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g¿1x2 d311 x2 1 x22 10 4 d1 1 x2 1 x22 10 a 1 x2 1 x 2 b 9 d dx a 1 x2 1 x 2 b 10 a 1 x2 1 x 2 b 9 c 2x11 x2 2 12x211 x 2 2 11 x 2 2 2 d 10 a 1 x2 1 x 2 b 9 c 2x 2x3 2x 2x 3 11 x 2 2 2 d 10 a 1 x2 9 1 x 2 b 4x c 11 x 2 2 2 d 1011 x2 2 9 11 x 2 2 9 4x 11 x 2 2 2 40x11 x2 2 9 11 x 2 2 11 d dx a 1 x2 1 x 2 b (Chain rule and quotient rule) (Expand) (Simplify) IN SUMMARY Key Idea • The chain rule: If y is a function of u, and u is a function of x (i.e., y is a composite function), dy dy du then provided that and exist. dx dy du du dx , du dx Therefore, if h1x2 1f g21x2, then h¿1x2 f ¿1g1x22 g¿1x2 (Function notation) or d3h1x24 dx d3f 1g1x22 4 d3g1x24 d3g1x24 dx (Leibniz notation) Need to Know • When the outer function is a power function of the form y 3g1x24 n , we have a special case of the chain rule, called the power of a function rule: d dx 3g1x24n d3g1x24n d3g1x24 d3g1x24 dx n3g1x24 n1 g¿1x2 104 2.5 THE DERIVATIVES OF COMPOSITE FUNCTIONS NEL
Exercise 2.5 PART A 1. Given f 1x2 Vx and g1x2 x 2 1, find the following value: a. f 1g1122 c. g1 f 1022 e. f 1g1x22 b. g1 f 1122 d. f 1g1422 f. g1 f 1x22 2. For each of the following pairs of functions, find the composite functions 1 f g2 and 1g f 2. What is the domain of each composite function? Are the composite functions equal? a. f 1x2 x 2 b. f 1x2 1 c. x f 1x2 1 x C K 1p 2 x 2 2 y ku n , g1x2 Vx g1x2 x 2 1 g1x2 Vx 2 a. f 1x2 12x 32 4 d. f 1x2 3 3. What is the rule for calculating the derivative of the composition of two differentiable functions? Give examples, and show how the derivative is determined. 4. Differentiate each function. Do not expand any expression before differentiating. b. g1x2 1x 2 42 3 e. y Vx 2 3 c. 1 h1x2 12x 2 3x 52 4 f. f 1x2 1x 2 162 5 PART B 5. Rewrite each of the following in the form y u n or and then a. 1 y 2 c. y 1 e. y x 3 x 2 4 5x 2 x differentiate. b. 1 y 1 d. y 3 f. y x 1 9 x 2 1x 2 x 12 4 6. Given h g f, where f and g are continuous functions, use the information in the table to evaluate h112 and h112. x f(x) g(x) f ’(x) g’(x) 1 1 18 5 15 0 2 5 1 11 1 1 4 3 7 2 4 9 7 3 3 13 10 11 1 7. Given f 1x2 1x 32 2 , g1x2 1 , and h1x2 f 1g1x22, determine h1x2. x NEL CHAPTER 2 105
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Chapter 1 INTRODUCTION TO CALCULUS
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y x 3 1 , if 3 6 t 6 0 t 6. A func
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What Is Calculus? Two simple geomet
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If a and b were radicals, squaring
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Exercise 1.1 PART A 1. Write the co
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Consider a curve y f 1x2 and a poi
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EXAMPLE 2 Tech Support For help gra
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The slope of the tangent at 13, 42
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INVESTIGATION 4 Tech Support For he
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4. Simplify each of the following d
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18. Copy the following figures. Dra
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A. Determine the average velocity o
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Instantaneous Velocity The velocity
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Solution a. C11002 10V100 1000 1
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Exercise 1.3 C PART A 1. The veloci
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f 1x2 f 1a2 15. Use the alternate
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. Use your results for part a to ap
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The graph shown on your calculator
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limits because, in each case, only
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11. For each function, sketch the g
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EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 Using
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INVESTIGATION Here is an alternate
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In this section, we learned the pro
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A 11. Jacques Charles (1746-1823) d
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A second condition for continuity a
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d. The function F is not continuous
Page 53 and 54: x 3, if x 3 12. g1x2 e 2 k, if
Page 55 and 56: Key Concepts Review We began our in
Page 57 and 58: c. What is the present rate of chan
Page 59 and 60: 16. Evaluate the limit of each diff
Page 61 and 62: Chapter 2 DERIVATIVES Imagine a dri
Page 63 and 64: 2 4 5. Expand, and collect like te
Page 65 and 66: Section 2.1—The Derivative Functi
Page 67 and 68: For the Parabola f1x2 x 2 The slop
Page 69 and 70: Note that f 1t2 Vt is defined for
Page 71 and 72: EXAMPLE 6 Reasoning about different
Page 73 and 74: Exercise 2.1 PART A 1. State the do
Page 75 and 76: T 15. Match each function in graphs
Page 77 and 78: EXAMPLE 1 Applying the constant fun
Page 79 and 80: We conclude this section with the s
Page 81 and 82: EXAMPLE 6 Connecting the derivative
Page 83 and 84: x 1 8. Determine the slope of the
Page 85 and 86: Section 2.3—The Product Rule In t
Page 87 and 88: EXAMPLE 3 Selecting an efficient st
Page 89 and 90: EXAMPLE 6 Selecting a strategy to d
Page 91 and 92: PART B dy 5. Determine the value of
Page 93 and 94: 9. Determine the equation of the ta
Page 95 and 96: EXAMPLE 1 Applying the quotient rul
Page 97 and 98: Exercise 2.4 PART A 1. What are the
Page 99 and 100: Section 2.5—The Derivatives of Co
Page 101 and 102: with respect to x is equal to the p
Page 103: EXAMPLE 7 Combining derivative rule
Page 107 and 108: Technology Extension: Derivatives o
Page 109 and 110: Key Concepts Review Now that you ha
Page 111 and 112: 9. For what values of x does the cu
Page 113 and 114: 24. At a manufacturing plant, produ
Page 115 and 116: Chapter 3 DERIVATIVES AND THEIR APP
Page 117 and 118: 4. Determine the area of each figur
Page 119 and 120: Section 3.1—Higher-Order Derivati
Page 121 and 122: Motion on a Straight Line An object
Page 123 and 124: d. The object moves in a positive d
Page 125 and 126: EXAMPLE 5 Analyzing motion under gr
Page 127 and 128: Exercise 3.1 C K PART A 1. Explain
Page 129 and 130: T 12. A dragster races down a 400 m
Page 131 and 132: H. Repeat part D for the following
Page 133 and 134: Only t 3 is in the given interval,
Page 135 and 136: IN SUMMARY Key Ideas • The maximu
Page 137 and 138: T PART B 4. Using the algorithm for
Page 139 and 140: Mid-Chapter Review 1. Determine the
Page 141 and 142: Section 3.3—Optimization Problems
Page 143 and 144: For extreme values, set V¿1x2 0.
Page 145 and 146: Exercise 3.3 C PART A 1. A piece of
Page 147 and 148: a. Determine the dimensions that sh
Page 149 and 150: The revenue function is R1x2 17 0
Page 151 and 152: The minimal cost is approximately $
Page 153 and 154: T C 8. The fuel cost per hour for r
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CAREER LINK WRAP-UP Investigate and
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10. For each of the following cost
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s1t2 t 26. Find the absolute maxi
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Chapter 4 CURVE SKETCHING If you ar
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5. Determine the derivative of each
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Section 4.1—Increasing and Decrea
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dy Since this is a polynomial funct
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IN SUMMARY Key Ideas • A function
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9. Each of the following graphs rep
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Solution dy First determine dx . dy
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Note that there is no value of x fo
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EXAMPLE 4 Graphing the derivative g
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4. Find the x- and y-intercepts of
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Section 4.3—Vertical and Horizont
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The behaviour of the graph can be i
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EXAMPLE 2 Expressing a polynomial f
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When lim f 1x2 k or lim f 1x2 k,
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For horizontal asymptotes, 3x f 1x2
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Now let's consider the straight lin
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Exercise 4.3 PART A 1. State the eq
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12. Use the features of each functi
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11. a. What does f ¿1x2 7 0 imply
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From these investigations, we can m
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Setting dy dx 0, we obtain 3 1x 2
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The second derivative of f 1x2 is f
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Exercise 4.4 K PART A 1. For each f
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Section 4.5—An Algorithm for Curv
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Therefore, by the second derivative
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Interval 1, 12 11, 0.82 10.8, 22 12
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K A 1x PART B 4. Use the algorithm
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Key Concepts Review In this chapter
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5. For each of the following, check
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17. If f 1x2 x3 8 , determine the
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Chapter 5 DERIVATIVES OF EXPONENTIA
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Radian Measure A radian is the meas
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5. Convert the following angles to
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Section 5.1—Derivatives of Expone
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From the investigation, you should
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EXAMPLE 4 Connecting the derivative
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A C PART B 6. Determine the equatio
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Section 5.2—The Derivative of the
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In general, for the exponential fun
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Solution a. January 1, 1900, is exa
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Section 5.3—Optimization Problems
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The average revenue to the company
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Exercise 5.3 PART A 1. Use graphing
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12. Find the maximum and minimum va
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9. The rapid growth in the number o
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INVESTIGATION 2 A.Using your graphi
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With practice, you will learn how t
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We evaluate f 1x2 at the critical n
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A T 6. Determine the absolute extre
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Derivatives of Composite Functions
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Review Exercise 1. Differentiate ea
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18. The position of a particle is g
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Chapter 5 Test dy 1. Determine the
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8. Use algebraic methods to evaluat
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f 1x2 142e 5x1 24. For each functi
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Review of Prerequisite Skills In th
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CAREER LINK Investigate CHAPTER 6:
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What is a Vector? A vector is a mat
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EXAMPLE 1 B A E D C Connecting vect
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A K B 5. Given the vector AB ! as s
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Section 6.2—Vector Addition In th
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the overall magnitude would be equa
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Solution c b a a - b + c a - b c a
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W N S E The vectors are drawn so th
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Exercise 6.2 PART A 1. The vectors
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10. a. In ! the ! example ! involvi
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0 0 The previous example illustrate
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Solution a. 30˚ y x 2x-y 2x -y 30
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Solution Draw a sketch and determin
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8. The two vectors a ! and b ! are
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Section 6.4—Properties of Vectors
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Demonstrating the distributive law
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IN SUMMARY Key Idea • Properties
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Mid-Chapter Review 1. ABCD is a par
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Section 6.5—Vectors in R 2 and R
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Right-Handed System of Coordinates
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Solution a. A16, 0, 02 is a point o
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There is one further observation th
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C A 14. a. What is the equation of
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R 2 EXAMPLE 1 Representing vectors
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EXAMPLE 2 Representing the sum and
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EXAMPLE 5 Calculating the magnitude
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A 10. Parallelogram is determined b
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EXAMPLE 1 Representing vectors in R
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. Using components, a ! b ! c !
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IN SUMMARY Key Ideas • In R 3 , i
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Section 6.8—Linear Combinations a
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In R 2 , it is possible to take any
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Geometrically, the linear combinati
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IN SUMMARY Key Ideas • In R 2 , O
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Review Exercise 1. Determine whethe
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15. A rectangular prism measuring 3
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Chapter 6 Test 1. The vectors a ! ,
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Section 7.1—Vectors as Forces The
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of 6 cm, also pointing east. The ve
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is drawn approximately to scale, an
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The magnitude of the vertical compo
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! T1 sin 105° 1961sin 45°2 T ! 1
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49 34 30 cos CBA 1 cos CBA 2 120
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C T f 3 40 N O 35 N 45° 30 N f 2 2
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Solution We start by drawing positi
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v a v + w w 20 triangles, Solving t
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A T C 9. A small airplane has an ai
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Perhaps the most important observat
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If we look at the right-angled tria
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Solution Consider the following dia
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A T 10. What is the value of the do
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Expanding, we get b 2 1 2a 1 b 1
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. 0 a ! 0 2 @b ! 112 2 122 2 142
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One of the most important propertie
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4. a. From the set of vectors e11,
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Mid-Chapter Review 1. a. If 0 a ! 0
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Section 7.5—Scalar and Vector Pro
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EXAMPLE 1 Reasoning about the chara
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x EXAMPLE 3 z P (2, 1, 4) g b a O(0
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EXAMPLE 4 Calculating a specific di
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IN SUMMARY Key Idea • A projectio
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B C 12. In the diagram shown, ^ ABC
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a 3 b, k = 1 O b 3 a, k = -1 b a is
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If we carry out an identical proced
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EXAMPLE 2 Calculating cross product
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K A C T 4. Calculate the cross prod
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EXAMPLE 1 Using the dot product to
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. We start by constructing position
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EXAMPLE 4 Using the cross product t
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Page 419 and 420:
Review Exercise 1. Given that a !
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18. For the vectors m ! 1 V3, 2, 3
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Chapter 7 Test 1. Given the vectors
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Both of these vectors can be multip
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c. If the point Q121, 232 lies on t
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In this section, the vector and par
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C K A T 6. a. If the equation of a
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EXAMPLE 1 Representing the Cartesia
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It is not possible, in this case, t
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AP ! 1x 4, y 1222 1x 4, y 22.
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u cos 1 a 3 12 1 10 21 25 2 b u
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A T 10. For each pair of lines, det
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is its direction vector. If represe
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EXAMPLE 4 Representing the equation
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C A T 6. a. Determine parametric eq
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13. The Cartesian equation of a lin
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s t s(1, 2, 1) t(0, 2, 1), s, tR P
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After deriving vector and parametri
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EXAMPLE 4 Representing the equation
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A K T 7. a. Determine parameters co
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To derive the equation of this plan
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A number of observations can be mad
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When we considered lines in R 2 , w
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IN SUMMARY Key Idea • The Cartesi
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Section 8.6—Sketching Planes in R
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EXAMPLE 2 Representing the graph of
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EXAMPLE 5 Graphing planes whose Car
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IN SUMMARY Key Idea • A sketch of
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Page 481 and 482:
Review Exercise 1. Determine vector
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20. Calculate the acute angle that
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Chapter 8 Test 1. a. Given the poin
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4 13 s2 211 4s2 12 8s2 8 0 1
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Method 2: Again, this result can be
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We can now select any two of the th
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IN SUMMARY Key Ideas • Line and p
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A T 11. a. Show that the lines and
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equations. Since the lines would be
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Solution 1: Interchange equations 1
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Substituting into the second equati
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1: Multiply equation 1 by k, and ad
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K C PART B 4. Solve each system of
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Section 9.3—The Intersection of T
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If we had solved the system using e
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then x 3s 1. Now it is a matter o
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Exercise 9.3 C K PART A 1. A system
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Mid-Chapter Review 1. Determine the
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Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
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Thus, 7y 5t 3. Dividing by 7, we g
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Inconsistent Systems for Three Equa
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Therefore, we can choose ! ! ! m 1
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Exercise 9.4 PART A 1. A student is
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9. Solve each system of equations u
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Section 9.5—The Distance from a P
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EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
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Exercise 9.5 PART A 1. Determine th
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Section 9.6—The Distance from a P
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It is also possible to use the dist
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p 1 p 2 U V d L 1 : r = (-2, 1, 0)
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The required distance between the t
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K A T PART B 2. Determine the follo
Page 553 and 554:
Review Exercise 1. The lines 2x y
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14. You are given the lines , and r
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Chapter 9 Test 1. a. Determine the
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P(1, -2, 4) Q P9 2x - 3y - 4z + 66
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29. A line that passes through the
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Solution a. Differentiate both side
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Exercise PART A 1. State the chain
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dA b. To determine differentiate A
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1 16 dh dt Therefore, at the momen
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11. Two cyclists depart at the same
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Solution a. y ln15x2 Using the cha
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x1x 42 0 x 4 or x 0 But x 0 is
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The Derivatives of General Logarith
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The Derivative of a Composite Funct
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To solve this, we take the natural
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Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
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Properties of a Matrix in Row-Echel
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Exercise PART A 1. Write an augment
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12. Determine the equation of the p
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1 0 0 18 4 (row 3) row 1 £ 0 1 0
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Finally, convert the entry in the s
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Review of Technical Skills Appendix
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3 Evaluating a Function 1. Enter th
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7 Making a Table of Differences To
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4. Cursor along the curve to any po
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5. Display and analyze the results.
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14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
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Use either the calculator keypad or
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20 Graphing the Derivative of a Fun
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4. Create a function. Right-click t
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composition: the process of combini
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G geometric vectors: vectors that a
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ationalizing the denominator: the p
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Answers Chapter 1 0 0 0 0 0 Review
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d. about 1 e. about 7 8 f. no tang
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d. 13. m 3; b 1 14. a 3, b 2, c
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Review Exercise, pp. 56-59 1. a. 3
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19. a. y 7 b. y 5x 5 c. y 18x 9
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20. a. 34.3 m> s b. 39.2 m> s c. 54
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17. a. 100 bacteria b. 1200 bacteri
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11. a. b. 160x y 16 0 60x y 61
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3. 1 2x 4. a. x 2 15x 6 b. 6012x
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14. t 1 s; away 15. a. s1t2 kt 2
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14. a. triangle side length 0.96 cm
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Review Exercise, pp. 156-159 11. 20
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c. i. ii. iii. d. i. ii. iii. 5 4 3
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. local minimum: 11.41, 39.62, loca
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2. increasing: x 6 1 and x 7 2, dec
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to a zero of its derivative, the nu
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7. (-2, 10) 10 8 y 10. a. 8 y e. 4
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4. hole at x 2; large and negative
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sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
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Review Exercise, pp. 263-265 1. a.
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x 2. 5. a. 19 000 fish> year b. 23
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Section 6.1, pp. 279-281 6. a. b. c
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17. P G 4. Answers may vary. For ex
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11. a. AG ! a ! b ! c ! , b. AG
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d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
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Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
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. Since the equilibrant is directed
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7. a. scalar projection: 0, vector
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2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
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cos A 1 cos B 1 cos C 1 area of t
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. r ! 11, 1, 02 t10, 1, 12, tR; x
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2. 12, 3, 42 3. P must lie on plane
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25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
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Mid-Chapter Review, pp. 518-519 1.
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8. a. b. y 1 z 1 4 , 2 c. x 3t
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10. vector equation: r ! 10, 0, 42
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. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
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16. 17. 18. 19. 2 3 4 m>min 144 m>m
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ln x 1 e. ln 2 4x 1 x ln 13x 2 2
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Index A Absolute extrema, 130, 132,
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normal, 461-469, 486, 512, 524 para
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