Page 64 - 2024-calc4e-SE proofs-4e.indd

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

214 Chapter 2 • The Derivative and Its Properties

f (t) 86. Investing in Fine Art The value V of a painting t years after it
82. Let F(t) = f (t) · g(t) and G(t) = .
g(t) is purchased is modeled by the function
y y g(t) (7, 6) 100t + 50
2
6 (4, 6) V (t) = + 400 1 ≤ t ≤ 5
t
(3, 4)
4 (a) Find the rate of change in the value V with respect to time.
(5, 3)
(b) What is the rate of change in value after 2 years?
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
2 y f(t)
(5, 2) (2, 2) (c) What is the rate of change in value after 3 years?
(d) Interpret the answers in (b) and (c).
4 2 4 6 t
87. Drug Concentration The concentration of a drug in a
2 (5, 2) patient’s blood t hours after injection is given by the
0.4t
′
′
(a) F (0) (b) F (3) function f (t) = (in milligrams per liter).
2
2t + 1
′
(c) F (−4) (d) G (−2)
′
(a) Find the rate of change of the concentration with respect to
d 1 time.
′
(e) G (−1) (f) at t = 3
dt f (t) (b) What is the rate of change of the concentration after 10 min?
After 30 min? After 1 hour?
Applications and Extensions
(c) Interpret the answers found in (b).
PAGE
211 83. Vertical Motion An object is propelled vertically upward (d) Graph f for the first 5 hours after administering the drug.
from the ground with an initial velocity of 39.2 m/s.
The distance s (in meters) of the object from the (e) From the graph, approximate the time (in minutes) at which
ground after t seconds is given by the position the concentration of the drug is highest. What is the highest
concentration of the drug in the patient’s blood?
2
function s = s(t) = −4.9t + 39.2t.
88. Population Growth A population of 1000 bacteria is introduced
(a) What is the velocity of the object at time t? into a culture and grows in number according to the formula
(b) When will the object reach its maximum height?
4t
(c) What is the maximum height? P(t) = 1000 1 + , where t is measured in hours.
100 + t 2
(d) What is the acceleration of the object at any time t?
(e) How long is the object in the air?
(a) Find the rate of change in population with respect to time.
(f) What is the velocity of the object upon impact with the
(b) What is the rate of change in population at t = 1, t = 2, t = 3,
ground? What is its speed?
and t = 4?
(g) What is the total distance traveled by the object?
(c) Interpret the answers found in (b).
84. Vertical Motion A ball is thrown vertically upward from (d) Graph P = P(t), 0 ≤ t ≤ 20.
a height of 6 ft with an initial velocity of 80 ft/s. The distance s (e) From the graph, approximate the time (in hours) when the
(in feet) of the ball from the ground after t seconds is given by population is the greatest. What is the maximum population
2
the position function s = s(t) = 6 + 80t − 16t . of the bacteria in the culture?
(a) What is the velocity of the ball after 2 s?
89. Economics The price-demand function for a popular e-book is
(b) When will the ball reach its maximum height?
100, 000
(c) What is the maximum height the ball reaches? given by D(p) = 2 , 4 ≤ p ≤ 20, where D = D(p) is
p + 10p + 50
(d) What is the acceleration of the ball at any time t?
the quantity demanded at the price p dollars.
(e) How long is the ball in the air?
(f) What is the velocity of the ball upon impact with the ground? (a) Find D (p), the rate of change of demand with respect to
′
What is its speed? price.
(g) What is the total distance traveled by the ball? (b) Find D (5), D (10), and D (15).
′
′
′
(c) Interpret the results found in (b).
85. Environmental Cost The cost C, in thousands of dollars, for
the removal of a pollutant from a certain lake is given by the 90. Intensity of Light The intensity of illumination I on a surface
5x is inversely proportional to the square of the distance r from the
function C(x) = , where x is the percent of pollutant
110 − x surface to the source of light. If the intensity is 1000 units when
removed. the distance is 1 m from the light, find the rate of change of the
intensity with respect to the distance when the source is 10 meters
(a) What is the domain of C?
from the surface.
(b) Graph C.
(c) What is the cost to remove 80% of the pollutant?
(d) Find C (x), the rate of change of the cost C with respect to
′
the amount of pollutant removed.
(e) Find the rate of change of the cost for removing 40%, 60%,
80%, and 90% of the pollutant.
(f) Interpret the answers found in (e).
© 2024 BFW Publishers PAGES NOT FINAL
For Review Purposes Only, all other uses prohibited
Do Not Copy or Post in Any Form.