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Sometimes lim f 1x2 cannot be found by direct substitution. This is particularly xSa interesting when direct substitution results in an indeterminate form Q 0 In such 0 R. cases, we look for an equivalent function that agrees with f for all values except at x a. Here are some examples. EXAMPLE 5 Selecting a factoring strategy to evaluate a limit Evaluate lim xS3 x 2 2x 3 . x 3 Solution 0 Substitution produces the indeterminate form The next step is to simplify the 0 . function by factoring and reducing to see if the limit of the reduced form can be evaluated. x 2 2x 3 lim lim xS3 x 3 xS3 1x 121x 32 x 3 lim 1x 12 xS3 The reduction is valid only if x 3. This is not a problem, since lim requires values as x approaches 3, not when x 3. Therefore, x lim 2 2x 3 lim 1x 12 4. xS3 x 3 xS3 xS3 EXAMPLE 6 Selecting a rationalizing strategy to evaluate a limit Evaluate lim xS0 x 1 1 . x Solution A useful technique for finding a limit is to rationalize either the numerator or the denominator to obtain an algebraic form that is not indeterminate. Substitution produces the indeterminate form 0 0 , so we will try rationalizing. lim xS0 x 1 1 x lim xS0 lim xS0 lim xS0 lim xS0 x 1 1 x x 1 1 x1x 1 12 x x1x 1 12 1 x 1 1 x 1 1 x 1 1 (Rationalize the numerator) (Simplify) (Evaluate) 1 2 42 1.5 PROPERTIES OF LIMITS NEL
INVESTIGATION Here is an alternate technique for finding the value of a limit. 1x 12 A. Find lim by rationalizing. xS1 x 1 1x 12 B. Let u Vx, and rewrite lim in terms of u. We know x u 2 , Vx 0, xS1 x 1 and u 0. Therefore, as x approaches the value of 1, u approaches the value of 1. 1u Use this substitution to find 2 12 by reducing the rational expression. u 1 lim uS1 EXAMPLE 7 Selecting a substitution strategy to evaluate a limit Evaluate lim xS0 1x 82 1 3 2 . x Solution This quotient is indeterminate Q 0 when x 0. Rationalizing the numerator 0 R 1 3 1x 82 2 is not so easy. However, the expression can be simplified by substitution. Let u 1x 82 1 3 . Then u 3 x 8 and x u 3 8. As x approaches the value 0, u approaches the value 2. The given limit becomes 1x 82 1 3 2 lim lim xS0 x uS2 lim uS2 lim uS2 u 2 u 3 8 1 u 2 1u 221u 2 2u 42 u 2 2u 4 (Factor) (Simplify) (Evaluate) 1 12 EXAMPLE 8 Evaluating a limit that involves absolute value 0 x 2 0 Evaluate lim Illustrate with a graph. xS2 x 2 . Solution Consider the following: 1x 22 f 1x2 0 x 2 0 x 2 µ x 2 ,if x 7 2 1x 22 x 2 , if x 6 2 1, if x 7 2 e 1, if x 6 2 NEL CHAPTER 1 43
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 and 38: limits because, in each case, only
Page 39 and 40: 11. For each function, sketch the g
Page 41: EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 Using
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
Page 91 and 92: 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 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
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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
Page 463 and 464:
To derive the equation of this plan
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A number of observations can be mad
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When we considered lines in R 2 , w
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IN SUMMARY Key Idea • The Cartesi
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Section 8.6—Sketching Planes in R
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EXAMPLE 2 Representing the graph of
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EXAMPLE 5 Graphing planes whose Car
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IN SUMMARY Key Idea • A sketch of
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Page 481 and 482:
Review Exercise 1. Determine vector
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20. Calculate the acute angle that
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Chapter 8 Test 1. a. Given the poin
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4 13 s2 211 4s2 12 8s2 8 0 1
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Method 2: Again, this result can be
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We can now select any two of the th
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IN SUMMARY Key Ideas • Line and p
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A T 11. a. Show that the lines and
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equations. Since the lines would be
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Solution 1: Interchange equations 1
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Substituting into the second equati
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1: Multiply equation 1 by k, and ad
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K C PART B 4. Solve each system of
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Section 9.3—The Intersection of T
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If we had solved the system using e
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then x 3s 1. Now it is a matter o
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Exercise 9.3 C K PART A 1. A system
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Mid-Chapter Review 1. Determine the
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Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
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Thus, 7y 5t 3. Dividing by 7, we g
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Inconsistent Systems for Three Equa
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Therefore, we can choose ! ! ! m 1
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Exercise 9.4 PART A 1. A student is
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9. Solve each system of equations u
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Section 9.5—The Distance from a P
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EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
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Exercise 9.5 PART A 1. Determine th
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Section 9.6—The Distance from a P
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It is also possible to use the dist
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p 1 p 2 U V d L 1 : r = (-2, 1, 0)
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The required distance between the t
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K A T PART B 2. Determine the follo
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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
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Solution a. Differentiate both side
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Exercise PART A 1. State the chain
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dA b. To determine differentiate A
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1 16 dh dt Therefore, at the momen
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11. Two cyclists depart at the same
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Solution a. y ln15x2 Using the cha
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x1x 42 0 x 4 or x 0 But x 0 is
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The Derivatives of General Logarith
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The Derivative of a Composite Funct
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To solve this, we take the natural
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Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
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Properties of a Matrix in Row-Echel
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Exercise PART A 1. Write an augment
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12. Determine the equation of the p
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1 0 0 18 4 (row 3) row 1 £ 0 1 0
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Finally, convert the entry in the s
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Review of Technical Skills Appendix
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3 Evaluating a Function 1. Enter th
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7 Making a Table of Differences To
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4. Cursor along the curve to any po
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5. Display and analyze the results.
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14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
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Use either the calculator keypad or
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20 Graphing the Derivative of a Fun
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4. Create a function. Right-click t
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composition: the process of combini
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G geometric vectors: vectors that a
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ationalizing the denominator: the p
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Answers Chapter 1 0 0 0 0 0 Review
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d. about 1 e. about 7 8 f. no tang
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d. 13. m 3; b 1 14. a 3, b 2, c
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Review Exercise, pp. 56-59 1. a. 3
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19. a. y 7 b. y 5x 5 c. y 18x 9
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20. a. 34.3 m> s b. 39.2 m> s c. 54
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17. a. 100 bacteria b. 1200 bacteri
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11. a. b. 160x y 16 0 60x y 61
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3. 1 2x 4. a. x 2 15x 6 b. 6012x
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14. t 1 s; away 15. a. s1t2 kt 2
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14. a. triangle side length 0.96 cm
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Review Exercise, pp. 156-159 11. 20
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c. i. ii. iii. d. i. ii. iii. 5 4 3
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. local minimum: 11.41, 39.62, loca
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2. increasing: x 6 1 and x 7 2, dec
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to a zero of its derivative, the nu
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7. (-2, 10) 10 8 y 10. a. 8 y e. 4
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4. hole at x 2; large and negative
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sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
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Review Exercise, pp. 263-265 1. a.
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x 2. 5. a. 19 000 fish> year b. 23
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Section 6.1, pp. 279-281 6. a. b. c
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17. P G 4. Answers may vary. For ex
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11. a. AG ! a ! b ! c ! , b. AG
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d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
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Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
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. Since the equilibrant is directed
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7. a. scalar projection: 0, vector
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2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
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cos A 1 cos B 1 cos C 1 area of t
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. r ! 11, 1, 02 t10, 1, 12, tR; x
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2. 12, 3, 42 3. P must lie on plane
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25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
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 ,
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16. 17. 18. 19. 2 3 4 m>min 144 m>m
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ln x 1 e. ln 2 4x 1 x ln 13x 2 2
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Index A Absolute extrema, 130, 132,
Page 709:
normal, 461-469, 486, 512, 524 para
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