Edvantage Math
AP® Calculus AB
AP Calculus AB
Authors
Bruce McAskill, BSc, BEd, MEd, PhD Deanna Catto, BSc, BEd Mathew Geddes, BSc, BEd, MMT Steve Bates, BEd, MEd
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ISBN: 9781774301708 Editorial Development Lead: Lionel Sandner Editor: Ryan Chowdhry Care has been taken to trace ownership of copyright material contained in this text. The publishers will gladly accept any information that will enable them to rectify any reference or credit in subsequent printings. The Big Ideas at the beginning of each chapter are quoted from AP Calculus: Course and Exam Description published by the College Board, New York, NY. Advanced Placement, AP and College Board are registered trademarks of the College Board.
ii
Table of Contents Unit 1: Differential Calculus
1
Chapter 1: Analyzing Functions Using Limits 1.1. Rates of Change and Limits …………………………………………………………………………………………………………. 1.2. Exploring Limits ……………………………………………………………………………………..…………………………...………. 1.3. Asymptotic Behavior……………………………………………………………………………………………………………………. 1.4. End Behavior of Functions ………………………………………………….……………………………………….………………. 1.5. Function Continuity…………………………………………………………………….……………………………….………………. 1.6. Intermediate Value Theorem……………………………………………………………………….……………….……………… 1.7. Chapter Review………………………………………………………………………………………………………….…………..…….
2 3 10 24 35 46 60 66
Chapter 2: The Derivative 2.1. Slope of a Function ……………………………………………………………………………………………………….….…………. 2.2. Definition of the Derivative…………………………………………………………………………………………….……………. 2.3. Alternate Definition of the Derivative at a Point ……………………………………………………………….…………. 2.4. Differentiability of a Function…………………………………………………………………………………..………….….…… 2.5. Symmetric Difference Quotient…………………………………………………..………………………………………..……… 2.6. Graphing the Derivative of a Function …………………………………………………………………………….…….…….. 2.7. Chapter Review……………………………………………………………………………………………………..……………….…….
71 72 84 94 103 113 121 134
Chapter 3: Differentiation Techniques 3.1. Derivatives of Polynomial Functions …………………………………………………………………………………….……… 3.2. Higher Order Derivatives……………………………………………………………………………………………………...……. 3.3. Velocity and Other Rates of Change……………………………………………………………………………………………… 3.4. Derivatives of Trigonometric Functions ……………………………………………………………………………………….. 3.5. Introducing the Chain Rule ………………………………………………………………………………………………….………. 3.6. Chapter Review…………………………………………………………………………………………………………………………….
143 144 155 166 187 196 204
Chapter 4: Advanced Differentiation Techniques 4.1. The Chain Rule …………………………………………………………………………………………………….………………………. 4.2. Composite Functions and Function Notation ………………………………………………………………………………. 4.3. Implicit Differentiation…………………………………………………………………………………………………………………. 4.4. Derivatives of Inverse …………………………………………………………………………………………………….……………. 4.5. Derivatives of Exponential and Logarithmic Functions ………………………………………………………………… 4.6. Logarithmic Differentiation………………………………………………………………………………….………………………. 4.7. Derivatives of Inverse Trigonometric Functions …………………………………………………………………………… 4.8. L’Hôspital’s Rule ………………………………………………………………………………………………………………….………. 4.9. Chapter Review…………………………………………………………………………………………………………………………….
213 214 221 229 239 246 258 268 282 287
Unit 1 Project: Roller Coaster Design………………………………………………………………………….…………………..…………
295
iii
Unit 2: Applications of Derivatives
297
Chapter 5: Analyzing Functions Using Derivatives 5.1. The First Derivative Test …………………………………………………………………………………………..…………………. 5.2. Modeling and Optimization ……………………………………………………………………………………….………………… 5.3. The Second Derivative Test ………………………………………………………………………………………….……………… 5.4. Curve Sketching …………………………………………………………………………………………………………………..……… 5.5. The Mean Value Theorem …………………………………………………………………………………………………………… 5.6. Chapter Review ……………………………………………………………………………………………………………………………
298 299 309 316 327 336 342
Chapter 6: Solving Problems Using Derivatives 6.1. Related Rates Involving Shape and Space ………………………………………………………………………….………… 6.2. Related Rates Involving Motion …………………………………………………………………………………………………… 6.3. Related Rates Involving Periodic Functions ……………………………………………………………………….………… 6.4. Linearization and Differentials ……………………………………………………..……………………………………...…….. 6.5. Chapter Review ……………………………………………………………………………………………………………………………
347 348 356 364 371 378
Unit 2 Project: Shrinking Lollipop .…………………………………………………………………………………………………………….
383
Unit 3: Integral Calculus
385
Chapter 7: Antidifferentiation 7.1. The Antiderivative – Working Backwards ………………………………………………………..…………..……………… 7.2. Antiderivative of Trigonometric Functions ……………………………………………………………..…………………… 7.3. Antidifferentiation Using Substitution .………………………………………………………………………………………… 7.4. Advanced Antidifferentiation Techniques …………………………………………………………………………………… 7.5. Antidifferentiation Involving Exponential and Logarithmic Functions ………………………………………….. 7.6. Antidifferentiation Involving Inverse Trigonometric Functions ……………………………………………………. 7.7. Chapter Review ……………………………………………………………………………………………………………………………
386 387 392 396 403 409 415 422
Chapter 8: The Integral 8.1. Solving Differential Equations Analytically …………………………………………………………………………………… 8.2. Solving Differential Equations Graphically …………………………………………………………………………………… 8.3. Approximating Area Under a Curve Using Riemann Sums ……………………………………..……………………. 8.4. Calculating Area Under Functions Graphically …………………………………………..………………………………… 8.5. Chapter Review ……………………………………………………………………………………………………………………………
425 426 435 446 458 468
Chapter 9: Integral and the Fundamental Theorem of Calculus 9.1. Integrating Using the Fundamental Theorem …………………………………………………….……………………… 9.2. Derivative of a Definite Integral …………………………………………………………………………………………………… 9.3. The Average Value of a Function ……………………………………………………………………………….………………… 9.4. The Integral as Net Change ………………………………………………………………………………………….……………… 9.5. Chapter Review ……………………………………………………………………………………………………………………………
473 474 487 498 505 515
Chapter 10: Area and Volume 10.1. Area Between Two Curves …………………………………………………………………………………………………………… 10.2. Area Between Two Curves Using Horizontal Slices …………………………………….………………………………… 10.3. Volumes of Solids ………………………………………………………………………………………………………………………… 10.4. Finding Volume Using the Washer Method …………………………………………………………….…………………… 10.5. Finding Volume Using CrossSectional Area ………………………………………………………….………..…………… 10.6. Chapter Review …………………………………………………………………………………………………………………………...
521 522 533 543 551 560 570
iv
Unit 1: Differential Calculus This unit focuses on the following AP Big Idea from the College Board:
Big Idea 1: The idea of limits is essential for discovering and developing important ideas, definitions, formulas, and theorems in calculus
By the end of this unit, you should be able to: Express limits symbolically using correct notation Interpret limits expresses symbolically Estimate limits of functions Determine limits of functions Deduce and interpret behavior of functions using limits Analyze functions for intervals of continuity or points of discontinuity Determine the applicability of important calculus theorems using continuity
By the end of this unit, you should know the meaning of these key terms:
Continuity at a point Continuous function Difference rule End behavior models (leftend and rightend)
Extended function Horizontal asymptote Infinite discontinuity Intermediate Value Theorem (for continuous functions) Jump discontinuity Lefthand limit Limit Onesided limit Oscillating discontinuity Product rule Quotient rule Righthand limit Sandwich Theorem Sum rule Twosided limit Vertical asymptote
When the Mars Rover lands, it’s important to know its velocity at all times.
1
Chapter 1: Analyzing Functions Using Limits Students often wonder “What is Calculus?” Calculus uses the logic of mathematics to analyze situations involving continuous change by breaking them down into the smallest components (differential calculus) and then rebuilding them (integral calculus). As such, Calculus is considered the science and the art of change. It provides a mechanism to understand the world around us, make predictions, and interpret behavior. One type of behavior involves a limit. What is a limit and how is it useful? The mathematical answer is that a limit is the yvalue that a function,
, approaches as the value
of x approaches some number. A few “real life” examples can be found in:
Physics  pressure at a point is calculated as the average pressure (force per unit area) applied to an area that is shrinking to zero (i.e., shrinking to a point).
Ecology sustainable population (or carrying capacity) is often extrapolated by determining, over an extended period of time (i.e., infinite), the upper bound on some population that can be sustained by a given ecosystem.
Chemistry  if you drop an ice cube into a glass of warm water and measure the temperature against time, the temperature will eventually approach the room temperature where the glass is stored. Determining the final temperature is a limit as time approaches infinity.
A limit describes the behavior of a function at a particular point based on the points around it.
2
1.1 Rates of Change and Limits – Determining Limits Graphically
The concept of a limit is CENTRAL to calculus. Limits can be used to describe how a function behaves as the independent variable moves towards a certain value. Consider the functions
and
.
Graph on , using an appropriate range. On the same grid graph the denominator. (Be careful…when does x = 1?)
When does What are the values of How do.
. and
and
at the point of intersection?
behave near x = 0?
What do you think the value of the quotient function
is at the point of
intersection?
3
Look graphically and numerically: Using your calculator, graph both
and
, then complete the tables of values below. x
x
0.3
0.03
0.2
0.02
0.1
0.01
0
0
0.1
0.01
0.2
0.02
0.3
0.03
Consider what happens to the yvalue as the xvalue gets closer and closer to 0.
What value will the quotient approach as the values of the functions become more and more alike?
How do the functions behave as x approaches 0? (use the ZOOM function on your calculator to explore)
Definition: If functions substituting
and
are both 0 at
as the result would be
then
cannot be found by
. Substitution produces a meaningless
expression known as an indeterminate form. Other indeterminate forms include:
.
To determine the value of an indeterminate form, other methods, such as graphing, analyzing numerically or using algebraic manipulations, must be used.
4
Example 1: Determining a limit graphically and numerically Find the value that
approaches as x approaches
.
Use a graph and a table of values to justify your answer. How to Do It What to Think About Graphing the function shows that the function approaches the value of 1 from both the left side How does the function behave of zero and the right side of zero. as x gets closer to zero from the left? From the right?
What is the value of the function at ?
Observe the table of values: x
y
0.03
0.99985
0.02
0.9993
0.01
0.998
0
ERROR
0.01
0.9998
0.02
0.9993
0.03
0.99985
What value does the function approach as x gets closer to zero from the left? From the right?
Therefore, as
Definition:
is read, “the limit of f of x as x approaches a equals L”. It means
that as x gets closer and closer to a (from the leftside and the rightside of a), the function’s y – value gets closer and closer to the number L.
5
Your Turn Compare
and
approaches as
graphically then graph
and find the value that
. Confirm using a table of values.
x
y
0.03 0.02 0.01 0 0.01 0.02 0.03
Example 2: The existence of a limit versus the existence of the function at a point. Given How to Do It
determine
. Is
equal to
?
What to Think About How does the function behave as x gets closer to 2 from the left? From the right?
What is the value of the function at ? Is the function continuous at x = 2?
6
Your Turn For each of the following functions, determine if a)
at
b)
at
Did You Know: The modern notation of placing the arrow below the limit symbol is attributed to the English mathematician G.H. Hardy, who used it in his 1908 book A Course of Pure Mathematics.
Summary: The value of a limit is the y – value that the function approaches as
.
The existence of a limit as never depends on if the function is defined or undefined at . The limit exists if the function approaches the same y – value on both sides of .
7
Practice For questions 1 to 8, determine the limit graphically. 1.
8
2.
3.
4.
5.
6.
7.
8.
For questions 9 to 12, use a graph to show that the limit does or does not exist
9.
10.
11.
12.
9
1.2 Exploring Limits Warm Up Consider the following functions:
a)
b)
c)
Without a graphing calculator sketch the graph of each function. Make sure to consider the points where the function does not exist. a) b) c)
Using the graphs, determine each of the following limits and confirm algebraically. a)
b)
c)
Why does factoring help to determine the limit algebraically?
10
Did You Know A removable discontinuity (or point discontinuity) occurs when a function has a hole at one point c on an open interval such that , but or does not exist. You will explore the concept of continuity in Section 1.5.
Example 1: Determining the Limit of a Function Find the limit of each of the following functions, if it exists. a)
How to Do It a)
b)
c)
d)
What to Think About What is the value of the function if you substitute x = 2? What is the value of the function if you substitute x = 1?
b) Can the function be simplified if the numerator or denominator is factored?
c) d)
What happens on the graph of b) at the point (1,3)?
What is the value if the limit involves division by zero?
11
Your Turn Find the limit of each of the following functions, if it exists. a)
d) lim
b)
e)
x 16
x 4 x 16
c)
Look at b) and c) graphically. What do you think makes a limit exist?
12
Definition: The existence of a limit depends on the onesided limits about x = a. So when
, then
exists and
, where
.
When the righthand limit equals the lefthand limit at a point, the limit exists. Example 2: Onesided and Twosided Limits Given the graph of f determine each limit
How to Do It a)
What to Think About What does the notation “ mean?
”
b)
c)
d)
What does the notation “ mean?
”
Is the left side limit the same as the rightside limit?
e)
f)
What does the abbreviation “DNE” represent?
g)
13
Did You Know A jump discontinuity occurs when, for some point c on an open interval, but
. The function may or may not exist at
Your Turn 1) Given the graph of g determine each limit. a)
e)
b)
f)
c)
g)
d)
h)
2) Given the graph of f determine each limit. a)
b)
c)
14
.
,
3) Given the following function, sketch the graph of the function and then use one sided limits j
to show that the
does not exist.
15
Example 3: The Greatest Integer Function or
is called the greatest integer function and is defined to be the largest
integer to the “left” of x on the number line (or the greatest integral part of x). What is the result when the greatest integer function is applied? a)
b)
How to Do It a)
c)
e)
c)
b)
d)
e)
f)
g)
What to Think About What is the greatest integer less than the number 2.3?
d)
f)
What is the greatest integer less than 1.2? Try plotting this on a number line.
Did You Know: The greatest integer function is commonly used when calculating utility bills (gas, electricity, water, etc.) as well as postage rates.
16
Your Turn 1. Graph the greatest integer function (also called a step function):
2. Use the graph to help determine the following limits: a
a)
d)
b)
e)
c)
f)
Summary: For the greatest integer function, (
n is an integer)
17
Practice 1. What is
?
Determine each limit by substitution and support the result graphically. 2.
3.
18
4.
5.
6.
19
7. Explain why you cannot use substitution to determine. Use a graph to support your explanation a.
b.
Did You Know: The signum function is a mathematical function that extracts the sign of a real number. It is often represented as
Determine each limit. 8.
9.
20
10.
Which of the statements are true about the function
graphed below and which are
false?
11.
16.
12.
17.
13.
18.
14.
19. There is a removable discontinuity
at 15.
.
20. There is a jump discontinuity at .
21
Given that
and
determine the following limits.
21.
23.
22.
24.
Complete the following parts for each of questions 25 and 26. a) Draw the graph of f. b) Determine c) Does 25. c = 1,
22
and
.
exist? If so, what is it? If it does not exist explain why.
26. c = 2,
Complete the following parts for question 27. a) Draw the graph of f. b) At what points c in the domain of f does
exist?
c) At what point “c” does the limit exist and why?
27.
23
1.3 Asymptotic Behavior
Sketch the graph of
and determine each limit: a)
d)
b)
e)
c)
Now sketch the graph of
and determine each limit:
a)
d)
b)
e)
c)
What can you conclude are the conditions necessary for a vertical or a horizontal asymptote to exist?
24
or
Example 1: Finding a Horizontal Asymptote Find the horizontal asymptote(s) for each function. Use a graph and limits to help. a)
b)
How to Do It a)
Therefore, a horizontal asymptote occurs at y = 0.
What to Think About Numerically, what happens to the value of the rational expression as x becomes very large?
What value does the function approach as the denominator gets positively or negatively very large?
Does the +1 have an effect on the value of the denominator as ?
Therefore, 2 horizontal asymptotes occur at y = 1 and y = 1
25
Your Turn Find the horizontal asymptote(s) for each function using a graph of the function and limits. a)
b)
, where
26
.
.
Example 2: Finding a Vertical Asymptote Find the vertical asymptote(s) for each function. Use a graph and limits to help a)
b)
How to Do It a)
What to Think About Are there any values of x that make the function undefined?
As x approaches this value, what happens to the value of the denominator?
Therefore, a vertical asymptote occurs at x = 1. b)
Therefore, a vertical asymptote occurs at x = 1.
What is the limit of the function at the value(s) where the denominator equals zero?
What is the shortcut for determining if there is a vertical asymptote?
27
Your Turn Find the vertical asymptote(s) for each function using a graph of the function and limits. a)
28
b)
Example 3: Finding the asymptotes of a function. Determine the asymptote(s) of
. Justify your answer.
How to Do It
What to Think About Are there values that make the denominator equal to zero? Does the numerator also equal zero at any of these points?
Use this result to help sketch the function
What happens at x = 2?
Is it necessary to show both sides of an infinite limit to justify a vertical asymptote?
At x = 0 there is a vertical asymptote.
Do either – 4 or + 2x affect the numerator or denominator as ?
A removable discontinuity exists at
and There is a horizontal asymptote at y = 1.
29
Your Turn Determine the asymptote(s) of each function. Justify your answer. a)
b)
30
Practice For questions 1 to 4 determine each limit graphically and numerically. 1.
2.
31
3.
4.
32
For questions 5 to 8: Determine the asymptote(s) of f(x). Justify your answer using limits. Support your solution graphically and/or numerically. 5.
6.
33
7.
8.
34
1.4 End Behavior of Functions
Imagine standing 2 meters away from any wall in your classroom. Now, take a step that is exactly of the way towards the wall. Next, repeat that process – take another step that is exactly of the way towards the wall. Then, keep repeating this step process until you touch the wall with your foot. Graph what this looks like using the number of steps, n, as the independent variable and the distance from the wall, d, as the dependent variable.
Express this process as limit. What number of steps would you use in the limit?
In theory, if the pattern is followed forever, will you ever reach the wall?
Did You Know The end behavior of a function describes the shape of the curve as It can be used to predict behavior in the distant future or distant past.
.
35
Example 1: End Behavior Function Models Let and . Show that while f and g are quite different for small values of x, they are essentially identical for x large. How to Do It
What to Think About Compare the graphs of the functions for small and large values of x.
Consider the graphs,
What happens to the shape of the graphs as increases?
Right End Left End g(x) is a right and left endbehaviour model for f(x) since they behave in the same manner as .
36
Does
represent a good model
for the behavior of end or both?
at either
Definition: End Behavior Model The function g is a good approximation for the behavior of f and is called a right end behavior model for f, if and only if
.
The function g is a good approximation for the behavior of f and is end behavior model for f, if and only if
called a left
.
If g approximates f well at either end, then they behave the same way as . By dividing them and taking the limit, the quotient must equal 1 because they behave the same.
Your Turn Let
. Show graphically and analytically that
model for f and that
is a rightend behavior
is a leftend behavior model for f.
37
Example 2: Finding End Behavior Models Find a function that is an end behavior model for each function. a)
b)
How to Do It
What to Think About
a)
Which are the dominant terms in the numerator and denominator?
Since the highest degree terms in the numerator and denominator become larger, faster than the rest of the terms as
,
. Let
. When x is very large, what happens to the rest of the terms
Consider
Does the function have a limit as ? It follows that
is a rightend behaviour
model for f. It can also be shown in the same way that is a leftend behavior model for f. b) Let
. .
.
Therefore,
is a rightend behavior model for f.
It can also be shown that behavior model for f.
38
is a leftend
What does this limit tell you about the function?
Your Turn For each function determine the end behavior model to find the limits as
. Identify any
horizontal asymptotes. a)
b)
c)
39
Example 3: Determining End Behavior Using the Squeeze Theorem What is the right end behavior of
?
How to Do It Sketch the function
What to Think About Can you sketch a graph of the function?
What limit are you using to determine the end behavior?
What functions can be used that squeeze (or sandwich) f(x) as ? Determine the horizontal asymptote graphically On the same grid graph y = f(x),
, and
What inequality expresses how the two functions squeeze f(x)?
Can you evaluate the limits that squeeze f(x)? Confirm Analytically What conclusion can you now make?
Therefore,
40
Your Turn Use the Squeeze Theorem to show that
Did You Know: The squeeze theorem was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, but it was Carl Friedrich Gauss who formulated the modern version.
41
Practice For questions 1 to 4 match the function with the graph of its end behavior model. a) 1.
2.
3.
b)
c)
d) 4.
42
For questions 5 to 10: a) Determine a power function,
, end behavior model for f; and,
b) Identify any horizontal asymptotes, if they exist. 5.
8.
6.
9.
7.
10.
43
For questions 11 and 12 sketch a graph of a function
that satisfies the stated
conditions. Include any asymptotes.
44
11.
,
,
12.
,
,
,
,
,
,
,
For questions 13 – 18 use the Squeeze Theorem to determine each limit. 13.
16.
14.
17.
15.
18.
45
1.5 Function Continuity
For what intervals is the function f continuous? Use interval notation to express your answer.
For which points is f discontinuous?
For each point of discontinuity determine the value of the function.
Find the limit at each point of discontinuity or state that it does not exist.
Compare the value of the function at each discontinuity to the limit of the function. What do you notice?
Determine
and compare this to the value of the function at x = 3.
What is the relationship between the limit and the value of the function at each point where the graph is continuous?
46
Definition: Continuity at a Point Interior Point: A function y = f(x) is continuous at an interior point c of its domain if
EndPoint: A function y = f(x) is continuous at its left endpoint a of its domain if
Similarily, a function y = f(x) is continuous at its right endpoint b of its domain if
NB: If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Sometimes a point c may be classified as a point of discontinuity even though it is not in the domain of f. Can you provide an example?
Example 1: Finding Points of Continuity and Discontinuity Find the intervals of continuity and the points of discontinuity of the greatest integer function. How to Do It
What to Think About Where is the function discontinuous?
Can you generalize the statement? The greatest integer function is discontinuous at every integer because the limit as does not equal the value of the function at x = c. In general, if
, n an integer, then
What does this mean in terms of the intervals of continuity and points of discontinuity?
The intervals of continuity are defined as The points of discontinuity are defined as
47
Your Turn Find the points of discontinuity and intervals of continuity for each function. a)
b)
48
Example 2: Types of Discontinuities Write the equation of each function. Use a piecewise function if necessary. Then, state the type of discontinuity or state that it is continuous. How to Do It a)
What to Think About b) What function will generate each graph?
What algebraic expression results in a hole in a graph?
Type: continuous
Type: removable discontinuity
c)
d) Which graphs can be represented using a piecewise function?
Type: jump discontinuity Type: removable discontinuity
e)
f)
Type: infinite discontinuity
Type: oscillating
What happens to these functions as ?
49
Your Turn For each function identify the point(s) of discontinuity and the type of the corresponding discontinuity. Determine the interval(s) of continuity.
50
a)
d)
b)
e)
c)
f)
Example 3: Extending a function to create a continuous function Extend
to create a new function, g(x) that is continuous at x = 3.
How to Do It
What to Think About Can you factor the denominator? What is the domain of f?
What is happening around x = 3?
Can you graph it and use your calculator zoom to determine numerically if there is a removable discontinuity at x = 3?
Is a factor of the numerator of f?
What is the limit of the simplified function as
?
Use synthetic division to confirm this.
51
Extended function
Your Turn a) For each function find the value of a that makes the function continuous.
b) Determine the extended function for
52
.
Definition: A function is continuous on an interval if and only if it is continuous at every point of the interval. A function is called a continuous function if it is continuous at every point of its domain. Note: A continuous function need not be continuous on every interval, since there may be a point of discontinuity at a point outside of the domain of the function. For example,
is not continuous on
since it has a point of discontinuity at
x = 0; but the function is still considered to be a continuous function, because x = 0 is not part of the domain of the function!
Example 4: Continuous Functions Is each of the following functions considered a continuous function? Justify your answer. a)
b)
How to Do It
What to Think About Are there any values of x where the function does not exist?
a) Domain of the function is This is a continuous function since it is continuous everywhere except , which is not part of the domain.
Is the xvalue part of the domain of the function?
What do you know about each part of the piecewise function?
b)
Which value(s) of the domain need Since x + 1, x , and 2x 1 are all polynomials they are to be checked? continuous everywhere on their respective domains. 2
53
x = 2 must be checked. What are the limits of each part of the function at x = 2 and what is the value of the function at x=2
Therefore f(x) is not continuous at x = 2. Since x = 2 is part of the domain of f and there is a point of discontinuity at that point, this is not a continuous function.
What do the results tell you?
Your Turn Determine if each function is a continuous function. Justify your reasoning. a)
b)
54
Summary: Properties of Continuous Functions If functions f and g are continuous at x = a, then the following combinations are continuous at x = b. 1. Sum rule:
f+g
2. Difference rule:
fg
3. Product rule: 4. Constant multiples:
, for any number c
5. Quotient rule:
Practice For questions 1 to 6 identify the interval(s) of continuity, the point(s) of discontinuity, and each type of discontinuity. 1.
4.
2.
5.
3.
6.
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For questions 7 to 8 use the following function f and the corresponding graph of f.
7. a) Does exist? 8. a) Does
b) Does
c) Does
exist?
exist? b) Does
exist?
c) Does
exist?
exist?
d) Is f continuous at x = 1? d) Is f continuous at x = 1?
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For questions 9 to 11: a) determine each point of discontinuity, b) identify which discontinuities are removable or not removable and state your reasons. 9.
10.
11.
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For questions 12 to 13 identify the extended function, g(x), that makes the given function continuous at the indicated point. 12.
13.
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For questions 14 and 15 sketch a possible graph for a function f that has the stated properties. 14. f(5) exists but
does not.
15. f(x) is continuous for all x except x = 2, where f has a nonremovable discontinuity.
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1.6 Intermediate Value Theorem a) On the grid provided below draw a graph of f(x) = x +1 using a scale and domain [a, b] of your choice. Label the left endpoint of your graph as (a, f(a)) and the right end point as (b, f(b))
b) Place a ruler on your graph so that it is parallel to the xaxis and passes through (a, f(a)). c) Keeping your ruler parallel to the xaxis slide it upwards until you reach the point (b, f(b)). d) Is your graph continuous on the interval [a, b]?
e) Was there any yvalue (i.e., any value of f(x)) between f(a) and f(b) that your ruler did not pass through?
f) Do you think the same obervations would be made for any function? Can you think of a function where parts d) and e) do not hold true? Explain
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Example 1: Applying the IVT to Determine if a Function Has a Root on a Given Interval Does
have at least one root in [0, 1]?
How to Do It y is a polynomial function. Therefore, it is continuous on the closed interval [0, 1] as all polynomials are continuous on . Since the function is continuous and the interval is closed the Intermediate Value Theorem applies. The leftmost yvalue is yvalue is
What can you conclude from the conditions?
and the rightmost
.
Therefore, there must be at least one value c that results in
What to Think About What conditions do you know exist?
What do you know about the yvalues of the end points and what does this tell you?
by the intermediate value
theorem. Therefore, there must be at least one root in [0, 1].
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Your Turn Is any real number on the closed interval [1, 2] exactly 1 less than its cube? (Hint: Is there a solution to ?)
Example 2: Applying the EVT to Determine if a Function Has a Maximum or Minimum Let
. Although f is continuous on
, it has no minimum value on the interval.
Does this contradict the EVT? Justify your answer. How to Do It The function is continuous on
, but the
What to Think About What conditions are stated?
interval is not closed. Since the interval is not closed the Extreme Value Theorem does not apply in this case. Therefore, the statement does not contradict the EVT
62
What can you conclude from the conditions?
How does this affect the given statement?
Your Turn For each graph identify the xvalue at which a maximum or minimum occurs. Explain how your answer is consistent with the Extreme Value Theorem. a)
b)
c)
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Practice 1. The function k(x) is continuous on the interval [8, 11]. If k(8) = 3 and k(11) = 2, can you conclude that k(x) is ever equal to 0? Justify your answer.
2. Is there a solution to
3. Let
4. If reasoning.
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? Justify your answer.
. Show that there is a number c such that f(c) = 1.
, is there a number in the interval
where f(x) = 0.4? Justify your
5. Suppose that function f has every value between y = 0 and y = 1 on the interval [0, 1]. Must f be continuous on that interval? Explain your reasoning.
6. Between which of the following two values does the equation solution? a) Between 2 and 1 b) Between 3 and 2 c) Between 0 and 1 d) Between 1 and 0
7. Prove that the function the same be said for the function
have a
intersects the xaxis on the interval [0, 2]. Can ? Explain your reasoning.
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1.7 Chapter Review For questions 1 to 3, use your calculator to graph each function and estimate the limit numerically to two decimal places or state that the limit does not exist. 1.
2.
3.
For questions 4 to 11, evaluate the limit if it exists. If not, determine whether the onesided limit exists.
66
4.
5.
6.
7.
8.
9.
10.
11.
12. Determine the left and righthand limits of the function f(x) shown below at x = 0, 2, and 5.
13. Determine: a) b) c)
whether g(x) is continuous at x = 3.
d) the points of discontinuity of g(x). e) whether any points of discontinuity are removable.
67
14. Find all points of discontinuity for the given functions. a)
b)
For questions 15 & 16 find i) a rightend behavior model and ii) a leftend behavior model for the function. 15.
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16.
17. What value of k makes f a continuous function?
18. Show that
has a root somewhere in the interval [1, 2].
19. Find all asymptotes for each of the following: a)
b)
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70
Chapter 2: The Derivative The average velocity of a dragster can be calculated given its positiondistance time graph or a table of values.
Time 0 1 2 3 4 5 6 7
Distance 0 30 65 160 290 480 720 1000
What is the car’s average speed between 0 and 5 seconds as it begins to reach top speed? How could you calculate it?
What is the car’s speed exactly at 4.3 seconds? How would you calculate it?
By the end of this chapter you will be able to calculate the instantaneous speed of the dragster.
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2.1 Slope of a Function Warm Up On a road trip through Tennessee, a group of college students were 80 miles from South Padre Island at 3pm. By 4:15pm they were only 10 miles away. What was their average speed?
Do you believe this speed accurately reflects their trip? Why?
How might you estimate the instantaneous velocity of the dragster at exactly 3.5 seconds given the finite data in the graph?
Note to student: Refer to applet that follows the curve of a function and solves for the instantaneous rate of change. https://www.geogebra.org/m/STHtFSzX#material/aUjja5NV
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In general, the average rate of change of a function over an interval, is the amount of change divided by the length of the interval, written as a difference quotient. Average rate of change =
or
or
Example 1: Finding Average Rate of Change Find the average rate of change of How to Do It
over the interval [1, 3] What to Think About What are values of the function at the endpoints?
How are these used to calculate the difference quotient?
Graphically, what does the value represent?
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Your Turn Find the average rate of change of
over the interval [2, 5]
Graph any function f(x) and illustrate the slope between any two points evaluated at x = a and x = b.
Definition: A line through two points on a curve is called a secant line. The slope of the secant line of a continuous function on the closed interval [a, b]
The slope of the secant line represents the average rate of change of f on [a, b]. .
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Example 2: Growth in Technology According to Moore’s Law, the computing power, related to the number of transistors in a dense integrated circuit, doubles every 2 years indicating that the speed of computer processing has been doubling during that time frame. This trend has been used to set goals within the industry.
Given the data listed above, determine the average rate of change in transistor growth between 1993 and 2000 once the Pentium processor was introduced. How to Do It
What to Think About What are the applicable units for this rate of change? When is the processor computing speed growing at its slowest speed? Fastest speed? Why?
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Your Turn a) The rise of bitcoin as an alternate currency has been a highly volatile and intriguing investment. Over the long term, we can assess its average rate of change. Here is a graphical representation of its value. Estimate the average rate of change from June 2017 to January 2018. Estimate the average rate of change from January 2018 to March 2018. How does this compare to the Standard & Poor’s 500 which is an American stock market index based on the market capitalizations of 500 large companies having common stock listed on the NYSE or NASDAQ.
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Slope of a Function at a point To calculate the exact value of the instantaneous change of a function we need to use the concept of infinitesimals and limits. In the development of calculus, Newton and Leibniz used the concept of infinitesimals but did not define it explicitly. It is now defined as an indefinitely small quantity approaching zero. The idea is to break down a changing function into tiny pieces, considering each piece to occur over an instantaneously or infinitesimal change in the independent variable. Over an infinitely small interval we can consider the change to have a constant slope. Leibniz notation
Did You Know  Leibniz Notation
Gottfried Leibniz (16461716) discovered calculus through his use of inifinitesimals, which is expressed in differentials. Thus, the derivative in Leibniz notation is expressed as and read as “dy by dx.”
Note to student: Refer to applet with slider that draws a second point on the curve back to the point of tangency https://www.geogebra.org/m/WGCsKBeM
Definition: Slope of a Curve at a Point The slope of a curve limit
, at a point is the value
provided the
exists.
The tangent to the curve at P is the line through P with this slope. The slope of the tangent indicates the instantaneous rate of change at that point.
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Example 3: Exploring Slope and Tangent a) Find the slope of
between x = a and x = a+h, find the slope of the tangent any
point. b) Given the function
, find the slope of the tangent at any point x = a. Use a limit as
to calculate the slope at x = a. c) Find the slope for x = 2 How to Do It Therefore, at x = 2 the
What to Think About What is the significance 2x as it relates to the original function?
slope is equal to
What is the significance of the slope of the tangent to the function at x = 2?
Draw a rough sketch to illustrate your calculations.
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Your Turn 2 Given the function y x 3 ,
a) Determine the slope of the tangent any point x=a.
b) Evaluate the function x 3
Practice Find the average rate of change of the function over each interval. 1.
2.
3.
4.
5.
6.
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7. You’ve developed a new algorithm for machine learning that is quickly receiving interest in the AI marketplace. Your customer base for your new company has risen at a record pace over the last few months as illustrated in the table below. What is the average growth rate per month for your consumer interest over the months of September and December?
Month
Aug 2017
Sept 2017
Oct 2017
Nov 2017
Dec 2017
Jan 2018
Feb 2018
Mar 2018
# of Customers
256
512
1,024
2,048
4,096
8,192
16,384
32,768
8. Describe what happens to the line tangent to the function of
over the interval [1,
5] between consecutive points with a whole value of x.
9. Describe what happens to the tangent at x = a as a changes for the function
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10. The height, in feet, of a roller coaster entering a full corkscrew is given by the graph below. Estimate its instantaneous rate of change at 6 seconds.
11. The Dow Jones Industrial Average rises and falls during the day. Use the data below to estimate its instantaneous rate of change at 2:05pm.
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12. The curve results at
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has been analyzed with certain calculations yielding the following . What do you expect to happen in the graph of the function
?
13. Determine the slope of the tangent to the graph
at
14. Determine the slope of the tangent to the graph
at
.
.
15. Determine the slope of the tangent to the graph
16. Determine the slope of the tangent to the graph
at
at
.
.
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2.2 Definition of the Derivative Warm Up Sketch a graph of the function
and calculate the slope of the tangent x = 2
Calculate the slope of the curve at x = a using a limit. Use this information and the value of the function at x= 2 to write the equation of a line that is tangent to the curve.
Definition: Tangent line equation at a point
with a slope of
Note: Newton notation for the slope function uses an apostrophe.
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Definition: The derivative of a function, of the domain.
represents the slope of the curve of
at any point
Note: The process of determining the derivative of a function is called differentiation
Example 1: Exploring Slope and Tangent a) Given the function , find the derivative function. b) What is the equation of the tangent at x = 1? How to Do It a)
What to Think About Did you apply any shortcuts to expand the brackets?
What is the significance of the expression ?
b)
What is the pointslope form of a linear equation?
Draw a rough sketch to illustrate your calculations.
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Your Turn a) Given the function , find the slope of the tangent any point x = a.
Did You Know  Newton Notation
Sir Isaac Newton (16421727) invented calculus through his method of fluxions. His notation is useful when the independent variable is time. Newton’s notation for derivatives is expressed as y’ and read as “y prime.” y’ or represents the slope of the equation of the tangent for the graph of y or b) What is the equation of the tangent x = 2?
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respectively.
Example 2: Differentiating Functions a) Use the definition of the derivative to differentiate (find the slope of)
.
b) What is the equation of the tangent when the slope of the tangent equals How to Do It
?
What to Think About What does differentiate mean?
How can we deal with the complex fraction?
Which method do you prefer?
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How to Do It b)
What to Think About What does it mean if there are two xvalues where the slope is
At x = 2
and at x = 2
Therefore, there are two equations where the slope of the tangent is and
88
What does this look like graphically?
?
Your Turn Differentiate
at
to calculate the slope of the tangent at that point. Use the
result to determine the equation of the tangent at that point.
Example 3: Differentiating Trigonometric Functions at a point What is the slope of the function justify your result.
at
? Use the definition of the derivative to
How to Do It
What to Think About Differentiate the function using the concept of limits.
Do you recognize this limit?
What does this look like graphically?
Your Turn What is the slope of the function
at x 0 ?
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Example 4: Differentiating Piecewise Functions Consider the function
. Evaluate
using two different methods.
How to Do It
What to Think About How do you write the function using piecewise notation?
What are two ways to identify the slope of the tangent at x = 2?
OR Why must the derivative of an absolute value function be written in a piecewise fashion? Consider this graphically and algebraically.
The graph is linear at x = 2 and has a slope of 2 which would indicate the derivative at that point.
90
Your Turn Consider the function
. Evaluate
using two different methods.
Practice In questions #14, at the indicated point, identify the slope of the curve, the equation of the tangent, and draw a graph representing the relationship between the tangent and function 1.
at
2.
at
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3.
at
4.
at
In questions #56, describe what happens to the derivative for the entire domain of the following functions 5.
7.
6.
At Wrigley Field in Chicago, IL it is customary for a fan to throw an opposing team’s home run ball back onto the field. Suppose an eager fan in the left field stands throws a ball that follows a flight pattern where its height is measured as a function of time where , how fast is the ball falling 2.5 seconds after it leaves their hand? NB: The rate of change of height (i.e., how height changes over time) is the slope of the tangent to the curve at each point in time and is defined as the velocity of the ball.
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. This represents
8.
Cliff jumping is a very popular pastime in Mexico. Suppose that a diver jumps from a 100ft cliff, what will his speed be when he enters the water if his height is measured as a function of time and ? The rate of change of height represents the velocity of the diver.
9.
At what point is the tangent to the graph of horizontal?
10.
Find an equation for each tangent to the curve
that has a slope of 1.
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2.3 Alternate Definition of the Derivative at a Point Warm Up If a derivative does exist at a certain point, we can use the slope formula in another way to find the derivative or slope at a point. Consider the slope formula as change of y over the change of x
.
What is the slope of the secant line in this case?
f(x)
f(a)
a
x
What would happen if the value of x approached the value a?
How can you write these infinite operations in one mathematical expression?
Which notation is Leibniz and which is Newton?
94
Example 1: Determine the Derivative of a Function Using the Alternate Definition Differentiate How to Do It
using the alternate definition. What to Think About What does the alternate definition solve for?
How do you simplify an expression that involves a radical binomial? OR
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Definition: The Alternative Definition of a Derivative at a Point
Your Turn Differentiate
96
using the alternate definition.
Example 2: Use the Alternate Definition to Determine a Derivative Differentiate How to Do It
using the alternate definition. What to Think About What are the two different algebraic manipulations used?
OR
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Your Turn Differentiate
using the alternate definition.
Did You Know “The Controversy” 17091716 Two 17th century mathematicians both claimed that they invented calculus. Isaac Newton began his work on the development of calculus in 1666 with manuscripts that highlighted his method of fluxions and fluents, which he called the derivative of a continuous function. Newton published his work in 1687 in his book the Philosophiae Naturalis Principia Mathematica. Leibniz first produced his findings in 1684 and 1686 with his writings on differential calculus and integral calculus in his book Nova Methodus pro Maximis et Minimus. However, there was much circumstantial evidence that Leibniz had access to Newton’s manuscripts. In the end, both men have been noted as independent inventors of calculus early in the 18th century after a decade old controversy about the intellectual property related to the invention of modern calculus.
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Example 3: The Function and its Derivative Expressed as a Limit a) Given
represented as a limit, determine
b) Express the derivative of
.
as a limit.
How to Do It
What to Think About What is the relationship between the original function and the derivative as a limit?
a) b)
Your Turn Given the derivative in the form of a limit, determine the original function: a)
d)
b)
e) c)
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Practice For questions #14, find the derivative of the given function at the indicated point using the definition
100
1.
2.
3.
4.
For questions 5 – 8, find the derivative of the given function at the indicated point using the alternative definition
5.
7.
6.
8.
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For questions 9 and 10 determine the derivative of each function. 9.
11. What is
12. Estimate
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10.
?
for the graph of the piecewise function (Hint: Pythagorean Theorem)
2.4 Differentiability of a Function Warm Up Zooming in to “See” Differentiability Graph the following functions and assess the value of their derivative at x = 0.
What do you notice happens to your graphs as you zoom in at
Is it possible to calculate the slope at
?
in
What would the slope be at a point that is a corner?
Local Linearity – all functions that are differentiable at a point x = c are well modeled by a unique tangent line in a neighborhood of c and are thus considered locally linear.
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Definitions:
The Righthand derivative of a function f(x) at x
The Lefthand derivative of a function f(x) at x
Example 1  OneSided Derivatives Can Differ at a Point Show that the following function has lefthand and righthand derivatives at x 0 .
How to Do It
What to Think About What does the graph of the function look like?
What can you conclude about the derivative x = 0?
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Your Turn Show that the following function has lefthand and righthand derivatives at .
Graphically what does this look like?
What is the lefthand derivative at x = 0? What happens when the lefthand and righthand derivatives are not the same? What is the righthand derivative at x = 0?
What is the derivative at x = 0?
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Four ways in which a function fails to be differentiable: Example 2: Corners Graph
and analyze its derivative
How to Do It
What to Think About What is the relationship between the left and righthand derivatives at the corner?
Therefore, the derivative does not exist at the corner.
Your Turn Graph
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and analyze its derivative.
Example 3: Cusp Graph
and analyze its derivative graphically
How to Do It
What to Think About What is the relationship between the left and righthand derivatives at the cusp?
How does the graph of the derivative relate to the graph of the function?
Did You Know Your graphing calculator can graph the derivative function. In the graphing screen, choose 8: nderiv( from the MATH menu.
Therefore, the derivative does not exist at the cusp.
Use x as the independent variable, then enter Y1 or the function of your choice.
Finally, choose “for all x”. on a TI83 calculator enter .
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Your Turn Graph
and analyze its derivative graphically (specifically at x = 2)
Example 4: Vertical Tangent Graph
and analyze its derivative graphically
How to Do It
What to Think About What is the relationship between the left and righthand derivatives at the vertical tangent?
Does the function have a tangent line at ?
There is a vertical asymptote at x = 0, and
108
Your Turn Graph
and analyze its derivative graphically
Example 5: Discontinuity a) Graph b) Graph How to Do It
and analyze its derivative graphically and analyze its derivative graphically What to Think About What is the relationship between the left and righthand derivatives at the discontinuity?
109
Your Turn Graph
and analyze its derivative graphically.
Note: Continuity at a point does not guarantee differentiability as there may be a corner, cusp or vertical tangent. Conversely, however, if a function is differentiable at a point, then it must be continuous at the point.
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Practice For the following graphs, identify the points at which the graph is (a) differentiable (b) continuous but not differentiable (c) neither continuous nor differentiable. Write your solution using set notation or interval notation 1.
2.
3.
4.
5.
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For what values of x do the following functions not have a derivative? 6.
7.
8.
9.
For questions 10 to 13 compare the lefthand derivative and righthand derivatives to justify that the function is differentiable at point P.
112
10.
,P=0
12.
,P=0
11.
13.
,P=1
,P=0
2.5 Symmetric Difference Quotient Warm Up
y = f(x) f(a  h) ah
a f(a + h)
a+h
b
For each interval and point determine whether the derivative is positive, negative, zero, or does not exist?
x=ah
x=a+h
x=b
Draw a secant joining the points on the curve with xvalues a – h and a + h. What is the slope of this secant?
What would happen when the value of h approaches 0?
Express this using proper mathematical notation.
What does this expression approximate?
Note to student: Refer to applet illustrating symmetric difference quotient https://www.geogebra.org/m/ZKUZTaww 113
Definition: The Symmetric Difference Quotient can be used to find the slope at
Example 1: Using the Symmetric Difference Quotient to Estimate the Derivative at a Point. Given the following table, estimate each of the derivatives
How to Do It
0
1
2
3
4
5
5
2
1
3
7
4
What to Think About
Which two points are special cases in determining the derivative of the function?
Can you still apply the symmetric difference quotient at the endpoints?
114
Your Turn Given the following table, estimate each of the derivatives 0
1
2
3
4
5
3
4
6
0
2
3
Example 2: Compute a Numerical Derivative Using Your Calculator Compute the derivative of
at
.
How to Do It
What to Think About (MATH: 8)
This notation indicates that you are taking the derivative of the cubic function, with respect to x when x = 2
Did you know that your calculator can approximate the derivative at a point?
NB: The calculator may only be accurate to 5 decimal places in which case the true answer is
115
Your Turn a) Compute the derivative of
b) Compute the derivative of confirm?
using your calculator.
at
. What value do you expect the calculator to
NOTE: The calculator uses the symmetrical difference quotient to calculate the derivative, therefore the calculator LIES! Be smarter than your calculator!
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Practice 1.
a) Given the numerical values shown, find approximate values for the derivative of. S at each of the xvalues given.
x
0
1
2
3
4
5
6
7
8
18
13
10
9
9
11
15
21
30
b) Over what interval doe the rate of change of to be positive?
appear
c) Where is it negative?
d) Where does the rate of change of greatest?
seem to be
117
2. Given
fill in the table below then use it to estimate
(NB: Round your answers to 3 decimal places)
x 2 3 4
What do you notice? Can you guess a formula for
3. Sketch a possible graph of
118
?
given the following information.
.
Use your calculator to find the derivative of the function at the indicated point. Is the function differentiable at the indicated point? 4.
5.
6.
7.
8.
9.
Find all values of x for which the given function is differentiable 11. 10.
12.
13.
(Hint: Synthetic division)
14.
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15. Let f be the function defined as
.
Justify that the function is continous at x = 1 and that it is differentiable at x = 1.
16. True or False. If f has a derivative at x = a, then the function is continuous. If f is continuous at x = a then the function is differentiable at x = a. Justify your answer.
17. Which of the following is true about the graphs of and at x = 0? a. b. c. d. e.
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It has a corner. It has a vertical tangent It has a cusp It has a discontinuity f(0)=DNE
2.6 Graphing the Derivative of a Function Warm Up
Graph the piecewise function
Graph the derivative of f on the second grid.
What would the vertical axis be labelled?
How could you draw a function if you were given the graph of its derivative?
Note to student: Refer to applet that identifies the value of the derivative https://mathinsight.org/applet/derivative_function 121
Example 1: Sketch f ’ given the function f in blue Given the graph of a function sketch a graph of the derivative. How to Do It
What to Think About When f is increasing what do we know about f’?
When f is decreasing what do we know about f’?
What happens to the slope of f over ?
What happens to the slope at the vertex?
What happens to the slope of f over ?
122
Your Turn Sketch a graph of f ‘ for each function shown below
123
Example 2  Sketch f given the derivative function. Given the graph of
, graph f. Assume
How to Do It f’
What to Think About If , then what can you assume about the graph of f?
If then what can we assume is true of f ?
f
If f’ = 0 then what can we assume of the original graph?
What is the connection between the degree of the polynomial function and the degree of its derivative?
124
Your Turn Sketch f given the graph of f ‘
125
Practice Match the graph of the function with its derivative.
126
1.
a.
2.
b.
3.
c.
4.
d.
5. Given the function shown below sketch a possible graph of f(x).
6. Sketch a possible graph of f(x).
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7. The declination of a celestial body is its angular distance north or south of the equator. The declination of the Sun changes from 23.5 N to 23.5 S and back again during the course of the year. Answer the following questions by estimating the slopes of the graph given below. (Graph is for Northern Hemisphere)
a) On what day is the angle of the sun increasing at the greatest rate?
b) On what day does the angle of the sun have no rate of change?
c) On what day is the angle of the sun decreasing at the greatest rate?
128
8. Over the course of a 12hour day, the value of a stock on the TSX was charted on the graph below. a) What is the value of the stock when the value of the derivative is at its largest? Most negative? Stock Value
Hours
b) What is the value of the stock when it’s derivative is zero?
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9. A father and son have a foot race at the family picnic. The father orchestrates a photo finish where the race is a tie and he meets his son at the finish line. The position of each runner in 10s of metres is given in relation to their time in seconds in the graph below. The father’s progress is graphed in red. a) Describe the speed of each runner throughout the race.
b) At what point are they the farthest apart?
c) When are they running at the same speed?
d) When does the father start running faster than his son and closing the gap?
130
10. Given the graph of each function f below, sketch the graph of f’ below it
a.
b.
c.
d.
e.
f.
131
11. Given the graph of the derivative, f’, below, sketch the graph of the function f.
132
a.
b.
c.
d.
13. Identify in order of least to greatest the following values.
133
2.7 Chapter Review Find the average rate of change of each function over the given interval. 1.
2.
Estimate the slope of the curve at the indicated point using three separate difference quotients 3.
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4. The following graph shows the number of active Uber drivers in the USA. Find the average rate of change related to this increase between July 2013 and July 2014.
5. An object in free fall begins to pick up speed according to the data below. Use the concept of infinitesimals to estimate the speed of the object at 3 seconds.
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6. The intensity of a sound as it moves away from its source will resemble an inverse square law. This is useful in measuring the strength of a radio signal. Estimate how fast the signal is losing its strength at 3 metres away from its source
7. Use the definition of the derivative to determine the slope of the tangent to the graph .
136
8. Use the definition of the derivative to determine the slope of the tangent to the graph .
9. Use the definition of the derivative to determine the slope of the tangent to the graph .
10. At what point is the tangent to the graph of
horizontal?
137
11. At what point does the tangent to the curve
have a slope of
?
12. Determine whether the function has a tangent at the indicated points. If it does, indicate the slope. If not, indicate why.
13. Find the derivative of
138
at x = 5 using the alternate definition.
14. Find the derivative of definition.
at x = 1 at the indicated point using the alternate
15. Identify points on the graph where the function is not differentiable
139
16. Sketch the graph of the function that satisfies the following conditions
17. Determine the derivative at the point x = 3.5 using the symmetric difference quotient and data below x
140
0
1
2
3
4
5
6
1
2
8
5
4
For questions 18 & 19, use your calculator to find the derivative of the function at the indicated point. 18.
19.
20. Given the graph of the function f, below, sketch the graph of the derivative f’.
141
142
Chapter 3: Differentiation Techniques Although the concept of a derivative as the slope of the tangent to a function can be attributed back to mathematicians such as Euclid, Archimedes and Apollonius in the 3rd century BC, their differentiation techniques were limited to the use of infinitesimals. Modern techniques of differentiation are generally credited to Leibniz and Newton, though these are largely built on earlier works submitted by other great minds in the field of mathematics. Newton was the first to apply his techniques to theoretical physics and Leibniz’s notation is still widely used today. Today derivatives are used by: governments in population censuses; economists to ascertain the marginal cost; and, pharmaceutical scientists as they test the body’s reaction to certain medications. Any time you are discussing a rate of change, you are discussing derivatives.
143
3.1 Derivatives of Polynomial Functions Warm Up: the function Use your knowledge of the definition of the derivatives to find the derivative of each function:
Can you find a pattern to avoid using the definition?
Definition of the Derivative (the derivative function):
The Derivative at a Point (2 formulas)
144
RightHand Derivative at a
LeftHand Derivative at a
The Difference Quotient
The Symmetric Difference Quotient
The Power Rule, as with all other derivative formulas, has its origins using limits.
Power rule Proof Using Limits
Proof #2 of the Power Rule using Limits
145
Example 1: Application of the Power Rule for Differentiation Find
for the function
How to Do It
What to Think About Application of the power rule How does this connect the graph of the function and its derivative?
Your Turn a.
e.
b.
f.
c.
g.
d.
h.
Sum rule Difference rule Constant multiple Constant rule
146
Example 2: Application of the Power Rule Within the Sum, Difference & Constant Rules of Differentiation Find
if
How To Do It
What to Think About Application of the power rule How does this follow the graph of the function and its derivative?
Your Turn Find
a)
if: b)
147
Example 3: Determining Tangent Lines Find the equation of the tangent to the curve
at the point
. Support
your answer graphically. How to Do It
What to Think About What does
represent?
Are there any points where can’t exist? Why?
To illustrate, use 2nd Draw 5: Tangent To get the Y1 function on the calculator use VARS, YVARS, 1:Function, 1: Y1
148
To illustrate, use 2nd Calc 6: and enter value for x
Your Turn Find the equation of the tangent to the curve
at the point
. Support your
answer graphically.
149
Example 4: Finding Horizontal Tangents Does the curve have any horizontal tangents? If so, at which xvalues? What are the equations of the horizontal tangents? How to Do It What to Think About What condition must exist for a graph to have a horizontal tangent?
Remember Your graphing calculator can graph the derivative function.
In the graphing screen, choose 8: nderiv( from the MATH menu.
Use x as the independent variable, then enter Y1 or the function of your choice.
Finally, choose “for all x”. On a TI83 calculator enter
How can you use the graph of the derivative to find where a horizontal tangent line exists on f?
There are horizontal tangents at
150
.
Your Turn Does the curve have any horizontal tangents? If so, at which xvalues? What are the equations of the horizontal tangents?
Example 5: Finding Horizontal Tangents using technology Determine the values of x where the curve tangents.
has horizontal
How to Do It
What to Think About What is the slope of a horizontal tangent?
Graph the derivative of the function and identify its zeroes using 2nd Calc Zeroes The derivative has zeroes at , therefore the original function has horizontal tangents at
Your Turn Determine the function values where the curve has horizontal tangents.
151
Practice Find
for each equation:
1.
4.
2.
5.
3.
6.
Determine the values of x for which each curve has a horizontal tangent 7.
152
8.
9.
11.
10.
12.
13. What is the value of the slope of the tangent to the equation
14. Find the equation of the line tangent to the equation
15. What is the value of the slope of the tangent to the equation
at
at
at
?
?
?
153
16. Find the equation of the line perpendicular to the tangent to the curve the point
17. Find the points on the curve
at
when the slope of the tangent is 5. What
feature does this point correspond to on the original function?
18. Find the intercepts of the tangent to the curve (2,16).
19. Where are the tangent lines of
and
when the point of tangency is
parallel?
20. What condition must occur for a function to have a vertical tangent line? Consider both the derivative and the original function.
154
3.2 Higher Order Derivatives Warm Up The heights in feet of a water skier’s jumps are modelled by the equation . How would you determine an equation that represents how the height is changing over time in seconds? What are the units of this equation?
How would you determine an equation that represents how fast the change in height is changing? What are the units of this equation?
What realworld measurement do each of these new equations represent?
155
Example 1: Second and Higher Order Derivatives Find the first four derivatives of How to Do It
What to Think About
Newton Notation
How do you denote the different derivatives in both Leibniz and Newton notation?
Leibniz Notation
Did You Know The 4th derivative is known as the “jounce” and is useful in determining the cosmological equation of state, which is characterized by a dimensionless number w. It is defined as the ratio of its pressure to its energy density. The 5th and 6th derivatives, while useful in applications of theoretical physics, have no universally accepted name. Recently, the use of the fifth derivative and curve fitting were used to perform DNA analysis and population matching.
156
Your Turn Find the first four derivatives of the following functions. a) c)
b)
d)
Example 2: Differentiating a Product Does the derivative of
equal
?
How to Do It
What to Think About Is it possible to rewrite the function in order to have only power terms?
Note: the derivative of a product is not the product of the derivatives!
Your Turn Find the derivative of
157
Product Rule – the product of two differentiable functions u and v is differentiable
Example 3: Differentiating a Product of Functions Find
if
How to Do It
What to Think About Does it matter which function is u and which v?
For the product of polynomials, how can we verify the derivative?
Your Turn Differentiate
158
Example 4: Working with Numerical Values Let
be the product of the functions f and g. Find ,
,
, and
How to Do It
What to Think About What differentiation rule must you use for this function?
Your Turn Let be the product of the functions f and g. Find ,
if
,
if
and
Quotient Rule – at a point where
, the quotient
of two differentiable
functions is differentiable
159
Example 5: Differentiating a Quotient Find the derivative of a)
and b)
How to Do It
. What to Think About What differentiation rule must you use for this function?
a)
or
Does it matter which term goes first in the numerator?
b)
Is there an advantage to keeping your denominator in a factored form?
Your Turn Find the derivative of the following function
160
Example 6: The Quotient Rule with an Application of Numerical Values Let
be the quotient of the functions f and g.
Find
if
.
How to Do It
What to Think About How would you determine the derivative in terms of x for this function?
Your Turn Let
be the quotient of the functions f and g. Determine , and
if
,
,
.
161
Practice For questions 1 to 4 determine the first four derivatives of each function. 1. 3.
2.
4.
For questions 5 to 12 determine 5.
162
6.
7.
10.
8.
11.
9.
12.
163
Given that u and v are functions that are continuous and differentiable, and that , determine the values of the derivative at x = 3. 13.
15. Determine the derivative of
164
14.
using the quotient rule.
For questions 16 & 17, determine the equation of the tangent line to the curve at the given point. 16.
at (1,16)
17.
at (1,2)
18. Explain why
19. Justify why
has a horizontal tangent at y=0
20. How many horizontal tangents does
have?
165
3.3 Velocity & Other Rates of Change Warm Up Analyzing a car’s Performance Suppose you drive your car for 3 hours and travel exactly 180 mi. What is you average velocity?
If the car’s velocity function is during the first 3 hours, is there a time that the car’s velocity is exactly 60 mph? Justify your answer.
Instantaneous Velocity = first derivative of the position function = s'(t) = How does Liebniz notation help to determine the units of a derivative?
Definitions Let s(t) be a position function for an object. Then the velocity function is represents the change in position over time.
When s’(t) = 0, the object is not moving. The object is stopped or at rest. When s’(t) > 0, the object is moving in the positive direction (forward/to the right/up). When s’(t) < 0, the object is moving in the negative direction (backward/to the left/down).
The net change in position over [a,b] is called the displacement and equals s(b) – s(a). The average velocity over [a,b] equals Some objects move onedimensionally. This motion is called rectilinear motion.
166 s'(t) < 0_________________
Example 1: Relating Velocity & Position Given the position function
(t is measured in sec, s is measured in ft)
a) find the velocity function.
d) find the velocity at 2 secs and at 4 secs.
b) find the displacement over the first 2 secs.
e) determine when the object is at rest.
c) find the average velocity after 2 secs.
f) determine when the object is moving in a positive direction and in a negative direction. What to Think About
How to Do It a) b)
What is the relationship between the position function and its velocity?
c) d)
What is the relationship between the position function and displacement?
What is the relationship between average velocity and displacement? e)
f) The object is moving in the positive direction
What is the relationship between the velocity at a point and the position function?
What do zero, positive and negative values of the derivative imply about the function?
The object is moving in the negative direction
167
Your Turn Baby Kaya’s first steps were filmed last week in a 5 second video. The position of her right foot, relative to her left, was recorded as the following function (t is measured in sec, s is measured in cm) a) What is the velocity at time t?
b) When is the right foot at rest?
c) What is the velocity after 1 seconds? 1.5 seconds? 3 seconds?
d) When is the right foot moving forward (in a positive direction)?
e) When is the right foot moving backward (in a negative direction)?
168
Example 2: Relating Velocity to a Positiontime graph Given the position function determine the following. a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero. b) Draw a diagram to represent the rectilinear motion of the object for . How to Do It a) t 0 1 2 3 4 5
s 0 4 2 0 4 20
v + 0 0 + +
b)
from example 1, use the previous solution to c) graph the positiontime graph. d) find the displacement after the first 5 secs. e) find the total distance traveled over the first 5 secs. What to Think About What direction is the object moving during the first second? When does the object change directions? In between stopping points, is it possible for an object to change direction?
Does the positiontime graph indicate which direction the object is moving?
c) Graph a distance / time graph Does the positiontime graph indicate when an object is stopped?
[0,5] by [0,20]
169
d)
How do displacement and total distance differ?
e)
Your Turn Consider baby Kaya’s first steps
from the last example.
a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero. Does this table of values describe all important intervals? Please add any additional points that would be useful.
t 0 1 2 3 4 5
s
v
t
s
v
b) Draw a diagram to represent the rectilinear motion of the object for
170
.
c) graph the positiontime graph.
d) Find the displacement after the first 5 secs.
e) Find the total distance traveled over the first 5 secs.
171
Example 3: FreeFalling Object The vertical height y (in feet) of a coin thrown upwards from the observation deck of the Empire State Building, in time t seconds, is described by the position function: . a) What is the initial velocity at t = 0? b) What is the height of the coin when the velocity is zero? c) What is the speed of the coin when it returns to earth? How to Do It
What to Think About
a)
What does the word initial imply?
b)
What is important about the height when the velocity is zero in this situation?
c)
When does the coin return to earth (hit the ground)?
How does speed relate to velocity? Therefore the speed is 224 ft/sec.
172
Your Turn On a construction site, a dynamic blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of ft after t seconds. a) How high does the rock go?
Did You Know Free Falling to Maximum Velocity Skydivers are familiar with the term “terminal velocity” which states that terminal velocity is the highest velocity attainable by an object as it falls through a fluid (air being the most common example)
b) What is the velocity of the rock when it is 256 ft above the ground on the way up? on the way down? As a skydiver falls towards the earth, his velocity continually gets faster. This change in velocity is known as acceleration. Acceleration is the rate of change of the velocity
c) What is the speed of the rock when it is 256 ft above the ground on the way up? on the way down?
d) When does the rock hit the ground?
173
Example 4: Relating Position to Velocity and Acceleration Consider the position function for a particle at time, t. a) Determine v(t) and a(t) b) Graph s(t), v(t) and a(t) over the interval 0 < t < 8. Use a vertical scale between –40 and 60. c) Describe the motion of the particle. How to Do It a)
What to Think About What is the relationship between velocity, acceleration and position?
b)
What points on the velocity function correspond to minimum and maximum points on the position function?
What feature of the velocity function do the zeros of the acceleration function correspond to?
When the acceleration function is zero and the velocity function is at an extreme value, what is happening on the position function?
174
c)
as
the particle moves forward How do you determine which direction the particle is moving?
On the interval
the particle moves
On the interval
backwards as On the interval moving forwards as On the interval moving backwards as On the interval moving backwards as
the particle speeds up . the particle speeds up
How do you determine if the particle is speeding up or slowing down?
. the particle slows down .
175
Summary A particle is speeding up when its velocity and acceleration move in the same direction (both v and a have the same sign).
A particle is slowing down when its velocity and acceleration move in the opposite directions (v and a have different signs).
Your Turn Consider the horizontal position function of a hummingbird as it approaches a feeder a) Determine v(t) and a(t)
b) Graph s(t), v(t) and a(t) and describe the horizontal motion of the hummingbird over the interval 0 < t < 6.
176
Practice 1. You are training your new puppy Hugo to walk with a leash. His velocity in the first few seconds is shown in the graph below. a) When is he moving forwards?
b) When is he moving backwards?
c) When did he stop?
d) When is his acceleration negative?
e) When is his acceleration positive?
f) When is he moving at his greatest speed?
g) When is he speeding up?
h) When is he slowing down?
177
2. Slacklining is the act of walking along a suspended length of taut, flat ribbon that is tensioned between two anchors. It resembles tightrope walking and is growing in popularity due to its workout benefits, simplicity and versatility. A slackliner in his backyard is walking across the rope on a summer evening. The velocity, v(t), of his movements during time t, 0 ≤ t ≤ 11, is illustrated in the graph below. a) At what time(s) t does the walker change direction?
b) At what time(s) t is the speed of the walker greatest?
c) When is the walker moving forward? Moving backward?
d) When is the walker speeding up? Slowing down?
178
3. The height of a football of an NFL punt is given by the function . a) What is the ball’s velocity as a function of time?
b) What is the ball’s acceleration as a function of time?
c) What was the maximum height of the ball?
d) When did the ball hit the turf?
e) When was the ball at a speed of 4.4mph?
179
4. A golfer in a longdistance drive contest hits a ball whose height is given by the equation . What is the velocity of the ball at 2 seconds?
5. Chris hits a ball out of the sand trap with a height of where h is the height and d represents the horizontal distance the ball travels. What is the maximum height the ball reaches?
180
6. A bungee jumper leaps off a bridge with an initial velocity of 3.6mph and reaches heights of . a) What is his velocity at time t?
b) When does he reach the peak of his jump?
c) When does he hit the water below?
d) What is his velocity when he hits the water?
181
7. In fighting a fire with a hose, a firefighter has access to a tank that holds liters at a time, t seconds after turning it on full blast. a) What is the average velocity of the water in the first 10 seconds of use?
b) What is the instantaneous velocity of the water at t=10 seconds?
8. On the show American Ninja Warrior, during the first hooks of the Sky Hang obstacle, the velocity of a contestant’s body is represented by the function . Find the contestant’s velocity every time their acceleration is zero.
182
9. An NFL football offensive lineman, while doing a wave drill moving forwards and backwards to work on his foot speed, had his position from the goal line marked with the function a) What is the displacement during the first 5 seconds?
b) What is the average velocity in the first 5 seconds?
c) What is the instantaneous velocity at 3 seconds?
d) What is his acceleration at 3 seconds?
e) When did he change direction?
183
10. A teenager flyboarding in San Diego has his height in feet above the water represented by the function for . In order to propel the rider upwards, the instructor must increase the throttle. a) When does the instructor increase the throttle?
b) When does the instructor let off the throttle?
c) What is the total vertical distance traveled by the flyboarder over the first 15 seconds?
d) What is the average velocity of the flyboarder between 0 & 7 seconds?
e) What is the instantaneous velocity of the fly boarder at time 4 seconds?
f) How far below the surface does the fly boarder end up after he lost his balance the first time?
g) What is the maximum height the flyboarder reaches above the surface of the water?
184
11. Given the graph of functions a, b, and c, indicate which represents the function and its first two derivatives.
a
b c
12. Given the functions a, b, c and d, find indicate which represents the original function and its first three derivatives.
b a
d
c
185
13. What is the instantaneous rate of change of the function
at x = 3?
14. What is the instantaneous rate of change of the area of a circle when its radius is 2 in?
15. Identify the derivatives of h, such that have the derivative
and
. Identify the family of functions, .
16. What is the derivative of the product of the three differentiable functions abc?
186
3.4 Derivatives of Trigonometric Functions Warm Up Sketch the graph of
Create a sketch of the derivative graph for
by considering the slope for
several points along the graph above.
Repeat the process again, this time by considering the slope for several points along the graph that you just created.
What can you conclude about the derivative of the sine function?
187
See applet to help identify the value of the derivative of the sine function https://www.intmath.com/differentiationtranscendental/differentiationtrigonometricfunctionsinteractiveapplet.php Derivatives of basic trigonometric functions:
Example 1: Derivatives Involving Trigonometric Functions Differentiate How to Do It
What to Think About What differentiation rule must be used?
Your Turn Find the derivative of: a)
b)
188
c)
Example 2: The Motion of a Weight on a Spring (Simple Harmonic Motion) A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is . What is its velocity and acceleration at time t?
How to Do It
What to Think About What are other examples of periodic motion?
.
Did You Know
Simple Harmonic Motion is periodic motion that can be modeled with a sinusoidal position function. Your Turn A dog in a swing with an appetite for extreme sports sees his height above the ground follow the parameters of the simple harmonic function . What is his velocity and acceleration at time t?
Given his thrillseeking nature and desire for the rush of adrenaline, your dog emphatically requests that you push him through each swing from the back. Mathematically, how do we account for this external factor?
189
Definition: Jerk A sudden change in acceleration is called “jerk”. Therefore, a Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is
Example 3: Deriving the Jerk a) Determine the jerk caused by the constant acceleration of gravity. b) Determine the jerk of the simple harmonic motion in Example 2. How to Do It a)
Fortunately, there is no jerk applicable to gravity!
b)
Your Turn Determine the jerk felt by the dog in the swing
190
What to Think About Does it make sense that gravity has a jerk of zero?
Did You Know Jerk is felt as a change in force such as the boost that a space shuttle gets during its launch as it exits the atmosphere. For those of us that will never visit the moon, we can still experience the jerk. Roller coasters are designed to stimulate the system by elevating the acceleration through controlling its rate of change, magnitude, and duration. These changes will prey upon your body’s ability to handle gforces as it demands the designers to have an impeccable understanding of the jerk!
Example 4: Determine How to Do It
What to Think About How can you rewrite the tangent function in order to derive it?
Which differentiation rule must we apply to find the derivative?
Your Turn Find
.
191
Derivatives of Other Basic Trigonometric Functions
Example 5: Finding Tangent and Normal Lines Find the equations for the lines that are tangent and normal to the graph of at x = 2. Use a calculator to round all significant values to the nearest hundredth and support your conclusion graphically. Use the window [5,5] by [4,4] How to Do It
192
What to Think About How do you enter calculator?
into your
Your Turn Find the equations for the lines that are tangent and normal to the graph of x=
at
.
Practice For questions 1 to 10 determine 1.
4.
2.
5.
3.
6.
193
7. A bungee jumper is bouncing in simple harmonic motion with its position . What is the jerk at time t?
8. What is the equation of the line that is tangent to the graph of
9. Over the domain
, find the point(s) on the curve
at
?
that have
tangents with a slope of 2.
10. What is the equation of the horizontal tangent to the curve
194
?
11. What is the value of a in which the function
will be differentiable at
x = 0?
12. Evaluate the derivative of for
at
. (Hint: use your trigonometric identities
.
A child playing with a yoyo has its position at
.
13. Is it traveling upwards or downwards at
14. Is it speeding up or slowing down at
?
?
195
3.5 Introducing the Chain Rule Warm Up A composite function is a nested function of the form
. Identify the inside and
the outside functions in each composition:
a)
Outside Function y=
Inside Function u=
b)
y=
u=
c)
y=
u=
d)
y=
u=
e)
y=
u=
Find
Find
where Why can’t you simply use the power rule on the original function?
196
Definition: The Chain Rule
Example 1: Taking the Derivative of a Composite function Given the function
what is
?
How to Do It Using simplification & power rule
What to Think About What differentiation rule can you use to find the derivative of this function?
Using Chain Rule
Your Turn Determine the derivative of the following functions a)
c)
b)
197
Example 2: Application of the Chain Rule in the Context of a Trigonometric Derivative a) Graph the function and its derivative using your graphing calculator b) Graph to see the graph of the derivative of this function How to Do It What to Think About What should you input into your a) calculator?
From the graphs what do you think the derivatives are?
What should you input into your calculator?
b)
From the graphs what do you think the derivatives are?
Why does the amplitude of the derivative graph change and what does it mean?
198
Your Turn a) Plot the function
and its derivative using your graphing calculator and show the
graph on the axes below:
b) Now graph
and its derivative
c) Why does the amplitude increase by a factor of 3? Explain in your own words.
199
New Rules for Composite Trigonometric Functions:
Practice For questions 1 to 8 determine 1.
2.
3.
200
.
4.
5.
6.
7.
8.
201
9. An Olympic swimmer doing the butterfly has his center of gravity found by the function
.
What is the velocity of the swimmer as a function of time?
For questions 10 and 11 determine 10.
11.
202
Determine the derivative of each function 14.
12.
13.
15. The pedals on a spin bike during a recreational spin class rise and fall such that their position is satisfied by the equation where r is the length of the pedal arm, s is the number of revolutions per second and t is the time in seconds. If you double the number of revolutions per second on the bike, how does this affect the rotational velocity and acceleration of the pedals?
203
3.6 Chapter Review Differentiate each equation: 1.
5.
2.
6.
3.
7.
4.
204
For questions 8 to 10 determine the values of x for which the function has horizontal tangents 8. 10.
9.
For questions 11 and 12 determine the slope of the tangent at the indicated point 11. 12.
13. Show that
has no tangents with a slope of 2.
205
For questions 14 & 15 determine the equation of the tangent line and the normal line at the 14.
15. For what value(s) of x are the tangent lines of and parallel?
For questions 16 and 17 determine the first four derivatives of the given equation. 16. 17.
For questions 18 to 21 differentiate the given equation. 18.
206
19.
20.
21.
For questions 22 to 25, given that u and v are continuous and differentiable functions, and that , , , and , determine the values of the derivatives at . 22.
24.
23.
25.
207
Use the following information to answer questions 26 to 33: A particle is moving along the xaxis with its velocity function, v(t), shown in the figure below. 26. When is the particle moving to the right?
27. When is the particle moving to the left?
28. When did it stop?
29. When is the particle’s acceleration negative?
30. When is the particle’s acceleration positive?
31. When is the particle moving at its greatest speed?
32. When is the particle speeding up?
33. When is the particle slowing down?
208
Use the following information to answer questions 34 to 39: A particle in rectilinear motion follows the position function in the first 7 seconds of its journey. 34. What is the velocity at time t?
35. What is the velocity after 2 seconds? 5 seconds?
36. When is the particle at rest?
37. When is the particle moving to the right (in a positive direction)?
38. What is the displacement during the 7 seconds?
39. What is the total distance traveled from time t = 0 to t = 7 sec?
209
Use the following information to answer questions 40 to 42: The outside sphere on a pendulum on a desk continues in a simple harmonic motion in which its height is measured as . 40. Determine its velocity and acceleration at time t.
41. Find the velocity, acceleration at time
42. Identify the direction of the pendulum and whether it is speeding up or slowing down.
210
Differentiate each function: 43.
44.
45.
46.
47.
211
48.
49.
50.
51.
212
Chapter 4: Advanced Differentiation Techniques Noise cancelling headphones take the noise accumulated, measured in decibels, outside of the ear phones and constructs a function to represent it. It order to cancel it, that function is embedded within a function that builds its opposite. It would require an application of the chain rule to determine the instantaneous change in decibels required to cancel out random sounds.
Entering a corkscrew turn in a hanging roller coaster, the center of gravity of one of the riders follows the pathway defined by … it requires an application of implicit differentiation to determine how fast the height of the person is changing (and the G forces applied on the body!)
213
4.1 The Chain Rule Warm Up Determine the derivative of the following functions a) b)
Which rule must be applied to differentiate each of the above functions?
Write each function as a composition of functions. Write each with an outer function of and an inner function of . a) b)
Example 1: Using the New Rules and Beyond Determine How to Do It
214
for
and What to Think About Which differentiation rule is needed for this type of function?
Your Turn Differentiate the following functions a)
c)
b)
Example 2: Using Multiple Laws of Differentiation Differentiate How to Do It
What to Think About Which laws of differentiation are applicable?
215
Your Turn Differentiate each composite function. a)
c)
b)
d)
Example 3: Rectilinear Motion An object moves along the xaxis so that its position at any time is given by . What are the velocity and acceleration of the object as a function of t? How to Do It
216
What to Think About How are position, velocity and acceleration functions related?
Your Turn An object moves along the xaxis so that its position at any time is given by . Find the velocity and acceleration of the object as a function of t.
Example 4: Equation of the Tangent Line What is the equation of the line tangent to the curve How to Do It
at the point where
?
What to Think About What is the tangent line equation in point slope form?
Which special triangle will apply in the evaluation of the point of tangency?
217
Your Turn a) What is the equation of the line tangent to the curve
at the point where
?
b) Show that the slope of every line tangent to the curve
is positive.
Did You Know Biologists employ the Chain Rule when they determine the speed of growth of a bacteria population
218
Example 5: Composition of Multiple Functions What is the derivative of
?
How to Do It
What to Think About How many times will the chain rule be applied?
OR Let
,
,
. Then
Could we use a substitution method to make the problem simpler?
Your Turn Differentiate the following functions a)
b)
219
Practice For questions 1 to 8 determine 1.
. 4.
2. 5.
3.
6. What is the largest value of the slope of the curve
220
4.2 Composite Functions and Function Notation Warm Up Given
and
, determine
a)
b)
Given
and
, determine
a)
b)
Does the order of function matter when using this notation?
221
Example 1: Derivative of a Linear Composite Function Using two different methods, determine
if
How to Do It
if
,
More Notation for the Chain Rule: Given the composite function Then:
222
and
What to Think About Is it advantageous to determine the expression for the composite function before differentiating?
OR
Your Turn Determine
,
and
Example 2: Derivative of a NonLinear Composite Function Given the following information determine the value of at
given
and How to Do It
What to Think About What differentiation rule is required for this function?
OR Which notation would you prefer to use Leibniz or Newton? OR
What are the advantages of both?
Your Turn Determine
at the given value of x. at
223
Example 3: Calculate the Derivative Given a Table of Values Suppose: Function
and the following table of values is true: x=
1
2
3
5
4
7
3
3
2
7
7
8
5
9
12
Determine: a) b) h’(3) c) How to Do It a)
What to Think About As is a composite function, what must be applied to derive it?
b) How can you write solely in terms of a composite function in terms of and
c)
224
Your Turn Suppose the functions f and g and their derivatives have the following values at x
f(x)
g(x)
2
8
2
3
6
4
and
f’(x)
g’(x)
3π
5
Evaluate the derivatives with respect to x of the following combinations at the given value of x.
a)
at x = 3
b)
c)
at
d)
at
e)
at
at
225
Example 4: Calculate the Derivative Given Specific Values Suppose
,
,
, and
How to Do It
. What is
?
What to Think About Do you need to know what the functions are to solve this type of question?
Your Turn If
,
,
, what is
b) If
?
, , what is
226
, and
, ?
, and
Did You Know In a memoir by Gottfried W. Leibniz, it is apparent that he attempted to use the Chain Rule to differentiate a polynomial inside of a square root. Although the memoir contained various errors, it is believed to the place in which the Chain rule originated. The Chain Rule is a relationship between three rates of changes, stating that the first is a product of the other two. In Leibniz notation is
where y is a
function of u, and u is a function of x.
Practice Determine the value of
at the given value of x.
1.
3.
2.
4.
227
Given that the functions f and g and their derivatives have the following values at x = 1 and x = 2. x 1
8
2
3
2
3
4
5
Calculate the derivative of the following.
228
5.
7.
6.
8.
4.3 Implicit Differentiation Warm Up Graph the following two functions on the grids provided. Determine the equations of their tangents at x =2 and sketch them on the graph.
What do you notice about the slopes of these two functions?
229
Example 1: Implicit Differentiation of a Square Root Relation Given the relation , how would you determine the slope of the tangent at the points (4,2) and (4,2)? Graphically support your findings. How to Do It
What to Think About Is this equation a function? What are the obstacles in differentiating a relation?
How can you differentiate a relation that has more than one point at an x value? OR Is it possible to simply move lefttoright along the equation and differentiate each term with respect to x?
How is differentiating y2 with respect to x similar to the chain rule?
What are the advantages of deriving this relation using implicit differentiation?
How would you graph this relation using a graphing calculator?
230
Your Turn Given the relation
, how would you determine the slope of the tangent at the points
(16,2) and (16, 2)? Graphically support your findings.
Example 2: Implicit Differentiation Involving a Circle Find the slope of the circle How to Do It
at the point (4, –5) using implicit differentiation. What to Think About How do you write y explicitly in terms of x in order to derive it?
What does this look like graphically?
231
Your Turn Find the slope of the circle
at the point
using implicit differentiation.
Note: Implicit differentiation required differentiating each term with respect to x. Whenever a variable is involved in the equation that is a function of x, you can consider it as a function inside the relation and apply the CHAIN RULE. This is another way to think of implicit differentiation.
232
Example 3: Implicit Differentiation Involving an Ellipse Find the tangent and normal to the ellipse
How to Do It
at the point (2,3)
What to Think About What differentiation rules must you apply to find the derivative implicitly?
What is the relationship between the slope of the tangent and the slope of the normal?
233
Your Turn What are the tangent and normal to the ellipse
at the point (1,3)?
Example 4: Using Implicit Differentiation in a Proof Show that the slope How to Do It
is defined at every point on the graph of What to Think About Which differentiation technique must be applied?
What is the link between continuity and differentiability?
234
Your Turn Find a)
of the following relations b)
Example 5: Implicit Differentiation Involving a Hyperbola Find if How to Do It What to Think About Moving lefttoright and differentiating as you go, which rules do you need to apply to differentiate with respect to x?
What differentiation rule must you apply when taking the second derivative?
How could you simplify the second derivative?
235
Your Turn Find
if
Practice For questions 1 to 5 determine 1.
4.
2.
5.
3.
236
For questions 6 and 7 determine the slope of the curve at the indicated point. 6. 7.
For questions 8 to 10 determine the equation of the tangent line to the curve at the indicated point. 10. 8.
9.
237
11. Determine
12. On the circle
238
for the function
, what is the value of
at the point
?
4.4 Derivatives of Inverse Functions Warm Up What are three characteristics of an inverse function?
Graph the linear function
in blue and its inverse in red. What do you notice
about the slopes of their tangents?
Suppose (a,b) is a point on the curve y = f (x) or, in other words, f (a) = b. Since an inverse function has the x and y coordinates interchanged, then (b,a) is a point on the curve y = f 1(x) where f 1 (x) represents the inverse function. Use the following applet to explore the relationship of the slopes at their inversed points. http://webspace.ship.edu/msrenault/geogebracalculus/derivative_inverse_functions.html
239
Definition: The relationship between the slope of the inverse functions at their “inversed” points is they are reciprocals of one another. The relationship between the slopes of the tangent line to f at (a,b) and the slope of the tangent line to f 1 at (b,a)? OR
Example 1 – Determine the Slope of the Inverse at a Point If f (3) = 7 and How to Do It
, what are
and
? What to Think About What is the point on the function?
What is the corresponding point on the inverse function?
240
Your Turn x
f(x)
2
5
5
8
Given the information in the table above, if g is the inverse of f, what is the slope of the tangent line to g at the point where x = 5?
241
Example 2: Determine the Slope of the Inverse at a Given Point Let
, and let g be the inverse function. Evaluate
How to Do It Analyze the inverse function at (0, ?). Therefore we analyze the original function at (?, 0). By inspection x = 1 (or you can use synthetic or long division. The point on the original functions is (0, 1) and point on the inverse function is (1,0)
.
What to Think About What are the two different ways you can solve this problem?
OR The equation of the inverse is Use implicit differentiation
Note that this is the y coordinate of the inverse.
242
What does it mean to solve by inspection?
Your Turn a) Let
b) Let
, and let g be the inverse function. Evaluate
, and let g be the inverse function. Evaluate
.
.
243
Practice 1. Suppose that
, find the value of
.
2. Given the following function values for f, determine
x
f(x)
2
6
5
4
For questions 3 to 8 determine the derivative of the inverse at the indicated value for x. 3.
244
4.
5.
6.
7.
8.
245
4.5 Derivatives of Exponential and Logarithmic Functions Warm Up Graph
. Then graph in bold the derivative of this function using the numerical
derivative feature of your calculator:
What do you notice about the function and its derivative?
246
Did You Know e is an irrational number. This constant, called Euler’s Number, named after Leonhard Euler, to 9 decimal places equals: 2.71828128. More formally
247
Derivative of base e exponential functions
Example 1: Derivative of the Exponential Function as Part of a Composite Function Find the derivative of How to Do It
What to Think About What is the embedded function?
Your Turn Derive the following functions a)
e)
b)
f)
c)
g)
d)
248
Did You Know e & Compound Interest Compound Interest is found given the formula
where F = Future
Value, P = Present Value, r = rate, n =
number of payments. Substitute
and
you will get:
This is why e appears in many banking formulas. It also hints at the idea that a banker, prior to a mathematician, discovered the value of e! Necessity is the mother of all inventions.
Example 2: How Fast Does a Flu Spread? The spread of a flu virus in a major city is modeled by the equation
where a) b) c) d)
is the total number of citizens infected t days after the flu was first discovered.
Graph the function with the dimensions Estimate the initial number of citizens infected with the flu. How fast was the flu spreading after 3 days? Graph the derivative with the dimensions spread at its maximum rate? What is this rate?
How to Do It a) Graph the function
.
; when will the flu
What to Think About in the window shown.
What point on the original graph identifies the maximum rate of the spread of the flu?
249
b) Estimate the initial number of citizens infected with the flu.
What are two methods to find the maximum rate of the speed of the spread of the flu?
c) How fast was the flu spreading after 3 days?
c) Graph the derivative; when will the flu spread at its maximum rate? What is this rate?
How fast is the flu spreading by day 30?
250
Your Turn
The growth of the Spanish Influenza (1918), in its first 4 months spread at a rate that can be modelled by the equation
where P is the number of citizens and t is the time in
months. a) Graph the function
with the dimensions
.
251
b) Estimate the initial number of citizens infected with the flu.
Did You Know The Natural Logarithm In Latin, “logarithmus naturali”, denoted by the button on your calculator, is the inverse of e.
c) How fast was the flu spreading after 3 days?
d) Graph the derivative with the dimensions ; when will the flu spread at its maximum rate? What is this rate?
The Scottish mathematician, John Napier (1550 – 1617), nicknamed the Marvelous Merchiston, was studying the motion of someone covering a distance by dividing the time into short intervals of length. He coined the name for this relationship using the Greek words logos (ratio) and arithmos (number). This term became Latinized into the current version, logarithm. Napier is best known as the discoverer of the concept of logarithms which provided a platform for later scientific advances in astronomy, dynamics, and other branches of physics. Napier’s Counting Tables
252
Example 3: Prove the Derivative of the Natural Log Derive How to Do It
What to Think About How can we rewrite the natural logarithm into a form that we can differentiate?
As the function y is not written explicitly in terms of x while in exponential form, what type of differentiation must be applied?
Definition
Example 4: Application of the Derivative of the Natural Logarithm Find the derivative of How to Do It
What to Think About What is the embedded function?
OR
Identify any restrictions
253
Your Turn Determine the derivative of each function: a)
e)
b)
f)
c)
g)
d)
254
Practice Determine 1.
7.
2.
8.
3. 9.
4.
10.
5.
11. 6.
255
12. At what point on the graph of
is the tangent line perpendicular to the line
?
13. A line with slope m passes through the origin and is a tangent to the curve What is the value of m?
14. Find the equation of the tangent line to the curve
256
.
and passes through the origin.
15. Find the equation of the normal line to the curve origin.
that passes through the
Laws of Logarithms
Recall Proof:
257
4.6 Logarithmic Differentiation Warm Up Use the applet below to visually identify the graph of the following exponential functions http://webspace.ship.edu/msrenault/geogebracalculus/derivative_exponential_functions. html
All of the graphs have one common coordinate. What is this coordinate?
What happens to the graph of the derivative for the functions mentioned above as you change its base?
Why would the derivatives of their graphs be slightly different?
Logarithmic Differentiation – the process of taking the natural logarithm of both sides of an equation, differentiating, and then solving for the desired derivative. This is applicable when ___________________________________________________________________
258
Example 1: Both the Base and the Exponent are Variables What is the derivative of
?
How to Do It
What to Think About When is logarithmic differentiation applied?
How is this function different from the function ?
Your Turn What is the derivative of
?
259
Example 2: The Derivative of an Exponential Function with a Constant Base and a Variable Exponent What is the derivative of
?
How to Do It
What to Think About How does this function differ from the previous example?
Your Turn What is the derivative of
?
We now have a new “shortcut” formula for taking the derivative of a function in the form where a is a constant. Definition
260
Example 3: Using the Laws of Logarithms At what point on the graph of the function
does the tangent line have a slope of 10?
Round all values to the nearest hundredth. How to Do It
What to Think About What will we have to equate to each other in order to solve this problem?
261
Your Turn At what xvalue on the graph of the function 8? Round all values to the nearest hundredth.
262
does the tangent line have a slope of
Definition
Example 4: Develop the Derivative of a Common Logarithm Develop the derivative of How to Do It
What to Think About How can we determine the derivative of a logarithm without being given the formula?
What type of differentiation is required?
Your Turn What is the derivative of
?
Did You Know At a rock concert, the sound pressure can be measured
with the logarithmic function
where W
represents the sound (in watts) and W0 represents the lowest threshold of sound that humans can detect. The application of the derivative of a common logarithm would determine the change in sound pressure (in decibels) over time.
263
Example 5: The Derivative of a Logarithmic Composite Function What is the derivative of
?
How to Do It
What to Think About Which is the embedded function?
Your Turn Determine the derivative of each function. a)
d)
b)
e)
c)
264
Example 6: A Tangent Through the Origin A line with slope m passes through the origin and is tangent to the graph of What is the value of m? How to Do It Two points on the graph are (0, 0) and
.
What to Think About What are two methods to determine the slope of the tangent line?
Your Turn A line with slope m passes through the origin and is tangent to the graph of the natural logarithm function . What is the value of m?
265
Practice Determine
266
for each equation.
1.
7.
2.
8.
3.
9.
4.
10.
5.
11.
6.
12.
13.
14.
15. Prove that the graph of
approaches a horizontal tangent as
16. A glass of cold milk from the refrigerator is left on the counter on a warm summer day. Its temperature y (in degrees Fahrenheit) after sitting on the counter t minutes is a) What is the temperature of the refrigerator? How can you tell?
b) What is the temperature of the room? How can you tell?
267
4.7 Derivatives of Inverse Trigonometric Functions WarmUp Given the function
, find
Support your findings graphically using the Draw Inverse function on your calculator.
Determine the equation of the tangent to
at
Determine the equation of the tangent to
at
.
.
Compare the slopes of the equations and make an inference about the relationship between the two graphs.
268
Example 1: Derivative of the Arcsine Function a) Determine the inverse of the sine function written explicitly in terms of x b) Determine the derivative of the inverse of the sine function c) Graph the sine function and its inverse. Identify their domain and range How to Do It
What to Think About How do you write the inverse of sine explicitly in terms of x?
a)
We don’t know the derivative of its inverse written explicitly in terms of x, so we need another approach How would you write your derivative solely in terms of x?
Need this with respect to x using graphical representation of sine inverse.
How does this relate to the Pythagorean Theorem?
1 x
y a
269
b)
c)
Domain Range
is the reciprocal of
270
Definitions
Your Turn a) Determine the inverse of the cosine function written explicitly in terms of x
b) Determine the derivative of the inverse of the cosine function
c) Graph the cosine function and its inverse. Identify their domain and range
Domain Range
271
Definitions
Example 2: Application of the Derivative of a Trigonometric Inverse Determine: How to Do It
Your Turn Determine
272
What to Think About Which is the embedded function?
Example 3: Derivative of the Arctangent Function a) Determine the inverse of the tangent function written explicitly in terms of x b) Determine the derivative of the inverse of the tangent function c) Graph the restricted tangent function over the domain
and its inverse.
d) Identify their domain and range How to Do It
What to Think About How do you write the inverse of tangent explicitly in terms of x?
a)
We don’t know the derivative of its inverse written explicitly in terms of x, so we need another approach b)
We need to restate this with respect to x – substitute using graphical representation of tangent inverse
c
How would you write your derivative solely in terms of x?
x
y 1
273
How does this relate to the Pythagorean Theorem?
c)
Domain Range
274
Definition
Your Turn a) Determine the inverse of the cotangent function written explicitly in terms of x
b) Determine the derivative of the inverse of the cotangent function
275
c) Graph the restricted tangent function over the domain their domain and range
d) Domain Range
Definition
276
and its inverse. Identify
Example 4: Inverse Trigonometric Functions and Rectilinear Motion A particle moves along a line so that its position at any time the velocity of the particle when ? How to Do It
is
. What is
What to Think About Can you use the Chain Rule?
What is the relationship between the position function and the velocity function?
Your Turn A particle moves along a line so that its position at any time the velocity of the particle when
is
. What is
?
277
Example 5: Inverse Trigonometric Functions and Their Tangent Lines a) Determine an equation for the line tangent to the graph of
at the point
.
b) Determine an equation for the line tangent to the graph of
at the point
.
How to Do It a)
b)
278
What to Think About Before you try to solve it algebraically what is the relationship between their slopes?
Definition If the equation of the tangent line to f at equation of the tangent line to f 1 at
is is
Your Turn a) Determine an equation for the line tangent to the graph of
then the .
at the point
.
b) Determine an equation for the line tangent to the graph of
at the point
.
279
Definition: Derivative of the inverse trigonometric functions.
Practice For questions 10 to 12 determine 1.
2.
280
3.
4.
6.
5.
7. A particle moves along an xaxis such that its position is designated by the function , where t represents the time in seconds. Find the velocity at the time indicated.
281
4.8 L’Hôpital’s Rule
L’Hopital’s rule is clinically proven for use on limits of indeterminate form, specifically those with conditions
or
. Many notable
mathematicians have rigorously proven the
effectiveness of its application in resolving indeterminate limits through centuries of intense scrutiny.
Instructions for use: differentiate the numerator and denominator of the limit separately, and then take the limit of the new quotient. Repeat as required. WARNING! Use only as directed. Do not apply to limits other than those listed above. Application in place of other forms of algebraic simplification may prove to be harmful to your solution. DISCLAIMER: the brilliance of L’Hopital is not responsible for any damage stemming from improper use, including, but not limited to the following:
282
Lost points on AP Calculus assessments
Damage caused my miscalculation of the velocity of an object.
Example 1: Applying L’Hopital ‘s Rule Evaluate How to Do It
What to Think About Do you recall this limit?
Which indeterminate form do you have?
Why do you believe L’Hopital’s Rule to be true?
Your Turn What is
?
283
Example 2: A Second Application of L’Hopital ‘s Rule Evaluate How to Do It
Your Turn Determine
284
What to Think About Which indeterminate form do you have?
Did You Know For Sale – Mathematical Respect £300! As it turns out, L’Hopital’s rule was not actually discovered by Guillaume de L’Hopital, but the Swiss mathematician Johann Bernoulli! In 1691, the acclaimed French mathematician met with the talented, yet widely unknown entity, Bernoulli. L’Hopital, recognizing the young mathematician’s obvious talents and wanting to learn from him, employed him to give private lessons for an initial retainer of 300 pounds under the caveat that none of their work would be disclosed and exclusive right to any of Bernoulli’s discoveries during that time. Under this business agreement, the rule was discovered in 1964 by Bernoulli, but not published until 1969 in L’Hopital’s calculus textbook entitled Analise Des Infiniments Petits (which included a translated copy of the arrangement they entered into collegially). L’Hopital did acknowledge the input of Bernoulli in a letter to Leibniz regarding the rule that bears his name!
Practice Determine each limit 1.
3.
2.
4.
285
5.
8.
6.
9.
7.
286
4.9 Chapter Review For questions 1 to 9 determine 1.
5.
2.
6.
3.
7.
4.
8.
287
9. Determine the equations of the tangent line to the curve
10. Does the graph of
at x = 1
have horizontal asymptotes over the interval
?
Use the following information to answer questions 11 to 16. Given that the functions f and g and their derivatives have the following values at x = 3 and x=4 x
11.
288
3
−6
4
3
4
3
−4
−4
16
−1
7
5
67
6
11
−7
−4
−8
3
−3
12.
8
1
13.
15.
14.
16.
For questions 17 to 19 determine 17.
18.
19.
289
For questions 20 and 21 determine 20.
: 21.
For questions 22 to 24 determine the equation of the tangent line to the curve at the indicated point.
290
22.
at
23.
at
24.
at
x 2
6
5
2
Use the table above to answer questions 25 and 26. 25. Given that g is the inverse of f, determine
26. Given that g is the inverse of
, determine
291
Differentiate the following functions: 27.
32.
28.
33.
29.
34.
30.
35.
31.
292
36. On what interval(s) is
for
37. For what values of x is
?
parallel to
38. What is the equation of the tangent line to the curve
?
at the point
?
39. You are attempting an inside loop with your new RC model airplane. It is following the route represented by . At what rate is it climbing when
?
293
Evaluate the following limits 40.
43.
41.
44.
42.
294
Unit 2 Project  Design a Roller Coaster
You have been hired to design a new roller coaster for Belmont Park in San Diego. Your coaster must satisfy the following safety conditions: no descent can be over design must begin with a
, the
incline, never get as high as its previous high point and end at
the same height it starts. Write a summary and present your results to the class as if you were trying to convince them to invest in and build your coaster, since your design is the best! 1.
In small groups discuss some possible themes for your design. From your experiences discuss what elements you think make a good roller coaster. Come up with a plan to design the world’s newest roller coaster! You must include the following: Functions you use must be differentiable and continuous at all points along the curve of the roller coaster. Use at least 5 distinct functions.
295
You need to show the graph and a proof that the function you generate is continuous and differentiable at the points where each pair of curves meet. You may use any type of function including linear, trigonometric, exponential, logarithmic, inverse trigonometric, and polynomial. 2.
Research the best roller coasters in the world, answer some of the following questions (and get some inspiration for designing your own roller coaster):
3.
Choose a theme for your coaster and transfer your design to a piece of poster paper. It can be drawn freehand, but it should look like what you have drawn on Desmos. Tape or glue several pieces of poster paper together horizontally. You are encouraged to use Desmos or any other graphing application to print your functions. Make sure that you use proper scaling to ensure the shape is accurate when enlarged
4.
Cut out the roller coaster so that the top of it is your track. Then tape or glue one end of the poster paper to the other end. You will end up with a 3dimensional representation of your roller coaster. Below each of your curves, write its function. Use a heavy dot to show the transition points. On the poster paper itself, show that the curves are continuous and differentiable at those points.
Answer the following questions:
296
What makes them great? Where are they? What kind of names do they have? How much does it cost to ride them? How much did it cost to build them? How fast do they go? If you could ride one roller coaster in the world, which one would you most like to ride and why?
Where is the path increasing and decreasing? Where is it concave up and down? For each fall, where is the steepest descent and how steep is the angle? Draw a graph of the slope of the path verses the distance along the ground from the start (graph the first derivative). Draw a graph of the rate of change of the slope (graph the second derivative). The thrill of the roller coaster is defined as the sum of the angle of steepest descent in each fall (in radians) plus the number of tops. Calculate the thrill of each coaster. Approximate how fast your best roller coaster will travel, showing your assumptions and calculations.
Unit 2: Applications of Derivatives This unit focuses on the following AP Big Idea from the College Board:
Big Idea 2: Applications of the derivative include analyzing the graph of a function (for example, determining whether a function is increasing or decreasing and finding concavity and extreme values), and solving a variety of realworld applications including optimization.
By the end of this unit, you should be able to: Use derivatives to analyze properties of a function Solve problems involving optimization Understand and apply the Mean Value Theorem Connect the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval through the Mean Value Theorem.
By the end of this unit, you should know the meaning of these key terms:
Increasing Decreasing Maximum Minimum Horizontal inflection point Vertical inflection point Critical point Concavity
At what rate does the surface area of a spherical object in its volume
Extreme Value Inflection point Mean Value Theorem Rolle’s Theorem Optimization First derivative test Second derivative test
change in relation to the change
?
297
Chapter 5: Analyzing Functions Using Derivatives We understand our world by observing change. In almost every field of study, an observer can collect data and analyze the change behavior of the situation. Knowing how something is changing over time allows the researcher to make prediction about the future and the past. Mathematicians can create models (or functions) of the rates of change data; and these can be used to discover the conditions that would optimize the situation. Optimization is a key application of the derivative.
298
5.1 The First Derivative Test Warm Up Given the graph of f’
Determine the following characteristics of f Characteristics Interval Characteristics of f or values of f f is increasing since on f is decreasing on f is neither increasing or decreasing at
since since
Sketch a graph of f that passes through (0,0) and (2,–4) with zeroes at x = 0 and 3.
Is there a way to determine where the maximum and minimum values of a function occur?
Definition: An extreme value of a function f is called a relative (local) maximum if there is an open interval containing c on which is a maximum value. An extreme value of a function f is called a relative (local) minimum if there is an open interval containing c on which is a minimum value. If or if is undefined at c, then c is called a critical point of f.
299
Example 1: Determining the extreme values of a function Given
, find the maximum and minimum values of f using the derivative.
How to Do It
What to Think About What information do we need to determine the behavior of f?
Determine the derivative of f
What are the critical points? What intervals are defined by the critical points? Therefore, f has critical points at
.
Choose a test point in the intervals defined by the critical points.
How do you determine whether f is increasing or decreasing over the interval?
Interval Test value
+

Could you use the graph of to determine the intervals where f is increasing or decreasing?
+
Since changes from positive to negative at has a relative maximum value at .
,
Since changes from negative to positive at has a relative minimum value at .
,
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What are the extreme values of at each critical point?
Theorem: The First Derivative Test Let c be a critical point of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then is a: o relative (local) maximum if changes from positive to negative at c. o relative (local) minimum if changes from negative to positive at c. Let a be the left endpoint of a function f that is continuous on a closed interval o is a relative (local) maximum if . o
is a relative (local) minimum if
.
Let b be the right endpoint of a function f that is continuous on a closed interval o is a relative (local) maximum if . o
is a relative (local) minimum if
.
Your Turn Find the extreme values for each of the following functions. a)
b)
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Example 2: Determining the extreme values of a function on a closed interval Given defined on the closed interval find the extreme values of . How to Do It
What to Think About What are the zeroes (xintercepts) of ?
Sketch the graph function
What is the yintercept of ?
What are the endpoints of
?
Determine the critical points
What does the rate of change information provided by the derivative function indicate about the behavior of at each value of x?
Therefore,
has critical points at
.
To determine when is positive or negative, choose a test point in the intervals defined by the critical points.
302
Interval
éë 2,0
)
(0,1.5)
(1.5,2ùû

+
Why is there not an extreme value at every critical point?
Test value

Using the first derivative test, we conclude the following: o
does not have an extreme value at does not change sign.
o
has a relative minimum at from negative to positive.
since
changes
has a relative maximum at the left endpoint .
o
has a relative maximum at the right endpoint since .
There is one minimum at
?
since
o
There are two maxima, one at .
What happens at
and one at Are there any values on where f has an overall maximum or minimum?
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Definition: An extreme value of a function f is called an absolute (global) maximum if there is a point c in the domain D of f such that (global) minimum occurs at c if
for all for all
. Similarly, an absolute .
Your Turn Find and identify each extreme value for the following functions. Justify your reasoning. a)
on
b)
Theorem: Extreme Value Theorem If a function f is continuous on a closed interval [a, b] then there exists both an absolute maximum and minimum on the interval.
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Did You Know In 1637 mathematician Pierre de Fermat, in his work Methodus ad Disquirendam Maximam et Minimam developed a method to find local maxima, minima, and tangets to various curves that was equivalent to differential calculus. This was five years before Sir Isaac Newton was born!
Practice In questions 1 3, identify each xvalue at which any absolute extrema occur. 1. 2. 3.
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In questions #47, find the extreme values of the function on the given interval. 4.
6.
5.
7.
In questions #812, find the extreme values of the function. 8.
11.
9. 12.
10.
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In questions #1315, identify all critical points and determine local extreme values. 13.
15.
14.
16. If then
( )
is a local minimum on the continuous function f over the open interval a,b . Determine a counter example to this statement.
17. Find the absolute maximum on the function
.
307
18. Which of the following functions contains exactly two extreme values? a.
b.
c.
d.
e.
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5.2 Modelling and Optimization Warm Up A ranger has 200 m of fencing with which to enclose three adjacent rectangular pends. They border a straight river and do not require fencing along the river. What dimensions should be used so that the enclosed area should be a maximum? Draw a diagram and label the quantities relevant to the problem.
What are the possible quantities for the variables used for the dimensions? (What are the restrictions on the variables?)
Write a system of equations to describe the situation and the quantity to be maximized or minimized.
Reduce the system to one equation involving the quantity to be maximized or minimized and one independent variable.
What kind of equation is the reduced system?
Change from standard form to vertex form.
What is the maximum and when does it occur?
How else could you find the extreme value of the situation?
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Example 1: Optimization problems An open topped box is created by taking a rectangular piece of cardboard measuring 16cm by 30cm and cutting out congruent squares from each corner so that the sides can be folded upwards to create the box. What size should the squares be to maximize the volume of the box? How to Do It
What to Think About What does this situation look like?
Draw a labelled diagram. Let x be the side length of each square cut. the width of the box is 16 – 2x, the length of the box is 20 2x, and the height of the box is x.
What are the variables?
What equation can be used to determine the Volume?
16 x
The first derivative test allows us to determine extreme values.
30
Are there any restrictions on the variables?
What happens at the remaining critical point?
310
Since
changes from positive to negative at
,V
has a relative maximum value. To maximize the volume of the box squares of side length
cm should be cut.
Your Turn The yearbook pages will have an area of 672 cm2 with top and side margins of 3 cm and bottom margin of 4 cm. What page dimensions will create the greatest possible area available for pictures.
311
Example 2: Optimizing using the coordinate plane Find the area of the largest rectangle that can be inscribed in a semicircle of radius 2 cm. How to Do It
What to Think About How can a coordinate plane be used to help model the situation?
What equations can be used to model the situation?
What are the restrictions?
Your Turn Find the area of the largest rectangle that can be inscribed between the xaxis and under the curve .
312
Practice 1. A rectangular plot is to be fenced using two kinds of fencing. The opposite sides will use chainlink fencing selling for $3 a metre, while the other two sides will use wood panels that cost $2 a metre. What are the dimensions of the rectangular plot of greatest area that can be fenced at a cost of $6,000?
2. A rectangular plot of land is bounded on one side by a river and on the other three sides by barbedwire fence made with two strands of wire. If you have 2400 m of barbed wire what is the largest area that can be enclosed? What are the dimensions of the largest rectangular plot of land in this situation?
3. Your company has obtained a contract to build a squarebased, opentopped rectangular tank for storing water. The contract states that the tank must have a volume of 500 ft3; be made from stainless steel; and, be as light as possible. What are the dimensions of the tank that meet these requirements?
313
4. Two sides of a triangle have lengths x and y. The angle, in radians, between the two sides is θ. What is the measure of the angle that results in the largest possible area of
1 the triangle? (NB: A = absinC ) 2
5. A rectangle has its two lower corners on the xaxis and it two upper corners on the curve y = 9 x 2 . What are the dimensions that create a rectangle with the largest area?
6. An open box is to be made from a 3 cm by 8 cm piece of metal by cutting squares of equal size from each corner and bending up the sides to make a box. What is the maximum volume of the box?
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7. Find the lengths of the sides of an isosceles triangle that has a perimeter of 12 inches and has the maximum area.
8. What are two positive numbers with a product of 200 such that the sum of one number and twice the second number is as small as possible?
9. Challenge: The trough in the figure is to be made to the dimensions shown. Only the angle θ can be varied. What value of θ will maximize the trough’s volume?
1m
θ
θ
1m
1m
15 m
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5.3 The Second Derivative Test Warm Up a) Given
, graph g’ and use the first derivative test to sketch
.
G
b) Let
. If
, use the first derivative test to sketch f.
At which point does f change from curving up to curving down?
Definition: Concavity is the word used to describe the curvature of a function. Intervals can be described as concave up (“smile”) or concave down (“frown”). The concavity of a function changes at a point of inflection.
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Example 1: Determining the inflection points of a function. Given
, find the point(s) of inflection of f.
How to Do It
What to Think About How does the graph of f behave?
Use the first derivative test to determine the characteristics of f
What function tells us how the slope of f changes over time?
The second derivative of f is the first derivative of . We can find the critical points of to find the extreme rates of change of f.
f has a relative maximum at x = 0.
What characteristics on f do the critical points of indicate?
f has a relative minimum at x = 2.
What connection can you find between the second derivative and the behavior of f? Test value
f f¢ f f¢ f ¢¢

+

+
+
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Concavity, Points of Inflection, and the Second Derivative Let be a critical point of a function that corresponds to a differentiable function f that is continuous on an open interval containing c. o If , then is increasing and f is concave up. o If , then is decreasing and f is concave down. o If , then has a critical point and f may have a point of inflection at c. If and , or vice versa, then changes sign at .
Thus, is a point of inflection on f. Note: when changes sign, changes from increasing to decreasing or vice versa and f changes from concave up to concave down or vice versa.
Your Turn Find the points of inflection and intervals of concavity for each of the following functions. a)
318
for
b)
Example 2: Determining the extreme values of a function using the Second derivative test. a) Given
, find the extreme values of f using the second derivative test.
b) Find the inflection points of f.
How to Do It a) Determine the intercepts and critical points of f.
What to Think About How can we use concavity to determine the behavior of f at each critical point?
If , then how are behaving?
and f
If , then how are behaving?
and f
What about at the critical point?
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b) Determine the inflection point(s).
Possible inflection points Test Value f f”
How do you know if f will have an inflection point?
x = 0.586 x = 3.414 x = 1
x=1
x = 10
+
+
Therefore, (0.586, 0.191) and (3.414, 0.384) are inflection points.
Theorem: Second Derivative Test for finding Local Extrema Let f be a function such that and the second derivative of f exists on an open interval containing c. 1. 2.
320
Your Turn Find and identify each extreme value of Justify your reasoning.
using the second derivative test.
Practice In questions 1 & 2, use the First Derivative Test to determine the extreme values of the function. 1.
2.
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In questions 3 & 4, use the Concavity Test to determine the intervals on which the function is concave up and concave down. 3.
4.
Find the points of inflection for the following functions 5.
For questions 7 to 9 use the graph to estimate where 7.
322
6.
,
and
8.
9.
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10. The graphs of the first and second derivative of a function On the same diagram, sketch a possible graph of the function f.
11. If a differentiable function f has an interior point c such that must f have a local maximum or minimum at
324
? Explain.
are shown below.
in its domain,
12. Sketch the following function on x
13. If
2 0 4 0
3 3 0 −2
4 0
Æ Æ
5 −4 −5 0
, must there be a point of inflection at
? Explain.
325
14. If
, find the interval on which the function
15. How many inflection points does the function
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is concave up.
have?
5.4 Curve Sketching Warm Up Use the following information to sketch a possible graph of g. a) g is an even function and is continuous on x g g' g” x g g' g”
+ + 0 2 DNE DNE
1 0 0 0
.
+ 2
3
1 DNE DNE
3
g
b) Determine the absolute extrema on
1 0 What is happening at each point of inflection?
. Justify your reasoning.
327
Definition: A horizontal inflection point occurs when changes sign at a point c and both . A vertical inflection point is a special case where exists but both do not exist, where f has a vertical tangent line at point c. Note f’’ must also change sign at c.
Example 1: Determining the inflection points of a function. Given the graph of
below
a) Over what intervals is the graph of f is increasing? b) At what value(s) of
should f have a relative maximum?
c) Over what interval(s) is f concave up? d) Sketch a possible graph for f . How to Do It
What to Think About
What are the critical points of f?
x
+
—
+
+
—
+
—
+
a) and since . b) f has a relative maximum at x = 5 since f ’ changes from positive to negative c) f is concave up since d)
What does the behaviour of us about f?
tell
What do the extreme values of tell us about ?
What does the behaviour of us about ?
tell
What function tells us how the slope of f changes over time? What assumptions was made when sketching the graph f?
328
Your Turn Given the graph of to the right: a) Over what intervals is the graph of f
f ¢( x )
increasing?
b) At what value(s) of x should f have a relative minimum?
c) What behavior does f have at
?
d) Over what interval(s) is f concave up?
e) What are the inflection points of f?
f) Sketch a rough graph of a possible f.
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Example 2: Sketch the curve without a calculator Given , from example 2 in section 5.3, sketch the curve by including all characteristics of the function. How to Do It
What to Think About What are the critical points?
What does the first derivative tell you about f?
What does the second derivative tell you about f?
Where do possible inflection points occur?
Therefore (0.586, 0.191) and (3.414, 0.384) are inflection points of f. Consider the end behaviour of the function (determined using your number sense):
How do you know if f will have an inflection point?
Which term of f dominates the behavior as ?
Which term of f dominates the behavior as ?
330
Your Turn Sketch the curve of
by including all characteristics of the function.
331
Practice 1. Sketch a continuous curve appropriate coordinates.
332
with the following properties. Remember to label
2. Sketch a continuous curve appropriate coordinates. x
with the following properties. Remember to label
y
Curve Falling, concave up
3
2
Horizontal tangent Rising, concave up
6
5
Inflection point Rising, concave down
8
8
Horizontal tangent Falling, concave down
333
3. Use the graph of f ’, where the domain is which the continuous function f is: a. increasing
, to estimate the intervals on
b. decreasing
c. Estimate the xcoordinates of all local extreme values.
334
4. f is continuous on
and satisfies the following: x f
x f
1 2 3 4 0 2 0 3 2 0 Does not exist 1 0 1 Does not exist 0
+ +
+
a. What are the absolute extrema of f and where do they occur?
b. What are the points of inflection?
c. Sketch a possible graph of f.
335
5.5 The Mean Value Theorem Warm Up Sketch a differentiable function that passes through the points (–2, 4) and (5, 4). Is there at least one point on the graph where the derivative is zero?
What is the average rate of change on the interval?
Is it possible to graph a differentiable function that passes through both points without at least one point where the derivative is zero?
Sketch a differentiable function that passes through the points (–9, 8) and (7, –4). Is there at least one point on the graph where the derivative is zero?
What is the average rate of change on the interval?
Draw the secant line between both endpoints.
Is there at least one point on the graph where the derivative is equal to the slope of the secant line?
Is it possible to graph a differentiable function that passes through both points without at least one point where the slope of the curve is equal to the slope of the secant line?
336
How could you write these two situations using a mathematical equation?
Theorem: Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one xvalue
such that
.
Theorem: Rolle’s Theorem (a special case of the MVT) If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and then there exists at least one xvalue such that .
Did You Know Joseph Louis Lagrange lived from 1736 – 1813. By the age of 19, he was appointed professor of mathematics at the Royal Artillery School in Turin. Lagrange’s mean value theorem has a clear physical interpretation. If we assume that f(t) represents the position of a body moving along a line, depending on time t, then the ratio of
is the average velocity of the body in the
period of time b – a. Since f’(t) is the instantaneous velocity, this theorem means that there exists a moment of time in which the instantaneous speed is equal to the average speed.
337
Example 1: Using the Mean Value Theorem Show that for
on the interval [1, 2], there exists at least one point such that
. Justify your reasoning. How to Do It Since f (x) is a polynomial function, it is continuous and differentiable at all points in its domain. Thus, we can apply the Mean Value Theorem. So, there exists at least one point
such that:
c
338
What to Think About What conditions must be met to use the Mean Value Theorem?
Your Turn Two stationary police vehicles equipped with radar are 8 kilometers apart on a highway. As a logging truck passes the first police car, its speed is clocked at 90 km/hr. Four minutes later, when the truck passes the second police car, its speed is clocked at 80 km/hr. Prove that the truck driver must have exceeded the speed limit of 110km/hr at some point during the 4 minutes.
339
Example 2: Necessary conditions for the Mean Value Theorem and Rolle’s Theorem “If the graph of a function has three xintercepts, then it must have at least two points at which its tangent line is horizontal”. Is this statement always true, sometimes true or never true? How to Do It This statement is only true when the function is continuous and differentiable over a closed interval that contains the three xintercepts.
What to Think About What conditions results in the Mean Value Theorem not applying?
Counterexample
Your Turn True or False? (justify your answer) a) The Mean Value Theorem can be applied to
on the interval
.
b) If the graph of a polynomial function has three xintercepts, then it must have at least two points at which its tangent line is horizontal.
c) If
340
in the domain of f, then f is a constant function.
Practice For questions #14, state whether the mean value theorem is applicable on the given interval and the value of c for which it applies 1.
2.
3.
4.
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5.6 Chapter Review 1.
Find the local extrema of f using the first derivative test. Sketch the graph and indicate where the function is increasing and decreasing. a.
b.
2.
The gas mileage in your car can be modeled by the function
in which y is the gas
mileage depending on the car’s speed, x, measured in km/hr. This function is applicable for speeds . What is the maximum gas mileage and what is the speed that maximizes it?
342
3.
Find the local extrema of f using the second derivative test. Sketch the graph and indicate the points of inflection as well as intervals of concavity. a.
b.
343
4.
A man in a wingsuit jumps out of a plane from 15000 feet as part of his “flying” experience. Initially he keeps his body horizontal to slow his descent before maneuvering to a more vertical position so that he may increase his speed. His altitude is modeled by . As he approaches the ground, he once again flattens his body position to utilize the functionality of the wingsuit and slow his speed prior to landing. The rate of his descent, as it relates height (feet) to time (minutes), is given by the differential equation maximum?
344
. At what point was his rate of descent at a
5.
A farmer wants to put a fence around a rectangular field and then subdivide it into three smaller rectangular fields by placing two parallel fences to one of the sides. If he has a total of 1000 yards of fencing, what dimensions will give him the maximum area?
6.
Find the height of a cylinder of maximum surface area that can be inscribed in a sphere of radius x.
345
7.
The interior of a 400meter track consists of a rectangle with semicircles at two opposite ends. Finds the dimensions that will maximize the area of the rectangle.
8.
If interval of
346
, find the number c that satisfies the mean value theorem on the
Chapter 6: Solving Problems Using Derivatives In the previous chapter we looked at using derivatives to determine information about graphs of functions and optimization problems. These are not the only applications for derivatives. We will also see how derivative can be used to solve the following types of problems:
Rates of Change – In this section you will take a look at problems that involve two quantities or variables. If the rate of change of one quantity, such as the height of water in a conical tank, changes with respect to time, how do you find the rate of change of the second quantity, the radius of the water in the tank, with respect to time?
Linear Approximations – In this section you will explore how to use the derivatives to compute a linear approximation to a function. You can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do when you have the function, it is much easier to solve certain problems using a simpler “approximation”.
Differentials – In this section you will compute the differential for a function. You will use differentials to explore how much a change in one variable affects the amount of change in a second variable. For example Newton’s second law states that, for a rocket, the thrust is proportional to the exhaust velocity and fuel burn rate, or
.
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6.1 Related Rates Involving Shape and Space Warm Up You throw a rock into a calm pond. A circular ripple pattern emerges with the outermost circle’s radius growing at a rate of 8 cm per second.
Using Leibniz notation, write the rate of change of the radius as a derivative.
What is the formula for the area of the outermost circle?
Write an expression for the rate of change of the area?
What is the rate of change?
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Steps to solve problems involving related rates: 1. Draw a picture and label and name the variables 2. Consider which variables change with time. Consider which aspects are constant. 3. Consider any restrictions or constraints on your variables. 4. Write an equation (a mathematical model) that relates the variables. 5. Simplify the expression by substituting any constant values. 6. Differentiate with respect to time. You will likely need to use implicit differentiation. 7. Substitute the given rate information and solve for the unknown.
8. Ensure that your answer is reasonable and eliminate any extraneous solutions.
Example 1: Determine the rate of change in Volume problems. a) A spherical balloon is being filled with air. The radius of the balloon is increasing at 2 cm/s. What is the rate of increase in the volume of air when the radius is 3 cm? b) Water is being poured into a cylindrical water tower in Shaunavon, Saskatchewan that is 8 m tall and 1 m in radius at a rate of 0.6 m3/s. At what rate does the surface of the water rise? How to Do It a) A sphere has volume,
. Both the volume
What to Think About What equation can be used as a mathematical model for this situation?
and the radius are changing over time. Which quantities are changing, and which are not?
With respect to which variable should we differentiate?
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b) A cylinder has volume,
.
Both the volume and the height of the water are
Which equation can be used as a mathematical model for this situation?
changing over time; however, the radius remains constant.
Which quantities are changing, and which are not?
The water level is rising at the rate of ~0.19 m/min
Your Turn You are working at a local outdoor recreation center. To get ready for the summer, your boss had you clean the bottom of the swimming pool. The pool opens tomorrow, and you need to fill the pool back up. The hose you are using lets water in at a rate of
. If the pool is in the
shape of a rectangular prism, that is 12 m long, 6 m wide and always 2 m deep, what is the rate at which the height of the water is changing when the water is 0.5m deep? Would the rate change when the height is at 1.5m? Explain.
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Example 2: Determine the rate of change in Volume problems. A conical vat in Charlie’s Chocolate Factory is being filled with liquid chocolate at a constant rate of 6 ft3/min. If the vat stands point down with a height of 8 ft and a radius of 4 ft, find the rate at which the level of chocolate is rising when the depth is 2 ft? What about when it is at 4 ft? How to Do It
What to Think About What mathematical model can be used to represent this situation?
Draw a diagram of the situation. A cone has volume,
. Which quantities are changing, and which are not?
Which quantities rates of change are unknown?
We can relate r in terms of h.
Why do we need to write the radius in terms of the height?
At 2 ft
At 4 ft
Could we solve the given problem by substituting for height in terms of the radius? Why or why not?
Does it make sense that the rate that the height increases is slowing down as the cone fills up?
351
Your Turn A trough is 20 feet long and 4 feet across the top. Its ends are isosceles triangles with heights of 3 feet. Water is running into the trough at a rate of 2.1 ft3/min. What is rate at which the height of the water in the trough is increasing when the water is 1.5 ft deep? Would the rate change when the height is at 2 ft deep? Explain.
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Practice 1.
As a spherical balloon is being inflated, its radius (cm) after t minutes is given by , where . What is the rate of change with respect to t of each of the following at t = 8. a. r(t) b. The volume of the balloon c. The surface area
2.
A stone is dropped into the lake, causing water waves to create concentric circles. If, after t seconds, the radius of the waves is 35t cm, find the rate of change with respect to t of the area of the circle caused by the wave at: a. t = 1 sec b. t = 2 sec c. t =3 sec
3.
A circular metal pancake griddle is being heated, its radius changes at a rate of 0.02cm/min. When the radius is 10 cm at what rate is the area changing?
353
354
4.
A balloon is being filled up with gas at a rate of 4cm3/min. Find the rate at which the radius is changing when the diameter is 16 cm.
5.
Salt leaks out of a hole in a container and forms a conical pile whose height is always double the radius. If the height of the pile is increasing at a rate of 5 cm/min, what is the rate at which the sand is leaking out of the container when the height is 8cm.
6.
A spherical snowball is melting, and the radius is decreasing at a constant rate, changing from 11 inches to 7 inches in 45 minutes. How fast was the volume changing when the radius was 9 inches.
7.
The ends of a water trough 8ft long are equilateral triangles whose sides are 3 feet long. If water is being pumped into the trough at a rate of 4ft3/min, find the rate at which the water level is rising when the depth is 8 inches.
8.
A point P(x,y) moves on the graph of the equation , the xcoordinate is changing at a rate of 3 units/second. How fast is the y coordinate changing at the point (1,7).
9.
A spherical balloon is deflating at a rate of 9 ft3/hr. At what rate is the radius changing when the volume is 375 ft3?
10. A stone is dropped into the pond and it creates circular waves whose radii increases at a rate of 0.4 m/sec. At what rate is the circumference of a wave changing when the radius is 5 m?
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6.2 Related Rates Involving Motion Warm Up Consider a 13foot ladder leaning against a vertical wall. If the bottom of the ladder begins to slide along the ground away from the wall at a rate of 0.5 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 5 feet from the ground? Draw a diagram that represents this situation. Label the quantities.
Which quantities are changing? Which quantities are constant?
Can you create an equation that models this situation?
What would the derivative of this equation be if this situation is changing over time?
Can you answer the question posed?
What are the units of your solution? Why is your solution negative?
356
Example 1: Solving related rates problems when one quantity is constant A taxi leaves Walmart and begins travelling east for 3 Km at a speed of 60 Km per hour. The driver turns north at an intersection and travels at a rate of 90 Km per hour. When the taxi is 4 km away from the intersection, how fast is the distance between the taxi and Walmart increasing? How to Do It Draw a diagram of the situation.
What to Think About Which quantities are changing, and which are constant?
How is the distance the taxi has travelled east changing when driving north?
Determine a mathematical model that represents the situation.
What mathematical model can be used to represent how the distance between the taxi and Walmart changes over time?
The taxi’s distance from Walmart is increasing at 72 km/hr.
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Your Turn a) An armadillo is walking away from a 20 m tall tree at a rate of 3 m/min. At what rate is the distance from the top of the tree changing when it is 3 m from the base of the tree?
b) A boat is being pulled into a boat launch that is 2 m above the level of the surface of the lake. Assume the rope is attached to the bow of the boat and is 50 cm above the water. If the rope is pulled in at a rate 0.2 m/sec, how fast is the boat approaching the launch when it is 10 m from the dock?
358
Example 2: Solving related rates problems when all quantities are changing In a highspeed chase, a police cruiser approaches a rightangled intersection from the north. The officers are chasing an escaped fugitive on a speeding motorcycle that has turned the corner and is now heading east. When the cruiser is 0.6 miles north of the intersection and the motorcycle is 0.8 miles east, the police officers determine with radar that the distance between them and the motorcycle is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the fugitive? How to Do It Draw a diagram
Model the situation with the Pythagorean relationship
What to Think About Which quantities are changing in this situation?
At the instant of measurement, which quantities are known, and which others can be determined?
Why is the rate of change of the distance between the intersection and the police cruiser considered negative? Differentiate both sides with respect to x.
359
Your Turn You are the captain of a small sailboat drifting at sea at midday and spot a ferry 15 km west of you. You are travelling at 6 km/h north and the ferry is travelling east at 8 km/h. At 2 pm, how fast is the distance between the boats changing?
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Practice 1.
A ladder 25 feet long leans against a vertical building. If the top of the ladders slides down the wall at a rate of 4ft/sec, how fast is the ladder sliding away from the wall when the top of the ladder is 7 feet from the ground?
2.
Sebastian starts at point A and runs east at a rate of 12ft/sec. One minute later Orla runs north at a rate of 10 ft/sec. At what rate is the distance between them changing 1 minute after Orla starts.
3.
A man 6 feet tall walks at a rate of 6ft/sec away from a lamp that is 18 feet above the ground. When he is 10 feet from the base of the lamp at what rate is his shadow changing?
361
4.
Jake is driving east at 60km/h and Samantha is driving north at 75 km/h. Both cars are approaching the intersection of the two roads. At what rate is the distance changing between them when Jake’s car is 0.5km from the intersection and Samantha’s car is 0.4 km from the intersection?
5.
A point P(x,y) moves along the graph
such that
, where t is time. Find
at the point (2,4).
6.
362
Sebi is flying a kite at cattle point and holds the string 4ft above ground level and lets the string out at a rate of 2ft/sec as the kite moves horizontally at height of 110 ft. Assuming the string is taut, find the rate at which the kite is moving when 150 ft of string has been let out.
7.
A hot air balloon is rising vertically at a rate of 2 ft/sec. An observer waiting for their turn is seating 100yards from a point directly below the balloon. At what rate is the distance between the balloon and the observer changing when the balloon is 500 feet high.
8.
When two electrical resistors R1 and R2 are connected in parallel the total resistance R is given by
. If R1 and R2 are increasing at rates of 0.02 ohm/sec and 0.01
ohm/sec, respectively, at what rate is R changing at the instant when R1= 90 ohms and R2= 30 ohms?
363
6.3 Related Rates Involving Periodic Functions Warm Up The minute and hour hand are moving at different rates as they travel around the clock.
What is the rate of change of the minute hand as it travels around the clock?
What is the rate of change of the hour hand as it travels around the clock?
What is the rate of change of the angle between the minute hand and the hour hand as they travel around the clock at 4:42pm?
What is the rate of change of the angle between the minute hand and the hour hand as they travel around the clock at 3:00pm?
At what time(s) does the rate of change switch from negative to positive or vice versa?
364
Example 1: Related rates involving changing angles The London Eye ferris wheel has a radius of 60 m and rotates at about 2 revolutions per hour. How fast is the rider rising or falling when the rider is 70 m above the ground? How to Do It Create a diagram and label it
What to Think About What equation can you create that represents a mathematical model of the situation?
How fast is the ferris wheel rotating?
Your Turn An airplane approaches at an altitude of 5 miles towards a point directly overhead of an observer. The speed of the plane 550 mph. Find the rate at which the angle of elevation is changing when the angle is 40?
365
Example 2: A clock has a minute hand 20 cm long and an hour hand of 12 cm. Determine the rate the distance between the hour and minute hands are changing at 4:42 pm. How to Do It Clock angle formulae https://sites.google.com/site/mymathclassroom/trigonometry/clockangleproblems/clockangleproblemformula
The angle between the hands is changing at a rate of . At 42 minutes past the hour, the minute hand will be at an angle of .
What to Think About What equation can be used to relate the changing quantities?
How can you find the angle between the minute and hour hands?
At 4:42 pm the hour hand will be at an angle of . So, the hands are
apart.
Why must the rate of change of q be in terms of radians for the answer to make sense?
366
Your Turn You spot two freighters moving towards each other offshore. From your perspective you measure the angle between the ships as 130 with the first ship anchored in the harbor 500m away from you. The second ship is travelling on a circular arc that keeps it always 900m away from you. If the angle between the ships is changing at 30 per minute, how fast is the distance between the ships changing?
367
Practice 1.
2.
368
A balloon rises at a rate of 4 m/s from a point on the ground 45 m from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30m above the ground.
Joshua is standing on the Johnston street bridge and is reeling in a fish at a rate of 1.5 feet/sec from a point 18 m above the water. At what rate is the angle between the fishing line and the water changing when there is 30 feet of line out.
3.
The beacon of a lighthouse located at a perpendicular distance of 300m from point N on a straight shoreline revolves 5 rev/min and shines a spotlight on the shore. N
300m
How fast is the spot of light sweep along the shore at a point 400 m from N?
369
4.
Two sides of a triangle have lengths 15 m and 25 m. The angle between them is increasing at
. How fast is the length of the third side changing when the
angle between the sides is
5.
370
?
As the sun rises, the shadow cast by a 20 m tree is decreasing at a rate of 50 cm/h. At what rate is the angle of elevation from the shadow to the sun increasing when the shadow is is 12 m in length?
6.4 Linearization and Differentials Warm Up Let
. Show that the line tangent to the graph of
at the point
is
.
Set
and
. Zoom in on the two graphs at
Consider and How good of an approximation is
Is
. Find for
and ?
. Sketch what you see.
. What happens to the accuracy of the approximations as one moves further away from the point of tangency?
an underestimation or an overestimation? Explain.
371
Definition: for a function f that is differentiable at x = a: The differential, , is the principal change in a function relative to an infinitesimal change in the independent variable . The linearization is a linear approximation model used to estimate a past or future value of an unknown function. The linearization uses a specific point on the function and the rate of change information at that point; thus, it is equivalent to the tangent line at the specific point. In practice, we often find such that . This is generally a reasonable approximation provided is sufficiently small.
Example 1: Determining the linearization and using it to approximate a function. a) Find the linear approximation b) How accurate is the approximation
(or linearization) of
at for values of
.
near ?
c) Sketch what the linearization looks like at values close to 1. How to Do It a)
372
What to Think About What is the linearization equivalent to?
b)
How well does the linearization approximate the curve near the point of tangency?
c)
What happens to the approximation at ?
? At
373
Your Turn a) What is the linearization to the graph of
b) How good is the approximation for
374
at
?
?
Example 2: Using differentials Find and evaluate dy for each of the following functions at the given point. a) b)
at x = 1 and dx = 0.01 at x = 10 and dx = 1
How to Do It a)
What to Think About What does the differential dy represent?
b)
Your Turn For
, find dy when
and
375
Practice 1.
Find and evaluate dy for each of the following functions at the given point. a.
b.
c.
2.
376
at x = 1 and dx = 0.01
at x = 1 and dx = 0.1
at x = 1 and dx = 0.03
Find the linear approximation to the function approximation for ?
at x = 0? How good is the
3.
Find the linearization to the graph of approximation for
4.
Find the linear approximation
at
? How good is the
?
(or linearization) of
a. How accurate is the approximation
at for values of
. near ?
b. Sketch what the linearization looks like at values close to 1.
5.
Find the linear approximation accurate is the approximation
(or linearization) of for values of
at
. How
near ?
377
6.5 Chapter Review
378
1.
Water is being emptied out of a spherical fishbowl of radius 10 cm. If the depth of the water in the tank is 6 cm and is decreasing at a rate of 3cm/sec, at what rate is the radius of the top of the surface of the water decreasing?
2.
A water tank has the shape of a right circular cone of altitude 12 ft and base radius 4 ft, with the vertex at the bottom of the tank. If water is being taken out of the tank at a rate of 10ft3/min, for fast is the water level falling when the depth is 5 ft?
3.
At what rate is the area of an equilateral triangle increasing if its base is 12 cm long and is increasing at a rate of 0.4 cm/sec?
4.
A balloon is being filled at a rate of 121π cm3/sec. At what rate is the balloon increasing when its radius is 12 cm.
5.
The base of a rectangle is increasing at 4cm/s while the height is decreasing at 3 cm/s. At what rate is the area changing when its base is 20 cm and its height is 12 cm?
6.
A square is expanding. When each edge is 16 cm its area is increasing at a rate of 80cm3/sec. At what rate is the length of each ledge changing?
379
7.
Two roads intersect at right angles. At 9 am a car passes through the intersection headed due west at 60 km/h. at 10 am a truck heading south at 75 km/h passes through the same intersection. Assume they maintain their speeds and directions. At what rate are they separating at 1 pm?
8.
A circle is inscribed in a square as shown. The circumference of the circle is increasing at a constant rate of 8 cm/sec. As the circle expands the square expands to maintain its tangency.
a. Find the rate at which the perimeter of the square is increasing.
b. At the instant when the area of the circle is 36π square inches, find the rate of increase of the area between the circle and the square.
380
9.
Water is draining from a conical tank with a height of 10 feet and diameter 8 feet into a cylindrical tank. The depth of the water in the conical tank is changing at a rate of (h – 8) feet per minute.
a. Volume of a cone
,write an expression for the volume of water in the
conical tank in terms of h.
b. At what rate is the volume of the water in the conical tank changing when h = 4?
381
10. The radius of a basketball is increasing a rate of 0.01 cm/sec. a. When the basketball has a radius of 10 cm, what is the rate of increase of its volume.
b. At the time when the volume of the basketball is 48π cubic centimeters, what is the rate of increase of the area cross section through the center of the basketball.
c. When the radius and the basketball are increasing at the same rate, what will the radius be?
382
Unit 3 Project: Shrinking Lollipop This project compares the predicted rate of change of the radius of a lollipop to the actual rate of change. Test the hypothesis that the rate of change of the volume of a lollipop is proportional to its surface area. It seems reasonable that the rate of change of the volume of a lollipop would be proportional to its surface area, since the candy dissolves from the surface as you suck the lollipop. In mathematical terms, we predict that
where k is a constant of proportionality. But is this prediction true? What can we learn, by mathematical deduction, about k, and how can we test this prediction experimentally? 1. Prepare a table with time in the first column, radius in the second column. 2. Measure (mm) and record the radius (diameter/2) of a lollipop (Tootsie Roll Pops work well). 3. Suck on the lollipop for one minute and measure the radius. Record your data. 4. Repeat step 3 until the lollipop is gone or you reach candy of a different consistency (e.g. the inner nougat of the Tootsie Roll Pop). Note: it is important to suck on the lollipop the same from one trial to the next. If you suck especially vigorously on one trial but only lightly on the next, you won’t be able to interpret your results. 5. Graph your results: radius (yaxis) vs time (xaxis). What kind of function do you see: linear? parabolic? ? ? 6. Use the regression function on your calculator or Desmos to determine an equation. 7. Now consider the prediction. Use a chain rule decomposition of (Hint:
to help find k.
)
8. Do the experimental results support your prediction? Questions: 1. What happens to the shape of a square lollipop (or other shape with edges and corners) as it dissolves? Why? 2. How does the theoretical compare to the calculus you have performed?
383
384
Unit 3: Integral Calculus This unit focuses on the following AP Big Idea from the College Board: Big Idea 3: Integrals are used in a wide variety of practical and theoretical applications. It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus
By the end of this unit, you should be able to: Recognize antiderivatives of basic functions. Interpret the definite integral as the limit of a Riemann sum in integral notation. Express the limit of a Reimann sum in integral notation. Approximate a definite integral. Calculate a definite integral using areas and properties of definite integrals. Analyze functions defined by an integral. Calculate antiderivatives. Evaluate definite integrals. Interpret the meaning of a definite integral within a problem. Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume. Analyze differential equations to obtain general and specific solutions. Interpret, create, and solve differential equations from problems in context.
By the end of this unit, you should know the meaning of these key terms:
Antidifferentiation Area under a curve Average value of a function Definite integral Differential equation
?
Fundamental Theorem of Calculus Indefinite integral Integral Riemann sum Rotational volume
? 385
Chapter 7: Antidifferentiation Differentiation allows us to determine how a function is changing. Most of the time, in realworld experiments, we collect data that measures change. We then try to determine a function, which is a derivative that represents the data. Once the derivative is known, we can use antidifferentiation to find the original function. Integration is like filling a tank from a tap. The input (before integration) is the flow rate from the tap. Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank. Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap). As the flow rate increases, the tank fills up faster and faster. With a flow rate of up at .
, the tank fills
We have integrated the flow to get the volume.
Example: with the flow rate in litres per minute, and the tank starting at t = 0 After 3 minutes
:
the flow rate has reached and the volume has reached
And after 5 minutes
liters
:
the flow rate has reached and the volume has reached
386
liters/min,
liters/min, liters
7.1 The Antiderivative – Working Backwards Warm up: Work backwards to find the original Function. What to Think About
What pattern do you observe?
What happens when the coefficient does not fit the pattern?
Generalized pattern
Are there any other functions that you could differentiate to get
?
Can you determine a general formula to follow?
Finding the Integral (the antiderivative) using the “Reverse Power Rule” If Integral notation,
. , asks you to find the antiderivative.
The constant of integration C must always be added when antidifferentiating because the derivative of a constant is zero. Without more information about the original function we cannot know the value of the constant term.
387
Example 1: Finding the Antiderivative using the Reverse Power Rule. Find the antiderivative of the following: How to Do it
What to Think About How do you apply the reverse power rule?
a)
Why must you include a constant? b)
Can you write the derivative as a power term?
Your Turn a)
388
b)
Example 2: Determining the Antiderivative of a Function Find the antiderivative of the following How to Do it
What to Think About How can you write the function with an exponent?
a)
b)
Your Turn a)
Derivative Notation
b)
Differential Notation
Integral Notation
389
Practice 1.
5.
2. 6.
3. 7.
4.
390
8.
9.
11.
10.
12.
391
7.2 Antiderivative of Trigonometric Functions Warm up Consider all the primary Trigonometric differentiation formulas:
These differentiation formulas lead to the following antiderivatives:
392
Example 1: Find the Antiderivative by Reversing Trigonometric Differentiation. How to do it a)
What to think about What would you differentiate to get ?
b) If you remember the derivatives can you find the antiderivatives? c)
Your Turn a)
b)
393
Example 2: Using Trigonometric Identities Before Finding the Antiderivative. How to do it
What to think about Can you rewrite the function using a trig identity?
a)
Why does using a trigonometric identity help you to antidifferentiate? b)
Your Turn a)
b)
394
Practice 1.
7.
2. 8.
3.
9.
4.
5.
10.
6.
395
7.3 Antidifferentiation Using Substitution Warm up Take the Derivative of the following Functions:
After reviewing the derivatives using the chain rule how can you use this idea to find the antiderivative?
How else can be written?
396
“Reverse chain rule”
Example 1: Reversing the Chain Rule Using “u/du” Substitution How to do it a)
What to Think About Are you trying to find the antiderivative of a composite function?
Is the derivative of the embedded function present?
b)
Why do you need to expand this function first?
Which rule did you use to find the antiderivative?
397
Your Turn a)
b)
398
Example 2: Finding the Antiderivative using “u/du” Substitution How to do it a)
What to Think About Can you simplify the expression using a u/du substitution?
Why is a u/du substitution not possible for this function? b)
What would we need to do to find the antiderivative?
Have you ever heard of the binomial expansion theorem?
399
Your Turn a)
b)
c)
Pascal’s Triangle and the Binomial Expansion Theorem
The binomial expansion theorem is a quick way to expand a binomial raised to a large integer exponent. Pascal’s triangle provides a quick way to determine the coefficients.
400
Practice Evaluate each integral 1.
2.
3.
4.
5.
6.
401
402
7.
8.
9.
10.
11.
12.
7.4 Advanced Antidifferentiation Techniques Warm Up Differentiate each expression
Trigonometric Integration Formulas
403
Example 1: Evaluate Integrals Involving Trigonometric Functions and u/du Substitution. Evaluate the following integrals: How to Do It What to Think About Why is the u/du helpful for this a) question?
b)
Is a u/du always necessary? Sometimes, it is best to ask the question, “What would I differentiate to get this function?”
c)
Your Turn a)
404
b)
Example 2: Evaluate the Indefinite Integral How to Do It a)
What to Think About What would you need to differentiate to get ?
How does the constant answer?
b)
p effect the final
Why do you not have to use a reverse of the quotient rule?
Your Turn Determine each indefinite integral: a)
b)
405
Example 3: Evaluating an Integral using Reverse Substitution Sometimes the “u/du” method appears not to work. We can use a “sneaky” u/du reverse substitution method. In these cases, a u/du substitution is unavoidable. How to Do It
What to Think About There is a problem after using u/du. What is it?
What needs to be substituted to write the integral in terms of u?
406
Your Turn Evaluate the integral
Practice 1.
3.
2.
4.
407
5.
8.
6.
9.
7.
10.
11.
408
7.5 Antidifferentiation Involving Exponential and Logarithmic Functions Warm Up Integrate the following:
What did you notice? What about the last one? Did it follow the rule?
The Antiderivative of the Reciprocal Function:
409
Example 1: Removing Constants Before Finding the Antiderivative. How to Do It Evaluate each integral
What to Think About It is often beneficial to remove a constant before finding the antiderivative.
a)
Why do we have to include the absolute value? b)
c)
Your Turn
Evaluate each integral a)
410
b)
Example 2: Finding the Antiderivative of a Reciprocal Function. How to Do It
What to Think About
Evaluate each integral
What would you differentiate to get
a)
1 ? x +1
Why may we omit the absolute value in this case?
b)
Your Turn Evaluate each integral a)
c)
b)
411
Example 3: Evaluate the Indefinite Integral How to Do It
What to Think About Is there a constant to be factored out?
Evaluate each integral a)
What would you differentiate to get ?
b) What would you differentiate to get the exponential component of this function?
c)
Why must you first break this up into three separate fractions first?
Your Turn a)
412
b)
Practice 1.
6.
2.
7.
3.
8.
4.
9.
5.
413
10.
13.
11.
14.
12.
414
7.6 Antidifferentiation Involving Inverse Trigonometric Functions Warm Up
If you figured out it is not the same as the rest, you are correct. What can you differentiate to get
?
What happened with the last one? Did it follow the same rule as all the rest?
The Corresponding Integrals
415
Example 1: Recognizing an Inverse Trigonometric Derivative Evaluate the following integrals How to Do It
What to Think About Is there a constant to be factored out?
a)
Do you recognize the function?
Why can’t we use u/du? b) How is this different than the previous example?
How can you recognize this in the future?
Your Turn a)
416
b)
Example 2: Using u/du and Recognition to Find the Antiderivative Evaluate the following integrals How to Do It
What to Think About What would you differentiate to get this function?
a) Can you use a u/du? If so, what u is most appropriate so that du simplifies the function?
What function can you differentiate to get the simplified function? b)
Your Turn a)
b)
417
Practice 1.
418
5.
2.
6.
3.
7.
4.
8.
9.
13.
10.
14.
11.
15.
12.
419
16.
19.
20. 17.
18.
21.
420
Other Useful Formulas:
421
7.7 Chapter Review Evaluate each indefinite integral.
422
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
423
424
13.
14.
15.
16.
Chapter 8: Solving Differential Equations There are many ways to model realworld situations mathematically. Differential equations describe how phenomena change. A differential equation is a mathematical model used to measure the change of a quantity that is determined by another quantity. There are a variety of solution techniques for these types of equations depending on the complexity of the equations involved. Differential equations are composed of several terms, just as conventional algebraic equations are. Consider the example of Yellow fever in the body.
In vaccinology, scientists are interested in the changing of cells, molecules and virus concentrations with respect to time. The following equation is one that represents a model of the virus (V) for Yellow Fever.
dV = π v ( V )H*c v V k v VA t dt Each of the variables represents a specific quantity that is being measured related to the Yellow Fever antibody.
425
8.1
Solving Differential Equations Analytically
Warm Up Consider: What does
mean?
Multiply both sides by
How can you solve for y?
Draw a possible graph of y.
Definition An equation involving a derivative is called a differential equation.
426
There is one differential equation that you probably know. It is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us.
To see that this is in fact a differential equation we need to rewrite it. First, remember that acceleration, a, can be written in one of two ways. or Where v is the velocity of the object and s is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position. Now Newton’s Second Law can be written as a differential equation in terms of either the velocity, v, or the position, s, of the object as follows. or Example 1: Evaluate a Differential Equation by Separating the Variables How to Do It
What to Think About How can you separate the variables so that all x terms are on one side of the equation?
How can you solve for y?
Why is there no constant of integration on the left side of the equation?
What does the constant of integration determine about the behavior of y?
What do you need to know to find C?
427
Your Turn Solve the differential equation
Example 2: Evaluate Differential Equations Implicitly to Solve for “y” How to Do It a)
What to Think About How can you separate the variables so that all x terms are on one side and all y terms are on the other?
What do you need to do to remove the differentials from the equation?
Why is there no constant of integration on the left side of the equation?
What does C1 equal?
428
b)
How can you separate the variables so that all x terms are on one side and all y terms are on the other?
Why do we not need to use the absolute value sign anymore?
Your Turn Solve each differential equation. a)
b)
429
Example 3: Interpret the Solution to a Differential Equation Graphically If we go back to the differential equation
, what does the solution mean graphically?
How to Do It
What to Think About What does the represent
What does the C represent?
The solution describes a family of curves. To find a particular solution, we would need to know at least one point on the curve.
How can you know which is the correct graph?
Your Turn Graph the solutions to the differential equations and create at least two different curves. a)
430
b)
Example 4: Find the Particular Solution Given an Initial Value for the Function: Suppose the slope of the tangent line to each point on the curve is x and the curve passes through the point How to Do It
. Find the specific curve. What to Think About How can you separate the variables?
How does the initial value help us find the exact function y?
Your Turn Given that
and the solution curve passes through the point
, what is y?
431
Practice
432
1.
5.
2.
6.
3.
7.
4.
8.
9. A particle moves along a line with an acceleration of at a time t. When velocity is 3 and its position . When , what is the position s.
, its
10. An automobile accelerates from a standing start with a constant acceleration of . How far does it travel in the first 10s?
11. Suppose that the acceleration a(t) of a particle at time t, is given by a(t) = 6t 3 and where s(t) is the position function. Find v(t) and s(t).
433
12. A tree farm sells a bush after 7 years of growth. The growth rate of the tree during the first seven years is approximated by
, where t is the time in years and h is
the height in cm. The seedlings are 10 cm tall when planted t = 0. a. Find the height after t years
b. How tall are the bushes when they are sold?
434
8.2 Solving Differential Equations Graphically Warm up Given that
, find the slope at each point
Use each point
At each of the points
.
to calculate
draw a short line through the point with slope
.
y
x
Can you sketch a graph of y through the point
? What does
tell us
about the function y?
435
Slope Field
A slope field uses the idea of local linearity; that is, if a function is differentiable at a point, then the tangent line approximates the function close to that point. By drawing a slope field, we can graphically “see” the family of functions or the general solution to the given differential equation. The slope field is like the wind blowing through a field of grass. You can see the change behavior that the differential equation is describing. Then by choosing a starting point, an initial value, one can “determine” the particular solution. Some differential equations can only be solved by considering the slope field.
Example 1: Sketching a Slope Field Consider the differential equation
through the point
. What does the
slope field look like? What curve passes through the initial value? How to Do It Determine the slope at each point on the grid.
What to Think About What does
tell us about the
slope at each point? Note: For any point the slope will be 2 less than the xvalue. Sketch the slope lines at each point for all xvalues in the domain. The result is shown below.
How can we use the slope field to draw a curve through the initial point?
436
Your Turn Sketch a graph of the slope field for
Example 2: Determine a Specific Solution to a Differential Equation. Determine the slope field for
. Sketch a solution function that passes through
?
Confirm algebraically the solution to the differential equation.
How to Do It a) Draw the graph of your solution through the given point on the grid.
What to Think About What does
tell us about the
slope at each point?
How can we use the slope field to draw a curve through the initial point?
Why does the solution show only “half” of the curve?
437
b) Solve the differential equation given the initial condition that the solution passes through the point.
Why must we choose between the positive and negative versions of the y function
How do we decide which is the solution?
Your Turn a)
438
What is the slope field for the differential equation
?
b)
Draw the solution to this differential equation that passes through the point
,
then solve the differential equation algebraically given this initial condition.
Example 3: Solving a NonSeparable Differential Equation Graphically: Draw solution curves for the differential equation following points: How to Do It
,
that pass through each of the
,
What to Think About Where is the slope zero? Why?
Can you find an algebraic solution to this differential equation?
439
Your Turn Sketch the slope field for
440
then sketch the curve through
Practice 1. Draw slope fields for the following differential equations a)
b)
c)
d)
441
2. a) Draw the slope field for the differential equation b) Draw the unique curve that passes through the point (2, 1).
c) Solve the differential equation with this initial condition.
442
3. Match the slope field (a – d) to one of the differential equations (i – iv). a) b)
c)
i.
d)
ii.
iii.
4. Where would the slope field for the differential equation
iv.
have vertical
segments?
443
5. Draw the solution of each differential equation. Then, determine the corresponding differential equation below each function. a.
c.
444
b.
d.
6. Consider the differential equation a. On the axis provided, sketch a graph of the slope field for the given differential equation at the 6 points indicated.
b. Find
in terms of x and y. Determine the concavity of all solution curves in
Quadrant I. Explain.
c. Let condition
be the particular solution to the differential equation with the initial . Does f have a relative minimum, relative maximum or neither.
Explain
445
8.3 Approximating Area Under a Curve Using Riemann Sums Warm up Consider a car travelling with the following velocitytime information. a.
How far has the car travelled between
b.
How far has the car travelled between
Can you determine the exact distance travelled when the function is a curve?
Definition The process of evaluating a product in which one factor varies is called finding the definite integral. The definite integral will give you the area under the curve. We can approximate the definite integral in several ways.
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Example 1: Estimating the Area Under a Curve Geometrically. Graph the function
on the grid and calculate the area under the curve on
How to Do It
.
What to Think About Can you determine an exact answer in this case?
Is there another way other than counting we can find the area under the curve?
Method 1: Counting the squares under the curve from gives us an area of
What shapes can we use to determine the area?
Method 2: Using Triangles
What other shape can be used to represent the area? Method 3: Using a Trapezoid
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Your Turn Graph
448
and calculate the area under the graph from
to
Example 2: Rectangular Approximation Methods to Determine the Area Under a Curve. Graph the function and make a table of values. Approximate the area under the curve from by using 4 quadrilaterals of equal width. How to Do It Divide the region into 4 rectangles (4 subintervals)
What to Think About How do you determine the width of each subinterval?
Can you use the entire interval to determine the width of each subinterval?
What does i represent? Method 1: Left Rectangular Approximation Consider each rectangle on subinterval i,
, to
have width equal to
and
What does n represent?
height equal to . This method uses a rectangle with the height of the function at the left side of each subinterval.
What height should we choose for each rectangle?
What does the ai represent in the area equation?
How do you determine each rectangle?
for
Does the left rectangular approximation method overestimate or underestimate the area in this example?
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Method 2: Right Rectangular Approximation Consider each rectangle on subinterval i,
, to
What height should we choose for each rectangle?
have height equal to . This method uses a rectangle with the height of the function at the right side of each How do you determine subinterval. each rectangle?
for
Does the right rectangular approximation method overestimate or underestimate the area in this example?
Provide any counterexamples where this is not always true
Method 3: Trapezoidal Approximation Consider each trapezoid on subinterval i, have average height to equal to
What do
represent?
, to . This
method uses the average of the left rectangular height and the right rectangular height of the function on each side of each subinterval. This calculates the area of the trapezoid under the curve in each subinterval.
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and
Is the trapezoidal approximation an overestimation or an underestimation for this function?
Method 4: Midpoint Rectangular Approximation Consider each rectangle on subinterval i , , to have height equal to
. This method uses a
Compared to the trapezoidal approximation, what is different about the average used in the midpoint approximation?
rectangle with the height of the function at the midpoint of the subinterval.
Which is the most accurate approximation, LRAM, RRAM, Trapezoidal, or MRAM?
Rule
Formula NOTE: These formulas work only for n equal subintervals.
Left Endpoint
Right Endpoint
Midpoint
Trapezoidal
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Your Turn A particle starts at Where is the particle at of equal width.
and moves along the xaxis with velocity
Calculate the lefthand area (LRAM):
b.
Calculate the righthand area (RRAM):
c.
Calculate the trapezoid area:
d.
Calculate the midpoint area (MRAM):
e.
Solve the differential equation
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.
? Approximate the area under the curve using four quadrilaterals
a.
Then, determine
for time
that passes through the point
.
Example 3: Evaluate A machine fills a milk carton with milk at a constant rate. The rates (in cases per hour) are recorded at hourly intervals during a 11hour period, from 6:00 am to 5:00 pm Use the trapezoidal approximation method with n = 11 to determine approximately how many cases of milk are filled by the machine over the 11hour period. How to Do It Time Rate (cases/h)
What to Think About Why is the value 17 used for 5:00 pm?
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Your Turn The economy is continuously changing, we can analyze it with certain measurements. The following table records the annual inflation rate as measured each month for 13 consecutive months. Use the trapezoidal rule with n = 12 to find the overall inflation rate for the year.
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Practice 1. The function f is continuous on the interval [2, 8] and has values that are given in the table. Using the subintervals [2, 5], [5, 7] and [7, 8], what is the trapezoidal approximation of ?
x 2 5 7 8 f(x) 10 30 40 20 2. The following table shows the speedometer readings of a truck, taken at tenminute intervals during one hour of the trip. Use the table and the midpoint rule to estimate the distance that the truck traveled in the hour. Watch your units! Time (min)
0
10 20 30 40 50 60
Speed (km/h) 40 45 50 60 70 65 60
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3. If three equal subdivisions of [4, 2] are used, what is the trapezoidal approximation of ?
4. Use the Midpoint Rule with n = 5 to approximate
.
5. Use the Trapezoidal Rule with n = 2 to approximate the integral
6. Use the Right Endpoint Rule with n = 4 to approximate the integral
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.
.
7. The temperature, in degrees Celsius
, of the water in a pond is a differentiable
function W of time t. t days 0
20
3
31
6
28
9
24
12
22
15
21
The table above shows the water temperature as recorded every 3 days over a 15day period. a)
Use the data from the table to find an approximation for
. Show the
computations that lead to your answer. Indicate units of measure.
b)
Determine the LRAM over the time interval 0 ≤ 𝑡 ≤ 15 days with 5 subintervals .
c)
Determine the RRAM over the time interval 0 ≤ 𝑡 ≤ 15 days with 5 subintervals.
d)
Determine the MRAM over the time interval 0 ≤ 𝑡 ≤ 12 days with 2 subintervals .
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8.4 Calculating Area Under Functions Graphically Warm up Graph the function the curve from
and make a table of values. Find the area under .
What happens when part of the graph is below the xaxis?
What is the difference between net area, and total area?
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Example 1 Determining the Area Under PieceWise Functions Geometrically Evaluate the following integral given the graph of How to Do It
below What to Think About Where should you start?
Do we need to find the equation of each linear section?
What happens when some of the area is below the ?
Geometrically we can divide the graph up into convenient intervals and find the area of each using geometric shapes. Net area
Total Area
Does the integral ask you to determine the net area or the total area?
What is the difference between the net area and the total area?
459
Your Turn Given the piecewise function below, write an integral statement and evaluate.
a.
Find the net area on [0,7].
b.
Find the total area on [0,7].
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Example 2 Finding the Area Under a Curve of Known Shape: Evaluate How to Do It
Since we know this creates a semicircle of radius 3 centered at the origin. The area can be calculated using the area of a semicircle equation
What to Think About What shape is the curve of the function?
What formula can be used to determine the area under the curve over the interval [3,3]?
Your Turn Using the graph calculate the following integral.
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Example 3 Solving RealWorld Integration Problems using Geometry. A car moves along the highway at a constant rate of 65 miles per hour from 6:00 am to 8:00 am. Express the total distance travelled as an integral and evaluate. What to Think About Under which type of function will the shape under the curve be a rectangle?
Consider the units that determine the area function, why is the result in miles?
Your Turn Find the output from a pump producing 30 gallons per minute during the first 2 hours of operation. Express your answer using correct units.
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Example 4 Evaluating Definite Integrals using a TI Calculator: Evaluate How to Do It Using your calculator, we can calculate the integral
What to Think About Which buttons on your calculator should you press to evaluate a definite integral?
Your Turn Use your calculator to evaluate the following integral.
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Example 5 Solving RealWorld Problems using a Calculator: The rate of consumption of oil in Canada during the 1990’s (in billions of barrels per year) is modeled by the function , where t is the number of years after January 1, 1990. Find the Total consumption of oil in Canada from January 1, 1990 to January 1, 2000. How to Do It The total amount consumed from
is
What to Think About Why are the constants of integration 0 and 10?
Could you use different constants to get the same result?
What are the units obtained after integrating?
Your Turn The rate at which our homes consume electricity is measured in kilowatts. Most homes consume electricity at a rate of 1 kilowatt for 1 hour. Suppose that the average consumption æ πt ö rate of your home is modeled by the function C ( t ) = 3.6 2.4sinç ÷ , where is measured è 12 ø in kilowatts and t is measured in hours past midnight. Find the average daily consumption for your home, measured in kilowatt hours.
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Practice 1. Determine the area under each graph
2. Evaluate the integral using geometrical shapes. a.
b.
c.
d.
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3. Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. a.
b.
c.
d.
e.
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4.
The graph of f below consists of a line segment and a semicircle. Evaluate each definite integral using geometric formulas
a.
b.
c.
d.
e.
f.
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8.5 Chapter Review 1.
Consider the curve given by
a) Show that
b) Find all points on the curve whose ycoordinate is 1. Write an equation for the line tangent to the curve at each of the points.
2.
Consider the differential equation equation
468
. Find the solution to the differential
3.
A function is differentiable for all real numbers. The point and the slope at each point is given by
is on the graph of .
a. Find the second derivative and evaluate it at the point
b. Find
by solving
.
using the initial condition
.
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4.
Consider the differential equation a. On the axis provided, sketch a possible slope field.
b. Find the solution given the initial condition
5.
For the following use the LRAM, RRAM, Trapezoidal, and MRAM to approximate the definite integral for the stated value of n. a.
470
b.
c.
6.
Evaluate the definite integral using geometrical shapes. a.
b.
c.
d.
7.
Given a.
and
find:
c.
b.
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8.
The graph of
a.
b.
c.
d.
e.
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is below. Evaluate each definite integral by using geometric formulas.
Chapter 9: Integrals and the Fundamental Theorem of Calculus
n=5
n = 15
n = 50
n=
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9.1 Integrating Using the Fundamental Theorem
Warm up Approximate
using a left rectangular approximation method with:
a) 3 equal subintervals
x 1 2 3
b) 4 equal subintervals 4 5 6
c) 6 equal subintervals
7 8 9 10
d) 12 equal subintervals
11 12 13
e) Use your calculator to determine the exact value of
What happens to the approximations as we increase the number of rectangles?
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Example 1: Making the Connection Between the Area Under a Curve and the Antiderivative. a) Write the area
in terms of
and
given
, then
calculate the value of the integral. b) Determine How to Do It a)
What to Think About How can you split up the area under the curve from 0 to 8 at x = 2 into two known shapes?
What is the connection between and
?
b)
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Your Turn Write the area
in terms of
and
the value of the integral over the interval [2, 4]
476
given
then calculate
Definition: The Definite Integral (more formally) Let f be a continuous function defined on a closed interval . Let the interval be subdivided into n intervals of equal length
. As we increase
the number of subdivisions, , and , we can use an infinite number of infinitely thin rectangular slices to calculate a more accurate approximation for the area under a curve by using a limit. The Riemann sum of all these rectangles calculates the area under the curve as follows,
where is an xvalue chosen arbitrarily in the subinterval. Mathematicians chose new notation using the differential, , to write the definite integral of f over as:
If
is integrable over a closed interval
, then the area under the curve
from a to b is the integral of f from a to b,
The above statement is read “integral of f from a to b”. a lower limit of integration b upper limit of integration is the function of the integrand.
the integral sign
is the variable of integration. See: https://www.desmos.com/calculator/tgyr42ezjq for an exploration of this concept.
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Theorem: The Fundamental Theorem of Calculus If a function f(x) is continuous on a closed interval [a, b] and F(x) is the antiderivative of f(x) on the interval [a, b], then
Note: It is not necessary to include the constant of integration C in the antiderivative because
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Example 2: Using the Fundamental Theorem of Calculus Evaluate the following: a)
b)
How to Do It a)
What to Think About How do you use the antiderivative and the fundamental of theorem of calculus to solve?
Why is the answer in b) negative? b) How does this relate to the graph?
What conclusion can you make?
How do you know before you start if the answer will be negative or positive?
Your Turn Evaluate the following: 1.
2.
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Example 3: Connecting the FTC to the Net Area Under the Curve Find
graphically and confirm your answer using the FTC.
How to Do It Graphically the area is 0 because of the symmetry of the graph of the function (odd)
What to Think About What do you notice about the shape?
Is there any symmetry?
Can you guess what the net area will be?
Compare the areas on either side of the xaxis, what conclusion can you make?
Does your answer make sense?
How do you know before you start if the answer will be negative or positive or zero?
480
Definition: If f is a function whose domain contains x whenever it contains x, then a. f is even if for all x in the domain of f
b.
f is odd if
for all x in the domain of f
Your Turn Solve the following graphically then confirm using the Fundamental Theorem.
a.
b.
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SUMMARY – Graphical connections to the Fundamental Theorem of Calculus The observations made can be summarized as follows: 1.
If
is above the xaxis between a and b then .
2.
If
is below the xaxis between a and b then
a
b
a
b
.
3.
If
is above and below the xaxis between a and b then which is the net area.
A3
A1
a
A2
b
4.
5.
6.
a
b
Note: when b > a, and we calculate the integral from b to a, dx < 0.
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c
Example 4: Changing the Limits of Integration When using the Substitution Method Solve How to do it
by substitution. What to think about
How would you integrate using substitution if this was an indefinite integral?
What values will u take, when integrating over
?
Why must you change the constants of integration when using substitution?
Your Turn Solve
483
Practice Evaluate each integral:
484
1.
5.
2.
6.
3.
7.
4.
8.
9. Given:
. Let
. Change the limits of integration and write the
new integral in terms of u, then solve.
10. Evaluate the integral
11. Suppose
and leave the answer in exact form
, then evaluate
485
12. Given
,
, and
a.
b.
c.
13. Given
a.
b.
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and
, find
find
9.2 Derivative of a Definite Integral Warm Up Today we are going to analyze the Fundamental Theorem of Calculus more closely. Consider what the theorem states:
What is the relationship between
For the integral
identify
and
?
and
Can you state the Fundamental Theorem of Calculus an alternative way?
487
The Fundamental Theorem of Calculus can be stated another way
This represents the notion that the integral of a rate of change function gives us the net change of the integral function over the interval of integration.
Example 1: Evaluate the Derivative of a Definite Integral a) How to Do It a)
b) What to Think About What does
What does
ask you to do?
ask you to do?
How does your answer relate to the original integrand?
b) What does
What does
ask you to do?
ask you to do?
How does your answer relate to the original integrand??
488
There is one other form of the FTC often called the Second Fundamental Theorem of Calculus:
If the function g is defined as
a x b then
for a < x < b.
In other words, g is an antiderivative of f and therefore f is the derivative of g.
The second fundamental theorem of calculus is sometimes referred to as the NewtonLeibniz axiom
Your Turn What does each expression represent? a)
b)
489
Example 2: Evaluate Using the Second Fundamental Theorem with the Chain Rule a)
How to Do It a)
b)
What to Think About What is different about this expression?
When differentiating which rule must you follow when there is a function in the place of a constant of integration?
b) What part of the expression indicates that the chain rule must be used?
490
Theorem: Extension of the Second Fundamental Theorem of Calculus
Your Turn What does each expression represent? a)
b)
491
Example 3: Connecting the FTC to the Area Under the Curve Consider the Area Function Evaluate the following: a) b) c) d) e) f) How to Do It a)
What to Think About What do the constants of integration mean graphically?
b) What does graphically?
492
represent
c)
Why is the answer negative?
d) What does the derivative of the area function represent graphically?
e)
How is the value of this expression represented graphically?
f)
Why is this solution not negative?
493
Your Turn Let
where f(t) is the function graphed here. a) Evaluate
.
y 3 2 1
b) Estimate
.
3
2
1
1 1 2 3
c) On what interval is g(x) increasing?
d) Where does g have its maximum value?
494
2
3
x
Practice Evaluate each expression:
1.
6.
For
, find
.
7.
For
8.
For
, find
.
9.
For
, find
.
2.
, find
.
3.
4.
5.
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10.
496
for
.
a)
Find the value of x where F attains its minimum value.
b)
Find open intervals over which F is only increasing or only decreasing.
c)
Find open intervals over which F is only concave up or only concave down.
11.
Let
where
is the function graphed to the right.
a) At what values of x do the local maximum and minimum of g occur?
4 3 2 1 1
2
3
4
5
6
7
8
9
10
1 2 3 4
b) Where does g attain its absolute maximum?
c) On what intervals is g concave downward?
497
9.3 The Average Value of a Function Warm Up The area of the two shaded regions are equal. a) Find the area under the curve g(x) on the interval [1,3].
b) Find the value of g(c), such that the area of the rectangle on the interval [1,3] is equal to the area under the curve of on the interval [1,3]
c) How is g(c) related to the area of g(x)? c
d) Is the c the midpoint of the interval?
The area under any curve,
, can
be represented as a rectangle with height g(c) over the interval [a,b]. Can you write this relationship as an equation?
498
Definition: The Average Value of a function If f is integrable on the interval [a,b] then
is the average value of the
function on the interval.
Example 1: Finding the Average Value of a Function on an Interval a) Find the average value of the function on the interval b) Find c such that
is the average value of
.
on the interval [0,2].
How to Do It a) The graphs below show equivalent areas.
What to Think About Graphically, what does the average value represent?
Why does dividing the integral by the width of the interval gives us the average value?
Let
be the average value of f over the specified interval.
499
b)
What value will the function have at c?
Your Turn a) What is the average value of the function
b) Find c such that
500
is the average value of
on the interval
on the interval [0,4].
?
Theorem: Mean Value Theorem for Integrals If f is a continuous function on
, then there exists a number c in
such that:
The area under the graph of f from a to b is equal to the product of the interval length and the average value . The average value is the height of the area represented as a rectangle. The average value (mean value) of the rate of change function can be written and shown in two ways:
You will recognize the first as the average value of average rate of change of f on
on
, otherwise known as the
. The second, you will recognize from the mean
value theorem as the average rate of change of f on . Using the fundamental theorem of calculus, we can see how these two expressions are equivalent.
Thus, if we are looking for the average rate of change for a function and we are given the derivative function, we use the mean value theorem for integrals
. If we are looking for the average rate of change for a function and we are given the original function, we use the mean value theorem for derivatives
.
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Example 2: Finding the Average Rate of Change Find the average velocity for given each situation below a) The position function b) The velocity function How to Do It a) The average velocity is the average rate of change of position over time.
What to Think About What formula should you use to calculate the average value?
Why do you not use an integral?
What formula should you use to calculate the average value? b) The average velocity can be found using the mean
value theorem for integrals. What do you get when you integrate a rate of change?
How do you determine the average value given the net change over an interval?
502
Your Turn Show that the average velocity of a car over a time interval [a,b] is the same as the average of its velocities during the trip.
Practice 1. Let f be a continuous function on [0, 3]. If, of
then the greatest possible value
is
2. Find the average value of the function
3. Let
on the interval
be a continuous function on
smallest possible values of
. If
, what are the greatest and
?
503
4. Find the average value of the function on the given interval a.
b.
5.
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Given the graph of f a.
Evaluate
b.
Determine the average value of f on the interval [0,8]
c.
Determine the answer to a, b is the graph if translated 3 units upwards
9.4 The Integral as Net Change Warm Up Given the graph of the velocity of a particle moving on the xaxis. The particle starts at x = 2 when t = 0.
a) What is the net distance traveled by the particle during the trip?
b) Find where the particle is at the end of the trip.
c)
Find the total distance traveled by the particle.
505
Example 1: Motion Along a Line The velocity of a particle is given by
cm/sec. If the particle is moving along
a horizontal xaxis for , a) Draw the graph of the velocity function and describe the motion of the particle. b) Find the net distance traveled by the particle in the first 4 seconds. c) Suppose the initial position of the particle is . What is the particle’s position at ,
.
d) Use your calculator to find the total distance traveled by the particle in the first 4 seconds How to Do It What to Think About How do we know from the graph a) when it is moving left or right?
What is the point where the direction changes?
The particle is moving to the left from to the right
b)
506
and
.
When you integrate the rate of change function what do you get?
c) at
at
sec
What does
sec
Why must we consider this when determining the particle’s position at a different time?
represent?
d) Which buttons do you need to press to do this on your calculator?
Your Turn The velocity of a particle is given by
cm/sec. If the particle is moving along
a horizontal xaxis for a) What is the net distance traveled by the particle in the first 6 seconds?
b) Suppose the initial position of the particle is ,
. What is the particle’s position at
.
507
c) What is the total distance traveled by the particle in the first 4 seconds?
Making Predictions using the FTC Since the integral of a rate of change function gives us the net change of the original function, we use the facts that
and
to determine future or past values of the function.
We can rearrange the fundamental theorem of calculus to create the following formula given the velocity function and one value, , to find a future value
.
In other words, the value of x at b is the value of x at a plus the net change of x from a to b.
or given
, to find a past value
.
508
Did You Know Three hundred years before Newton and Leibniz first developed calculus Nicole Oresme (1323 – 1382), in his Treatise on the Configuration of Qualities and Motions proved geometrically that under uniform acceleration, the distance traveled is equal to the distance traveled at constant average velocity.
Example 2: Using your Calculator to Solve RealWorld Applications using Net Change People enter a line for a turnstile at a rate modeled by the function given by
where is measured in people per second and t is measured in seconds. As people go through the turnstile the exit the line at a constant rate of 0.85 person per second. There are 22 people in line at time . a) How many people enter the line for the turnstile during the first 5 minutes? b) During the first 5 minutes, there are always people in line. How many people are in line at ? c) After 5 minutes, what is the first time that there are no people in line? d) During the first 5 minutes, at what time will the number of people in line be a minimum? Find the number of people in line to the nearest whole number at this time. Justify your answer. How to Do It What to Think About How do you determine the net a) . change of people entering the line in the first 5 minutes?
509
b) Let P be the number of people in line. y
How many people are in line to begin with? How many people enter the line in the first 5 minutes? How many people exit the line in the first 5 minutes?
c) Since 51 people are in line at the 5 minute mark, we simply need to determine how long it takes these 51 people to pass through the turnstiles at a rate of .85 person per second.
What equation can be used to model this situation?
Are there other strategies to get this answer?
Can we determine this by starting at t = 0?
d)
How can the rate of change function be represented as a piecewise function?
We look for critical points such that Where do minimums occur? Since changes from negative to positive, has a minimum at by the first derivative test.
How can you justify that a minimum has occurred?
How do you determine the net change in people in line?
510
Can you write equation?
be as an
511
Your Turn Oil is pumped into an underground tank at a constant rate of 5 liters per second. Oil leaks out of the tank at a rate of
liters per second, for
hours. At t = 0, the tank contains 115
liters of oil. a) How many liters of oil leak out of the tank in the first 40 seconds?
b) How many liters of oil are in the tank at t = 40 seconds?
c) Write an expression for the total amount of oil in the tank at any time t
d) At what time t for Justify your answer.
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seconds, is the amount of oil in the tank a maximum.
Practice 1. A particle moves along the xaxis so that its velocity at time t is oven by
. At time t = 0, the particle is at a position x = 2. a) Find the acceleration of the particle at time t = 1. Is the speed of the particle increasing at t = 1? Explain your answer.
b) Find all times in the open interval (0,5) when the particle changes direction. Justify your answer.
c) What is the total distance traveled by the particle from t = 0 to t = 5?
d) What is the particles position at t = 4?
513
2. A 100gallon aquarium in a dentist’s office contains a rare collection of fish at t = 0. During a 12hour period from 8:00 am t = 0 to 8:00pm t = 12 the doctor is cleaning the tank by refilling the tank at a rate of
and during the same time interval there is
water being removed from the tank at a rate of a) How many gallons of water is being pumped into the tank during the time interval hours?
b) Is the amount of water in the tank rising or falling at time t = 8 hours? Give a reason for your solution.
c) How much water is in the tank at t = 12 hours?
d) At what time t between 8 am and 8 pm is the volume of the water in the fish tank th least?
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9.5 Chapter Review Find the average value over each interval. 1.
2.
3.
4.
Show that the average value of is equal to the average rate of change of f over the interval [a, b]
515
Evaluate the following definite integrals 5.
8.
9. 6.
10.
7. 11.
Evaluate the derivative of the following. 12.
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13.
14.
15.
Change the limits of integration and then evaluate the function. 16.
18.
17.
19.
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20. Use the function f in the figure and the function g defined by
to answer the
following:
a) Complete the chart. x g(x)
0
1
2
3
4
b) Plot the points from the table.
c) What are the absolute extreme(s) on [0,8]? Explain.
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5
6
7
8
21. The rate of consumption of an electric vehicle was recorded during a road trip along the coastal Chapman Peak’s Drive in Cape Town, South Africa. The table of selected values represents a twicedifferentiable function E(t) over time t for the interval of minutes. t (minutes) 0 10 15 25 30 40
kWh (per minute) .21 .28 .3 .34 .3 .26
a) Determine
and explain its meaning as it relates to E.
b) What can you state about the rate of consumption on the interval
c) Determine the value of
?
using a trapezoidal sum of 5 subintervals. Explain
the significance of
519
22. The rate of growth of the radius of a tree is given by the function
where the radius r is
written in terms of the # of days, d. This is often due to environmental changes both seasonal and those due to natural disasters. a) Is the rate of the radial growth of the tree increasing or decreasing at d=62?
b) How much has the tree’s radial distance grown between days 30 and 80?
c) Suppose that a forest ranger measures the tree’s radius to be 7cm at the onemonth point and then determines that this function represents the tree’s radial growth starting after its first month. What is the tree’s expected radius after 4 years? What was the average rate of growth over this time frame? Determine the age of the tree at this point given the following process:
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Find the tree’s radius Subtract ¼”1” to account for bark (dependent upon the species of tree) Research the average ring width online
Chapter 10: Area and Volume Every day you see examples of how integration is used in a wide variety of contexts. For example:
Consumer Testing – A car magazine tests two kinds of engine. One engine had acceleration modeled by , t seconds after starting from rest. The acceleration of a turbocharged model could be approximated by ,t seconds after starting from rest. How much faster is the turbocharged model at the end of a 10 second test run.
Population Predictions – A
government estimates that the population of the country (in 1000 people per year) will grow at the rate of . If an education program is instituted, they believe the population growth will change to
over
a 5year period. How many fewer people will be in the country if the education program is implemented and is successful?
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10.1 Area Between Two Curves Warm Up Find the area between the xaxis and the curve y = x 2 from x =1 to x = 4
Find the area under the curve from
to
.
What would you do to find the area between the line, y = – 1 and the functions above?
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Example 1: Area Between Two Functions Over a Closed Interval Write an expression that represents the area over the interval [a, b]: a) Between and the xaxis b) Between
and the xaxis
c) Between
and a
How to Do It
b
What to Think About What integral can you use to calculate the area under the curve?
a)
a
b
b)
How can you combine the first two areas to determine the area between these two curves?
a
b
How can you combine the difference of the areas into one integral expression?
c)
a
b
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Area of the Region Between Two Curves To determine the area between two curves, partition (slice) the interval into n equal subintervals of . Use a rectangular area for each subinterval with as the width and as the height of the ith subinterval. By taking the limit as , we can find the exact area using the definite integral. Note, the region will be bounded on an interval between two xvalues where .
Your Turn Find the area bounded by each region. 1.
2.
and the xaxis between
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Example 2: Finding the Area Between Two Curves Find the area of the region between and How to Do It Sketch the curves.
. What to Think About What do the graphs look like?
What interval contains the region?
How do you determine the intersection points?
Sketch a rectangular slice with width . What is the area of this rectangular partition?
Does it matter that one curve is below the xaxis?
What are the limits of integration for this region?
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Your Turn 1. Find the area between
2.
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and
Determine the area of the region enclosed by
.
Example 3: Finding the Area Between Two Curves that Intersect at More than Two Points Find the area of the region between and . How to Do It What to Think About What do the graphs look like? Sketch the curves. At which xvalues do the graphs intersect?
Sketch a slice of the region on the interval [3, 1]. What is the area of this rectangular partition with width ?
Sketch a slice of the region on the interval [1,1]. What is the area of this rectangular partition?
What integrals can be used to find the area of each region?
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Your Turn 1. Find the area between
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and
.
Practice Sketch the region bounded by the graphs of the given equations, show a typical vertical slice and find the area of the region. 1.
2.
3.
and
,
between x = 1, x = 2
and
and
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4.
5.
6.
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and
and
and
from
7.
8.
9.
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10.
and x=0
11.
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10.2 Area Between Two Curves Using Horizontal Slices Warm Up Use your graphing calculator to find the area bounded by the graphs of and . Sketch the region and a sample rectangular partition.
Is there a way to determine where maximum and minimum values of a function occur?
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Example 1: Finding the Area Bounded by a Top Curve and Two Different Bottom Curves Find the area bounded , the xaxis and . What to Think About Sketch the curves. Does the top curve stay the same for the entirety of the bounded region?
Does the bottom curve stay the same for the entirety of the bounded region?
What is the area of a slice in each region with width ? The area is bounded by
as the top curve
over the interval [0,4] and two different bottom curves. Therefore, we will consider two separate regions and add them together. What integral can be used to determine the area of the region defined by each bottom curve?
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Your Turn Find the area of the region bounded by
,
and
.
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Example 2: Finding Area using Horizontal Slices Find the area bounded , the xaxis and How to Do It
.
What to Think About What would a horizontal rectangular slice look like?
Sketch the curve first.
What is the vertical width of the slice?
What is the horizontal length of the slice?
What is the area of a rectangular slice? The area is bounded by
which is
equivalent to , the left curve and which is equivalent to , the right curve. The width of each slice is , so the interval of the limits of integration are in terms of y from [0,2]. The horizontal length of each slice will be equal to the difference in the xvalues of the right and the left curve.
What integral can be used to determine the area of the region between the right curve and the left curve?
What are the limits of integration?
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Area of the Region Between Two Curves using Horizontal Partitioning To determine the area between two curves, using horizontal partitions (slices) the interval into n equal subintervals of . Use a rectangular area for each subinterval with as the width and as the length of the ith subinterval. By taking the limit as , we can find the exact area using the definite integral. Note, the region will be bounded on an interval that ranges between two yvalues where .
Your Turn Find the area bounded
and
.
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Example 3: Finding Area with a Graphing Calculator Find the area bounded and . How to Do It To graph on your calculator, you must solve each equation for y.
What to Think About How can you graph the given relations on your calculator?
Which slicing method, horizontal or vertical, makes the most sense to use in this case?
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What integral can be used to determine the area of the region between the right curve and the left curve?
What are the limits of integration?
The area is bounded by
on the right and
on the left. Calculate the intersection points of the functions to determine the limits of integration.
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Your Turn Find the area of the region bounded by the following curves using horizontal slices.
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,
and
Practice Find the area of the enclosed region. 1. and between y = 2, y = 3
2.
3.
4.
and
and
and
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5.
6.
7.
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and
and
and
and y = 3.5
10.3 Volumes of Solids Warm Up You are given a roll of sushi and want to determine the volume of the entire roll.
a. Cut the sushi up into slices of width
. Sketch the shape of each slice.
b. Write an expression for the volume of one slice with width
and radius
c. Imagine the top of the sushi roll can be represented by the line
.
. The 3D
sushi roll is a cylinder that is created by rotating around the xaxis so that the axis goes through the center of the roll. Sketch the roll over the interval [a, b] and label one slice with , .
How can you determine the volume using infinitely thing compact discs (cylinders)?
d. Create an equation to find the volume of the entire roll.
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Definition: Let be a continuous function on
. The volume V of the solid of revolution
generated by revolving the region bounded by the graphs of
and the x
axis around the xaxis is Note: is the radius R of the solid of revolution from the xaxis to the outermost function.
Example 1: Calculating the Volume of a Solid Revolution About the xaxis If , find the volume of the solid generated by revolving the region under the graph of f from x = 1 to x = 1 about the xaxis. How to Do It What to Think About Draw a picture and label it What does this shape look like?
What is the shape a slice that is perpendicular to the xaxis?
What is the width of the slice? The radius of the disk is
What other dimension do I need to determine the volume of the slice?
What is the expression for the volume of a slice?
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Your Turn: Find the volume of the solid generated by revolving the region under the graph of f about the xaxis. a)
b)
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Example 2: Calculating the Volume of a Solid of Revolution About the yaxis Find the volume of the solid generated by revolving the region between the graph of , y = 1, y = 1 and x = 0 about the yaxis. How to Do It What to Think About Draw a picture and label it What is the shape a slice that is perpendicular to the yaxis?
What is the width of the slice? What is the radius of the slice?
What is the expression for the volume of each slice?
The radius of the disk is What are the limits of integration?
Definition: The volume V of the solid of revolution generated by revolving the region bounded by the graphs of and the yaxis around the yaxis is
Note: g(y) is the radius R of the solid of revolution from the yaxis to the outermost function. 546
Your Turn What is the volume of region bounded by around the yaxis?
, if the region is rotated
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Example 3: Calculating the Volume of a Solid of Revolution About a Line Find the volume of the solid generated by revolving the region between the graph of and y = 1, around the line y = 1. How to Do It What to Think About Draw a picture and label it Draw the radius for one slice of the solid of revolution. What expression can be used to find the distance between the axis of rotation and the function to be rotated?
What is the radius of each slice?
What is the expression for the volume of a slice?
The radius of the disk is Is it possible to write the radius in another way? Does it make a difference to the solution?
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Your Turn The region, bounded by lines. Find each volume. a. the line y = 3
and
, is rotated about each of the following
b. the line x = 9
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Practice Set up and evaluate the integral that gives the solid formed by revolving about the indicated axis or line. 1.
2.
[0,1] and the x – axis.
[1,4] and the x – axis
3.
4.
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about the line x = 6
10.4 Finding Volume Using the Washer Method Warm Up You are building an electric gokart for a competition and your car is over the allowable weight. To make it lighter, you decide to hollow out the cylindrical steel frame. Consider a steel bar of length 1.4 meters and radius 20mm that weighs 9.87 kg/m. Determine the volume of the steel bar, its weight and then its density in kg/m3. Show the integral and calculations that lead to your solution. Include units in your answer.
The bar must weigh under 10 kg. You hollow out a cylindrical hole down the middle of the bar. What is the minimum radius required to remove enough material? Write an integral expression for the volume of the cylindrical hole created.
Use two integrals and the radii above to write an expression for the volume of the steel remaining in the bar.
Is there a way to determine the volume of a solid that has a hole in the middle?
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Definition: The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions and around the xaxis is
Note: There are two radii that must be considered in this situation. The larger radius, R, is determined as the difference between the outermost function and the axis of rotation. The smaller radius, r, is determined as the difference between the innermost function and the axis of rotation.
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Example 1: Finding the Volume of a Solid with a Hole in the Middle – the Washer Method Find the volume of the solid generated by the region bounded by the graphs of , the line How to Do It Draw a picture and label it
,
,
rotated around the xaxis. What to Think About What shape best describes a slice?
What is the volume of the whole slice if there was no hole in the middle?
What is the volume of the hole in the middle of each slice?
What is the expression for the volume of each slice?
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Your Turn Find the volume of the solid generated by the region bounded by the graphs of the line
rotated about the x – axis.
Example 2: The Washer Method – Rotations Around the yaxis. Find the volume of the solid generated by the region bounded by the graphs of the line
,
rotated about the y – axis.
How to Do It Draw a picture and label it
What to Think About What is the outer radius of each slice?
What is the inner radius of each slice?
What is the expression for the volume of a slice?
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Definition: The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions and around the yaxis is
Note: There are two radii that must be considered in this situation. The larger radius, R, is determined as the difference between the outermost function and the axis of rotation. The smaller radius, r, is determined as the difference between the innermost function and the axis of rotation.
Your Turn Find the volume of the solid generated by the region bounded by the graphs of about the y – axis. HINT: Find the intersection points first
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Example 3: Finding the Volume of Revolution by Rotating Around a Line Find the volume of the solid generated by the region bounded by the graphs of , the line How to Do It Draw a picture and label it
,
,
rotated around the line
.
What to Think About What is the outer radius of each slice?
What is the inner radius of each slice?
What is the expression for the volume of a slice?
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Definition: The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions and around the is
The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions and around the is
Note: There are two radii that must be considered in this situation. The larger radius, , is determined as the difference between the outermost function and the axis of rotation. The smaller radius, , is determined as the difference between the innermost function and the axis of rotation.
Your Turn Find the volume of the solid generated by the region bounded by the graphs of line,
,
,
rotated around the line
, the
. Write the integral used and
evaluate using a calculator.
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Practice 1.
2.
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about the line x = 6
about the line x = 4
3.
about the line y = 3
4. a) About the x – axis
b) About the y – axis
c) About the line x = 4
d) About the line y = 2
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10.5 Finding Volume Using CrossSectional Area Warm Up So far, we have been considering solids created by rotating a region around a line. Imagine taking a crosssectional slice of one of these solids, what does one of the slices look like? Write an expression for its volume.
Imagine another type of solid that is formed with a known crosssection. How can we find the volume of one of these solids? For example, consider a solid with a flat circular base, that has crosssections perpendicular to the xaxis, that are squares of side length equal to the distance across the circular base at that point. Draw or build a solid that has these features.
What would this shape look like if the crosssectional area was a semicircle, or an isosceles right triangle, or some irregular area?
See http://mathdemos.org/mathdemos/sectionmethod/sectiongallery.html https://www.geogebra.org/m/bwRfChS9
Definition: The volume V of the solid with cross sectional area A(x) taken perpendicular to the xaxis
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Example 1: Finding the Volume of a Solid with Known CrossSection Perpendicular to the xaxis Find the volume of a solid with a circular base of radius 3, that has crosssectional areas perpendicular to the xaxis that are isosceles right triangles with one leg in the plane of the base. How to Do It
What to Think About What is the equation of the base region?
What is the area expression for the crosssectional area of this shape?
Why does the integrand have the constant
?
Why does the integrand have the constant 2?
Why does the integrand have an exponent of 2?
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Your Turn Find the volume of a solid with a circular base of radius 3, that has crosssectional areas perpendicular to the xaxis that are squares.
Definition: The volume V of the solid with cross sectional area A(x) taken perpendicular to the y axis,
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Example 2: Finding the Volume of a Solid of Known CrossSection Perpendicular to the yaxis Find the volume of a solid with base region bounded by , and , that has crosssectional areas perpendicular to the yaxis that are semicircles. How to Do It
What to Think About What does this solid look like? What is the equation of the base region when considering crosssections perpendicular to the yaxis?
What is the area expression for the crosssectional area of this shape?
Why does the integrand have the constant
?
Why is the function divided by 2?
Why does the integrand have an exponent of 2?
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Your Turn Find the volume of a solid with base region bounded by , and , that has crosssectional areas perpendicular to the yaxis that are equilateral triangles with one leg in the plane of the base region.
Example 3: Finding the Volume of a Solid of Known CrossSection Between Two Curves Find the volume of a solid bounded above by and below by with crosssectional areas perpendicular to the xaxis that are squares where the sidelength is equal to the distance between the two boundary curves over the interval How to Do It
.
What to Think About What is the area expression for the crosssectional area of this shape?
Why is the function not divided by 2?
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Your Turn Find the volume of a solid bounded above by and below by with crosssectional areas perpendicular to the xaxis that are semicircles where the sidelength is equal to the distance between the two boundary curves over the interval .
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Practice 1.
The base of a solid is the circular region bounded by the graph of volume of the solid: a) If every crosssection perpendicular to the xaxis is a square.
. Find the
b) If every crosssection perpendicular to the x – axis is an isosceles triangle and the height equal to the base.
c) If every crosssection perpendicular to the xaxis is a semicircle.
d) If every crosssection perpendicular to the xaxis is an equilateral triangle.
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2.
A solid has as its base the region bounded by and . Find the volume of the solid if every crosssection is perpendicular to the xaxis. a) Is an isosceles right triangle with hypotenuse in the xy – plane
b) Is a square
3.
The base of the solid is bounded by the graphs of and . Find the volume of the solid if every crosssection perpendicular to the yaxis is a semicircle with diameter in the xy – plane.
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4.
The base of the solid is bounded by the graphs of
and
and the yaxis. Find the volume of the solid if every crosssection perpendicular to the xaxis is an isosceles right triangle with one leg in the xyplane.
5.
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The base of the solid is bounded by the graphs of and . Find the volume of the solid if every crosssection perpendicular to the xaxis is a square with side extending from
6.
The base of the solid is bounded by the graphs of
and
. Find the volume of the solid if every crosssection perpendicular to the xaxis is an isosceles right triangle with one of the legs extending from
7.
The base of the solid is a region in the fourth quadrant bounded by the x – axis and the y – axis, and the line
. If the crosssection of the solid is perpendicular to the x
– axis are semicircles. What is the volume?
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10.6 Chapter Review 1.
Let R be the region in the first quadrant enclosed by the graphs of
and
R a. Find the area of R
b. The region R is the base of a solid. The cross sections perpendicular to the yaxis are squares. Find the volume of the solid.
c. The region R is rotated about the line y = 8. Find the volume of the solid.
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2.
Let h and k be the functions given by
and
. Let R be
the region in the firsst quadrant enclosed by the yaxis and the graphs of h and k, and S be the region in the first quadrant enclosed bythe graphs h and k.
S
R a. Calculate the area of R
b. Calculate the area of S
c. Determine the volume of the solid when S is revolved about the horizontal line y = –2.
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3.
Let R be the shaded region in the first quadrant enclosed by the graphs of and
and the yaxis.
R
a. Find the area of the region R
b. The volume of the solid generated when the region R is revolved about the x axis.
c. Find the volume of the solid generated when the region R is revolved about the line .
d. The region R is the base of a solid. For the solid, each crosssection perpendicular to the xaxis is a semicircle. Find the volume of the solid.
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4.
Let R be the region bounded by the xaxis, the graphs of
and the line
a. Find the area of region R
b. Find the volume of the solid generated when R is evolved about the yaxis.
c. The region R is the base of a solid. For the solid, each crosssection perpendicular to the xaxis is an isosceles triangle with one of its legs bound by the two graphs. Find the volume of the solid.
R
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