chapter_4-1

chapter_4-1 - Chapter 4 Exponential and Logarithmic...

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144 Chapter 4 Exponential and Logarithmic Functions 4.1 Exponential Functions; Continuous Compounding 2. 31 0.01 0.1 21 2 13 3 20.086 0.368 1.010 0.905 7.389 0.607 1 1.396 0.717 ≈≈ ee e e 4. 12 11 and 34 ⎛⎞ == ⎜⎟ ⎝⎠ xx yy pass through (0, 1). The x axis is a horizontal asymptote. 2 y lies above 1 y for 0. x < 6. (a) 37 ( 128) 8 −= (b) 23 32 27 64 3 8 2.304 64 25 45 ⎛⎞⎛⎞ ⎜⎟⎜⎟ ⎝⎠⎝⎠ 8. (a) 1 1 7 1 1 7 (2 3 ) (8 9) 1 = (b) () 43 2 4 27 8 3 2 0.008 −− += + = 10. (a) 2 3 51 5 5 5 (b) 4/3 22 1/2 3/2 2 ππ π π π π = = = 12. (a) 1.2 2.7 3.9 4.1 4.1 0.2 (3 )(3 ) 3 1 0.803 (3 ) 3 3 ==≈ (b) 1/4 2/3 2 2 16 125 2 2 8 81 8 3 75 5 ⎛⎞⎛ ⎞ ⎜⎟⎜ ⎟ ⎝⎠⎝ ⎠ 14. (a) 1/3 3/2 (1/3)(3/2) x (b) 3/4 (2/3)( 3/4) 1 x x = = = 16. (a) 3 3 2/3 7/3 ( 2 )(3 ) 6 6 6 tt t t t + =− (b) 2/3 3/4 1/12 ( ) t t + 18. (a) 233 2 3 3 69 3 63 9 ( ) ( ) ( ) xy z x y z xz y = = = (b) 1/6 3 2 3 1/6 2 1/6 44 1 / 6 2/3 1/3 () ( ) xy x y zz x zy = = 20. 3 2 3 4 12 144 12 xx x x x = = Thus 2 x = . 22. 3 1 2 8 28 2 x x ⎛⎞ ⎛⎞ = = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ or 3 x = . Thus 3 x or 3 x .
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Chapter 4. Exponential and Logarithmic Functions 145 24. 23 2 (3.2) (3.2) xx −− = implies 2 35 5 3 x x −=− = = 26. 2 2 2 1 11 3 13 2 2 1 1,000 10 (10 ) 10 10 10 4 2 ⎛⎞ = ⎜⎟ ⎝⎠ = = −= = x x x x x x 28. 2 2 2 4 213 4 62 4 2 2 1 3 9 (3 ) 3 33 62 4 32 1 0 (3 1)( 1) 0 = = = = +− = x x 1 3 =− x or x = 1 30. 10 –10 –5 15 32. 10 –10 –5 15 34. For x yC b = to contain (2, 3) and (3, 9) we must have 2 3 Cb = and 3 9 Cb = . Dividing the second equation by the first gives 22 9 3 3 Cb b b Cb b == = = . Substituting 3 b = in the first equation gives 2 9 CC or 1 3 C = . Thus 1 1 (3 ) 3 3 y . 36. () 1 kt r Bt P k =+ (a) With annual compounding (the time period is 1 year) 1. k = 10 0.1 (10) (5,000) 1 1 (5,000)(2.5937) 12,968.71 B = = (b) With semi-annual compounding 2. k = 20 0.1 (10) (5,000) 1 2 (5,000)(2.6533) 13,266.49 B = = (c) With daily compounding 365. k = 3650 0.1 (10) (5,000) 1 365 (5,000)(2.7179) 13,589.55 B = = (d) With continual compounding, . rt Pe = (0.1)(10) (10) (5,000) (5,000)(2.7183) 13,591.41 Be = = = 38. ( ) (20,000), 0.07, and 20. r t = 0.07 20 20,000 ( ) , so $4,931.94. rt P e × =
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146 Chapter 4. Exponential and Logarithmic Functions 40. (0.07)(5) 10,000 7,046.88 e = (0.07)(5) 20,000 2(7,046.88) 14,093.76 e = = 42. (a) 0/200 (0) 7 50 7 50(1) 57 =+ = pe The price for 0 units is $57. (b) /200 () ( 7 5 0 ) == + x Rx xp x e 200/200 1 (200) 200(7 50 ) 200(7 50 ) 5079 Re e The revenue for 200 units is $5079. (c) 100/200 1/2 (100) 100(7 50 ) 100(7 50 ) 3733 e 50/200 1/4 (50) 50(7 50 ) 50(7 50 ) 2297 e 3733 2297 = 1436 The revenue for 100 units is $1463 more than the revenue for 50 units. 44. (a) 2000 corresponds to t = 0. 0.018(0) 0 (0) 2 5 2 5 2 1 2 =⋅ =⋅ =⋅= P The population in 2000 was 2 million people. (b) 2010 corresponds to t = 10. 0.018(10) 0.18 ( 10 ) 25 25 2 .6 7 P The population in 2010 is expected to be about 2.67 million people. 46. (a) 0.87 ; = t Ct Ae C (1) = 4 0.87(1) 0.87 0.87 4 4 49 . 5 4 8 = = =≈ Ae Ae (b) 0.87 ( ) 9.548 = t e 0.87(0) (0) 9.548 9.548 Ce 0.87(2) (2) 9.548 1.676 The initial concentration is 9.548 mg/ml and the concentration after 2 hours is 1.676 mg/ml. (c) 1.676 9.548 7.872 3.936 20 2 −− Over the first two hours the average rate of change is 3.936 mg/ml per hour. 48. (a) 0.015 ( ) 4,000 = t St e 0.015(0) (0) 4,000 4,000 Se Sales are 4,000 units when the advertising ends.
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This note was uploaded on 05/20/2011 for the course MAC 2233 taught by Professor Royer during the Spring '08 term at FIU.

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chapter_4-1 - Chapter 4 Exponential and Logarithmic...

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