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Elite Ninja Math S. Budd Lamar High School pdf September 29, 2010


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Mr. Budd, compiled September 29, 2010


Contents I

Calculus

1

1 Elite Differentiation 1.1 Implicit Differentiation . . . . . . . . . . . . . . . . . 1.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . 1.2.1 Inverse Functions . . . . . . . . . . . . . . . . 1.2.2 Differentiating Inverse Functions . . . . . . . 1.2.3 Antidifferentiating Inverse Functions . . . . . 1.3 Related Rates . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Introduction to Related Rates . . . . . . . . . 1.3.2 Triangle problems . . . . . . . . . . . . . . . 1.3.3 Volume problems . . . . . . . . . . . . . . . . 1.4 Calculus in 2–D . . . . . . . . . . . . . . . . . . . . . 1.4.1 Intro to Parametric Equations . . . . . . . . 1.4.2 Calculus of Parametrics . . . . . . . . . . . . 1.4.3 Displacement, Velocity, Acceleration Vectors 1.5 L’Hˆ opital’s Rule . . . . . . . . . . . . . . . . . . . . 1.5.1 L’Hˆ opital’s Rule . . . . . . . . . . . . . . . . 1.5.2 L’Hˆ opital’s Rule : 0 · ∞, 1∞ , 00 and ∞ − ∞ 1.5.3 Indeterminate Products . . . . . . . . . . . . 1.5.4 Indeterminate Differences . . . . . . . . . . . 1.5.5 Indeterminate Powers . . . . . . . . . . . . . 1.5.6 Special Limit: lim sinθ θ . . . . . . . . . . . . .

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3 5 9 9 11 12 15 15 16 17 23 23 24 24 31 31 32 32 33 33 34

2 Elite Antidifferentiation 2.1 Integration by Parts . . . . . . . . . . . . . . . . . 2.1.1 The Anti-Product Rule . . . . . . . . . . . 2.1.2 Derivative Known, Antiderivative Unknown 2.1.3 Rapid, Repeated Integration by Parts . . . 2.1.4 The Back Around Technique . . . . . . . . 2.2 Partial Fractions . . . . . . . . . . . . . . . . . . . 2.2.1 Partial Fractions . . . . . . . . . . . . . . . 2.2.2 Nonrepeating Linear Factors . . . . . . . . 2.2.3 Repeating Factors . . . . . . . . . . . . . . 2.2.4 Quadratic Factors . . . . . . . . . . . . . .

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iv

CONTENTS . . . . . . . . .

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3 Elite Integration 3.1 Accumulation Functions . . . . . . . . . . . . . . . . . . . . . 3.1.1 Accumulation Functions . . . . . . . . . . . . . . . . . 3.1.2 Fundamental Theorem of Calculus, part II . . . . . . . 3.1.3 Curve Sketching with Accumulation Functions . . . . 3.2 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Improper Integrals . . . . . . . . . . . . . . . . . . . . 3.2.2 More Improper Integrals . . . . . . . . . . . . . . . . . 3.3 Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Definite Integral . . . . . . . . . . . . . . . . . . . . . 3.3.2 Elite Ninja Volume . . . . . . . . . . . . . . . . . . . . 3.3.3 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Mare Orea . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Total Distance . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Volumes of Rotation: Sweet, Sweet Loaves of Calculus 3.4 Non-Circular Slicing . . . . . . . . . . . . . . . . . . . . . . . 3.5 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Distance Traveled . . . . . . . . . . . . . . . . . . . . 3.6 MVT for Integrals . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Average Value . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . 3.6.3 Mean Value Theorem for Integrals . . . . . . . . . . .

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59 . 61 . 61 . 61 . 64 . 69 . 69 . 72 . 75 . 76 . 76 . 76 . 77 . 78 . 80 . 91 . 95 . 95 . 96 . 99 . 99 . 103 . 106

4 Polar Coordinates and Complex Numbers 4.1 Area in Polar Coordinates . . . . . . . . . . . . 4.1.1 Area . . . . . . . . . . . . . . . . . . . . 4.1.2 Arc Length . . . . . . . . . . . . . . . . 4.2 Complex Numbers and Mathematical Induction 4.2.1 Complex Numbers . . . . . . . . . . . . 4.2.2 Graphing Complex Numbers . . . . . . 4.2.3 Mathematical Induction . . . . . . . . . 4.2.4 De Moivre’s Theorem . . . . . . . . . . 4.3 DeMore DeMoivre . . . . . . . . . . . . . . . . 4.3.1 DeMoivre’s Theorem and Power Series . 4.3.2 Roots of Complex Numbers . . . . . . .

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2.3

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2.2.5 Heaviside Shortcut for Nonrepeating Linear Factors 2.2.6 Improper Algebraic Fractions . . . . . . . . . . . . . 2.2.7 Telescoping Series . . . . . . . . . . . . . . . . . . . Separable Differential Equations . . . . . . . . . . . . . . . 2.3.1 Separable Differential Equations . . . . . . . . . . . 2.3.2 Slope Fields . . . . . . . . . . . . . . . . . . . . . . . Accumulation Functions . . . . . . . . . . . . . . . . . . . . 2.4.1 Creating Quick, Cheap Antiderivatives . . . . . . . . 2.4.2 Parametrics Revisited . . . . . . . . . . . . . . . . .

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CONTENTS

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4.3.3

Deriving Trigonometric Formulas . . . . . . . . . . . . . . 124

5 Taylor Series 5.1 Series Basics . . . . . . . . . . . . . . . . . . 5.1.1 Some Vocabulary . . . . . . . . . . . . 5.1.2 Remember Geometric Series? . . . . . 5.1.3 nth Term Test for Divergence . . . . . 5.1.4 Geometric Series . . . . . . . . . . . . 5.2 Manipulation of Taylor and Maclaurin Series 5.2.1 Telescoping Series . . . . . . . . . . . 5.2.2 Introduction to Power Series . . . . . 5.2.3 Taylor Series . . . . . . . . . . . . . . 5.2.4 Manipulation of Series . . . . . . . . . 5.3 Interval of Convergence . . . . . . . . . . . . 5.3.1 Ratio Test . . . . . . . . . . . . . . . . 5.3.2 Radius and Interval of Convergence . 5.3.3 Testing Endpoints . . . . . . . . . . .

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127 129 130 130 131 132 135 135 135 136 136 141 141 142 143

6 Series of Constants 6.1 The Integral Test and p-Series . 6.1.1 Integral Test . . . . . . 6.1.2 p–Series . . . . . . . . . 6.2 Comparison Tests . . . . . . . . 6.2.1 Direct Comparison Test 6.2.2 Limit Comparison Test 6.3 Alternating Series . . . . . . . 6.3.1 Alternating Series . . . 6.3.2 Problems . . . . . . . . 6.4 Convergence Test Review . . .

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145 147 147 148 151 151 152 155 155 155 157

7 Differential Equations 7.1 Euler’s Method . . . . . . . . . . . . . . 7.2 Logistic Equation . . . . . . . . . . . . . 7.3 Homogeneous Differential Equations . . 7.4 Linear DiffEq’s and Integrating Factors 7.5 Differential Equation Review Problems .

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I.B. Topics

8 Vectors 8.1 Vector 8.1.1 8.1.2 8.1.3 8.1.4

Basics . . . . . . . Vectors . . . . . . Representation . . Simple Arithmetic Magnitude . . . .

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vi

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Line . . . . . .

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186 187 187 189 194 196 196 197 197 203 204 204 205 206 206 211 211 215 215 216 217 217 221 221 223 223 225 227 227 228 229 230 235 235 235 236 237 237 238

9 Probability 9.0 Re-introduction to Probability . . . . . . . . . . . . 9.0.1 The Basics . . . . . . . . . . . . . . . . . . . 9.0.2 Probability and Set Theory: Venn Diagrams 9.0.3 Conditional Probability . . . . . . . . . . . . 9.0.4 Sampling Without Replacement . . . . . . .

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8.2 8.3

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8.5

8.6

8.1.5 Making Unit Vectors . . . . . . . . . . . 8.1.6 Scalar Product . . . . . . . . . . . . . . 8.1.7 Angle . . . . . . . . . . . . . . . . . . . 8.1.8 Lines in Two Dimensions . . . . . . . . 8.1.9 Applications . . . . . . . . . . . . . . . 8.1.10 Lines in Three Dimensions . . . . . . . 8.1.11 Angle Between Lines . . . . . . . . . . . 8.1.12 Intersection of Lines . . . . . . . . . . . 8.1.13 Applications in Three Dimensions . . . 8.1.14 Matrix Addition . . . . . . . . . . . . . 8.1.15 Matrix Multiplication . . . . . . . . . . 8.1.16 Matrix Division . . . . . . . . . . . . . . 8.1.17 Determinants . . . . . . . . . . . . . . . 8.1.18 Solving Systems of Equations . . . . . . 8.1.19 Singularity . . . . . . . . . . . . . . . . Cross Product . . . . . . . . . . . . . . . . . . . 8.2.1 Cross Product . . . . . . . . . . . . . . Planes . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Planes . . . . . . . . . . . . . . . . . . . 8.3.2 Angle Between Planes . . . . . . . . . . 8.3.3 Intersection of a Line and a Plane . . . 8.3.4 Angle Between a Line and a Plane . . . Plane Crashes . . . . . . . . . . . . . . . . . . . 8.4.1 Intersection of Two Planes . . . . . . . 8.4.2 Singularity . . . . . . . . . . . . . . . . 8.4.3 Intersection of Three Planes . . . . . . . 8.4.4 Solving Systems of Equations . . . . . . Non-Unique Solutions . . . . . . . . . . . . . . 8.5.1 Unique Solution . . . . . . . . . . . . . 8.5.2 No Solution . . . . . . . . . . . . . . . . 8.5.3 Infinite Solutions . . . . . . . . . . . . . 8.5.4 Augmented Matrices and Echelon Forms Distances . . . . . . . . . . . . . . . . . . . . . 8.6.1 Projection of a Vector . . . . . . . . . . 8.6.2 Distance From a Point to a Plane . . . . 8.6.3 Distance Between Parallel Planes . . . . 8.6.4 Distance Between a Plane and a Parallel 8.6.5 Distance Between a Point and a Line . . 8.6.6 Distance Between Parallel Lines . . . .

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CONTENTS

9.1 9.2

9.3

9.4

9.5

9.6

9.7

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9.0.5 Statistical Independence . . . . . . . . . . . . . . . . 9.0.6 Counting Methods . . . . . . . . . . . . . . . . . . . Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Permutations . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Combinations . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Binomial Expansion . . . . . . . . . . . . . . . . . . Probability Distribution Functions . . . . . . . . . . . . . . 9.3.1 Discrete Random Variables . . . . . . . . . . . . . . 9.3.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Other Measures . . . . . . . . . . . . . . . . . . . . . 9.3.4 Binomial Distribution . . . . . . . . . . . . . . . . . Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Binomial Distribution with Huge n and Tiny p . . . 9.4.2 Poisson Distribution . . . . . . . . . . . . . . . . . . Continuous Random Variables . . . . . . . . . . . . . . . . . 9.5.1 Continuous Random Variables . . . . . . . . . . . . 9.5.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Other Measures . . . . . . . . . . . . . . . . . . . . . Normal Distribution . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Standardizing a Normal Distribution . . . . . . . . . 9.6.2 Normal Approximation to the Binomial Distribution Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Central Tendency . . . . . . . . . . . . . . . . . . . . 9.7.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Frequency . . . . . . . . . . . . . . . . . . . . . . . . 9.7.4 Cumulative Frequency . . . . . . . . . . . . . . . . . 9.7.5 Population vs. Sample . . . . . . . . . . . . . . . . . 9.7.6 Discrete vs. Continuous . . . . . . . . . . . . . . . . 9.7.7 Groups . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.8 Histograms . . . . . . . . . . . . . . . . . . . . . . . 9.7.9 Continuous Data . . . . . . . . . . . . . . . . . . . . 9.7.10 Percentiles and Quartiles . . . . . . . . . . . . . . .

10 Error of Series 10.1 Remainder Reminder . . . 10.1.1 Geometric Series . 10.1.2 Alternating Series 10.1.3 Integral Test . . . 10.2 Error Analysis for Series .

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viii

CONTENTS

Mr. Budd, compiled September 29, 2010


Part I

Calculus

1


Unit 1

Elite Differentiation 1. Implicit Differentiation 2. Inverse Functions 3. Calculus of Vectors 4. Calculus of Parametric Functions 5. Arc Length

Advanced Placement Asymptotic and unbounded behavior. • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) Parametric and vector functions. The analysis of planar curves includes those given in parametric form and vector form. Applications of derivatives. • Analysis of planar curves given in parametric form and vector form, including velocity and acceleration vectors. • L’Hˆ opital’s Rule , including its use in determining limits and convergence of improper integrals. 3


4

HL Unit 1 (Elite Differentiation)

Computations of derivatives. • Derivatives of parametric and vector functions. Applications of integrals. To provide a common foundation, specific applications should include the distance traveled by a particle on a line, and the length of a curve (including a curve given in parametric form). Techniques of antidifferentiation • Improper integrals (as limits of definite integrals).

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 1: Implicit Differentiation

1.1

5

Implicit Differentiation

International Baccalaureate 7.8 Implicit differentiation. Advanced Placement Computation of derivatives. • Chain rule and implicit differentiation. Textbook §2.8 Implicit Differentiation and Inverse Trigonometric Functions [15] Resources Exploration 4-8a: “Implicit Relation Derivatives” and Exploration 4-8b: ”Ovals of Cassini Project” in [11] Example 1.1.1 For x = tan (f (x)), find f 0 (x) in terms of x only. d Note that your answer is dx arctan x. Example 1.1.2 Write an equation √ of the line tangent to the graph of x2 + y 2 = 16 at the point 2 3, 2 √  √  Ans: y − 2 = − 3 x − 2 3 Implicit Differentiation is Latin for “Remember the Freakin’ Chain Rule”. Remember: anytime you differentiate a y-term with respect to x, you must dy multiply by . dx d dy O(y) = O0 (y) · dx dx Example 1.1.3 Identify the variable, then Remember the Freakin’ Chain Rule: d 2 x dx d 2 (b) y dy (a)

Mr. Budd, compiled September 29, 2010


6

HL Unit 1 (Elite Differentiation) d 2 [f (x)] dx d 2 (d) y dx d 2 (e) x dt (c)

Example 1.1.4 Remember the Product Rule: d (xy) dx d (−4xy) (b) dx (a)

Example 1.1.5 For x2 + y 2 = 16: (a) For what coordinate pairs is the graph increasing? (b) For what coordinate pairs does the curve have a vertical tangent? (c) Remember the Quotient Rule: For what values of x or y is d2 y dx2 > 0 ?

Example 1.1.6 Do Foerster’s Exploration 4-8 in your mighty, mighty groups of four.

Example 1.1.7 Second Derivative Problems (adapted from BC98) p d2 y dy = 1 − y 2 , what is ? [Ans: −y] dx dx2 p dy d2 y (b) If = − 1 − y 2 , what is ? [Ans: −y] dx dx2 (c) Name two functions for which f 00 (x) = −f (x). Is it true for p p dy dy those functions that = 1 − y 2 and = − 1 − y2 ? dx dx (a) If

Example 1.1.8 Find

dy for y = cos (xy) dx Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 1: Implicit Differentiation

7

The derivative at a specific point You may plug in the values for x and y at any point after differentiating. If you dy only need a slope at a specific value, and don’t need a formula for , it may dx behoove you to plug in values right after differentiation. It’s a lot easier to deal with 12, 6, and −24 than with 8y, 3x2 y, and 4xy. Note that sometimes, you must find one of the coordinate pairs from the other.

The explicit option Some equations can be solved explicitly. Just solve for y, and take the derivative per usual. Example 1.1.9 Solve x2 + y 2 = 16 explicitly for y, then find dy , and show that the answer is the same as we get using implicit dx differentiation.

The derivative at a specific point Example 1.1.10 [4] The slope of the tangent to the graph of ln (x + y) = x2 at the point where x = 1 is

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

The explicit option Example 1.1.11 Solve ln (x + y) = x2 explicitly for y, then find dy when x = 1. dx

Problems 1.A-1 (from [3]) Consider the curve x + xy + 2y 2 = 6. The slope of the line tangent to the curve at the point (2, 1) is Ans: − 31 Mr. Budd, compiled September 29, 2010


8

HL Unit 1 (Elite Differentiation)

1.A-2 (from [3]) If x2 + 2xy − 3y = 3, then the value of 1.A-3 (from [3]) If y 2 − 3x = 7, then

d2 y = dx2

1.A-4 (from [3]) If tan (x + y) = x, then

dy = dx

dy at x = 2 is [Ans: −2] dx h i Ans: − 4y93   Ans: − sin2 (x + y)

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

d2 y at the dx 2  Ans: 10 27

1.A-8 If x2 − y 2 = 25, for what coordinate pairs will the curve have vertical tangents? [Ans: (5, 0), (−5, 0)] 1.A-9 If y 2 − x2 = 25, for what coordinate pairs will the curve have vertical tangents? [Ans: none] 1.A-10 (adapted from AB ’98) If x2 +xy = −10, then when x = 2,

 dy = Ans: dx

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

3 2



dy = dx

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

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

dy h ? Ans: dx

1+y sin(xy) 1−x sin(xy)

1.A-15 (HL 5/02) A curve has equation xy 3 + 2x2 y = 3. Find the equation of the tangent to this curve at the point (1, 1). [Ans: y − 1 = − (x − 1)] 1.A-16 (HL 5/03) A curve has equation x3 y 2 = 8. Find the equation of the normal to the curve at the point (2, 1). [Note: the normal  sticks out of the curve like hairs, and is perpendicular to the tangent.] Ans: y − 1 = 34 (x − 2) 1.A-17 For ey = x, find

dy in terms of x only. dx

 Ans:

1 x



Mr. Budd, compiled September 29, 2010

i


HL Unit 1, Day 2: Inverse Functions

1.2

9

Inverse Functions

International Baccalaureate 3.4 The inverse functions x 7→ arcsin x, x 7→ arccos x, x 7→ arctan x; their domains and ranges; their graphs. 7.1 Derivatives of arcsin x, arccos x, arctan x. 7.2 The chain rule for composite functions; Application of chain rule to related rates of change.

Advanced Placement Applications of derivatives. • Use of implicit differentiation to find the derivative of an inverse function. Computation of derivatives. • Knowledge of derivative of basic functions, including inverse trigonometric functions. • Chain rule and implicit differentiation. Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables.

1.2.1

Inverse Functions

To get an inverse relation, switch the x’s and y’s, then solve for y. Example 1.2.1 (adapted from D&S) Let f (x) = x3 −9x2 +31x−39 and let g be the inverse of f . What is the value of g 0 (0)?  Ans:

1 4



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

Mr. Budd, compiled September 29, 2010


10

HL Unit 1 (Elite Differentiation) Example 1.2.2 (adapted from AB ’07) The functions f and g are differentiable for all real numbers, and g is strictly increasing. Table 1.1 gives values of the functions and their first derivatives at selected values of x. Table 1.1: From AB 2007 Exam x 1 2 3 4

f (x) 6 9 10 −1

g(x) 4 2 −4 3

f 0 (x) 2 3 4 6

g 0 (x) 5 1 2 7

(a) If g −1 is the inverse function of g, write an equation for the line tangent to the graph of y = g −1 (x) at (a) x = 3 (b) x = 4 (c) x = 5  Ans: y − 2 = 1 (x − 3); y − 3 =

1 2

(x − 4);



(b) What is the problem with asking for an equation for the line tangent to the graph of y = f −1 (x) at x = 9? [Ans: the inverse of f ain’t a function] In order for a function f to have an inverse function f −1 , i.e., an inverse relation that is also a function, then f −1 must pass the vertical line test, and f must pass a horizontal line test. One way to ensure that f passes a horizontal line test is if f is monotonically increasing or monotonically decreasing. The inverse relation won’t always be a function. Sometimes we must take selective pieces of the graph to make sure that the inverse is a function. • For example, with

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

• For arcsin x, we use only a portion of the graph that will give a full spectrum of sines, i.e., from −1 to 1. We use that portion of the graph in the fourth and first where the cosine is always positive. The range h quadrants π πi of arcsin x is − , . 2 2 • For arccos x, we use a portion of the graph that gives a full spectrum of cosine values, so the domain is x ∈ [−1, 1]. The range is the angles in quadrants I and II where the sine is positive. Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 2: Inverse Functions

11

• The domain ofarctan  x is all real numbers, but the range is limited to the π π open interval − , . 2 2 • If f (x) passes a horizontal line test, then its inverse relation passes a vertical line test, and is therefore a function. q 2 (−3)

Example 1.2.3 Find Example 1.2.4 √

(a) Find arcsin (b) Find arccos

3 2 √1 2

(c) Find arctan −1 (d) Find arcsin sin 4π 3



(e) Graph y = arcsin (sin x) Example 1.2.5 Use a right triangle to find cos arcsin 35



Example 1.2.6 Use a right triangle to find sec (arctan 2x)

Example 1.2.7 For x = cos y, find that your answer is

1.2.2

d dx

dy in terms of x only. Note dx

arccos x.

Differentiating Inverse Functions

Example 1.2.8 Find an inverse function, g(x) = f −1 (x), for f (x) = x3 , and find g 0 (x) two ways. Example 1.2.9 Find the derivatives of (a) arcsin x (b) arccos x (c) arctan x

1 d arcsin x = √ dx 1 − x2 Mr. Budd, compiled September 29, 2010


12

HL Unit 1 (Elite Differentiation) 1 d arccos x = − √ dx 1 − x2 d 1 arctan x = dx 1 + x2

Example 1.2.10 Find

d arcsin x5 dx

Example 1.2.11 Find

d dx

√  x 1 − x2 + arcsin x √   Ans: 2 1 − x2

1.2.3

Antidifferentiating Inverse Functions Z

1 dx = arcsin x + C 1 − x2

Example 1.2.12 Antidifferentiate: Z 1 √ dx 1 − 9x2



Z

Ans:

1 3

arcsin 3x + C



1 dx = arctan x + C 1 + x2

Example 1.2.13 Antidifferentiate: Z 1 √ dx x (1 + x)

[Ans: 2 arctan

x + C]

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 2: Inverse Functions

13

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

1.B-2 For x = sin y, find

x 1.B-3 (adapted from [3]) An equation for a tangent to the graph of y = arctan 3  π  at the point 3, is: Ans: y − π4 = 16 (x − 3) 4 √ 3 0 1.B-4 [3]  If g(x)  = x − 1 and f is the inverse function of g, then f (x) = 2 Ans: 3x 1.B-5 (adapted from [3])

d [arctan 2x] = dx

h Ans:

2 1+4x2

i

1.B-6 (adapted from AB ’07) The functions f and g are differentiable for all real numbers, and g is strictly increasing. Table 1.1 gives values of the functions and their first derivatives at selected values of x. If g −1 is the inverse function of g, write an equation for the line graph  tangent to the of y = g −1 (x) at x = 6. Ans: y − 4 = 17 (x − 6) 1.B-7 (adapted from D&S)

d 4x arcsin = dx 3

h

Ans:

4 9−16x2

i

1

1.B-8 (adapted from D&S) If f (x) = x− 4 , what is the derivative of the inverse  of f (x)? Ans: − x45 Z   3x2 √ 1.B-9 Find dx using the u-substitution u = x3 . Ans: arcsin x3 + C 6 1−x

Mr. Budd, compiled September 29, 2010


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HL Unit 1 (Elite Differentiation)

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 3: Related Rates

1.3

15

Related Rates

Advanced Placement Derivative as a function. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Applications of derivatives. • Modeling rates of change, including related rates problems. Textbook §3.8 Related Rates [15] Resources §10-4 Related Rates in Foerster; Exploration 10-4

1.3.1

Introduction to Related Rates

Example 1.3.1 (adapted from Acorn BC ’00) A point (x, y) is moving along a curve y = f (x). At the instant when the slope of 1 the curve is − , the x-coordinate of the point is increasing at the 5 rate of 3 units per second. The rate of change, in units per second, of the y-coordinate of the point is   Ans: − 35 √ Example 1.3.2 (BC ’98) When x = 8, the rate at which 3 x is 1 increasing is times the rate at which x is increasing. What is the k value of k? [Ans: 12] Example 1.3.3 [3] If xy 2 = 20, and x is decreasing at the rate of 3 units per second, the rate at which y is changing when y = 2 is approximately [Ans: 0.6 units/sec] Mr. Budd, compiled September 29, 2010


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HL Unit 1 (Elite Differentiation)

1.3.2

Triangle problems

Example 1.3.4 (adapted from AB ’93) The top of a 13-foot ladder is sliding down a vertical wall at a constant rate of 2 feet per minute. When the top of the ladder is 5 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

 Ans:

5 6

 feet per minute

Example 1.3.5 (AB 2002 Form B) Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 10 km/hr. Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in Figure 1.1. Figure 1.1: From 2002 AP Calculus AB Exam

(a) Find the distance, in kilometers, between Ship A and Ship B when x = 4 km and y = 3 km. (b) Find the rate of change, in km/hr, of the distance between the two ships when x = 4 km and y = 3 km. (c) Let θ be the angle shown in Figure 1.1. Find the rate of change of θ, in radians per hour, when x = 4 km and y = 3 km. Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 3: Related Rates

1.3.3

17

Volume problems

Example 1.3.6 (adapted from AB ’98) The radius of a circle is decreasing at a constant rate of 0.2 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second?

[Ans: −0.2C] Example 1.3.7 (AB 2002) A container has the shape of an open right circular cone, as shown in Figure 1.2. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at the 3 cm/hr. constant rate of − 10 (The volume of a cone of height h and radius r is given by V = 1 2 πr h.) 3 Figure 1.2: From 2002 AP Calculus AB Exam

(a) Find the volume V of water in the container when h = 5 cm. 3 Indicate units of measure. Ans: 125 12 π cm (b) Find the rate of change of the volume of water in the container, with respect to time,when h = 5 cm. Indicate units of measure.  3 Ans: − 15 8 π cm /hr (c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportion- 3 ality? Ans: − 10 Mr. Budd, compiled September 29, 2010


18

HL Unit 1 (Elite Differentiation) Example 1.3.8 [3] The function V whose graph is sketched in Figure 1.3 gives the volume of air, V (t), (measured in cubic inches) that a man has blown into a balloon after t seconds.   4 3 V = πr 3 The rate at which the radius is changing after 6 seconds is approximately what? Figure 1.3: From [3]

[Ans: 0.1 in/sec]

Problems 1.C-1 (adapted from Acorn AB ’00) If r is positive and increasing, for what value of r is the rate of increase of r3 forty-eight times that of r? [Ans: 4] 1.C-2 [3] Let y = 2ecos x . Both x and y vary with time in such a way that y increases at the constant rate of 5 units per second. The rate at which x π is changing when x = is [Ans: −2.5 units/sec] 2 1.C-3 [3] One ship traveling west is W (t) nautical miles west of a lighthouse and a second ship traveling south is S(t) nautical miles south of the lighthouse at time t (hours). The graphs of W and S are shown in Figure 1.4. At what approximate time is the distance between the ships increasing at t = 1? (nautical miles per hour = knots) [Ans: 4 knots] 1.C-4 (adapted from AB ’97) A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 3: Related Rates

19

Figure 1.4: From [3]

an eastbound train traveling at 80 meters per second. At how many meters per second is the train moving away from the observer 3 seconds after it passes through the intersection? [Ans: 76.8] 1.C-5 (from HL 11/03) An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At at given instant the observer notes that the angle θ is 1 1 π radians and is increasing at radians per second. Find the speed, in 3 60 kilometers per hour, at which the airplane ismoving towards the observer. 1 Ans: 15 km/s = 240 km/hr Figure 1.5: Airplane flying towards an observer.

1.C-6 Figure 1.6 shows an isosceles triangle ABC with AB = 10 cm and AC = BC. The vertex C is moving in a direction perpendicular to (AB) with speed 2 cm per second. Calculate the rate of of the angle CAB  increase 1 at the moment the triangle is equilateral. Ans: 10 radians per second 1.C-7 (adapted from D&S) Two cars start at the same place and at the same time. One car travels east at a constant velocity of 40 miles per hour and a second car travels north at a constant velocity of 48 miles per hour. Approximately how fast is the distance between them changing after half an hour? Round your answer to the nearest mile per hour. [Ans: 62 mph] Mr. Budd, compiled September 29, 2010


20

HL Unit 1 (Elite Differentiation)

Figure 1.6: Growing isosceles triangle.

1.C-8 (adapted from D&S) A missile rises vertically from a point on the ground 65, 000 feet from a radar station. If the missile is rising at the rate of 17, 500 feet per minute at the instant when it is 38, 000 feet high, what is the rate of change, in radians per minute, of the missile’s angle of elevation from the radar station at this instant? [Ans: .201 radians per minute] 1.C-9 [3] When the area of an expanding square, in square units, is increasing three times as fast as its side is increasing, in linear units, the side is   Ans: 23 1.C-10 [3] Water is flowing into a spherical tank with 6 foot radius at the constant rate of 30π cubic ft per hour. When the water is h feet deep, the volume of water in the tank is given by V =

πh2 (18 − h) 3

What is the rate at which the depth of the water in the tank is increasing at the moment when the water is 2 feet deep? [Ans: 1.5 ft per hr] 1.C-11 [3] The edge of a cube is increasing at the uniform rate of 0.2 inches per second. At the instant when the total surface area becomes 150 square inches, what is the rate of increase, in cubic inches per the  second, of volume of the cube? Ans: 15 in3 /sec 1.C-12 (adapted from AB ’98) If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 6 inches per minute, what relationship must exist between b and h for the area A of the triangle to be decreasing? Ans: b > 21 h 1.C-13 [3] Sand is being dumped on a pile in such a way that it always forms a cone whose base radius is always 3 times its height. The function V whose graph is sketched in Figure 1.7 gives the volume of the conical sand pile, V (t), measured in cubic feet, after t minutes. At what approximate rate is the radius of the base changing after 6 minutes. [Ans: 0.22 ft/min]

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 3: Related Rates

21

Figure 1.7: From [3]

Mr. Budd, compiled September 29, 2010


22

HL Unit 1 (Elite Differentiation)

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 4: Calculus in 2–D

1.4

23

Calculus in 2–D

Advanced Placement Parametric and vector functions. The analysis of planar curves includes those given in parametric form and vector form. Applications of derivatives.

• Analysis of planar curves given in parametric form and vector form, including velocity and acceleration vectors.

Computations of derivatives.

• Derivatives of parametric and vector functions.

Textbook 11.1 Vector-Valued Functions [15] Resources §1.8 Parametric Equations in Anton [1]. §4-7 Derivative of a Parametric Function in Foerster [12]. Exploration 4-7a: “Parametric Function Graphs” and Exploration 6-8 (1e): “A Limit by l’Hospital’s Rule” in [11]

1.4.1

Intro to Parametric Equations

Example 1.4.1 Sketch the trajectory over the time interval 0 ≤ t ≤ 10 of the particle whose parametric equations of motion are x = t − 3 sin t and y = 4 − 3 cos t.[1]

Example 1.4.2 Find the graph of the parametric equations x = cos t, y = sin t, (0 ≤ t ≤ 2π). [1]

Example 1.4.3 Graph the parametric curve x = 2t − 3, y = 6t − 7 by eliminating the parameter, and indicate the orientation on the graph. [1]

Example 1.4.4 Use a graphing utility to graph the equation x = 3y 5 − 5y 3 + 1. [1] Mr. Budd, compiled September 29, 2010


24

HL Unit 1 (Elite Differentiation)

1.4.2

Calculus of Parametrics dy dy/dt = dx dx/dt

Example 1.4.5 Given x = 3 cos (2πt) and y = 5 sin (πt) (a) Plot the xy-graph. Use a t-range that generates at least one complete cycle of x and y. A t-step of 0.05 is reasonable. Sketch the result. (b) Describe the behavior of the xy-graph as t increases. (c) Find an equation for

dy dx

in terms of t.

dy dx

(d) Calculate when t = 0.15. Show how the answer corresponds to the graph. dy is indeterminate when t = 0.5. Find the approx(e) Show that dx dy imate limit of dx as t approaches 0.5. How does the answer relate to the graph?

(f) Make a conjecture about what geometrical figure the graph represents. Then confirm your conjecture by eliminating the parameter t and analyzing the resulting Cartesian equation. (g) How do the range and the domain of the parametric equation relate to the range and domain of the Cartesian equation?

Example 1.4.6 (BC93) If x = t2 + 1 and y = t3 , then

d2 y = dx2  Ans:

1.4.3

3 4t



Displacement, Velocity, Acceleration Vectors

Remember that • velocity is the derivative of position, position is the antiderivative of velocity; • acceleration is the derivative of velocity, velocity is the antiderivative of acceleration; • speed is the magnitude of velocity. Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 4: Calculus in 2–D

25

Figure 1.8: from 2003 BC Exam

Example 1.4.7 (BC03) A particle starts at point A on the positive x-axis at time t = 0 and travels along the curve from A to B to C to D, as shown in Figure 1.8. The coordinates of the particles position dx = (x(t), y(t)) are differentiable functions of t, where x0 (t) = dt √     πt π t+1 dy −9 cos sin and y 0 (t) = is not explicitly given. 6 2 dt At time t = 9, the particle reaches its final position at point D on the positive x-axis. dy dx (a) At point C, is positive? At point C, is positive? Give a dt dt reason for each answer. (b) The slope of the curve is undefined at point B. At what time t is the particle at point B? (c) The line tangent to the curve at the point (x(8), y(8)) has equa5 tion y = x − 2. Find the velocity vector and the speed of the 9 particle at this point.

[Ans: no, no; t = 3; h−4.5, −2.5i, 5.147;] Example 1.4.8 (BC03B) A particle moves in the xy-plane so that the position of the particle at any time t is given by x(t) = 2e3t + e−7t and y(t) = 3e3t − e−2t . (a) Find the velocity vector for the particle in terms of t, and find the speed of the particle at time t = 0. dy dy in terms of t, and find lim . (b) Find t→∞ dx dx (c) Find each value t at which the line tangent to the path of the particle is horizontal, or explain why none exists. (d) Find each value t at which the line tangent to the path of the particle is vertical, or explain why none exists. Mr. Budd, compiled September 29, 2010


26

HL Unit 1 (Elite Differentiation) 

√ Ans: 6e3t − 7e−7t , 9e3t + 2e−2t , 122; 32 ; none;

1 10

ln

7 6



Example 1.4.9 (1994 BC-3) A particle moves along the graph of y = cos x so that the x-component of acceleration is always 2. At time t = 0, the particle is at the point (π, −1) and the velocity vector of the particle is (0, 0). (a) Find the x- and y-coordinates of the position of the particle in terms of t. (b) Find the speed of the particle when its position is (4, cos 4).

Example 1.4.10 Derive   1 2 ~r(t) = (v0 cos θ) t i + h + (v0 sin θ) t − gt j 2 Example 1.4.11 A baseball player at second base throws the ball 90 feet to the player at first base. The ball is thrown at 50 miles per hour at an angle of 15 above the horizontal. At what height does the first baseman catch the ball if the ball is thrown from a height of 5 feet? [15]

Example 1.4.12 The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground and leaves the bat at a speed of 100 miles per hour. What is the minimum angle if the hit is a home run?

Problems 1.D-1 (adapted from BC98) In the xy-plane, the graph of parametric equations x  = 3t 5and  y = 5t + 2, for −3 ≤ t ≤ 3, is a line segment with what slope? Ans: 3 1.D-2 (adapted from BC97) If x = sin (2t) and y = e2t , then

h dy = Ans: dx

e2t cos(2t)

i

1.D-3 (adapted from BC97) For what values of t does the curve given by the parametric equations x = t3 + t2 − 1 and y = t4 + 2t 2 − 8t have a vertical tangent? Ans: 0 and − 32 only 1.D-4 (adapted from BC98) If f is a vector-valued function defined by f (t) = (e−t , sin t), then f 00 (t) = [Ans: (e−t , − sin t)] Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 4: Calculus in 2–D

27

1.D-5 (adapted from BC98) A particle moves on a plane curve so that at any 3 time t > 0 its x-coordinate is t3 −t and its y-coordinate is (2t − 3) . What is the acceleration vector of the particle at t = 2? [Ans: (12, 24)] 1.D-6 (BC04) An object moving along a curve in the xy-plane has position  dx dy (x(t), y(t)) at time t ≥ 0 with = 3 + cos t2 . The derivative dt dt is not explicitly given. At time t = 2, the object is at position (1, 8). dy is −7. Write an equation for the line dt tangent to the curve at the point (x(2), y(2)).

(a) At time t = 2, the value of

(b) Find the speed of the object at time t = 2. (c) For t ≥ 3, the line tangent to the curve at (x(t), y(t)) has a slope of 2t + 1. Find the acceleration vector of the object at time t = 4. [Ans: y − 8 = −2.983 (x − 1); 7.383; h2.303, 24.814i] 1.D-7 (BC04B) A particle moving along a curve in the plane has position (x(t), y(t)) at time t, where dx p 4 dy = t + 9 and = 2et + 5e−t . dt dt for all real values of t. A time t = 0, the particle is at point (4, 1). (a) Find the speed of the particle and its acceleration vector at time t = 0. (b) Find an equation for the line tangent to the path of the particle at time t = 0. √   Ans: 58, h0, −3i; y − 1 = 73 (x − 4) 1.D-8 (BC03B) A particle moves in the xy-plane so that the position of the particle at any time t is given by x(t) = 2e3t + e−7t and y(t) = 3e3t − e−2t . (a) Find the velocity vector for the particle in terms of t, and find the speed of the particle at time t = 0. dy dy (b) Find in terms of t, and find lim . t→∞ dx dx (c) Find each value t at which the line tangent to the path of the particle is horizontal, or explain why none exists. (d) Find each value t at which the line tangent to the path of the particle is vertical, or explain why none exists. 

√  1 Ans: 6e3t − 7e−7t , 9e3t + 2e−2t , 122; 32 ; none; 10 ln 76 Mr. Budd, compiled September 29, 2010


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HL Unit 1 (Elite Differentiation)

1.D-9 (BC02) Figure 1.9 shows the path traveled by a roller coaster car over the time interval 0 ≤ t ≤ 18 seconds. The position of the car at time t seconds can be modeled parametrically by x(t) = 10t + 4 sin t, y(t) = (20 − t) (1 − cos t), where x and y are measured in meters. Figure 1.9: from 2002 BC Exam

(a) Find x0 (t) and y 0 (t). [Ans: x0 (t) = 10 + 4 cos t, y 0 (t) = (20 − t) sin t + cos t − 1] (b) Find the slope of the path at time t = 2. Show the computations that lead to your answer. (c) Find the acceleration vector of the car at the time when the car’s horizontal position is x = 140. (d) Find the time t at which the car is at its maximum height, and find the speed, in m/sec, of the car at this time. [Ans: 1.794; h−3.529, 1.226i; 3.024, 6.028] 1.D-10 (BC02B) A particle moves in the xy-plane so that its position at any time t, for −π ≤ t ≤ π, is given by x(t) = sin (3t) and y(t) = 2t. (a) Sketch the path of the particle. Indicate the direction of motion along the path. (b) Find the range of x(t) and the range of y(t). (c) Find the smallest positive value of t for which the x-coordinate of the particle is a local maximum. What is the speed of the particle at this time?   Ans: ; −1 ≤ x(t) ≤ 1, −2π ≤ y(t) ≤ 2π; π6 , 2 1.D-11 (BC01) An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t with   dx dy = cos t3 and = 3 sin t2 dt dt for 0 ≤ t ≤ 3. At time t = 2, the object is at position (4, 5). (a) Write an equation for the line tangent to the curve at (4, 5). Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 4: Calculus in 2–D

29

(b) Find the speed of the object at time t = 2. [Ans: y − 5 = 15.604 (x − 4); 2.275] 1.D-12 (BC00) A moving particle has position (x(t), y(t)) at time t. The position of the particle at time t = 1 is (2,  6) and the velocity vector at any time 1 1 t > 0 is given by 1 − 2 , 2 + 2 . t t (a) Find the acceleration vector at time t = 3. (b) Find the position of the particle at time t = 3. (c) For what time t > 0 does the line tangent to the path of the particle at (x(t), y(t)) have a slope of 8? (d) The particle approaches a line as t → ∞. Find the slope of this line. Show the work that leads to your conclusion. h  10 32  q 3 i 2 2 , − 27 ; 3 , 3 ; 2; 2 Ans: 27

Mr. Budd, compiled September 29, 2010


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HL Unit 1 (Elite Differentiation)

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 5: L’Hˆ opital’s Rule

1.5

31

L’Hˆ opital’s Rule

International Baccalaureate f (x) using l’Hopital’s Rule and/or g(x) the Taylor series. Cases where the derivatives of f (x) and g(x) vanish for x = a. 10.5 The evaluation of limits of the form lim

x→a

Advanced Placement Applications of derivatives.

• L’Hˆ opital’s Rule , including its use in determining limits and convergence of improper integrals.

Textbook §3.2 Indeterminate Forms and L’Hˆopital’s Rule [15] Resources §4.5 Indeterminate Forms and L’Hˆ opital’s Rule in Stewart [20]. §68 Limits of Indeterminate Forms: l’Hospital’s Rule in Foerster [12]. Exploration 6-8: “A Limit by l’Hospital’s Rule” in Instructor’s Resource Book [11].

1.5.1

L’Hˆ opital’s Rule Basics:

0 0

and

∞ ∞

As improper integrals are essentially limits of definite integrals, I will take this opportunity to introduce (or re-introduce) L’Hˆopital’s Rule , which is a technique for evaluating recalcitrant limits. Theorem 1.1 (L’Hˆ opital’s Rule ). If f (x) = g (x) , 0 x→c h (x)

0, then lim f (x) = lim x→c

0

g(x) h(x)

and if lim g(x) = lim h(x) = x→c

x→c

provided the latter limit exists.

Corollaries of the rule lead to the same conclusion if x → ∞ or if both g(x) and h(x) approach infinity. Basically, L’Hˆ opital’s Rule is a way for me to evaluate limits which, upon substitution, yield either 00 or ∞ ∞ . To evaluate the limits: once I have shown that I have either of the two allowable indeterminate forms, i.e., 00 or ∞ ∞ , I take the derivative of the top and the bottom, and evaluate my new limit.

Example 1.5.1 Prove that lim

θ→0

sin x x

= 1 without using L’Hˆopital’s

Rule . Mr. Budd, compiled September 29, 2010


32

HL Unit 1 (Elite Differentiation) Example 1.5.2 [19]

ex − x − 1 x→0 x2 lim

 Ans:

1 2



Example 1.5.3 Given x = 3 cos (2πt) and y = 5 sin (πt) (a) Plot the xy-graph. Use a t-range that generates at least one complete cycle of x and y. A t-step of 0.05 is reasonable. Sketch the result. (b) Describe the behavior of the xy-graph as t increases. (c) Find an equation for

dy dx

in terms of t.

dy dx

(d) Calculate when t = 0.15. Show how the answer corresponds to the graph. dy (e) Show that dx is indeterminate when t = 0.5. Find the approxdy imate limit of dx as t approaches 0.5. How does the answer relate to the graph?

(f) Make a conjecture about what geometrical figure the graph represents. Then confirm your conjecture by eliminating the parameter t and analyzing the resulting Cartesian equation. (g) How do the range and the domain of the parametric equation relate to the range and domain of the Cartesian equation?

1.5.2

L’Hˆ opital’s Rule and Indeterminate Forms other than 0 ∞ and 0 ∞

1.5.3

Indeterminate Products

0·∞ If you have the form 0 · ∞, you need to change 0 to change the ∞ to 10 (to get 0 · 01 = 00 ).

1 ∞

(to get

1 ∞

·∞=

∞ ∞)

or

Example 1.5.4 Show (a) lim be−b = 0 b→∞

(b) lim b2 e−b = 0 b→∞

Mr. Budd, compiled September 29, 2010


HL Unit 1, Day 5: L’Hˆ opital’s Rule

33

(c) lim b3 e−b = 0 b→∞

Example 1.5.5 Introduction to Mathematical Induction • Recall that lim b1 e−b = 0 b→∞

• Show that, assuming lim bn e−b = 0, then lim bn+1 e−b = 0 b→∞

b→∞

must also be true. What are the implications?

1.5.4

Indeterminate Differences

∞−∞ Example 1.5.6  lim

x→0

1 1 − sin x x



Use Least Common Denominator (LCD) to add fractions.

1.5.5

Indeterminate Powers

∞0 Example 1.5.7 [10] Find limx→∞ x1/x

00 Example 1.5.8 Show lim xx = 1

x→0

1∞ Example 1.5.9 Find 1

lim x 1−x

x→1

 Ans:

1 e



Mr. Budd, compiled September 29, 2010


34

HL Unit 1 (Elite Differentiation)

1.5.6

Special Limit: lim sinθ θ θ→0

Example 1.5.10 Prove that lim

θ→0

sin x x

= 1 with and without using

L’Hˆ opital’s Rule . This is a classic problem. First show geometrically that sin x ≤ x ≤ tan x, so 1 1 sin x cos x ≤ ≤ , so that cos x ≤ ≤ 1. Then use the Squeeze that sin x x sin x x sin x Theorem to show that as x → 0, → 1. x Using that information, I realize that sin x and x are essentially the same for values very close to 0. Hence,  it makes sense that, from a previous example (in section 1.5.4) lim sin1 x − x1 = 0 x→0

Example 1.5.11 Show that lim

x→0

1−cos x x

= 0 without using L’Hˆopital’s

Rule .

Example 1.5.12 Use the limit definition of derivative to show that d sin 7x = 7 cos x. dx

Problems 1.E-1 Using the limit definition of the derivative, show that the derivative of cos x is − sin x. Use the fact that cos (θ1 + θ2 ) = cos θ1 cos θ2 − sin θ1 sin θ2 . (a) Find cos (x + h) using the sum of angles formula for cosine. (b) Use this to find cos (x + h) − cos x. cos (x + h) − cos x . h (d) Find the limit of the above difference quotient, using known limits. (c) Use this to find

1.E-2 Attempt L’Hˆ opital’s Rule on the following limits. What happens? Use your graphing calculator to find the limit. (a) lim √ x→∞

(b)

lim

x→π/2−

x x2

+1 tan x sec x

Mr. Budd, compiled September 29, 2010


Unit 2

Elite Antidifferentiation 1. Antidifferentiation by Parts 2. Improper Integrals 3. Antidifferentiation by Partial Fractions 4. Quick, Cheap, Dirty Antiderivative 5. Separable Differential Equations

Advanced Placement Asymptotic and unbounded behavior. • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) Parametric and vector functions. The analysis of planar curves includes those given in parametric form and vector form. Applications of derivatives. • Analysis of planar curves given in parametric form and vector form, including velocity and acceleration vectors. • L’Hˆ opital’s Rule , including its use in determining limits and convergence of improper integrals. 35


36

HL Unit 2 (Elite Antidifferentiation)

Computations of derivatives. • Derivatives of parametric and vector functions. Applications of integrals. To provide a common foundation, specific applications should include the distance traveled by a particle on a line, and the length of a curve (including a curve given in parametric form). Techniques of antidifferentiation • Improper integrals (as limits of definite integrals).

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 1: Integration by Parts

2.1

37

Integration by Parts

International Baccalaureate R R 7.10: Integration by parts. Examples: x Rsin x dx andR ln x dx. Repeated integration by parts: examples: x2 ex , dx ex sin x dx. Advanced Placement Techniques of antidifferentiation. • Antiderivatives by parts. Textbook §6.2 Integration by Parts [15] Resources §5.6 Integration by Parts in Stewart [20]. Exploration 9-3: “Integration by Parts Practice” in Instructor’s Resource Book [11]. §9-2 Integration by Parts - A Way to Integrate Products and §9-3 Rapid Repeated Integration by Parts in Foerster [12]. §8.2 Integration by Parts in Salas [19].

2.1.1

The Anti-Product Rule

From the product rule, we can derive the formula for integration by parts:

Z

Z u dv = uv −

v du

or

Z

f (x)g 0 (x) dx = f (x)g(x) −

Example 2.1.1 Find

R

Z

g(x)f 0 (x) dx

x sin 4x dx. [20]

Example 2.1.2 Do the integrating:

R

5xe3x dx [12]

 Ans:

5 3x 3 xe

− 59 e3x + C



Mr. Budd, compiled September 29, 2010


38

HL Unit 2 (Elite Antidifferentiation) R

Example 2.1.3 Do the integrating:  Ans:

2.1.2

1 2 4x

x2 cos 4x dx [12]

sin 4x + 18 x cos 4x −

1 32

sin 4x + C



Derivative Known, Antiderivative Unknown

Example 2.1.4 Calculate

R1 0

arctan y dy. [20] 

Ans:

π 4

ln 2 2



Try this classic example Example 2.1.5 Find

R

ln x dx. [Ans: x ln x − x + C]

2.1.3

Rapid, Repeated Integration by Parts

Example 2.1.6 Do the integrating:  Ans: Example 2.1.7 Find

R

R

1 2 4x

x2 cos 4x dx [12]

sin 4x + 81 x cos 4x −

1 32

sin 4x + C



x2 ex dx. [20]   Ans: x2 ex − 2xex + 2ex + C

Rearrangement of Terms Example 2.1.8 Do the integrating:

R

2

x3 ex dx [12] h Ans:

Example 2.1.9

R

1 2 x2 2x e

2

− 12 ex + C

i

2

x (ln x) dx h

Ans:

x2 2

2

(ln x) −

x2 2

ln x +

x2 4

+C

i

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 1: Integration by Parts

2.1.4

39

The Back Around Technique R

Example 2.1.10 Evaluate

ex sin x dx. [20]



Example 2.1.11 Do the integrating:

R

 Ans:

Ans:

1 2



(sin x − cos x) + C

e6x cos 4x dx [12]

1 6x 13 e

sin 4x +

3 6x 26 e



cos 4x + C

Example 2.1.12 Z

sin2 x dx

  Ans: − 12 sin x cos x + 21 x + C

Problems 2.A-1 [12] Wanda Y. Knott evaluates cos x dx. She gets

R

x2 sin x −

x2 cos x dx, letting u = x2 and dv = Z 2x sin x dx.

For the second integral, she lets u = sin x and dv = 2x dx. Show Wanda why her second choice of u and dv is inappropriate. R 2.A-2 [12] Amos Take evaluates x2 cos x dx by parts, letting u = cos x and dv = x2 dx. Show Amos that although his choice for dv can be integrated, it is a mistake to choose the parts as he did. 2.A-3 Find

R3 1

 Ans:

t ln t dt

9 2

 ln 3 − 2

√   Ans: x arcsin x + 1 − x2 + C

2.A-4 Antidifferentiate arcsin x h

2.A-5 [19] Evaluate

R

x3 3x dx

Ans: 3x

2.A-6 [19] Evaluate

R

3x2 cos x3 dx

2.A-7 [19] Evaluate

R

e3x cos 2x dx



x3 ln 3

3x2 (ln 3)2

+

6x (ln 3)3

6 (ln 3)4



+C

i

  Ans: sin x3 + C  Ans:

3 3x 13 e

cos 2x +

2 3x 13 e

sin 2x



Mr. Budd, compiled September 29, 2010


40

HL Unit 2 (Elite Antidifferentiation)

2.A-8 Find a function f such that [17] Z Z f (x) sin x dx = −f (x) cos x + x3 cos x dx. h Ans:

x4 4

i

R 2.A-9 The integral cos2 x dx can be integrated by clever use of trigonometric properties, as well as by parts. Substitute 12 (1 + cos 2x) for cos2 x and integrate. Compare this answer with that which you obtain by using integration by parts, and show that the two answers are equivalent. [12] 2.A-10 Derive the formula Z Z n−1 1 cosn−2 x dx cosn x dx = cosn−1 x sin x + n n similarly to how you antidifferentiated cos2 x by parts. Then use the formula to find generate the same answer you previously achieved for R cos2 x dx.

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 2: Partial Fractions

2.2

41

Partial Fractions

Advanced Placement Techniques of antidifferentiation.

• Antiderivatives by simple partial fractions (nonrepeating linear factors only). International Baccalaureate HL 8.10: Further integration. Included: integration using partial fraction decomposition.

Textbook §6.4 Integration of Rational Functions Using Partial Fractions [15]

2.2.1

Partial Fractions

Partial fractions decomposition is used to antidifferentiate integrands with a polynomial in the denominator. It is more of an algebraic technique than a calculus one. After factoring the denominator, the one fraction is replaced with several fractions with the individual denominators. From [10]:

General Description of the Method Success in writing a rational function f (x)/g(x) as a sum of partial fractions depends on two things:

• The degree of f (x) must be less than the degree of g(x). That is, the fraction must be proper. If it isn’t, divide f (x) by g(x) and work with the remainder term. • We must know the factors of g(x). In theory, any polynomial with real coefficients can be written as a product of real linear factors and real quadratic factors. In practice, the factors may be hard to find. Mr. Budd, compiled September 29, 2010


42

HL Unit 2 (Elite Antidifferentiation)

Here is how we find the partial fractions of a proper fraction f (x)/g(x) when the factors of g are known. Method of Partial Fractions (f (x)/g(x) Proper) m

1. Let x − r be a linear factor of g(x). Suppose (x − r) is the highest power of x − r that divides g(x). Then, to this factor, assign the sum of the m partial fractions: Am A1 A2 + ··· + + m. x − r (x − r)2 (x − r) Do this for each distinct linear factor of g(x). n 2. Let x2 + px + q be a quadratic factor of g(x). Suppose x2 + px + q is the highest power of this factor that divides g(x). Then, to this factor, assign the sum of the n partial fractions: B1 x + C1 Bn x + Cn B2 x + C 2 + ··· + 2 + n. 2 2 2 x + px + q (x + px + q) (x + px + q) Do this for each distinct quadratic factor of g(x) that cannot be factored into linear factors with real coefficients. 3. Set the original fraction f (x)/g(x) equal to the sum of all these partial fractions. Clear the resulting equation of fractions and arrange the terms in decreasing powers of x. 4. Equate the coefficients of corresponding powers of x and solve the resulting equations for the undetermined coefficients.

2.2.2

Nonrepeating Linear Factors

Example 2.2.1 Integrate by resolving into partial fractions: [12] Z x−2 dx (x − 5) (x − 1)

2.2.3

Repeating Factors

Example 2.2.2 [12] Z

4x2 + 18x + 6 (x + 5) (x + 1)

2 dx

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 2: Partial Fractions

2.2.4

43

Quadratic Factors

Example 2.2.3 [12] Z

2.2.5

4x2 + 6x + 11 dx (x2 + 1) (x + 4)

Heaviside Shortcut for Nonrepeating Linear Factors

Example 2.2.4 Integrate by resolving into partial fractions: [12] Z x−2 dx (x − 5) (x − 1) Example 2.2.5 [12] Z

2.2.6

11x2 − 22x − 13 dx x3 − 2x2 − 5x + 6

Improper Algebraic Fractions

Example 2.2.6 [12] Z

2.2.7

x3 − 9x2 + 24x − 17 dx x2 − 6x + 5

Telescoping Series

Example 2.2.7 Show that

∞ P

1 converges, and find its k=1 k (k + 1)

limit.

Problems big giant blue-green Calculus book §6.4: #3-35 (every other odd), 37, Exploratory Exercises #1; §8.2: # 7

Mr. Budd, compiled September 29, 2010


44

HL Unit 2 (Elite Antidifferentiation)

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 3: Separable Differential Equations

2.3

45

Separable Differential Equations

International Baccalaureate 7.10 Solution of first order differential equations by separation of variables. 10.6 First order differential equations: geometric interpretation using slope fields. Advanced Placement Applications of derivatives. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. Applications of antidifferentiation. • Solving separable differential equations and using them in modeling. Textbook §7.2 Separable Differential Equations and §7.3 Direction Fields and Euler’s Method [15] Resources Explorations 7-4a: “Introduction to Slope Fields”, 7-4b: “Slope Field Practice”, and 7-6b: “A Predator-Prey Problem” in [11]

2.3.1

Separable Differential Equations

Definition 2.1. Differential Equation A differential equation is an equation that contains the derivative of a function. [12]

Example 2.3.1 The solution to the differential equation 2xe−y , where y(0) = 1, is

dy = dx

  Ans: y = ln x2 + e

dy Example 2.3.2 (AB05) Consider the differential equation = dx 2x − . y Mr. Budd, compiled September 29, 2010


46

HL Unit 2 (Elite Antidifferentiation) (a) Let y = f (x) be the particular solution to the differential equation with the initial condition f (1) = −1. Write an equation for the line tangent to the graph of f at (1, −1) and use it to approximate f (1.1). (b) In your mighty, mighty groups of four: Find the particular solution y = f (x) to the given differential equation with the initial condition f (1) = −1. (c) How close was the tangent line approximation? √   Ans: y + 1 = 2 (x − 1), −0.8; y = − 3 − 2x2

Example 2.3.3 (adapted from [3]) If the  graph of y = f (x) contains 2x sin x2 dy the point (0, 1) and if = , then f (x) = dx y

h i p Ans: 3 − cos (x2 )

Example 2.3.4 (AB97) Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t ≥ 0. After her parachute opens, dv her velocity satisfies the differential equation = −2v − 32, with dt initial condition v(0) = −50. (a) Use separation of variables to find an expression for v in terms of t, where t is measured in seconds. (b) Terminal velocity is defined as lim v(t). Find the terminal t→∞ velocity of the skydiver to the nearest foot per second. (c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?

  Ans: v = −34e−2t − 16, −16, 1.070

2.3.2

Slope Fields

At each point, plot a short line segment with a slope determined by the differential equation, using the coordinates of the point for x and for y. Use the line segments to determine the flow, or current of the solution curve. Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 3: Separable Differential Equations

47

Figure 2.1: 1998 Calculus BC

Example 2.3.5 (BC98) Consider the differential equation given by xy dy = . On the axes provided in Figure 2.1, sketch a slope field dx 2 for the given differential equation at the nine points indicated. dy Example 2.3.6 (AB05) Consider the differential equation = dx 2x − . y (a) On the axes provided in Figure 2.2, sketch a slope field for the given differential equation at the twelve points indicated. (b) Sketch the solution that passes through the point (1, −1), i.e., √ sketch y = − 3 − 2x2 .

Mr. Budd, compiled September 29, 2010


48

HL Unit 2 (Elite Antidifferentiation)

Figure 2.2: 2005 Calculus AB

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 3: Separable Differential Equations

49

Problems 2.C-1 (adapted from AB93) If x = 3, y =

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

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

2.C-3 (AB05B) Consider the differential equation given by

Figure 2.3: 2005B AB Exam

(a) On the axes provided (Figure 2.3), sketch a slope field for the given differential equation at the twelve points indicated. (b) Write an equation for the line tangent to the graph of f at x = −1. (c) Find the solution y = f (x) to the given differential equation with the initial condition f (−1) = 2. h i Ans: y − 2 = 2 (x + 1);y = x24+1 2.C-4 (adapted from [3]) The point (1, 4) lies on the graph of an equation y = dy √ f (x) for which = 6x2 y where x ≥ 0 and y ≥ 0. When x = 0 the dx value of y is [Ans: 1] 2.C-5 (adapted from [3]) If the graph of y = f (x) is defined for all x ≥ 0, dy √ contains the point (0, 4), has = 3 xy and f (x) > 0 for all x, then dx h 2 i f (x) = Ans: x3/2 + 2 Mr. Budd, compiled September 29, 2010


50

HL Unit 2 (Elite Antidifferentiation)

2.C-6 (AB98) Let f be a function with f (1) = 4 such that for all points (x, y) 3x2 + 1 on the graph of f the slope is given by . 2y   (a) Find the slope of the graph of f at the point where x = 1. Ans: 21 (b) Write an equation for the line tangent to the graph of f at x = 1 and use it to approximate f (1.2). Ans: y − 4 = 21 (x − 1), 4.1 dy 3x2 + 1 (c) Find f (x) by solving the separable differential equation = dx 2y  √  with the initial condition f (1) = 4. Ans: x3 + x + 14 (d) Use your solution to find f (1.2).

[Ans: 4.114]

dy 2.C-7 (adapted from [3]) If = 2xy and if y = 3 when x = 0, then y = dx h i 2 Ans: 3ex ` dy + 2xy = 0 that contains 2.C-8 (adapted from [3]) A solution of the equation dx h i  2 the point 0, e2 is Ans: y = e2−x dy 2.C-9 (BC05) Consider the differential equation given by = 2x − y. On the dx axes provided in Figure 2.4, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point (0, 1). Figure 2.4: 2005 Calculus BC

2.C-10 (BC98) Shown in Figure 2.5 is a slope field for which of the following differential equations? Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 3: Separable Differential Equations

51

Figure 2.5: BC98

(A) (B) (C) (D) (E)

dy dx dy dx dy dx dy dx dy dx

=1+x = x2 =x+y =

x y

= ln y   dy Ans: =x+y dx

2.C-11 (BC00) Consider the differential equation given by

dy 2 = x (y − 1) . dx

(a) On the axes provided (Figure 2.6), sketch a slope field for the given differential equation at the eleven points indicated. (b) Use the slope field for the given differential equation to explain why a solution could not have the graph shown in Figure 2.7 (c) Find the particular solution y = f (x) to the given h differential equa-i tion with the initial condition f (0) = −1. Ans: y = 1 − x22+1 (d) Find the range of the solution found in part 11c. [Ans: −1 ≤ y < 1]

Mr. Budd, compiled September 29, 2010


52

HL Unit 2 (Elite Antidifferentiation)

Figure 2.6: 2000 BC Exam

Figure 2.7: 2000 BC Exam

Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 4: Accumulation Functions

2.4

53

Accumulation Functions

Advanced Placement Parametric and vector functions. The analysis of planar curves includes those given in parametric form and vector form. Fundamental Theorem of Calculus. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Textbook §4.4 The Fundamental Theorem of Calculus: “The Second Fundamental Theorem of Calculus” [15]

2.4.1

Creating Quick, Cheap Antiderivatives

Rb Remember that a F 0 (x) dx gives you F (b) − F (a), which is the change in the antiderivative. If we want to know where the antiderivative ends up, we add the change (given by the definite integral), to where we started.

Z

b

F 0 (x) dx

= F (b) − F (a)

a

Z F (b)

= F (a) +

b

F 0 (x) dx

a

In an initial value problem, you are generally given the derivative of some function you need, as well as an initial value (or a value at some time other than 0). To find the value you need, add the starting value to the change in values, as determined from the definite integral. e.g., Z b x(b) = x(0) + v(t) dt 0

You can use fnInt on the calculator to approximate the antiderivative at any point. What is nice is that we can do this for any function, even ones we cannot analytically antidifferentiate.

Example 2.4.1 (BC97) If f is the antiderivative of

x2 such 1 + x5

that f (1) = 0, then f (4) is what? Mr. Budd, compiled September 29, 2010


54

HL Unit 2 (Elite Antidifferentiation) [Ans: 0.376]

Example 2.4.2 Let A(x) = A(0)

R√

t3 + 1 dt. If A(2) = 3, then find

[Ans: ] 2

Example 2.4.3 Let P 0 (x) = √12π e−x /2 . If P (0) = 0.5, then find P (1), P (−1), P (2), P (8). Check your answer using normalcdf(-8,X) where x = 1, −1, 2, 8. What are you finding?

2.4.2

Parametrics Revisited Figure 2.8: from 2003 BC Exam

Example 2.4.4 (BC03) A particle starts at point A on the positive x-axis at time t = 0 and travels along the curve from A to B to C to D, as shown in Figure 2.8. The coordinates of the particles position dx (x(t), y(t)) are differentiable functions of t, where x0 (t) = = dt √     πt π t+1 dy sin and y 0 (t) = is not explicitly given. −9 cos 6 2 dt At time t = 9, the particle reaches its final position at point D on the positive x-axis. dx dy (a) At point C, is positive? At point C, is positive? Give a dt dt reason for each answer. (b) The slope of the curve is undefined at point B. At what time t is the particle at point B? Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 4: Accumulation Functions

55

(c) The line tangent to the curve at the point (x(8), y(8)) has equa5 tion y = x − 2. Find the velocity vector and the speed of the 9 particle at this point. (d) How far apart are points A and D, the initial and final positions, respectively, of the particle? [Ans: no, no; t = 3; h−4.5, −2.5i, 5.147; 39.255] Example 2.4.5 An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t ≥ 0 with   dx dy 4 = 12t − 3t2 and = ln 1 + (t − 4) . dt dt At time t = 0, the object is at position (−13, 5). At time t = 2, the object is at point P . Find the coordinates of point P . Trying doing the x-coordinate without a calculator. [Ans: (3, 13.671)]

Problems 2.D-1 (BC ’97) If f is the antiderivative of is what?

x2 such that f (1) = 0, then f (4) 1 + x5 [Ans: 0.376] √

2.D-2 (BC Acorn ’04-05) If the function f is defined by f (x) = x3 + 2 and g is an antiderivative of f such that g(3) = 5, then what is g(1)? [Ans: −1.585] 2.D-3 (adapted from BC ’03) A particle moves along the x–axis so that at any  time t ≥ 0, its velocity is given by v(t) = cos 2 − t2 . The position of the particle is 3 at time t = 0. What is the position of the particle at the second time when its velocity is equal to zero? [Ans: 3.563] 2.D-4 (adapted from [3]) The rate at which ice is melting in a pond is given by √ dV = 1 + 2t , where V is the volume of ice in cubic feet, and t is the dt time in minutes.   What amount of ice has melted in the first 4 minutes? Ans: 9.645 ft3 2.D-5 (adapted from [3]) Oil is leaking from a tanker at the rate of R(t) = 500e−0.2t gallons per hour, where t is measured in hours. The amount of oil that has leaked out, starting at the end of the second hour, until the end of the tenth hour is Mr. Budd, compiled September 29, 2010


56

HL Unit 2 (Elite Antidifferentiation)

2.D-6 [3] A particle moves along the x-axis with velocity at time t given by v(t) = t + 2 sin t. If the particle is at the origin when t = 0, its position at the time when v = 5 is x = [Ans: 17.277] 2

2.D-7 (adapted from AB ’98) Let F (x) be an antiderivative of 10, then what is F (9)? 2.D-8 (adapted from AB Acorn ’04-05) If f 0 (x) = cos



πex 2

(ln x) . If F (2) = x [Ans: 13.425]

 and f (0) = 2, then

f (1) =

[Ans: 1.351]

1 1 2 2.D-9 Let C(x) be the antiderivative of n(x) = √ e−x /2 such that C(0) = . 2 2π (a) Find C(2). (b) Find limx→∞ C(x) [Ans: 0.97725; 1] 2.D-10 (adapted from BC ’03) A particle starts at point A on the positive x-axis at time t = 0 and travels along the x-axis. The particle’s √ is a dif position  x(t)  πt π t+1 dx 0 = −9 cos sin . ferentiable function of t, where x (t) = dt 6 2 At time t = 9, the particle reaches its final destination at point D on the positive x-axis. How far apart are points A and D, the initial and final positions, respectively, of the particle? [Ans: 39.255 apart] 2.D-11 (adapted slightly from AB ’07) The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 ≤ t ≤ 7, where t is measured in hours. In this model, rates are given as follows: √ (a) The rate at which water enters the tank is f (t) = 100t2 sin t gallons per hour for all seven hours. (b) The rate at which water leaves the tank is: 250 gallons per hour for the first three hours, and 2000 gallons per hour for the next four hours. At t = 0, the amount of water in the tank is 5000 gallons. (a) On the same graph, sketch the rate at which water enters the tank, together with the rate at which water leaves the tank. At what times are the two rates the same? Z 7 (b) Using correct units, explain the meaning of f (t) dt in terms of water in the tank.

0

(c) How many gallons of water are there in the tank at t = 0? t = 3? t = 7? Mr. Budd, compiled September 29, 2010


HL Unit 2, Day 4: Accumulation Functions

57 [Ans: ; ; 5000, 5126.591, 4513.807]

2.D-12 (adapted from AB ’07B) A particle moves along the x–axis so that its velocity v at time t ≥ 0 is given by v(t) = sin t2 . The position of the particle at time t is x(t) and its position at time t = 0 is x(0) = 5. (a) Find the position of the particle at t = 0 and at t = 3. [Ans: 5, 5.774] (b) Find the first time at which the particle changes direction. Does it change from right to left, or left to right? Find the position of the particle at that time. [Ans: 1.772, 5.895] 2.D-13 Suppose x(b) is a function in terms of b such that Z b x(b) = x(0) + v(t) dt 0

What would be

dx , i.e., x0 (b), i.e., the derivative of x? db

2.D-14 (BC04) An object moving along a curve in the xy-plane has position  dx dy (x(t), y(t)) at time t ≥ 0 with = 3 + cos t2 . The derivative dt dt is not explicitly given. At time t = 2, the object is at position (1, 8). Find the x-coordinate of the position of the object at time t = 4. [Ans: 7.133] 2.D-15 (BC04B) A particle moving along a curve in the plane has position (x(t), y(t)) at time t, where dx p 4 dy = t + 9 and = 2et + 5e−t . dt dt for all real values of t. A time t = 0, the particle is at point (4, 1). Find the x-coordinate of the position of the particle at time t = 3. [Ans: 17.931] 2.D-16 (BC01) An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t with   dx dy = cos t3 and = 3 sin t2 dt dt for 0 ≤ t ≤ 3. At time t = 2, the object is at position (4, 5). Find the position of the object at time t = 3. [Ans: (3.954, 4.906)] 2.D-17 (BC00) A moving particle has position (x(t), y(t)) at time t. The position of the particle at time  t = 1 is (2,6) and the velocity vector at any time 1 1 t > 0 is given by 1 − 2 , 2 + 2 . Find the position of the particle at t t time t = 3. Try doing this with a quick, cheap antiderivative,  but without  32 using your calculator. Ans: 10 3 , 3

Mr. Budd, compiled September 29, 2010


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HL Unit 2 (Elite Antidifferentiation)

Mr. Budd, compiled September 29, 2010


Unit 3

Elite Integration 1. Fundamental Theorem of Calculus, part Deux 2. Mean Value Theorem for Integrals, Average Value 3. Area and Volume 4. Volume: Flat-Based Solids 5. Arc Length

Advanced Placement Applications of integrals. To provide a common foundation, specific applications should include the distance traveled by a particle on a line, and the length of a curve (including a curve given in parametric form). Techniques of antidifferentiation â&#x20AC;˘ Improper integrals (as limits of definite integrals).

59


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HL Unit 3 (Elite Integration)

Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 1: Accumulation Functions

3.1

61

Accumulation Functions

Advanced Placement Fundamental Theorem of Calculus. • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Textbook §4.5 The Fundamental Theorem of Calculus: [15]

3.1.1

Accumulation Functions

Example 3.1.1 (AB98)

Rx 0

sin t dt =

[Ans: 1 − cos x]

Example 3.1.2 (AB97) Let f (x) =

Rx a

h(t) dt, where h has the

Figure 3.1:

graph shown in Figure 3.1. Sketch the graph of f .

[Ans: Figure 3.2]

3.1.2

Fundamental Theorem of Calculus, part II d dx

Z

x

f (t) dt = f (x) a

Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Figure 3.2:

As soon as you see an accumulation function, differentiate it. Stop immediately, do not continue reading the problem until you have taken the derivative of the accumulation function. It is very easy to do, and the chances are extremely high that you will need the derivative. You can start reading the problem again, realizing that you know the derivative.

Example 3.1.3 (AB98) If F (x) = F 0 (2)?

Rx√ 0

t3 + 1 dt, then what is

[Ans: 3]

Example 3.1.4 [20] The error function Z x 2 2 erf(x) = √ e−t dt π 0 is used in probability, statistics, and engineering. Show that the 2 function y = ex erf(x) satisfies the differential equation y 0 = 2xy + 2 √ . π Example 3.1.5 (BC93) If F and f are differentiable functions Rx such that F (x) = 0 f (t) dt, and if F (a) = −2 and F (b) = −2 where a < b, which of the following must be true? (A) f (x) = 0 for some x such that a < x < b. (B) f (x) > 0 for all x such that a < x < b. (C) f (x) < 0 for all x such that a < x < b. (D) F (x) ≤ 0 for all x such that a < x < b. Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 1: Accumulation Functions

63

(E) F (x) = 0 for some x such that a < x < b.

[Ans: A] Using the Chain Rule, d dx

Z

v(x)

f (t) dt = f (v(x)) v 0 (x)

a

R x2 Example 3.1.6 (BC97) Let f√(x) = 0 sin t dt. At how many points in the closed interval [0, π] does the instantaneous rate of change of f equal the average rate of change of f on that interval?

[Ans: Two]

Example 3.1.7 [19] Let Z F (x) = 2x + 0

x2

sin 2t dt 1 + t2

Determine (a) F (0) (b) F 0 (0) (c) F 00 (0)

[Ans: 0;2;2] To carry extend part II even further, d dx

Z

v(x)

f (t) dt = f (v(x)) v 0 (x) − f (u(x)) u0 (x)

u(x)

Example 3.1.8 Let F (x) = answer by simplifying F (x).

R 3x 1 dt. Find F 0 (x) and explain your x/2 t

[Ans: 0; ln 6] Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Example 3.1.9 [20] Find the derivative of g(x) =

R 3x u2 − 1 du. 2x u2 + 1

  2(4x2 −1) 3(9x2 −1) Ans: 9x2 +1 − 4x2 +1

3.1.3

Curve Sketching with Accumulation Functions

Example 3.1.10 (BC97) The graph of f is shown in Figure 3.3. If Figure 3.3:

g(x) =

Rx a

f (t) dt, for what value of x does g(x) have a maximum?

[Ans: c]

Example 3.1.11 (BC97) Refer to the graph in Figure 3.4. At Figure 3.4: The function f is defined on the closed interval [0, 8]. The graph of its derivative f 0 is shown.

what value does the absolute minimum of f occur? The absolute maximum? Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 1: Accumulation Functions

65 [Ans: 0]

Example R x 3.1.12 (AB Acorn ’02) If the function g is defined by g(x) = 0 sin(t2 ) dt on the closed interval −1 ≤ x ≤ 3, then g has a local minimum at x = [Ans: 2.507] Example 3.1.13 (AB 2002) The graph of function f shown in Figure 3.5: From AP Calculus AB 2002 Exam

Figure 3.5. Let g be the function given by g(x) = (a) (b) (c) (d)

Rx

f (t) dt.   Find g(−1), g 0 (−1), and g 00 (−1). Ans: − 32 ; 0; 3 For what values of x in the open interval (−2, 2) is g increasing? Explain your reasoning. [Ans: −1 < x < 1] For what values of x in the open interval (−2, 2) is the graph of g concave down? Explain your reasoning. [Ans: 0 < x < 2] Sketch the graph of g on the closed interval [−2, 2]. 0

Problems 3.A-1 (AB ’97) Let f be the function given by f (x) =

x − 3.

(a) Sketch the graph of f and shade the region R enclosed by the graph of f , the x-axis, and the vertical line x = 6. (b) Find the area of the region R. (c) Rather than using the line x = 6, consider the line x = w, where w can be any number greater than 3. Let A(w) be the area of the region enclosed by the graph of f , the x-axis, and the vertical line x = w. Write an integral expression for A(w). Mr. Budd, compiled September 29, 2010


66

HL Unit 3 (Elite Integration) (d) Find the rate of change of A with respect to w when w = 6. √ Rw√ √   Ans: ; 2 3; 3 x − 3 dx; 3

3.A-2 (AB ’93) What is

d Rx cos(2πu) du? dx 0

3.A-3 (adapted from [3]) Suppose F (x) =

[Ans: cos(2πx)] R x2 0

F 0 (−1) =

1 dt for all real x, then 2 + t3   Ans: − 32

5 3.A-4 (adapted from [3]) Consider the function F defined so that F (x) + = 2   Rx πt 0 cos dt. The value of F (2) + F (2) is [Ans: −3] 2 3 3.A-5 (adapted from [3]) If the function G is defined for all real numbers by  R 3x √ G(x) = 0 cos t2 dt, then G0 ( π) = [Ans: −3] R 2 sin x √ 3.A-6 (adapted from [3]) Suppose F (x) = 0 9 + t3 dt for all real x, then 0 F (π) = [Ans: −6]  π Rx 3.A-7 (adapted from [3]) If for all x > 0, G(x) = 1 cos ln t dt, then the 2   π 00 value of G (e) is Ans: − 2e Rx 3.A-8 [3] Which of the following are true about the function F (x) = 1 ln (2t − 1) dt? I. F (1) = 0 II. F 0 (1) = 0 III. F 00 (1) = 1 [Ans: I and II only] 3.A-9 (AB ’94) Let F (x) =

Rx 0

sin t

 2

dt for 0 ≤ x ≤ 3.

(a) Use the trapezoidal rule with four equal subdivisions of the closed interval [0, 1] to approximate F (1). [Note: you may use a calculator, but don’t use a program.] (b) On what intervals is F increasing? (c) If the average rate of change of F on the closed interval [1, 3] is k,  R3 find 1 sin t2 dt in terms of k. √    √ 2π, 3 ; 2k Ans: 0.316; (0, π), 3.A-10 [20] The sine integral function Z Si (x) = 0

x

sin t dt t

is important in electrical engineering. Using your calculator to help you: Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 1: Accumulation Functions

67

(a) Draw the graph of Si . (b) At what values of x does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation: Z x 0

sin t dt = 1 t

 Rx 2 3.A-11 (BC Acorn ’02) Let g be the function given by g(x) = 1 100 t2 − 3t + 2 e−t dt. Which of the following statements about g must be true? I. g is increasing on (1, 2). II. g is increasing on (2, 3). III. g(3) > 0. [Ans: II only] 3.A-12 (AB 2002B) The graph of a differentiable function f on the closed interval Figure 3.6: From AP Calculus AB 2002B Exam

[−3, 15] is shown in Figure 3.6. R x The graph of f has a horizontal tangent line at x = 6. Let g(x) = 5 + 6 f (t) dt for −3 ≤ x ≤ 15. (a) Find g(6), g 0 (6), and g 00 (6).

[Ans: 5; 3; 6]

(b) On what intervals is g decreasing? Justify your answer. [Ans: [−3, 0] and [12, 15]] (c) On what intervals is the graph of g concave down? Justify your answer. [Ans: (6, 15)] R 15 (d) Find a trapezoidal approximation of −3 f (t) dt using six subintervals of length ∆t = 3. [Ans: 12] 3.A-13 (BC Acorn R x’02) The graph of f in Figure 3.7 consists of four semicircles. If g(x) = 0 f (t) dt, where is g(x) nonnegative? [Ans: [−3, 3]] Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Figure 3.7: Graph of f

Figure 3.8: Graph of g

3.A-14 (BC98) Let g(x) =

Rx a

f (t) dt, where a ≤ x ≤ b. Figure 3.8 shows the  

      graph of g on [a, b]. Sketch the graph of f on [a, b].  Ans:     

Mr. Budd, compiled September 29, 2010

            


HL Unit 3, Day 2: Improper Integrals

3.2

69

Improper Integrals

International Baccalaureate R∞ 10.1 Improper integrals of the type a f (x) dx. f (x) 10.5 The evaluation of limits of the form lim using l’Hopital’s Rule and/or x→a g(x) the Taylor series. Cases where the derivatives of f (x) and g(x) vanish for x = a. Advanced Placement Applications of derivatives. • L’Hˆ opital’s Rule , including its use in determining limits and convergence of improper integrals.

Techniques of antidifferentiation. • Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only). • Improper integrals (as limits of definite integrals). Textbook §6.6 Improper Integrals [15] Resources §4.5 Indeterminate Forms and L’Hˆ opital’s Rule in Stewart [20]. Exploration 9-10: “Introduction to Improper Integrals” in Instructor’s Resource Book [11]. §9-10 Improper Integrals and §6-8 Limits of Indeterminate Forms: l’Hospital’s Rule in Foerster [12]. §10.1 When Is an Integral Improper? in Ostebee & Zorn [17].

3.2.1

Improper Integrals

Integrals are improper when they have an infinite dimension: • the width R ∞ is infinite because the limits of the integral are either ∞ or ∞, e.g. 1 x1 dx, or • the height is infinite because somewhere inside the interval of integration the integrand (the function that you’re integrating) is infinite, usually R1 because of division by 0, e.g. −1 x1 dx. Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

To handle improper integrals, we change the improper integral into the limit of a definite and quite proper integral. For example, Z ∞ 1 dx x 1 becomes a most excellently proper b

Z lim

b→∞

1

1 dx x

Changing the improper integral into the limit of a definite integral is the most important thing for you to do when dealing with improper integrals. Not doing this is the easiest way to lose points on the Free Response section of the AP Calculus BC exam. If the limit of the definite integral exists, we say that the improper integral converges to that limit. If the limit of the definite integral does not exist, e.g. it is infinite, then we say that the improper integral diverges. Be aware of the type of curve that is a candidate for convergence. If a function goes to infinity as x goes to infinity, it cannot converge, so we don’t need to bother with testing for convergence. R∞

Example 3.2.1 Graph the integrand for or not the integral might converge. [12]

0

x0.2 dx and tell whether

Infinite Width ∞

Z

b

Z f (x) dx = lim

b→∞

a

and Z

b

f (x) dx a

Z f (x) dx = lim

−∞

Example 3.2.2 Make sense of

a→−∞

b

f (x) dx a

R ∞ dx . [17] 1 x2

Example 3.2.3 (BC ’06) An object moving along a curve in the xy–plane is at position (x(t), y(t)) at time t, where  dx dy 4t = sin−1 1 − 2e−t and = dt dt 1 + t3 for t ≥ 0. At time t = 2, the object is at the point (6, −3). Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 2: Improper Integrals

71

(a) Let m(t) denote the slope of the line tangent to the curve at the point (x(t), y(t)). Write an expression for m(t) in terms of t and use it to evaluate lim m(t). [Ans: 0] t→∞

(b) Explain the meaning of your answer in terms of the asymptotic behavior of the curve. (c) The graph of the curve has a horizontal asymptote at y = c. Write an expression involving an improper integral that represents this value c. i h R ∞ 4t Ans: −3 + 2 1+t 3 dt

Infinite Height Z a

b

Z f (x) dx = lim− k→c

k

a

Z f (x) dx+ lim+ k→c

Example 3.2.4 Evaluate

b

f (x) dx, f is discontinuous at x = c in [a, b] k

R 3 dx if possible. [20] 0 x−1

Be careful: What answer would you get if you do not realize that there is a vertical asymptote at x = 1? Now, if the discontinuity is a vertical asymptote at one of the bounds of our definite integral, I don’t need to split it up.

Example 3.2.5 Discuss

R 1 dx . [17] 0 x2

Split Infinitives Example 3.2.6 How about

R ∞ dx ? [17] 0 x2

The key is that if you split an integral into two parts, I1 and I2 , both parts must converge for the full integral to converge. This becomes important for another classic example:

Example 3.2.7

Ra 1 dx −a x Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Geometrically, we can make arguments that this integral is 0. As convincing as these arguments are, they are ultimately not correct. We split the integral into two parts, both of which are divergent; hence the overall integral diverges. How about:

Example 3.2.8

3.2.2

R∞ −∞

1 dx 1 + x2

More Improper Integrals

OK, back to improper integrals. R∞ Example 3.2.9 Tell whether 0 x2 e−x dx converges or not, and, if so, to what limit it converges.

Now You In this problem you will explore Z ∞ f (k) = xk e−x dx, 0

where k is a constant with respect to the integration. Use your calculator to graph the integrand for k = 1, k = 2, and k = 3. [12] a. Find f (1), f (2), and f (3) by evaluating the improper integral. Along the way you will have to show, for instance, that lim b3 e−b = 0.

b→∞

b. From the pattern you see in the answers to (a), make a conjecture about what f (4), f (5), and f (6) are equal to. c. Integrate by parts once and thus show that f (k) = k · f (k − 1). Use the answer to confirm your conjecture in (b). d. The result of the above work forms a basis for the definition of the factorial function. Explain why this definition is consistent with the definition k! = (k) (k − 1) (k − 2) . . . (2) (1) . Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 2: Improper Integrals

73

e. Confirm that the integral for 3! approaches 6 by integrating numerically from t = 0 to Rt = b for some fairly large value of b. Analytically find the antiderivative x3 e−x dx. Use this antiderivative to find a value of b for

R

b 3 −x which 0 x e dx − 6 ≤ 10−6 . f. The improper integral can be used to define factorials for noninteger values of x. Write an integral equal to 0.5!. Evaluate it numerically, using the value of b from part (e). How can you tell from the graphs that your answer will be closer than 10−6 to the correct answer. How does your answer compare with the value in the National Bureau of Standards Handbook of Mathematical Functions, namely, 0.5! = 0.8862269255? g. Quick! Without further integration, calculate 1.5!, 2.5!, and 3.5!. (Remember part c.) h. Show that 0! = 1, as you probably learned in algebra. i. Show that (−1)!, (−2)!, (−3) , . . . are infinite but (−0.5)!, (−1.5!), and (−2.5) are finite. j. Show that the value of 0.5! in (f) can be expressed rather simply in terms of π.

Problems big giant blue-green Calculus book p. 558: # 23, 29, 53, 59, 61, 63, 65 3.B-1 (BC97) Z 1

x 2

(1 + x2 )

dx

is

 Ans:

1 4



3.B-2 p-Integral Problem: An integral of the form Z ∞ 1 Ip = dx, xp 1 where p stands for a constant, is called a p-integral. For some values of the exponent p, the integral converges and for others it doesn’t. Use your R∞ 1 dx, which converges, and for calculator to examine graphs for 1 x1.01 R∞ 1 dx, which diverges. The two graphs look practically identical, 1 x0.99 but only one of the integrals converges. In this problem your objective is to find the value of p for which the p-integral converges and those for which it diverges. [12] Mr. Budd, compiled September 29, 2010


74

HL Unit 3 (Elite Integration) (a) Show that Ip converges if p = 1.001, but not if p = 0.999. (b) Does Ip converge if p = 1? Justify your answer. (c) Complete the statement “Ip converges if p ? .”

?

, and diverges if p

3.B-3 Volume of an Unbounded Solid Problem: Graph the region under y = from x = 1 to x = b. [12]

1 x

(a) Does the region’s area approach a finite limit as b approaches infinity? Explain (b) The region is rotated about the x-axis to form a solid. Does the volume of the solid approach a finite limit as b → ∞? If so, what is the limit? If not, explain why not. (c) True or false: “If a region has infinite area, then the solid formed by rotating that region about an axis has infinite volume.” 3.B-4 A function f defined on (−∞, ∞) is a probability density function if (i) f (x) ≥ 0 for all x in (−∞, ∞), and R∞ (ii) −∞ f (x) dx = 1. Find the number k (at least approximately) so that each of the following is a probability density function. Then graph f . [16] (a) f (x) =

k 1 + x2 2

(b) f (x) = ke−x

/2

 Ans:

1 π;

0.3989422804



Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums

3.3

75

Infinite Sums of Infinitesimal Slices

International Baccalaureate 7.5 Definite integrals. Area between a curve and the x-axis or y-axis in a given interval, areas between Rb Rb curves. a y dx and a x dy. Rb Volumes of revolution. Revolution about the x-axis or the y-axis. V = a πy 2 dx, Rb V = a πx2 dy. 7.6 Kinematic problems involving displacement, s, velocity, v, and acceleration, a. Area under velocity–time graph represents distance. Advanced Placement Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals. Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables (including change of limits for definite integrals) Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Textbook §5.1 Area between Curves and §5.2 Volume: Slicing, Disks, and Washers [15] Resources §8.2 Finding Volumes by Integration in Ostebee and Zorn [17]. §6.2 Volumes in Stewart [20]. §7.4 Volumes in Finney [10]. Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

3.3.1

Definite Integral

The definite integral can be seen as several things: 1. An infinite sum. Area is an infinite sum of infinitesimally thin rectangles. 2. A product. Displacement is velocity times time. How is a product really a sum? 3. An accumulated change. If oil is leaking out of a tank at certain rate Rb R(t), then a R(t) dt represents how much oil has leaked out from t = a to t = b. This is the accumulation of all the oil that has leaked out at a rate R(t). Change in velocity is the accumulation of accleration over a certain time period.

3.3.2

Elite Ninja Volume

1 . Let R be the x unbounded region between the graph of y = f (x) and the x-axis, to the right of the line x = 1.

Example 3.3.1 Consider the function f (x) =

(a) Determine whether the area of this region is finite. (b) Consider the solid S generated when R is rotated about the x-axis. Determine whether the volume of this solid is finite.

3.3.3

Area

Example 3.3.2 (adapted from AB ’00) Let R be the region in the 2 first quadrant enclosed by the graphs of y = e−x , y = 1 − cos x, and the y-axis. (a) Find the area of R. (b) If the line x = k divides the region R into two regions of equal area, what is the value of k?

[Ans: 0.591; 0.310]

Example 3.3.3 (adapted from Ostebee & Zorn [17]) Find the area of the region R bounded by the curves x = 0, y = 2, and y = ex . Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums

77

(a) Find the area of R (two ways). (b) If the line x = h divides the region R into two regions of equal area, what is the value of h? (c) If the line y = k divides the region R into two regions of equal area, what is the value of k? [Ans: 0.386; 0.219 ; 1.683] When you get to use the calculator, feel free to use abs Example 3.3.4 (adapted from AB ’83) Do the following problem without a calculator, then with a calculator. Find the area bounded by the curve f (x) = 3x2 − 12x + 9 and the x-axis, between the lines x = 0 and x = 2. [Ans: 6]

3.3.4

Mare Orea

Slicing dx Example 3.3.5 Find the area of the region bounded by the graphs 1 of y = ex/2 , y = 2 , x = 2, and x = 3. Try this with, and without x a calculator. 

 Ans: 2 e3/2 − e −

1 6

= 3.360



Example 3.3.6 Find the area of the region bounded by the graphs 4 of y = x, y = , x = 1, and x = 4. Try this with, and without a x calculator.  Ans:

9 2

+ 4 ln 4 − 8 ln 2 =

9 2

= 4.500



π Example 3.3.7 (adapted from AB ’98) If 0 ≤ k ≤ and the area 2 π under the curve y = cos x from x = k to x = is 0.2, then k = 2 [Ans: 0.927] Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Slicing dy Example 3.3.8 Find the area of the region bounded by the graphs 1 of y = 2 , y = x, and y = 2. Try this two ways. x  Ans:

7 2

√  −2 2

Total Distance Remember that Distance : Displacement :: Area : Definite Integral Example 3.3.9 (AB ’02B) A particle moves along the x-axis so that its velocity v at any time t, for 0 ≤ t ≤ 16, is given by v(t) = e2 sin t − 1. At time t = 0, the particle is at the origin. (a) Sketch the graph of v(t) for 0 ≤ t ≤ 16. (b) During what intervals of time is the particle moving to the left? Give a reason for your answer. (c) Find the total distance traveled by the particle from t = 0 to t = 4. Do this using absolute value, and without using absolute value. (d) Is there any time t, 0 < t ≤ 16, at which the particle returns to the origin? Justify your answer. [Ans: ; (π, 2π), (3π, 4π), (5π, 16]; 10.542, no]

3.3.5

Total Distance

Distance : Displacement :: Area : Definite Integral There is a difference between the total distance traveled and the displacement. When you go backward, distance is counted positively, but displacement is counted negatively. How is that similar to the relationship between area and the definite integral? Speed is the magnitude (absolute value of velocity), and total distance traveled is the accumulation (i.e., definite integral) of speed. Displacement is the accumulation of velocity. Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums D=

Rb a

79

|v(t)| dt

Just like area, total distance traveled must be positive. In a graph of velocity, the definite integral yields the displacement, i.e., change in position. The area yields the total distance traveled. Example 3.3.10 (adapted from AB 1997) A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by the function whose graph is shown in Figure 3.9 Figure 3.9: Vertical velocity of a bug

(a) At value of t does the bug change direction? (b) What is the total distance the bug traveled from t = 0 to t = 8? (c) What is the net displacement of the bug between t = 0 and t = 8? (d) What is the total distance traveled downward by the bug? upward? (e) What is the bug’s velocity at t = 5? The acceleration at t = 5? (f) What other questions could we ask about the bug? Example 3.3.11 (AB ’03) A particle moves along the x-axis so that its velocity at time t is given by  2 t v(t) = − (t + 1) sin 2 At time t = 0, the particle is at position x = 1. (a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not? Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration) (b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time t = 0 until time t = 3. (d) During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.



Ans: 1.588, yes;

2π; 4.334; 2.265



Example 3.3.12 (adapted slightly from AB ’83) A particle moves along the x-axis so that at time t its position is given by x(t) = t3 − 6t2 + 9t + 11. (a) What is the velocity of the particle at time t? (b) During what time intervals is the particle moving to the left? (c) What is the total distance traveled by the particle from t = 0 to t = 2? Do this two ways. R2 (d) What does 0 v(t) dt represent?   Ans: 3t2 − 12t + 9; 1 < t < 3; 6; x(2) − x(0)

3.3.6

Volumes of Rotation: Sweet, Sweet Loaves of Calculus

In general, b

Z V =

A(x) dx a

or Z V =

d

A(y) dy c

where A(x) or (A(y)) represents the cross-sectional area of the solid at a particular value of x (or y) 2

For volumes of rotation where cross sections are discs, then A(x) = π [r(x)] or 2 A(y) = π [r(y)] . Mr. Budd, compiled September 29, 2010


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Example 3.3.13 (adapted from AB 1997) Let R be the √ region bounded by the y-axis, the line y = 2, and the curve y = x. (a) Find the area of region R. (b) Find the volume of the solid generated when region R is rotated about the y-axis.

 Ans:

8 32π 3; 5



Things to keep in mind for volume of rotation problems: • If the axis of rotation is the x-axis, or parallel to the x-axis, slice dx. If the axis of rotation is the y-axis, or parallel to the y-axis, slice dy. Rb • V = a πr2 d • If slicing dy, the radius will be a high x minus a low x. If slicing dx, the radius will be a high y minus a low y. If the axis of rotation is either the x- or y-axis, one of these values will be zero. 1 Example 3.3.14 [4] A region in the plane is bounded by y = √ , x the x-axis, the line x = m, and the line x = 2m where m > 0. A solid is formed by revolving the region about the x-axis. The volume of this solid (A) is independent of m (B) increases as m increases (C) decreases as m decreases 1 (D) increases until m = , then decreases 2 (E) is none of the above

[Ans: A]

Example 3.3.15 Derive, from scratch, the formula for the volume of a sphere of radius r.

Example 3.3.16 Derive, from scratch, the formula for the volume of a cone with height h and base radius r. Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration) Example 3.3.17 [17] The following table gives the circumference (in inches) of a pole at several heights (in feet). Height Circumference

0 16

10 14

20 10

30 5

40 3

50 2

60 1

Assuming that cross sections of the pole taken parallel to the ground are circles, estimate the volume of the pole using: (a) T3 (b) M3 (c) T6 (d) S2·3 =

M3 + 2T3 , a weighted average of M3 and T3 . 3

Nonstandard axes of rotation Remember that each radius is a high y minus a low y (or high x minus low x).

Example 3.3.18 Let R be the region bounded by the graphs of y = ex/2 , y = 1, and x = ln 2. (a) Set up, but do not solve, a definite integral that could be used to find the area of R. (b) Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about the line y = 1. (c) Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about the line x = ln 2. (d) Preview: what changes if the axes of rotation are y = −1 or x = 0?

For washers, i h i h 2 2 2 2 A(x) = π (R(x)) − π (r(x)) = π (R(x)) − (r(x)) or

h i h i 2 2 2 2 A(y) = π (R(y)) − π (r(y)) = π (R(y)) − (r(y))

Remember that each radius is [high y - low y]. Mr. Budd, compiled September 29, 2010


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Example 3.3.19 Let R be the region bounded by the graphs of y = ex/2 , y = 1, and x = ln 2. Set up, but do not solve, a definite integral that could be used to find the volume of the region obtained by rotating R about (a) (b) (c) (d) (e)

the the the the the

y-axis; line y = −1; line x = −1; line x = 1; √ line y = 2

Example 3.3.20 ([4]) Let R be the region in the first quadrant bounded above by the graph of f (x) = 2 arctan x and below by the graph of y = x. What is the volume of the solid generated when R is rotated about the x-axis? [Ans: 7.151] Example 3.3.21 (AB ’02) Let f and g be the functions given by f (x) = ex and g(x) = ln x. Find the volume of the solid generated 1 when the region enclosed by the graphs of f and g between x = 2 and x = 1 is revolved about the line y = 4. [Ans: 23.609] Example 3.3.22 (AB ’97) Let f be the function given by f (x) = 3 cos x. As shown in Figure 3.10, the graph of f crosses the y-axis at point P and the x-axis at point Q. (a) Write an equation for the line passing through points P and Q. (b) Write an equation for the line tangent to the graph of f at point Q. Show the analysis that leads to your equation. (c) Find the x-coordinate of the point on the graph of f , between points P and Q, at which the line tangent to the graph of f is parallel to line P Q. (d) Let R be the region in the first quadrant bounded by the graph of f and the line segment P Q. Write an integral expression for the volume of the solid generated by revolving the region R about the x-axis. h

Ans: y − 3 = − π6 (x − 0); y − 0 = −3 (x − π/2); 0.690; π

R π/2 h 0

2

(3 cos x) − − π6 x + 3

Mr. Budd, compiled September 29, 2010

2 i

i dx


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HL Unit 3 (Elite Integration)

Figure 3.10: AB Exam 1997

Problems 3.C-1 (BC Acorn 2000) A particle moves along the x-axis so that at any time t ≥ 0 its velocity is given by v(t) = ln (t + 1) − 2t2 + 4t − 1. (a) What is the total distance traveled by the particle from t = 0 to t = 2? [Ans: 2.178] (b) What is the net displacement of the particle between t = 0 and t = 2? [Ans: 1.963] 3.C-2 (AB ’87) A particle moves along the x-axis so that its acceleration at any time t is given by a(t) = 6t − 18. At time t = 0 the velocity of the particle is v(0) = 24, and at time t = 1 its position is x(1) = 20. (a) Write an expression for the velocity v(t) of the particle at any time t. (b) For what values of t is the particle at rest? (c) Write an expression for the position of the particle at any time t. (d) Find the total distance traveled by the particle from t = 1 to t = 3.   Ans: 3t2 − 18t + 24; 2, 4; t3 − 9t2 + 24t + 4; 6 3.C-3 (AB ’93) A particle moves on the x-axis so that its position at any time t ≥ 0 is given by x(t) = 2te−t . Find the total distance traveled by the particle from t = 0 to t = 5. Try this using fnInt, then try this on your calculator, without calculating a definite integral. [Ans: 1.404] Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums

85

3.C-4 (AB 1997) A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t) = 3t2 − 2t − 1. The position x(t) is 5 for t = 2. (a) Write a polynomial expression for the position of the particle at any time t ≥ 0. (b) For what values of t, 0 ≤ t ≤ 3, is the particle’s instantaneous velocity the same as its average velocity on the closed interval [0, 3]? (c) Find the total distance traveled by the particle from t = 0 until time t = 3. Try it with a calculator, then without a calculator.   Ans: t3 − t2 − t + 3; t = 1.786; 17 3.C-5 (adapted from AB ’86) A particle moves along the x-axis so that at any 1 time t ≥ 1 its acceleration is given by a(t) = . At time t = 1, the velocity t of the particle is v(1) = −2 and its position is x(1) = 4. (a) Find the velocity at time t = 9. (b) What is the position at time t = 9? (c) What is the total distance traveled from t = 1 to t = 9? (d) How far did the particle travel backwards, starting from t = 1? [Ans: ln 9 − 2; −0.225; 4.553; 4.389] 3.C-6 (AB ’00) Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. Figure 3.11, which consists of two line segments, shows the Figure 3.11: from AP Calculus AB 2000 exam

velocity, in meters per second, of Runner A. The velocity, in meters per 24t second, of Runner B is given by the function v defined by v(t) = . 2t + 3 Find the total distance run by Runner A and the total distance run by Runner B over the time interval 0 ≤ t ≤ 10 seconds. Indicate units of measure. Then find Runner B’s distance without using a calculator. Recall what to do when antidifferentiating and the degree on top is the   Ans: 85 m; 83.336 m; 120 − ln 23 same or higher as on bottom. 3 Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

3.C-7 (AB ’93) Set up a definite integral to find the area shaded regioni h of the Rb in Figure 3.12. Ans: a (d − f (x)) dx Figure 3.12: AP Calculus AB (1993)

3.C-8 (AB ’96) [NO CALCULATOR] Let R be the region in the first quadrant 1 under the graph of y = √ for 4 ≤ x ≤ 9. x (a) Find the area of R. (b) If the line x = k divides the region R into two regions of equal area, what is the value of k?   Ans: 2; 25 4 3.C-9 (adapted from AB ’97) [NO CALCULATOR] The area of the region enclosed by the graph of y = x2 − 2x + 2 and the line y = 10 is [Ans: 36] 3.C-10 (adapted from AB ’98) [NO CALCULATOR] What is the area of the region between the graphs of y = x3 and y = −x from x = 0 to x = 3?   99 Ans: 4 3.C-11 (AB ’02B) Let R be the region bounded by the y–axis and the graphs of x3 y= and y = 4 − 2x. Find the area of R. [Ans: 3.215] 1 + x2 3.C-12 (adapted from BC Acorn) Find the area of the region R bounded by the curves x = 1, y = 1, and y = e3x . (a) Find the area of R, by slicing dx, then again by slicing dy. (b) If the vertical line x = h divides the region R into two regions of equal area, what is the value of h? (c) If the horizontal line y = k divides the region R into two regions of equal area, what is the value of k? [Ans: 5.362; 0.814 ; 5.065] Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums

87

x2 y2 3.C-13 Find the area enclosed by the ellipse + = 1. What is area in the 16 9 form kπ? What do you suppose is the area enclosed by any generic ellipse y2 x2 + 2 = 1? [Ans: 12π] 2 a b 3.C-14 (AB ’02) Let f and g be the functions given by f (x) = ex and g(x) = ln x. Find the area of the region enclosed by the graphs of f and g between 1 [Ans: 1.223] x = and x = 1. 2 √ 3.C-15 (AB ’03) Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. Find the area of R. [Ans: 0.443] 3.C-16 (adapted from AB ’03B) Let f be the function given by f (x) = 4x2 − x3 , and let ` be the line y = 18 − 3x, where ` is tangent to the graph of f . Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f , the line `, and the x-axis. (a) At what point is ` tangent to f (x)? (b) Find the area of R. (c) Find the area of S.  Ans: (3, 0);

64 3

;7.917



3.C-17 (AB ’05)

1 Let f and g be the functions given by f (x) = + sin (πx) and g(x) = 4−x . 4 Let R be the shaded region in the first quadrant enclosed by the y–axis Mr. Budd, compiled September 29, 2010


88

HL Unit 3 (Elite Integration) and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Find the area of S. [Ans: 0.0648; 0.410]

3.C-18 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. Find the area of R. [Ans: 0.429] 3.C-19 (adapted from AB ’03B) Let R be the region bounded by the graph of f (x) = 4x2 − x3 and the x-axis. Find the volume of the solid generated when R is revolved about the x-axis. [Ans: 490.208] 3.C-20 [20] A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log) are listed in the table. x (m) 0 1 2 3 4 5

A (m2 ) 0.68 0.65 0.64 0.61 0.58 0.59

x (m) 6 7 8 9 10

A (m2 ) 0.53 0.55 0.52 0.50 0.48

Estimate the volume of the log using:   Ans: 5.8 m3   (b) the Trapezoid Rule with n = 5. Ans: 5.7 m3   (c) the Trapezoid Rule with n = 10. Ans: 5.75 m3   M5 + 2T10 , a weighted average of M5 and T10 . Ans: 5.767 m3 (d) S2·5 = 3 (a) the Midpoint Rule with n = 5.

3.C-21 [10] We wish to estimate the volume of a flower vase using only a calculator, a string, and a ruler. We measure the height of the vase to be 6 inches. We then use the string and the ruler to find the circumference of the vase (in inches) at half-inch intervals. (We list them from starting at the top Circumferences 5.4 10.8 4.5 11.6 4.4 11.6 left, moving down) 5.1 10.8 6.3 9.0 7.8 6.3 9.4 (a) Sketch the vase. Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 3: Infinite Sums

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(b) Find the areas of the cross sections that correspond to the given circumferences. (c) Express the volume of the vase as an integral with respect to y over the interval [0, 6]. (d) Approximate the integral using the Trapezoidal Rule with n = 12. i h R6 2 1 3 [C(y)] dy; 34.7 in Ans: 2.3, 1.6, 1.5, . . .; 4π 0 3.C-22 (adapted from AB ’00) Let R be the region in the first quadrant enclosed 2 by the graphs of y = e−x , y = 1 − cos x, and the y-axis. (a) Find the volume of the solid generated when the region R is revolved about the x–axis. (b) Find the volume of the solid generated when the region R is revolved about the line y = −1. (c) Suppose R is revolved around the line y = k, where k > 1 so that the line is above the region. Find k if the volume of this solid is the same as the volume of the solid in the previous part, where R is revolved around the line y = −1. [Ans: 1.747; 5.460; 1.941] √ 3.C-23 (AB ’03) Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. Find the volume of the solid generated when R is revolved about the horizontal line y = 1. [Ans: 1.424] 3.C-24 (AB Acorn ’04-05) Let S be the region enclosed by the graphs of y = 2x and y = 2x2 for 0 ≤ x ≤ 1. Write a definite integral for the volsolid generated when S is revolved about the line y = 3. hume of the i 2 R1 2 Ans: π 0 3 − 2x2 − (3 − 2x) dx 3.C-25 (adapted from AB ’02B) Let R be the region bounded by the y–axis and x3 the graphs of y = and y = 4 − 2x. Find the volume of the solid 1 + x2 generated when R is revolved about (a) the x–axis. (b) the line y = 4. [Ans: 31.885;48.906] 3.C-26 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. Find the volume generated when R is revolved about the x–axis. [Ans: 4.267] 3.C-27 (AB ’05) Mr. Budd, compiled September 29, 2010


90

HL Unit 3 (Elite Integration)

1 Let f and g be the functions given by f (x) = + sin (πx) and g(x) = 4−x . 4 Let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. Find the volume of the solid generated when S is revolved about the horizontal line y = −1. [Ans: 4.558] 3.C-28 (AB 1993) [No calculator! ]The region enclosed by the x-axis, the line √ is the x = 3, and the curve y = x is rotated about the x-axis. What  volume of the solid generated? Ans: 92 π

Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 4: Non-Circular Slicing

3.4

91

Flat-Bottomed Volumes: Non-Circular Slicing

Advanced Placement Interpretations and properties of definite integrals • Computation of Riemann sums using left, right, and midpoint evaluation points. • Definite integral as a limit of Riemann sums over equal subdivisions. Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Textbook §5.1 Area between Curves and §5.2 Volume: Slicing, Disks, and Washers [15] Figure 3.13: AB 1998

Example 3.4.1 (AB 1998) The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x + 2y = 8, as shown in the Figure 3.13. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

Mr. Budd, compiled September 29, 2010


92

HL Unit 3 (Elite Integration) [Ans: 16.755]

Example 3.4.2 (AB 1997) √ The base of a solid S is the region enclosed by the graph of y = ln x, the line x = e, and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is

[Ans: 1]

Example 3.4.3 (BC 1997) The base of a solid is the region in the first quadrant enclosed by the graph of y = 2−x2 and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume of the solid is given by

h

Ans:

R2 0

(2 − y) dy

i

Example 3.4.4 Repeat for equilateral triangles, isosceles right triangles, etc.

Problems 3.D-1 (AP ’96) Let R be the region in the first quadrant under the graph of 1 y = √ for 4 ≤ x ≤ 9. Find the volume of the solid whose base is x the region R and whose cross-sections cut by planes perpendicular to the  x-axis are squares. Ans: ln 49 3.D-2 (AB ’00) Let R be the region in the first quadrant enclosed by the graphs 2 of y = e−x , y = 1 − cos x, and the y-axis. The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a square. Find the volume of the solid. [Ans: 0.461] 3.D-3 (adapted from AB ’02B) Let R be the region bounded by the y–axis and x3 the graphs of y = and y = 4 − 2x. The region R is the base of 1 + x2 a solid. For this solid, each cross section perpendicular to the x–axis is a semicircle. Find the volume of this solid. [Ans: 3.533] 3.D-4 (AB ’03) Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 4: Non-Circular Slicing

93

√ Let R be the region bounded by the graphs of y = x and y = e−3x and the vertical line x = 1. The region R is the base of a solid. For this solid, each cross section perpendicular to the x–axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. [Ans: 1.554] 3.D-5 (AB ’05B) Let f and g be the functions given by f (x) = 1 + sin (2x) and g(x) = ex/2 . Let R be the region in the first quadrant enclosed by the graphs of f and g. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x–axis are semicircles with diameters extending from y = f (x) to y = g(x). Find the volume of this solid. [Ans: 0.078] 3.D-6 [20] A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0.   (a) Use the Midpoint Rule to estimate the volume of the liver. Ans: 1110 cm3 (b) Use the Trapezoidal Rule to estimate the volume of the liver.  Ans: 1053 cm3

Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 5: Arc Length

3.5

95

Arc Length

Advanced Placement Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. To provide a common common foundation, specific applications should include finding the length of a curve (including a curve given in parametric form). Textbook §5.4: Arc Length and Surface Area: “Arc Length”. §9.3: Arc Length and Surface Area in Parametric Equations. [15] Resources §8-7 Length of a Plane Curve - Arc Length in Foerster [12]. Exploration 8-7: “Length of a Plane Curve (Arc Length)” [11].

3.5.1

Arc Length

First Explorations Exploration 8-7 in [11]

Z L=

Z L=

Z q dL =

2

(dx) + (dy)

Z q Z 2 2 dL = (dx) + (dy) =

2

s

 1+

dy dx

2 dx

Example 3.5.1 Find the arc length of y = x2 between x = −1 and x = 2.

Note: the integrand for arc length comes from the Pythagorean theorem, so that many of these integrals require trigonometric substitution.

Example 3.5.2 Without using calculus, find the arc length of the “curve” y = mx between x = a and x = b. Then confirm that the formula for arc length works. Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

3.5.2

Distance Traveled: Arc Length in Parametric Form

Z L=

Z q dL =

2

2

Z

s

(dx) + (dy) =

dx dt

2

 +

dy dt

2 dt

Example 3.5.3 A bug crawls with a path given by x(t) = r cos t and y(t) = r sin t. (a) What does the bug’s path look like? (b) How far does the bug travel between t = 0 and t = 2π? (c) How far does the bug travel between t = 0 and t = 4π?

Example 3.5.4 (adapted from [12]) Find the perimeter of the graph given by x(θ) = 4+5 cos θ, y(θ) = 6+3 sin θ. Approximate the graph with a figure for which you have a formula to find the perimeter.

Note: an important part of this problem is to decide what the limits of integration are.

Problems 1 1 3.E-1 Let C be the curve described by the equation y = x3 + x−1 over the 3 4 interval 1 ≤ x ≤ 3. (a) Sketch the curve C, and estimate its length. (b) Compute the length of C exactly. [Hint: If you do your algebra correctly, the square root disappears.]  Ans:

106 12



ex + e−x from x = 1 to x = 4. Find the 2   1 1 x −x 2 x −x 2 length of C exactly. [Hint: (e + e ) = 1 + (e − e ) .] 2 2   Ans: 21 e4 − e + e−1 − e−4

3.E-2 Let C be the curve y =

3.E-3 (BC98) Write an integral expression for the length of the path described 1 1 by the parametric equations x = t3 and y = t2 , where 0 ≤ t ≤ 1 3 2 h i R1√ Ans: 0 t4 + t2 dt Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 5: Arc Length

97

3.E-4 (BC97) Find, without a calculator, the length of the path described by the   π parametric equations x = cos3 t and y = sin3 t, for 0 ≤ t ≤ . Ans: 32 2 3.E-5 (BC93) Write an integral expression for the length of the curve determined by the equations x = t2 and y = t from t = 0 to t = 4. h i R4√ Ans: 0 4t2 + 1 dt 3.E-6 (BC04B) A particle moving along a curve in the plane has position (x(t), y(t)) at time t, where dy dx p 4 = t + 9 and = 2et + 5e−t . dt dt for all real values of t. A time t = 0, the particle is at point (4, 1). (a) Find the total distance traveled by the particle over the time interval 0 ≤ t ≤ 3. [Ans: 45.227] 3.E-7 (BC02B) A particle moves in the xy-plane so that its position at any time t, for −π ≤ t ≤ π, is given by x(t) = sin (3t) and y(t) = 2t. (a) Is the distance traveled by the particle from t = −π to t = π greater than 5π? Justify your answer. [Ans: 17.973 > 5π] 3.E-8 (BC01) An object moving along a curve in the xy-plane has position (x(t), y(t)) at time t with   dy dx = cos t3 and = 3 sin t2 dt dt for 0 ≤ t ≤ 3. At time t = 2, the object is at position (4, 5). (a) Find the total distance traveled by the object over the time interval 0 ≤ t ≤ 1. [Ans: 1.458]

Mr. Budd, compiled September 29, 2010


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Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 6: MVT for Integrals

3.6

99

Average Value and Mean Value Theorem for Integrals

Advanced Placement Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals. Textbook §4.4 The Definite Integral: “Average Value of a Function” [15]

3.6.1

Average Value

Suppose we were to average the following numbers: 17, 7, 5, 11. We would add 17 + 7 + 5 + 11, and then divide by 4. Note that 4 is the same as 1 + 1 + 1 + 1. So, what we have is essentially 17 + 7 + 5 + 11 = 10 1+1+1+1 Also note that 17 is 7 more than the average, 7 is 3 less, 5 is 5 less, and 11 is 1 more. Overall there is a total of 8 above the average, and 8 below the average. In general, to find the average, we take n P

yn

i=1 n P

1

i=1

. Now consider averaging a continuous function f (x) over an interval [a, b]. There are an infinite number of points to average, so how do we add an infinite number Mr. Budd, compiled September 29, 2010


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of numbers?

Rb

f (x) dx Rb 1 dx a

a

Definition 3.1 (Average Value). The average value of a function f over the interval [a, b] is given by Rb Z b f (x) dx 1 a f (x) dx = b−a b−a a The average value is basically the average height of the function over an interval. We get this by taking the “area” (the definite integral), and dividing by the width (b − a). Example 3.6.1 (adapted from AB97) What is the average value of cos x on the interval [−2, 6]?  Ans:

sin 6+sin 2 8



Example 3.6.2 Graphically estimate the average value of sec2 (πx)  1 over the interval 0, . Confirm your answer analytically. 3 Example 3.6.3 (adated from BC Acorn ’02) If f is a continuous R a+h function for all real x, then what is lim h1 a f (x) dx? h→0

[Ans: f (a)]

Substitution of Variables Example 3.6.4 (adapted from AB ’98) Graphically estimate the √ mean value of y = x2 x3 + 1 on the interval [−1, 2]. Confirm or reject analytically.

Average Rate of Change given the derivative If you are given f 0 (x), the derivative of f with respect to x, then the average rate of change of f is the same as the average value of f 0 . The average rate of change is the average value of the rate of change, i.e., the average value of the derivative. Mr. Budd, compiled September 29, 2010


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dy 1 Example 3.6.5 (adapted from BC93) If = , what is the dx x average rate of change of y with respect to x on the closed interval [1, e]?

h

Ans:

1 e−1

i

Average Value of f : Rb a

f (x) dx b−a

Average Rate of Change of f :

f (b) − f (a) = b−a

Rb a

f 0 (x) dx b−a

Connection between the average value of a function and the Fundamental Theorem of Calculus

Recall the proof of the Fundamental Theorem of Calculus that shows that Rb 0 G (x) dx = G(b) − G(a). The proof involved setting up a Riemann sum a such that the sample points, ci , for each subinterval were chosen as Rthe points that satisfied the Mean Value Theorem for the antiderivative G(x) = G0 (x) dx for each subinterval [xi−1 , xi ]. That is,

G0 (ci ) =

G(xi ) − G(xi−1 ) xi − xi−1

Recognizing that xi − xi−1 = 4xi , the MVT becomes

G0 (ci ) =

G(xi ) − G(xi−1 ) 4xi Mr. Budd, compiled September 29, 2010


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Replacing this in the formula for the Riemann sum, we have Rn

=

n X

G0 (ci )4xi

i=1

= =

n X G(xi ) − G(xi−1 ) i=1 n X

4xi

4xi

G(xi ) − G(xi−1 )

i=1

=

G(x1 ) − G(x0 ) + G(x2 ) − G(x1 ) + G(x3 ) − G(x2 ) .. . + G(xn ) − G(xn−1 )

Rn

=

G(xn ) − G(x0 )

=

G(b) − G(a)

Note that if we choose our sample points this way, then the value we get for Rn , G(b) − G(a), is independent of n, the number of subintervals. So that we can have three, six, one hundred, one million, or even one subinterval, and we get the same value for the Riemann sum, which is therefore the same value for the definite integral. Suppose that we choose one rectangle instead of three, or six, or one hundred. If we choose our one sample point as above, so that the Mean Value Theorem is satisfied for the antiderivative, then our one rectangle will have exactly the same area as the definite integral. Both shapes have the same “area” and width, but the rectangle has a constant height, and the definite integral has a (possibly) variable height. Since the one constant height gives the same signed area as the variable height, that one height behaves similarly to all the combined variable heights. This one constant height that mimics our changing heights is said to be the average or mean height. The proof of the Fundamental Theorem basically involves taking sample points such that the height of each rectangle is equal to the average value of the function over the corresponding subinterval. Finding the average value in some ways involves replacing the funky shape under f (x) with a rectangle of exactly the same “area” and width. The rectangle has a height equal to the average value of the function over the interval. When comparing the funky shape to the rectangle, parts sticking out above the rectangle Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 6: MVT for Integrals

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exactly match the parts drooping below the rectangle in size.

3.6.2

Mean Value Theorem

Theorem 3.1 (Mean Value Theorem). If f (x) is a function that is continuous over [a, b] and differentiable over (a, b), then ∃ c ∈ (a, b) 3 f 0 (c) =

f (b) − f (a) b−a

A nice, smooth function has a spot where the tangent line is parallel to the secant line, i.e., where the instantaneous rate of change matches the average rate of change.  Example 3.6.6 The function L(t) = 10 000 e−0.2 − e−0.2t gives the number of gallons that have leaked out of a tanker, where t the time in hours after noon. L(3) − L(1) . Explain the meaning of this value. [Ans: 1349.596] 3−1 (b) Find the average rate at which oil leaked out of the tanker from 3 p.m. to 9 p.m. Indicate units. [Ans: 639.188 gal/hr] (a) Find

(c) Find L0 (t). Using correct units, explain the meaning of L0 (t). (d) At what time between 1 p.m. and 3 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (e) At what time between 3 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? 

Ans: 2 000e−0.2t , rate of leakage; t = 1.96671 (1:58); t = 5.70352 (5:42)



Example 3.6.7 (adapted slightly from AB ’02) Let f be a function that is differentiable for all real numbers. Table 3.1 gives the values of f and its derivative f 0 for selected points x in the closed interval −1.5 ≤ x ≤ 1.5. The second derivative of f has the property that f 00 (x) > 0 for −1.5 ≤ x ≤ 1.5. (a) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and f 00 (c) = r. Give a reason for your answer. Mr. Budd, compiled September 29, 2010


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Table 3.1: AB ’02 x f (x) f 0 (x)

−1.5 −1 −7

−1.0 −4 −5

−0.5 −6 −3

0 −7 0

0.5 −6 3

1.0 −4 5

1.5 −1 7

(

2x2 − x − 7 for x < 0 2x2 + x − 7 for x ≥ 0 The graph of g passes through each of the points (x, f (x)) given in the table above. Is it possible that f and g are the same function? Give a reason for your answer.

(b) Let g be the function given by g(x) =

(c) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to approximate the value of f (1.2). Is this approximation greater than or less than the actual value of f (1.2)? Give a reason for your answer. (d) Write an equation of the line tangent to the graph of f at the point where x = 1.5. Use this line to approximate the value of f (1.2). Which of these approximations do you suppose is more accurate, and why?

[Ans: 6; no; −3 < f (1.2); −3.1 < −3 < f (1.2)]

Example 3.6.8 (from Acorn ’02) Let f be a function such that f 00 (x) < 0 for all x in the closed interval [1, 2]. Selected values of f are shown in Table 3.2. Which of the following must be true about Table 3.2: Acorn ’02 # 18 x f (x)

1.1 4.18

1.2 4.38

1.3 4.56

1.4 4.73

f 0 (1.2)? (a) f 0 (1.2) < 0 (b) 0 < f 0 (1.2) < 1.6 (c) 1.6 < f 0 (1.2) < 1.8 (d) 1.8 < f 0 (1.2) < 2.0 (e) f 0 (1.2) > 2.0 Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 6: MVT for Integrals

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Example 3.6.9 (adapted slightly from AB ’06B) A car travels on a straight track. During the time interval 0 ≤ t ≤ 60 seconds, the car’s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. Table 3.3 shows selected values of these functions. Table 3.3: AB ’06B t (sec) v(t) (ft/sec) a(t) (ft/sec2 )

0

15

25

30

35

50

60

−20

−30

−20

−14

−10

0

10

1

5

2

1

2

4

2

Z

60

(a) Using appropriate units, explain the meaning of v(t) dt in Z 60 30 terms of the car’s motion. Approximate v(t) dt using a 30

trapezoidal approximation with the three subintervals determined by the table. Z 60 (b) Using appropriate units, explain the meaning of |v(t)| dt Z 60 30 in terms of the car’s motion. Approximate |v(t)| dt using 30

a trapezoidal approximation with the three subintervals determined by the table. (c) For 0 < t < 60, must there be a time t when v(t) = −5? Justify your answer. (d) For 0 < t < 60, must there be a time t when a(t) = 0? Justify your answer. (e) For 0 < t < 60, must there be a time t when a0 (t) = 0? Justify your answer.

Example 3.6.10 (adapted slightly from AB ’03) A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. Table 3.4 gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B(x) is a twice–differentiable function that represents the diameter at that point. Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Table 3.4: AB ’04B Distance x (mm) Diameter B(x) (mm)

0

60

120

180

240

300

360

24

30

28

30

26

24

26

1 360

Z

360

B(x) dx 2 0 Z 360 B(x) 1 dx in terms of the blood vessel. Approximate the value of 360 0 2 using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer. Is your answer reasonable? 2 Z 360  B(x) (b) Using correct units, explain the meaning of π dx 2 0 in terms of the blood vessel. (a) Using correct units, explain the meaning of

(c) Explain why there must be at least one value x, for 0 < x < 360, such that B 00 (x) = 0.

[Ans: 14 mm; volume of the b.v. from x = 0 to x = 360; B 0 (c1 ) = B 0 (c2 ) = 0, . . . ]

3.6.3

Mean Value Theorem for Integrals

Theorem 3.2. Mean Value Theorem for Integrals If f (x) is continuous over the interval [a, b], then there exists a number c ∈ [a, b] such that b

Z f (c) (b − a) =

f (x) dx a

or f (c) =

1 b−a

Z

b

f (x) dx a

Example 3.6.11 (BC93) If f is continuous on the closed interval Rb [a, b], then there exists c such that a < c < b and a f (x) dx = (A)

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


HL Unit 3, Day 6: MVT for Integrals

107

f (b) − f (a) b−a (C) f (b) − f (a) (B)

(D) f 0 (c) (b − a) (E) f (c) (b − a)

[Ans: E]

Example 3.6.12 Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. Z 3 1 (a) Find R(t) dt. Explain the meaning of this value. [Ans: 1349.596] 3−1 1 (b) Find the average rate at which oil leaked out of the tanker from 3 p.m. to 9 p.m. Indicate units. [Ans: 639.188 gal/hr] (c) At what time between 1 p.m. and 3 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (d) At what time between 3 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval?



Ans: 2 000e−0.2t , rate of leakage; t = 1.96671 (1:58); t = 5.70352 (5:42)



Connection between the Mean Value Theorem for Integrals and the average value of a function Note that f (c) is the average (or mean) value of the function f (x) over the interval [a, b]. The Mean Value Theorem for Integrals is little more than applying the Mean Value Theorem to the antiderivative. If we look at the Mean Value Theorem for G(x), the antiderivative of G0 (x), over [a, b]: G(b) − G(a) G0 (c) = b−a This is similar to the approach we take in the proof of the Fundamental Theorem, except that we only have one subinterval. Now we learned (from the Mr. Budd, compiled September 29, 2010


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HL Unit 3 (Elite Integration)

Fundamental Theorem) that G(b) − G(a) is the same as these gives us: Rb 0 G (x) dx G0 (c) = a , b−a

Rb a

G0 (x) dx. Using

which is the signed area divided by the width, so that G0 (c) is the average value of G0 (x) over the interval [a, b]. If we let f (x) = G0 (x), i.e., G(x) is the antiderivative of f (x), then Rb f (c) =

a

f (x) dx b−a

at some point c ∈ (a, b).

Problems   π is Ans: π4 4

2

3.F-2 ([3]) The average rate of change of the function f (x) = x − 2 |x + 2|

over the interval −3 < x < −1 is [Ans: −3] 3.F-1 ([3]) The average value of sec2 x over the interval 0 ≤ x ≤

1 over the interval 1 ≤ x ≤ e is 3.F-3 ([3]) The average (mean) value of x h i 1 Ans: e−1 3.F-4 ([3]) The average value of f (x) = e2x + 1 on the interval 0 ≤ x ≤ [Ans: e]

1 is 2

3.F-5 (adapted from [3]) What is the average value of −2t3 + 6t2 + 4 over the interval −1 ≤ t ≤ 1? [Ans: 6] 3.F-6 (adapted from BC97) Let f be a twice differentiable function such that f (1) = −2 and f (4) = 7. Which of the following must be true for the function f on the interval 1 ≤ x ≤ 4? I. The average rate of change of f is 3. 5 II. The average value of f is . 3 0 III. The average value of f is 3. [Ans: I and III only] 3.F-7 Draw a function that has an average value of 0 over the closed interval [−π, π]. Mr. Budd, compiled September 29, 2010


HL Unit 3, Day 6: MVT for Integrals

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3.F-8 (AB ’97) Let f be the function given by f (x) = x3 − 6x2 + p, where p is an arbitrary constant. (a) Write an expression for f 0 (x) and use it to find the relative maximum and minimum values of f in terms of p. Show the analysis that leads to your conclusion. [Ans: max: f (0) = p, min: f (m = 4) = p − 32] (b) For what values of the constant p does f have three distinct roots? [Ans: 0 < x < 32] (c) Find the value of p such that the average value of f over the   closed interval [−1, 2] is 1. Ans: 23 4 3.F-9 (AB ’96) The rate of consumption of cola in the United States is given by S(t) = Cekt , where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980. (a) The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion gallons per year. Find C  and k. Ans: C = 6, k = 15 ln 2 (b) Find the average rate of consumption of cola over the 10-year time period beginning January 1, 1983. Indicate units of measure. [Ans: 19.680 billion gallons/year] (c) Use the trapezoidal rule with four equal subdivisions to estimate R7 S(t) dt. [Ans: 27.668] 5 R7 (d) Using correct units, explain the meaning of 5 S(t) dt in terms of cola consumption. [Ans: amount, in billions of gallons, of cola consumed in the two year period from 1/1/85 to 1/1/87] 3.F-10 (AB) A particle moves along the x-axis so that its velocity at time t, 0 ≤ t ≤ 5, is given by v(t) = 3 (t − 1) (t − 3). At time t = 2, the position of the particle is x(2) = 0. Find the average velocity of the particle over the interval 0 ≤ t ≤ 5. [Ans: 4] x 3.F-11 (AB ’88) Without a calculator, find the average value of y = 2 on x h +2 i  √  √2 the interval 0, 6 . Ans: ln 6 3.F-12 Using a calculator √ only to graph the function, exactly find the  average  value of y = − 1 − x2 + 1 over the interval [−1, 1]. Ans: 4−π 4  πx  3.F-13 (AB ’85) Let f (x) = 14πx2 and g(x) = k 2 sin for k > 0. 2k (a) Find the average value of f on [1, 4]. (b) For what value of k will the average value of g on [0, k] be equal to the average value of f on [1, 4]? Mr. Budd, compiled September 29, 2010


110

HL Unit 3 (Elite Integration) [Ans: 98π; 7π]

3.F-14 [4] Which of the following statements is true? I. If the graph of a function is always concave up, then the left-hand Riemann sums with the same subdivisions over the same interval are always less than the right-hand sums. Rb II. If the function f is continuous on the interval [a, b] and a f (x) dx = 0, then f must have at least one zero between a and b. III. If f 0 (x) > 0 for all x in an interval, then the function f is concave up in that interval. [Ans: II only] 3.F-15 Oil is leaking from a tanker at the rate of R(t) = 2 000e−0.2t gallons per hour, where t is measured in hours after noon. Z 5 1 (a) Find R(t) dt. Explain the meaning of this value. [Ans: 1127.128] 5−1 1 (b) Find the average rate at which oil leaked out of the tanker from 5 p.m. to 9 p.m. Indicate units. [Ans: 506.451 gal/hr] (c) Find the average rate at which oil leaked out of the tanker from 1 p.m. to 9 p.m. Indicate units. [Ans: 816.790 gal/hr] (d) At what time between 1 p.m. and 5 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (e) At what time between 5 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? (f) At what time between 1 p.m. and 9 p.m. is the instantaneous rate of leakage the same as the average rate of leakage over that same time interval? [Ans: t = 2.86737 (2:52); t = 6.86737 (6:52); t = 4.477603 (4:29)] 3.F-16 Sometimes the Mean Value Theorem for Integrals is called the Average Value Theorem. Explain why, in that case, the Mean Value Theorem proper could be called the Average Value Theorem for Derivatives.

Mr. Budd, compiled September 29, 2010


Unit 4

Polar Coordinates and Complex Numbers 1. Introduction to Polar Coordinates 2. Arc Length and Area in Polar Coordinates 3. Complex Numbers and DeMoivreâ&#x20AC;&#x2122;s Theorem 4. Roots of Complex Numbers

Advanced Placement Applications of derivatives. 1. Analysis of planar curves given in polar form. Computation of derivatives. 1. Derivatives of parametric, polar, and vector functions. Applications of integrals. Specific applications should include finding the area of a region (including a region bounded by polar curves), and the length of a curve.

111


112

HL Unit 4 (Polar Coordinates and Complex Numbers)

Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 1: Area in Polar Coordinates

4.1

113

Area in Polar Coordinates

Advanced Placement Applications of derivatives. • Analysis of planar curves given in polar form. Computation of derivatives. • Derivatives of parametric, polar, and vector functions. Applications of integrals. Specific applications should include finding the area of a region (including a region bounded by polar curves), and the length of a curve. Textbook §9.5 Calculus and Polar Coordinates [15] Resources §8-9 Lengths and Areas for Polar Coordinates in Foerster [12].

4.1.1

Area Z A= a

b

1 2 r dθ 2

Example 4.1.1 Find the region enclosed by the limacon r = 5 + 4 cos θ. [12]

[Ans: 103.672]

Example 4.1.2 For the limacon r = 1 + 3 sin θ [12] (a) Find the area of the region inside the inner loop. (b) Find the area of the region between the outer loop and the inner loop.

[Ans: 2.528; 14.751] Mr. Budd, compiled September 29, 2010


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Figure 4.1: From 2003 BC exam

Example 4.1.3 (from BC03) Figure 4.1 shows the graphs of the p 5 line x = y and the curve C given by x = 1 + y 2 . Let S be the 3 shaded region bounded by the two graphs and the y-axis. The line and the curve intersect at point P . (a) Find the coordinates of point P and the value of

dx for curve dy

C at point P . (b) Set up and evaluate an integral expression with respect to y that gives an area of S. (c) Curve C is part of the curve x2 −y 2 = 1. Show that x2 −y 2 = 1 1 can be written as the polar equation r2 = . cos2 θ − sin2 θ (d) Use the polar equation to set up an integral expression with respect to the polar angle θ that represents the area of S.  Ans:

4.1.2

5 3 4, 4



, 53 ; 0.347;



Arc Length Z L= a

b

s

dr dθ

2 + r2 dθ

Example 4.1.4 Find the length of the limacon r = 1 + 3 sin θ. [12]

[Ans: 19.377] Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 1: Area in Polar Coordinates

115

Figure 4.2: From 2003 BC exam

Example 4.1.5 Find the length of one petal on the polar rose r = cos 2θ [17]

Problems 4.A-1 (BC03B) Figure 4.2 shows the graphs of the circles x2 + y 2 = 2 and 2 (x − 1) + y 2 = 1. The graphs intersect at the points (1, 1) and (1, −1). Let R be the (barely) shaded region in the first quadrant bounded by the two circles and the x-axis. (a) Set up an expression involving one or more integrals with respect to x that represents the area of R. (b) Set up an expression involving one or more integrals with respect to y that represents the area of R. √ (c) Show that the polar equations of the circles are r = 2 and r = 2 cos θ, respectively. (d) Set up an expression involving one or more integrals with respect to the polar angle θ that represents the area of R. [Ans: How could you check your answers?] 4.A-2 (adapted from BC05) A curve is drawn in the xy–plane and is described by the equation in polar coordinates r = θ + sin (2θ) for 0 ≤ θ ≤ π, where r is measured in meters and θ is measured in radians.   dr (a) Find the derivative of r with respect to θ. Ans: dθ = 1 + 2 cos (2θ) (b) Find the area bounded by the curve and the x–axis.

[Ans: 4.382]

(c) Find the angle θ that corresponds to the point on the curve with x–coordinate −2. [Ans: 2.786] Mr. Budd, compiled September 29, 2010


116

HL Unit 4 (Polar Coordinates and Complex Numbers) π 2π dr (d) For <θ< , is negative. What does this fact say about r? 3 3 dθ What does this fact say about the curve? π (e) Find the value of θ in the interval 0 ≤ θ ≤ that corresponds to the 2 point on the curve in the first quadrant with greatest distance from  the origin. Justify your answer. Ans: π3

4.A-3 (adapted from BC07) Two polar curves are given by r = 2 and r = 3 + 2 cos θ.   4π (a) At what values of θ do the curves intersect? Ans: 2π 3 , 3 (b) Let R be the region that is inside the graph of r = 2 and also inside the graph of r = 3 + 2 cos θ. Find the area of R. (c) A particle moving with nonzero velocity along the polar curve given by r = 3 + 2 cos θ has position (x(t), y(t)) at time t, with θ = 0 when dr dr = . Find t = 0. This particle moves along the curve so that dt dθ π dr at θ = and interpret your answer in terms of the the value of dt 3 motion of the particle. [Ans: −1.732] dy dy dy (d) For the particle described above, = . Find the value of dt dθ dt π at θ = and interpret your answer in terms of the motion of the 3 particle. [Ans: 0.5]

Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 2: Complex Numbers and Mathematical Induction

4.2

117

Complex Numbers and Mathematical Induction

International Baccalaureate 1.4 Proof by mathematical induction. Forming conjectures to be proved by mathematical induction. √ 1.5 Complex numbers: the number i = −1; the terms real part, imaginary part, conjugate, modulus, and argument. Cartesian form z = a + ib. Modulus-argument form z = r (cos θ + i sin θ). Included: Awareness that z = r (cos θ + i sin θ) can be written as z = r cis θ. The complex plane. Included: The complex plane is also known as the Argand diagram. 1.6 Sums, products, and quotients of complex numbers. 1.7 De Moivre’s theorem. Included: proof by mathematical induction for n ∈ Z + . Powers and roots of a complex number. 1.8 Conjugate roots of polynomial equations with real coefficients.

Resources Chapter 8 Mathematical Induction and §9.1 Complex Numbers and §9.2 Geometrical Representation of Complex Numbers and §9.3 Polar Form of Complex Numbers in Cirrito [6]

4.2.1

Complex Numbers

Example 4.2.1 [6] State the real and imaginary parts of w = 3−9i Example 4.2.2 Find i73

Example 4.2.3 [6] Given that z = 3 + i and w = 1 − 2i evaluate the following: (a) z + w (b) 2z − 3w (c) zw (d) w2 (e) z

[Ans: 4 − i; 3 + 8i; 5 − 5i; −3 − 4i; 3 − i] Mr. Budd, compiled September 29, 2010


118

HL Unit 4 (Polar Coordinates and Complex Numbers)

Example 4.2.4 [6] Express the complex number u + iv.

1 − 4i in the form 1 + 5i

 Ans: − 19 26 −

9 26 i



Example 4.2.5 (HL01) (z + 2i) is a factor of 2z 3 − 3z 2 + 8z − 12. Find the other two factors.

4.2.2

Graphing Complex Numbers

Complex numbers are represented on an Argand diagram and can be used to represent vectors.

Example 4.2.6 [6] Represent each of the following complex numbers on an Argand diagram. (a) z = 1 + 3i (b) z = −2 + i (c) z = −2i

Polar Forms of Complex Numbers Complex numbers can be represented in a pseudo-Cartesian form x + iy where x and y are the real and imaginary parts, respectively. Complex numbers can also be represented in a polar form r (cos θ + i sin θ) = r cis θ where r is called the modulus (or magnitude) and θ is referred to as the argument. The principal argument of a complex number is considered to be in the interval (−π, π].

Example 4.2.7 [6] Find the modulus and principle argument of the following complex numbers, and write them in polar form: (a) 1 + i Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 2: Complex Numbers and Mathematical Induction

119

(b) −1 + 2i √ (c) −1 − 3i √

Example 4.2.8 [6] Convert

 2 cis

3π 4

 into cartesian form.

[Ans: −1 + i] 2

Example 4.2.9 (HL01) Given that z = (b + i) , where b is real and positive, find the exact value of b when arg z = 60◦ . Example 4.2.10 (HL 5/09) (a) Show that the complex number i is a root of the equation x4 − 5x3 + 7x2 − 5x + 6 = 0 (b) Find the other roots of this equation.

4.2.3

Mathematical Induction

Example 4.2.11 Prove by induction that 1 2 + 2 2 + 3 2 + . . . + n2 =

1 n (n + 1) (2n + 1) 6

Example 4.2.12 Prove the power rule by mathematical induction. You may assume that you know the product rule.

4.2.4

De Moivre’s Theorem

Theorem 4.1 (De Moivre’s Theorem). n

[r (cos θ + i sin θ)] = rn [cos(nθ) + i sin(nθ)]

Example 4.2.13 (HL 5/03) (a) Prove, using mathematical induction, that for a positive integer n, n

(cos θ + i sin θ) = cos nθ + i sin nθ where i2 = −1. Mr. Budd, compiled September 29, 2010


120

HL Unit 4 (Polar Coordinates and Complex Numbers) (b) The complex number z is defined by z = cos θ + i sin θ. 1 (a) Show that = cos (−θ) + i sin (−θ). [Why can’t we used z what we just proved earlier?] (b) Deduce that z n + z −n = 2 cos nθ.

Example 4.2.14 [6] Find

5 3 + i using De Moivre’s Theorem. √   Ans: −16 3 + 16i

Example 4.2.15 [6] Find (−1 + i)

−4

using De Moivre’s Theorem.

  Ans: − 41

Problems 4.B-1 (HL 5/03) The complex number z satisfies the equation √

z=

2 + 1 − 4i. 1−i

Express z in the form x + iy, where x, y ∈ Z. 4.B-2 (HL Spec ’08) Given that |z| = where z ∗ is the conjugate of z.

[Ans: −5 − 12i]

10, solve the equation 5z +

10 = 6 − 18i, z∗

4.B-3 (HL Spec ’08) Solve the simultaneous equations iz1 + 2z2

=

3

z1 + (1 − i) z2

=

4

giving z1 and z2 in the form x + iy, where x and y are real. 4.B-4 (HL Spec ’08) Find b where

2 + bi 7 9 = − + i. 1 − bi 10 10 2

4.B-5 (HL Spec ’08) Given that z = (b + i) , where b is real and positive, find the value of b when arg z = 60◦ . 4.B-6 (HL 5/08) Express

a 1 √ 3 in the form where a, b ∈ Z. b 1−i 3 Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 2: Complex Numbers and Mathematical Induction

121

4.B-7 (HL 5/08) Find, in its simplest form, the argument of (sin θ + i (1 − cos θ)) where θ is an acute angle. √ m n 4.B-8 (HL 5/08) z1 = 1 + i 3 and z2 = (1 − i) .

2

(a) Find the modulus and argument of z1 and z2 in terms of m and n, respectively. (b) Hence, find the smallest positive integers m and n such that z1 = z2 . 4.B-9 (HL 5/09) Consider the complex numbers z = 1 + 2i and w = 2 + ai, where a ∈ R. Find a when (a) |w| = 2 |z|; (b) Re (zw) = 2Im (zw). 4.B-10 (HL 5/09) If z is a non-zero complex number, we define L(z) by the equation L(z) = ln |z| + i arg (z), 0 ≤ arg (z) < 2π (a) Show that when z is a positive real number, L(z) = ln z. (b) Use the equation to calculate i. L(−1); ii. L(1 − i); iii. L(−1 + i). (c) Hence show that the property L(z1 z2 ) = L(z1 ) + L(z2 ) does not hold for all values z1 and z2 . 4.B-11 Use mathematical induction to prove that lim bn e−b = 0 for natural b→∞

numbers n. 4.B-12 Use mathematical induction to prove the power rule.  4.B-13 (HL 5/07) Prove by induction that 12n + 2 5n−1 is a multiple of 7 for n ∈ Z +. 4.B-14 (HL 5/08) The function f is defined by f (x) = xe2x  Use mathematical induction to prove that f (n) (x) = 2n x + n2n−1 e2x for all n ∈ Z + , where f (n) (x) represents the nth derivative of f (x).   1 1 1 4.B-15 (HL 5/09) Let A =  0 1 1  0 0 1   n (n + 1) 1 n   2 Prove by induction that An =  0 1  for n ∈ Z + n 0

0

1

Mr. Budd, compiled September 29, 2010


122

HL Unit 4 (Polar Coordinates and Complex Numbers)

Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 3: DeMore DeMoivre

4.3

123

DeMore DeMoivre

International Baccalaureate 1.4 Proof by mathematical induction. Forming conjectures to be proved by mathematical induction. √ 1.5 Complex numbers: the number i = −1; the terms real part, imaginary part, conjugate, modulus, and argument. Cartesian form z = a + ib. Modulus-argument form z = r (cos θ + i sin θ). Included: Awareness that z = r (cos θ + i sin θ) can be written as z = r cis θ. The complex plane. Included: The complex plane is also known as the Argand diagram. 1.6 Sums, products, and quotients of complex numbers. 1.7 De Moivre’s theorem. Included: proof by mathematical induction for n ∈ Z + . Powers and roots of a complex number. 1.8 Conjugate roots of polynomial equations with real coefficients.

4.3.1

DeMoivre’s Theorem and Power Series

Example 4.3.1 Write the series for (a) cos θ (b) sin θ and therefore i sin θ (c) ex and therefore eiθ Example 4.3.2 Show Euler’s Formula, eiθ = cos θ + i sin θ using series. Show how this gives DeMoivre’s Theorem.

Example 4.3.3 Find (a) [3 (cos40◦ + i sin 40◦ )] [4 (cos80◦ + i sin 80◦ )] 7

(b)

(2 cis 15◦ )

3

(4 cis 45◦ ) √ !10 1+i 3 √ (c) . 1−i 3 h √ √ Ans: −6 + 6i 3; 3 − i; − 12 +

i

3 2 i

Mr. Budd, compiled September 29, 2010


124

HL Unit 4 (Polar Coordinates and Complex Numbers)

4.3.2

Roots of Complex Numbers 2

Example 4.3.4 (HL Spec ’08) Given that (a + bi) = 3 + 4i obtain a pair of simultaneous equations involving a and b. Hence find the two square roots of 3 + 4i.

Example 4.3.5 (HL 5/07) The complex number z is defined by   √  π 2π 2π π z = 4 cos + i sin + 4 3 cos + i sin 3 3 6 6 (a) Express z in the form reiθ , where r and θ have exact values. (b) Find the cube roots fo z, expressing in the form reiθ , where r and θ have exact values. √ Example 4.3.6 Find the square roots of −1 − i 3. Designate the principal root.

Example 4.3.7 Find the three cube roots of (a) −1 (b) 8i (c) −1 + i Designate the principal root.

4.3.3

Deriving Trigonometric Formulas

Example 4.3.8 Use binomial expansion to prove the identities: (a) cos 5θ = 16 cos5 θ − 20 cos3 θ + 5 cos θ; sin 5θ = 16 cos4 θ − 12 cos2 θ + 1. (b) sin θ Example 4.3.9 (HL 5/03) The complex number z is defined by z = cos θ + i sin θ. Previously we showed that z n + z −n = 2 cos nθ. 5 (a) Find the binomial expansion of z + z −1 . 1 (b) Hence show that cos5 θ = (a cos 5θ + b cos 3θ + c cos θ), where 16 a, b, c are positive integers to be found. Mr. Budd, compiled September 29, 2010


HL Unit 4, Day 3: DeMore DeMoivre

125

Problems 4.C-1 (HL 5/03) (a) Prove, using mathematical induction, that for a positive integer n, n

(cos θ + i sin θ) = cos nθ + i sin nθ where i2 = −1. [5 marks] (b) The complex number z is defined by z = cos θ + i sin θ. 1 i. Show that = cos (−θ) + i sin (−θ). z ii. Deduce that z n + z −n = 2 cos nθ. [5 marks]  5 (c) i. Find the binomial expansion of z + z −1 . 1 (a cos 5θ + b cos 3θ + c cos θ), where ii. Hence show that cos5 θ = 16 a, b, c are positive integers to be found. [5 marks] 4.C-2 Show that eiθ + e−iθ ; 2 eiθ − e−iθ (b) sin θ = 2i (a) cos θ =

4.C-3 (HL 5/07) (a)

i. Factorize t3 − 3t2 − 3t + 1, giving your answer as a product of a linear factor and a quadratic factor. ii. Hence find all the exact solutions to the equation t3 − 3t2 − 3t + 1=0

(b) Using de Moivre’s theorem and the binomial expansion i. show that cos 3θ = cos3 θ − 3 cos θ sin2 θ; ii. write down a similar expression for sin 3θ. (c)

3 tan θ − tan3 θ . 1 − 3 tan2 θ ◦ ◦ ii. Find the values of θ, 0 ≤ θ ≤ 180 , for which this identity is not valid. i. Hence show that tan 3θ =

(d) Using the results from parts (a) and (c) above, find the exact values of tan 15◦ and tan 75◦

Mr. Budd, compiled September 29, 2010


126

HL Unit 4 (Polar Coordinates and Complex Numbers)

Mr. Budd, compiled September 29, 2010


Unit 5

Taylor Series 1. Geometric Series 2. Polynomials for Sine, Cosine, Exponential Functions 3. Taylor and Maclaurin Series 4. Manipulation of Taylor Series 5. Interval of Convergence 6. The Integral Test and p-Series 7. Comparison Tests 8. Alternating Series 9. Error Analysis

Advanced Placement Concept of Series. A series is defined as a sequence of partial sums and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. Series of Constants • Motivating examples, including decimal expansion. • Geometric series with applications. • The harmonic series. 127


128

HL Unit 5 (Taylor Series) • Alternating series with error bound. • Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series. • The ratio test for convergence and divergence. • Comparing series to test for convergence or divergence.

Taylor series • Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) • Maclaurin series and the general Taylor series centered at x = a. • Maclaurin series for the functions ex , sin x, cos x, and

1 . 1−x

• Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series. • Functions defined by power series. • Radius and interval of convergence of power series. • Lagrange error bound for Taylor polynomials. International Baccalaureate 1.1 Geometric sequences and series; sum of finite and infinite geometric series. Applications of the above. Included: sigma notation, applications of sequences and series to compound interest and population growth. 12.1 Convergence of infinite series. Tests for convergence: ratio test; limit comparison test; integral test. P Included: conditions for the application of these tests, the divergence theorem, if un is a convergent series then lim un = 0. n→∞ 12.2 Alternating series. Conditional convergence. Included: knowledge that the absolute value of the truncation error is less than the next term in the series; absolute convergence of an infinite series. 12.3 Power series: radius of convergence. Determination of the radius of convergence by the ratio test. Included: power series in (x − k), k 6= 0. 12.5 Use of Taylor series expansions, including the error term. Maclaurin series as a special case. Taylor polynomials. Taylor series by multiplication. Included: application to the approximation of functions; bounds on the error term. Included: 2 finding the Taylor approximations for functions such as ex arctan x by multiplying 2 the Taylor approximations for ex and arctan x.

Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 1: Series Basics

5.1

129

Series Basics

Advanced Placement Concept of Series. A series is defined as a sequence of partial sums and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. Series of Constants

• Motivating examples, including decimal expansion. • Geometric series with applications.

Taylor series

• Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) • Maclaurin series for the functions ex , sin x, cos x,

1 . 1−x

International Baccalaureate 1.1 Geometric sequences and series; sum of finite and infinite geometric series. Applications of the above. Included: sigma notation, applications of sequences and series to compound interest and population P growth. 12.1 Included: the divergence theorem, if un is a convergent series then lim un = n→∞ 0.

Textbook §8.1 Sequences of Real Numbers and §8.2 Infinite Series [15] Resources §12-1 Introduction to Power Series and 12-2 Geometric Sequences and Series as Mathematical Models in Foerster [12]. Exploration 12-2: “Two Geometric Series” in [11]. §11.2 Infinite Series, Convergence, and Divergence in Ostebee and Zorn [17]. Resources §12-3 Power Series for an Exponential Function and §12-4 Power Series for Other Elementary Functions in [12]. Explorations 12-3: “A Power Series for a Familiar Function” and 12-4: “Power Series for Other Familiar Functions” in [11]. Mr. Budd, compiled September 29, 2010


130

HL Unit 5 (Taylor Series)

5.1.1

Some Vocabulary

Terms to know • sequence • series • terms • term index • partial sum • sequence of partial sums • convergence of a sequence • convergence of a series • arithmetic sequence or series • geometric sequence or series

5.1.2

Remember Geometric Series?

Be aware of the difference between a geometric sequence and a geometric series. The sequence is the list of terms, the series is the sum of terms. Example 5.1.1 Write 0.35 as a fraction. Example 5.1.2 Write an expression for the first n terms in a geometric series, i.e., the nth partial sum of the geometric series. h Ans: Sn =

Sn =

n X k=0

arn =

a (1 − rn ) 1−r

,

a(1−r n ) 1−r

,

i r 6= 1

r 6= 1

Example 5.1.3 For what values of r will the geometric series S = lim Sn converge? To what will S converge? n→∞

Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 1: Series Basics

131

1 1 1 1 − + · · · converges. Example 5.1.4 [17] The series − + 3 6 12 24 To what limit?

Example 5.1.5 (HL 5/00) The probability distribution of a dis 2 2 crete random variable X is given by P(X = x) = k , for 3 x = 0, 1, 2, . . .. Find the value of k.

1 3



θ 1−cos θ

i

 Ans:

Example 5.1.6 (HL 5/04) AOB is a sector on the unit circle, where ˆ = θ. AOB The lines (AB1 ), (A1 B2 ), (A2 B3 ) are perpendicular to OB. A1 B1 , A2 B2 are all arcs of circles with centre O. Calculate the sum to infinity of the arc lengths AB + A1 B1 + A2 B2 + A3 B3 + . . .

h Ans:

Example 5.1.7 (HL 5/05) The sum of the first n terms of an arithmetic sequence {un } is given by the formula Sn = 4n2 − 2n. Three terms of this sequence, u2 , um and u32 , are consecutive terms in a geometric sequence. Find m.

[Ans: 7]

5.1.3

nth Term Test for Divergence

The series

∞ P n

tn has no chance of converging unless lim tn = 0. In other words, n→∞

the series has no chance of converging unless the sequence converges to 0. • If lim tn 6= 0, then you will continuously add some value for ever and n→∞ ever. Mr. Budd, compiled September 29, 2010


132

HL Unit 5 (Taylor Series) • If the sequence does not converge to 0, then the series will not converge to anything, i.e., the series will diverge. This is called the nth term test for divergence. Note that we can positively conclude divergence, but not convergence, with this test. • If the sequence does converge to 0, then the series may or may not converge. • If the series does converge (to anything), then the sequence must converge to 0.

These rules are for any series, not just geometric series.

Example 5.1.8 [17] Does

5.1.4

∞ P

k converge? k + 1000 k=1

Geometric Series

Example 5.1.9 [17] Evaluate

∞ 4 + 2k P . 3k k=0

[Ans: 9]

Example 5.1.10 [17] For the geometric series

∞ 3 P , calculate the k k=0 2

tail R10 = S − S10  Ans:

3 1024



Problems 5.A-1 Which of the following series converge to 2? ∞ P

2n n=1 n + 3 ∞ P −8 II. n n=1 (−3) ∞ P 1 III. n n=0 2 I.

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


HL Unit 5, Day 1: Series Basics

133

5.A-2 [17] What is wrong with the following argument? Let S = 1 + 2 + 4 + 8 + · · · . Then 2S = 2 + 4 + 8 + · · · = S − 1, so S = −1.

Mr. Budd, compiled September 29, 2010


134

HL Unit 5 (Taylor Series)

Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 2: Manipulation of Taylor and Maclaurin Series

5.2

135

Manipulation of Taylor and Maclaurin Series

Advanced Placement Taylor series • Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) • Maclaurin series and the general Taylor series centered at x = a. • Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series. International Baccalaureate 10.5 Taylor polynomials and series, including the error term. n Maclaurin series for ex , sin x, cos x, arctan x, ln (1 + x), (1 + x) . Use of substitution to obtain other series. Differentiation and integration of series (valid only on the interval of convergence of the initial series. Textbook §8.7 Taylor Series and §8.8 Applications of Taylor Series [15] Resources §12-5 Taylor and Maclaurin Series, and Operations on These Series in [12]. Exploration 12-4: “Power Series for Other Familiar Functions” in [11].

5.2.1

Telescoping Series

Example 5.2.1 Show that

∞ P

1 converges, and find its k=1 k (k + 1)

limit.

5.2.2

Introduction to Power Series

Example 5.2.2 Find a polynomial which is its own derivative. What must the degree be if n − 1 = n? What happens when you antidifferentiate this polynomial? Compare graphs of y = ex and various partial sums of your polynomial. Mr. Budd, compiled September 29, 2010


136

HL Unit 5 (Taylor Series)

Example 5.2.3 Let f (x) = a0 + a1 x + a2 x2 + a3 x3 + . . .

6 (or something else) and P (x) = 1−x

(a) Find f 0 (x), f 00 (x), f 000 (x), and f (4) (x), the first, second, third, and fourth derivatives of f (x). (b) Evaluate f and its first four derivatives at x = 0. (c) Repeat for P (x). (d) Find the values of a0 , a1 , . . . , a4 , such that f (0) = P (0), f 0 (0) = P 0 (0), f 00 (0) = P 00 (0), and so forth. (e) Compare graphs of f (x) and partial sums of P (x).

5.2.3

Taylor Series

Example 5.2.4 [12] Show by equating derivatives that the Taylor series for ln x expanded about x = 1 is ln x = (x − 1) −

1 1 1 2 3 4 (x − 1) + (x − 1) − (x − 1) + · · · 2 3 4

Example 5.2.5 Make a general formula for finding a power series 2 3 P (x) = a0 + a1 (x − c) + a2 (x − c) + a3 (x − c) . . . for any function 0 0 00 f (x), such that P (c) = f (c), P (c) = f (c), P (c) = f 00 (c), and so on. Example 5.2.6 [12] Expand f (x) = sin x as a Taylor series about π x= . 3 h Ans:

√ 3 2

+

1 2

x−

π 3



3/2 2!

x−

 π 2 3

1/2 3!

 π 3 3

x−

+

√ 3/2 4!

x−

 π 4 3

+ ···

i

A Taylor series expanded about x = 0 is called a Maclaurin series.

5.2.4

Manipulation of Series

Example 5.2.7 From the power series for series for ln x.

1 1−x ,

derive a power

Example 5.2.8 Find a power series for arctan x. What is the exact value of 1 − 31 + 15 − 71 + · · · ? Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 2: Manipulation of Taylor and Maclaurin Series h Ans: x −

x3 3

+

137 x5 5

x7 7

π 4

i

+ ···



+ ···;

2

Example 5.2.9 Find a power series for ex .

 Ans: 1 + x2 + 21 x4 +

1 6 3! x

2

Example 5.2.10 (BC97) Let P (x) = 7 − 3 (x − 4) + 5 (x − 4) − 3 4 2 (x − 4) + 6 (x − 4) be the fourth-degree Taylor polynomial for the function f about 4. Assume f has derivatives of all orders for all real numbers. (a) Find f (4) and f 000 (4).

[Ans: 7, −12] 0

(b) Write the second-degree Taylor h polynomial for f about 4 and i 2 0 use it to approximate f (4.3) Ans: −3 + 10 (x − 4) − 6 (x − 4) , −0.54 Rx (c) Write theh fourth-degree Taylor polynomial for g(x) = 4 f (t) dti 2 3 4 about 4. Ans: 7 (x − 4) − 23 (x − 4) + 53 (x − 4) − 12 (x − 4) (d) Can f (3) be determined from the information given? Justify your answer.

Problems 5.B-1 (BC98) What is the approximation of the value of sin 1 obtained   by using the fifth-degree Taylor polynomial about x = 0 for sin x? Ans: 101 120 5.B-2 (BC acorn ’02) The third-degree Taylor polynomial about x = 0 of ln (1 − ix) h Ans: −x −

is

x2 2

x3 3

5.B-3 [12] Expand each function as a Taylor series about the given value of x. Write enough terms to reveal clearly that you have seen the pattern. (a) f (x) = sin x, about x = π4 h √ √  Ans: 22 + 22 x − π4 − (b) f (x) = cos x, about x =

2 2·2!

x−

 π 2 4

2 2·3!

x−

 π 3 4

+

2 2·4!

x−

 π 4 4

π 4

(c) hf (x) = ln x, about x = 1 2 Ans: (x − 1) − 12 (x − 1) +

1 3

3

(x − 1) −

1 4

4

(x − 1) + · · ·

i

(d) f (x) = log x, about x = 10 Mr. Budd, compiled September 29, 2010

+

2 2·5!

x−

 π 5 4

− ···

i


138

HL Unit 5 (Taylor Series) 7/3

(e) f (x) = (x − 5) about x = 4 h 2 Ans: −1 + 37 (x − 4) − 37·4 2 2! (x − 4) + (f) f (x) = (x + 6)

4.2

7·4·1 33 3!

3

(x − 4) −

7·4·1·(−2) 34 4!

4

(x − 4) +

7·4·1·(−2)·(−5) 35 5!

(x −

about x = −5

5.B-4 [12] Find the Maclaurin series for cos (3x) by equating derivatives. Compare the answer, and the ease of getting the answer, with the series you obtain by substituting 3x for x in the cosine series. 5.B-5 [12] Find the Maclaurin series for ln (1 + x) by equating derivatives. Compare the answer, and the ease of finding the answer, with the series you obtain by substituting (1 + x) for x in the Taylor series for ln x, expanded about x = 1. 5.B-6 [12] Accuracy for ln x Series Value: Estimate ln 1.5 using S4 (1.5), the fourth partial sum of the Taylor series. How close is your answer to the real answer? How does the error in the series value compare with the first term of the tail of the series, t5 , which is the first term left out in the partial sum? [Ans: 0.40546510 . . . − 0.40104166 . . . = 0.00442344 . . . < 0.00625] 5.B-7 [12] Accuracy for ln x Series: Find the interval of values of x for which the fourth partial sum of the Taylor series for ln x gives values that are within 0.001 unit of ln x. ∞ P

n

(x − 1) . n n=1 Let f be the function given by the sum of the first three nonzero terms of this series. The maximum value of |ln x − f (x)| for 0.3 ≤ x ≤ 1.7 is [Ans: 0.145]

5.B-8 (BC98) The Taylor series for ln x, centered at x = 1, is

5.B-9 Find a series for f (x) = converge?

(−1)

n+1

1 . For what values of x does the series 1 − x   Ans: 1 + x + x2 + x3 + . . . + xn + . . .; |x| < 1

5.B-10 Use your series for f (x) to find a series for g(x) =

1 and for r(x) = 1+x

1 . 1 − − (x − 1) h 2 3 Ans: 1 − x + x2 − x3 + . . . + (−1)n xn + . . .; 1 − (x − 1) + (x − 1) − (x − 1) + . . . + (−1)n (x − R 1 dx. What is 5.B-11 Use your series for g(x) to find a series for h(x) = 1+x h i x2 x3 n+1 xn h(x)? Ans: C + x − 2 − 3 + . . . + (−1) + . . .; ln |1 + x| + C n 1 . How might you 5.B-12 Use your series for g(x) to find a series for j(x) = 1 + x2   get a series for arctan x? Ans: 1 − x2 + x4 − x6 + . . . + (−1)n x2n + . . .; antidifferentiate. Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 2: Manipulation of Taylor and Maclaurin Series

139

5.B-13 (BC acorn ’02) The third-degree Taylor polynomial about x = 0 of ln (1 − ix) h Ans: −x −

is

x2 2

x3 3

∞ (−1)n xn P is the Taylor series about x = 0 for which of the following n! n=0 functions? [Ans: e−x ]

5.B-14 [4]

5.B-15 (BC98) The graph of the function represented by the Maclaurin series n x2 x3 (−1) xn 1−x+ − + ... + + . . . intersects the graph of y = x3 at 2! 3! n! x= [Ans: 0.773] ∞ P

an xn is a Taylor series that converges to f (x) for all real x, n=0 P∞ then f 0 (1) = [Ans: n=1 nan ]

5.B-16 (BC98) If

5.B-17 (BC97) Let f be the function given by fh (x) = ln (3 − x). The third-degreei Taylor polynomial for f about x = 2 is Ans: − (x − 2) −

(x−2)2 2

(x−2)3 3

5.B-18 (BC93) The coefficient of x6 in the Taylor series expansion about x = 0   2 for f (x) = sin x is Ans: − 61  5.B-19 (BC97) If f is a function such that f 0 (x) = sin x2 , then the coefficient 1 of x7 in the Taylor series for f (x) about x = 0 is Ans: − 42   5.B-20 [4] The coefficient of x3 in the Taylor series for e2x at x = 0 is Ans: 43 5.B-21 (BC98) Let f be a function that has derivatives of all orders for all real numbers. Assume f (0) = 5, f 0 (0) = −3, f 00 (0) = 1, and f 000 (0) = 4. (a) Write the third-degree Taylor polynomial for f about x = 0 and use  it to approximate f (0.2). Ans: 5 − 3x + 21 x2 + 23 x3 , 4.425 (b) Write the fourth-degree Taylor polynomial for g, where g(x) = f (x2 ), about x = 0. Ans: 5 − 3x2 + 12 x4 Rx (c) Write the third-degree Taylor polynomial for h, where h = f (t) dt, 0  3 2 about x = 0. Ans: 5x − 2 x + 16 x3 (d) Let h be defined as above. Given that f (1) = 3, either find the exact value of h(1) or explain why it cannot be determined.

Mr. Budd, compiled September 29, 2010


140

HL Unit 5 (Taylor Series)

Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 3: Interval of Convergence

5.3

141

Interval of Convergence

Advanced Placement Series of constants. • The ratio test for convergence and divergence. Taylor series • Radius and interval of convergence of power series. International Baccalaureate 10.2 Convergence of infinite series. Tests for convergence: ratio test. 10.4 Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

Textbook §8.6 Power Series [15] Resources §12-6 Interval of Convergence for a Series - The Ratio Technique in [12]. Exploration 12-6: “Introduction to the Ration Technique” in [11]

5.3.1

Ratio Test

Theorem 5.1 (Ratio Test).

un+1

< 1. 1. un converges absolutely if lim

n→∞ un

un+1

un+1

P

=∞ 2. un diverges if lim

> 1 or lim

n→∞ n→∞ un

un

un+1

=1 3. The Ratio Test fails if lim

n→∞ un

∞ P

Example 5.3.1 [14] Determine the convergence or divergence of ∞ 1 P . n! Mr. Budd, compiled September 29, 2010


142

HL Unit 5 (Taylor Series) [Ans: converges]

Example 5.3.2 [14] Determine the convergence or divergence of ∞ nn P n! [Ans: diverges]

5.3.2

Radius and Interval of Convergence

The interval of convergence for a power series is the values of x for which the series converges. In finding the open-ended interval of convergence, we don’t worry about convergence at the endpoints of the interval. If the open-ended interval of convergence for a power series centered at c is x ∈ (c − k, c + k), i.e., |x| < k, then k is called the radius of convergence.

Example 5.3.3 Find the interval and radius of convergence for the 1 . series for 1−x Frequently, we use the ratio technique to find the interval of convergence.

Example 5.3.4 [4] The radius of convergence of the series x + 2x2 6x3 n!xn + + · · · + + · · · is 22 33 nn (a) ∞ (b) e2 (c) e e (d) 2 (e) 0

[Ans: e]

Example 5.3.5 Find the open-ended interval of convergence for the Taylor series of ln x expanded about x = 1. (Find the open-ended interval of convergence means you don’t need to test the end-points). Mr. Budd, compiled September 29, 2010


HL Unit 5, Day 3: Interval of Convergence

143

Example 5.3.6 Find the radius of convergence of ex . ∞ P n Example 5.3.7 Show that, for non-skipping power series bn (x − c) , where bn are the coefficients, the radius of convergence, R is given by

bn

R = lim

n→∞ bn+1

5.3.3

Testing Endpoints

Example 5.3.8 (BC93) The interval of convergence of

∞ (x − 1)n P 3n n=0

is (a) −3 < x ≤ 3 (b) −3 ≤ x ≤ 3 (c) −2 < x < 4 (d) −2 ≤ x < 4 (e) 0 ≤ x ≤ 2

[Ans: −2 < x < 4]

Problems 5.C-1 Find the radius of convergence for the Maclaurin series for ex , sin x, and cos x. ∞ (−1)n+1 (x − 2)n P ? What is n · 3n n=1 the open-ended interval of convergence? [Ans: 3, (−1, 5)]

5.C-2 [4] What is the radius of convergence of

n ∞ n+1 P (x − 3) · ? What is 2n n=1 2n + 1 the open-ended interval of convergence? [Ans: 2, (1, 5)]

5.C-3 [4] What is the radius of convergence of

xn ? n n=0 (n + 1) 3 [Ans: 3, (−3, 3)]

5.C-4 [4] What is the radius of convergence of the power series What is the open-ended interval of convergence?

∞ P

x x2 x3 5.C-5 [4] What is the radius of convergence of the series + 2 + 3 + · · · + 4 4 4 xn + ··· [Ans: 4] 4n Mr. Budd, compiled September 29, 2010


144

HL Unit 5 (Taylor Series)

5.C-6 [4] Let f be the function defined by the power series f (x) = an−1 where a0 = 1 and an = for n ≥ 1. n

∞ P

an x2n

n=0

(a) Write the first five terms of the series and the general term. (b) Determine the radius of convergence for the series above. Show your reasoning. (c) Show that f 0 (x) = 2xf (x). h 6 4 Ans: 1 + x2 + x2 + x6 +

x8 24

+ ··· +

5.C-7 The Maclaurin Series for f (x) is given by (−1)n x2n + ··· (2n + 2)!

x2n n!

+ · · · , converges for all x

i

1 x2 x4 x6 − + − + ··· + 2! 4! 6! 8!

(a) For what values of x does the given series converge? (b) Let g 0 (x) = 1 − x2 f (x). Write the Maclaurin series for g 0 (x), showing the first three nonzero terms and the general term. (c) Write g 0 (x) in terms of a familiar function without using series. Then write f (x) in terms of the same familiar function. (d) Given that g(0) = 3, write g(x) in terms of a familiar function without series.   2 4 (−1)n (x2n ) 1−cos x 0 Ans: all x, 1 − x2! + x4! − · · · + − · · · , g (x) = cos x, f (x) = , g(x) = sin x + 3 2 (2n)! x

Mr. Budd, compiled September 29, 2010


Unit 6

Series of Constants 1. The Integral Test and p-Series 2. Comparison Tests 3. Alternating Series Advanced Placement Concept of Series. A series is defined as a sequence of partial sums and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. Series of Constants • Motivating examples, including decimal expansion. • Geometric series with applications. • The harmonic series. • Alternating series with error bound. • Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series. • The ratio test for convergence and divergence. • Comparing series to test for convergence or divergence. Taylor series 145


146

HL Unit 6 (Series of Constants) • Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) • Maclaurin series and the general Taylor series centered at x = a. • Maclaurin series for the functions ex , sin x, cos x, and

1 . 1−x

• Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series. • Functions defined by power series. • Radius and interval of convergence of power series. • Lagrange error bound for Taylor polynomials. International Baccalaureate 1.1 Geometric sequences and series; sum of finite and infinite geometric series. Applications of the above. Included: sigma notation, applications of sequences and series to compound interest and population growth. 12.1 Convergence of infinite series. Tests for convergence: ratio test; limit comparison test; integral test. P Included: conditions for the application of these tests, the divergence theorem, if un is a convergent series then lim un = 0. n→∞ 12.2 Alternating series. Conditional convergence. Included: knowledge that the absolute value of the truncation error is less than the next term in the series; absolute convergence of an infinite series. 12.3 Power series: radius of convergence. Determination of the radius of convergence by the ratio test. Included: power series in (x − k), k 6= 0. 12.5 Use of Taylor series expansions, including the error term. Maclaurin series as a special case. Taylor polynomials. Taylor series by multiplication. Included: application to the approximation of functions; bounds on the error term. Included: 2 finding the Taylor approximations for functions such as ex arctan x by multiplying 2 the Taylor approximations for ex and arctan x.

Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 1: The Integral Test and p-Series

6.1

147

The Integral Test and p-Series

Advanced Placement Series of Constants • The harmonic series. • Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series. International Baccalaureate 10.2 Convergence of infinite series. Tests for convergence: integral test. Included: conditions for the application of these tests. P 1 P 1 ; is convergent for p > 1 and divergent otherwise. When The p–series, np np p = 1, this is the harmonic series. Use of integrals to estimate sums of series. Textbook §8.3 The Integral Test and Comparison Tests [15] Resources §12-7 Convergence of Series at the Ends of the Convergence Interval in [12]. Exploration 12-7a: “Improper Integrals to Test for Convergence” in [11]

6.1.1

Integral Test

Theorem 6.1. [20] Suppose f is a continuous, positive, decreasing function on ∞ P [1, ∞) and let an = f (n). Then the series an is convergent if and only if the n=1 Z ∞ improper integral f (x) dx is convergent. 1

In other words: Z

1. If

f (x) dx is convergent, then Z

f (x) dx is divergent, then 1

f (n) is convergent.

n=1

1

2. If

∞ P

∞ P

f (n) is divergent.

n=1

Notice the three requirements: f (x) must be Mr. Budd, compiled September 29, 2010


148

HL Unit 6 (Series of Constants)

1. continuous; 2. positive; 3. decreasing. All three criteria must be shown before the integral test may be applied.

Example 6.1.1 Consider the series

∞ 1 P . p n=1 n

(a) Will the series converge if p < 0? Explain. (b) Will the series converge if p = 0? Explain. (c) Will the series converge if 0 < p < 1? Explain. (d) Will the series converge if p = 1? Explain. (e) Will the series converge if p > 1? Explain.

6.1.2

p–Series

Example 6.1.2 Determine the convergence of 4+1+

4 1 4 + + + ··· 9 4 25

Problems R b dx p b→∞ 1 x

6.A-1 (BC98) If lim

is finite, then which of the following must be true?

(A)

∞ 1 P converges p n=1 n

(B)

∞ 1 P diverges p n=1 n

(C) (D) (E)

∞ P

1 converges p−2 n n=1 ∞ P

1 converges p−1 n n=1 ∞ P

1

n=1

np+1

diverges

Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 1: The Integral Test and p-Series

149

6.A-2 (HL May ’06) Use the integral test to determine the set of values of k for which the series ∞ X 1 k

n=2

n (ln n)

(a) is convergent; (b) is divergent. 6.A-3 [20] Determine the convergence of 1+

1 1 1 1 + + + + ··· 8 27 64 125 [Ans: C; p = 3 > 1]

6.A-4 (HL May ’06) Given that

∞ P n=1

un is convergent, it can be shown that

∞ P n=1

u2n

is also convergent. State, with a reason, whether or not the converse of this result is true.

Mr. Budd, compiled September 29, 2010


150

HL Unit 6 (Series of Constants)

Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 2: Comparison Tests

6.2

151

Comparison Tests

Advanced Placement Series of Constants • Comparing series to test for convergence or divergence. International Baccalaureate 10.2 Convergence of infinite series. Tests for convergence: comparison test; limit comparison test. Included: conditions for the application of these tests.

Textbook §8.3 The Integral Test and Comparison Tests [15] Resources §12-7 Convergence of Series at the Ends of the Convergence Interval in [12]. Exploration 12-7a: “Improper Integrals to Test for Convergence” in [11]

6.2.1

Direct Comparison Test

Theorem 6.2 (Direct Comparison Test for Positive Series). [17] Suppose that ∞ ∞ P P for all k ≥ 1, 0 ≤ ak ≤ bk . Consider the two series ak and bk . k=1

1. If

∞ P

bk converges, so does

k=1

∞ P

ak , and

k=1 ∞ X

ak ≤

k=1

2. If

∞ P

k=1

ak diverges, so does

k=1

∞ P

∞ X

bk

k=1

bk .

k=1

Out of the a-series and b-series, one is the series that you are testing, the other is a series whose convergence is easily known (e.g., geometric series or p-series)

Example 6.2.1 [20] Does

∞ ln n P converge? Determine with two n=2 n

different tests. Mr. Budd, compiled September 29, 2010


152

HL Unit 6 (Series of Constants)

Example 6.2.2 [17] Consider S =

∞ P k=0

2k

1 +1

(a) Does the series converge? (b) Find the sum of the first ten, twenty, and one hundred terms. (c) Use a geometric series to determine how closely does S100 ≈ 1.264499781 approximates the true limit S? Why would you need an overapproximation? ∞ P 1 ? (d) What happens if we test k −1 2 k=0

Example 6.2.3 (HL May ’07) Consider the infinite series

∞ P

1 . n (n + 2) n=1

(a) Show that the series is convergent. 1 (b) (a) Express in partial fractions. n (n + 2) ∞ P 1 . (b) Hence find n (n + 2) n=1

h

6.2.2

Ans:

1 2n

1 3 2(n+2) ; 4

i

Limit Comparison Test

Theorem 6.3 (Limit Comparison Test for Positive Series). [20] Suppose that ∞ ∞ P P ak and bk are series with positive terms. If k=1

k=1

lim

k→∞

an =c bn

where c is a finite number and c > 0, then either both series converge or both diverge. Why can c not be negative? Note that if c is zero or infinite, then you picked the wrong series for comparison. Basically, if the limit of the ratio of the terms is neither 0 nor the reciprocal of 0, then the series are comparable. Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 2: Comparison Tests

153

Example 6.2.4 [20] Determine the convergence of ∞ X k=0

1 2k − 1

Example 6.2.5 [20] Determine the convergence of

∞ P n=1

sin

  1 n

∞ P

5 converge? Try different 2 + 4n + 3 2n n=1 ∞ ∞ ∞ P 1 P 2 P 1 comparisons, such as n2 , n2 , and n3 .

Example 6.2.6 [20] Does

Problems 6.B-1 Does

∞ P

1 converge? k=1 10k + 1

6.B-2 (HL May ’06) Given that ∞ P n=1

∞ P

[Ans: no] un is convergent, where un ≥ 0, prove that

n=1

u2n is also convergent.

Mr. Budd, compiled September 29, 2010


154

HL Unit 6 (Series of Constants)

Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 3: Alternating Series

6.3

155

Alternating Series

Advanced Placement Series of Constants • Alternating series with error bound. International Baccalaureate 10.3 Series that converge absolutely. Series that converge conditionally. Alternating series. Conditions for convergence. The absolute value of the truncation error is less than the next term in the series.

Textbook §8.4 Alternating Series [15]

6.3.1

Alternating Series

Example 6.3.1 (BC98) What are all values of x for which the ∞ (x + 2)n P √ series converges? n n=1 [Ans: −3 ≤ x < −1]

6.3.2

Problems ∞ n3n P n n=1 x [Ans: |x| > 3]

6.C-1 (BC acorn ’02) What are all values of x for which the series converges? 6.C-2 (BC97) What are all values of x for which the series verges?

∞ (x − 2)n P conn3n n=1 [Ans: −1 ≤ x < 5]

6.C-3 [4] Let f be the function defined by f (x) = ln (x + 1). (a) Find f (n) (0) for n = 1 for n = 3, where f (n) is the nth derivative of f. (b) Write the first three nonzero terms and the general term for Taylor series expansion of f (x) about x = 0. (c) Determine the radius of convergence for the series above. Show your reasoning. Mr. Budd, compiled September 29, 2010


156

HL Unit 6 (Series of Constants) R 0.5 (d) Use the series above to evaluate 0 f (x) dx with an error no greater than 0.001.   x2 x3 x4 (−1)n+1 xn Ans: f 0 (0) = 1, f 00 (0) = −1, f 000 (0) = 2, x − + − + ··· + + · · · , 1, 0.108 2 3 4 n

6.C-4 (HL00) (a) Use the ratio test to calculate the radius of convergence of the power ∞ (x − 5)n P series 3 n2 n=1 (b) Using your result from above, determine all points x where the power series above converges. [Ans: radius of 1, 4 ≤ x ≤ 6] π ∞ P 6.C-5 (HL May ’07) Find the interval of convergence of the series xn . sin n n=1 [Ans: −1 ≤ x < 1]

Mr. Budd, compiled September 29, 2010


HL Unit 6, Day 4: Convergence Test Review

6.4

157

Convergence Test Review

6.D-1 (BC98) Which of the following series converge? ∞ P

n n + 2 n=1 ∞ cos nπ P II. n n=1 ∞ 1 P III. n=1 n I.

∞ (−1)kn P n n=1

6.D-2 (BC98) For what integer k, k > 1, will both converge?

[Ans: II only]  n ∞ P k and n=1 4 [Ans: 3]

6.D-3 (BC93) Which of the following series diverge? ∞ P

2 2+1 k k=3  k ∞ P 6 II. k=1 7 I.

III.

∞ (−1)k P k k=2

[Ans: None] 6.D-4 [4] Which of the following series are convergent? 1 1 1 + 2 + ... + 2 + ... 22 3 n (−1)n 1 1 II. 1 − + − . . . + + ... 2 3 n n 8 2 III. 2 + 1 + + . . . + 2 + . . . 9 n I. 1 +

[Ans: I and II only] 6.D-5 (HL01) Test the convergence or divergence of the following series (a)

∞ P

(b)

n=1 ∞ P n=1

sin

1 ; n

(n + 10)

 cos nπ  n1.4

6.D-6 (HL99) Test the convergence or divergence of the following infinite series, indicating the tests used to arrive at your conclusion: Mr. Budd, compiled September 29, 2010


158

HL Unit 6 (Series of Constants)

(a) (b)

∞ k+1 P k k=1 3 ∞ P 1 k=2

(c)

∞ P

3

k (ln k)

(−1)k+1

k=1

k k2 + 1

6.D-7 (HL98) Determine whether each of the following series converges or diverges. State clearly which test of convergence or divergence you use. (a)

∞ P

(b)

k=1 ∞ P

(−1)k

k=1

(c)

∞ P

1 3k 2 − 2k

(−1)k

k=1

2k 4k − 3

(k + 1)! (k + 1)

3

[Ans: D (nth term), C (Comp), D (Ratio)]

Mr. Budd, compiled September 29, 2010


Unit 7

Differential Equations 1. Logistic Equation 2. Slope Fields 3. Euler’s Method 4. Homogeneous Differential Equations

Advanced Placement

I. Derivatives Applications of derivatives. • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. • Numerical solution of differential equations using Euler’s method. II. Integrals Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth. • Solving logistic differential equations and using them in modeling. 159


160

HL Unit 7 (Differential Equations)

International Baccalaureate (HL 8.9) Anti-differentiation with a boundary condition to determine the constant term. (HL 8.10) Further integration: integration by parts. (HL 8.11) Solution of first order differential equations by separation of variables. Included: transformation of a homogeneous equation by the substitution y = νx

Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 1: Euler’s Method

7.1

161

Euler’s Method

International Baccalaureate dy = f (x, y) using Euler’s method. yn+1 = yn + h × 10.6 Numerical solution of dx f (xn , yn ); xn+1 = xn + h, where h is a constant. Advanced Placement Applications of derivatives. • Numerical solution of differential equations using Euler’s method. Resources Explorations 7–5a: “Introduction to Euler’s Method” and 7–6a: “Logistic Function- The Roadrunner Problem” in [11]. §12.3 Euler’s Method: Solving DE’s Numerically in [17] Example 7.1.1 (BC98) Consider the differential equation given by xy dy = . Refer to Figure 2.1 and problem 2.3.2. dx 2 (a) Let y = f (x) be the particular solution to the given differential equation with the initial condition f (0) = 3. Use Euler’s method starting at x = 0, with a step size of 0.1, to approximate f (0.2). Show the work that leads to your answer. (b) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = 3. Use your solution to find f (0.2) dy Example 7.1.2 (BC02) Consider the differential equation = dx 2y − 4x. Let f be the function that satisfies the given differential equation with the initial condition f (0) = 1. Use Euler’s method, starting at x = 0 with a step size of 0.1, to approximate f (0.2). Show the work that leads to your answer.

[Ans: 1.4] Example 7.1.3 (BC acorn) Let y = f (x) be the solution to the dy differential equation = arcsin (xy) with initial condition f (0) = dx 2. What is the approximation for f (1) if Euler’s method is used, starting at x = 0 with a step size of 0.5? Mr. Budd, compiled September 29, 2010


162

HL Unit 7 (Differential Equations)   Ans: 2 + π4

Problems §16.3: 40-42 in textbook [15]. 7.A-1 [4] Suppose a continuous function f and its derivative f 0 have values as given in the following table. Given that f (1) = 2, use Euler’s method to approximate the value of f (2). x f 0 (x) f (x)

1.0 0.4 2.0

1.5 0.6

2.0 0.8

[Ans: 2.5] dy 7.A-2 [4] The slope field for the differential equation = 0.5xy, 0 ≤ x ≤ 5 and dx 0 ≤ y ≤ 4, is shown in Figure 7.1. Figure 7.1: [4]

(a) Sketch the solution curve that satisfies the initial condition y(0) = 2 on the slope field. (b) Solve the differential equation and find the particular solution thati h 2 contains the point (0, 2). Ans: y = 2ex /4 (c) Use the function in part 2b to calculate the exact value of y when x = 2. [Ans: 2e ≈ 5.437] Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 1: Euler’s Method

163

(d) Starting at the point (0, 2), use Euler’s method with 2 steps of size 1 to estimate y(2). How does this value compare with the exact value in part 2c. [Ans: 3] 1 dy = and y(0) = 0. An approxidx x+1 mation of y(1) using Euler’s method with two steps and step size 4x   = 0.5 is Ans: 65

7.A-3 [4] Given the differential equation

dy = x + y and y(0) = 2. An approxidx mation of y(1) using Euler’s method with two steps and step size 4x = 0.5 is Ans: 19 4

7.A-4 [4] Given the differential equation

dy = dx 4x + y. An approximation to y(2) using Euler’s Method with two equal steps is [Ans: 6]

7.A-5 [4] The curve passing through (1, 0) satisfies the differential equation

7.A-6 [4] Let f be a function such that f 0 (x) =

2x + sin x and f (2) = −1. x2 + 1

(a) Use Euler’s method with 3 equal steps to approximate f (2.3). [Ans: −0.719] (b) Use f 00 (x) to show that your approximation in 6a is an overestimate of f (2.3). (c) Use the tangent line to the graph of y = f (x) at the point (2, −1) to approximate f (2.3). [Ans: −0.705] (d) Use a definite integral to express the exact value of f (2.3). [Ans: −0.726]

Mr. Budd, compiled September 29, 2010


164

HL Unit 7 (Differential Equations)

Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 2: Logistic Equation

7.2

165

Logistic Equation

Advanced Placement Applications of antidifferentiation. • Finding specific antiderivatives using initial conditions, including applications to motion along a line. • Solving separable differential equations and using them in modeling. In particular, studying the equation y 0 = ky and exponential growth. • Solving logistic differential equations and using them in modeling. International Baccalaureate 7.5 Anti-differentiation with a boundary condition to determine the constant term. 7.10 Solution of first order differential equations by separation of variables.

The Idea With unlimited growth, the rate of change of a population is proportional to the population dP ∝P dt dP =P dt In actuality, unlimited growth is only experienced in rare cases where resources such as food and space are unlimited. The logistics model of growth is based on the rate of change being proportional to the population, but also on the difference between the population and some maximum population, L, known as the carrying capacity, or limiting population. dP ∝ P, L − P dt dP = kP (L − P ) dt   dP P = KP 1 − dt L

where K = kL. Mr. Budd, compiled September 29, 2010


166

HL Unit 7 (Differential Equations)

What happens to the population if P > L? If P = 0? If 0 < P < L? Example 7.2.1 Draw an appropriate slope field for a logistic differential equation.

Carrying Capacity, or Limiting Population Examine the following problem using a program for Euler’s method, and look to see how you could do it simply, without using a program. Example 7.2.2 (adapted from BC98) The population  P (t) of a P dP =P 5− , species satisfies the logistic differential equation dt 3000 where the initial population P (0) = 2000 and t is the time in years. What is lim P (t)? t→∞

[Ans: 15000]

Maximum rate of growth Example 7.2.3 Show that the maximum rate of growth occurs when P = 12 L.

Solution Example 7.2.4 Show that the solution to the differential equation   P dP = KP 1 − dt L is P (t) =

L L − P0 , where a = 1 + ae−Kt P0

Problems 7.B-1 [4] If a population of wolves grows according to the differential equation dN = 0.05N − 0.0005N 2 dt Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 2: Logistic Equation

167

where N is the number of wolves and t is measured in years, then lim N (t) = t→∞

[Ans: 100] 7.B-2 (adapted from BC ’03) The number of moose in a national park is modeled dM by the function M that satisfies the logistic differential equation = dt   M 0.5M 1 − , where t is the time in years and M (0) = 50. Find 180 (a) lim M (t); t→∞

(b) lim M 0 (t); t→∞

(c) the maximum rate of growth in the moose population. [Ans: 180 moose; 0; 22.5 moose per year] 7.B-3 [4] Suppose a population of bears grows according to the logistic differential equation dP = 2P − 0.01P 2 dt where P is the number of bears at time t in years. Which of the following statements are true? I. The growth rate of the bear population is greatest at P = 100. II. If P > 200, the population of bears is decreasing. III. lim P (t) = 200 t→∞

[Ans: I, II, III] dy = f (x, y) is given in Figure 7.B-4 [4] A slope field for a differential equation dx 7.2. Which of the following statements are true? dy at the point (3, 3) is approximately 1. dx II. As y approaches 8 the rate of change of y approaches zero. I. The value of

III. All solution curves for the differential equation have the same slope for a given value of x. [Ans: I and II only] 7.B-5 (adapted from BC ’04) A population is modeled by a function P that satisfies the logistic differential equation   dP P P = 1− dt 5 12 Mr. Budd, compiled September 29, 2010


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Figure 7.2: Slope field for problem 4

(a) If P (0) = 3, what is lim P (t)? If P (0) = 3, what is lim P (t)? If t→∞

t→∞

P (0) = 0, what is lim P (t)? t→∞

(b) If P (0) = 3, for what value of P is the population growing the fastest? (c) Find P (t) if P (0) = 3. (d) A different population is modeled by a function Y that satisfies the separable differential equation   dY Y t = 1− dt 5 12 Find Y (t) if Y (0) = 3. (e) What is lim ? t→∞

h

Ans: 12, 12, 0; 6; P (t) =

12 1+3e−t/5

t

t2

;Y (t) = 3e 5 − 120 ; 0

i

Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 3: Homogeneous Differential Equations

7.3

169

Homogeneous Differential Equations

International Baccalaureate 7.5 Anti-differentiation with a boundary condition to determine the constant term. 10.6 Solution of first order differential equations by separation of variables. Included: transformation of a homogeneous equation by the substitution y = νx Textbook §16.2 Separation of Variables in First-Order Equations: “Homogeneous Differential Equations” [15] Resources §2.10 Homogeneous Equations in [5] Homogeneous equations are those that can be written as y dy =F dx x Also, y 0 = f (x, y) is homogeneous if f (x, y) is such that f (tx, ty) = f (x, y) y . Then x y dy =F dx x dν ν+x = F (ν) dx

Use the substitution y = νx or ν =

Example 7.3.1 After showing that it is homogeneous, solve the differential equation [5] dy y 2 + 2xy = dx x2 cx2 1−cx

i

2x Ans: − x+y = ln c |x + y|

i

h

Ans: y =

Example 7.3.2 After showing that it is homogeneous, solve [5] dy x + 3y = dx x−y h

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170

HL Unit 7 (Differential Equations) Example 7.3.3 Pick a function T (x) and solve the differential equation dy T (x) = y − x dx using the substitution y = νx Example 7.3.4 (HL May ’06 F1.06) Consider the differential equady 3x2 + y 2 tion = where x, y > 0. dx xy (a) Show that the differential equation is homogeneous (b) Find the general solution of the differential equation, giving your answer in the form y 2 = f (x). (c) Solve the differential equation, given that y = 2 when x = 1.   Ans: y 2 = 6x2 ln x + kx2 ; y 2 = 6x2 ln x + 4x2

Problems dy y = by separation of variables, then using the substitution dx x y = xν. [Ans: y = Ax]

7.C-1 Solve

7.C-2 (HL 05/02) The function y = f (x) satisfies the differential equation 2x2

dy = x2 + y 2 dx

(x > 0)

(a) Using the substitution y = νx, show that 2x

dν 2 = (ν − 1) . dx

(b) Hence show that the solution of the original differential equation is 2x y =x− , where c is an arbitrary constant. (ln x + c) (c) Find the value of c given that y = 2 when x = 1. 7.C-3 (HL 11/02) Consider the differential equation

dy 3y 2 + x2 = , for x > 0. dx 2xy

3ν 2 + 1 dν = . dx 2ν (b) Hence find the solution of the differential equation, given that y = 2 when x = 1. (a) Use the substitution y = νx to show that ν + x

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HL Unit 7, Day 3: Homogeneous Differential Equations

171   √ Ans: y = x 5x − 1

7.C-4 (HL 11/03) Use the substitution y = xν to show that the general solution  dy to the differential equation, x2 + y 2 +2xy = 0, x > 0 is x3 +3xy 2 = k, dx where k is a constant. 7.C-5 Find the solution to

dy x+y = . [5] dx x

[Ans: y = cx + x ln |x|]   Ans: y = cx2

7.C-6 Find the solution to 2y dx − x dy = 0. [5]

  dy x2 + 3y 2 = . [5] Ans: x2 + y 2 − cx3 = 0 dx 2xy i h  x + ln |x| = c 7.C-8 Find the solution to x2 + 3xy + y 2 dx−x2 dy = 0. [5] Ans: x+y

7.C-7 Find the solution to

    7.C-9 Find the solution of 3xy + y 2 dx+ x2 + xy dy = 0. [5] Ans: c = x2 y 2 + 2x3 y 7.C-10 (HL Nov ’06) (a) Show that

d dx



 ln

1+x 1−x

 =

2 , |x| < 1. 1 − x2

(b) Find the solution to the homogeneous differential equation x2 given that y =

dy = x2 + xy − y 2 dx

1 when x = 1. Give your answer in the form y = g(x). 2   3x3 − x Ans: y = 3x2 + 1

7.C-11 (HL Nov ’08) Show that the solution of the homogeneous differential equation dy y = + 1, x > 0 dx x given that y = 0 when x = e, is y = x (ln x − 1)

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HL Unit 7, Day 4: Linear DiffEq’s and Integrating Factors

7.4

173

Linear DiffEq’s and Integrating Factors

International Baccalaureate 7.5 Anti-differentiation with a boundary condition to determine the constant term. 10.6 Solution of y 0 + P (x)y = Q(x), using the integrating factor.

Resources §8.8 Differential Equations: First-Order Linear Equations in [19]

Example 7.4.1 For the following µ(x)/P (x) function pairs, find the following: •

d (µ(x) · y) dx

• µ(x) (P (x)y) Z • P (x) dx • e

R

P (x) dx

2

(a) µ(x) = ex , P (x) = 2x 3

(b) µ(x) = ex , P (x) = 3x2 (c) µ(x) = x, P (x) =

1 x

(d) µ(x) = sec x, P (x) = tan x (e) µ(x) =

x+1 1 , P (x) = x+2 (x + 1) (x + 2)

2

Example 7.4.2 [19] Show that y = 1 + Ce−x /2 , C any constant, is a solution of the first-order differential equation y 0 + xy = x

The integrating factor for solving y 0 + P (x)y = Q(x) I(x) = e

R

P (x) dx

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Note that I 0 (x) = I(x) ·

d dx

R

 P (x) dx = I(x)P (x), so

y 0 + P (x)y = Q(x) I(x)y 0 + I(x)P (x)y = I(x)Q(x) I(x)y 0 + I 0 (x)y = I(x)Q(x) d (I(x)y) = I(x)Q(x) dx Z I(x)y =

I(x)Q(x) dx

Example 7.4.3 Use an integrating factor to solve

dy + xy = x dx

Example 7.4.4 Pick a function T (x) and solve the differential equation dy T (x) = y − x dx using an integrating factor Example 7.4.5 (HL Spec ’05) Consider the differential equation dy xy + = 1, where |x| < 2 and y = 1 when x = 0. dx 4 − x2 (a) Use Euler’s method with h = 0.25 to find an approximate value of y when x − 1, giving your answer to two decimal places. (b) By first finding an integrating factor, solve this differential equation. Give your answer in the form y = f (x). (c) Calculate, correct to two decimal places, the value of y when x = 1. (d) Sketch the graph of y = f (x) for 0 ≤ x ≤ 1. Use your sketch to explain why your approximate value of y is greater than the true value of y. √  Ans: 1.84; y = 4 − x2 arcsin

x 2



+

1 2



 ; 1.77; think concavity

Example 7.4.6 (HL May ’08 Z2) A curve that passes through the point (1, 2) is defined by the differential equation  dy = 2x 1 + x2 − y dx Mr. Budd, compiled September 29, 2010


HL Unit 7, Day 4: Linear DiffEq’s and Integrating Factors

175

(a) (a) Use Euler’s method to get an approximate value of y when x = 1.3, taking steps of 0.1. Show intermediate steps to four decimal places in a table. (b) How can a more accurate answer be obtained using Euler’s method? (b) Solve the differential equation giving your answer in the form y = f (x). h

2

Ans: 2.141; ; y = x2 + e1−x

i

Problems 7.D-1 Solve

y dy = using an integrating factor. dx x

[Ans: y = Ax]

7.D-2 (HL Nov ’08) Show, that the solution of the homogeneous differential equation dy y = + 1, x > 0 dx x given that y = 0 when x = e, is y = x (ln x − 1). Do this (a) using the substitution ν =

y , then again, but x

(b) using an integrating factor. 7.D-3 (HL 11/99) Solve the differential equation dy = y tan x + 1, dx

0≤x≤

if y = 1 when x = 0.

π , 2 [Ans: y = tan x + sec x]

7.D-4 [19] Consider again the logistic differential equation dy = ky (L − y) dx (a) Show that

dy − kLy = −ky 2 ; dx

(b) Show that the substitution ν =

1 yields the differential equation y

dν + kM ν = k; dt (c) Solve the ‘nu’ differential equation (hee, hee) to get ν = L and hence y = 1 + ae−kLt

ekLt + C2 , LekLt

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7.D-5 (HL M08 F1.05, whatever that means) Solve the following differential equation dy +y =x+1 (x + 1) (x + 2) dx h  i x+2 1 giving your answer in the form y = f (x) Ans: x+1 ln |x + 2| + x+1 +C 7.D-6 (HL May ’08 Z1) Consider the differential equation x

x3 dy − 2y = 2 dx x +1

(a) Find an integrating factor for this differential equation. (b) Solve the differential equation given that y = 1 when x = 1, giving your answer in the form y = f (x).   Ans: x12 ; y = x2 arctan x + 1 − π4 7.D-7 (HL May ’07 Z1) (a) Use integration by parts to show that Z sin x cos xe− sin x dx = −e− sin x (1 + sin x) + C

Consider the differential equation

dy − y cos x = sin x cos x. dx

(b) Find an integrating factor. (c) Solve the differential equation, given that y = −2 when x = 0. Give your answer in the form y = f (x).   Ans: e− sin x , − sin x − 1 − esin x 7.D-8 (HL May ’06) (a) Find an integrating factor for solving the differential equation dy + y tan x = sec x dx (b) Solve this differential equation given that y = 2 when x = 0. Give your answer in the form y = f (x).   x+2 Ans: sec x; y = tan sec x dy 7.D-9 (HL May ’09) The variables x and y are related by − y tan x = cos x. dx Solve the differential equation given that y = 0 when x = π. Give the solution in the form y = f (x). Ans: 21 (sin x + (x − π) sec x)

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HL Unit 7, Day 5: Differential Equation Review Problems

7.5

177

Differential Equation Review Problems

7.E-1 (BC01) Let f be the function satisfying f 0 (x) = −3xf (x), for all real numbers x, with f (1) = 4 and lim f (x) = 0. x→∞

R∞

(a) Evaluate [Ans: −4]

1

−3xf (x) dx. Show the work that leads to your answer.

(b) Use Euler’s method, starting at x = 1 with a step size of 0.5, to approximate f (2). [Ans: 2.5] (c) Write an expression for y = f (x) by solving the differential equation h i dy 2 = −3xy with the initial condition f (1) = 4. Ans: y = 4e3/2 e−3x /2 dx 7.E-2 (BC99) Let f be the function whose graph goes through the point (3, 6) 1 + ex and whose derivative is given by f 0 (x) = . x2 (a) Write an equation of the line tangent to the graph of f at x = 3 and use f (3.1). h it to approximate i 3 Ans: y − 6 = 1+e (x − 3), f (3.1) ≈ 6.234 9 (b) Use Euler’s method, starting at x = 3 with a step size of 0.05, to approximate f (3.1). Use f 00 to explain why this approximation is less than f (3.1). [Ans: 6.236] R 3.1 0 (c) Use 3 f (x) dx to evaluate f (3.1). [Ans: 6.238] 7.E-3 [5] Solve the differential equation y 2 + 2xy dy = dx x2 [Ans: ]

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HL Unit 7 (Differential Equations)

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Part II

I.B. Topics

179


Unit 8

Vectors 1. Dot Product 2. Cross Product 3. Lines 4. Planes 5. Finding Distances with Dot Product 6. Finding Distances with Cross Product International Baccalaureate 5.1 Vectors as displacements in the plane and in three dimensions; Distance between points in three dimensions. Components of a vector; column representation; Components are with respect to the unit vectors i, j, k (standard basis). The sum and difference of two vectors; the zero vector; the vector, −v. Multiplication by a scalar, kv. Magnitude of a vector, |v|. Unit vectors; base vectors i, j, and k. ~ = a; AB ~ = OB ~ − OA ~ = b − a. Position vectors OA 5.2 The scalar product of two vectors v · w = |v| |w| cos θ; v · w = v1 w1 + v2 w2 + v3 w3 . Algebraic properties of the scalar product v · w = w · v; u · (v + w) = 2 u · v + u · w; (kv) · w = k (v · w);v · v = |v| . Perpendicular vectors; parallel vectors. Included: for non-zero perpendicular vectors v·w = 0; for non-zero parallel vectors v · w = ± |v| |w|. The angle between two vectors. 5.3 Vector equation of a line r = a+λb. Vector equation of a plane r = a+λb+µc. Use of normal vector obtain r·n = a·n. Cartesian equations of a line and plane. Inx − x0 y − y0 z − z0 cluded: cartesian equation of a line in three dimensions = = ; l m n cartesian equation of a plane ax + by + cz = d. 5.4 Coincident, parallel, intersecting, and skew lines, distinguishing between these 181


182

HL Unit 8 (Vectors)

cases. Points of intersection. 5.5 The vector product of two vectors |v × w| = |u| |w| sin θ. The formula for 1 the area of a triangle in the form |v × w|. Included: geometric interpretation of 2 the magnitude of v × w as the area of a parallelogram. Included: the determinant representation. 5.6 Vector equation of a plane 5.7 Intersection of: two lines; a line with a plane; two planes; three planes. Angle between: two lines; a line and a plane; two planes. 5.7 Distances in two and three dimensions between points, lines, and planes.

Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 1: Vector Basics

8.1

183

Vector Basics

International Baccalaureate 5.1 Vectors as displacements in the plane and in three dimensions; Distance between points in three dimensions. Components of a vector; column representation; Components are with respect to the unit vectors i, j, k (standard basis). The sum and difference of two vectors; the zero vector; the vector, −v. Multiplication by a scalar, kv. Magnitude of a vector, |v|. Unit vectors; base vectors i, j, and k. ~ = a; AB ~ = OB ~ − OA ~ = b − a. Position vectors OA

8.1.1

Vectors

Scalars • Have magnitude but no direction. • Just one number (which may be positive or negative.) • Example: mass, speed. Vectors • Have direction and magnitude. • Can be represented by two numbers (two dimensions), three numbers (three dimensions), or more. • Example: force, weight, velocity, acceleration.

Example 8.1.1 [6] Classify the following situations as needing to be described with a vector or a scalar. (a) A classroom chair is moved from the front of the room to the back of the room. (b) The balance in a bank account. (c) The electric current passing through an electric light tube. (d) A dog, out for a walk, is being restrained by a lead. Mr. Budd, compiled September 29, 2010


184

HL Unit 8 (Vectors) (e) An aircraft starts its takeoff run. (f) The wind conditions before a yacht race. (g) The amount of liquid in a jug. (h) The length of a car. (i) The time that it takes to boil an egg. (j) The number of goals scored in a soccer match.

8.1.2

Representation

Vectors can be represented 1. on a graph as a directed line segment (directed with an arrow at one end). 2. as a sum of components based on the unit vectors i and j(i moves one space to the right, j one space up), e.g., −i + 3j.   −1 3. as a column matrix or column vector, e.g. . 3 4. as an ordered pair, e.g. h−1, 3i.

8.1.3

Simple Arithmetic

Scalar multiples are intuitive. Example 8.1.2 [6] If a = 2i − j and b = −i + 3j, find (a) −a (b) −2a (c) 3b See the effects graphically. [Ans: −2i + j; −4i + 2j; −3i + 9j] Addition is intuitive. Example 8.1.3 Find a + b. Add analytically and graphically. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 1: Vector Basics

185 [Ans: i + 2j]

Subtraction is adding −1 times the vector being subtracted. Example 8.1.4 Find (a) b − a (b) 3b − 2a [Ans: −3i + 4j;−7i + 11j] 

   3 −2 Example 8.1.5 If p = −1 and q =  0 , find: 4 3 (a) p + q 3 (b) q − p 2 

   1 −6 Ans: −1;  1  7 0.5 There are two types of multiplication: • scalar multiplication (dot product), where the answer is a scalar; • vector multiplication (cross product), where the answer is a vector.

There is not really division with vectors. [Note:

8.1.4

q 1 is really q] 2 2

Magnitude

To get the magnitude of a vector, use the distance formula: p |ha, bi| = a2 + b2 p |ha, b, ci| = a2 + b2 + c2 Mr. Budd, compiled September 29, 2010


186

HL Unit 8 (Vectors) Example 8.1.6 Find the lengths of the vectors: (a) 3i − 4j (b) −i + 2j − 5k   3 (c) −1 2 √  √  Ans: 5; 30; 14

8.1.5

Making Unit Vectors

To make a unit vector in the direction of v, simply divide v by its own magniv tude, . |v| Example 8.1.7 (MM 5/99) The vectors ~i, ~j are unit vectors along the x-axis and y-axis respectively. The vectors ~u = −~i + 2~j and ~v = 3~i + 5~j are given. (a) Find ~u + 2~v in terms of ~i and ~j. A vector w ~ has the same direction as ~u + 2~v , and has a magnitude of 26. (b) Find w ~ in terms of ~i and ~j.

Problems 8.A-1 (MM 5/00) The vectors u, v are given by u = 3i + 5j, v = i − 2j. Find scalars a, b such that a (u + v) = 8i + (b − 2) j. [Ans: a = 2, b = 8]

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HL Unit 8, Day 1: Vector Basics

187

[ Dot Product]The Dot Product and the Angle Between Vectors International Baccalaureate 5.2 The scalar product of two vectors v · w = |v| |w| cos θ; v · w = v1 w1 + v2 w2 + v3 w3 . Algebraic properties of the scalar product v · w = w · v; u · (v + w) = u · v + u · w; 2 (kv) · w = k (v · w);v · v = |v| . Perpendicular vectors; parallel vectors. Included: for non-zero perpendicular vectors v · w = 0; for non-zero parallel vectors v · w = ± |v| |w|. The angle between two vectors.

8.1.6

Scalar Product

There are at least two ways that vectors can be multiplied. One of the ways is signified with a dot, ·, and the other, with a cross, ×. Dot multiplication yields a scalar answer:

ha, b, ci · hd, e, f i = ad + be + cf This is called dot multiplication, or, more formally, scalar multiplication. We will discuss cross multiplication, or vector multiplication, later. Note that the dot product is commutative, i.e., a · b = b · a. This may seem trivial, but the cross product not commutative. Example 8.1.8 For the following pairs of vectors, find the scalar product.     −1 −1 (a) and 3 2 (b) −5j + 4k and −5i − j − 3k [Ans: 7; −7]

8.1.7

Angle

Example 8.1.9 Use the law of cosines to show the formula for the angle between two vectors. Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors)

cos θ =

a·b |a| |b|

Note that if a and b are unit vectors, then the scalar product is the cosine of the angle between them. Example 8.1.10 For the following pairs of vectors, find the angle between the vectors, correct to the nearest degree.     −1 −1 (a) and 3 2 (b) −5j + 4k and −5i − j − 3k [Ans: 8; 101] Example 8.1.11 If two vectors are perpendicular, what is their dot product? [Ans: 0] Two vectors are said to be orthogonal if their dot product is 0. This is a fancy word for perpendicular. Two vectors are parallel if they are scalar multiples of each other, e.g., −i + 3j and 2i − 6j. If v and w are parallel, then v · w = ± |v| |w|, so long as neither is the zero vector. Example 8.1.12 [6] Three towns are joined by straight roads. Oakham is the state capital and is considered as the ‘origin’. Axthorp is 3 km east and 9 km north of Oakham, and Buddville is 5 km east and 5 km south of Axthorp. Considering i as a 1 km vector pointing east and j as a 1 km vector pointing north: (a) Find the position vector of Axthorp relative to Oakham. (b) Find the position vector of Buddville relative to Oakham. A light rail station (R) is situated two thirds of the way along the road from Oakham to Axthorp. −−→ −−→ (c) Find the vectors OR and BR. (d) Prove that the light rail station is the closest point to Buddville on the Oakham to Axthorp road. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 1: Vector Basics

8.1.8

189

Lines in Two Dimensions

Forms of a Line from Algebra I: • Point-Slope: y − y1 = m (x − x1 ) • Slope-Intercept: y = mx + b • Standard: Ax + By = C The first two have the advantage of using constants/ parameters with graphic meanings. The graphic advantages of using standard form are not all that apparent. The problem with these is that if we extend them into three dimensions, e.g. Ax + By + Cz = D, then they are no longer lines, but planes.   0 Example 8.1.13 Let a be the position vector , and let d be 28   6 the vector . On graph paper, graph the points that correspond −8 to the following position vectors: (a) a (b) a + d (c) a + 2d (d) a + 3d (e) a + 12 d (f) a − d (g) a − 2d (h) a − 3d (i) a − 12 d (j) a + 32 d (k) a + 0.75d (l) a + 0.875d (m) a plus every single possible multiple of d How can you draw an infinite number of points without drawing an infinite number of points? Mr. Budd, compiled September 29, 2010


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A line can be written in the form r = r0 + td     x x where r = , or y , a generic point on the line, r0 is a specific, known y z point on the line, and d gives the direction of the line. It is very easy to confuse the slope of a line and the direction vector of a line, but remember that slope is a scalar. Another vector equation of a line (in 2-D) uses the normal vector, n, to the line, instead of the direction vector, d. The normal vector is orthogonal to the line and its direction vector. If r represents a generic point on the line, and r0 is a specific point on the line, then the vector (r − r0 ) must lie in the line and have the same direction as the line. Since (r − r0 ) is in the direction of the line, it must be orthogonal to the normal vector n, so that n · (r − r0 ) = 0, or n · r = n · r0 . This gives way to the standard form of a line. A line (in 2-D) can also be written in the form ax + by = c   a where gives the normal vector of the line, i.e., a vector that is orthogonal b to the direction vector. To find c, just pick  a point (x0 , y0 ) on the line: c = a ax0 + by0 , so ax + by = ax0 + bx0 . If is a normal vector, then you can b obtain a quick direction   vector  simply by switching a and b, and making one of b −b them negative: or . −a a The descriptions above regarding normal vectors are only valid in two dimensions. In three dimensions, normal vectors are used to describe planes, not lines. Vector-Based Forms of a Line • Vector Form: r = r0 + λd • Parametric Form: x = x0 + λl, y = y0 + λm, z = z0 + λn • Cartesian Form:

y − y0 z − z0 x − x0 = = l m n

λ is considered the parameter. The parametric form is virtually identical to the vector form; each coordinate is listed separately, usually so that it can be Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 1: Vector Basics

191

written on one line. If each coordinate equation (x, y, and z) is solved for the parameter, λ, you get the Cartesian equation.

Example 8.1.14 Rewrite r =

    0 6 +t in all of the other 28 −8

forms of a line.

Problems 8.A-2 (MM 5/02) Two boats A and B start moving from the same point P. Boat A moves in a straight line at 20 km h−1 and boat B moves in a straight line at 32 km h−1 . The angle between their paths is 70◦ . Find the distance between the boats after 2.5 hours. [Ans: 78.5 km] 8.A-3 (HL 5/02) Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians. [Ans: 0.702]   1 8.A-4 (MM Spec ’99) Find the size of the angle between the two vectors 2   6 and . Give your answer to the nearest degree. [Ans: 117◦ ] −8 8.A-5 (HL 5/97) The coordinates of P and Q are (3, −1) and (λ, −4 − λ), respectively, where λ is a constant. If O is the origin, find all values of λ for −−→ −−→ which OP is perpendicular to OQ. [Ans: −1] 8.A-6 (MM 11/96) Triangle OAB has one vertex at the origin O. Vertices    A 5 2 ~ = ~ = and B are given by the position vectors OA and OB . 3 −5 Find the size of ∠AOB, giving your answer correct to the nearest tenth of a degree. [Ans: 99.2◦ ] 8.A-7 (MM Spec ’99) A line passes through the point (4, −1) and its direction   2 is perpendicular to the vector . Find the equation of the line in the 3 form ax + by = p, where a, b and p are integers to be determined. [Ans: 2x + 3y = 5] 8.A-8 (MM 5/00) Find a vector equation of the line passing through (−1, 4) and (3,−1). your answer   Give     in the form r = p + td, where t ∈ R. x −1 4 Ans: = +t y 4 −5   3 8.A-9 A line is perpendicular to and passes through (6, 2). −4 Mr. Budd, compiled September 29, 2010


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(a) Find the equation of the line in the form ax + by = g. Then find

|g| . |n|

[Ans: 3x − 4y = 10; 2]      6 4 (b) Write an equation of the line in the form r = r0 +td. Ans: r = +t 2 3 −−→ (c) Write a vector OR, in terms of t, for the position of a generic point on the line. [Ans: h6 + 4t, 2 + 3ti] (d) If point S is the closest point on the line to the origin, how is the −→ vector OS related to the direction vector? (e) Find the value of t that gives S, the closest point on the line to the origin. Ans: − 56   (f) Find the coordinates of S. Ans: 65 , − 85 (g) Confirm the shortest distance from the line to the origin. 8.A-10 (MM 5/02) Three of the coordinates of the parallelogram ST U V are S (−2, −2), T (7, 7), U (5, 15). −→ (a) Find the vector ST and hence find the coordinates of V . [Ans: (−4, 6)] (b) Find a vector equation of the line (U = p+λd where  V ) in theform  r    x 5 9 λ ∈ R. Ans: e.g., = +t y 15 9   1 (c) Show that the point E with position vector is on the line (U V ), 11   and find the value of λ for this point. Ans: e.g., − 49   a The point W has position vector , a ∈ R. 17

−−→

(d) i. If EW = 2 13, show that one value of a is −3 and find the other possible value of a.

[Ans: 5] −−→ −→ ii. For a = −3, calculate the angle between EW and ET . [Ans: 157◦ ] 8.A-11 (SL 5/08) The point O has coordinates (0, 0, 0), point A has coordinates (1, −2, 3), and point B has coordinates (−3, 4, 2).   −4 −−→ (a) i. Show that AB =  6 . −1 ˆ ii. Find B AO. [Ans: 45.8◦ ]       x −3 −4 (b) The line L1 has equation y  =  4  + s  6 . z 2 −1 Write down the coordinates of two points on L1 . [Ans: (−3, 4, 2), (−7, 10, 1)] −−→ (c) The line L2 passes through A and is parallel to OB. Mr. Budd, compiled September 29, 2010


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i. Find a vector equation for L2 , giving your  answer   in the  form  1 −3 Ans: r = −2 + t  4  r = a + tb. 3 2 ii. Point C (k, −k, 5) in on L2 . Find the coordinates of C. [Ans: (−2, 2, 5)]       x 3 1 (d) The line L3 has equation y  = −8 + p −2, and passes z 0 −1 through the point C. Find the value of p at C. [Ans: −5]

Mr. Budd, compiled September 29, 2010


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Lines International Baccalaureate 5.3 Vector equation of a line r = a+tb. Lines in the plane and in three-dimensional space. Knowledge of the following forms for equations of lines. Parametric form: x = x0 + λl, y = y0 + λm, z = z0 + λn. x − x0 y − y0 z − z0 Cartesian equation of a line in three dimensions = = . l m n Examples of applications: interpretation of t as time and b as velocity, with |b| representing speed. The angle between two lines.

8.1.9

Applications

Differentiate r = r0 +td with respect to time. What is the derivative of position? One way to think about the line is that r represents the position vector of an object that starts at position r0 and moves with velocity vector d, so that after one unit of time r = r0 + d. After two units of time, r = r0 + 2d. If you look at all the possible points for all the possible times, then you have a straight line (as long as the velocity remains constant).   1 Example 8.1.15 (MM Spec ’99) In this question the vector 0   0 km represents a displacement due east, and the vector km rep1 resents a displacement due north. Figure 8.1: MM Spec

Figure 8.1 shows the path of the oil-tanker Aristides relative to the Mr. Budd, compiled September 29, 2010


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port of Orto, which is situated at the point (0, 0). The position of Aristides is given by the vector equation       x 0 6 = +t y 28 −8 at time t hours after 12:00. (a) Find the position of the Aristides at 13:00. [2 marks] (b) Find (i) the velocity vector; (ii) the speed of the Aristides. [4 marks] (c) Find a cartesian equation for the path of the Aristides in the form ax + by = g. [4 marks] Another ship,thecargo-vessel Boadicea, is stationary, with po18 sition vector km. 4 (d) Show that the two ships will collide, and find the time of collision. [4 marks] To avoid collision, the Boadicea starts to move at 13:00 with   5 velocity vector km h−1 . 12 (e) Show that the position of the Boadicea for t ≥ 1 is given by       x 13 5 = +t y −8 12 (f) Find how far apart the two ships are at 15:00.       6 6 Ans: ; km h−1 , 10 km h−1 ; 4x + 3y = 84 20 −8 [Ans: 26 km apart]

[2 marks] [4 marks]

[Ans: 15:00]

Speed is the magnitude of velocity. Speed = |v| Example 8.1.16 (MM 5/01) In this question, a unit vector represents a displacement of 1 metre. A miniature car moves in a straight line, starting at the point (2, 0). After t seconds, its position, (x, y), is given by the vector equation       x 2 0.7 = +t y 0 1 Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors) (a) How far from the point (0, 0) is the car after 2 seconds? [Ans: 3.94 m]   (b) Find the speed of the car. Ans: 1.22 ms−1 (c) Obtain the equation of the car’s path in the form ax + by = c. [Ans: 10x − 7y = 20] Another miniature vehicle, a motorcycle, starts at the point (0, 2), and travels in a straight line with constant speed. The equation of its path is y = 0.6x + 2, x ≥ 0 Eventually, the two miniature vehicles collide. (d) Find the coordinates of the collision point. [Ans: (5.86, 5.52)] (e) If the motorcycle left point (0, 2) at the same moment the car   left point (2, 0), find the speed of the motorcycle. Ans: 1.24 ms−1

8.1.10

Lines in Three Dimensions

Example 8.1.17 (HL 5/99) The coordinates of the points P , Q, R, and S are (4, 1, −1), (3, 3, 5), (1, 0, 2c), and (1, 1, 2), respectively. −−→ −→ (a) Find the value of c so that the vectors QR and P R are orthogonal. (b) Find an equation of the line ` which passes through the point −→ Q and is parallel to the vector P R.

[Ans: 1; r = 3 (1 − t) i + (3 − t) j + (5 + 3t) k]

8.1.11

Angle Between Lines

Intersecting lines form two angles (one obtuse and one acute), unless they are perpendicular. One of the angles between two lines is the same as the angle between the two direction vectors. Generally, however, we want the acute angle between two lines, which is given by cos θ =

|d1 · d2 | |d1 | |d2 |

If the dot product of the direction vectors is negative, then it will give the obtuse angle. The absolute value ensures that you will get the acute angle. Mr. Budd, compiled September 29, 2010


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8.1.12

197

Intersection of Lines

Example 8.1.18 (MM 5/03) The vector equations of two lines are given below.         5 3 −2 4 r1 = +λ , r2 = +t . 1 −2 2 1 The lines intersect at the point P . Find the position vector of P . Addendum: Find the angle between the lines.    2 Ans: 3 Example 8.1.19 (HL 5/02) The vector equations of the lines L1 and L2 are given by L1 : r

= i + j + k + λ (i + 2j + 3k) ;

L2 : r

= i + 4j + 5k + µ (2i + j + 2k) .

The two lines intersect at point P . Find the position vector of P . Addendum: Find the angle between the lines. Example 8.1.20 (HL 5/01) The triangle ABC has vertices at the points A (−1, 2, 3), B (−1, 3, 5), and C (0, −1, 1). Let `1 be the line −−→ parallel to AB which passes through D (2, −1, 0) and `2 be the line −→ parallel to AC which passes through E (−1, 1, 1). (a) Find the equations of the lines `1 and `2 . (b) Hence show that `1 and `2 do not intersect. [Ans: `1 : x = 2, y = −1 + λ, z = 2λ, `2 : x = −1 + µ, y = 1 − 3µ, z = 1 − 2µ] Non-parallel lines that do not intersect are skew lines.

8.1.13

Applications in Three Dimensions

Example 8.1.21 (Spec ’06) In this question, distance is in kilometres, time is in hours. −1 A balloon is moving at a constant  height with a speed of 18km h , 3 in the direction of the vector 4. 0 At time t = 0, the balloon is at point B with coordinates (0, 0, 5). Mr. Budd, compiled September 29, 2010


198

HL Unit 8 (Vectors) (a) Show that the position vector b of the balloon at time t is given by       x 0 10.8 b = y  = 0 + t 14.4 z 5 0 At time t = 0, a helicopter goes to deliver a message to the balloon. The position vector h of the helicopter at time t is given by       x 49 −48 h = y  = 32 + t −24 z 0 6 (b) (a) Write down the coordinates of the starting position of the helicopter. (b) Find the speed of the helicopter. (c) The helicopter reaches the balloon at point R. (a) Find the time the helicopter takes to reach the balloon. (b) Find the coordinates of R.



Ans: (49, 32, 0), 54 km h−1 ; t =

5 6

 hr, (9, 12, 5)

This is a very rare case in which the two parameters are the same for each line. If two lines do use the same parameter, it should be in a situation like this, where the parameter is time.

Problems   1 8.A-12 (MM 5/00) In this question, the vector km represents a displacement 0   0 due east, and the vector km a displacement due north. [Mr. Budd’s 1 note: Do this on a separate sheet, and include a graph on graph paper ] Two crews of workers are laying an underground cable in a north-south direction across a desert. At 06:00 each crew sets out from their base camp which is situated at the origin (0, 0). One crew is in a Toyundai vehicle and the other in a Chryssault vehicle.   18 The Toyundai has velocity vector km h−1 , and the Chryssault has 24   36 velocity vector km h−1 . −16 (a) Find the speed of each vehicle.

  Ans: 30 km h−1 ;39.4 km h−1 Mr. Budd, compiled September 29, 2010


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199

     9 18 Ans: ; 12 −8 ii. Hence, or otherwise, find the distance between the vehicles at √ 06:30. Ans: 481 km i. Find the position vectors of each vehicle at 06:30.

(c) At this time (06:30), the Chryssault stops and its crew begin their day’s work, laying cable in a northerly direction. The Toyundai continues travelling in the same direction at the same speed until it is exactly north of the Chryssault. The Toyundai crew then begin their day’s work, laying cable in a southerly direction. At what time does the Toyundai crew begin laying cable? [Ans: 07:00] (d) Each crew lays an average of 800 m of cable in an hour. If they work non-stop until their lunch break at 11:30, what is the distance between them at this time? [Ans: 24.4 km] (e) How long would the Toyundai take to return to base camp from its lunch-time position, assuming it travelled in a straight line and with the same average speed as on the morning journey? (Give your answer to the nearest minute.) [Ans: 54 minutes] 8.A-13 (HL 5/97) (a) The line L1 is parallel to the vector v = 3i+j+3k and passes through the point (2, 3, 7). Find a vector equation of the line. (b) The parametric equations of another line L2 are x = t, y = t, and z = −t, −∞ < t < ∞ Show that i. L1 is not parallel to L2 ; ii. L1 does not intersect L2 . 8.A-14 (SL 5/06) The position vector of point A is 2i + 3j + k and the position vector of point B is 4i − 5j + 21k. −−→ (a) i. Show that AB = 2i − 8j + 20k. −−→ ii. Find the unit vector u in the direction of AB −→ iii. Show that u is perpendicular to OA. Let S be the midpoint of [AB]. The line L1 passes through S and is −→ parallel to OA. (b)

i. Find the position vector of S. [Ans: 3i − j + 11k] ii. Write down the equation of L1 . [Ans: r = (3i − j + 11k) + t (2i + 3j + k)] The line L2 has equation r = (5i + 10j + 10k) + s (−2i + 5j − 3k).

(c) Explain why L1 and L2 are not parallel. (d) The lines L1 and L2 intersect at the point P . Find the position vector of P . [Ans: 7i + 5j + 13k] Mr. Budd, compiled September 29, 2010


200

HL Unit 8 (Vectors) (e) Calculate the angle between the lines L1 and L2 .

[Ans: 69.7◦ ]

8.A-15 (SL 5/07 TZ2) In this question, distance is in metres, time is in minutes. Two model airplanes are each flying in a straight line. At 13:00 the first model airplane   isatpoint(3, 2, 7). Its position vector x 3 3 after t minutes is given by y  = 2 + t  4 . z 7 10 √   (a) Find the speed of the model airplane. Ans: 5 5 metres per minute At 13:00 the second model airplane is at the point (−5, 10, 23). After two minutes, it is at the point (3, 16, 39).   x (b) Show that its position vector after t minutes is given by y  = z     4 −5  10  + t 3. [Note: “Show that” 6= “Check that”. You should 8 23 solve the problem from scratch, as if you don’t know what the answer already is.] (c) The airplanes meet at point Q. i. At what time do the airplanes meet? [Ans: 13:08] ii. Find the position of Q. [Ans: (27, 34, 87)] (d) Find the angle θ between the paths of the two airplanes. [Ans: 0.167 radians] 8.A-16 (SL 5/07) Points P and Q have position vectors −5i + 11j − 8k and −4i + 9j − 5k, respectively, and both lie on line L1 . −−→ (a) i. Find P Q. [Ans: i − 2j + 3k] ii. Hence show that the equation of L1 can be written as r = (−5 + s) i + (11 − 2s) j + (−8 + 3s) k The point R (2, y1 , z1 ) also lies on L1 . (b) Find the value of y1 and z1 . [Ans: −3,13] The line L2 has equation r = 2i + 9j + 13k + t (i + 2j + 3k). (c) The lines L1 and L2 intersect at point T . Find the position vector of T. [Ans: −i + 3j + 4k] (d) Calculate the angle between the lines L1 and L2 . [Ans: 64.6◦ ] 8.A-17 (adapted from SL 5/08) The point O has coordinates (0, 0, 0), point A has coordinates (1, −2, 3), and point B has coordinates (−3, 4, 2).       x −3 −4 (a) The line L1 has equation y  =  4  + s  6 . z 2 −1 Write down the coordinates of two points on L1 . [Ans: (−3, 4, 2), (−7, 10, 1)] Mr. Budd, compiled September 29, 2010


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−−→ (b) The line L2 passes through A and is parallel to OB. Find a vector equation for L2 , giving your answer in the form r = a + tb.       x 3 1 (c) The line L3 has equation y  = −8 + p −2, and intersects z 0 −1 line L2 at point C. Find the coordinates of point C. [Ans: (−2, 2, 5)] (d) Calculate the acute angle between the lines L2 and L3 . [Ans: 9.76◦ ] 8.A-18 (5/06 TZ2) The following diagram shows a solid figure ABCDEF GH. Each of the six faces is a parallelogram. The coordinates of A and B are Figure 8.2: SL 5/06 TZ 2

A (7, −3, −5), B (17, 2, 5). (a) Find 

 10 Ans:  5  10

−−→ i. AB

−−→

ii. AB

[Ans: 15]

The following information is given.     −6 −2 −−→   −→   6 , AE = −4 AD = 3 4 Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors) (b)

−−→

[Ans: 9] i. Calculate AD .

−→

[Ans: 6] ii. Calculate AE . −−→ −→ iii. Calculate AD · AE. −−→ −−→ iv. Calculate AB · AD. −−→ −→ v. Calculate AB · AE. vi. Hence, write down the size of the angle between any two  inter secting edges. Ans: π2

(c) Calculate the volume of the solid ABCDEF GH. (d) The coordinates of G are (9, 4, 12). [Ans: (−1, −1, 2)]

[Ans: 810]

Find the coordinates of H.

(e) The lines (AG) and  intersect at point P .  (HB) 2 −→ Given that AG =  7 , find the acute angle at P . 17

[Ans: 71.6◦ ]

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Introduction to Matrices International Baccalaureate 4.1 Definition of a matrix: the terms element, row, column, and order. 4.2 Algebra of matrices: equality; addition; subtraction; multiplication by a scalar. Matrix operations to handle or process information. Multiplication of matrices. Identity and zero matrices. 4.3 Determinant of a square matrix. Calculation of 2 × 2 and 3 × 3 determinants. The inverse of a 2 × 2 matrix. Conditions for the existence of the inverse of a matrix. 4.4 Solution of linear equations using inverse matrices (a maximum of three equations in three unknowns).

8.1.14

Matrix Addition

In order to add two matrices, they must be of the same order, i.e., have the same number of rows and the same number of columns. Addition is intuitive. Scalar multiplication is intuitive. There is no such thing as subtraction of matrices, but we pretend like there is, just as for vectors.

The Zero Matrices A zero matrix is a matrix full of zeros. There is a zero matrix for every dimension of matrices.

A+0=A

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8.1.15

Matrix Multiplication

The key to matrix multiplication is to multiply rows by columns. Suppose you are finding the product Am×n Bn×p , which we will designate as Cm×p . The element of C in the ith row and jth column will be the ith row of A times the jth row of B. For this reason each row of A must have the same number of columns as the columns of B have rows.

Example 8.1.22 Use your handy dandy grapher to randomly generate a 3 × 2 matrix, and store it as A. Then store a 2 × 3 matrix into B. By hand, AB and BA, then check your answers.

Matrix multiplication is not necessarily commutative! AB 6= BA For this reason, we must distinguish the order of multiplication. Pre-multiplication is different from post-multiplication, and if you pre-multiply one side of an equation by A, you should not post-multiply the other side by A

The Identity Matrices The identity matrix is a square matrix with 1s across the main diagonal (top left to bottom right), and 0s everywhere else. It is the matrix form of 1.

AI = IA = A

8.1.16

Matrix Division

There is no such thing as division with matrices. Instead, you may pre-multiply or post-multiply by the inverse. A nonsingular matrix A has an inverse matrix, designated A−1 such that AA−1 = A−1 A = I Mr. Budd, compiled September 29, 2010


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Example 8.1.23 [21] From AB = C find a formula for A−1 .



Ans: A−1 = BC−1



The inverse of a 2 × 2 matrix is given by 

   1 d −b b = d ad − bc −c a

a c

Example 8.1.24 (HL 5/02) 

2 (a) Prove using mathematical induction that 0 for all positive integer values of n.

n  n 1 2 = 1 0

 2n − 1 , 1

(b) Determine whether or not this result is true for n = −1.

The inverse of a product: −1

(AB)

= B−1 A−1

Example 8.1.25 From PA = LU find a formula for A−1 .

  Ans: U−1 L−1 P

8.1.17

Determinants det

 a c

 b = ad − bc d

 Example 8.1.26 (SL 11/06) Let A =

3 k

  2 2 and B = 4 1

 2 . 3

Find in terms of k, (a) 2A − B; (b) det (2A − B).

[Ans: 22 − 4k] Mr. Budd, compiled September 29, 2010


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8.1.18

Solving Systems of Equations

8.1.19

Singularity

Problems     a b 1 0 and B = . Giving your answers c 0 d e in terms of a, b, c, d, and e,

8.A-19 (SL Spec ’06) Let A =

(a) write down A + B; (b) find AB. 



a+1 c+d    5 4 8 = . 7 a 15 Ans:

 8.A-20 (SL 5/06) Let

b 7

  3 9 + 8 −2

  b a + bd ; e c

(a) Write down the value of a. (b) Find the value of  −4 8.A-21 (SL 5/06) Let 3 2 q.

b.   8 2 −5 1 q

 be 0

[Ans: 5] [Ans: −5] 

0 −4

8.A-22 (MM 5/97) Let the 2 × 2 matrix M =

 =

−22 −9

 √ a 1 − a2



24 . Find the value of 23 [Ans: 3] √  1 − a2 , with −1 ≤ −a

a ≤ 1. Find the matrices (a) M 2 (b) M 3   1 Ans: 0

  0 a , √ 1 1 − a2

1 − a2 −a



8.A-23 (MM 11/96) Evaluate the matrix product:  π π √ ! cos − sin −2 3  6 6 .   π π 2 sin cos 6 6 √ ! −2 3 Then graph the vector , along with your answer. 2 Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 1: Vector Basics 8.A-24 (HL 5/97) Three  1 p A = 1 0 0 0

207

3 × 3 matrices   q 2 r  , B = 3 2 0

A, B, and C are given by   0 0 −1 −1 1 0 , and C =  2 0 0 1 0 0

 10 −6 2

[Ans: p = −1, q = 10, r = −6]

Find p, q, and r so that AB = C.   2 −1 8.A-25 (HL 5/97) Let A = . 1 0 (a) Calculate A2 and A3 .

[2 marks] n

(b) Conjecture a matrix for A , n ∈ N , in terms of n.

[3 marks]

(c) Use mathematical induction to prove your conjecture in part (b).[5 marks] 8.A-26 (HL 5/02) Find the determinant of  1 1 2

the matrix  1 2 2 1 1 5 [Ans: 0]

 8.A-27 (MM 5/99) If A =

2p −4p

3 p

 and det A = 14, find the possible values of [Ans: −7, 1]

p. 8.A-28 (HL 5/96) Solve the equation    1 4 1 1 det  x 2 2 = det x 2 x 2 1

 3 3  Ans:

8.A-29 [21] Find the inverses of   0 2 (a) A1 = 3 0   2 0 (b) A2 = 4 2   cos θ − sin θ (c) A3 = sin θ cos θ



 Ans:

cos θ − sin θ

5 6,

 −1

sin θ cos θ



8.A-30 [21] If the inverse of A2 is B, show that the inverse of A is AB. (Thus A is invertible whenever A2 is invertible.)   0 2 8.A-31 (SL 11/07) Let A = . 2 0 Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors) (a) Find i. A−1 ; ii. A2 .       0 .5 4 0 p Ans: ; Let B = .5 0 0 4 0   2 6 (b) Given that 2A + B = , find the value of p and of q. 4 3

 2 . q

(c) Hence find A−1 B. (d) Let X be a 2 × 2 matrix such that AX = B. Find X.   0 Ans: 2, 3; 1   2 1 8.A-32 (SL 5/08) Let M = . 2 −1

3 2



1

(a) Find the determinant of M. (b) Find M−1 .     x 4 (c) Hence solve M = . y 8 1



4 1 2

Ans: −4;

1 4 − 21



 ; x = 3, y = −2

8.A-33 (HL Spec ’99) Solve, by any method, the following system of equations: 3x − 2y + z = −4 x + y − z = −2 2x + 3y = 4 [Ans: x = −1, y = 2, z = 3] 8.A-34 (SL Spec ’05)  1 (a) Write down the inverse of the matrix A = 2 1

−3 2 −5

 1 −1 3

(b) Hence solve the simultaneous equations x − 3y + z = 1 2x + 2y − z = 2 x − 5y + 3z = 3 [Ans: x = 1.2, y = 0.6, z = 1.6] Mr. Budd, compiled September 29, 2010


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8.A-35 (adapted from [22]) Consider the system of equations x + y + 2z = 2 2x + y − z = 4 x−y− z=5 (a) Check that the determinant is not 0. (b) Find the Cartesian equation of `, the line of intersection of x+y+2z =i h y x−2 =z 2 and 2x + y − z = 4. Ans: 3 = −5   15 3 (c) Find the intersection of ` and the plane x−y−z = 5. Ans: 23 7 ,− 7 , 7 (d) Find the solution using an inverse matrix. 8.A-36 ([22]) Find if and where the following planes meet: P1 :

r1 = 2i − j + λ (3i + k) + µ (i + j − k)

P2 : r2 = 3i − j + 3k + r (2i − k) + s (i + j)       0 1 2       −1 −1 −1 −u +t P3 : r3 = 2 0 2  Ans: 8.A-37 (HL 5/02) The matrix A is given by  2 1 A = 1 k 3 4

94 68 64 29 , − 29 , 29



 k −1 2   Ans: 3, − 31

Find the values of k for which A is singular.

8.A-38 (HL 5/99) Find the value of a for which the following system of equations does not have a unique solution. 4x − y + 2z = 1 = −6

2x + 3y

x − 2y + az =

7 2

[Ans: 1]    1 6 x 8.A-39 (adapted from HL 5/07) Let A = and X = . Given that 4 3 y AX = kX, where k ∈ R, find the values of k for which there is not a unique solution for x. [Ans: 7,−3] 

Mr. Budd, compiled September 29, 2010


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Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 2: Cross Product

8.2

211

Cross Product

International Baccalaureate 5.5 The vector product of two vectors v × w. The vector product is also known as the cross product. The determinant representation. Geometric interpretation of |v × w|. Areas of triangles and parallelograms.

8.2.1

Cross Product

The cross product or vector product gives a third vector • whose direction is perpendicular to the other two; • whose magnitude is equal to the area of the parallelogram defined with the original two multiplying vectors as sides. Why then can’t we have a cross product for two-dimensional vectors? How could you use the cross-product to get the area of the triangle defined by three points? 

    x1 x2 i  y1  ×  y2  = det x1 z1 z2 x2

j y1 y2

 k z1  z2

Example 8.2.1 Is the cross product commutative, i.e., is u × v = v × u?

Example 8.2.2 (adapted from HL 5/02) The points A, B, C, D have the following coordinates A : (1, 3, 1)

B : (1, 2, 4)

C : (2, 3, 6)

D : (5, −2, 1) .

−−→ −→ (a) Evaluate the vector product AB × AC, giving your answer in −−→ −→ terms of the unit vectors i, j, k. [Confirm that the AB × AC −−→ −→ is orthogonal to AB and AC.] (b) Find the area of the triangle ABC Mr. Budd, compiled September 29, 2010


212

HL Unit 8 (Vectors) h Ans: −5i + 3j + k;

35 2

i

Example 8.2.3 (adapted from HL 5/01) The triangle ABC has vertices at the points A (−1, 2, 3), B (−1, 3, 5), and C (0, −1, 1). −−→ −→ (a) Find the size of the angle θ between the vectors AB and AC. (b) Hence find the area of the triangle ABC. (c) Otherwise find the area of the triangle ABC. [Ans: 147◦ or 2.56; 2.29] Example 8.2.4 (adapted from 5/08) The points A, B, C have position vectors i + j + 2k, i + 2j + 3k, 3i + k respectively and lie in the plane π. Find (a) a vector normal to π. (b) the area of triangle ABC; (c) the shortest distance from C to the line AB; (d) confirm your result by (a) writing a formula for a generic point R on the line, including the parameter; −→ −−→ (b) find out what R must be so that CR is orthogonal to AB;

−→

(c) find CR . √   Ans: 2j − 2k; 2; 2 Example 8.2.5 (adapted from HL 5/01) Find a vector normal to the plane containing the two lines x−1=

1−y x+1 2−y z+2 = z − 2 and = = 2 3 3 5 [Ans: −7i − 2j + 3k]

Problems 8.B-1 (HL 5/04) Given that a = (i + 2j + k), b = (i − 3j + 2k), and c = (2i + j − 2k), calculate (a − b) · (b × c). [Ans: 23] Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 2: Cross Product

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8.B-2 (HL 5/03) Given that a = i+2j−k, b = −3i+2j+2k, and c = 2i−3j+4k, find (a × b) · c. [Ans: 41] 8.B-3 (HL Spec ’00) The position vectors of points P and Q are: p

=

3i + 2j + k

q

= i + 3j − 2k

(a) Find the vector product p × q. (b) Using your answer to part (a), or otherwise, find the area of the −−→ −−→ parallelogram with two sides OP and OQ. √   Ans: −7i + 7j + 7k;7 3 = 12.1 8.B-4 (HL Spec ’00) For the vectors a = 2i + j − 2k, b = 2i − j − k, and c = i + 2j + 2k, show that (a) a × b = −3i − 2j − 4k. (b) (a × b) × c = − (b · c) a. 8.B-5 (HL 5/99) (a) Find a vector perpendicular to the two vectors: −−→ OP = −−→ OQ =

i − 3j + 2k −2i + j − k

−−→ −−→ (b) If OP and OQ are position vectors for the points P and Q, use your answer to part (a), or otherwise, to find the area of the triangle OP Q. h

Ans: i − 3j − 5k;

35 2

i

8.B-6 (HL 5/97) The coordinates of points P and Q are (−1, 5, 7) and (1, 2, 3), respectively, and O is the origin. −−→ −−→ (a) Find the vectors OP and OQ. −−→ −−→ (b) Calculate OP × OQ. −−→ (c) Calculate the area of the parallelogram which has adjacent sides OP −−→ and OQ. √   Ans: −i + 5j + 7k, i + 2j + 3k; i + 10j − 7k; 5 6 8.B-7 (HL 5/96) Find the area of the parallelogram determined by the vectors √  a = 3i − j + 2k and b = 2i + j − 4k. Ans: 285 Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors)

8.B-8 (HL 5/00) Find a vector that is normal to the plane containing the lines L1 and L2 , whose equations are: L1 : r

= i + k + λ (2i + j − 2k) ;

L2 : r

=

3i + 2j + 2k + µ (j + 3k) . [Ans: 5i − 6j + 2k]

8.B-9 (HL 5/96) The line L passes through the point P (2, 3, 1) and has direction u = i + 2j − 3k. A second line M passes through the point Q (4, 2, 0) and has direction v = 3i − j + k. (a) Write down, in parametric form and using t as the parameter, the equation of the line M . [Ans: x = 4 + 3t; y = 2 − t; z = t] −→ (b) Find the vector w = RP , where R is any point on the line M . [Ans: (−2 − 3t) i + (1 + t) j + (1 − t) k] (c) Writehdown the vector u × w and hence express |u × w| in terms ofi p t. Ans: (5 + t) i + 5 (1 + 2t) j + (5 + 7t) k; 15 (5 + 12t + 10t2 ) (d) Deduce that |u × w| is minimised when t = − mum value.

3 and find this mini5 √   Ans: 21

Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 3: Planes

8.3

215

Planes

International Baccalaureate 5.6 Vector equation of a plane r = a + λb + µc. Use of normal vector to obtain r · n = a · n. Cartesian equation of a plane ax + by + cz = d. 5.7 Intersection of: a line with a plane. Angle between: a line and a plane.

8.3.1

Planes

Cartesian Equation If P is a specific, known point in the plane, and R is a generic point in the plane, −→ then the vector P R is in the plane. If the vector n in normal to the plane, then −→ P R = r − p must be orthogonal to n. Hence n · (r − p)

=

0

n·r

=

n·p

If n = ha, b, ci, and p = hx1 , y1 , z1 i, then ha, b, ci · hx, y, zi = ha, b, ci · hx1 , y1 , z1 i ax + by + cz

= ax1 + by1 + cz1

ax + by + cz

= g

The general equation for a plane in space is similar to that of a line in two dimensions: ax + by + cz = g where ha, b, ci is the normal vector to the plane. Compare this to the formula ax + by = g in 2-D where ha, bi is a vector normal to the line. Example 8.3.1 (adapted from HL 5/01) Find an equation of the plane containing the two lines x−1=

x+1 2−y z+2 1−y = z − 2 and = = 2 3 3 5 [Ans: −7x − 2y + 3z = −3] Mr. Budd, compiled September 29, 2010


216

HL Unit 8 (Vectors)

Vector Representation of a Plane Start with a point, and move in two directions.

r = p + λd1 + µd2

Example 8.3.2 (HL 5/99) The coordinates of the points P , Q, R, and S are (4, 1, −1), (3, 3, 5), (1, 0, 2), and (1, 1, 2), respectively. (a) Find an equation of the line ` which passes through the point Q −→ and is parallel to the vector P R. [Ans: r = 3 (1 − t) i + (3 − t) j + (5 + 3t) k] (b) Find a vector equation of the plane π which contains the line ` and passes through the point S.

      −2 −3 3 Ans: 3 + λ −1 + µ −2 −3 3 5 

8.3.2

Angle Between Planes

The angle between two planes is almost the angle between the two normals, just as the angle between two lines is almost the angle between the two direction vectors. Two planes make two different angles, one acute and one obtuse. To make sure you have the acute angle, • if your angle is more than 90◦ , subtract it from 180◦ , or • easy fix: cos θacute =

|n1 · n2 | |n1 | |n2 |

Example 8.3.3 (HL 5/98) Find the acute angle between the planes 2x + 3y − z = 7

and

7x − y + 3z = −5

to the nearest tenth degree.

[Ans: 73.8◦ ] Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 3: Planes

8.3.3

217

Intersection of a Line and a Plane

To find the intersection of a line and a plane, make sure you have: • the line in parametric form; • the plane in Cartesian form. Then plug the three parametric equations for the line into the Cartesian equation for the plane. Solve for λ. Then use λ to find the coordinates from the parametric equations. Example 8.3.4 (adapted from HL 5/99) Find the coordinates of z−2 x−4 = −y − 2 = the point where the line given by the 2 3 intersects the plane with equation 2x + 3y − z = 2. [Ans: (2, −1, −1)]

8.3.4

Angle Between a Line and a Plane

The normal vector forms a right angle with the plane. This angle is cut by the line. The acute angle between the normal vector and the line added to the acute angle between the plane and the line must be ninety degrees. Therefore |n · d| π , or the acute angle between the line and the plane is − arccos 2 |n| |d| sin θ =

|n · d| |n| |d|

Problems y+1 = 8.C-1 (HL 5/00) The plane 6x − 2y + z = 11 contains the line x − 1 = 2 z−3 . Find l. [Ans: −2] l 8.C-2 (HL Spec ’00) Three points A, B, and C have coordinates (2, 1, −2), −→ −−→ −−→ (2, −1, −1), and (1, 2, 2) respectively. The vectors OA, OB, and OC, where O is the origin, form three concurrent edges of a parallelepiped OAP BCQSR as shown in Figure 8.3. Mr. Budd, compiled September 29, 2010


218

HL Unit 8 (Vectors)

Figure 8.3: HL Spec ’00 Paper 2

(a) Find the coordinates of P , Q, R, and S. [Ans: (4, 0, −3), (3, 3, 0), (3, 1, 1), (5, 2, −1)] (b) Find an equation of the plane OAP B.

[Ans: 3x + 2y + 4z = 0]

(c) Calculate the volume, V , of the parallelepiped given that

−→ −−→ −−→

V = OA × OB · OC

[Ans: 15] 8.C-3 (HL 5/99) The coordinates of the points P , Q, R, and S are (4, 1, −1), (3, 3, 5), (1, 0, 2), and (1, 1, 2), respectively. (a) Find an equation of the line ` which passes through the point Q and −→ is parallel to the vector P R. (b) Find a cartesian equation of the plane π which contains the line ` and passes through the point S. [Ans: r = 3 (1 − t) i + (3 − t) j + (5 + 3t) k; 9x − 15y + 4z = 2] 8.C-4 (HL 5/03) The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane 2x + y − z = 6. (a) Find n, the normal to the plane.

[Ans: h2, 1, −1i]

(b) Let p be the position vector for (1, 1, 9). Write an equation in terms of λ for a, where a is p plus an unknown multiple (λ) of a vector perpendicular to the plane. [Ans: (1 + 2λ) i + (1 + λ) j + (9 − λ) k] (c) Use the fact that A is in the plane to solve for λ, and then find the coordinates of A. [Ans: 2; (5, 3, 7)] (d) Check by showing that A satisfies the equation of the plane and that the vector from (1, 1, 9) to A is parallel to the normal of the plane. z y+1 = and the plane r · (i + 2j − k) = 1 2 3 intersects at the point P . Find the coordinates of P . [Ans: (2, 3, 1)]

8.C-5 (HL 5/04) The line x − 1 =

5−z x−3 = y+1 = and the plane 2x − y + 3z = 10 2 3 intersect at the point P . Find the coordinates of P . [Ans: p7, 1, −1]

8.C-6 (HL 5/05) The line

Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 3: Planes

219

8.C-7 (HL 11/06) Let P be the point (1, 0, −2) and Π be the plane x+y−2z+3 = 0. Let P 0 be the reflection of P in the plane Π. Find of  the coordinates  the point P 0 . Ans: − 53 , − 83 , 10 3

Mr. Budd, compiled September 29, 2010


220

HL Unit 8 (Vectors)

Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 4: Plane Crashes

8.4

221

Plane Crashes

International Baccalaureate 4.3 Determinant of a square matrix. The terms singular and non-singular matrices. Calculation of 2 × 2 and 3 × 3 determinants. The result det AB = det A det B. The inverse of a matrix: conditions for its existence. 4.4 Solution of systems of linear equations . Conditions for the existence of a unique solution, no solution, and an infinity of solutions. These cases can be investigated using row reduction, including the use of augmented matrices. Unique solutions can also be found using inverse matrices. 5.6 Vector equation of a plane r = a + λb + µc. Use of normal vector to obtain r · n = a · n. Cartesian equation of a plane ax + by + cz = d. 5.7 Intersections of: two planes; three planes. Inverse matrix method and row reduction for finding the intersection of three planes. Awareness that three planes may intersect in a point, or in a line, or not at all.

Resources §1.6 Inverses and Transposes in Strang [21] §17F The Intersection of Two or More Planes in [22]

8.4.1

Intersection of Two Planes

The system of two equations and three unknowns is underdetermined. If there is a solution at all, there must be an infinite number of answers. The intersection of two planes could be • empty set (parallel planes) • a plane (both planes are the same) • a line (typical situation) It should be fairly easy to recognize a situation where the planes are the same, or even if the planes are parallel.

Let one variable be λ To find the line of intersection, pick x, y, or z to be the parameter λ, then use elimination to find the other two variables in terms of λ (which is x, y, or z). Mr. Budd, compiled September 29, 2010


222

HL Unit 8 (Vectors) Example 8.4.1 (HL 5/01) Find the equation of the line of intersection of the two planes −4x + y + z = −2 and 3x − y + 2z = −1. Then find the angle of intersection.  Ans: x =

3y+3 11

= 3z + 3



Crossing normals The direction vector of the line lies in both planes; it is therefore orthogonal to both normal vectors. (How can you get a vector that is orthogonal to two other vectors?) The trick here is finding a point on the line. You could assume a value (0 works well) for either x, y, or z, which reduces the number of variables in your system of equations to match the number of equations (two variables, two equations). Example 8.4.2 (HL 5/98) The equations of the planes P1 and P2 are given by P1 : r · (3i − j + 2k) = −1 P2 : r · (−2i + j − 5k) = 4 where r = xi + yj + zk is the position vector for a point on the plane. Let L be the line of intersection of the two planes P1 and P2 . (a) Show that L is parallel to 3i + 11j + k. (b) Show that the point A (0, −1, −1) lies on the line L. Hence, or otherwise, find the equation of L. [Ans: r = −j − k + λ (3i + 11j + k)] Example 8.4.3 Try this technique with the example from the other technique.

Row Reduction This is worthwhile because it is similar to how we will find the intersection of three matrices using matrices. Example 8.4.4 Use row reduction solve the previous examples. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 4: Plane Crashes

8.4.2

223

Singularity

A matrix is singular if its determinant is 0. Singular matrices have several properties: • the determinant is zero. • they do not have inverse matrices. • one or more rows is a linear combination of the other rows. 

3 −2 −3 4



Example 8.4.5 (HL 5/03) Given that A = and I =   1 0 , find the values of λ for which (A − λI) is a singular matrix. 0 1 [Ans: 1 or 6] Note: these values of λ are called the eigenvalues of A 

α Example 8.4.6 (HL 5/08) Let M be the matrix  0 −1 Find all the values of α for which M is singular.

2α α −1

 0 1 . α

[Ans: 0, ±1]

8.4.3

Intersection of Three Planes

A system of three equations in three variables can be thought of as looking for the intersection of three planes. There are several options: • Consistent, independent: The three planes intersect at a single point. There is a unique solution. • Consistent, dependent: One plane is a linear combination of the others. Typically, a third plane contains the line of intersection of the other two. Other possibilities: Two or three of the same plane. • Inconsistent: The three planes do not intersect. One plane is parallel a linear combination of the other two. Typically, the three planes form a prism, but never intersect all at once. Or you could have parallel planes. Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors)

Elimination • Pick a pair of equations, eliminate a variable. • Pick another pair of equations, eliminate the same variable. • You should now have two equations in two unknowns.

Geometric considerations • Pick a pair of planes, find the line of intersection. • Find the intersection between the line and the third plane.

Inverse Matrices • If there is a unique solution, it is given by A−1 b. • If there is not a unique solution, it is because det (A) = 0, i.e., A is singular.

Example 8.4.7 (HL 5/03) The variables x, y, z satisfy the simultaneous equations x + 2y + z

=k

2x + y + 4z

=6

x − 4y + 5z

=9

where k is a constant. Show that these equations do not have a unique solution. If there is a unique solution to Ax = b, it is given by x = A−1 b.

Augmented Matrix In terms of the augmented matrix [A|b], • Consistent, independent: The coefficient matrix A is nonsingular and has an inverse. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 4: Plane Crashes

225

• Consistent, dependent: The coefficient matrix is singular. At least one row of the augmented matrix is a linear combination of the other two rows. • Inconsistent. The coefficient matrix is singular. While one row of the coefficient matrix is a linear combination of the other two rows, that is not true of the augmented matrix. We will discuss this more later.

8.4.4

Solving Systems of Equations

Problems 8.D-1 (HL 5/04) (a) The point P (1, 2, 11) lies in the plane π1 . The vector 3i − 4j + k is perpendicular to π1 . Find the Cartesian equation of π1 . [Ans: 3x − 4y + z = 6] (b) The plane π2 has equation x + 3y − z = −4. i. Show that the point P also lies in the plane π2 . ii. Find a vector equation of the line of intersection of π1 and π2 . [Ans: i + 2j + 11k + λ (i + 4j + 13k)] [Ans: 53.7◦ ]

(c) Find the acute angle between π1 and π2 .

8.D-2 (HL Spec ’99) Solve, by any method, the following system of equations: 3x − 2y + z = −4 x + y − z = −2 2x + 3y = 4 [Ans: x = −1, y = 2, z = 3] 8.D-3 (SL Spec ’05)  1 (a) Write down the inverse of the matrix A = 2 1

−3 2 −5

 1 −1 3

(b) Hence solve the simultaneous equations x − 3y + z = 1 2x + 2y − z = 2 x − 5y + 3z = 3 [Ans: x = 1.2, y = 0.6, z = 1.6] Mr. Budd, compiled September 29, 2010


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HL Unit 8 (Vectors)

8.D-4 (adapted from [22]) Consider the system of equations x + y + 2z = 2 2x + y − z = 4 x−y− z=5 (a) Check that the determinant is not 0. (b) Find the Cartesian equation of `, the line of intersection of x+y+2z =i h y x−2 =z 2 and 2x + y − z = 4. Ans: 3 = −5   15 3 (c) Find the intersection of ` and the plane x−y−z = 5. Ans: 23 7 ,− 7 , 7 (d) Find the solution using an inverse matrix. 8.D-5 ([22]) Find if and where the following planes meet: P1 :

r1 = 2i − j + λ (3i + k) + µ (i + j − k)

P2 : r2 = 3i − j + 3k + r (2i − k) + s (i + j)       0 1 2       −1 −1 −1 −u +t P3 : r3 = 2 0 2  Ans:

94 68 64 29 , − 29 , 29



Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 5: Non-Unique Solutions

8.5

227

Non-Unique Solutions

International Baccalaureate 4.3 Determinant of a square matrix. The terms singular and non-singular matrices. Calculation of 2 × 2 and 3 × 3 determinants. The result det AB = det A det B. The inverse of a matrix: conditions for its existence. 4.4 Solution of systems of linear equations . Conditions for the existence of a unique solution, no solution, and an infinity of solutions. These cases can be investigated using row reduction, including the use of augmented matrices. Unique solutions can also be found using inverse matrices. 5.7 Intersections of: two planes; three planes. Resources §1.6 Inverses and Transposes in Strang [21] §17F The Intersection of Two or More Planes in [22]

8.5.1

Unique Solution

Consistent, Independent: One Solution Example 8.5.1 [22] Consider the system of equations x + 3y − z = 0 3x + 5y − z = 0 x − 5y + z = 8 (a) Solve by elimination. Check on calculator with inverse matrices. (b) Find the intersection of two planes. (For later purposes, let z be the parameter.) (c) Find the intersection of the line with the third plane. (d) Give a geometric interpretation to the solution. (e) Solve using elementary row operations on an augmented matrix. Example 8.5.2 (HL 11/99) 

a (a) Find the values of a and b given that the matrix A = −8 −5   1 2 −2 is the inverse of the matrix B =  3 b 1 . −1 1 −3

−4 5 3

 −6 7 4

Mr. Budd, compiled September 29, 2010


228

HL Unit 8 (Vectors) (b) For the values of a and b you found, solve the system of linear equations x + 2y − 2z = 5 3x + by + z = 0 −x + y − 3z = a − 1

[Ans: 7, 2; (−1, 2, 1)]

Example 8.5.3 [22] Use elementary row operations to solve the system x + 3y − z =

0

3x + 5y − z =

0

x − 5y + (2 − m) z =

9 − m2

i h  m2 −9 m2 −9 9−m2 , 2(m+1) , m+1 Ans: 2(m+1)

8.5.2

No Solution

Example 8.5.4 [6] Consider the system of equations 3x + y + 4z

=

8

3x − y − z

=

4

x + y + 3z

=

2

(a) Check that the determinant is 0. (b) Eliminate y’s and see whether or not the system is consistent. (c) Find the Cartesian equation of the line hof intersection of 3x +i z−4 y + 4z = 8 and 3x − y − z = 4. Ans: x = y+8 5 = −2 (d) Show that this line is parallel to all three planes. (e) Determine whether the line lies in the third plane, x+y+3z = 2. (f) Show that one normal is a linear combination of the other two, but that the plane is not a linear combination of the others. (g) Examine the system using rref. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 5: Non-Unique Solutions

8.5.3

229

Infinite Solutions 

1 4

 6 and 3

Example 8.5.5 (adapted from HL 5/07) Let A =   x X= . Given that AX = kX, where k ∈ R, find the values of y k for which there is an infinity of solutions for x. Investigate the system of equations generated for each k.

[Ans: 7,−3] Example 8.5.6 [6] Consider the system of equations 3x − y − z

=

1

x + 2y + z

=

4

x − 5y − 3z

= −7

(a) Check that the determinant is 0. (b) Eliminate x’s and see whether or not the system is consistent. (c) Find a Cartesian equation of the line of intersection of 3x − y −i h z = 1 and x + 2y + z = 4.

Ans: x =

y−5 −4

=

z+6 7

(d) Show that this line is parallel to all three planes. (e) Determine whether the line lies in the third plane, x−5y −3z = −7. (f) Show that one equation is a linear combination of the other two. (g) Solve the system using rref. Example 8.5.7 (HL 5/96) (a) Derive an equation that a, b, and c must satisfy for the system of equations −3x + y + 2z = a −11x + 2y + 6z = b 7x + y − 2z = c to have a solution.

[Ans: 5a − 2b − c = 0]

(b) Derive a solution when a = 2 and b = 7. Is the solution unique? Explain your answer clearly. [10 marks]  Ans: c = −4, no: x = λ, y = −1 − 2λ, z =

1 2

 (3 + 5λ)

Mr. Budd, compiled September 29, 2010


230

HL Unit 8 (Vectors) Example 8.5.8 Use rref to solve x − 3y + 2z = 8 3x − 9y + z = 4



8.5.4

Ans:

x+2 3

 = y; z = 5

Augmented Matrices and Echelon Forms

Example 8.5.9 Use ref to find the row echelon form of the augmented matrix for Example 8.5.1. Use rref to find the row-reduced echelon form. Use rref to find the row-reduced echelon form for the intersection of the two planes.

Problems 8.E-1 (HL 5/02) The matrix A is given by  2 1 A = 1 k 3 4

 k −1 2   Ans: 3, − 31

Find the values of k for which A is singular.

8.E-2 (HL 5/99) Find the value of a for which the following system of equations does not have a unique solution. 4x − y + 2z = 1 2x + 3y

= −6

x − 2y + az =

7 2 [Ans: 1]

8.E-3 (HL 11/06) Consider the system of equations x + 2y + kz

=

0

x + 2y + z

=

3

kx + 8y + 5z

=

6

(a) Find the set of values of k for which this system of equations has a unique solution. Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 5: Non-Unique Solutions

231

(b) For each value of k that results in a non-unique solution, find the solution set.  Ans: k =

1 3

 ⇒ Ø; k = 3 ⇒ (−6 − 7λ, 3 + 2λ, λ)



   1 6 x 8.E-4 (adapted from HL 5/07) Let A = and X = . Given that 4 3 y AX = kX, where k ∈ R, find the values of k for which there is not a unique solution for x. [Ans: 7,−3] 8.E-5 Classify each set of planes as (i) intersecting in a single point, in which case give its coordinates, or (ii) never intersecting, or (iii) intersecting in a line, in which case give a Cartesian equation. (a) x+y−z

=

10

2x − 3y + z

=

5

x − 4y + 2z

=

6 [Ans: (ii)]

(b) x+y+z

=

10

2x − y

=

9

−x + 3y + 4z

=

14 [Ans: (i) (5, 1, 4)]

(c) x + 2y − z

=

10

3x − y + z

=

11

2x + y + 4z

=

−1 [Ans: (i) (5, 1, −3)]

(d)

h Ans: (ii)

x−17 −8

=y=

2x + y + 3z

= −5

x − 2y + 2z

= −9

3x + 4y + 4z

= −1

z+13 5

or x = 17 − 8y =

−8z−19 5

or

5x+19 −8

= 5y − 13 = z

Mr. Budd, compiled September 29, 2010

i


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HL Unit 8 (Vectors)

8.E-6 [6] Find the value of k for which the system of equations 8x + 3y + z

=

12

x + 2z

=

3

2x + y − z

= k

represents 3 planes that intersect in a  common line  and find  the  vector  3 −2 Ans: 2, ~r = −4 + λ −5 equation of the line of intersection. 0 1 8.E-7 (HL 5/03) The variables x, y, z satisfy the simultaneous equations x + 2y + z

=k

2x + y + 4z

=6

x − 4y + 5z

=9

where k is a constant. (a) Show that these equations do not have a unique solution. (b) Find the value of k for which the equations are consistent (that is, they can be solved). (c) For this value of k, find the general solution of these equations.   11−7λ Ans: 1; z = λ, y = 2λ−4 3 , x= 3 8.E-8 (HL 5/02) (a) Find the determinant of the matrix   1 1 2 1 2 1 2 1 5 [Ans: 0] (b) Find the value of λ for which the following system of equations can be solved.      1 1 2 x 3 1 2 1 y  =  4  2 1 5 z λ (c) For this value of λ, find the general solution to the system of equations. [Ans: 5; z = µ, y = 1 + µ, x = 2 − 3µ and others] Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 5: Non-Unique Solutions

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8.E-9 (MM 11/96) The matrix equation      3 −1 x 9 = , −6 k y −18

k ∈ R,

has more than one solution. (a) Find the value of k.     c −d (b) Given that and are both solutions, find the values of c d c and d. [Ans: k = 2; c = 1.8, d = −3.6] 8.E-10 (HL 5/00) The system of equations represented by the following matrix equation has an infinite number of solutions.      7 x 2 −1 −9 1 2 3  y  = 1 z 2 1 −3 k Find the value of k.

[Ans: 5]

8.E-11 (adapted from HL 5/99) Consider the system of equations 2x − 2y + kz= 0 x + 4z= 0 kx + y + z = 0 (a) Find all values of k for which the system of equations has a nonzero solution [that is, (x, y, z) 6= (0, 0, 0)]. (b) For those values of k, find the solution. h Ans: k = − 76 ;

x 28

=

y 31

=

z −7

i

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HL Unit 8, Day 6: Distances

8.6

235

Distances

8.6.1

Projection of a Vector

Example 8.6.1 Derive the formula for a vector in the direction of v that has a magnitude of |u| cos θ, where θ is the angle between u and v. Hence find the formula for the projection of u onto v. Also find a formula for the magnitude of this projection.

Example 8.6.2 (adapted from MM 5/03) Consider the vectors c = 3i + 4j and d = 5i − 12j. (a) Calculate the scalar product c · d. (b) Calculate the vector projection of c in the direction of d. (c) Calculate the scalar projection of the vector c in the direction of d.

  33 Ans: −33; ;− 13 Example 8.6.3 Repeat for two other vectors.

8.6.2

Distance From a Point to a Plane

Example 8.6.4 (HL 5/02) The points A, B, C, D have the following coordinates A : (1, 3, 1)

B : (1, 2, 4)

C : (2, 3, 6)

D : (5, −2, 1) .

The plane containing the points A, B, C is denoted by Π and the line passing through D perpendicular to Π is denoted by L. The point of intersection of L and Π is denoted by P . −−→ −→ (a) Evaluate the vector product AB × AC, giving your answer in terms of the unit vectors i, j, k. [Ans: −5i + 3j + k] (b) Find the cartesian equation of Π. (c) Find the cartesian equation of L.

h

[Ans: −5x + 3y + z = 5] i y+2 Ans: x−5 −5 = 3 = z − 1

(d) Determine the coordinates of P . (e) Find the perpendicular distance of D from Π.

[Ans: (0, 1, 2)] √   Ans: 35

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Shortest distance between a point and a plane:

−−→

P Q · n

|n| −−→ where n is the normal of the plane and the vector P Q is a vector between the point and any point on the plane.

Example 8.6.5 [6] Find the distance of the point (3, 2, 1) from the plane x + y + 2z = 10. h

Ans:

√3 6

i

Shortest distance between a point (p, q, r) and a plane ax + by + cz = d can also be written as:

ap + bq + cr − d

a2 + b2 + c2

Example 8.6.6 (adapted from HL 5/99) Find the shortest distance between the point P (4, 1, −1) and the plane       −2 3 −3 π : r = 3 + λ −1 + µ −2 5 3 −3

h

8.6.3

Ans:

√15 322

or 0.836

i

Distance Between Parallel Planes

The shortest distance between two parallel planes is the same as the distance between a plane and any point on the other plane. Example 8.6.7 [6] Show that the planes x − y − 2z = 1 and 3x − 3y + 6z = −5 are parallel, and find the distance between them. h

Ans:

8 √

3 6

= 1.09

i

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HL Unit 8, Day 6: Distances

8.6.4

237

Distance Between a Plane and a Parallel Line

The shortest distance between a plane and a line is the same as the distance between a plane and any point on the line.

Example 8.6.8 [6] Show that the plane P with equation 4x+y−z = 9 and the line r = i + 2j + λ (i − 3j + k) are parallel, and find the distance between them. h

8.6.5

Ans:

√1 2

i

Distance Between a Point and a Line

An interesting phenomenon about the standard, or Cartesian, form of a line, ax + by = c, is that the shortest distance from the line to the origin is given by |c| √ . 2 a + b2

Example from MM 5/02) A vector equation of a  8.6.9  (adapted   x 1 −2 line is = +t , t ∈ R. y 2 3 (a) Find the equation of this line in the form ax + by = c, where a, b, and c ∈ Z. [Ans: 3x + 2y = 7] (b) hHence find ithe shortest distance between the line and the origin. Ans: √713 −−→ (c) Write a vector OR, in terms of t, for the position of a generic point on the line. [Ans: h1 − 2t, 2 + 3ti] (d) If point S is the closest point on the line to the origin, how is −→ the vector OS related to the direction vector? (e) Find the value of t that gives S, the closest point on the line 4 to the origin. Ans: − 13 (f) Confirm the shortest distance from the line to the origin.

h

Ans: 3x + 2y = 7;

√7 13

i

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HL Unit 8 (Vectors)

8.6.6

Distance Between Parallel Lines

Example 8.6.10 (HL 5/00) Find the coordinates of the point which is nearest to the origin on the line L : x = 1 − λ, y = 2 − 3λ, z = 2 

Ans:

3 1 10 , − 10 , 2



Example 8.6.11 (HL 5/01) The triangle ABC has vertices at the points A (−1, 2, 3), B (−1, 3, 5), and C (0, −1, 1). Let `1 be the line −−→ parallel to AB which passes through D (2, −1, 0) and `2 be the line −→ parallel to AC which passes through E (−1, 1, 1). (a) Find the equations of the lines `1 and `2 . (b) Hence show that `1 and `2 do not intersect. (c) Find the shortest distance between `1 and `2 .

h

Ans:

√9 21

i or 1.96

Problems 8.F-1 (adapted from 5/03) Find the perpendicular distance from D : (5, −2, √ 1) to Π : −5x + 3y + z = 5. Ans: 35 8.F-2 [6] Find the distance of the point (4, −2, 7) from the planes: √   Ans: 2 3 h i Ans: √2521 h i Ans: √1710

(a) x + y + z = 3 (b) x − 2y − 4z = 5 (c) x = 3z (d) 2x − y − z = 3

[Ans: 0]

8.F-3 [6] Find the distance between each pair of planes.     2 2 (a) r · −3 = 7 and r · −3 = 35 6 6

[Ans: 4]

(b) 2x − y − 2z = 6 and 4x − 2y − 4z = 15 (c) r · (i + 2j − 5k) = −4 and r · (i + 2j − 5k) = 6

h

[Ans: 0.5] √ i Ans: 330

8.F-4 [6] Given the points A (−1, 4, 0), B (3, 1, 2), and C (−2, 1, 0): (a) Show that triangle ABC is isosceles and calculate its area. [Ans: BA = BC, 8.14] Mr. Budd, compiled September 29, 2010


HL Unit 8, Day 6: Distances

239

(b) Find a vector equation of line AB. [Ans: −i + 4j + λ (4i − 3j + 2k)] (c) Show that line AB is parallel to the plane P with equation x+2y+z = 1. √ (d) Show that the distance between line AB and plane P is 6. 8.F-5 [6] Given the points A (4, 2, 3), B (0, 10, 5), C (6, 0, 1), D (5, 2, 5), and E (8, 5, −1): (a) Show that the Cartesian equation of plane ABC is 3x + y + 2z = 20. (b) Find a vector equation of the line DE and show that it is parallel to plane ABC. [Ans: 5i + 2j + 5k + λ (3i + 3j − 6k)] h √ i (c) Find the distance between line DE and plane ABC. Ans: 214 h √ i (d) Write down the distance of point D from plane ABC. Ans: 214 (e) Find in its simplest form the ratio length of AB: length of CD. [Ans: 2:1] 8.F-6 (HL 11/06) (a) The line l1 passes through the point A (0, 1, 2) and is perpendicular to h the plane x − 4y −i3z = 0. Find the Cartesian equations of l1 . z−2 Ans: x = y−1 −4 = −3 (b) The line l2 is parallel to l1 and passes through the point P (3, −8, −11). Find the vector equation of the line l2 . −−→ (c) The point Q is on the line l1 such that P Q is perpendicular to l1 and l2 . Find the coordinates of Q. [Ans: (3, −11, −7)] (d) Hence find the distance between l1 and l2 .

[Ans: 6]

8.F-7 (HL 5/97) The line L1 is parallel to the vector v = 3i + j + 3k and passes through the point (2, 3, 7). The parametric equations of another line L2 are x = t, y = t, and z = −t, −∞ < t < ∞ Let O be the origin and P be the point (−1, 2, 4). (a) Find a vector w which is parallel to the line L. −−→ (b) Find the vector P O. (c) Find the shortest distance between the lines L1 and L2 by using the formula

−−→

P O · (v × w)

d=

|v × w|

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HL Unit 8 (Vectors)

Mr. Budd, compiled September 29, 2010


Unit 9

Probability 1. Bayes’ Theorem 2. Combinations 3. Permutations 4. Continuous and Normal Distributions 5. Discrete and Binomial Distributions International Baccalaureate 7.1 Sample space, U ; the event A. The probability of event A as P(A) =

n (A) . n (U )

The complementary events A and A0 (not A); the relation P(A) + P(A0 ) = 1. 7.2 Combined events, A ∩ B and A ∪ B. The relation P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive events; the relation P(A ∩ B) = 0. 7.3 Conditional probability; the relation P(A | B) =

P(A ∩ B) . P(B)

Independent events; the relations P(A | B) = P(A) = P(A | B 0 ) . Use of Bayes’ Theorem for two events. 241


242

HL Unit 9 (Probability)

7.4 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems. Applications. 7.5 Counting principles, including permutations and combinations. Included: the number of ways of selecting and arranging r objects from n. 7.6 Discrete probability distributions. Expectation, mode, median, variance, and standard deviation. 7.7 The binomial distribution, its mean and variance. Included: situations and conditions for using a binomial model. 7.8 Continuous probability distributions. Expectation, mode, median, variance and standard deviation. Included: the concept of a continuous random variable; definition and use of probability density functions. 7.9 The normal distribution. Standardization of a normal distribution; the use of the standard normal distribution table.

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HL Unit 9, Day 0: Re-introduction to Probability

9.0

243

Re-introduction to Probability

International Baccalaureate 6.5 Concepts of trial, outcome, equally likely outcomes, sample space U , and event. n(A) The probability of an event A as P(A) = . n(U ) The complementary events A and A0 (not A); P(A) + P(A0 ) = 1. 6.6 Combined events, the formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Appreciation of the non-exclusivity of “or”. P(A ∩ B) = 0 for mutually exclusive events. Use of P(A ∪ B) = P(A) + P(B) for mutually exclusive events. 6.7 Conditional probability; the definition P(A | B) =

P(A ∩ B) . P(B)

6.8 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems.

9.0.1

The Basics

Definitions Definition 9.1. A (probability) experiment E is a specific set of actions, the results of which cannot be predicted with certainty. [18] Definition 9.2. A set S (or U or Ω) is a sample space for an experiment E if each element of S represents a unique outcome of E , and if each outcome of E is represented by a unique element of S. [18] A single experiment may have several different sample spaces.

Example 9.0.1 Suppose we perform the experiment of tossing a coin twice. (Note: in this case, the single experiment is tossing the coin twice.) Indicate appropriate sample space if (a) we wish to record what happens on each toss, or (b) we wish to record the number of heads obtained [18] Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability)

If we wish to record what happens on each toss, an appropriate sample space would be S = {T T, T H, HT, HH}. However, in the same experiment, we may use a different sample space. If we wish to record the number of heads, an appropriate sample space would be Sˆ = {0, 1, 2}. Note that the outcome of 1 in Sˆ corresponds to two different outcomes in sample space S. Definition 9.3. An event is a subset of a sample space. A simple event is a subset containing a single element. In many discussions, we will liberally interchange the words “event” and “subset.” [18]

Example 9.0.2 Consider the experiment of tossing a coin twice, with sample space S = {T T, T H, HT, HH}. Represent the following verbally described events as subsets of S. [18] (a) exactly one head; (b) head on the second toss; (c) at least one head; (d) head on both tosses. [18] [Ans: A = {T H, HT }, B = {T H, HH}, C = {T H, HT, HH}, D = {HH} (a simple event)] Given the events A, B, and C, determine which of these events are subsets of other events. [18] [Ans: A ⊂ C, B ⊂ C, A ⊂ A, B ⊂ B, C ⊂ C, A 6⊂ B] Definition 9.4. (Complement of a Set) The complement of a set A with respect to S (or, simply, the complement of A), denoted by A0 (some use Ac or A), is the set of all those elements of S that do not belong to A. [18] Definition 9.5. (Intersection of Sets) [18] The intersection of two sets A and B, denoted by A ∩ B, is the set of all those elements which belong to both A and B. In symbols (where “x ∈” means “x is an element of”), we say A ∩ B = {x | x ∈ A and x ∈ B} Definition 9.6. (Union of Sets) The union of two sets A and B, denoted by A ∪ B, is the set of all those elements that are either in A alone, or in B alone, or in both A and B.[18] Often one says that A ∪ B is the set of all elements that belong to A or to B where “or” is used in the inclusive sense of “and/or.” Still another valid statement is “A ∪ B is the set of those elements that belong to at least one of the sets A and B.”[18]

A ∪ B = {x | x ∈ A or x ∈ B} Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 0: Re-introduction to Probability

245

Example 9.0.3 Consider the experiment of tossing a coin twice, with the associated sample space U = {T T, T H, HT, HH}, with events A = {T H, HT }, B = {T H, HH}, C = {T H, HT, HH}. Determine the following events: (a) A0 , B 0 , and C 0 ; (b) A ∪ B; (c) A ∩ B; (d) A ∩ C 0 . [18] [Ans: {T T, HH}, {T T, HT }, {T T }; {T H, HT, HH}; {T H}; Ø] Definition 9.7. (Mutually Exclusive Events) The pair consisting of A and B is mutually exclusive (disjoint) if A ∩ B = Ø.[18]

Example 9.0.4 Which pairs of events of the following events are mutually exclusive? A = {T H, HT }, B = {T H, HH}, C 0 = {T T }, A0 = {T T, HH}, A ∩ B = {T H}. [18] [Ans: A and C 0 ; A and A0 ; B and C 0 ; C 0 and A ∩ B; A0 and A ∩ B] Definition 9.8. A partition of a set E is a collection of subsets E1 , E2 , . . . , Em of E with the following properties: 1. Ei ∩ Ej = Ø for i 6= j. That is the collection E1 , E2 , . . . , Em is mutually exclusive. 2. E = E1 ∪ E2 ∪ . . . ∪ Em . That is, E equals the union of the Ei ’s.[18] Definition 9.9. The symbol P(A) indicates the probability of an event A of some sample space S. Here, the probability P(A) is a real number. [18]

Axioms Axioms are the basic theoretical assumptions, upon which we prove everything else. Axiom 9.1. For any event A of a sample space S, 0 ≤ P(A) ≤ 1. [18] Axiom 9.2. P(S) = 1. [18] Axiom 9.3 (Addition Rule). [18] For disjoint (mutually exclusive) events A1 , A2 , A3 , . . ., P(A1 ∪ A2 ∪ A3 ∪ . . .) = P(A1 ) + P(A2 ) + P(A3 ) + . . . Mr. Budd, compiled September 29, 2010


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Theorems Theorem 9.1. For the complement A0 of any event A, P(A) + P(A0 ) = 1. [18] Example 9.0.5 A pair of dice is tossed. What is the probability  a sum that is at least 3? [18] of getting Ans: 35 36 Theorem 9.2. For the empty set Ø, P(Ø) = 0. [18] Theorem 9.3. If A and B are events in a sample space S, and A ⊂ B, then P(A) ≤ P(B). [18] Example 9.0.6 A coin, which is not necessarily fair, is tossed twice. Prove that the probability of getting two heads is less than or equal to the probability of getting a head on the second toss. Theorem 9.4. For any finite sequence of events (subsets) A1 , A2 , . . . , Am of sample space S, P(A1 ∪ A2 ∪ A3 ∪ . . . ∪ Am ) ≤ P(A1 ) + P(A2 ) + . . . + P(Am )

Examples Example 9.0.7 A particular unbalanced coin is tossed twice. For the sample space S = {T T, T H, HT, HH}, the following probabilities are assigned, based on repeating the double toss several million times, and observing the corresponding relative frequencies: P({T T }) = .16;

P({T H}) = P({HT }) = .24;

P({HH}) = .36

(a) Compute the probability of getting exactly one head. (b) Repeat for the event “head on the second toss.” (c) Repeat for the event “at least one head.” [18] [Ans: .48,.60,.84]

Equally Likely Events: Classic Probability There are certain cases in which sample space is made up of equally likely events. This is not always the case: rolling a seven on a pair of dice is not as likely as rolling a two. However, the prospect of rolling a six and an ace is as likely as rolling an ace and an ace. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 0: Re-introduction to Probability

247

Figure 9.1: Sets A and B [18]

Theorem 9.5 (Equally Likely Simple Events Theorem). Suppose all n of the simple events associated with a finite sample space S = {e1 , e2 , . . . , en } are equally likely. That is, P({e1 }) = P({e2 }) = . . . = P({en }). Then, 1. P({ei }) = 1/n for i = 1, 2, . . . , n, and 2. P(A) = a/n for any event A to which a elements belong. Put differently, if we let N (A) and N (S) stand for the number of elements in A and S, respectively, then we can write P(A) = N (A) /N (S).

Example 9.0.8 A pair of dice is tossed. Determine the probability that (a) a two appears on at least one die, (b) the sum of the numbers on both dice equals six, and (c) the same number of dots is obtained on both dice. 

9.0.2

Ans:

11 5 6 36 , 36 , 36



Probability and Set Theory: Venn Diagrams

Theorem 9.6 (General Addition Rule for Two Events). P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Consider Figure 9.1, which partitions the sample space S into subsets R1 , R2 , R3 , and R4 . Since these are mutually exclusive events, we can use the Addition Rule (Axiom 9.3). Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability)

Figure 9.2: Venn diagram for Example 9 [18]

P(A ∪ B)

=

P(R1 ∪ R2 ∪ R3 )

=

P(R1 ) + P(R2 ) + P(R3 )

=

P(R2 ) + P(R1 ) + P(R1 ) + P(R3 ) − P(R1 )

=

P(R2 ∪ R1 ) + P(R1 ∪ R3 ) − P(R1 )

=

P(A) + P(B) − P(A ∩ B)

Example 9.0.9 The probability that a student will attend the first meeting of a class is .75, the probability that a student will attend the second meeting is .65, and the probability that a student will attend both meetings is .60. (a) What is the probability that a student will attend at least one of the first two meetings of the class? (b) What is the probability that a student will miss both th first and second meetings of the class? (c) What is the probability that a student will attend exactly one of the first two meetings of the class? [Ans: .80,.20,.20]

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HL Unit 9, Day 0: Re-introduction to Probability

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Figure 9.3: Sets A, B, and C [18]

Theorem 9.7 (General Addition Rule for Three Events). P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)

P(A ∪ B ∪ C)

=

P(R1 ∪ R2 ∪ R3 ∪ R4 ∪ R5 ∪ R6 ∪ R7 )

=

P(R1 ) + P(R2 ) + P(R3 ) + P(R4 ) + P(R5 ) + P(R6 ) + P(R7 )

=

P(R1 ) + P(R2 ) + P(R3 ) + P(R4 ) + P(R5 ) + P(R6 ) + P(R7 ) +P(R1 ) − P(R1 ) + P(R1 ) − P(R1 ) + P(R1 ) − P(R1 ) +P(R2 ) − P(R2 ) + P(R3 ) − P(R3 ) + P(R5 ) − P(R5 )

=

P(R1 ) + P(R2 ) + P(R3 ) + P(R4 ) +P(R1 ) + P(R2 ) + P(R5 ) + P(R6 ) +P(R1 ) + P(R3 ) + P(R5 ) + P(R7 ) −P(R1 ) − P(R2 ) − P(R1 ) − P(R3 ) −P(R1 ) − P(R5 ) + P(R1 )

=

[P(R1 ) + P(R2 ) + P(R3 ) + P(R4 )] + [P(R1 ) + P(R2 ) + P(R5 ) + P(R6 )] + [P(R1 ) + P(R3 ) + P(R5 ) + P(R7 )] − [P(R1 ) + P(R2 )] − [P(R1 ) + P(R3 )] − [P(R1 ) + P(R5 )] +P(R1 )

=

P(A) + P(B) + P(C) −P(A ∩ B) − P(A ∩ C) − P(B ∩ C) +P(A ∩ B ∩ C) Mr. Budd, compiled September 29, 2010


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9.0.3

Conditional Probability

Definition 9.10. Given that A and B are events of sample space S, and P(A) > 0, then the conditional probability of B given A is defined by the equation P(B | A) =

P(A ∩ B) P(A)

Conditional probability is a probability we use when a given condition is known to exist. If a condition (such as event B has happened) exists, then the sample space has been changed. If I have the condition that event B happens, then there is no probability that B 0 will happen. I need to exclude B 0 from my sample space. My sample space has been reduced from the whole sample space S to the given condition B. The new probability of A given B is equal to the probability of A within the new sample space of B. Therefore, the conditional probability of A given B is the ratio between the overlap between A and B, i.e., that portion of A that is in B, divided by the total probability of my new “sample space”, B. Example 9.0.10 (MM 5/00) In a survey, 100 students were asked ‘do you prefer to watch television or play sport?’ Of the 46 boys in the survey, 33 said they would choose sport, while 29 girls made this choice.

Television Sport Total

Boys

Girls

33 46

29

Total

100

By completing this table or otherwise, find the probability that (a) a student selected at random prefers to watch television; (b) a student prefers to watch television, given that the student is a boy.  Ans:

19 13 50 ; 46



Example 9.0.11 (SL 5/06 TZ2) In a class, 40 students take chemistry only, 30 take physics only, 20 take both chemistry and physics, and 60 take neither. (a) Find the probability that a student takes physics given that the student takes chemistry. Mr. Budd, compiled September 29, 2010


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(b) Find the probability that a student takes physics given that the student does not take chemistry. (c) Are the events “taking chemistry” and “taking physics” mutually exclusive? Looking ahead: are these events independent?

Example 9.0.12 A medical team has developed a possible vaccine Table 9.1: Example 12 Got cold Got vaccinated Yes No Total Yes 48 32 80 No 52 28 80 Total 100 60 160

for the common cold. They test it on a group of 160 volunteers divided into an 80-person experimental group and an 80 person control group. The members of the experimental group are vaccinated, while the members of the control group are not. After 12 months all 160 people are asked if they got a cold during the past year. The results are summarized in Table 9.1 (e.g., 48 vaccinated people got a cold.) The probability experiment is to randomly select one of the 160 people. If for this experiment S = {the 160 people}, A = {got a cold}, and B = {vaccinated}, then find: [2] (a) P(A | B) (b) P(A | B 0 ) (c) P(A | B) + P(A0 | B) (d) P(B | A) (e) P(B | A0 ) 0

Example 9.0.13 (HL 5/05) Given that (A ∪ B) = Ø, P(A0 | B) =   1 6 , and P(A) = , find P(B). Ans: 37 3 7

Multiplication Rules for Two, Three, and k Events Rearranging the definition of conditional probability gives us the multiplication rule. Mr. Budd, compiled September 29, 2010


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Theorem 9.8 (Multiplication Rule). If A and B are any two events of a sample space S and P(A) > 0, then P(A ∩ B) = P(A) · P(B | A)

The multiplication rules for three, and for any number of events is as follows. Theorem 9.9 (Multiplication Rule for Three Events). If A and B and C are any three events of a sample space S and P(A ∩ B) > 0, then P(A ∩ B ∩ C) = P(A) · P(B | A) · P(C | A ∩ B) Theorem 9.10 (General Multiplication Rule). If A1 , A2 , . . . , Ak are any k events (k ≥ 2) of a sample space S, and P(A1 ∩ A2 ∩ . . . ∩ Ak−1 ) > 0, then P(A1 ∩ A2 ∩ . . . ∩ Ak )

=

P(A1 ) · P(A2 | A1 ) · P(A3 | A1 ∩ A2 ) · . . . ·P(Ak | A1 ∩ A2 ∩ . . . ∩ Ak−1 )

Theorem 9.11. Suppose P(A) > 0. Then P(B 0 | A) exists, and is given by P(B 0 | A) = 1 − P(B | A).

9.0.4

Sampling Without Replacement

The key thing to remember is that, if you are picking from n items, after the first pick you have n − 1 items, after the second pick you have n − 2 items, and so on.

Example 9.0.14 Two cards are drawn from a well-shuffled, standard deck of playing cards. If the first card is not replaced between selections, then what is the probability that: (a) both cards will be hearts, (b) both will be queens, (c) one will be a king and the other a queen?

There are at total of 52 card on the first draw, and 51 cards on the second draw. The probability of getting a first heart will be 13 52 and the probability of getting a second hear on the second draw is 12 . The probability of getting two hearts 51 Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 0: Re-introduction to Probability would be

13 52

·

12 51

=

12 204

253

= 0.059. In equation form,

P(H1 ∩ H2 )

= = =

P(H1 ) P(H2 | H1 ) 13 12 × 52 51 12 = 0.059 204

Note that the probability of drawing a second hear is dependent on drawing a first heart, therefore we have to use P(H2 | H1 ) instead of just P(H2 ). We would use P(H2 ) in a different situation, in which we replaced the first card in the deck and reshuffled, so that the two events were independent of one another. For drawing two consecutive queens, we’d have a similar equation: P(Q1 ∩ Q2 )

= = =

P(Q1 ) P(Q2 | Q1 ) 3 4 × 52 51 3 = 0.0045 663

For drawing a king and then a queen, we’d have: P(K1 ∩ Q2 )

= = =

P(K1 ) P(Q2 | K1 ) 4 4 × 52 51 4 = 0.0060 663

For a queen and then a king: P(Q1 ∩ K2 )

= = =

P(Q1 ) P(K2 | Q1 ) 4 4 × 52 51 4 = 0.0060 663

Overall, the probability of a king and a queen would be P(K1 ∩ Q2 )+P(Q1 ∩ K2 ) = 0.0060 + 0.0060 = 0.0120. Note that getting a king and then a queen is mutually exclusive from getting a queen and then a king, so that I can find the probability of both simply by adding the two. Example 9.0.15 What is the probability of getting a spade on the first draw? What is the probability of getting a spade on the second Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability) draw, accounting for the fact that the first draw could be a spade, or could not be a spade? Example 9.0.16 (HL 97) A bag contains 5 white and 7 black balls. If two balls are drawn at random without replacement, what is the probability that one of them is black and the other is white.

 Ans:

9.0.5

35 66



Statistical Independence

Definition of Independence Definition 9.11. Events A and B are independent if and only if P(A ∩ B) = P(A) · P(B) This is the mathematical definition. Intuitively, A and B are independent if the probability of A does not change, regardless of whether B happens, or B does not happen. In other words, A and B are independent if(f) P(A) = P(A | B) = P(A | B 0 ), and equally P(B) = P(B | A) = P(B | A0 ). Example 9.0.17 (MM 95) A and B are events such that P(A) = 0.4 and P(B) = 0.3. (a) If A and B are mutually exclusive events, find P(A ∪ B). [Ans: 0.7] (b) If A and B are independent events, find (a) P(A ∪ B); (b) P(A | B). (c) P(A | B 0 ).

[Ans: 0.58] [Ans: 0.4] [Ans: 0.4]

Example 9.0.18 (MM 11/02) For events A and B, the probabilities 3 4 are P(A) = , P(B) = . 11 11 Calculate the value of P(A ∩ B) if 6 ; 11 (b) events A and B are independent. (a) P(A ∪ B) =

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HL Unit 9, Day 0: Re-introduction to Probability

255  Ans:

1 12 11 ; 121



Example 9.0.19 (HL 11/00) Given that events A and B are independent with P(A ∩ B) = 0.3 and P(A ∩ B 0 ) = 0.3, find P(A ∪ B).

[Ans: 0.8] Example 9.0.20 (HL 5/02) Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice. (a) (a) Calculate   the probability that Alan obtains a score of 9. Ans: 19 (b) Calculate the probability that Alan and Belle both   obtain 1 a score of 9. Ans: 81 (b) (a) Calculate the probability that Alan and Belle obtain the same score. Ans: 146 362 (b) Deduce the probability that Alan’s score exceeds Belle’s   score. Ans: 575 362 Theorem 9.12. 1. Events A and B are independent if and only if P(B | A) = P(B). (Here, assume that P(A) > 0 so that P(B | A) exists.) 2. Events A and B are independent if and only if P(A | B) = P(A). (Here, assume that P(B) > 0 so that P(A | B) exists.) 3. Events A and B are independent if and only if P(B | A0 ) = P(B). (Here, assume that P(A0 ) > 0 so that P(B | A0 ) exists.) 4. Events A and B are independent if and only if P(A | B 0 ) = P(A). (Here, assume that P(B 0 ) > 0 so that P(A | B 0 ) exists.) Here, the if and only if connectors tell you that the implications of the statements are bidirectional, that is, for part 1, P(B | A) = P(B) implies independence, and independence implies P(B | A) = P(B), as well as P(B | A0 ) = P(B), as well as P(A | B) = P(A), as well as P(A | B 0 ) = P(A). Theorem 9.13. If the two events A and B are independent, then 1. the two events A and B 0 are also independent, and 2. the two events A0 and B are also independent, and 3. the two events A0 and B 0 are also independent. Mr. Budd, compiled September 29, 2010


256

HL Unit 9 (Probability) 2 Example 9.0.21 Let E and F be events such that P(E) = and 3 1 P(E ∩ F ) = 3 (a) Calculate P(F | E). What must P(F ) be if events E, F are to be independent? 1 (b) Suppose P(E 0 ∩ F 0 ) = . Are E and F independent? 12 5 (c) Suppose instead that P(E ∪ F ) = . Are E and F independent 6 in that case?  Ans:

1 2;

 N; Y

Example 9.0.22 One card is drawn, and placed face down, so that its identity is unknown. Accounting for the fact that the first card could be a spade, or might not be a spade, find the probability that the next card drawn is a spade. Are the events “getting a spade on the first draw” and “getting a spade on the second draw” independent of one another? Definition 9.12 (Independence of Three Events). Three events A, B, and C are independent if and only if the following four equations hold: P(A ∩ B)

=

P(A) · P(B) ;

P(A ∩ C)

=

P(A) · P(C) ;

P(B ∩ C)

=

P(B) · P(C) ;

P(A ∩ B ∩ C)

=

P(A) · P(B) · P(C)

The first three conditions are that the three events are pairwise independent, i.e., each pair is independent. Events A1 , A2 , . . . , Ak are pairwise independent if P(Ai ∩ Aj ) = P(Ai ) · P(Aj ) for every distinct pair i 6= j. Definition 9.13 (Independence of k Events). Events A1 , A2 , . . . , Ak are independent if and only if the probability of the intersection of any 2, 3, . . . , or k of these events equals the product of their respective probabilities.

9.0.6

Counting Methods

Tree Diagrams Example 9.0.23 (adapted from SL Nov ’06) Mr. Budd, compiled September 29, 2010


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(a) A single bag contains 8 red marbles and 7 blue marbles. Two marbles are drawn without replacement. What is the probability that two marbles of different colors are chosen? (b) Bag 1 contains 4 red marbles and 5 blue marbles. Bag 2 contains 4 red marbles and 2 blue marbles. A marble is chosen from Bag 1, followed by a marble from Bag 2. (a) What is the probability that two marbles of different colors are chosen? (b) Suppose a blue marble is drawn. What is the probability that it came from Bag 1? What is a fundamental difference in the two situations? Example 9.0.24 Suppose Alice and Bob are playing a tennis match in which the winner is the one who wins three sets first (“best three out of five sets match”). Use a tree diagram to determine all possible ways the match can be decided. (a) Why do teams play best-of series? (b) If Alice has a 60% chance of winning each game, what is the probability that she wins a best of three match? (c) The newspaper reports that Alice won the match, but doesn’t give details. What is the probability that she swept? The tree diagram is seen in Figure 9.4. Example 9.0.25 (MM 5/04) Dumisani is a student at the Boodle School for International Students. The probability that he will be 7 woken by his alarm clock is . If he is woken by his alarm clock the 8 1 probability that he will be late for school is . If he is not woken by 4 his alarm clock the probability he will be late for school is 35 . (a) Create a tree diagram for this problem. (b) Calculate the probability that Dumisani will be late for school. (c) Given that Dumisani is late for school, what is the probability that he was woken by his alarm clock? [Ans: 0.294; 0.745] 2 2 Example 9.0.26 (HL 5/01) Given that P(X) = , P(Y | X) = , 3 5 1 and P(Y | X 0 ) = , find 4 Mr. Budd, compiled September 29, 2010


258

HL Unit 9 (Probability)

Figure 9.4: Tree diagram for best three out of five

(a) P(Y 0 ); (b) P(X 0 ∪ Y 0 ).

 Ans:

13 11 20 ; 15



Fundamental Counting Principle

Theorem 9.14 (Fundamental Counting Principle). Given a combined operation consisting of k component operations. Suppose a first operation can be done in n1 ways, and for each of these ways a second operation can be done in n2 ways, and for each of these ways a third operation can be done in n3 ways, and so on, for each of the k component operations. Then, a combined operation consisting of the k component operations can be done in n1 · n2 · n3 · . . . · nk ways.[18] Mr. Budd, compiled September 29, 2010


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Problems 1 3 9.R-1 (MM 5/04) Let A and B be events such that P(A) = , P(B) = , and 2 4   7 P(A ∪ B) = . Calculate P(A ∩ B). Ans: 38 8 9.R-2 (MM 99) Two unbiased six-sided dice are thrown; one is red and the other is green. Two numbers, between 1 and 6 inclusive, are shown. Find the probability that (a) the number on the red die is even; (b) the number on the green die is greater than the number on the red die.   5 Ans: 12 ; 12 9.R-3 (MM 97) For the events C, D, P(C) = 0.7, P(D) = 0.3, and P(C ∪ D) = 0.9. Find P(C ∩ D). [Ans: 0.1] 9.R-4 (MM 97) Two standard six-faced dice are thrown. Let X be the total score obtained. Find (a) P(X = 12); (b) P(X ≤ 3);  Ans:

1 1 36 , 12



9.R-5 (MM 96N) Two fair dice are rolled. Let x be the sum of the two numbers showing uppermost. Find the probability that (a) x = 10; (b) x ≤ 7.  Ans:

1 7 12 ; 12



9.R-6 (MM 96) For the events A and B, P(A) = 0.3 and P(B) = 0.4. Find P(A0 ∩ B 0 ) if A and B are mutually exclusive events. [Ans: 0.3] 9.R-7 (MM 94) When two standard 6-faced dice are tossed, T is the total score obtained. (a) Evaluate P(T > 8) (b) If the two dice are tossed twice, what is the probability that T exceeds 8 exactly once?   5 65 Ans: 18 ; 162 Mr. Budd, compiled September 29, 2010


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9.R-8 (MM 93S) Two unbiased dice are thrown and the total score is observed. X is the event that the total score is even, and Y is the event that the total score is a factor of 12. (a) Find P(X). (b) Find P(Y ). (c) Find P(X ∪ Y ).  Ans:

1 1 5 2; 3; 9



9.R-9 (SL 91) An unbiased coin, which can show ‘head’ or ‘tail’, is tossed four times. What is the probability that the fourth toss is a ‘head’   for the 3 second time? Ans: 16 9.R-10 (MM 5/05) The table below shows the subjects studied by 210 students at a college.

History Science Art Totals

Year 1 50 15 45 110

Year 2 35 30 35 100

Totals 85 45 80 210

(a) A student from the college is selected at random. Let A be the event the student studies Art. Let B be the event the student is in Year 2. i. Find P(A). ii. Find the probability that the student is a Year 2 Art student.  8 Ans: 21 ; 16 (b) Given that a History student is selected at random, calculate the  probability that the student is in Year 1. Ans: 10 17 (c) Two students are selected at random from the college. Calculate the probability that one of the students is in Year 1, and the other is in Year 2. Ans: 200 399 9.R-11 (MM 11/04) A packet of seeds contains 40% red seeds and 60% yellow seeds. The probability that a red seed grows is 0.9, and that a yellow seed grows is 0.8. A seed is chosen at random from the packet. Create a tree diagram to answer the following problems. (a) Calculate the probability that the chosen seed is red and grows. (b) Calculate the probability that the chosen seed grows. (c) Given that the seed grows, calculate the probability that it is red. [Ans: 0.36; 0.84; 0.429] Mr. Budd, compiled September 29, 2010


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3 1 9.R-12 (MM 5/04) Let A and B be events such that P(A) = , P(B) = , and 2 4   7 P(A â&#x2C6;Ş B) = . Calculate P(A | B). Ans: 12 8 9.R-13 (MM 5/03) A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. (a) The first two apples are green. What is the probability that the third apple is red? (b) What is the probability that exactly two of the three apples are red?   693 Ans: 22 23 ; 2300 = 0.301 9.R-14 (MM 5/01) A bag contains 10 red balls, 10 green balls, and 6 white balls. Two balls are drawn at random from the bag without replacement.  What  is the probability that they are of different color? Ans: 44 65 9.R-15 (MM Spec â&#x20AC;&#x2122;00) In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20 males. (a) Using this information, complete the table below. Males

Females

Unemployed Employed Totals

Totals

200

(b) If a person is selected at random from this group of 200, find the probability that this person is i. an unemployed female; ii. a male, given that the person is employed.  Ans: Table, 15 ,

9 14



9.R-16 (HL 99) A bag contains 2 red balls, 3 blue balls, and 4 green balls. A ball is chosen at random from the bag and is not replaced. A second ball is chosen. Find the probability of choosing one green ball and one blue ball in any order. Ans: 13 9.R-17 (SL 82) A research worker has a cage containing 3 male animals and 5 female animals. He chooses two animals at random (without replacement). Find the probability p that the two chosen animals are of different sexes.  Ans: 15 28 9.R-18 (SL 85) Two cards are drawn simultaneously from a normal pack of 52 playing cards. Find, giving answers correct to three significant figures in each case, [10 marks] (a) the probability that both cards are Aces, Mr. Budd, compiled September 29, 2010


262

HL Unit 9 (Probability) (b) the probability that one of the cards is the Ace of Clubs, (c) the probability that neither card is an Ace or a Club, (d) the probability that at least one of the cards is an Ace or a Club, and (e) the probability that one of the cards is the Ace of Clubs given that one of the cards at least is a Club.  Ans:

1 105 116 17 1 221 ; 26 ; 221 ; 221 ; 195



9.R-19 (HL 5/04) Robert travels to work by train every weekday from Monday to Friday. The probability that he catches the 8:00 train on Monday is 0.66. The probability that he catches the 8:00 train on any other weekday is 0.75. A weekday is chosen at random. (a) Find the probability that he catches the train on that day. (b) Given that he catches the 8:00 train on that day, find the probability that the chosen day is a Monday. [Ans: 0.732; 0.180] 9.R-20 (HL 96) Three suppliers A, B, and C produce respectively 45%, 30%, and 25% of the total number of a certain component that is required by a car manufacturer. The percentages of faulty components in each supplierâ&#x20AC;&#x2122;s output are, again respectively, 4%, 5%, and 6%. What is the probability that a component selected at random is faulty? [Ans: 0.048] 9.R-21 (MM 5/05) The table below shows the subjects studied by 210 students at a college.

History Science Art Totals

Year 1 50 15 45 110

Year 2 35 30 35 100

Totals 85 45 80 210

Are the events A and B independent? Justify your answer.

[Ans: N]

1 3 9.R-22 (MM 5/04) Let A and B be events such that P(A) = , P(B) = , and 2 4 7 P(A â&#x2C6;Ş B) = . 8 (a) Calculate P(A | B). (b) Are the events A and B independent? Give a reason for your answer.  Ans:

1 2;

 yes

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263

9.R-23 (MM 93S) For events A and B, P(A) =

3 4 6 , P(B) = , and P(A ∪ B) = . 11 11 11

(a) Find P(A ∩ B).

[2 marks]

(b) Find P(A | B).

[2 marks]

0

(c) Find P(A | B ).

[4 marks]

(d) Determine whether the events A and B are independent, giving a reason for your answer. [4 marks]   1 Ans: 11 ; 14 ; 27 ; N 9.R-24 (MM 5/03) Two fair dice are thrown and the number showing on each is noted. The sum of these two numbers is S. Find the probability that (a) S is less than 8;

[2 marks]

(b) at least one die shows a 3;

[2 marks]

(c) at least one die shows a 3, given that S is less than 8. [3 marks]   7 1 Ans: 12 ; 11 36 ; 3 9.R-25 (MM 5/98) The events A and B are independent, and P(A ∩ B) = 0.6, P(B) = 0.8. Find (a) P(A | B); (b) P(A | not B). [Ans: 0.75; 0.75] 9.R-26 (MM 11/96) A department store has two burglar alarm systems. In the event of an attempted break-in, the systems function properly with probabilities 0.95 and 0.90, respectively. The two systems function independently. When an attempt is made to break in, what is the probability of at least one of the alarm systems functioning properly? [Ans: 0.995] 9.R-27 (MM 96) For the events A and B, P(A) = 0.3 and P(B) = 0.4. (a) Find P(A ∪ B) if A and B are independent events. (b) Find P(A0 ∩ B 0 ) if A and B are mutually exclusive events. [Ans: 0.58, 0.3] 9.R-28 (MM 11/95) A and B are independent events such that P(A) = 0.5 and P(A0 ∪ B 0 ) = 0.7. Find (a) P(B); Mr. Budd, compiled September 29, 2010


264

HL Unit 9 (Probability) (b) P(B | A0 ). [Ans: 0.6, 0.6]

9.R-29 (SL 93) A and B are independent events with P(A ∪ B) = 0.8 and P(B) = 0.2. Find P(A). [Ans: 0.75] 9.R-30 (SL 91) Two standard unbiased dice are thrown. The random variable X is defined to be the sum of the two values shown on the dice. (a) Write down P(X = 3).

[3 marks]

(b) Calculate P(X is divisible by 4).

[4 marks]

The two dice are thrown twice and X1 and X2 are the two values of X obtained. (c) Show that P(X1 = X2 ) =

146 362

[5 marks]

(d) Hence, or otherwise, determine P(X1 > X2 ).

[3 marks]

(e) Find P(X1 = 3 | X1 > X2 ).  Ans: 9.R-31 (SL 89) If P(A) = calculate

1 3

and P(B) =

2 5

[5 marks]  1 1 575 2 18 ; 4 ; ; 362 ; 575

and A and B are independent events,

(a) P(A ∪ B), (b) P(A0 ∪ B 0 ). 

Ans:

3 13 5 ; 15



9.R-32 (SL 84) There are two roads only from town A to town B; each of the roads is liable to be closed because of landslides. On average the low road is closed on one day in six and on average the high road is closed on one day in five. Assuming that the closures of the roads are independent of one another, find (a) the probability that it is possible to travel from town A to town B on a given day, and (b) the probability that both the roads will be open on a given day.   2 Ans: 29 30 ; 3 9.R-33 (HL 5/03) The independent events A, B are such that P(A) = 0.4 and P(A ∪ B) = 0.88. Find (a) P(B); (b) the probability that either A occurs or B occurs, but not both. [Ans: 0.8; 0.56] Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 0: Re-introduction to Probability

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9.R-34 (HL 5/00) In a game a player rolls a biased tetrahedral (four-faced) die. The probability of each possible score is shown in Figure 9.2. Find the probability of a total score of six after two rolls. Ans: 14 Table 9.2: Problem 34: Score 1 1 Probability 5

Tetrahedral Die 2 3 4 2 1 x 5 10

9.R-35 (HL 5/98) For two independent events A and B, P(A | B) = 41 and 1 . Find P(A) and P(B). P(A â&#x2C6;Š B) = 32 Ans: 41 , 18

Mr. Budd, compiled September 29, 2010


266

HL Unit 9 (Probability)

Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 1: Bayes’ Theorem

9.1

267

Bayes’ Theorem

International Baccalaureate 6.7 Conditional probability; the definition P(A | B) =

P(A ∩ B) . P(B)

Independent events; the definition P(A | B) = P(A) = P(A | B 0 ) . The term “independent” is equivalent to “statistically independent.” Use of P(A ∩ B) = P(A) P(B) for independent events. Use of Bayes’ Theorem for two events. 6.8 Use of Venn diagrams, tree diagrams and tables of outcomes to solve problems. Resources §7-12 ”Total Probability Theorem and Bayes’ Formula” in [18]

9.1.1

Bayes’ Theorem

Theorem 9.15 (Theorem of Total Probability). If the sample space S is partitioned into k events B1 , B2 , . . . , Bk [that is, S = B1 ∪ B2 ∪ . . . ∪ Bk , where Bi ∩ Bj = Ø for i 6= j], and E is an event of the sample space S, then the probability of the event E can be expressed as follows [18]: P (E) = P (B1 )P (E | B1 ) + P (B2 )P (E | B2 ) + . . . + P (Bk )P (E | Bk ) Example 9.1.1 Suppose we choose two cards without replacement from a standard deck of 52 cards. Find the probability that the second card is a spade; denote the probability of this event by P (S2 ) [18].   Ans: P (S2 ) = 41 Example 9.1.2 Suppose a box contains three coins, of which two are fair coins and one is a coin with two heads. A coin is selected at random from the box and tossed once. What is the probability that the result of that toss is a head? [18]   Ans: P (H) = 23 Mr. Budd, compiled September 29, 2010


268

HL Unit 9 (Probability) Example 9.1.3 The Great Idea [GI] Company manufactures a certain type of light bulb at factories I, II, and III. GI has in stock a large batch of bulbs, 25% of which were made in factory I, 35% in factory II, and 40% in factory III. Also, the GI track record shows that 3% of all bulbs produced by factory I are defective, 5% of all bulbs produced by factory II are defective, and 4% of all bulbs produced by factory III are defective. If a single bulb is picked at random from the batch, what is the probability that it is defective? [18]

[Ans: P (E) = 0.041] Theorem 9.16 (Bayes’ Formula). P (Bi | E) =

P (Bi )P (E | Bi ) P (E)

Example 9.1.4 Refer to the Great Idea Company problem on page 268, which deals with light bulbs manufactured at three factories of the GI Company. Suppose we have picked out a defective light bulb from the batch of light bulbs. What is the probability that the light bulb came from (a) factory I, (b) factory II, and (c) factory III? [18]

[Ans: 0.183; 0.427; 0.390] Theorem 9.17. Suppose that sample space S is partitioned into k events B1 , B2 , . . . , Bk , and that E is any event of sample space S. Then [18] P (B1 | E) + P (B2 | E) + · · · + P (Bk | E) = 1

Problems 9.A-1 (HL 5/04) A desk has three drawers. Drawer 1 contains three gold coins, Drawer 2 contains two gold coins and one silver coin, and Drawer 3 contains one gold coin and two silver coins. A drawer is chosen at random and from it a coin is chosen at random. (a) Find the probability that the chosen coin is gold. (b) Given that the chosen coin is gold, find the probability that Drawer 3 was chosen.   Ans: 23 ; 16 9.A-2 (HL 5/02) The probability that it rains during a summer’s day in a certain town is 0.2. In this town, the probability that the daily maximum Mr. Budd, compiled September 29, 2010


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temperature exceeds 25◦ C is 0.3 when it rains and 0.6 when it does not rain. Given that the maximum daily temperature exceeded 25◦ C on a particular   summer’s day, find the probability that it rained on that day. Ans: 19 9.A-3 (HL 5/99) In a bilingual school there is a class of 21 pupils. In this class, 15 of the pupils speak Spanish as their first language and 12 of these 15 pupils are Argentine. The other 6 pupils in the class speak English as their first language and 3 of these 6 pupils are Argentine. A pupil is selected at random from the class and is found to be Argentine. Find the probability that the pupil speaks Spanish as his/her first  12 language. Ans: 15 9.A-4 (HL Spec ’00) A girl walks to school every day. If it is not raining, the probability that she is late is 15 . If it is raining, the probability that she is late is 32 . The probability that it rains on a particular day is 14 . On one particular day the girl is late. Find the probability that  it was  raining on that day. Ans: 10 19 9.A-5 (HL 5/99) A new blood test has been shown to be effective in the early detection of a disease. The probability that the blood test correctly identifies someone with this disease is 0.99, and the probability that the blood test correctly identifies someone without that disease is 0.95. The incidence of this disease in the general population is 0.0001. A doctor administered the blood test to a patient and the test result indicated that this patient had the disease. What is the probability that the patient has the disease? [6 marks] [Ans: 0.00198] 9.A-6 (HL 5/97) The table below shows the number of components produced by machines A and B, and the probability of machines A and B producing faulty components. Machine A B

Number of components produced 2500 1500

Probability of producing faulty components 0.04 0.05

(a) If a component is chosen at random from the total number of components produced, what is the probability that it is faulty? (b) If a component is selected at random and it is found to be faulty, what is the probability that it is produced by machine A? 

Ans:

7 160 ,0.571



9.A-7 (HL 5/98) A student travels to school by bus, train, or taxi. She rolls an unbiased six-sided die to decide which method of transport to use. If she gets a 1, 2, or 3 on the die, she travels by bus and the probability of being Mr. Budd, compiled September 29, 2010


270

HL Unit 9 (Probability) late for school is 10%. If she gets a 4 or a 5 on the die, she travels by train and the probability of being late is 5%. If she gets a 6 on the die, she travels by taxi and the probability of being late is 2%. The cost of each method of transport for a journey to school is: bus $0.50; train $1.80; and taxi $9.00. This information is shown in the table below. Number on die 1, 2, or 3 4 or 5 6

Method of transport Bus Train Taxi

Probability of being late 10% 5% 2%

(a) Find the probability that the student is late for school on a day 7 chosen at random. Ans: 100 (b) On a day chosen at random, it is found that the student was late for school. Find the probability that she traveled to school by  bus on  that day. Ans: 57 9.A-8 (HL 5/96) Note: In this question all answers must be given exactly as rational numbers. A man can invest in at most one of two companies, A and B. The probability that he invests in A is 37 and the probability that he invests in B is 27 , otherwise he makes no investment. The probability that an investment yields a dividend is and 32 for company B.

1 2

for company A

The performances of the two companies are totally unrelated. Draw a probability tree to illustrate the various outcomes and their probabilities. What is the probability that the investor receives a dividend and, given that he does, what is the probability that it was from his  investment  9 in company A? [8 marks] Ans: 17 42 ; 17 9.A-9 (HL 96) Suppose that a woman must decide whether or not to invest in each company. The decisions she makes for each company are indepen3 dent and the probability of her investing in company A is 10 while the 6 probability of her investing in company B is 10 . [Editorâ&#x20AC;&#x2122;s note: she might invest in both.] Assume that there were the same probabilities of the investments yielding a dividend as in problem 8. (a) Draw a probability tree to illustrate the investment choices and whether or not a dividend is received. Include the probabilities for the various outcomes on your tree. (b) If she decides to invest in both companies, what is the probability that she receives a dividend from at least one of her investments? (c) What is the probability that she decides not to invest in either company? Mr. Budd, compiled September 29, 2010

Cost $0.50 $1.80 $9.00


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(d) If she does not receive a dividend at all, what is the probability that she made no investment? [10 marks]   7 28 ; 51 Ans: ; 56 ; 25 9.A-10 [18] Carla and Dave each toss a coin twice. The one who tosses the greatest number of heads wins a prize. Suppose that Dave has a fair coin [PD (H) = .5], while Carla has a coin for which the probability of head on a single toss is .4 [PC (H) = .4]. What is the probability that (a) Carla will win the prize, (b) a tie results, and (c) Dave will win the prize? [Ans: .24, .37, .39] 9.A-11 [18] An appliance store receives shipments of a certain type of radio from three of its warehouses: 20% come from warehouse I, 50% from warehouse II, and 30% from warehouse III. It is known from past experience that 5% of the radios from warehouse I are defective, while the corresponding percentages of defectives are 4% and 7% for warehouses II and III, respectively. (a) If a single radio is picked at random from the inventory at the store, what is the probability it is defective? [Equivalently, what is the fraction of defective radios among all the radios shipped to the appliance store from the three warehouses?] (b) What is the probability that a radio came from warehouse I if [equivalently, given that] it turns out to be defective? [Ans: 0.051; 0.196] 9.A-12 [18] A new test for detecting cancer has been developed. Suppose that 90% of the cancer patients in a large hospital reacted positively to the test, while 15% of the remaining patients in the hospital reacted positively to the test. [Assume that none of the remaining patients has cancer. That is, assume that the only patients in the hospital who have cancer are the cancer patients. Also, observe that a positive reaction to the test means that the test indicates that a person has cancer. That is, it is possible for a person who does not have cancer to show a positive reaction to the test. This is known as a false positive.] Suppose that 4% of the patients in the hospital are cancer patients [equivalently, 4% of the patients in the hospital actually have cancer]. What is the probability that a hospital patient picked at random who reacts positively to the test is a cancer patient? [Ans: 0.20]

Mr. Budd, compiled September 29, 2010


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9.2

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Combinatorics

International Baccalaureate 1.3 Counting principles,   including permutations and combinations. n The formula for also denoted by n Cr . r n The binomial theorem: expansion of (a + b) , n ∈ N. Resources Chapter 4 Counting Methods and Probability in Probability and Statistics by Ronald I. Rothenberg [18]

9.2.1

Permutations

Order does matter.

Example 9.2.1 [18] Determine the number of two-letter “words” that can be formed from four distinct letters, e.g., a, b, c, d, (a) if repetition is allowed;

[Ans: 16]

(b) if repetition is not allowed.

[Ans: 12]

Example 9.2.2 [18] Given four distinct objects, determine the number of permutations taken (a) one at a time, (b) three at a time, and (c) four at a time, respectively. Think of the objects as being the four letters a, b, c, and d, and the permutations as being the “words” that can be formed from the letters, where repetition of the letters is not allowed.

[Ans: 4; 24, 24]

Example 9.2.3 [18] Derive a general equation for n Pr , the number of permutations of n distinct objects taken r at a time.

The number of different ways of arranging (ordering) n distinct objects, taken r at a time is n! n Pr = (n − r)! Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability) Example 9.2.4 [18] Determine the number of three-digit numbers that can be formed from the digits 1, 2, 3, 4, and 5 if no digit is repeated in any one number. [Ans: 60]

The real Word Problems What if you have n objects which you want to arrange, but theyâ&#x20AC;&#x2122;re not all distinct? Example 9.2.5 How many different ways can you order the letters in: (a) (b) (c) (d) (e)

math budd lamar, ninja, elite calculus aardvark

Round table problems Example 9.2.6 Allison, Ashley, Eric, Hanmei, Helen, Hunil, Julie, Julio, Keith, Leslie, Lucky, Paul, Sarah, Scott, and Zuhdi sit around a green felt table for, how shall we say. . . an experiment in probability. Keith and Julie are known to get violent and must not sit together. Calculate the number of different ways these fifteen people can sit at a round table without Keith and Julie sitting together. [Ans: 74 724 249 600]

9.2.2

Combinations

Order doesnâ&#x20AC;&#x2122;t matter Example 9.2.7 For n objects arranged r at a time, i.e., n Pr , how many different arrangements are the same subset of r distinct items? Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 2: Combinatorics

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Example 9.2.8 List the combinations of the four letters a, b, c, and d, taken two at a time.

The number of combinations of n items taken r at a time is given by   n n! = n Cr = r r! (n − r)!

Example 9.2.9 Compute the number of, and list the combinations of the digits 1, 2, 3, and 4 taken three at a time. Determine the number of permutations that correspond to each combination. List the permutations for a particular combination.

[Ans: 4; 6; ]

Example 9.2.10 Given a miniature card deck consisting of three spades [ace, two, and three] and two hearts [ace and two]: (a) How many ways can three cards be dealt from this mini-deck of five cards? [Ans: 10] (b) List these three-card hands. (c) What is the probability i of obtaining two spades and a heart? h · 2 C1 =0.6 Ans: 3 C52 C 3 Note that a hand containing three cards is equivalent to a combination of five things taken three at a time. That is, order doesn’t matter.

Example 9.2.11 (HL 11/07) Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number of ways in which the remaining nine people may be allocated to the cars. (The arrangement of people within a particular car is not relevant).

[Ans: 1680] Mr. Budd, compiled September 29, 2010


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9.2.3

HL Unit 9 (Probability)

Binomial Expansion 2

(a + b)

3

(a + b)

4

(a + b)

=

(a + b) (a + b)

=

a2 + ab + ba + b2

=

a2 + 2ab + b2

=

(a + b) (a + b)

=

(a + b) a2 + 2ab + b2

=

a3 + 2a2 b + ab2 + a2 b + 2ab2 + b3

=

a3 + 3a2 b + 3ab2 + b3

=

(a + b) (a + b)

=

(a + b) a3 + 3a2 b + 3ab2 + b3

=

a4 + 3a3 b + 3a2 b2 + ab3 + a3 b + 3a2 b2 + 3ab3 + b4

=

a4 + 4a3 b + 6a2 b2 + 4ab3 + b4

2



3



“The calculations are already fairly complex and it is worth looking at these results for the underlying pattern. There are three main features to the pattern. Looking at the fourth power example above, these patterns are: 1. “The powers of a. These start at 4 and decrease: a4 , a3 , a2 , a1 , a0 . Remember that a0 = 1. 2. “The powers of b. These start at 0 and increase: b0 , b1 , b2 , b3 , b4 . Putting these two patterns together gives the final pattern of terms in which the sum of the indices is always 4: . . . a4 + . . . a3 b + . . . a2 b2 + . . . ab3 + . . . b4 . 3. “The coefficients complete the pattern. These coefficients arise because there is more than one way of producing most of the terms. Following the pattern begun above, produces a triangular pattern of coefficients known as Pascal’s Triangle. Blaise Pascal (1623-1662) developed early probability theory but is luck to have this triangle named after him as it had been studied by Chinese mathematicians long before he was born.”[6]

Pascal’s Triangle 1 1 1 1 1

1 2

3 4

1 3

6

1 4

1 Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 2: Combinatorics

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277

1 5 10 10 5 1 6 15 20 15 6 1

“The numbers in the body of the triangle are found by adding the two numbers immediately above and to either side. It is worth looking at the process developed on the previous page and comparing it with the same process that produces Pascal’s Triangle to confirm that the two are effectively the same. “In using the Pascal’s Triangle method to find the binomial coefficients, all that is necessary is to select the correct row. In the case of the example under discussion, the appropriate row is 1

4

6

4

1

and the complete result follows: 4

(a + b) = a4 + 4a3 b + 6a2 b2 + 4ab3 + b4

Example 9.2.12 (a) How many different 3 letter combinations can you make with R’s and G’s, if you are allowed to repeat letters? (b) How many different ways can you order: (a) (b) (c) (d)

3 2 1 3

R’s; R’s and 1 G; R and 2 G’s; G’s 3

(c) Find the expansion of (r + g)

(d) How many different 4 letter combinations can you make with R’s and G’s, if you are allowed to repeat letters? (e) How many different ways can you order: (a) (b) (c) (d) (e)

4 3 2 1 4

R’s; R’s and 1 G; R’s and 2 G’s; R and 3 G’s; G’s. 4

(f) Find the expansion of (r + g)

(g) Add up the numbers in each row of Pascal’s triangle. Mr. Budd, compiled September 29, 2010


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Binomial Theorem “An alternative to using Pascal’s Triangle to find the coefficients is to use com4 binatorial numbers. If expanding (a + b) the set of coefficients are:           4 4 4 4 4 = 1, = 4, = 6, = 4, =1 0 1 2 3 4 which is the same as those given by Pascal’s Triangle.[6]

k

(a + b)

        k k 0 k k−1 1 k k−r r k 0 k b + ... + b + ... + = a b + a a a b 0 1 r k k   k X k n−r r X n−r r = a b = b n Cr a r r=0 r=0

Example 9.2.13 [6] Expand 

 2x −

2 x

3 = =

2 2x − x

3

0  1  2  3 2 2 2 2 2 1 0 + 3 C1 (2x) − + 3 C2 (2x) − + 3 C3 (2x) − x x x x 8 24 − 3 8x3 − 24x + x x 3

3 C0 (2x)



Example 9.2.14 [6] Find the coefficient of the term in x6 in the 7 expansion of 3x2 − 6 3 3 7 The x6 term in 3x2 − 6 is the term containing 3x2 , because x2 = x6 . 3  7−3 The coefficient is 73 , so the term is 7 C3 3x2 (−6) = 35 × 27x6 × 1296 = 6 1224720x . The coefficient of the term is 1224720.  Example 9.2.15 Find the constant term in the expansion of

2 x2 − x 3

9 .

  Ans: − 1792 9 Mr. Budd, compiled September 29, 2010


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Example 9.2.16 (HL 99) Given that 5

6

(1 + x) (1 + ax) ≡ 1 + bx + 10x2 + . . . . . . + a6 x11 , find the values of a, b ∈ Z∗ .

[Ans: −2, −7]

6

Example 9.2.17 [6] Write the expansion of (1 + x) and hence find 1.016 . 6

6

(1 + x) = 1 + 6x + 15x2 + 20x3 + 15x4 + 6x5 + x6 . 1.016 = (1 + 0.01) = 2 3 4 5 6 1+6 (0.01)+15 (0.01) +20 (0.01) +15 (0.01) +6 (0.01) +(0.01) = 1+6 (0.01)+ 15 (0.0001)+200.000001+15 (0.00000001)+6 (0.0000000001)+0.000000000001 = 1 + 0.06 + 0.0015 + 0.000020 + 0.00000015 + 0.0000000006 + 0.000000000001 = 1.061520150601.

Problems 9.B-1 (MM 96) Xiang is one of eight people from whom three must be chosen in order to form a committee. The choice is made randomly. What  is the  probability that Xiang is one of the chosen three? Ans: 83 9.B-2 (SL 85) A computer prints out three digits chosen at random from the range 0 to 9 inclusive. Calculate the probability (a) that all three digits are different, and (b) that the third digit differs from the other two. [Ans: 0.72;0.81] 9.B-3 (SL 85) Two friends are invited to a dinner party. The dining table is a long, narrow table with six chairs on each side and no chairs at the ends. The seating plan has been organized randomly. What is the probability that the two friends will either be sitting next to each other or directly opposite each other? Ans: 336 720 9.B-4 (HL Spec ’08) A room has nine desks arranged in three rows of three desks. Three students sit in the room. If the students randomly choose a desk findthe probability that two out of the front three desks are chosen.  3 Ans: 14 Mr. Budd, compiled September 29, 2010


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9.B-5 (HL 11/07) In a promotion to try to increase the sales of a particular brand of cereal, a picture of a soccer player is put in each packet. There are ten different pictures available. Each picture is equally likely to be found in any packet of breakfast cereal. Charlotte buys four packets of breakfast cereal. (a) Find the probability that the four pictures in these packets are all different. (b) Of the ten players whose pictures are in the packets, her favourites are Alan and Bob. Find the probability that she finds at least one picture of a favourite player in these four packets. [Ans: 0.504; 0.590] 9.B-6 (HL 5/06) There are 10 seats in a row in a waiting room. There are six people in the room. (a) In how many different ways can they be seated? (b) In the group of six people, there are three sisters who must sit next to each other. In how many different ways can the group be seated? [Ans: 151200; 10080] 9.B-7 (HL 5/05) A team of five students is to be chosen at random to take part in a debate. The team is to be chosen from a group of eight medical students and three law students. Find the probability that (a) only medical students are chosen; (b) all three law students are chosen. [Ans: .121; .0606] 9.B-8 (HL Spec â&#x20AC;&#x2122;05) There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a committee. Calculate the probability that the committee contains (a) two girls and two boys; (b) students all of the same gender. [Ans: .368; .130] 9.B-9 (HL 11/03) A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the number of ways the committee may be chosen. [Ans: 55] 9.B-10 (HL 11/01) How many four-digit numbers are there which contain at least one digit 3? [Ans: 3168] 9.B-11 (HL 11/00) In how many ways can six different coins be divided between two students so that each student receives at least one coin? [Ans: 62] Mr. Budd, compiled September 29, 2010


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9.B-12 (HL 5/00) Mr. Blue, Mr. Black, Mr. Green, Mrs. White, Mrs. Yellow, and Mrs. Red sit around a circular table for a meeting. Mr. Black and Mrs. White must not sit together. Calculate the number of different ways these six people can sit at a table without Mr. Black and Mrs. White sitting together. [Ans: 72] 9.B-13 (HL 96) How many times must a pair of dice be thrown so that there is a better than even chance of obtaining a double, that is, the same number on both dice? [Ans: 4] 9.B-14 (HL 96) Using all the letters of CALCUTTA, (a) how many different arrangements of letters can be found? (b) how many of these arrangements begin and end with the letter C? [Ans: 5040,180] 9.B-15 [18] Given a collection of seven people, in which three are female and four are male: (a) How many committees with three people can be formed using people from the collection of seven people? [Ans: 35] (b) How many committees with one female and two males can be formed? [Ans: 18] (c) If a committee of three people is picked at random [say, by drawing lots], what is the probability that the committee contains one   female 18 and two males. Ans: 35 9.B-16 (HL 11/06) Express [Ans: −54, 30]

√ 3  3 − 3 in the form a 3 + b , where a, b ∈ Z.

 7 1 7 9.B-17 (HL 11/00) The coefficient of x in the expansion of x + 2 is . ax 3 Find the possible values of a. [Ans: 3, −3] 10

9.B-18 (HL 00) Find the coefficient of x7 in the expansion of (2 + 3x) , giving your answer as a whole number. [Ans: 2099520] 9.B-19 (HL 99) Given that 5

6

(1 + x) (1 + ax) ≡ 1 + bx + 10x2 + . . . . . . + a6 x11 , find the values of a, b ∈ Z∗ .

[Ans: −2, −7] Mr. Budd, compiled September 29, 2010


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9.B-20 (HL 97) Find the term independent of x in the binomial expansion of 

 Ans:

7 486

9 2 1 x â&#x2C6;&#x2019; 2 9x

9 .



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HL Unit 9, Day 3: Probability Distribution Functions

9.3

285

Probability Distribution Functions

International Baccalaureate 6.9 Concept of discrete and continuous random variables and their probability distributions. Expected value (mean), mode, median, variance, and standard deviation. Knowledge and use of the formula for E (X) and Var (X). Applications of expectations, for example, games of chance. 6.10 Binomial distribution, its mean and variance. Conditions under which random variables have binomial distribution. Resources Chapter 9 in [18]

9.3.1

Discrete Random Variables

Example 9.3.1 (a) Without me telling you how to find it, what do your instincts tell you about how to find the average or mean value from a roll of a die? (b) What changes if the die has two sides with a 1, three sides with a 3, and one side with a 5? Definition 9.14. [18] The expected value of a discrete random variable X with probability function P(X = x) = f (x) is given by X E (X) = x f (x) x

The expected value of X is calculated essentially the same way as the mean of X, with weighting based on probability instead of frequency. Weird notation rule: the X stands for what value the random variable is going to take. No mortal knows what this value is, as it will be determined in the future. x is a placeholder for all the different possible values that X might possibly take.  E X 2 would essentially be the average of the squares, and the formula would be  X 2 E X2 = x f (x) x

where f (x) is the probability function, f (x) = P(X = x). Mr. Budd, compiled September 29, 2010


286

HL Unit 9 (Probability)  Example 9.3.2 Find E (X) and E X 2 where X represents the roll obtained from (a) a normal die; (b) a die with two sides with one dot, three sides with three dots, and one side with five dots.

Theorem 9.18. [18] Suppose that X is a discrete random variable with probability function f (x), and that the random variable Y is a function of X given by Y = g(x). Then, X E(Y ) = E (g(x)) = g(x)f (x) x

where the summation is over all possible values of X. Theorem 9.19. [18] If a and b are constants, then E (aX + b) = aE (X) + b. Example 9.3.3 Show that E (a g(x) + b h(x)) = aE (g(x))+bE (h(x)). Do a lot of functions work like that, e.g., is sin (a g(x) + b h(x)) = a sin (g(x)) + b sin (h(x))? Example 9.3.4 [18] Suppose a pair of fair dice is tossed, and the random variable X is equal to the sum of the numbers that occur on the two dice. Determine E (X), the expected value of X.

[Ans: 7]

9.3.2

Variance

The variance of X is symbolized by Var (X). It is given by   2 Var (X) = E (X − µ) where µ = E (X) In actuality, we usually compute the variance with the following theorem: Theorem 9.20.  2 Var (X) = E X 2 − [E (X)] Example 9.3.5 Find Var (X) both ways where X represents the roll obtained from Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 3: Probability Distribution Functions

287

(a) a normal die; (b) a die with two sides with one dot, three sides with three dots, and one side with five dots. Check your answer on the calculator.

Example 9.3.6 (extension of MM 5/03) A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. (a) What is the probability that exactly two of the three apples are red? (b) Create a probability distribution chart. (c) Find the mean, median, and mode.

[Ans: 2.64, 3, 3]

(d) Find the standard deviation and variance.

Example 9.3.7 (SL Spec â&#x20AC;&#x2122;05) Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without replacement. Let X denote the number of red balls chosen. (a) Draw a tree diagram to represent the above information, including the probability of each event. Hence create a probability distribution table for X. (b) Calculate E (X), the mean number of red balls chosen. [Ans: 0.8] Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from Bag B. (c) Draw a tree diagram to represent the above information, including the probability of each event. Hence find the probability distribution for Y , where is the number of red balls chosen.  Ans: 2,16,12 30 A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B. (d) Calculate the probability that two red balls are chosen. (e) Given that two red balls are chosen, find the conditional prob-  ability that a 1 or 6 was rolled on the die. Ans: 0.3; 19 Example 9.3.8 (adapted from SL Nov â&#x20AC;&#x2122;99) Mr. Budd, compiled September 29, 2010


288

HL Unit 9 (Probability) (a) A single bag contains 8 red marbles and 7 blue marbles. Two marbles are drawn without replacement. What is expected number of red marbles drawn? (b) Bag 1 contains 4 red marbles and 5 blue marbles. Bag 2 contains 4 red marbles and 2 blue marbles. A marble is chosen from Bag 1, followed by a marble from Bag 2. What is expected number of red marbles drawn?

Example 9.3.9 (MM 5/96) A computer is programmed to generate a sequence x1 , x2 , x3 , . . . of random single digits, each of which takes a value from 0 to 9 inclusive. Each digit is equally likely. Let X be the discrete random variable which represents the value of a digit. Find the mean and variance of X.

[Ans: 4.5, 8.25] Theorem 9.21. [18] If a and b are constants, then Var (aX + b) = a2 Var (X).

9.3.3

Other Measures

The standard deviation is the square root of the variance. The median will be the x-value where the cumulative probability distribution function is 0.5. The mode will be the x-value where the probability function (for a discrete variable) is the largest.

Example 9.3.10 A die has two sides with one dot, three sides  with three dots, and one side with five dots. Find E (X), E X 2 , Var (X), standard deviation, median, and mode.

Example 9.3.11 (MM 11/96) A fair game is a game in which the expected profit of any player is zero. A player pays $3 to throw a pair of unbiased dice, and then receives a payment, $X, depending on the outcome of the throw, as follows: â&#x20AC;˘ X = 0, if the the total score on the dice is less than 7; â&#x20AC;˘ X = 3, if the total score is 7; â&#x20AC;˘ X = k, if the total score is 11 or 12; Mr. Budd, compiled September 29, 2010


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• X = 4, otherwise. The player’s profit is $Y , where Y = X − 3. (a) Show that E (Y ) =

1 12

(k − 14).

(b) Given that this is a fair game, find the value of k. [Ans: 14] Hence determine the profit earned by the player if his throw results in a total score of 11. [Ans: $11] (c) Find [Ans: −3] [Ans: 0]   Ans: 85 6   5 Ans: 12

(a) the mode of Y ; (b) the median of Y ; (c) the variance of Y ; (d) P(Y > 0).

Example 9.3.12 (adapted from HL 5/02) Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice. Let X denote the largest number shown on the four dice.  x 4 (a) Show that P(X ≤ x) = , for x = 1, 2, . . . , 6 6 (b) Find the median. (c) Copy and complete the following probability distribution table. x P(X = x)

1 1 1296

2 15 1296

3

4

5

(d) Calculate E (X)

9.3.4

6 671 1296  Ans:

575 1296



Binomial Distribution

Characteristics of the Binomial Probability Situation • a fixed number n of independent, identical trials of the same experiment. • each trial can be classified only as success or failure; • each trial has the same probability p of success, and a probability q = 1−p of failure; • the variable X represents the number of successes out of n trials. Mr. Budd, compiled September 29, 2010


290

HL Unit 9 (Probability) Example 9.3.13 What is the probability in 5 trials of getting SSSFF, i.e., three successes followed by two failures, if the probability of each success is p? What is the probability of SFSFS? How many different words can be made up of the letters SSSFF? Example 9.3.14 Develop a formula for getting x successes out of n trials, each with probability p of success. How is this related to n the expansion of (p + q) ?

If X ∼ B(n, p), i.e., X is a random variable representing the number of successes in n trials, with eachtrial having p probability of success, then P(X = x) = n−x n x = nx px q n−x x p (1 − p) In actuality, we do this on the calculator as binompdf(n,p,x). Example 9.3.15 (MM 93S) In a city with a large population, 65% of the population are vaccinated against infection by a certain disease. However, about 5% of those vaccinated nevertheless eventually acquire the disease, whilst about 20% of those unvaccinated acquire it. (a) If 5 unvaccinated people are selected at random from the city, find the probability that (a) none of them has the disease; (b) exactly two of them have the disease; (c) at least three of them have the disease.

[Ans: 0.328] [Ans: 0.205] [Ans: 0.0579]

(b) Show that the probability that a person has the disease is just over 10%. (c) Show that the probability that an infected person is vaccinated is about 0.32. Example 9.3.16 (HL 5/01) In a game, the probability of a player 1 scoring with a shot is . Let X be the number of shots the player 4 takes to score, including the scoring shot. (You can assume that each shot is independent of the others.)   9 (a) Find P(X = 3). Ans: 64 (b) Find the probability that the player will have at least three misses before scoring twice. Ans: 189 256 (c) Prove that the expected value of X is 4. (You may use the −2 result (1 − x) = 1 + 2x + 3x2 + 4x3 . . .) Mr. Budd, compiled September 29, 2010


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Example 9.3.17 [6] A fair die is rolled six times. If X denotes the number of fours obtained, find (a) E (X) [What would you expect?] (b) the mode of X (c) the standard deviation and variance

Mean and Variance Example 9.3.18 Using your calculator, (a) Put seq(X,X,0,5) into L1 (b) Put binompdf(5,0.2,L1) into L2 (c) Put binomcdf(5,0.2,L1) into L3 (d) What does binompdf do? What about binomcdf? (e) How many successes out of five would you expect, if the probability of success in each trial is 20%? (f) Use your calculator to find the mean and the variance. (g) Do this again, using a probability other than 0.2. Find a pattern for the expected value and the variance.

For the binomial probability distribution, E (X) = np Var (X) = np (1 − p) = npq

Example 9.3.19 (a) Show that the binomial distribution is a proper probability distribution in that all the probabilities add to 1. (b) Prove that E (X) = np. You may want to use the fact that sum of all the probabilities is 1 even for n − 1 trials.  (c) Similarly, prove that E X 2 = np ((n − 1) p + 1). Hence show that Var (X) = npq.

Example 9.3.20 [6] The random variable X has a binomial distribution such that E (X) = 8 and Var (X) = 4.8. Find P(X = 3).

[Ans: 0.0123] Mr. Budd, compiled September 29, 2010


292

HL Unit 9 (Probability)

Cumulative Distribution and Inequalities The cumulative probability distribution, i.e., P(X ≤ x) is given by binomcdf(n,p,x). • P(X ≤ x) • P(X < x) = P(X ≤ x − 1) (for x > 0) • P(X > x) = 1 − P(X ≤ x) • P(X ≥ x) = 1 − P(X < x) = 1 − P(X ≤ x − 1) (for x > 0)

Example 9.3.21 (MM 98) A factory produces light bulbs in large batches. In any batch 5% of the light bulbs are defective. The light bulbs are sold in packs of five. (a) Find the probability that a pack contains at least one defective bulb. [Ans: 0.226] (b) Find the probability that a pack contains more than two defective bulbs. [Ans: 0.00116] The quality of the batches is checked in the factory by taking a sample of 50 bulbs from a batch and testing them to find the number of defective bulbs in the sample. (a) Find the mean and standard deviation of the number of defective bulbs in a sample of 50 light bulbs. [Ans: 2.5, 1.54] (b) An entire batch is rejected if the number of faulty light bulbs found in a sample of 50 exceeds two standard deviations above the mean. Find the least number, n, of faulty bulbs which would justify rejection. [Ans: 6 > 5.58] (c) What is the probability that the number of faulty light bulbs exceeds two standard deviations above the mean?

Example 9.3.22 (HL 11/06) A bag contains a very large number of ribbons. One quarter of the ribbons are yellow and the rest are blue. Ten ribbons are selected at random from the bag. (a) Find the expected number of yellow ribbons selected. (b) Find the probability that exactly six of these ribbons are yellow. (c) Find the probability that at least two of these ribbons are yellow. (d) Find the most likely number of yellow ribbons selected. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 3: Probability Distribution Functions

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(e) What assumption have you made about the probability of selecting a yellow ribbon? [Ans: 2.5; .0162; .756; 2;] Example 9.3.23 (HL 5/06) Andrew shoots 20 arrows at a target. He has probability of 0.3 of hitting the target. All shots are independent of each other. Let X denote the number of arrows hitting the target. (a) Find the mean and standard deviation of X. (b) Find (a) P(X = 5); (b) P(4 ≤ X ≤ 8). [Ans: 4.2, 2.05; .179, .780] (c) P(X = 5 | 4 ≤ X ≤ 8) Bill also shoots arrows at a target, with probability of 0.3 of hitting the target. All shots are independent of each other. (a) Calculate the probability that Bill hits the target for the first time on his third shot. (b) Calculate the minimum number of shots required for the probability of at least one shot hitting the target to exceed 0.99. [Ans: 0.147; thirteen]

Preview of the Normal Distribution Example 9.3.24 Let X ∼ Bin(10, 0.5). (a) Find the mean and standard deviation. (b) Find the probability that X is (a) within one standard deviation of the mean; (b) within two standard deviations of the mean; (c) more than two standard deviations above the mean. (c) Repeat for n = 20, 40, 80, 160, . . . and/ or for p = 0.4, 0.6, 0.25, 0.75

Problems 9.C-1 (HL Spec ’05) A discrete random variable X has probability distribution given by P(X = x) = k (x + 1) , where x is 0, 1, 2, 3, 4 Mr. Budd, compiled September 29, 2010


294

HL Unit 9 (Probability)

(a) Show that k =

1 . 15  Ans:

(b) Find E (X).

8 3



9.C-2 (HL 5/00) The probability distribution of a discrete random variable X is given by  x 2 , for x = 1, 2, 3, . . . . P(X = x) = k 3 Find the  value of k. Ans: 21 9.C-3 (HL 5/99) A biased die with four faces is used in a game. A player pays 10 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters the player receives in return for each score. Score Probability Number of counters players receives

1

2

3

4

1 2

1 5

1 5

1 10

4

5

15

n

Find the value of n in order for the player to get an expected return of 9 counters per roll. [Ans: 30] 9.C-4 (HL 5/98) A discrete random variable X has the following probability distribution. x

P(X = x)

0

1 8

1

3k

2

1 6k

3

1 4

4

1 6k

 Ans:  Ans:

(a) Find the exact value of k. (b) Calculate P(0 < x < 4).

3 16  27 32



9.C-5 (HL 5/98) When a biased die is rolled the numbers from 1 to 6 appear according to the following probability distribution. Number on the uppermost face of the die Probability

1

2

3

4

5

6

2 9

1 9

2 9

1 9

2 9

1 9

Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 3: Probability Distribution Functions

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If X is the random variable which takes on values 1 through 6 of the above 26 table find the exact values of E (X) and Var (X). Ans: 10 3 , 9 9.C-6 (HL 5/97) An accounting firm randomly selects three employees from ten applicants to attend a convention. Six of the applicants are men and four of them are women. Assume X is the number of women selected and that each selection is independent of the others. Find the expected value, E (X), and the variance, Var (X). 9.C-7 (SL 89) A standard unbiased die is thrown 18 times. Find the probability that there are precisely 3 sixes, giving your answer correct to three significant figures. [Ans: 0.245] 9.C-8 (SL 90) When Alexis plays Boris at tennis, Alexis wins with a probability of 43 . If they play each other seven times, find the probability that Alexis wins five times and Boris twice. [Ans: 0.311] 9.C-9 (SL 90) A group of 294 people is selected at random. Assuming days of the week are equally likely as birthdays, find (a) the expected number of people in the group born on a Friday, and [Ans: 42] (b) the standard deviation of the number of people in the group born on a Friday. [Ans: 6] 9.C-10 (SL 82) During a national election 3 votes in every 5 voted for the winner, President X. The voters cast their votes independently. Find the probability that in a random sample of 5 voters, exactly 3 voted for President X. [Ans: 0.346] 9.C-11 (SL 93) An unbiased ten-faced die has the numbers 1, 2, . . . , 10 on the faces. The die is thrown ten times. (a) What is the probability of obtaining a 6 on the fourth throw? [Ans: 0.1] (b) What is the probability of obtaining a 6 exactly four times in the ten throws? [Ans: 0.0112] 9.C-12 (HL 5/03) When a boy plays a game at a fair, the probability that he wins a prize is 0.25. He plays the game 10 times. Let X denote the total number of prizes that he wins. Assuming that the games are independent, find (a) E (X); (b) P(X â&#x2030;¤ 2).

[Ans: 2.5] [Ans: 0.526]

9.C-13 (HL 5/02) When John throws a stone at a target, the probability that he hits the target is 0.4. He throws a stone 6 times. Mr. Budd, compiled September 29, 2010


296

HL Unit 9 (Probability) (a) Find the probability that he hits the target exactly 4 times. [Ans: 0.138] (b) Find the probability that he hits the target for the first time on his third throw. [Ans: 0.144]

9.C-14 (HL 5/01) X is a binomial random variable, where the number of trials is 5 and the probability of success of each trial is p. Find the value of p if P(X = 4) = 0.12. [Ans: 0.459] 9.C-15 (HL Spec â&#x20AC;&#x2122;99) In a school, 13 of the students travel to school by bus. Five students are chosen at random. Find the probability that exactly 3 of them travel to school by bus. [Ans: 0.165] 9.C-16 (HL 5/97) If 30% of college students do not graduate, find the probability that out of 6 randomly selected college students exactly 4 of them will graduate. [Ans: 0.324] 9.C-17 (HL 5/96) How many times must a pair of dice be thrown so that there is a better than even chance of obtaining a double, that is, the same number on both dice? [Ans: 4] 9.C-18 (MM 99) A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted. The disc is then replaced in the box. (a) In eight such selections, what is the probability that a black disc is selected i. exactly once? ii. at least once?

[Ans: 0.393] [Ans: 0.656]

(b) The process of selecting and replacing is carried out 400 times. What is the expected number of black discs that would be drawn? [Ans: 50] 9.C-19 (MM 97) Refer to problem ?? on page ??. When a certain biased sixfaced die is thrown the score is the random variable Y , and P(Y = y) = 2 k (y â&#x2C6;&#x2019; 3.5) + 0.1375. (a) Find k, and hence calculate P(Y = y) for y = 1, . . . , 6. [Ans: .2,.16,.14,.14,.16,.2] (b) Let Z be the random variable which represents the number of sixes obtained when the biased die is thrown 5 times. Calculate i. P(Z = 0); ii. P(Z = 1); iii. P(Z > 1).

[Ans: 0.402] [Ans: 0.402] [Ans: 0.196]

9.C-20 (MM 11/96) Refer to the fair game in problem 9.3.3 on page 288. A fair game is a game in which the expected profit of any player is zero. A player pays $3 to throw a pair of unbiased dice, and then receives a payment, $X, depending on the outcome of the throw, as follows: Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 3: Probability Distribution Functions

297

• X = 0, if the the total score on the dice is less than 7; • X = 3, if the total score is 7; • X = k, if the total score is 11 or 12; • X = 4, otherwise. The player’s profit is $Y , where Y = X − 3. (a) Show that E (Y ) =

1 12

(k − 14).

(b) Given that this is a fair game, find the value of k. Hence determine the profit earned by the player if his throw results in a total score of 11. [Ans: 14; $11] (c) When the player plays this game 8 times, W is the number of times his profit exceeds zero. Find P(W > 5), giving your answer to 3 decimal places. [Ans: 0.061] 9.C-21 (MM 96) Refer to problem 9.3.2 on page 288. A computer is programmed to generate a sequence x1 , x2 , x3 , . . . of random single digits, each of which takes a value from 0 to 9 inclusive. Each digit is equally likely. Let X be the discrete random variable which represents the value of a digit. If the computer generates a sequence of 8 random digits, find (a) the probability of obtaining no ‘4’ in the sequence;

[Ans: 0.430]

(b) the probability of obtaining exactly three ‘4’s in the sequence. [Ans: 0.0331] 9.C-22 (SL 93) In an advertising campaign a rail company presents to each child traveling by train an unmarked package. The package contains either a model engine or a model passenger carriage. The company claims that one third of the packages contain engines. Six packages are chosen at random (a) What is the probability that exactly two of them contain engines? [Ans: 0.329] (b) What is the probability that more than two of them contain engines? [Ans: 0.320] 9.C-23 (SL 89) If in tossing a supposedly unbiased coin ten times you obtained 8 heads and 2 tails, would you suspect the coin to be biased in favour of heads? Give reasons for your answer.

[13 marks]

9.C-24 (SL 82) Assume that equal numbers of birthdays occur in each of the twelve months of the year. [10 marks] (a) What is the probability that a person chosen at random has  a birth 1 day in June? Ans: 12 Mr. Budd, compiled September 29, 2010


298

HL Unit 9 (Probability) (b) Three people are chosen at random. Explaining your methods find the probabilities that: h  i 1 3 i. all three have birthdays in June Ans: 12 h  i 11 3 ii. none of them has a birthday in June Ans: 12 iii. exactly two of them have a birthday in June [Ans: 0.0191]   55 iv. no two of them will have a birthday in the same month. Ans: 72

9.C-25 (SL 82) A manufacturer makes an instrument which goes through a testing procedure. He rejects any instrument which shows a fault. The instruments are tested in batches of three. With 1000 such batches the following results were obtained: No. rejected in batch No. of batches

0 393

1 428

2 155

3 24

(a) Calculate the mean number rejected per batch of three and hence deduce the probability p that an instrument is rejected. [Ans: 0.81, 0.27] (b) Using the value of p from 25a as the basis of a binomial model, the expected numbers of batches with 0 to 3 rejects are given, to the nearest whole numbers, in the table: No. rejected in batch No. of batches

0 389

1 432

2 160

3 20

Explain how the entry â&#x20AC;&#x2DC;432â&#x20AC;&#x2122; was obtained.

Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 4: Poisson Distribution

9.4

299

Poisson Distribution

International Baccalaureate 6.10 Poisson distribution, its mean and variance. Conditions under which random variables have Poisson distribution.

9.4.1

Binomial Distribution with Huge n and Tiny p

Example 9.4.1 Suppose there are 2500 students, each of whom has a probability of 0.00002 of getting a paper cut on the Stanford 10 test document. What is the probability that 2 students get paper cuts?

Let

• µ = np be the mean of the binomial distribution

• p be tiny (1 − p ≈ 1) and n be huge (n → ∞), so that np − p ≈ np, np − 2p ≈ np

Example 9.4.2 Use L’Hˆ opital’s Rule to show that q n → e−µ as n→∞ Mr. Budd, compiled September 29, 2010


300

HL Unit 9 (Probability)

lim q n

n→∞

= = = = =

n

lim (1 − p)  µ n lim 1 − n→∞ n   µ n  lim exp ln 1 − n→∞   n µ  exp lim n ln 1 − n→∞ n!  ln 1 − nµ exp lim 1 n→∞

n→∞

=

=

= =

n

1 1−

 µ   n2   exp  lim n→∞  1 − 2 n   1 µ µ  1 − n n2 n2  exp  lim · 2 n→∞ 1 n  − 2 n   −µ exp lim µ n→∞ 1 − n exp (−µ) 

µ n

= e−µ

Example 9.4.3 Find a formula, for a binomial distribution with huge n and tiny p, which relies not on n and p, but solely on µ = np. Justify that it is a proper distribution function, then justify that the mean is still µ. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 4: Poisson Distribution

301

  n k n−k P(X = k) = p (1 − p) k n! n−k pk (1 − p) = k! (n − k)! n (n − 1) (n − 2) · · · (n − (k − 1)) k n−k = p (1 − p) k! nk k n ≈ p (1 − p) (since n >> k > 2 > 1) k! k (np) n = (1 − p) k! µk  µ n = 1− k! n µk e−µ = k!

9.4.2

Poisson Distribution

These notes stolen from Mathematics for the International Student by Paul Urban, et al. [22] The binomial distribution describes the probability of getting a certain number of successes out of a finite number of trials. If there are n trials, there are only n + 1 different possible outcomes. The Poisson distribution is similar, in that you are counting a number of successes. The difference is that you are counting the number of successes over a specified period of time. Theoretically, there is no bound on the number of possible successes. Examples of Poisson distributions • the number of incoming telephone calls to a given phone per hour • the number of misprints on a typical page of a book • the number of fish caught in a large lake per day • the number of light rail accidents per month Characteristics of the Poisson Distribution In reality, conditions for an ideal Poisson distribution are rarely met. We look for the following: Mr. Budd, compiled September 29, 2010


302

HL Unit 9 (Probability) • the average number of successes (or occurrences) (µ) is the same for each interval (e.g., you must be just as likely to catch a fish in March as in September) • the typical number of occurrences in a given interval should be much less than the theoretical maximum [Hmm... but shouldn’t the theoretical maximum be infinite?] • the number of occurrences in disjoint intervals are independent of each other (e.g., catching a fish on Tuesday doesn’t effect your chances of catching a fish on Wednesday)

The Poisson distribution is given by P(X = x) =

mx e−m for x = 0, 1, 2, 3 . . . x!

m here is a parameter that indicates how frequently successes occur. As it turns out, m happens to be the expected value, as well as the variance! Example 9.4.4 Show that, for a Poisson distribution, ∞ X

P(X = x) = 1

x=0

for any parameter m. Additionally, show that (a) E (X) = m  2 (b) Var (X) = E X 2 − X + X − (E (X)) = E (X (X − 1)) + 2 E (X) − (E (X)) = m If the random variable X follows a Poisson distribution, with a mean of m, we write X ∼ P0 (m). Example 9.4.5 When Sandra proof–read 80 pages of a text book she observed the following distribution of X, the number of errors per page: X frequency

0 3

1 11

2 16

3 18

4 15

5 9

6 5

7 1

8 1

9 0

10 1

(a) Find the mean number of errors per page. (b) Using the mean number of errors per page, make a probability distribution chart for x = 0, 1, 2, . . . , 10 Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 4: Poisson Distribution

303

(c) Compare the theoretical probability to the relative frequency. (d) Multiply the probabilities by 80. Example 9.4.6 LamarCars rents cars to juniors. They have four cars which are hired out on a daily basis. The number of requests each day is distributed according to the Poisson model with a mean of 3. Determine the probability that (a) none of its cars are rented (b) at least 3 of its cars are rented (c) some requests will be refused (d) all are hired out given that at least two are

[Ans: 0.0498; 0.577; 0.185; 0.210]

Unknown mean Example 9.4.7 (HL 11/06) The random variable X follows a Poisson distribution. Given that P(X â&#x2030;¤ 1) = 0.2, find (a) the mean of the distribution; (b) P(X â&#x2030;¤ 2)

[Ans: 2.99; 0.424] Example 9.4.8 (HL 5/05) Let X be a random variable with a Poisson distribution, such that P(X > 2) = 0.404. Find P(X < 2).

[Ans: 0.331]

Changing the time interval Example 9.4.9 (HL 5/01) Patients arrive at random at an emergency room in a hospital at the rate of 15 per hour throughout the day. Find the probability that 6 patients will arrive at the emergency room between 08:00 and 08:15.

[Ans: 0.0908] Mr. Budd, compiled September 29, 2010


304

HL Unit 9 (Probability)

Problems 9.D-1 (HL Spec ’05) The random variable X has a Poisson distribution with mean 4. Calculate (a) P(3 ≤ X ≤ 5); (b) P(X ≥ 3); (c) P(3 ≤ X ≤ 5|X ≥ 3). [Ans: 0.547; 0.762; 0.718] 9.D-2 (HL 5/99) A supplier of copper wire looks for flaws before despatching it to customers. It is known that the number of flaws follow a Poisson distribution with a mean of 2.3 flaws per metre. (a) Determine the probability that there are exactly 2 flaws in 1 metre of the wire. (b) Determine the probability that there is at least one flaw in 2 metres of the wire. [Ans: 0.265, 0.990] 9.D-3 (HL 11/02) A machine produces cloth with some minor faults. The number of faults per metre is a random variable following a Poisson distribution with a mean of 3. Calculate the probability that a metre of the cloth contains five or more faults. Give your answer to four significant figures. [Ans: 0.8153] 9.D-4 (HL 11/04) Let X be a random variable with a Poisson distribution such 2 that Var (X) = (E (X)) − 6. (a) Show that the mean of the distribution is 3. (b) Find P(X ≤ 3). [Ans: 0.647] 9.D-5 (HL 5/06) The number of car accidents occurring per day on the Southwest Freeway follows a Poisson distribution with mean 1.5. (a) Find the probability that more than two accidents will occur on a given Monday. (b) Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day. [Ans: 0.191; 0.246] 9.D-6 Sven’s Florist Shop receives the following distribution of phone calls, X in number, between 9:00 a.m. and 9:15 a.m. on Fridays. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 4: Poisson Distribution X frequency

0 12

305 1 18

2 12

3 6

4 3

5 0

6 1

(a) Find the mean of the X distribution.

[Ans: 1.5]

(b) Compare the actual data with that generated by the Poisson model [Ans: 11.6, 17.4, 13.0, 6.5, 2.4, 0.7, 0.2] 9.D-7 A Poisson distribution has a standard deviation of 2.67. (a) What is its mean? (b) Find P(X = 2) (c) Find P(X ≤ 3) (d) Find P(X ≥ 5) (e) Find P(X ≥ 3|X ≥ 1) [Ans: 7.13; 0.0204; 0.0752; 0.839; 0.974] 9.D-8 One gram of radioactive substance is positioned so that each emission of an alpha-particle will flash on a screen. The emissions over 500 periods of 10 second duration are given in the following table. X frequency

0 91

1 156

2 132

3 75

4 33

5 9

6 3

(a) Find µ, the mean of the distribution.

7 1 [Ans: 1.694]

(b) Fit a Poisson model to the data and compare the actual data to that from the model. (c) Find the standard deviation of the distribution. How close is it to √ µ? [Ans: 1.292] [Ans: 92, 156, 132, 74, 32, 11, 3, 1] 9.D-9 Let X ∼ P0 (m) (a) Find m given that P(X = 1) + P(X = 2) = P(X = 3) (b) If m = 2.7, find i P(X ≥ 3) ii P(X ≤ 4|X ≥ 2). h Ans:

√ 3+ 33 ; 2

0.506; 0.818

i

9.D-10 (HL 11/03) The random variable X has a Poisson distribution with mean λ. (a) Given that P(X = 4) = P(X = 2) + P(X = 3), find the value of λ. Mr. Budd, compiled September 29, 2010


306

HL Unit 9 (Probability) (b) Given that λ = 3.2, find i P(X ≥ 2) ii P(X ≤ 3|X ≥ 2). [Ans: 6; 0.829, 0.520]

9.D-11 (HL 5/03) Two typists were given a series of tests to complete. On average, Mr. Brown made 2.7 mistakes per test while Mr. Smith made 2.5 mistakes per test. Assume that the number of mistakes made by any typist follows a Poisson distribution. (a) Calculate the probability that, in a particular test, i. Mr. Brown made two mistakes; ii. Mr. Smith made three mistakes; iii. Mr. Brown made two mistakes and Mr. Smith made three mistakes. (b) In another test, Mr. Brown and Mr. Smith made a combined total of five mistakes. Calculate the probability that Mr. Brown made fewer mistakes than Mr. Smith. [Ans: 0.245, 0.214, 0.0524; 0.464] 9.D-12 (HL 5/04) The random variable X has a Poisson distribution with mean λ. Let p be the probability that X takes the value of 1 or 2. (a) Write down an expression for p. (b) Sketch the graph of p for 0 ≤ λ ≤ 4. (c) Find the exact value of λ for which p is a maximum. h Ans: p = λe−λ +

λ2 −λ ; 2 e

√ i ; 2

Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 5: Continuous Random Variables

9.5

307

Continuous Random Variables

International Baccalaureate 6.9 Concept of discrete and continuous random variables and their probability distributions. Definition and use of probability density functions. Expected value (mean), mode, median, variance, and standard deviation. Knowledge and use of the formula for E (X) and Var (X). Applications of expectations, for example, games of chance.

9.5.1

Continuous Random Variables

Probability Density Function The probability density function for a continuous variable is like the frequency distribution for a discrete variable. For a discrete random variable X, f (x) = P(X = x). But. . . for a continuous random variable, there is zero chance that X will be exactly equal to anything. For example, what are the chances that your height will be exactly 62.0000 . . . inches? For a continuous random variable, we must talk about the probability that X is between two values, i.e., P(61.5 ≤ X ≤ 62.5). If f (x) is the probability R 62.5 density function for our variable X, then P(61.5 ≤ X ≤ 62.5) = 61.5 f (x) dx. For any probability density function f (x),

R∞ −∞

f (x) dx = 1.

Example 9.5.1 (MM 5/99) A continuous random variable X has the probability density function ( kx, for 0 ≤ x ≤ 5; f (x) = 0, elsewhere. Find the value of (a) k; (b) P(2 ≤ X ≤ 3);

  2 Ans: 25   Ans: 15

Example 9.5.2 (HL 5/05) The probability density function f (x) of the continuous random variable X is defined on the interval [0, a] Mr. Budd, compiled September 29, 2010


308

HL Unit 9 (Probability) by 1  x   8 27 f (x) =    8x2  0,

for 0 ≤ x ≤ 3, for 3 < x ≤ a, otherwise.

 Ans:

Example 9.5.3 The basic shape of the bell curve is given by e−x R∞ 2 (a) Find 0 e−x /2 dx R∞ 2 (b) Find −∞ e−x /2 dx (c) Let f (x) = ke−x

2

2

/2

54 11



.

/2

. Find a value of k that will make h f (x) bei a probability density function. Ans: √12π

Expected Value Definition 9.15. The expected value E (X), of a continuous random variable X with probability density function f (x), is given by Z ∞ E (X) = x f (x) dx −∞

Example 9.5.4 Suppose that X is the lifetime of a Powermate bat-  3 tery, in months, and that the pdf is given by f (x) = 32 4x − x2 for 0 < x < 4, and f (x) = 0, elsewhere. Calculate E (() X), the mean of X.

[Ans: two months]

Example 9.5.5 (MM 5/99) A continuous random variable X has the probability density function   2x , for 0 ≤ x ≤ 5; f (x) = 25 0, elsewhere. Find the value of E (X). Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 5: Continuous Random Variables

309  Ans:

10 3



Example 9.5.6 Let X be a continuous random variable with prob2 e−x /2 ability density function f (x) = √ for all real numbers. Find 2π E (X). Theorem 9.22. Suppose that X is a continuous random variable, with probability density function f (x), and that the random variable Y is function of X given by Y = g(X). Then, Z ∞ g(x) f (x) dx E(Y ) = E (g(x)) = −∞

Example 9.5.7 Let X be a continuous random variable with prob2 e−x /2 ability density function f (x) = √ for all real numbers. Find 2π  E X2 .

9.5.2

Variance

Recall that the variance of X is symbolized by Var (X). It is given by   2 Var (X) = E (X − µ) where µ = E (X) Variance for discrete and continuous random variables are defined the same way. In actuality, we usually compute the variance as follows:  2 Var (X) = E X 2 − [E (X)] Example 9.5.8 (MM 5/99) A continuous random variable X has the probability density function   2x , for 0 ≤ x ≤ 5; f (x) = 25 0, elsewhere. Find the value of the variance.  Ans:

25 18



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HL Unit 9 (Probability) Example 9.5.9 Let X be a continuous random variable with prob2 e−x /2 ability density function f (x) = √ for all real numbers. Find 2π Var (X).

9.5.3

Other Measures

Recall from our discussion of discrete variables: • The standard deviation is the square root of the variance. • The median will be the x-value where the cumulative probability density function is 0.5. • The mode will be the x-value where the probability density function (for a continuous variable) is the largest. Example 9.5.10 (MM 5/99) A continuous random variable X has the probability density function   2x , for 0 ≤ x ≤ 5; f (x) = 25 0, elsewhere. Find the value of the median of X. h

Ans:

√5 2

i

Example 9.5.11 (HL 5/03) A business man spends X hours on the telephone during the day. The probability density function of X is given by (  1 8x − x3 , for 0 ≤ x ≤ 2 12 f (x) = 0, otherwise. (a) (a) Write down an integral whose value is E (X). (b) Hence evaluate E (X). [Ans: 1.24] (b) (a) Show that the median, m, of X satisfies the equation m4 − 16m2 + 24 = 0. (b) Hence evaluate m.

[Ans: 1.29] Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 5: Continuous Random Variables (c) Evaluate the mode of X.

311 [Ans: 1.63]

Example 9.5.12 (HL Spec ’05) The continuous random variable X has probability density function   1 x 1 + x2  , for 0 ≤ x ≤ 2, f (x) = 6 0, otherwise.

1. Sketch the grap of f for 0 ≤ x ≤ 2. 2. Write down the mode of X. 3. Find the mean of X. 4. Find the median of X.

 Ans: ; 2;

68 45 ;

 1.61

Cumulative Density Function If f (x) represents the probability density function of a continuous random variable X, then the cumulative probability density function is given by Z x F (x) = f (t) dt −∞

Example 9.5.13 If one has F (x), how could you get f (x)?

Problems 9.E-1 (HL 11/06) The continuous random variable X has probability density function (a) Find the exact value of k. (b) Find the mode of X. (c) Calculate P(1 ≤ X ≤ 2). √  Ans: e2 − 1; 1;

1 2

ln 52



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9.E-2 (HL 5/06) The time, T minutes, required by candidates to answer a question in a mathematics examination has probability density function   1 12t − t2 − 20 , for 4 ≤ t ≤ 10 f (t) = 72 0, otherwise. (a) Find i. µ, the expected value of T ; ii. σ 2 , the variance of T . (b) A candidate is chosen at random. Find the probability that the time taken by this candidate to answer the question lies in the interval [µ − σ, µ]. [Ans: 6.5, 2.15; 0.321] 9.E-3 (HL 5/07 TZ2) A continuous random variable X has the probability density function f given by  8  , for 0 ≤ x ≤ 2 2 + 4) π (x f (x) =  0, otherwise. (a) State the mode of X. (b) Find the exact value of E (X). 

Ans: 0;

4 π

 ln 2

9.E-4 (HL 5/04) Let f (x) be the probability density function for a random variable X, where ( kx2 , for 0 ≤ x ≤ 2 f (x) = 0, otherwise. (a) Show that k =

3 . 8

(b) Calculate i. E (X); ii. the median of X.  Ans:

3 2;

√  3 4

9.E-5 (HL 11/02) The probability density function f (x), of a continuous random variable X is defined by   1 x 4 − x2  , for 0 ≤ x ≤ 2 f (x) = 4 0, otherwise. Calculate the median of X.

h p √ i Ans: 4 − 8 Mr. Budd, compiled September 29, 2010


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9.E-6 (HL 5/98) The probability density function of a random variable X is given by ( 6x (1 + x) , 0 < x < k; f (x) = 0, otherwise.   Ans: 12   11 Ans: 32

(a) Find the value of k. (b) Find the exact value of the mean of X.

9.E-7 (HL 5/96) Let X be a continuous random variable with the probability density function ( x + c, 0 ≤ x ≤ 4 f (x) = 8 0, elsewhere. Find the value of c and the expected value of X.



Ans: 0,

8 3



9.E-8 (MM 5/98) The random variable X represents the life-times of a manufacturer’s domestic appliances. The probability density function for X is given by ( 2 kx2 (20 − x) , 0 ≤ x ≤ 20, f (x) = 0, otherwise. (a) Show that k = 9.375 × 10−6 . (b) Show that the probability that an appliance has a life-time exceeding 15 years is just over 10%. [Ans: 0.104] (c) Find the mean and standard deviation of the life-times. [Ans: 10, 3.78] 9.E-9 (MM 5/97) In the State of Euphoria a domestic coin used in coin-operated telephones has mass (160 + x) decigram, with x in the range 0 ≤ x ≤ 1. The random variable X represents x. The probability density function f for X is given by f (x) = kx2 (1 − x), 0 ≤ x ≤ 1. (a) Show that k = 12. Hence find E (X) and thus show that the mean mass is 160.6 decigram. (b) Find Var (X) and thus show that the standard deviation of the mass is 0.2 decigram. The coin-operated telephones in the State of Euphoria accept coins with masses within a range of one standard deviation of the mean. (c) Show that for accepted coins, 0.4 ≤ x ≤ 0.8. Hence find the probability that a randomly selected coin is accepted by a telephone. [Ans: 0.64] Following complaints at the low level of acceptance the telephones are adjusted to accept 95% of the coins. Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability) (d) Show that this is achieved by changing the range of acceptable coins to within 0.4 decigram of the mean.

9.E-10 (MM 5/95) The diagram in Figure 9.5 shows the graph of the probability density function of a continuous random variable X. The function has non-zero values for 0 < x â&#x2030;¤ 4, and is linear over this interval. Figure 9.5: Problem 10

Find  Ans:  Ans:  Ans:

(a) k; (b) E (X); (c) Var (X).

1 2  8 3  8 9



9.E-11 (MM 11/96) The graph of the probability density function of a certain continuous random variable X is the line segment joining the points (â&#x2C6;&#x2019;2, 0) and (4, k), as shown in Figure 9.6. Figure 9.6: Problem 11

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315  Ans:

(a) Find k. (b) Find E (X).

1 3



[Ans: 2]

(c) Find the median of X.

[Ans: 2.24]

9.E-12 (MM 5/96) A bus is scheduled to arrive at a particular bus-stop at 15:25. T is the continuous random variable which represents the number of minutes by which a bus is late at the bus-stop. A researcher uses a probability density function f to model the lateness of the bus, where ( 2 k (t − 5) , 0 ≤ t ≤ 5, f (t) = 0, otherwise. (a) Find the value of k, and hence sketch a graph of f .

[Ans: 0.024]   98 (b) Find the probability that the bus arrives before 15:27. Ans: 125 (c) Find E (T ), and hence show that, correct to 3 significant figures

where µ = E (T ).

P(|T − µ| < 0.1) = 0.0675,   Ans: 0.0675 E (T ) = 45

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9.6

317

Normal Distribution

International Baccalaureate 6.11 The normal distribution. Properties of the normal distribution. Appreciation that the standardized value (z) gives the number of standard deviations from the mean. Standardization of normal variables. Use of a calculator (or tables) to find normal probabilities; the reverse process. distribution; the use of the standard normal distribution table.

9.6.1

Standardizing a Normal Distribution

Standard Normal Distribution 2

Recall that the basic shape of the bell curve is given by e−x /2 . In order to have R∞ a proper probability density function f (x) such that −∞ f (x) dx = 1, we need 2

to scale e−x /2 by a certain amount. The probability density function for the bell curve is given by 2 e−x /2 √ f (x) = 2π This is called the standardized normal probability density function. What is very nice about it is that it has a mean of 0 and a variance of 1. Example 9.6.1 Z is the standardised normal random variable with mean 0 and variance 1. Write integral expressions for (a) P(Z < a); (b) P(a < Z < b); Before graphing calculators, there was no good way to quickly find these in−x2 /2 tegrals, has no antiderivative. So, tables were made for F (x) = Z xsince e 1 −x2 /2 √ e dx. 2π −∞ Example 9.6.2 Use the statistical tables to find (a) P(Z < 2); (b) P(−2 < Z < 2); (c) P(−1 < Z < 2); Repeat using fnInt. Repeat again using normalcdf. Mr. Budd, compiled September 29, 2010


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Standardizing the Nonstandard: z-Scores Very rarely are we dealing with situations where the mean is zero and the standard deviation is one. For a general normal distribution, "  2 # 1 x−µ 1 exp − f (x) = √ 2 σ σ 2π where µ is the mean and σ is the standard deviation. Remember that without a calculator, this is very difficult. And it’s not very feasible to make tables for all the different possibilities of µ and σ. Even on the calculator, this would be a very grim function to have to enter every single time. Instead, we use a peculiarity of normally distributed functions. All normal distributions have the same probability that X is within one standard deviation of the mean. In other words, P(µ − σ < X < µ + σ) = P(−σ < X − µ < σ) = P(|X − µ| < σ) is the same for all normally distributed functions. Likewise, the probability is the same that X is within two or three or 2.5 standard deviations from the mean for all normal distributions. Also, P(X − µ < 2σ) (X is below two standard deviations above the mean) is the same for any normal distribution, regardless of µ, and regardless of σ. We use this property of normal distributions to standardize our data, and then we only need to use a single, standard distribution function. Since P(−σ   < X − µ < σ) X −µ is the same for all normal distributions, so is P −1 < < 1 , just as σ   X −µ P < 2 is constant. σ If X is normally distributed with mean µ and standard deviation σ, then we X −µ can make a new variable Z = which follows the same standard normal σ distribution, regardless of what µ and σ are. Example 9.6.3 For a normal random variable X with mean of 4 and variance of 9, find the probability that: (a) 1 < X < 7; (b) X < 10; (c) 1.6 < X < 5.

[Ans: 0.683] [Ans: 0.977] [Ans: 0.419]

Example 9.6.4 (MM 5/96) If X is a normal random variable with mean 13.5 and variance 6.25, find Mr. Budd, compiled September 29, 2010


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(a) P(X > 14.95); (b) P(|X â&#x2C6;&#x2019; 13.5| < 1.2).

[Ans: 0.281, 0.369] Example 9.6.5 (SL 90) In a manufacturing process, circular metal cylinders are being produced as components of a certain product. For a cylinder to by usable, its length must be between 8.44 cm and 8.64 cm and its diameter between 1.52 cm and 1.62 cm. Cylinders are produced which have lengths which are normally distributed about a mean of 8.54 cm with a standard deviation of 0.05 cm while, independently, the diameters are normally distributed about a mean of 1.57 cm with a standard deviation of 0.02 cm. (a) Show that the percentage of cylinders produced whose lengths fall outside the specified limits is 4.5%. (b) Find the percentage of cylinders produced whose diameters fall outside the specified limits. [Ans: 1.24%] (c) Show that the percentage of cylinders that cannot be used is 5.74%. (d) Find the probability that in a sample of five cylinders taken at random, four are usable and one is not. [Ans: 0.226] (e) If a cylinder is not usable, what is the probability it has a length in excess of 8.64 cm? [Ans: 0.397]

Inverse Normal Example 9.6.6 (HL 5/01) Z is the standardised normal random variable with mean 0 and variance 1. Find the value of a such that P(|Z| â&#x2030;¤ a) = 0.75 by (a) using fnInt; (b) using normalcdf; (c) using the statistical tables.

[Ans: 1.15] Example 9.6.7 (MM 5/95) Figure 9.7 shows the probability density function of a normal random variable X. Each of the shaded regions has area 0.08. Mr. Budd, compiled September 29, 2010


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Figure 9.7: MM 5/95

(a) Find the mean of X. (b) Find the standard deviation of X. Example 9.6.8 (MM 11/96) X is a normal random variable with variance 4. Given that P(x > 13.6) = 0.877, find the mean of X.

[Ans: 15.9] Example 9.6.9 (MM 5/94) A random variable X is normally distributed with mean 13.5 and variance 1.44. If P(12.3 ≤ X < k) = 0.6055, find the value of k, giving your answer to three significant figures.

[Ans: 14.4] Example 9.6.10 (HL 5/06) The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 200 g . The weights of 88% of the loaves are less than 220 g . Find the standard deviation.

[Ans: 17.0 g]

9.6.2

Normal Approximation to the Binomial Distribution

For a binomial distribution with p = 0.5 as n → ∞, the probability function looks more like a normal distribution. If a discrete variable has a binomial distriMr. Budd, compiled September 29, 2010


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bution where n is very large and p is close to 0.5, then the binomial distribution can be approximated with a normal distribution.

Example 9.6.11 (MM 5/98) Refer to problem 9.3.4 on page 292. A factory produces light bulbs in large batches. In any batch 5% of the light bulbs are defective. The light bulbs are sold in packs of five. The quality of the batches is checked in the factory by taking a sample of 50 bulbs from a batch and testing them to find the number of defective bulbs in the sample. (a) Find the mean and standard deviation of the number of defective bulbs in a sample of 50 light bulbs. [Ans: 2.5, 1.54] (b) An entire batch is rejected if the number of faulty light bulbs found in a sample of 50 exceeds two standard deviations above the mean. Find the least number, n, of faulty bulbs which would justify rejection. [Ans: 6 > 5.58] (c) A normal approximation is used to find the probability that the number of defective bulbs in a sample of 50 is at least n. (a) Explain why it is necessary to use n â&#x2C6;&#x2019; 0.5 instead of n in this case. (b) Hence show that over a long period of time approximately 2.6% of the batches will be rejected.

Example 9.6.12 (MM 5/99) Refer to problem 18 on page 296. A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted. The disc is then replaced in the box. The process of selecting and replacing is carried out 400 times. Use a normal approximation to the binomial distribution to estimate the probability that a black disc is selected (a) at least 48 times; (b) exactly 48 times.

[Ans: 0.647; 0.0576] Compare these numbers to the actual values. Why was this in the curriculum, but not any more? Mr. Budd, compiled September 29, 2010


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Problems 9.F-1 (SL 91) X is a random variable with a normal distribution. If the mean is 1 and the standard deviation is 0.5, find P(0.225 < X < 0.575) [Ans: 0.137] 9.F-2 (MM Spec ’93) A random variable X is distributed normally with mean 20.2 and standard deviation 4.1. Find P(16.1 ≤ X ≤ 25.9).

[Ans: 0.759]

9.F-3 (MM 5/97) The normal random variable X has mean 265 and standard deviation 8. Find P(X > 255). [Ans: 0.894] 9.F-4 (MM 5/00) In a certain country, annual salaries for nurses are normally distributed with a mean of $40 000, and 95% of the nurses can earn between $33 000 and $47 000 per year. Show that the standard deviation of nurses’ salaries is $3570, correct to three significant figures. 9.F-5 (SL 87) X is a normally distributed random variable with mean 5 and standard deviation 3.2. Find the probability, giving your answers to 3 significant figures, that (a) a randomly chosen value of X is negative, and (b) out of two randomly chosen values of X, one is negative and the other is positive. [Ans: 0.0591;0.111] 9.F-6 (HL 11/06) A certain type of vegetable has a weight which follows a normal distribution with mean 450 grams and a standard deviation 50 grams. (a) In a load of 2000 of these vegetables, calculate the expected number with a weight greater than 525 grams. (b) Find the upper quartile of the distribution. [Ans: 134; 484 g] 9.F-7 (HL 5/97) A zoologist knows that the lengths of a certain type of tropical snake are normally distributed with mean length L meters and standard deviation 0.12 meters. If 20% of the snakes are longer than 0.70 meters, find the value of L. [Ans: 0.599] 9.F-8 (HL 5/98) The mean test scores for a mathematics class was 60 with a standard deviation of 10. Assuming that the test scores are normally distributed, find the proportion of students scoring more than 80 in a given test. [Ans: 0.0228] Mr. Budd, compiled September 29, 2010


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9.F-9 (HL 5/99) A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average weight of sugar in the bags is 1.02 kg. Assuming that the weights of the bags are normally distributed, find the standard deviation if 1.7% of the bags weigh below 1 kg. Give your answer correct to the nearest 0.1 gram. [Ans: 9.4 g] 9.F-10 (HL Spec ’00) The diameters of discs produced by a machine are normally distributed with a mean of 10 cm and a standard deviation of 0.1 cm. Find the probability of the machine producing a disc with a diameter smaller than 9.8 cm. [Ans: 0.228] 9.F-11 (HL 5/02) The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.12 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg. [Ans: 0.586] 9.F-12 (HL 5/03) The random variable X is normally distributed and P(X ≤ 10) = 0.670 P(X ≤ 12) = 0.937. Find E (X).

[Ans: 9.19]

9.F-13 (HL 5/04) The weights of adult males of a type of dog may be assumed to be normally distributed with mean 25 kg and standard deviation 3 kg. Given that 30% of the weights lie between 25 kg and x kg, where x > 25, find the value of x. [Ans: 27.5] 9.F-14 (MM 5/00) The lifespan of a particular species of insect is normally distributed with a mean of 57 hours and a standard deviation of 4.4 hours. (a) The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in Figure 9.8. This diagram represents the standard normal curve Figure 9.8: Problem 14

i. Write down the values of a and b.

[Ans: −0.455, 0.682] Mr. Budd, compiled September 29, 2010


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HL Unit 9 (Probability) ii. Find the probability that the lifespan of an insect of this species is A. more than 55 hours; [Ans: 0.675] B. between 55 and 60 hours. [Ans: 0.428] (b) 90% of the insects die after t hours. i. Represent this information on a standard normal curve diagram, similar to the one given in part 14a, indicating clearly the area representing 90%. ii. Find the value of t. [Ans: 62.6 hours]

9.F-15 (MM 5/96) Packets of sugar labeled ‘1 kg’ are filled by a machine which delivers M kg to a packet, where M is a random variable. It is known that M has a normal distribution with a standard deviation of 50 g. A filled packet of sugar is described as ‘underweight’ when it contains less than 1 kg of sugar. (a) In a large batch produced by the machine, the mean mass delivered to each packet is 1.025 kg. Show that almost one-third of the packets are underweight. [Ans: 0.3085] (b) The sugar supply to each packet is increased in order to reduce the proportion of underweight packets to 10%. What is now the mean mass delivered to each packet? [Ans: 1.06 kg] (c) Explain briefly why the manufacturer would not attempt to eliminate underweight packages from a batch by adjusting the mean. (d) The regulation is changed to allow no more than 2.5% of the packets to be underweight. If the same mean mass as in part 15b is still delivered to each packet, what standard deviation is required in order to achieve this? [Ans: 32.7 g] 9.F-16 (HL 5/00) A machine is set to produce bags of salt, whose weights are distributed normally, with a mean of 110 g and standard deviation of 1.142 g. If the weight of a bag of salt is less than 108 g, the bag is rejected. With these settings, 4% of the bags are rejected. The settings of the machine are altered and it is found that 7% of the bags are rejected. (a)

i. If the mean has not changed, find the new standard deviation, correct to three decimal places. [Ans: 1.355 g] The machine is adjusted to operate within this new value of the standard deviation. ii. Find the value, correct to two decimal places, at which the mean should be set so that only 4% of the bags are rejected. [Ans: 110.37] Mr. Budd, compiled September 29, 2010


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(b) With the new settings from part 16a, it is found that 80% of the bags of salt have a weight which lies between A g and B g, where A and B are symmetric about the mean. Find the values of A and B, giving your answers correct to two decimal places. [Ans: 108.63, 112.11] 9.F-17 Experiment at http://www.stat.wvu.edu/SRS/Modules/NormalApprox/normalapprox.html Let me know if you find a better web-site than this. 9.F-18 (MM 11/96) B is a binomial random variable with n = 30 and p = 0.8. (a) Find the mean and the variance of B.

[Ans: 24,4.8]

(b) Use a normal distribution to obtain an approximate value for P(20 ≤ B ≤ 26). [Ans: 0.853] 9.F-19 (HL 5/97) On average, 10% of light bulbs manufactured by a company are defective. Use the normal approximation to the binomial distribution to determine the probability that more than 12% of a random sample of 200 bulbs are defective. [Ans: 0.144] 9.F-20 (HL 5/99) The quality control department of a company making computer chips knows that 2% of the chips are defective. Use the normal approximation to the binomial probability distribution, with a continuity correction, to find the probability that, in a batch containing 1000 chips, between 20 and 30 chips (inclusive) are defective. [Ans: 0.535] 9.F-21 (MM 5/96) Refer to problems 9.3.2 on page 288 and 21 on page 297. A computer is programmed to generate a sequence x1 , x2 , x3 , . . . of random single digits, each of which takes a value from 0 to 9 inclusive. Each digit is equally likely. Let X be the discrete random variable which represents the value of a digit. The computer generates a sequence of 100 random digits. (a) Find the mean of the number of ‘4’s in the sequence.

[Ans: 10]

(b) Find the standard deviation of the number of ‘4’s in the sequence. [Ans: 3] (c) Use a normal approximation to the binomial distribution to find the probability of obtaining i. fewer than three ‘4’s; ii. exactly ten ‘4’s in the sequence.

[Ans: 0.0062] [Ans: 0.132]

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Statistics 1. Descriptive Statistics 2. Cumulative Frequency

International Baccalaureate 6.1: Concept of population and sample. Discrete data and continuous data. Frequency tables. 6.2: Presentation of data. Grouped data; mid-interval values; interval width; upper and lower interval boundaries. Included: treatment of both continuous and discrete data. Note: a frequency histogram uses equal class intervals. 6.3: Measures of central tendency: sample mean, x; median. Included: an awareness that the population mean, µ, is generally unknown, and that the sample mean, x, serves as an unbiased estimate of this quantity. 6.5: Measures of dispersion: range; standard deviation of the sample, sn . The unbiased estimate, s2n−1 , of the population standard deviation σ 2 . Included: an awareness that the population standard deviation, σ, is generally unknown, and n 2 knowledge that s2n−1 = s serves as an unbiased estimate of σ 2 . In examn−1 n inations: candidates are expected to use a statistical function on a calculator to find standard deviations. Be aware of calculator, text, and regional variations in notation for sample standard deviation.

327


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Descriptive Statistics

International Baccalaureate 6.1 Concepts of population, sample, random sample and frequency distribution of discrete and continuous data. 6.2 Presentation of data: frequency tables and diagrams, box and whisker plots. Treatment of both continuous and discrete data. Grouped data: mid-interval values, interval width, upper and lower interval boundaries, frequency histograms. 6.3 Mean, median, mode; quartiles, percentiles. Awareness that the population mean, µ, is generally unknown, and that the sample mean, x, serves as an unbiased estimate of this quantity. Range; interquartile range; variance, standard deviation. Awareness of the concept of dispersion and an understanding of the significance of the numerical value of the standard deviation. Obtain the standard deviation (and indirectly the variance) from a GDC and by other methods. Awareness that the population standard variance, σ 2 , is generally unknown, and that the s2n−1 serves as an unbiased estimate of σ 2 . 6.4 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

Discrete Data Recall Mean. Median. Mode. Standard deviation. Frequency. When we first talk about mean, median, mode, and standard deviation, we usually restrict ourselves to discrete data, and you are probably used to dealing with discrete data. Discrete data are taken from a limited set of values. The number rolled on a die, for example, cannot be any value between 1 and 6, but only one of {1, 2, 3, 4, 5, 6}. Data that are not discrete are continuous.

Example 9.7.1 Give some examples of discrete data. Give some examples of continuous data.

Sometimes the line between discrete data and continuous data gets blurry. While heights can take on any value, there is a limit to our precision of measurement, so that measured values could be considered discrete. Your age, for example, Mr. Budd, compiled September 29, 2010


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might be 17.247713 . . . years, but you probably just say 17. The way we handle continuous data is basically to make the data discrete by putting the data in groups, or classes.

9.7.1

Central Tendency

Central Tendency is basically, “Where is the data?” It gives, in some sense, a typical data value.

Mean The mean, µ, of a population is given by N P

x1 + x2 + · · · + xN = µ= N

xi

i=1

N

where N is the size of the population. This is the average as you are used to finding it.

Median As you know from TAKS, the median, µ ˜, is the middle item when the items are placed in order. If there are an even number of items, then it is the average of the middle two. Sometimes you don’t have all the individual items, as they may be grouped. Since you don’t know the individual items, the median must be calculated in a different manner. It is calculated as the 50th percentile. Percentiles will be discussed in Section 9.7.10.

Mode As you also know from TAAS, the mode is the most frequently occurring item. If a set of data is perfectly symmetric, the mean, median, and mode are the same value. If the data are skewed, then the mean, median, and mode will be different values. Mr. Budd, compiled September 29, 2010


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Other Other measures of central tendency include the geometric mean, v uN uY N t x, i i=1

and the harmonic mean, 1 N

1 N P i=1

, 1 xi

which is the reciprocal of the average reciprocal. The following example is taken from Probability and Statistics. Example 9.7.2 [18] Unfair Coin Tossed Thrice This example will be used frequently in this and the next sections. Given a coin for which the probability of head on a single toss is 0.6. Consider the experiment of tossing the coin three times. Let X stand for the number of heads obtained. Suppose the experiment is repeated 40 times, the data, i.e., values of X, in Table 9.3 are obtained. Find the mean, median, and mode.

Table 9.3: Unfair Coin Tosses - Number of Heads 1 2 2 1

9.7.2

0 1 1 2

2 2 2 2

2 1 0 0

3 3 2 2

1 3 1 1

2 2 3 3

1 1 2 2

2 2 1 3

2 3 2 2

Dispersion

Dispersion is, basically, “How spread out is the data?” There is a difference between the set of data {−100, 0, 100} and {−1, 0, 1}, but you would never know that from measures of the central tendency.

Range The range is the difference between the highest and lowest values in the population or sample. It is highly susceptible to outliers. Mr. Budd, compiled September 29, 2010


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Variance and Standard Deviation Example 9.7.3 The mean deviation is the the average difference between each item and the mean value, i.e., N P

(xi − µ)

i=1

N Is the average difference from the mean a good measure of dispersion? Why or why not? How could you improve it? The variance of a population is given by N P

σ2 =

(xi − µ)

2

i=1

N

It is an average; it is the average square difference between each item and the mean value. The difference is squared to make it positive. (If we didn’t square it, we’d be averaging a bunch of positive and negative numbers, resulting in 0). The difference between the item and the mean gives you an idea of how far the values are from the mean. Example 9.7.4 (SL 89) The variance of the numbers 5, 11, and k is equal to 14. Find the possible values of their mean.

[Ans: 6, 10] Example 9.7.5 Show that the variance is the average of the sqaures minus the square of the average. The standard deviation is the square root of the variance. v u N uP 2 u t i=1 (xi − µ) σ= N It is generally easier to conceptualize because it has the same units as both the sample items and the mean. Example 9.7.6 Refer to Table 9.3 and find the variance and standard deviation. Mr. Budd, compiled September 29, 2010


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Interquartile Range The interquartile range is the difference between the first and third quartiles, or the 25th and 75th percentiles. Percentiles will be discussed in more detail in Section 9.7.4. Remember that the median is the middle value, or average of the two middles (once the data is sorted). The first quartile is the median of the bottom half, and the third quartile is the median of the top half.

Example 9.7.7 [6] Find the interquartile range of {2, 6, 5, 4, 1, 0, 5, 2}.

[Ans: 3.5]

Others The mean absolute deviation is the average absolute value of the difference from the mean: N P |xi − µ| i=1

N The standard deviation is preferred to the mean absolute deviation only because there are lots of nice formulas that can be devised with the standard deviation.

9.7.3

Frequency

The frequency is the number of times a particular value occurs in a sample (or population). A frequency distribution for a sample is a tallying and recording of the frequencies for each value.

Example 9.7.8 Refer to Table 9.3 on page 331. Make a frequency chart.

[Ans: The frequency chart is shown in Table 9.4.] If we’re given the frequency data, where fi is the frequency for each individual value xi , i = 1, 2, . . . k, then we can find the sample (or population) mean by Mr. Budd, compiled September 29, 2010


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Table 9.4: Biased Coin Tossed Thrice Number of Frequency heads, x f 0 3 1 11 2 19 3 7 Total=40

using k

µ or x =

x1 f1 + x2 f2 + · · · + xk fk 1X = xi fi . f1 + f2 + · · · + fk n i=1

Similarly, 2

σ 2 or s2n =

2

k

2

(x1 − x) f1 + (x2 − x) f2 + · · · + (xk − x) fk 1X 2 = (xi − x) fi . f1 + f2 + · · · + fk n i=1

As always, for sn−1 , divide by n − 1 instead of n. Example 9.7.9 (MM 5/04) Table 9.5 below shows the marks gained in a test by a group of students. Table 9.5: Marks on a Test Mark Number of students

1 5

2 10

3 p

4 6

5 2

The median is 3 and the mode is 2. Find the two possible values of p. [Ans: 8, 9] Example 9.7.10 Table 9.6 displays the frequency of occurrences of scores in a competition. The mean score is 15. Find k. [Ans: 11] The relative frequency is the frequency divided by the total frequency, n. Mr. Budd, compiled September 29, 2010


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Table 9.6: Problem ?? - Scores in a Competition Score 12 13 14 15 16 17 Frequency 2 4 7 13 k 5

9.7.4

Cumulative Frequency

The cumulative frequency, F (x), of a score x, is basically, ‘What is the combined (or cumulative) frequency of all the scores less than or equal to x’ ?

Cumulative Distribution Function Definition 9.16. [18] The cumulative distribution function [or empirical cumulative distribution function or cumulative frequency distribution] at x0 is defined to be X 1 X fx Fˆ (x0 ) = fˆ (x) = N x≤x0

x≤x0

Example 9.7.11 Refer to the problem where an unfair coin is thrown three times, and the number of heads is recorded. (Page 331). (a) Develop a table for the cumulative frequency in terms of x. (b) Use your table to find the median and interquartile range. Check your answers with a calculator (c) Develop a table for the cumulative relative frequency in terms of x.

Table 9.7: Cumulative Frequency for Biased Coin Tossed Thrice Cumulative Cumulative No. of heads; x Frequency, f frequency relative frequency 0 3 3 3/40 = 0.075 1 11 3 + 11 = 14 14/40 = 0.350 2 19 14 + 19 = 33 33/40 = 0.825 3 7 33 + 7 = 40 40/40 = 1.000

The answer is given in Table 9.7. Mr. Budd, compiled September 29, 2010


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Figure 9.9: Graph of empirical cumulative distribution function for a discrete random variable.

Graphs If the data are discrete and not grouped, then the function cannot take on values between the isolated data points. Refer to the case of flipping a biased coin three times, counting the number of heads (page 331). Values that are recorded as 2 are 2, and cannot be 1.77893 or 2.431. The frequency of getting less than or equal to 2.73 heads is the exactly the same as the frequency of getting less than or equal to 2.998 heads, which is exactly the frequency of getting less than or equal to 2 heads. (Compare this to continuous data: We would expect more women to be shorter than 67.48 inches than are shorter than 66.82 inches.) In the case of discrete and ungrouped data, we do not connect the dots with lines, we make a series of horizontal lines that start at one data point and end at he next data point. See Figure 9.9 for an example. Note the appropriate use of closed and open circles on the graph.

9.7.5

Population vs. Sample

Sampling Usually, it can be difficult to get statistics for an entire population. Instead, we take a sample, which is a portion of the whole population. CBS cannot determine whom everyone in the state of Florida voted for on election night, so they do exit polling to get a sample of voters. Proper sampling requires that the sample be representative of the whole population, and is a whole field unto itself. If there are 1376 women at Gates College, then taking the heights of 50 women at Gates college is using a sample. Any statistics will be sample statistics, such as the sample mean, sample variance, and sample standard deviation. Usually we use these sample statistics as estimates of the population statistics, because Mr. Budd, compiled September 29, 2010


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the population is too big to measure, and we don’t have statistics for the entire population.

Sample mean The sample mean, represented by x, is just the mean (average) of the sample: n P

x1 + x2 + · · · + xn x= = n

xi

i=1

n

where n is the size of the sample. The sample mean happens to give us what is called an unbiased estimator of the population mean.

Sample variance and standard deviation In I.B. world, the sample variance s2n , and sample standard deviation, sn is calculated the same way as the population variance and standard deviation. That is, n P 2 (xi − x) p s2n = i=1 , sn = s2n . n This gives us an estimate of the variance (and hence standard deviation) for the entire population. The problem is that if a sample size is one (i.e., n = 1) then this gives a standard deviation of 0. In reality, if our sample size is one, we have no way of knowing what the dispersion is like. Mathematically, this estimate is said to be biased, and is not used in formulas that relate the population data to the sample data. An unbiased estimate of the population variance is given by: n P

s2n−1

=

2

(xi − x)

i=1

n−1

,

sn−1 =

q

s2n−1

The only difference is the division by n − 1 instead of n. If n = 1, then you can’t have an unbiased estimate of the population standard deviation.

Example 9.7.12 (HL 5/03) A teacher drives to school. She records the time taken on each of the 20 randomly chosen days. She finds Mr. Budd, compiled September 29, 2010


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20 X i=1

xi = 626 and

20 X

x2i = 19780.8 ,

i=1

where xi denotes the time, in minutes, taken on the ith day. Calculate an unbiased estimate of (a) the mean time taken to drive to school; (b) the variance of the time taken to drive to school.

[Ans: 31.3;9.84]

Differences in notation and terminology Now, this wouldn’t necessarily be a big deal, because we have two different formulas, one for the sn , the sample standard deviation (which isn’t very useful mathematically) and one for sn−1 , the unbiased estimate of σ. The problem is in notation and terminology. Table 9.8 summarizes several different notations for the two different standard deviations you get by dividing by n or by n − 1.

Useless Useful

Table 9.8: Notation for Sample Standard Deviation Source What I.B. calls it I.B. TI-83 [9] [18] “Sample Standard Deviation” sn σx s1 s˜ “Unbiased Estimate of σ” sn−1 sx s2 sˆ

[2] σ s

The titles of sn and sn−1 are in quotes because some texts [2],[9] call I.B.’s sn−1 the sample standard deviation. Their point is that if you are dividing by n and not n − 1, then you are treating the data as a population, and what you are finding is σ, the population standard deviation. When you divide by n − 1, you are treating the set of data as a sample, and this standard deviation should be labeled as such. I would tend to agree with these people, but I don’t write the I.B. exams. While I disagree with the I.B. terminology, their notation is perhaps the most clear, distinguishing between n and n − 1. A very big difference in notation that you need to be aware of is that when I.B. asks for sn , and you do this on your TI-83, you need to use σx and not sx . It is important for you to understand the differences in notation so that you can get the correct information from the calculator, or from a reference book. Remember that the unbiased estimate of σ is always larger than sn , because you’re dividing by a smaller number. Mr. Budd, compiled September 29, 2010


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Continuous Data 9.7.6

Discrete vs. Continuous

Discrete Data “Random variables that can only take on isolated values are called discrete random variables.” [9] The difference between discrete and continuous data is probably best expressed by giving examples of the two. The number of heads obtained when tossing three coins is discrete. The number of wins by the Rockets in a season is discrete.

Continuous Data The areas in square miles of the various states is continuous. Height is continuous. However, continuous data is limited by precision of measurement. For example, we can’t tell the difference between someone that is 63.28976 inches tall and someone that is 63.28967 inches tall. We might say that both are 63.29 inches tall. The continuous data is being forcibly grouped by limitations in precision. This grouping can make the continuous data appear or seem discrete. More on grouping later.

9.7.7

Groups

These are some definitions associated with grouping of data.

mid-interval value is the midpoint of a class interval. Also known as the class mark. interval boundaries are the smallest and largest actual values in the class. Also outside of the I.B. world as the upper and lower class boundaries. interval width difference between the class boundaries. Also known as the class width. Mr. Budd, compiled September 29, 2010


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Grouping due to precision

Frequencies for continuous data are a little more sketchy than for discrete data. For example, no two people have exactly the same height. Instead of assigning a frequency to a specific value, we assign a frequency to an interval of values. These intervals may be based upon our precision of measurement, but we may also choose to group the data into larger classes. The following example is taken from Probability and Statistics. [18]

Table 9.9: Heights of 50 Women at Gates College, in Inches 61 66 65 66 65

60 63 66 69 70

64 59 68 61 66

63 67 70 64 72

69 65 62 65 62

72 67 64 63 69

65 64 69 67 62

67 71 63 65 69

66 61 64 60 64

66 69 65 64 69

Example 9.7.13 Table 9.9 represents the heights of a sample of 50 women at Gates College, as recorded to the nearest inch. Make a frequency chart.

The frequency chart is seen Table 9.10. The heights arenâ&#x20AC;&#x2122;t discrete, theyâ&#x20AC;&#x2122;re continuous. Note that the heights are recorded to the nearest inch. That means that a woman recorded with a height of 50 inches might really be 50.472 inches or 49.7 inches. So a height recorded as 50 inches really means a height between 49.5 inches and 50.5 inches. Except for the fact that we are assigning a frequency to an interval of values instead of a single individual value, continuous data are treated similarly to discrete data. [There is some disagreement as to what the range would be. What arguments could you make for giving a range of 13? for 14?] The relative frequency, as for discrete data, is the frequency divided by the total frequency, n. Mr. Budd, compiled September 29, 2010


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Table 9.10: Heights of a Sample of Gates College Women, in Inches Recorded Height Actual Height Frequency 59 58.5-59.5 1 60 59.5-60.5 2 61 60.5-61.5 3 62 61.5-62.5 3 63 62.5-63.5 4 64 63.5-64.5 7 65 64.5-65.5 7 66 65.5-66.5 6 67 66.5-67.5 4 68 67.5-68.5 1 69 68.5-69.5 7 70 69.5-70.5 2 71 70.5-71.5 1 72 71.5-72.5 2 Total=50

Grouping by choice There may be times where we wish to form larger groups than what is allowed by precision of measurement. This consolidates the data, and, in some sense, makes the data more legible. The downside of consolidation, however, is the loss of information.

• In order for the class mark to be an actual recorded value, the class width should be an odd number. • Class intervals should have the same width. • The number of classes should be between 5 and 20.

Example 9.7.14 Group the heights of the sample of 50 women from Gates College. (Table 9.9). Nevermind, I’ll do it:

The range is either 13 or 14, depending on whom you ask. We  need at least five classes, so that would give them a class interval of 3> 14 5 . Even with six classes we’d need a class interval of 3> 14 , and one of the classes would be 6 unused. Seven classes would give a class size of two, which is even and therefore undesirable. (Why?) So: five classes is an excellent choice. Mr. Budd, compiled September 29, 2010


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There are two ways I can group the data into five classes. Five classes of width 3 gives fifteen values, and I have fourteen. So I can either start at 57.5 and go up to 72.5, or, I can start at 58.5 and go up to 73.5. Itâ&#x20AC;&#x2122;s just a question of putting the empty value at the end or the beginning.

Table 9.11: Heights of Gates College Women (Grouped Sample Data) Class Class Class Class Label Boundaries Mark Frequency 1 57.5-60.5 59 3 2 60.5-63.5 62 10 3 63.5-66.5 65 20 4 66.5-69.5 68 12 5 69.5-72.5 71 5 Total=50

Did I put the empty value at the end or at the beginning? What are the class boundaries? Find the mean based solely on the grouped data and compare it to the mean for the ungrouped data.

Statistics of Grouped Data To calculate means or standard deviations for the grouped data, treat each data in the class as if it were equal to the class mark. Calculating the median will be described in the Section 9.7.10. We know that there are 12 women with heights between 66.5 and 69.5 inches, and once the data is grouped, we have lost the information about how the values were distributed within that interval. In determining mean and standard deviation, we will assume that all 12 women have a height of 68 inches. Clearly, that isnâ&#x20AC;&#x2122;t true, but it will be our operating assumption, unless you got a better idea.

Example 9.7.15 Using the grouped data in Table 9.11, estimate: (a) the mean height; (b) the standard deviation and variance of the heights. Compare your answers to those obtained from the individual, ungrouped heights. Mr. Budd, compiled September 29, 2010


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343

Histograms

A histogram is a bar graph of the frequency distribution. It consists of a “collection of rectangles,” [18] one for each value (for ungrouped, discrete data), or one for each class (grouped, continuous data). The area of the rectangle is proportional to the frequency. If the class widths are equal, then the area is proportional to the height, and therefore the height is proportional to the frequency. That is why we want the class widths to be equivalent. Figure 9.10: Histogram for Gates Women’s Heights

For grouped, continuous data, the midpoint of the rectangle should be the class mark (see page 339), and the rectangles should start and end at the lower and upper class boundaries, respectively. Example 9.7.16 See Figure 9.10 for the histogram corresponding to the sample height of Gates women.

Figure 9.11: Histogram for Gates Women’s Heights

Different histograms can  be  created for different ordinates (y-axes) of frequency ˆ (f ), relative frequency f , and f ∗ = f . Some people choose f ∗ because then Nw

the area of each rectangle is equal to its relative frequency, and the sum of all the areas is 1. Also, some people express the relative frequency in terms of a decimal, and some in terms of a percentage. Mr. Budd, compiled September 29, 2010


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Cumulative Frequency Recall Median. Percentile. Quartile.

9.7.9

Continuous Data

Cumulative Distribution Function When dealing with a cumulative distribution function for grouped and/or continuous data, I find the cumulative frequency at each class boundary. For n classes, there will be n+1 boundaries. For each boundary, find the total number of occurrences that are less than that class boundary. The cumulative frequency at the lowest class boundary should always be 0. For a cumulative relative frequency distribution, I simply take each cumulative frequency and divide by the total number of events. The relative cumulative frequency of the uppermost class boundary should always be 1. Cumulative frequencies are given for class boundaries, not for class marks.

Example 9.7.17 Refer to Table 9.11, the frequency distribution for the grouped heights of women at Gates College. Develop a table which contains columns displaying the cumulative frequency and cumulative relative frequency at the class boundaries.

The final cumulative relative frequency table is seen in Table 9.13. This is sometimes called the empirical cumulative distribution function, because it represents a cdf (cumulative distribution function) that has been empirically determined, i.e., found by experience. This is used to distinguish it from a cumulative distribution function in probability, which is a theoretical and idealized function.

Example 9.7.18 (MM 95) In Aristia a company has almost 1000 employees. The distribution of employeesâ&#x20AC;&#x2122; annual salaries is summarized in the cumulative frequency table in Table 9.14. (a) Find the number of employees for whom (a) 20 < S â&#x2030;¤ 25;

[Ans: 334] Mr. Budd, compiled September 29, 2010


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Table 9.12: Generating Cumulative Relative Frequencies at the Class Boundaries for a Frequency Distribution Class Class Class Cumulative Cumulative mark frequency boundary frequency relative frequency 57.5 0 0 59 3 60.5 0+3=3 3/50 = 0.06 62 10 63.5 3 + 10 = 13 13/50 = 0.26 65 20 66.5 13 + 20 = 33 33/50 = 0.66 68 12 69.5 45 45/50 = 0.90 71 5 72.5 50 50/50 = 1.00

Table 9.13: Cumulative Relative Frequency Table for Sample of Heights of Women at Gates College Class Cumulative boundary, x relative frequency, FË&#x2020; (x) 57.5 0.00 60.5 0.06 63.5 0.26 66.5 0.66 69.5 0.90 72.5 1.00

(b) 30 < S â&#x2030;¤ 40.

[Ans: 136]

(b) Calculate an estimate for the mean salary, correct to 3 significant figures. [Ans: 24.7] (c) Note that we will find the median and quartiles in a way different than assuming all the values are at the class mark. Be aware that for continuous, grouped data, finding the median is not as easy as punching some buttons on the calculator.

Cumulative Frequency Polygon Once you have the cumulative frequency distribution, it is a simple matter to create the cumulative frequency polygon. Plot the class boundaries vs. the Mr. Budd, compiled September 29, 2010


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Table 9.14: Annual salaries in Aristia Salary S (in thousands N , the number of of Aristian Crowns) employees not exceeding 10 0 15 45 20 207 25 541 30 829 35 923 40 965 45 985

appropriate cumulative frequencies. You should plot n + 1 coordinates for each of the n + 1 class boundaries. Connect the dots with straight lines. A cumulative relative frequency polygon is done in a similar manner, simply plotting the relative frequency on the y-axis. Example 9.7.19 From the tabulated data in Table 9.13, construct a cumulative relative frequency polygon.

Figure 9.12: Cumulative relative frequency polygon for Gates College womenâ&#x20AC;&#x2122;s heights.

The relative cumulative frequency polygon is shown in Figure 9.12. Example 9.7.20 (HL 5/04) The heights of 60 children entering a school were measured. The cumulative frequency graph in Figure 9.13 illustrates the data obtained. (a) Determine the interval boundaries, and mid-interval values. Mr. Budd, compiled September 29, 2010


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(b) Make a frequency chart. (c) Estimate the mean height, and the variance of heights.

Figure 9.13: Heights of Entering Students

9.7.10

Percentiles and Quartiles

Definition 9.17. [18] The kth percentile pk is a value of x [where x denotes a score] such that k FË&#x2020; (pk ) = 100 The first quartile, Q1 , is the 25th percentile, the median is the 50th percentile, and Q3 , the third quartile, is the seventy-fifth percentile.

Percentiles from a graph We can use the cumulative relative frequency polygon to help us estimate percentiles. If we are looking for p10 , the tenth percentile, we can draw a horizontal line on the cumulative relative frequency polygon where y = 0.10. The x-value Mr. Budd, compiled September 29, 2010


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Figure 9.14: centiles.

Using a cumulative relative frequency polygon to estimate per-

where that horizontal line crosses the polygon is p10 , the score for the tenth percentile. See Figure 9.14, which shows how to estimate the median and third quartile for the sample of 50 women at Gates College. Example 9.7.21 Using Figure 9.13, estimate the median height of the sixty students entering a school.

Percentiles from a table While the cumulative relative frequency polygon is useful for estimating percentiles, a more precise measurement requires us to do some linear interpolation. When using grouped data, you cannot use 1-Var Stats on the calculator to determine the median and quartiles. The calculator treats the data discretely, which OK for estimating the mean and variance, but unacceptable in determining percentiles. Example 9.7.22 Refer to Table 9.13 on page 345, the cumulative relative frequency chart relating the heights of 50 women at Gates College. (a) Calculate p25 , the 25th percentile. (b) Calculate the median and the third quartile. In order to do this problem, we must look at the cumulative relative frequency chart (Table 9.13 on page 345). 25%, or 0.25, is between 0.06 and 0.26, so that p25 is somewhere between 60.5 and 63.5. Since 0.25 is much closer to 0.26, we Mr. Budd, compiled September 29, 2010


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would expect the first quartile to be fairly close to 63.5. To figure out exactly how close, interpolate between the two values. p25 − 60.5 0.25 − 0.06 = 63.5 − 60.5 0.26 − 0.06 In other words, p25 is proportionally the same distance between 60.5 and 63.5 as 0.25 is between 0.06 and 0.26. Since this is a proportion, I can set it up in several different manners, as with all proportions.

63.5 − p25 63.5 − 60.5

=

0.26 − 0.25 0.26 − 0.06

or

63.5 − p25 0.26 − 0.25

=

63.5 − 60.5 0.26 − 0.06

or

p25 − 60.5 0.25 − 0.06

=

63.5 − 60.5 0.26 − 0.06

or flipping:

0.25 − 0.06 p25 − 60.5

=

0.26 − 0.06 63.5 − 60.5

I can also think about it graphically, in terms of slopes of line segments. Since the point (p25 , 0.25) lies on the line segment joining (60.5, 0.06) and (63.5, 0.26), the slope from (p25 , 0.25) to (60.5, 0.06) (or (63.5, 0.26)) is the same as the slope between the two endpoints of the line segment, (60.5, 0.06) and (63.5, 0.26). Another way to think about it is as follows: There is a 0.20 difference between 19 0.06 and 0.26. 0.25 is .19 .20 = 20 of the way between 0.06 and 0.26. So that p25 19 must be 20 of the way from 60.5 to 63.5.

4x 3

=

0.19 0.20

4x

=

3 · 19 20

=

2.85

and, to get the 25th percentile, I need to add 4x to 60.5. p25

=

60.5 + 4x

=

60.5 + 2.85

=

63.35 ≈ 63.4 inches Mr. Budd, compiled September 29, 2010


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This is how one author [2] describes it: " Qj/m = b +

j×n m

# − Cf (w) f

“Where Qj/m is the x value below which are j mths of the data, b is the lower boundary of the implied range for the quantile category (the measurement category that contains the quantile), n is the sample size (or N is the population size), Cf is the cumulative frequency from all categories less than the quantile category, f is the frequency in the quantile category, and w is the width of the implied range of the quantile category.” [2] To find the median, we look between the cumulative relative frequencies of 0.26 and 0.66, corresponding to scores between 63.5 and 66.5:

p50 − 63.5 66.5 − 63.5

=

0.50 − 0.26 0.66 − 0.26

p50 − 63.5 3

=

0.24 0.40

p50 − 63.5

=

p50

=

63.5 + 1.8

p50

=

65.3 inches

24 40

or, alternatively, M d = p50

=

63.5 + 4x

=

63.5 + 3 ×

=

65.3 inches

where

4x 0.50 − 0.26 = 66.5 − 63.5 0.66 − 0.26

24 40

To find the third quartile, look between 0.66 and 0.90, corresponding to x values Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 7: Descriptive Statistics

351

between 66.5 and 69.5: p75 − 66.5 69.5 − 66.5

=

0.75 − 0.66 0.90 − 0.66

p75 − 66.5 3

=

0.09 0.24

p75 − 66.5

=

p75

=

66.5 + 1.125

p75

=

67.625 ≈ 67.6 inches

9 24

or, alternatively, Q3 = p75

=

66.5 + 4x

=

66.5 +

=

67.625 ≈ 67.6 inches

where

0.75 − 0.66 4x = 69.5 − 66.5 0.90 − 0.66

3·9 24

Interquartile Range The interquartile range is the difference between the first and third quartiles, or the 25th and 75th percentiles. It is used as a measure of dispersion, along with the variance, standard deviation, and range. (See page 331.)

Example 9.7.23 Calculate the interquartile range for the sample of 50 Gates women’s heights.

The interquartile range is Q3 − Q1 = 67.625 − 63.35 = 4.275 ≈ 4.3 inches.

Example 9.7.24 Using Figure 9.13, estimate the interquartile range of the sixty students entering a school. Mr. Budd, compiled September 29, 2010


352

HL Unit 9 (Probability)

Figure 9.15: Symmetric Box-andWhisker Plot

Figure 9.16: Skewed Box-andWhisker Plot

Box-and-Whiskers Plot The box extends from Q1 to Q3 with a dividing line at the median, covering the middle two quarters of data. The whiskers cover the upper and lower quarters of data.

Example 9.7.25 The box plot shown in Figure 9.15 shows a distribution summarized by a five-number summary: Q1 , Q2 , Q3 , xs , xl . Determine from the plot: interquartile range, median, range, and whether the distribution is symmetric or skewed.[2]

The interquartile range is 4 sec, from 13 sec to 17 sec. The median is 15 sec. The range is 10 sec, from 10 sec to 20 sec. The distribution is symmetric.

Example 9.7.26 The box plot shown in Figure 9.16 shows a distribution summarized by a five-number summary: Q1 , Q2 , Q3 , xs , xl . Determine from the plot: interquartile range, median, range, and whether the distribution is symmetric or skewed.[2]

The interquartile range is 0.2 kg, from 2.6 kg to 2.8 kg. The median is about 2.65 kg, and the range is 1.0 kg. The distribution is skewed. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 7: Descriptive Statistics

353

Example 9.7.27 (SL 5/07 TZ2) A set of data is 18, 18, 19, 19, 20, 22, 22, 23, 27, 28, 28, 31, 34, 34, 36 The box and whisker plot for this data is shown below.

(a) Write down the values of A, B, C, D, and E. (b) Find the interquartile range.

[Ans: 18, 19, 23, 31, 36; 12]

Example 9.7.28 Draw a box and whiskers plot for the height data from the sample of Gates College women.

Mr. Budd, compiled September 29, 2010


354

HL Unit 9 (Probability)

Problems 9.G-1 (HL 11/02) Consider the six numbers, 2, 3, 6, 9, a, and b. The mean of the numbers is 6 and the variance is 10. Find the value of a and of b, if a < b. [Ans: 5, 11] 9.G-2 (HL 5/01) A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown in Figure 2. Weight (g) Frequency

Problem 2: Weights of sugar packets 29.6 29.7 29.8 29.9 30.0 30.1 2 3 4 5 7 5

30.2 3

30.3 1

Find unbiased estimates of (a) the mean of the population from which this sample is taken. [Ans: 29.9] (b) the variance of the population from which this sample is taken. [Ans: 0.0336] 9.G-3 (HL Spec â&#x20AC;&#x2122;00) A machine fills bottles with orange juice. A sample of six bottles is taken at random. The bottles contain the following amounts (in mL) of orange juice: 753, 748, 749, 752, 750, 751. Find (a) the sample mean; (b) the sample variance; (c) an unbiased estimate of the population variance from which this sample is taken.   Ans: 750. or 750 (3 s.d.); 2.92; 3.50 9.G-4 (SL 11/06) The box and whisker diagram shown below represents the marks received by 32 students.

(a) Write down the values of the median mark. (b) Write down the value of the upper quartile. (c) Estimate the number of students who received a mark greater than 6. Mr. Budd, compiled September 29, 2010


HL Unit 9, Day 7: Descriptive Statistics

355 [Ans: 3; 6; 8]

9.G-5 (adapted from HL 5/04) The heights of 60 children entering a school were measured. The results are shown in Table 9.15. Estimate Table 9.15: Problem 5 - Heights of children Height (m) Number of Students 0.8 < h ≤ 0.9 9 0.9 < h ≤ 1.0 15 1.0 < h ≤ 1.1 15 1.1 < h ≤ 1.2 12 1.2 < h ≤ 1.3 6 1.3 < h ≤ 1.4 3

(a) the mean height

[Ans: 1.05 m]

(b) the standard deviation and variance of the heights. Why are the instructions ”Estimate” and not ”Find”? In Section 9.7.10, we will look at how to estimate the median. 9.G-6 (adapted from HL 5/00) A sample of 70 batteries was tested to see how long they last. The results are shown in Table 9.16 Table 9.16: Problem 6 - Battery Lifetimes Time(hours) Number of batteries (frequency) 0 ≤ t < 10 2 10 ≤ t < 20 4 20 ≤ t < 30 8 30 ≤ t < 40 9 40 ≤ t < 50 12 50 ≤ t < 60 13 60 ≤ t < 70 8 70 ≤ t < 80 7 80 ≤ t < 90 6 90 ≤ t < 100 1 Total 70

Find (a) the sample mean; (b) the sample variance; Mr. Budd, compiled September 29, 2010


356

HL Unit 9 (Probability) (c) an unbiased estimate of the mean of the mean of the population from which this sample is taken. (d) an unbiased estimate of the variance of the population from which this sample is taken. [Ans: 49.8; 459 (3 s.d.); 49.8; 466]

9.G-7 (HL 5/02) The 80 applicants for a Sports Science course were required to run 800 metres and their times were recorded. The results were used to produce the cumulative frequency graph in Figure 9.17. Estimate (a) the median; (b) the interquartile range. [Ans: 135,11] Figure 9.17: Problem 7 - Times on 800 metres

Mr. Budd, compiled September 29, 2010


Unit 10

Error of Series 1. Remainder Reminder 2. Lagrange Remainder Advanced Placement Concept of Series. A series is defined as a sequence of partial sums and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence. Series of Constants • Geometric series with applications. • Alternating series with error bound. • Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series. • The ratio test for convergence and divergence. Taylor series • Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) • Maclaurin series and the general Taylor series centered at x = a. 357


358

HL Unit 10 (Error of Series) • Maclaurin series for the functions ex , sin x, cos x, and

1 . 1−x

• Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series. • Functions defined by power series. • Radius and interval of convergence of power series. • Lagrange error bound for Taylor polynomials. International Baccalaureate 1.1 Geometric sequences and series; sum of finite and infinite geometric series. Applications of the above. Included: sigma notation, applications of sequences and series to compound interest and population growth. 12.1 Convergence of infinite series. Tests for convergence: ratio test; limit comparison test; integral test. P Included: conditions for the application of these tests, the divergence theorem, if un is a convergent series then lim un = 0. n→∞ 12.2 Alternating series. Conditional convergence. Included: knowledge that the absolute value of the truncation error is less than the next term in the series; absolute convergence of an infinite series. 12.3 Power series: radius of convergence. Determination of the radius of convergence by the ratio test. Included: power series in (x − k), k 6= 0. 12.5 Use of Taylor series expansions, including the error term. Maclaurin series as a special case. Taylor polynomials. Taylor series by multiplication. Included: application to the approximation of functions; bounds on the error term. Included: 2 finding the Taylor approximations for functions such as ex arctan x by multiplying 2 the Taylor approximations for ex and arctan x.

Mr. Budd, compiled September 29, 2010


HL Unit 10, Day 1: Remainder Reminder

10.1

359

Remainder Reminder

Advanced Placement Series of Constants • Alternating series with error bound. • Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series. Textbook §8.5 Alternating Series: “Alternating Series Remainder” [15] Resources §12-8 Error Analysis for Series in [12]. Explorations 12-8a: “Introduction to Error Analysis for Series” and 12-8b: “Error Analysis by Improper Integral” in [11].

10.1.1

Geometric Series

Example 10.1.1 [17] Consider S =

∞ P k=0

1 2k + 1

(a) Does the series converge? (b) Find the sum of the first ten, twenty, and one hundred terms. (c) Use a geometric sereis to determine how closely does S100 ≈ 1.264499781 approximates the true limit S? (d) How many terms, n, would you need so that Sn approximates the true value of S with an error of no more than 5 · 10−5 ?

10.1.2

Alternating Series

Example 10.1.2 What is the maximum error in approximating ∞ (−1)n+1 P with the fourth partial sum? n! n=1  Ans:

Example 10.1.3 How many terms are needed to find

1 120



∞ (−1)n+1 P n! n=1

with an error of no more than 0.001? Mr. Budd, compiled September 29, 2010


360

HL Unit 10 (Error of Series) [Ans: six]

Example 10.1.4 For the Taylor series for ln x expanded about x = 1, how many terms would be needed in the partial sum to compute ln 1.4 to five decimal places?

[Ans: n < 9.731 . . . < 10 terms]

Example 10.1.5 [20] Find than 0.001

10.1.3

R1

2

e−x dx with an error of no more

0

Integral Test

Example 10.1.6 Estimate the remainder of the p-series ∞ X

1 1.02 n n=1 after 20 terms.

[Ans: 47.04631 . . . < R20 < 47.09224 . . .]

Problems 10.A-1 (HL Nov ’06) Consider the series

∞ 1 P n=1 n!

(a) Use the ratio test to prove that the series is convergent. (b) Use a comparison test to show that S < 2. (c) Write down the exact value of S. 10.A-2 Show that the hypotheses of the alternating series test apply to the function, then find the number of terms needed in the partial sum to get the specified accuracy. [12] (a) cos 2.4 to 6 decimal places.

[Ans: n = 7 (8 terms)]

(b) e−2 to 7 decimal places. [Ans: n = 14 (15 terms)]  P∞ 10.A-3 For the p-series n=1 1/n3 find an upper bound for the tail of the series after 10 terms. Mr. Budd, compiled September 29, 2010


HL Unit 10, Day 1: Remainder Reminder

361

(a) How does this estimate of the error introduced by stopping at S10 compare with the value of t11 ? [Ans: R10 >> t11 ] (b) How many terms would be needed to ensure that the partial sum is correct to at least five decimal places? [Ans: n = 317 ⇒ 317 terms] (c) How many terms would be needed to ensure a range of values for R10 that is no bigger than 0.00001 wide? [Ans: 46 terms] 10.A-4 The series

1 1 1 1 + √ + √ + √ + ··· 2 3 4

is a p-series. Explain why the method for the previous problem would not be appropriate for estimating the remainder of this series.  P∞ 1.05 10.A-5 Amos Take wants to calculate the limit to which conn=1 1/n verges. With his grapher he calculates S99 = 4.69030101 . . .. Show Amos that although he has used many terms of the series, his answer is nowhere close to the value to which the series converges. Give Amos a better approximation, based on 99 terms. [Ans: R99 > 15.8865 . . .; S ≈ 20.5808 . . .]

Mr. Budd, compiled September 29, 2010


362

HL Unit 10 (Error of Series)

Mr. Budd, compiled September 29, 2010


HL Unit 10, Day 2: Error Analysis for Series

10.2

363

Error Analysis for Series

Advanced Placement Taylor series • Lagrange error bound for Taylor polynomials. International Baccalaureate 10.5 Taylor polynomials and series, including the error term. Applications to the approximation of functions; formulae for the error term, both in terms of the value th of the (n + 1) derivative at an intermediate point, and in terms of an integral of th the (n + 1) derivative Textbook §8.7 Taylor Polynomials and Approximations: “Remainder of a Taylor Polynomial” [15] Resources §12-8 Error Analysis for Series in [12]. Example 10.2.1 (BC98) The Taylor series for ln x, centered at n ∞ P n+1 (x − 1) . Let f be the function given by the x = 1, is (−1) n n=1 sum of the first three nonzero terms of this series. The maximum value of |ln x − f (x)| for 0.3 ≤ x ≤ 1.7 is [Ans: 0.145] If f (x) is expanded as a Taylor series about x = a and x is a number in the interval of convergence, then there is a number c between a and x such that the remainder Rn after the partial sum Sn is given by [12] f (n+1) (c) n+1 (x − a) (n + 1)!

If M is the maximum value of f (n+1) (x) over the interval between a and x, then M n+1 |Rn | ≤ |x − a| (n + 1)! Rn =

Note that the Lagrange form of the remainder looks a lot like the next term in the series; the (n + 1)th derivative is evaluated not at a, but at some unknown value of c within the interval from a (about which the Taylor series is expanded), and x, where you are actually evaluating the power series. Mr. Budd, compiled September 29, 2010


364

HL Unit 10 (Error of Series) Example 10.2.2 From Foerster [12] (a) Estimate e2 using the 11th partial sum (n = 10) of the Maclaurin series for ex . [Ans: S10 = 7.3889470 . . .] (b) Use the Lagrange form of the remainder to estimate the accuracy of using this partial sum. (Do not assume that you know e, but you do know that it is between 2 and 3) [Ans: |R10 | < 0.0004617] (c) How does this estimate of the remainder compare with the ac-  tual error? Ans: e2 − S10 = 0.00006138 (d) Find approximately the value of c for which the Lagrange form is equal to the remainder R10 . [Ans: 0.1794 ∈ (0, 2)] Example 10.2.3 How many terms are needed to find e2 correct to five decimal places? [Ans: 13] Example 10.2.4 [12] The value of e2 can be calculated by first finding the value of e−2 , then taking the reciprocal. After the first few terms the series for e−2 meets the hypotheses for the alternating series test. Thus the error for any partial sum is bounded by the first term of the tail of the series after that partial sum. Estimate the error in the estimate of e−2 using the eleventh partial sum (S10 ). Then estimate e2 by calculating the reciprocal of (S10 ). Is the actual error any smaller than the error in using S10 directly for e2 ? [Ans: |R10 | < 0.000051306 . . .]

(In general, an error of ε% in 1/f (x) gives a maximum error of

ε in 1 − ε/100

the value of f (x).) Example 10.2.5 [12] The error in calculating e2 could be estimated by bounding the tail of the series with a convergent geometric series t12 that has the first term equal to . Overestimate the error using a t11 geometric series. [Ans: |R10 | < 0.00006156 . . .] Example 10.2.6 How many terms are needed to calculate ln 0.5 with an error of no more than 0.001? Mr. Budd, compiled September 29, 2010


HL Unit 10, Day 2: Error Analysis for Series

365

Example 10.2.7 (HL00) (a) Find the Maclaurin series of the function g(x) = sin x2 using ∞ P x2n+1 the series expansion of sin x, i.e., sin x = (−1)n . (2n + 1)! n=0 (b) Using the Maclaurin series of g(x) = sin x2 evaluate the definite integral Z 1 sin x2 dx 0

correct to four decimal places.



Ans: x2 −

x6 3!

+

x10 5!

+ ··· =

∞ P

(−1)n

n=0

x4n+2 ; 0.3103, needing three terms (2n + 1)!



Problems 10.B-1 [12] For the following problems, • Find the indicated partial sum. • Use the Lagrange form of the remainder to estimate the number of decimal places to which the partial sum is accurate. • Confirm your answer by subtracting the partial sum from the actual value. • Calculate the value of c in the appropriate interval for which the Lagrange form of the remainder is equal to the actual remainder. (a) e3 using the 15th partial sum of the Maclaurin series (n = 14) [Ans: 0.00001346 . . . < 0.0002962 . . .] (b) ln 0.7 using 8 terms of the Taylor series expansion about x = 1. Ans: 2.9998 × 10−6 < 5.4195 . . . × 10−5 10.B-2 [12] Use the Lagrange form of the remainder to find the number of terms needed in the partial sum to estimate the function value to the specified accuracy. (a) ln 0.6 to 7 decimal places using the Taylor series expansion about x = 1. [Ans: 32 terms] (b) e10 to 5 decimal places using the Maclaurin series. [Ans: 44 terms (n = 43)] 10.B-3 (BC99) The function f has derivatives of all orders for all real numbers x. Assume f (2) = −3, f 0 (2) = 5, f 00 (2) = 3, and f 000 (2) = −8. Mr. Budd, compiled September 29, 2010


366

HL Unit 10 (Error of Series) (a) Write the third-degree Taylor polynomial for f about x = 2 and use it to approximate f (1.5). [Ans: −4.958]

(4)

(b) The fourth derivative of f satisfies the inequality f (x) ≤ 3 for all x in the closed interval [1.5, 2]. Use the Lagrange error bound on the approximation of f (1.5) found above to explain why f (1.5) 6= −5. (c) Write the fourth-degree Taylor polynomial, P (x), for g(x) = f (x2 +2) about x = 0. Use P to explain why g must have   a relative minimum at x = 0. Ans: −3 + 5x2 + 23 x4

10.B-4 (BC00) The Taylor series about x = 5 for a certain function f converges to f (x) for all x in the interval of convergence. The nth derivative of f at 1 (−1)n n! , and f (5) = . x = 5 is given by f (n) (5) = n 2 (n + 2) 2 (a) hWrite the third-degree Taylor polynomial for f iabout x = 5. 2 3 1 1 (x − 5) − 40 (x − 5) Ans: 21 − 16 (x − 5) + 16 (b) Find the radius of convergence of the Taylor series for f about x = 5. [Ans: 2] (c) Show that the sixth-degree Taylor polynomial for f about x = 5 1 approximates f (6) with error less than . 1000 10.B-5 (HL01) (a) Find Maclaurin’s series expansion for f (x) = ln (1 + x), for 0 ≤ x < 1. [4 marks] (b) Rn is the error term in approximating f (x) by taking the sum of the first (n + 1) terms of its Maclaurin’s series. Prove |Rn | ≤

1 , n+1

(0 ≤ x < 1) . [2 marks]

10.B-6 (HL specimen) Test whether the following is a convergent series:   ∞ X 1 (−1)n+1 n! n=1 If it is, then find an approximation for the sum to two decimal places; if it is not, explain why this is so. [Ans: 0.625 ≤ S ≤ 0.633] 10.B-7 (BC03) The function f is defined by the power series f (x) =

∞ n n X (−1) x2n x2 x4 x6 (−1) x2n =1− + − + ··· + + ··· (2n + 1)! 3! 5! 7! (2n + 1)! n=0

for all real numbers x. Mr. Budd, compiled September 29, 2010


HL Unit 10, Day 2: Error Analysis for Series

367

(a) Find f 0 (0) and f 00 (0). Determine whether f has a local maximum, a local minimum, or neither at x = 0. Give a reason for your answer. 1 1 (b) Show that 1 â&#x2C6;&#x2019; approximates f (1) with error less than . 3! 100 (c) Show that y = f (x) is a solution to the differential equation xy 0 +y = cos x.   1 1 Ans: local max (2nd deriv test): f 0 (0) = 0 and f 00 (0) < 0; < 120 100

Mr. Budd, compiled September 29, 2010


368

HL Unit 10 (Error of Series)

Mr. Budd, compiled September 29, 2010


Bibliography [1] Howard Anton. Calculus: A New Horizon, Brief Edition. John Wiley and Sons, Inc., New York, 1999. [2] Stephen Bernstein and Ruth Bernstein. Schaum’s Outline of Theory and Problems of Elements of Statistics I: Descriptive Statistics and Probability. McGraw–Hill, New York, 1999. [3] George W. Best and J. Richard Lux. Preparing for the (AB) AP Calculus Examination. Venture Publishing, Andover, Massachusetts, 1998. [4] George W. Best and J. Richard Lux. Preparing for the (BC) AP Calculus Examination. Venture Publishing, Andover, Massachussetts, 1998. [5] William E. Boyce and Richard C. DiPrima. Elementary Differential Equations and Boundary Value Problems. John Wiley and Sons, New York, 1986. [6] Nigel Buckle, Iain Dunbar, and Fabio Cirrito. Mathematics Higher Level (Core). IBID Press, Camberwell, Australia, second edition, 1999. [7] Fabio Cirrito, editor. Mathematical Methods. IBID Press, Melton, Australia, second edition, 1998. [8] Douglas Downing and Jeffrey Clark. Forgotten Statistics: A Self-Teaching Refresher Course. Barron’s Educational Series, Hauppage, New York, 1996. [9] Douglas Downing and Jeffrey Clark, editors. Statistics: The Easy Way. Barron’s Educational Series, Hauppage, New York, third edition, 1997. [10] Ross L. Finney, Flanklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus: Graphical, Numerical, Algebraic. Scott Foreesman Addison Wesley, New York, 1999. [11] Paul Foerster. Calculus: Concepts and Applications; Instructor’s Resource Book. Key Curriculum Press, Berkeley, California, 1998. 369


370

BIBLIOGRAPHY

[12] Paul A. Foerster. Calculus: Concepts and Applications. Key Curriculum Press, Berkeley, California, 1998. [13] Melvin Hausner. A Vector Space Approach to Geometry. Dover Publications, Mineola, New York, 1965. [14] Shirley O. Hockett and David Bock. How to Prepare for the Advanced Placement Examination, Calculus: Review of Calculus AB and Calculus BC. Barronâ&#x20AC;&#x2122;s Educational Series, New York, 1998. [15] Roland E. Larson, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry. D. C. Heath and Company, Lexington, Massachusetts, 1994. [16] Jeff Morgan and Selwyn Hollis. Single Variable CalcLabs with the TI82/83s. Brooks/ Cole Publishing Company, Pacific Grove, CA, 1999. [17] Arnold Ostebee and Paul Zorn. Calculus From Graphical, Numerical, and Symbolic Points of View. Saunders College Publishing, Fort Worth, 1997. [18] Ronald I. Rothenberg. Probability and Statistics. Harcourt Brace Jovanovich, Orlando, Florida, 1991. [19] Salas, Hille, and Garret J. Etgen. Calculus: One and Several Variables. John Wiley and Sons, Inc., New York, New York, 1999. [20] James Stewart. Calculus:Concepts and Contexts, Single Variable. Brooks/ Cole Publishing Company, Pacific Grove, California, 1998. [21] Gilbert Strang. Linear Algebra and Its Applications. Harcourt Brace Jovanovich College Publishers, Fort Worth, 1988. [22] Paul Urban, John Owen, David Martin, Robert Haese, Sandra Haese, and Mark Bruce. Mathematics for the International Student: Mathematics HL (Core). Haese and Harris Publications, Adelaide Airport, Australia, 2004.

Mr. Budd, compiled September 29, 2010

Elite Ninja Math  

Handouts for IBDP Mathematics HL