Chapter 3 | Derivatives
245
b. f ′( x ) = lim h →0 f ( x + h )− f ( x ) h 90. P ( x ) denotes the population of a city at time x in years. 91. C ( x ) denotes the total amount of money (in thousands of dollars) spent on concessions by x customers at an amusement park. 92. R ( x ) denotes the total cost (in thousands of dollars) of manufacturing x clock radios. 93. g ( x ) denotes the grade (in percentage points) received on a test, given x hours of studying. 94. B ( x ) denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in x years since 1990. 95. p ( x ) denotes atmospheric pressure at an altitude of x feet. 96. Sketch the graph of a function y = f ( x ) with all of the following properties: a. f ′( x ) >0 for −2≤ x <1 b. f ′(2) =0 c. f ′( x ) >0 for x >2 d. f (2) =2 and f (0) =1 e. lim x →−∞ f ( x ) =0 and lim x →∞ f ( x ) =∞ f. f ′(1) does not exist. 97. Suppose temperature T in degrees Fahrenheit at a height x in feet above the ground is given by y = T ( x ). a. Give a physical interpretation, with units, of T ′( x ). b. If we know that T ′ (1000) = −0.1, explain the physical meaning. 98. Suppose the total profit of a company is y = P ( x ) thousand dollars when x units of an item are sold. a. What does P ( b )− P ( a ) b − a for 0< a < b measure, and what are the units? b. What does P ′( x ) measure, and what are the units? c. Suppose that P ′(30) =5, what is the approximate change in profit if the number of items sold increases from 30 to31?
80. Use the graph to evaluate a. f ′(−0.5), b. f ′(0), c. f ′(1), d. f ′(2), and e. f ′(3), if it exists.
For
the
following
functions,
use
f ′( x + h )− f ′( x ) h
f ″( x ) = lim h →0
to find f ″( x ).
81. f ( x ) =2−3 x 82. f ( x ) =4 x 2
83. f ( x ) = x + 1 x For the following exercises, use a calculator to graph f ( x ). Determine the function f ′( x ), then use a calculator to graph f ′( x ).
84. [T] f ( x ) = − 5 x 85. [T] f ( x ) =3 x 2 +2 x +4. 86. [T] f ( x ) = x +3 x 87. [T] f ( x ) = 1 2 x 88. [T] f ( x ) =1+ x + 1 x 89. [T] f ( x ) = x 3 +1
For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units.
f ( x + h )− f ( x ) h
a.
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