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

Section 2.1 • Rates of Change and the Derivative 177

45. Market Share During a month-long advertising campaign, 52. Volume of a Cube A metal cube with each edge of length x
the total sales S of a magazine is modeled by the centimeters is expanding uniformly as a consequence of being
2
function S(x) = 5x + 100x + 10,000, where x, 0 ≤ x ≤ 30, heated.
represents the number of days since the campaign began.
(a) Find the average rate of change of the volume of the cube
(a) What is the average rate of change of sales from x = 10 with respect to an edge as x increases from 2.00 to 2.01 cm.
to x = 20 days?
(b) Find the instantaneous rate of change of the volume of the
(b) What is the instantaneous rate of change of sales cube with respect to an edge at the instant when x = 2 cm.
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when x = 10 days?
53. Rate of Change Show that the rate of change of a linear
46. Demand Equation The demand equation for an item
function f (x) = mx + b is the slope m of the line y = mx + b.
is p = p(x) = 90 − 0.02x, where p is the price in dollars
and x is the number of units (in thousands) made. 54. Rate of Change Show that the rate of change of a quadratic
2
function f (x) = ax + bx + c is a linear function of x.
(a) Assuming all units made can be sold, find the revenue
function R(x) = xp(x). 55. Agriculture The graph represents the diameter d
(b) Marginal Revenue Marginal revenue is defined as the (in centimeters) of a maturing peach as a function of the time t
additional revenue earned by selling an additional unit. If we (in days) it is on the tree.
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use R (x) to measure the marginal revenue, find the marginal
revenue when 1 million units are sold. d
47. Gravity If a ball is dropped from the top of the Empire State d  d(t)
Building, 1002 ft above the ground, the distance s (in feet) it falls
2
after t seconds is s(t) = 16t . Diameter (in centimeters)
(a) What is the average velocity of the ball for the first 2 s?
(b) How long does it take for the ball to hit the ground?
(c) What is the average velocity of the ball during the time it is
falling? 1 20 t
(d) What is the velocity of the ball when it hits the ground? Time (in days)
48. Velocity A ball is thrown upward. Its height h in feet is (a) Interpret the derivative d (t) as a rate of change.
′
2
given by h(t) = 100t − 16t , where t is the time elapsed in
seconds. (b) Which is larger, d (1) or d (20)?
′
′
(a) What is the velocity v of the ball at t = 0 s, t = 1 s, (c) Interpret both d (1) and d (20).
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′
and t = 4 s?
(b) At what time t does the ball strike the ground? 56. Business The graph represents the demand d (in gallons) for
(c) At what time t does the ball reach its highest point? olive oil as a function of the cost c (in dollars per gallon)
Hint: At the time the ball reaches its maximum height, it is of the oil.
stationary. So, its velocity v = 0.
d
49. Gravity A rock is dropped from a height of 88.2 m and
falls toward Earth in a straight line. In t seconds the rock
2
falls 4.9t m. d  d(c)
(a) What is the average velocity of the rock for the first 2 s? Demand (in gallons)
(b) How long does it take for the rock to hit the ground?
(c) What is the average velocity of the rock during its fall?
(d) What is the velocity v of the rock when it hits the ground?
5 30 c
50. Velocity At a certain instant, the speedometer of an automobile
Cost (in dollars per gallon)
1
reads V mi/h. During the next s the automobile travels 20 ft.
4
(a) Interpret the derivative d (c).
′
Approximate V from this information.
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PAGE (b) Which is larger, d (5) or d (30)? Give an interpretation
175 51. The table lists the outside temperature T , in degrees Fahrenheit, in ′ ′
Naples, Florida, on a certain day in January, for selected times x, to d (5) and d (30).
where x is the number of hours since 12 a.m.
x 5 7 9 12 13 14 16 17
T (x) 62 71 74 80 83 84 85 78
(a) Use the table to approximate T (11).
′
(b) Using appropriate units, interpret T (11) in the context of the
′
problem.

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