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Review of Prerequisite Skills Before beginning this chapter, review the following concepts from previous courses: • determining the slope of a line: m ¢y ¢x • determining the equation of a line • using function notation for substituting into and evaluating functions • simplifying algebraic expressions • factoring expressions • finding the domain of functions • calculating average rate of change and slopes of secant lines • estimating instantaneous rate of change and slopes of tangent lines Exercise 42 1. Determine the slope of the line passing through each of the following pairs of points: a. 12, 52 and 16, 72 d. 10, 02 and 11, 42 b. 13, 42 and 11, 42 e. 12.1, 4.412 and 12, c. 10, 02 and 11, 42 f. and a 7 4 , 1 4 b 4 , 1 4 b 2. Determine the equation of a line for the given information. a. slope 4, y-intercept 2 d. through 12, 42 and 16, 82 b. slope 2, y-intercept 5 e. vertical, through 13, 52 c. through 11, 62 and 14, 122 f. horizontal, through 13, 52 3. Evaluate for x 2. a. f 1x2 3x 5 c. f 1x2 3x 2 2x 1 b. f 1x2 14x 2213x 62 d. f 1x2 15x 22 2 4. For f 1x2 x determine each of the following values: x 2 4 , a. f 1102 b. f 132 c. f 102 d. f 1102 3 x, if x 6 0 5. Consider the function f given by f 1x2 e 3 x, if x 0 Calculate each of the following: a. f 1332 b. f 102 c. f 1782 d. f 132 2 REVIEW OF PREREQUISITE SKILLS NEL
y x 3 1 , if 3 6 t 6 0 t 6. A function s is defined for t 7 3 by s1t2 µ 5, if t 0 t 3 , if t 7 0 Evaluate each of the following: a. s122 b. s112 c. s102 d. s112 e. s11002 7. Expand, simplify, and write each expression in standard form. a. 1x 621x 22 d. 1x 121x 3212x 521x 22 b. 15 x213 4x2 e. 1a 22 3 c. x15x 32 2x13x 22 f. 19a 52 3 8. Factor each of the following: a. x 3 x c. 2x 2 7x 6 e. 27x 3 64 b. x 2 x 6 d. x 3 2x 2 x f. 2x 3 x 2 7x 6 9. Determine the domain of each of the following: b. e. 6x y 2x 2 5x 3 a. d. h1x2 x2 4 y Vx 5 x 1x 321x 42 c. y 3 f. y x 1 1x 221x 121x 52 10. The height of a model rocket in flight can be modelled by the equation h1t2 4.9t 2 25t 2, where h is the height in metres at t seconds. Determine the average rate of change in the model rocket’s height with respect to time during a. the first second b. the second second 11. Sacha drains the water from a hot tub. The hot tub holds 1600 L of water. It takes 2 h for the water to drain completely. The volume of water in the hot tub is modelled by V1t2 1600 t2 9, where V is the volume in litres at t minutes and 0 t 120. a. Determine the average rate of change in volume during the second hour. b. Estimate the instantaneous rate of change in volume after exactly 60 min. c. Explain why all estimates of the instantaneous rate of change in volume where 0 t 120 result in a negative value. 12. a. Sketch the graph of f 1x2 21x 32 2 4. b. Draw a tangent line at the point 15, f 1522, and estimate its slope. c. Estimate the instantaneous rate of change in f 1x2 when x 5. NEL CHAPTER 1 3
Page 1: Chapter 1 INTRODUCTION TO CALCULUS
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 and 38: limits because, in each case, only
Page 39 and 40: 11. For each function, sketch the g
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
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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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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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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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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
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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
Page 457 and 458:
After deriving vector and parametri
Page 459 and 460:
EXAMPLE 4 Representing the equation
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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
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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
Page 479 and 480:
Page 481 and 482:
Review Exercise 1. Determine vector
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20. Calculate the acute angle that
Page 485 and 486:
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
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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
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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
Page 565 and 566:
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
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y eliminating unknowns. In Example
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Properties of a Matrix in Row-Echel
Page 589 and 590:
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
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
Page 603 and 604:
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
Page 619 and 620:
G geometric vectors: vectors that a
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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
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d. 13. m 3; b 1 14. a 3, b 2, c
Page 629 and 630:
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
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
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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.
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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
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
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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
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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
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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