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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. 3 1 0.01 0.1 2 1 2 1 3 3 20.086 0.368 1.010 0.905 7.389 0.607 1 1.396 0.717 e e e e e e e e 4. 1 2 1 1 and 3 4 = = x x y y pass through (0, 1). The x axis is a horizontal asymptote. 2 y lies above 1 y for 0. x < 6. (a) 3 7 ( 128) 8 = − (b) 2 3 3 2 2 3 2 3 27 64 3 8 2.304 64 25 4 5 = = 8. (a) 3 2 11 7 11 7 (2 3 ) (8 9) 1 = = − (b) ( ) ( ) 3 2 3 2 2 3 4 3 2 4 27 8 3 2 0.008 + = + = 10. (a) 2 2 3 3 5 1 5 5 5 = = (b) ( ) 4/3 4/3 2 2 1/ 2 4/3 3/ 2 2 = = = 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) 1/ 2 ( ) x x x = = (b) 2/3 3/ 4 (2/3)( 3/ 4) 1/ 2 1/ 2 ( ) 1 x x x x = = = 16. (a) 3 2/3 3 2/3 7/3 7/3 ( 2 )(3 ) 6 6 6 t t t t t − + = − = − = − (b) 2/3 3/ 4 2/3 3/ 4 1/12 ( )( ) t t t t + = = 18. (a) 2 3 3 2 3 3 3 3 6 9 3 6 3 9 ( ) ( ) ( ) ( ) x y z x y z x y z x z y = = = (b) 1/6 3 2 3 1/6 2 1/6 4 4 1/6 1/ 2 2/3 1/3 ( ) ( ) ( ) x y x y z z x z y = = 20. 2 2 3 2 3 4 12 144 12 x x x x x = = = = Thus 2 x = . 22. 3 1 1 1 4 4 2 8 2 8 2 x x x x x x = = = = or 3 2 2 x = . Thus 3 x = or 3 x = − .
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Chapter 4. Exponential and Logarithmic Functions 145 24. 2 3 2 (3.2) (3.2) x x = implies 2 3 2 3 5 5 3 x x x x = = = 26. 2 2 2 1 1 1 3 1 3 2 2 1 1,000 10 (10 ) 10 10 10 1 3 4 2 = = = = = = ± x x x x x x 28. 2 2 2 1 3 4 2 1 3 4 6 2 4 2 2 1 3 9 (3 ) 3 3 3 6 2 4 3 2 1 0 (3 1)( 1) 0 = = = = = + = x x x x x x x x x x x x 1 3 = − x or x = 1 30. 10 –10 –5 15 32. 10 –10 –5 15 34. For x y Cb = 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 3 3 2 2 9 3 3 Cb b b Cb b = = = = . Substituting 3 b = in the first equation gives 2 3 3 9 C C = = or 1 3 C = . Thus 1 1 (3 ) 3 3 x x y = = . 36. ( ) 1 kt r B t 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 B t Pe = (0.1)(10) (10) (5,000) (5,000)(2.7183) 13,591.41 B e = = = 38. ( ) (20,000), 0.07, and 20. B t r t = = = 0.07 20 20,000 ( ) , so $4,931.94. rt B t Pe 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 = + = + = p e The price for 0 units is $57. (b) /200 ( ) (7 50 ) = = + x R x xp x e 200/200 1 (200) 200(7 50 ) 200(7 50 ) 5079 = + = + R e e The revenue for 200 units is $5079. (c) 100/200 1/2 (100) 100(7 50 ) 100(7 50 ) 3733 = + = + R e e 50/200 1/4 (50) 50(7 50 ) 50(7 50 ) 2297 = + = + R e e 3733 2297 = 1436 The revenue for 100 units is $1463 more than the revenue for 50 units.
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