Chapter 8

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8 Chapter Task Comparing Growth Rates in Bacteria Cultures In an experiment, bacteria were placed in a hostile environment and a bacterial count was made every hour. The results are given in the following table. Time Interval (t hours) 1 2 3 4 5 6 Bacterial Count (c) for Culture A 560 320 180 100 60 30 In a second experiment conducted simultaneously, more of the same bacteria were placed in an environment that encouraged their growth. A bacterial count was made every hour. The results are given in the table below. Time Interval (t hours) 1 2 3 4 5 6 Bacterial Count (c) for Culture B 42 68 110 156 212 380 ? When did the cultures have the same bacterial count, and at what rate was the population of each culture changing at this time? A. Graph the data for experiment 1. B. Determine the equation that best models the data. Explain the process you used to determine the equation. C. Graph the data for experiment 2. D. Determine the equation that best models these data. E. Use your models to estimate the bacterial count in both cultures after 10 h. F. Determine the average rate of change in population for each culture over the first 4 h. G. When will the cultures have the same bacterial count? Justify your answer in two ways. H. Estimate the rate at which populations of both cultures are changing when their bacterial counts are the same. Task Checklist ✔ Did you draw well-labelled graphs, including some values? ✔ Did you explain the process that you used to obtain an equation that best models the data? ✔ Did you justify your answer for part G in two different ways? ✔ Did you calculate the appropriate rate of change for part H? NEL Chapter 8 513
2(sec 2 x 2 tan 2 x) 9. 5 sin 2x sec x csc x 2(1) 5 sin 2x sec x csc x 2 5 sin 2x sec x csc x 2 sin x 5 sin 2x sec x 2 sin x cos x 5 sin 2x sec x cos x sin 2x 5 sin 2x sec x cos x sin 2x sec x 5 sin 2x sec x 11p 10. a) x 5 7p or 6 6 5p b) x 5 p or 4 4 4p c) x 5 2p or 3 3 11. a) y 522 or 2 5p 7p 11p b) x 5 p or 6 , 6 , 6 , 6 7p 11p 12. a) x 5 p or 2 , 6 , 6 p 5p 7p 11p b) x 5 0, p, or 2p 6 , 6 , 6 , 6 , 2p 4p 7p c) x 5 p or 4 , 3 , 3 , 4 d) x 5 0.95 or 4.09 3p 13. x 5 p p, or 2 , 2 Chapter Self-Test, p. 441 1 2 2 sin 2 x 1. 1 sin x 5 cos x cos x 1 sin x 1 2 2 sin 2 x 1 sin x 2 sin x cos x 1 sin x 5 cos x 2 sin x 1 2 2 sin 2 x 5 cos x 2 sin x cos x 1 sin x 1 2 2 sin 2 x 5 (cos x 2 sin x) 3(cos x 1 sin x) cos 2x 5 (cos x 2 sin x) 3(cos x 1 sin x) cos 2x 5 cos 2 x 2 sin 2 x cos 2x 5 cos 2x 2. all real numbers x, where 0 # x # 2p 3. a) x 5 p or x 5 11p 6 6 b) x 5 2p or x 5 5p 3 3 c) x 5 5p or x 5 7p 4 4 4. a 5 2, b 5 1 5. t 5 7, 11, 19, and 23 11p 6. Nina can find the cosine of by using 4 the formula cos (x 1 y) 5 cos x cos y 2 sin x sin y. The cosine of p is 21, and the 7p !2 cosine of is Also, the sine of p is 0, 4 2 . 7p and the sine of is 2 !2 Therefore, 4 2 . cos 11p 4 5 cos ap 17p 4 b 5 a21 3 !2 !2 b 2 a0 3 2 2 2 b 52 !2 2 2 0 52 !2 2 7. x 5 3.31 or 6.12 8. 9. 2 33 2 16 65 , 65 a) 2 4!5 9 c) 1 22 b) d) 9 27 10. p 5p a) or 3 , 3 , 3 , 3 2p b) or 3 , 3 , 3 , c) x 52pand p Chapter 8 0 Getting Started, p. 446 1. a) 1 5 5 1 2 25 d) ! 3 125 5 5 b) 1 e) 2 !121 5211 c) !36 5 6 2 3 27 f) a Å 8 b 5 9 4 2. a) 3 7 5 2187 d) 7 4 5 2401 b) (22) 2 5 4 e) 8 2 3 5 4 c) 10 3 5 1000 f) 4 1 2 5 !4 5 2 3. a) 8m 3 d) x 3 y 1 b) a 8 b 10 e) 2d 2 c 2 3 c) 40 x f) x 4. a) y 40 30 20 10 x –2 –1 0 1 2 3 4 –10 D 5 5xPR6, R 5 5 yPR 0 y . 06, y-intercept 1, horizontal asymptote y 5 0 y b) 40 30 20 10 x –4 –3 –2 –1 0 1 2 –10 3 2 !5 Å 6 4p 3 D 5 5xPR6, R 5 5 yPR0 y . 06, y-intercept 1, horizontal asymptote y 5 0 c) D 5 5xPR6, R 5 5 yPR y .226, y-intercept 21, horizontal asymptote y 522 5. a) i) y 5 x 1 6 3 ii) y 56!x 1 5 3 x iii) y 5 Å 6 iv) b) The inverses of (i) and (iii) are functions. 6. a) 800 bacteria b) 6400 bacteria c) 209 715 200 d) 4.4 3 10 15 7. 12 515 people 8. Similarities • same y-intercept • same shape • same horizontal asymptote • both are always positive Lesson 8.1, p. 451 1. a) x 5 4 y or f 21 (x) 5 log 4 x 1 –1 0 1 2 3 4 –1 –2 –3 –4 b) x 5 8 y or f 21 (x) 5 log 8 x 1 –1 0 1 2 3 4 –1 –2 –3 –4 80 60 40 20 x –3 –2 –1 0 1 2 3 4 –20 y y y Differences • one is always increasing, the other is always decreasing • different end behaviour • reflections of each other across the y-axis x x Answers NEL Answers 667
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444 NEL
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8 Getting Started Study Aid • For
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8.1 Exploring the Logarithmic Funct
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In Summary Key Ideas • The invers
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8.2 Transformations of Logarithmic
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Reflecting L. Describe the domain a
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EXAMPLE 2 Connecting a geometric de
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PRACTISING 4. Let f (x) 5 log 10 x.
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Solution B: Using a graphing calcul
Page 19 and 20: c) log 2 (24) 5 x Determine the exp
Page 21 and 22: EXAMPLE 4 Selecting a strategy to e
Page 23 and 24: CHECK Your Understanding 1. Express
Page 25 and 26: 17. The doubling function y 5 y 0 2
Page 27 and 28: log a (mn) 5 x 1 y m 5 a x so log a
Page 29 and 30: APPLY the Math EXAMPLE 4 Selecting
Page 31 and 32: EXAMPLE 6 Selecting strategies to s
Page 33 and 34: 9. Write each expression as a singl
Page 35 and 36: Study • See Lesson 8.2, Examples
Page 37 and 38: 8.5 Solving Exponential Equations G
Page 39 and 40: Solution C: Solving the equation gr
Page 41 and 42: EXAMPLE 4 Solve 2 x11 5 3 x21 to th
Page 43 and 44: 8. Solve for x. a) 4 x11 1 4 x 5 16
Page 45 and 46: Reflecting A. What strategies for s
Page 47 and 48: x 2 2 9 5 2 4 x 2 2 9 5 16 x 2 5 25
Page 49 and 50: 9. The loudness, L, of a sound in d
Page 51 and 52: APPLY the Math EXAMPLE 1 Solving a
Page 53 and 54: EXAMPLE 2 Representing exponential
Page 55 and 56: Solution Sound Loudness (dB) soft w
Page 57 and 58: 6. Calculate to two decimal places
Page 59 and 60: 8.8 Rates of Change in Exponential
Page 61 and 62: EXAMPLE 2 Selecting a strategy for
Page 63 and 64: Solution -2 Tangents to y = 10 x y
Page 65 and 66: y 8 y = log x - 1 6 2 d 4 2 y = - 1
Page 67 and 68: PRACTICE Questions Lesson 8.1 1. De
Page 69: 8 Chapter Self-Test 1. Write the eq
Page 73 and 74: d) D 5 5xPR 0 x . 06, R 5 5 yPR6 e)
Page 75 and 76: 17. log b x!x 5 log b x 1 log b !x
Page 77: value of x. Determine the average r
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Chapter 8

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