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4. Calculate each limit. a. lim x c. lim e. xS 5 b. lim 1x 72 d. lim 14 3x 2 2 f. xS3 x 2 xS10 xS 2 5. Determine lim 1, if x 4 f 1x2, where f 1x2 e 1, if x 4 . xS4 lim 4 xS1 lim xS3 2 x PART B 6. For the function f 1x2 in the graph below, determine the following: a. lim f 1x2 b. lim f 1x2 c. lim f 1x2 d. xS2 xS2 y 4 2 (2, 2) (–2, 0) x –2 2 4 6 8 –2 (2, –1) xS2 f 122 K 7. Use the graph to find the limit, if it exists. a. lim f 1x2 b. lim f 1x2 c. xS2 xS2 y y 4 4 lim f 1x2 xS3 y 3 3 3 2 2 1 0 1 2 3 4 x 2 1 0 1 2 3 4 x 1 0 1 2 3 4 x 8. Evaluate each limit. a. lim 19 x 2 2 b. lim c. limx 1 xS0 x 20 xS 1 2x 5 xS5 9. Find lim 1x 2 12, and illustrate your result with a graph indicating the xS2 limiting value. 10. Evaluate each limit. If the limit does not exist, explain why. 1 a. lim c. lim e. lim xS3 1x2 42 xS0 x4 xS3 x 2 1 1 b. lim d. lim f. lim xS2 1x2 42 xS1 x 3 xS3 x 3 38 1.4 THE LIMIT OF A FUNCTION NEL
11. For each function, sketch the graph of the function. Determine the indicated limit if it exists. x 2, if x 6 1 a. f 1x2 e x 2, if x 1 ; lim f 1x2 xS1 b. x 4, if x 2 f 1x2 e 2x 6, if x 7 2 ; lim f 1x2 xS2 A T c. d. 12. Sketch the graph of any function that satisfies the given conditions. a. f 112 1, lim f 1x2 3, lim xS1 xS1 b. f 122 1, lim f 1x2 0 xS2 c. if and lim xS1 d. f 132 0, lim xS3 13. Let f 1x2 mx b, where m and b are constants. If lim f 1x2 2 and lim f 1x2 4, find m and b. xS1 xS 1 PART C 14. Determine the real values of a, b, and c for the quadratic function f 1x2 ax 2 bx c, a 0, that satisfy the conditions f 102 0, f 1x2 5, and f 1x2 8. lim xS1 4x, if x 1 2 f 1x2 • 1 x , if x 6 1 2 1, if x 6 0.5 f 1x2 e x 2 0.25, if x 0.5 ; lim f 1x2 xS0.5 lim xS 2 ; lim f 1x2 xS 1 2 15. The fish population, in thousands, in a lake at time t, in years, is modelled by the following function: 3 1 12 t2 , if 0 t 6 p1t2 µ 2 1 18 t2 , if 6 6 t 12 This function describes a sudden change in the population at time t 6, due to a chemical spill. a. Sketch the graph of p1t2. b. Evaluate lim p1t2 and lim p1t2. tS6 tS6 c. Determine how many fish were killed by the spill. d. At what time did the population recover to the level before the spill? NEL CHAPTER 1 39
Page 1 and 2: Chapter 1 INTRODUCTION TO CALCULUS
Page 3 and 4: y x 3 1 , if 3 6 t 6 0 t 6. A func
Page 5 and 6: What Is Calculus? Two simple geomet
Page 7 and 8: If a and b were radicals, squaring
Page 9 and 10: Exercise 1.1 PART A 1. Write the co
Page 11 and 12: Consider a curve y f 1x2 and a poi
Page 13 and 14: EXAMPLE 2 Tech Support For help gra
Page 15 and 16: The slope of the tangent at 13, 42
Page 17 and 18: INVESTIGATION 4 Tech Support For he
Page 19 and 20: 4. Simplify each of the following d
Page 21 and 22: 18. Copy the following figures. Dra
Page 23 and 24: A. Determine the average velocity o
Page 25 and 26: Instantaneous Velocity The velocity
Page 27 and 28: Solution a. C11002 10V100 1000 1
Page 29 and 30: Exercise 1.3 C PART A 1. The veloci
Page 31 and 32: f 1x2 f 1a2 15. Use the alternate
Page 33 and 34: . Use your results for part a to ap
Page 35 and 36: The graph shown on your calculator
Page 37: limits because, in each case, only
Page 41 and 42: EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 Using
Page 43 and 44: INVESTIGATION Here is an alternate
Page 45 and 46: In this section, we learned the pro
Page 47 and 48: A 11. Jacques Charles (1746-1823) d
Page 49 and 50: A second condition for continuity a
Page 51 and 52: 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
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PART B dy 5. Determine the value of
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9. Determine the equation of the ta
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EXAMPLE 1 Applying the quotient rul
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Exercise 2.4 PART A 1. What are the
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Section 2.5—The Derivatives of Co
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with respect to x is equal to the p
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EXAMPLE 7 Combining derivative rule
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Exercise 2.5 PART A 1. Given f 1x2
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Technology Extension: Derivatives o
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Key Concepts Review Now that you ha
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9. For what values of x does the cu
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24. At a manufacturing plant, produ
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Chapter 3 DERIVATIVES AND THEIR APP
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4. Determine the area of each figur
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Section 3.1—Higher-Order Derivati
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Motion on a Straight Line An object
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d. The object moves in a positive d
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EXAMPLE 5 Analyzing motion under gr
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Exercise 3.1 C K PART A 1. Explain
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T 12. A dragster races down a 400 m
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H. Repeat part D for the following
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Only t 3 is in the given interval,
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IN SUMMARY Key Ideas • The maximu
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T PART B 4. Using the algorithm for
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Mid-Chapter Review 1. Determine the
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Section 3.3—Optimization Problems
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For extreme values, set V¿1x2 0.
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Exercise 3.3 C PART A 1. A piece of
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a. Determine the dimensions that sh
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The revenue function is R1x2 17 0
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The minimal cost is approximately $
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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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Page 263 and 264:
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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Page 345 and 346:
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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Page 353 and 354:
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
Page 411 and 412:
EXAMPLE 1 Using the dot product to
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. We start by constructing position
Page 415 and 416:
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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Page 427 and 428:
Page 429 and 430:
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
Page 453 and 454:
13. The Cartesian equation of a lin
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s t s(1, 2, 1) t(0, 2, 1), s, tR P
Page 457 and 458:
After deriving vector and parametri
Page 459 and 460:
EXAMPLE 4 Representing the equation
Page 461 and 462:
A K T 7. a. Determine parameters co
Page 463 and 464:
To derive the equation of this plan
Page 465 and 466:
A number of observations can be mad
Page 467 and 468:
When we considered lines in R 2 , w
Page 469 and 470:
IN SUMMARY Key Idea • The Cartesi
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Section 8.6—Sketching Planes in R
Page 473 and 474:
EXAMPLE 2 Representing the graph of
Page 475 and 476:
EXAMPLE 5 Graphing planes whose Car
Page 477 and 478:
IN SUMMARY Key Idea • A sketch of
Page 479 and 480:
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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Page 489 and 490:
Page 491 and 492:
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
Page 501 and 502:
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
Page 521 and 522:
Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
Page 525 and 526:
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
Page 531 and 532:
Exercise 9.4 PART A 1. A student is
Page 533 and 534:
9. Solve each system of equations u
Page 535 and 536:
Section 9.5—The Distance from a P
Page 537 and 538:
EXAMPLE 1 Calculating the distance
Page 539 and 540:
In triangle PQR, sin u d @ QP ! @
Page 541 and 542:
Exercise 9.5 PART A 1. Determine th
Page 543 and 544:
Section 9.6—The Distance from a P
Page 545 and 546:
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
Page 557 and 558:
Chapter 9 Test 1. a. Determine the
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P(1, -2, 4) Q P9 2x - 3y - 4z + 66
Page 561 and 562:
29. A line that passes through the
Page 563 and 564:
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
Page 581 and 582:
To solve this, we take the natural
Page 583 and 584:
Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
Page 587 and 588:
Properties of a Matrix in Row-Echel
Page 589 and 590:
Exercise PART A 1. Write an augment
Page 591 and 592:
12. Determine the equation of the p
Page 593 and 594:
1 0 0 18 4 (row 3) row 1 £ 0 1 0
Page 595 and 596:
Finally, convert the entry in the s
Page 597 and 598:
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.
Page 607 and 608:
14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
Page 611 and 612:
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
Page 645 and 646:
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
Page 659 and 660:
sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
Page 663 and 664:
Review Exercise, pp. 263-265 1. a.
Page 665 and 666:
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
Page 671 and 672:
11. a. AG ! a ! b ! c ! , b. AG
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d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
Page 675 and 676:
Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
Page 679 and 680:
. Since the equilibrant is directed
Page 681 and 682:
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
Page 689 and 690:
2. 12, 3, 42 3. P must lie on plane
Page 691 and 692:
25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
Page 695 and 696:
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 ,
Page 703 and 704:
16. 17. 18. 19. 2 3 4 m>min 144 m>m
Page 705 and 706:
ln x 1 e. ln 2 4x 1 x ln 13x 2 2
Page 707 and 708:
Index A Absolute extrema, 130, 132,
Page 709:
normal, 461-469, 486, 512, 524 para
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