dt. Using correct units, explain the meaning of v(t) dt. 0

(b) For each of v 0 (4) and v 0 (20), find the value or explain why it does not exist. Indicate units of measure. (c) Let a(t) be the car’s acceleration at time t, in meters per second per second. For 0 < t < 24, write a piecewise-defined function for a(t). (d) Find the average rate of change of v over the interval 8 ≤ t ≤ 20. Does the Mean Value Theorem guarantee a value of c, for 8 < c < 20, such that v 0 (c) is equal to this average rate of change? Why or why not?

Mr. Budd, compiled September 29, 2010

422

AP Unit 12 (Makeup)

12.E-135 (2005B AB-3) A particle moves along the x-axis so that its velocity v at time t, for 0 ≤ t ≤ 5, is given by v(t) = ln t2 − 3t + 3 . The particle is at position x = 8 at time t = 0. (a) Find the acceleration of the particle at time t = 4. (b) Find all times t in the open interval 0 < t < 5 at which the particle changes direction. During which time intervals, for 0 < t < 5, does the particle travel to the left? (c) Find the position of the particle at time t = 2. (d) Find the average speed of the particle over the interval 0 ≤ t ≤ 2. 12.E-136 (1999 AB–1) A particle moves along the y–axis with velocity given by v(t) = t sin t2 for t ≥ 0. (a) In which direction (up or down) is the particle moving at time t = 1.5? Why? (b) Find the acceleration of the particle at time t = 1.5. Is the velocity of the particle increasing at t = 1.5? Why or why not? (c) Given that y(t) is the position of the particle at time t and that y(0) = 3, find y(2). (d) Find the total distance traveled by the particle from t = 0 to t = 2. 12.E-137 (1997 AB–1) A particle moves along the x–axis so that its velocity at any time t ≥ 0 is given by v(t) = 3t2 − 2t − 1. The position x(t) is 5 for t = 2. (a) Write a polynomial expression for the position of the particle at any time t ≥ 0. (b) For what values of t, 0 ≤ t ≤ 3, is the particle’s instantaneous velocity the same as its average velocity on the closed interval [0, 3]? (c) Find the total distance traveled by the particle from time t = 0 until t = 3.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 6: MU: Data

12.6

423

MU: Data

Replaces problems 133-136, 161, 168-170, 172

Makeup 12.F-133 (2010–2) A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon (t = 0) and 8 p.m. (t = 8). The number of entries in the box t hours after noon is modeled by a differentiable function E for 0 ≤ t ≤ 8. Values of E(t), in hundreds of entries, at various times t are shown in the table below. t (hours) E(t) (hundreds of entries)

(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time t = 6. Show the computations that lead to your answer. (b) Use a trapezoidal sum with the four subintervals given by the table Z 1 8 to approximate the value of E(t) dt Using correct units, explain 8 0 Z 8 1 the meaning of E(t) dt in terms of the number of entries. 8 0 (c) At 8 p.m., volunteers began to process the entries. They processed the entries at a rate modeled by the function P , where P (t) = t3 −30t2 +298t−976 hundreds of entries per hour for 8 ≤ t ≤ 12. According to the model, how many entries had not yet been processed by midnight (t = 12)? (d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.

Mr. Budd, compiled September 29, 2010

424

AP Unit 12 (Makeup)

12.7

MU: Extrema and Optimization

Replaces problems 137-140, 157

Makeup 0 12.G-137 (2010B–2) The function g is defined for x > 0 with g(1) = 2, g (x) = 1 1 1 sin x + , and g 00 (x) = 1 − 2 cos x + . x x x

(a) Find all values of x in the interval 0.12 ≤ x ≤ 1 at which the graph of g has a horizontal tangent line. (b) On what subintervals of (0.12, 1), if any, is the graph of g concave down? Justify your answer. (c) Write an equation for the line tangent to the graph of g at x = 0.3. (d) Does the line tangent to the graph of g at x = 0.3 lie above or below the graph of g for 0.3 < x < 1? Why?

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 7: MU: Extrema and Optimization

425

12.G-138 (2007B AB–4) [NO CALCULATOR]

Let f be a function defined on the closed interval −5 ≤ x ≤ 5 with f (1) = 3. The graph of f 0 , the derivative of f , consists of two semicircles and two line segments, as shown above. (a) For −5 < x < 5, find all values of x at which f has a relative maximum. Justify your answer. (b) For −5 < x < 5, find all values of x at which the graph of f has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of f (x) over the closed interval −5 ≤ x ≤ 5. Explain your reasoning. 12.G-139 (2001 AB–3) A car is traveling on a straight road with velocity 55 ft/sec at time t = 0. For 0 ≤ t ≤ 18 seconds, the car’s acceleration a(t), in ft/sec2 , is the piecewise linear function defined by the graph below.

(a) Is the velocity of the car increasing at t = 2 seconds? Why or why not? Mr. Budd, compiled September 29, 2010

426

AP Unit 12 (Makeup) (b) At what time in the interval 0 ≤ t ≤ 18, other than t = 0, is the velocity of the car 55 ft/sec? Why? (c) On the time interval 0 ≤ t ≤ 18, what is the car’s absolute maximum velocity, in ft/sec, and at what time does it occur? Justify your answer.

12.G-140 (1985 AB-2) [NO CALCULATOR] A particle moves along the x-axis with acceleration given a(t) = cos t for t ≥ 0. At t = 0 the velocity v(t) of the particle is 2 and the position x(t) is 5. (a) Write an expression for the velocity v(t) of the particle. (b) Write an expression for the position x(t). (c) For what values of t is the particle moving to the right? Justify your answer. π (d) Find the total distance traveled by the particle from t = 0 to t = . 2

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 7: MU: Extrema and Optimization

427

12.G-141 (1992 AB-2) [NO CALCULATOR] A particle moves along the x-axis so that its velocity at time t, 0 ≤ t ≤ 5, is given by v(t) = 3 (t − 1) (t − 3). At time t = 2, the position of the particle is x(2) = 0. (a) Find the minimum accleration of the particle. (b) Find the total distance traveled by the particle. (c) Find the average velocity of the particle over the interval 0 ≤ t ≤ 5

12.G-142 (2006B AB-4) [NO CALCULATOR] The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the 3 1 function f . In the figure above, f (t) = − t3 + t2 + 1 for 0 ≤ t ≤ 4 and 4 2 f is piecewise linear for 4 ≤ t ≤ 24. (a) Find f 0 (22). Indicate units of measure. (b) For the time interval 0 ≤ t ≤ 24, at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer. (c) Find the total number of calories burned over the time interval 6 ≤ t ≤ 18 minutes.

Mr. Budd, compiled September 29, 2010

428

12.8

AP Unit 12 (Makeup)

MU: Implicit Differentiation

Replaces problems 141-144

Makeup 12.H-141 (2001 AB-6) [NO CALCULATOR] The function f is differentiable for all 1 is on the graph of y = f (x), and the real numbers. The point 3, 4 dy = y 2 (6 â&#x2C6;&#x2019; 2x). slope at each point (x, y) on the graph is given by dx d2 y 1 (a) Find and evaluate it at the point 3, . dx2 4 dy (b) Find y = f (x) by solving the differential equation = y 2 (6 â&#x2C6;&#x2019; 2x) dx 1 with the initial condition f (3) = . 4

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 9: MU: Area and Volume

12.9

429

MU: Area and Volume

Replaces problems 153-156, 173, 174

Makeup 12.I-153 (2010–4) [NO CALCULATOR] Let √ R be the region in the first quadrant bounded by the graph of y = 2 x, the horizontal line y = 6, and the y–axis, as shown in the figure below.

Mr. Budd, compiled September 29, 2010

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AP Unit 12 (Makeup)

12.I-154 (2010B–1) In the figure above, R is the shaded region in the first quadrant bounded by the graph of y = 4 ln (3 − x), the horizontal line y = 6, and the vertical line x = 2.

Mr. Budd, compiled September 29, 2010

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431

12.I-155 (2004B AB-1) Let R be the region enclosed by the graph of y = the vertical line x = 10, and the x–axis.

√

x − 1,

Let f and g be the functions given by f (x) = 1 + sin(2x) and g(x) = ex/2 . Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the x–axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles with diameters extending from y = f (x) to y = g(x). Find the volume of this solid. 12.I-157 (2008P AB–2) Mr. Budd, compiled September 29, 2010

432

AP Unit 12 (Makeup)

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 9: MU: Area and Volume

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12.I-158 (2007B AB–1)

Mr. Budd, compiled September 29, 2010

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12.I-159 (1999 AB-2)

The shaded region, R, is bounded by the graph of y = x2 and the line y = 4, as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated by revolving R about the x–axis. (c) There exists a number k, k > 4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of k. 12.I-160 (2007 AB–1) Let R be the region in the first and second quadrants bounded 20 above by the graph of y = and below by the horizontal line y = 2. 1 + x2 (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the x–axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles. Find the volume of this solid.

Mr. Budd, compiled September 29, 2010

AP Unit 12, Day 10: MU: Tangent Lines

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MU: Tangent Lines

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Mr. Budd, compiled September 29, 2010

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AP Unit 12 (Makeup)

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Bibliography [1] Stephen Bernstein and Ruth Bernstein. Schaum’s Outline of Theory and Problems of Elements of Statistics I: Descriptive Statistics and Probability. McGraw–Hill, New York, 1999. [2] George W. Best and J. Richard Lux. Preparing for the (AB) AP Calculus Examination. Venture Publishing, Andover, Massachusetts, 1998. [3] George W. Best and J. Richard Lux. Preparing for the (BC) AP Calculus Examination. Venture Publishing, Andover, Massachussetts, 1998. [4] Nigel Buckle, Iain Dunbar, and Fabio Cirrito. Mathematics Higher Level (Core). IBID Press, Camberwell, Australia, second edition, 1999. [5] Fabio Cirrito, editor. Mathematical Methods. IBID Press, Melton, Australia, second edition, 1998. [6] Douglas Downing and Jeffrey Clark. Forgotten Statistics: A Self-Teaching Refresher Course. Barron’s Educational Series, Hauppage, New York, 1996. [7] Douglas Downing and Jeffrey Clark, editors. Statistics: The Easy Way. Barron’s Educational Series, Hauppage, New York, third edition, 1997. [8] Ross L. Finney, Flanklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic. Scott Foreesman Addison Wesley, New York, 1999. [9] Paul Foerster. Calculus: Concepts and Applications; Instructor’s Resource Book. Key Curriculum Press, Berkeley, California, 1998. [10] Paul A. Foerster. Calculus: Concepts and Applications. Key Curriculum Press, Berkeley, California, 1998. [11] Nancy Baxter Hastings. Workshop Calculus with Graphing Calculators: Guided Exploration with Review, volume 2. Springer-Verlag, New York, 1999. 437

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[12] Nancy Baxter Hastings and Barbara E. Reynolds. Workshop Calculus with Graphing Calculators: Guided Exploration with Review, volume 1. Springer-Verlag, New York, 1999. [13] Melvin Hausner. A Vector Space Approach to Geometry. Dover Publications, Mineola, New York, 1965. [14] Shirley O. Hockett and David Bock. How to Prepare for the Advanced Placement Examination, Calculus: Review of Calculus AB and Calculus BC. Barronâ&#x20AC;&#x2122;s Educational Series, New York, 1998. [15] Ann R. Kraus. Test Item File to Accompany Calculus. D. C. Heath and Company, Lexington, Massachusetts, 1994. [16] Roland E. Larson, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry. D. C. Heath and Company, Lexington, Massachusetts, 1994. [17] Arnold Ostebee and Paul Zorn. Calculus From Graphical, Numerical, and Symbolic Points of View. Saunders College Publishing, Fort Worth, 1997. [18] Salas, Hille, and Garret J. Etgen. Calculus: One and Several Variables. John Wiley and Sons, Inc., New York, New York, 1999. [19] Wanda Savage. A. p. summer institute. Bound materials from the summer workshop, 2000. [20] James Stewart. Calculus:Concepts and Contexts, Single Variable. Brooks/ Cole Publishing Company, Pacific Grove, California, 1998. [21] Dale Varberg, Edwin J. Purcell, and Steven E. Rigdon. Calculus. Prentice Hall, Upper Saddle River, New Jersey, 2000.

Mr. Budd, compiled September 29, 2010