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13. a. 27 b. 1 c. 1 14. a. 0 b. 0 1 15. a. 0 b. 2 16. 2 17. does not exist y –4 –2 Section 1.6, pp. 51–53 1. Anywhere that you can see breaks or jumps is a place where the function is not continuous. 2. On a given domain, you can trace the graph of the function without lifting your pencil. 3. point discontinuity 10 8 6 4 2 jump discontinuity y 10 8 6 4 2 infinite discontinuity 10 8 6 4 2 y y hole –2 0 2 4 6 –2 –2 0 2 4 6 –2 4 2 –2 –1 0 1 2 3 4 –2 vertical –4 asymptote x x 0 2 x x 4. a. x 3 b. x 0 c. x 0 d. x 3 and x 3 e. x 3 and x 2 f. x 3 5. a. continuous for all real numbers b. continuous for all real numbers c. continuous for all real numbers, except 0 and 5 d. continuous for all real numbers greater than or equal to 2 e. continuous for all real numbers f. continuous for all real numbers 6. g1x2 is a linear function (a polynomial), and so is continuous everywhere, including x 2. 7. y 8. 9. 4 2 0 Yes, the function is continuous everywhere. –4 –2 The function is discontinuous at x 0. y 200 6 4 2 –6 –4 –2 0 –2 –4 –6 4 2 Discontinuities at 0, 100, 200, and 500 y x 0 2 4 –2 –4 2 4 6 x 400 600 x 10. no 11. Discontinuous at x 2 12. k 16 13. a. y 4 –4 b. i. 1 ii. 1 iii. does not exist c. f is not continuous since lim f 1x2 xS0 does not exist. 14. a. 2 b. 4 c. lim f 1x2 4 lim f 1x2 xS3 xS3 Thus, lim f 1x2 4. But, f 132 2. xS3 Hence, f is not continuous at x 2 and also not continuous on 3 6 x 6 8. 15. (1) A B 3 (2) 4B A 6 (if B 7 1, then A 7 2; if B 6 1, then A 6 2) 16. a 1, b 6 17. a. lim xS1g1x2 1 lim g1x2 1 lim g1x2 xS1 xS1 –4 b. x –2 0 2 4 lim xS1 g1x2 2 –2 –4 g1x2 does not exist. 4 2 y x –2 0 2 4 –2 –4 is discontinuous at x 1. NEL Answers 627
Review Exercise, pp. 56–59 1. a. 3 b. 7 c. 2x y 5 0 2. a. 1 3 c. 1 27 b. 1 2 d. 5 4 3. a. 2 b. 2 4. a. 1st second 5 ms, > 2nd second 15 ms > b. 40 ms > c. 60 ms > 5. a. 0.0601 g b. 6.01 g> min c. 6 g> min 6. a. 700 000 t b. 18 10 4 t per year c. 15 10 4 t per year d. 7.5 years 7. a. 10 b. 7; 0 c. t 3 and t 4 8. a. Answers may vary. For example: –2 2 1 b. Answers may vary. For example: 4 2 y x –1 0 1 2 –1 –2 y 9. a. b. x 1 and x 1 c. They do not exist. 10. not continuous at x 4 11. a. x 1 and x 2 b. lim f 1x2 2 , xS1 3 lim f 1x2 does not exist. xS2 12. a. lim f 1x2 does not exist. xS0 b. lim g1x2 0 xS0 c. lim h1x2 does not exist. xS3 13. a. x 1.9 1.99 1.999 2.001 2.01 2.1 x 2 x 2 x 2 14. –4 –2 b. 1 3 1 2 4 2 y x 0 2 4 –2 –4 0 16. a. 10 b. 1 c. 1 4 16 17. a. 4 c. 1 5 e. 1 8 b. 10a d. 1 3 f. 1 4 18. a. The function is not defined for x 6 3, so there is no left-side limit. b. Even after dividing out common factors from numerator and denominator, there is a factor of x 2 in the denominator; the graph has a vertical asymptote at x 2. c. lim xS1f 1x2 5 lim xS1 d. The function has a vertical asymptote at x 2. e. x S 0 0x x 0x 0 lim xS0 x 1 0x 0 lim xS0 x 1 0x 0 lim xS0 x lim 0x 0 xS0 x 0.344 83 0.334 45 0.333 44 0.333 22 0.332 23 0.322 58 x 0.9 0.99 0.999 1.001 1.01 1.1 x 1 x 2 1 0.526 32 0.502 51 0.500 25 0.499 75 0.497 51 0.476 19 x 0.1 0.01 0.001 0.001 0.01 0.1 –4 –2 x 0 2 4 –2 –4 x 3 3 x 0.291 12 0.288 92 0.2887 0.288 65 0.288 43 0.286 31 1 ; This agrees well with the values 23 in the table. 15. a. x 2.1 2.01 2.001 2.0001 f(x) 0.248 46 0.249 84 0.249 98 0.25 lim f 1x2 0.25 xS2 b. lim f 1x2 0.25 xS2 c. 0.25 f. lim f 1x2 1 xS1 f 1x2 5 lim xS1 lim xS1 f 1x2 Therefore, lim f 1x2 does not exist. xS1 lim xS1 f 1x2 628 Answers NEL
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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
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x 3, if x 3 12. g1x2 e 2 k, if
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Key Concepts Review We began our in
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c. What is the present rate of chan
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16. Evaluate the limit of each diff
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Chapter 2 DERIVATIVES Imagine a dri
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2 4 5. Expand, and collect like te
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Section 2.1—The Derivative Functi
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For the Parabola f1x2 x 2 The slop
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Note that f 1t2 Vt is defined for
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EXAMPLE 6 Reasoning about different
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Exercise 2.1 PART A 1. State the do
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T 15. Match each function in graphs
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EXAMPLE 1 Applying the constant fun
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We conclude this section with the s
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EXAMPLE 6 Connecting the derivative
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x 1 8. Determine the slope of the
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Section 2.3—The Product Rule In t
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EXAMPLE 3 Selecting an efficient st
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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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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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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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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
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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
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
Page 577 and 578: The Derivatives of General Logarith
Page 579 and 580: 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
Page 585 and 586: 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
Page 599 and 600: 3 Evaluating a Function 1. Enter th
Page 601 and 602: 7 Making a Table of Differences To
Page 603 and 604: 4. Cursor along the curve to any po
Page 605 and 606: 5. Display and analyze the results.
Page 607 and 608: 14 Graphing a Trigonometric Functio
Page 609 and 610: 2. Enter the second equation. Press
Page 611 and 612: Use either the calculator keypad or
Page 613 and 614: 20 Graphing the Derivative of a Fun
Page 615 and 616: 4. Create a function. Right-click t
Page 617 and 618: composition: the process of combini
Page 619 and 620: G geometric vectors: vectors that a
Page 621 and 622: ationalizing the denominator: the p
Page 623 and 624: Answers Chapter 1 0 0 0 0 0 Review
Page 625 and 626: d. about 1 e. about 7 8 f. no tang
Page 627: d. 13. m 3; b 1 14. a 3, b 2, c
Page 631 and 632: 19. a. y 7 b. y 5x 5 c. y 18x 9
Page 633 and 634: 20. a. 34.3 m> s b. 39.2 m> s c. 54
Page 635 and 636: 17. a. 100 bacteria b. 1200 bacteri
Page 637 and 638: 11. a. b. 160x y 16 0 60x y 61
Page 639 and 640: 3. 1 2x 4. a. x 2 15x 6 b. 6012x
Page 641 and 642: 14. t 1 s; away 15. a. s1t2 kt 2
Page 643 and 644: 14. a. triangle side length 0.96 cm
Page 645 and 646: Review Exercise, pp. 156-159 11. 20
Page 647 and 648: c. i. ii. iii. d. i. ii. iii. 5 4 3
Page 649 and 650: . local minimum: 11.41, 39.62, loca
Page 651 and 652: 2. increasing: x 6 1 and x 7 2, dec
Page 653 and 654: to a zero of its derivative, the nu
Page 655 and 656: 7. (-2, 10) 10 8 y 10. a. 8 y e. 4
Page 657 and 658: 4. hole at x 2; large and negative
Page 659 and 660: sin x b. 1 sin 2 tan x sec x x L
Page 661 and 662: 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
Page 667 and 668: Section 6.1, pp. 279-281 6. a. b. c
Page 669 and 670: 17. P G 4. Answers may vary. For ex
Page 671 and 672: 11. a. AG ! a ! b ! c ! , b. AG
Page 673 and 674: d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
Page 675 and 676: Section 6.7, pp. 332-333 1. a. 1i !
Page 677 and 678: 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
Page 683 and 684:
2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
Page 685 and 686:
cos A 1 cos B 1 cos C 1 area of t
Page 687 and 688:
. 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.
Page 697 and 698:
8. a. b. y 1 z 1 4 , 2 c. x 3t
Page 699 and 700:
10. vector equation: r ! 10, 0, 42
Page 701 and 702:
. 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
show all

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