Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.
Explain why the linear approximation becomes less accurate as you increase the distance between x and a. Use a graph to prove your argument.
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
51. f(x) = 1/x, a = 2
52. f(x) = tan x, a = pi/4
53. f(x) = sin x, a = pi/2
54. f(x) = x sin x, a = 2pi
55. f(x) = sin^2 x, a = 0
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.
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)
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.
62. [T] (1.01)^3
63. [T] cos(0.01)
64. [T] (sin(0.01))^2
65. [T] (1.01)^-3
66. [T] (1 + 1/10)^10
67. [T] sqrt(8.99)
For the following exercises, find the differential of the function.
68. y = 3x^4 + x^2 - 2x + 1
69. y = x cos x
70. y = sqrt(1 + x)
71. y = (x^2 + 2)/(x - 1)
For the following exercises, find the differential and evaluate for the given x and dx.
72. y = 3x^2 - x + 6, x = 2, dx = 0.1
73. y = 1/(x + 1), x = 1, dx = 0.25
74. y = tan x, x = 0, dx = pi/10
75. y = (3x^2 + 2)/sqrt(x + 1), x = 0, dx = 0.1
76. y = sin(2x)/x, x = pi, dx = 0.25
77. y = x^3 + 2x + 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.