(From
Hughes-Hallet et al., Calculus, 2nd ed., Wiley.)
1.
The cost of extracting T tons of ore from a copper mine is C
= f(T) dollars. Make reasonable guesses about the domain and the range
of f. What units are the inputs to f' measured in? What units
are the outputs from f' measured in? What does it mean in practical
terms to say that f'(2000) = 100?
2.
Let f(x) be the elevation in feet of the Mississippi river x
miles downstream from its source. Make reasonable guesses about the domain
and the range of the function f. What are the units of f'(x)?
What can you say about the sign of f'(x)?
3.
The temperature, T, measured in degrees Fahrenheit, of a cold yam
placed in a hot oven is given by T = f(t), where t is the
time in minutes since the yam was put in the oven. Make reasonable guesses
about the domain and the range of f. What is the sign of f'(t)?
What are the units of f'(20)? What is the practical meaning of the
statement f'(20) = 2?
4.
Suppose C(r) is the total cost of paying off a car loan borrowed
at an annual interest rate of r%. Make reasonable guesses about
the domain and the range of C. What are the units of C'(r)?
What is the practical meaning of C'(r)? What is its sign?
5.
An ice cream company knows that the cost C (in dollars) to produce
g
quarts of cookie dough ice cream is a function of g, say
C =
f(g). Make reasonable guesses about the domain and the range of f.
If f(200) = 70, what are the units of the 200? What are the units
of the 70? What is the equation telling you? If f'(200) = 3, what
are the units of the 200? What are the units of the 3? What is the equation
telling you? Estimate f(201).
6. Let f(t) be the number of centimeters of rainfall that has fallen (measured at the meteorology station atop Chauvenet Hall) since midnight on September 1, 2000, where t is the time in hours. Make reasonable guesses about the domain and the range of f. Interpret the following statements in practical terms, giving units:
(a)
f(10) = 3.1 (b) f-1(10) = 16 (c) f'(8) = 0.4 (d) (f-1)'(5)
= 2.
7. A company's revenue from car sales, C (measured in thousands of dollars), is a function of advertising expenditure, a (also measured in thousands of dollars). Suppose C = f(a). What does the company hope is true about the sign of f'? What does the statement f'(100) = 2 mean in practical terms? How about f'(100) = 0.5? Suppose the company plans to spend about $100,000 on advertising. If f'(100) = 2, should the company spend a little more or a little less than $100,000 on advertising? What if f'(100) = 0.5?