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Chapter 4 | Applications of Derivatives
4.2 EXERCISES 46. What is the linear approximation for any generic linear function y = mx + b ? 47. Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result. 48. Explain why the linear approximation becomes less accurate as you increase the distance between x and a . Use a graph to prove your argument. 49. When is the linear approximation exact? For the following exercises, find the linear approximation L ( x ) to y = f ( x ) near x = a for the function. 50. f ( x ) = x + x 4 , a =0
62. [T] (1.01) 3 63. [T] cos(0.01)
⎠ 2
64. [T] ⎛
⎝ sin(0.01) ⎞
65. [T] (1.01) −3
⎛ ⎝ 1+ 1 10
⎞ ⎠
10
66. [T]
67. [T] 8.99 For the following exercises, find the differential of the function. 68. y =3 x 4 + x 2 −2 x +1 69. y = x cos x
51. f ( x ) = 1 x , a =2 52. f ( x ) = tan x , a = π 4 53. f ( x ) = sin x , a = π 2 54. f ( x ) = x sin x , a =2 π 55. f ( x ) = sin 2 x , a =0
70. y = 1+ x
2 +2
71. y = x
x −1 For the following exercises, find the differential and evaluate for the given x and dx . 72. y =3 x 2 − x +6, x =2, dx =0.1
For the following exercises, compute the values given within 0.01 by deciding on the appropriate f ( x ) and a , and evaluating L ( x ) = f ( a )+ f ′( a )( x − a ). Check your answer using a calculator.
73. y = 1
x =1, dx =0.25
x +1 ,
74. y = tan x , x =0, dx = π 10 75. y = 3 x 2 +2 x +1
56. [T] (2.001) 6 57. [T] sin(0.02) 58. [T] cos(0.03) 59. [T] (15.99) 1/4 60. [T] 1 0.98 61. [T] sin(3.14)
, x =0, dx =0.1
x ) x , x = π , dx =0.25
76. y = sin(2
77. y = x 3 +2 x + 1 x , x =1, dx =0.05 For the following exercises, find the change in volume dV or in surface area dA . 78. dV if the sides of a cube change from 10 to 10.1. 79. dA if the sides of a cube change from x to x + dx . 80. dA if the radius of a sphere changes from r by dr .
For the following exercises, determine the appropriate f ( x ) and a , and evaluate L ( x ) = f ( a )+ f ′( a )( x − a ). Calculate the numerical error in the linear approximations that follow.
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