Page 39 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.2 • The Derivative as a Function; Differentiability 189

In Problems 51–54, find the derivative of each function. 73. Velocity The distance s (in feet) of an automobile from the
origin at time t (in seconds) is given by the position
2
51. f (x) = mx + b 52. f (x) = ax + bx + c function
1 1 3
53. f (x) = 54. f (x) = √ t if 0 ≤ t < 5
x 2 x s = s(t) =
125 if t ≥ 5
Applications and Extensions (This could represent a crash test in which a vehicle is accelerated
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until it hits a brick wall at t = 5 s.)
In Problems 55–66, each limit represents the derivative of a function f
at some number c. Determine f and c in each case. (a) Find the velocity just before impact (at t = 4.99 s) and just
after impact (at t = 5.01 s).
2
3
(2 + h) − 4 (2 + h) − 8
55. lim 56. lim (b) Is the velocity function v = s (t) continuous at t = 5?
′
h→0 h h→0 h
(c) How do you interpret the answer to (b)?
4
2
x − 1 x − 1
57. lim 58. lim
x→1 x − 1 x→1 x − 1 74. Population Growth A simple model for population growth
√ 1/3 states that the rate of change of population size P with respect to
9 + h − 3 (8 + h) − 2
59. lim 60. lim time t is proportional to the population size. Express this
h→0 h h→0 h statement as an equation involving a derivative.
√
1 2 75. Atmospheric Pressure Atmospheric pressure p decreases as
sin x − cos x −
2 2 the distance x from the surface of Earth increases, and the rate
61. lim 62. lim
x→π/6 π x→π/4 π of change of pressure with respect to altitude is proportional to
x − x −
6 4 the pressure. Express this law as an equation involving
a derivative.
2
3
2(x + 2) − (x + 2) − 6 3x − 2x
63. lim 64. lim 76. Electrical Current Under certain conditions, an electric
x→0 x x→0 x
current I will die out at a rate (with respect to time t) that is
2
2
(3 + h) + 2(3 + h) − 15 3(h − 1) + h − 3 proportional to the current remaining. Express this law as an
65. lim 66. lim equation involving a derivative.
h→0 h h→0 h
2
77. Tangent Line Let f (x) = x + 2. Find all points on the graph
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181 67. Units The volume V (in cubic feet) of a balloon is expanding of f for which the tangent line passes through the origin.
2
according to V = V (t) = 4t, where t is the time (in seconds). 78. Tangent Line Let f (x) = x − 2x + 1. Find all points on
Find the rate of change of the volume of the balloon with respect the graph of f for which the tangent line passes through the
to time. What are the units of V (t)? point (1, −1).
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68. Units The area A (in square miles) of a circular patch of oil is 79. Area and Circumference of a Circle A circle of radius r has
expanding according to A = A(t) = 2t, where t is the time (in area A = πr and circumference C = 2πr. If the radius changes
2
hours). At what rate is the area changing with respect to time? from r to r + 1r, find the:
What are the units of A (t)?
′
(a) Change in area.
69. Units A manufacturer of precision digital switches has a daily
cost C (in dollars) of C(x) = 10,000 + 3x, where x is the number (b) Change in circumference.
of switches produced daily. What is the rate of change of cost with (c) Average rate of change of area with respect to radius.
respect to x? What are the units of C (x)? (d) Average rate of change of circumference with respect to
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radius.
70. Units A manufacturer of precision digital switches has daily
(e) Rate of change of circumference with respect to radius.
x 2
revenue R (in dollars) of R(x) = 5x − , where x is the
2000 80. Volume of a Sphere The volume V of a sphere of radius r
number of switches produced daily. What is the rate of change of 4πr 3
revenue with respect to x? What are the units of R (x)? is V = . If the radius changes from r to r + 1r, find the:
′
3
3
x if x ≤ 0 (a) Change in volume.
71. f (x) = 2
x if x > 0 (b) Average rate of change of volume with respect to radius.
(a) Determine whether f is continuous at 0. (c) Rate of change of volume with respect to radius.
(b) Determine whether f (0) exists.
′
81. Use the definition of the derivative to show that f (x) = |x| is not
(c) Graph the function f and its derivative f .
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differentiable at 0.
√
3
2x if x ≤ 0 82. Use the definition of the derivative to show that f (x) = x is not
72. For the function f (x) = 2
x if x > 0 differentiable at 0.
83. If f is an even function that is differentiable at c, show that its
(a) Determine whether f is continuous at 0.
derivative function is odd. That is, show f (−c) = − f (c).
′
′
(b) Determine whether f (0) exists.
′
84. If f is an odd function that is differentiable at c, show that its
(c) Graph the function f and its derivative f .
′
′
derivative function is even. That is, show f (−c) = f (c).
′
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