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IN SUMMARY Key Ideas • For f1x2 e x , f ¿1x2 e x . d In Leibniz notation, dx 1ex 2 e x . • For f1x2 e g1x2 , f ¿1x2 e g1x2 g¿ 1x2. In Leibniz notation, d1 e g1x2 2 d1g1x22 d 1e g1x2 2 . dx d1g1x22 dx • The slope of the tangent at a point on the graph of y e x equals the value of the function at this point. Need to Know • The rules for differentiating functions, such as the product, quotient, and chain rules, also apply to combinations involving exponential functions of the form f1x2 e g1x2 . • e is called Euler's number or the natural number, where e 2.718. Exercise 5.1 K PART A 1. Why can you not use the power rule for derivatives to differentiate y e x ? 2. Differentiate each of the following: a. y e 3x c. y 2e 10t e. y e 56xx2 b. s e 3t5 d. y e 3x f. y e Vx 3. Determine the derivative of each of the following: a. y 2e x3 c. f 1x2 ex3 e. x b. y xe 3x d. f 1x2 Vxe x f. 4. a. If f 1x2 1 calculate f ¿112. 3 1e3x e 3x 2, b. If f 1x2 e 1x12 1 , calculate f ¿102. c. If h1z2 z 2 11 e z 2, calculate h¿112. h 1t2 et 2 3e t g 1t2 1 e 2t 5. a. Determine the equation of the tangent to the curve defined by y 2ex 1 e x at the point 10, 12. b. Use graphing technology to graph the function in part a., and draw the tangent at 10, 12. c. Compare the equation in part a. with the equation generated by graphing technology. Do they agree? e2t 232 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS, y e x NEL
A C PART B 6. Determine the equation of the tangent to the curve at the point where x 1. Graph the original curve and the tangent. 7. Determine the equation of the tangent to the curve defined by y xe x at the point A11, e 1 2. 8. Determine the coordinates of all points at which the tangent to the curve defined by y x 2 e x is horizontal. 9. If y 5 prove that 2 1ex 5 e x 5 2, 10. a. For the function y d determine 2 d , and 3 e3x dy y y , . dx , dx 2 dx 3 d b. From the pattern in part a., state the value of n y dx n . 11. Determine the first and second derivatives of each function. a. y 3e x b. y xe 2x c. y e x 14 x2 12. The number, N, of bacteria in a culture at time t, in hours, is N1t2 100030 e 30 t a. What is the initial number of bacteria in the culture? b. Determine the rate of change in the number of bacteria at time t. c. How fast is the number of bacteria changing when t 20? d. Determine the largest number of bacteria in the culture during the interval 0 t 50. e. What is happening to the number of bacteria in the culture as time passes? 13. The distance s, in metres, fallen by a skydiver t seconds after jumping (and before the parachute opens) is s 160 Q 1 4 t 1 et 4R. a. Determine the velocity, v, at time t. b. Show that acceleration is given by a 10 1 4 v. c. Determine v T lim v. This is the “terminal” velocity, the constant velocity tSq attained when the air resistance balances the force of gravity. d. At what time is the velocity 95% of the terminal velocity? How far has the skydiver fallen at that time? 14. a. Use a table of values and successive approximation to evaluate each of the following: i. lim a 1 1 ii. x b xSq b. Discuss your results. y– y 25 . lim xS0 1 x 11 x2 y e x CHAPTER 5 233 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
Page 181 and 182: Section 4.3—Vertical and Horizont
Page 183 and 184: The behaviour of the graph can be i
Page 185 and 186: EXAMPLE 2 Expressing a polynomial f
Page 187 and 188: When lim f 1x2 k or lim f 1x2 k,
Page 189 and 190: For horizontal asymptotes, 3x f 1x2
Page 191 and 192: Now let's consider the straight lin
Page 193 and 194: Exercise 4.3 PART A 1. State the eq
Page 195 and 196: 12. Use the features of each functi
Page 197 and 198: 11. a. What does f ¿1x2 7 0 imply
Page 199 and 200: From these investigations, we can m
Page 201 and 202: Setting dy dx 0, we obtain 3 1x 2
Page 203 and 204: The second derivative of f 1x2 is f
Page 205 and 206: Exercise 4.4 K PART A 1. For each f
Page 207 and 208: Section 4.5—An Algorithm for Curv
Page 209 and 210: Therefore, by the second derivative
Page 211 and 212: Interval 1, 12 11, 0.82 10.8, 22 12
Page 213 and 214: K A 1x PART B 4. Use the algorithm
Page 215 and 216: Key Concepts Review In this chapter
Page 217 and 218: 5. For each of the following, check
Page 219 and 220: 17. If f 1x2 x3 8 , determine the
Page 221 and 222: Chapter 5 DERIVATIVES OF EXPONENTIA
Page 223 and 224: Radian Measure A radian is the meas
Page 225 and 226: 5. Convert the following angles to
Page 227 and 228: Section 5.1—Derivatives of Expone
Page 229 and 230: From the investigation, you should
Page 231: EXAMPLE 4 Connecting the derivative
Page 235 and 236: Section 5.2—The Derivative of the
Page 237 and 238: In general, for the exponential fun
Page 239 and 240: Solution a. January 1, 1900, is exa
Page 241 and 242: Section 5.3—Optimization Problems
Page 243 and 244: The average revenue to the company
Page 245 and 246: Exercise 5.3 PART A 1. Use graphing
Page 247 and 248: 12. Find the maximum and minimum va
Page 249 and 250: 9. The rapid growth in the number o
Page 251 and 252: INVESTIGATION 2 A.Using your graphi
Page 253 and 254: With practice, you will learn how t
Page 255 and 256: We evaluate f 1x2 at the critical n
Page 257 and 258: A T 6. Determine the absolute extre
Page 259 and 260: Derivatives of Composite Functions
Page 261 and 262: CAREER LINK WRAP-UP Investigate and
Page 263 and 264: Review Exercise 1. Differentiate ea
Page 265 and 266: 18. The position of a particle is g
Page 267 and 268: Chapter 5 Test dy 1. Determine the
Page 269 and 270: 8. Use algebraic methods to evaluat
Page 271 and 272: f 1x2 142e 5x1 24. For each functi
Page 273 and 274: Review of Prerequisite Skills In th
Page 275 and 276: CAREER LINK Investigate CHAPTER 6:
Page 277 and 278: What is a Vector? A vector is a mat
Page 279 and 280: EXAMPLE 1 B A E D C Connecting vect
Page 281 and 282: 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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Review of Prerequisite Skills In th
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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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CAREER LINK Investigate CHAPTER 8:
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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
Page 477 and 478:
IN SUMMARY Key Idea • A sketch of
Page 479 and 480:
Page 481 and 482:
Review Exercise 1. Determine vector
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20. Calculate the acute angle that
Page 485 and 486:
Chapter 8 Test 1. a. Given the poin
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Page 489 and 490:
CAREER LINK Investigate CHAPTER 9:
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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
Page 517 and 518:
Exercise 9.3 C K PART A 1. A system
Page 519 and 520:
Mid-Chapter Review 1. Determine the
Page 521 and 522:
Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
Page 525 and 526:
Thus, 7y 5t 3. Dividing by 7, we g
Page 527 and 528:
Inconsistent Systems for Three Equa
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Therefore, we can choose ! ! ! m 1
Page 531 and 532:
Exercise 9.4 PART A 1. A student is
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9. Solve each system of equations u
Page 535 and 536:
Section 9.5—The Distance from a P
Page 537 and 538:
EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
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Exercise 9.5 PART A 1. Determine th
Page 543 and 544:
Section 9.6—The Distance from a P
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It is also possible to use the dist
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p 1 p 2 U V d L 1 : r = (-2, 1, 0)
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The required distance between the t
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K A T PART B 2. Determine the follo
Page 553 and 554:
Review Exercise 1. The lines 2x y
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14. You are given the lines , and r
Page 557 and 558:
Chapter 9 Test 1. a. Determine the
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P(1, -2, 4) Q P9 2x - 3y - 4z + 66
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29. A line that passes through the
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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
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The Derivatives of General Logarith
Page 579 and 580:
The Derivative of a Composite Funct
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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
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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
Page 597 and 598:
Review of Technical Skills Appendix
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3 Evaluating a Function 1. Enter th
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7 Making a Table of Differences To
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4. Cursor along the curve to any po
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5. Display and analyze the results.
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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
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
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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
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,
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normal, 461-469, 486, 512, 524 para
show all

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