Activity 1.9.3.
Let \(f\) be a function that measures a car’s fuel economy in the following way. Given an input velocity \(v\) in miles per hour, \(f(v)\) is the number of gallons of fuel that the car consumes per mile (i.e., “gallons per mile”). We know that \(f(60) = 0.04\text{.}\)
(a)
What is the meaning of the statement “\(f(60) = 0.04\)” in the context of the problem? That is, what does this say about the car’s fuel economy? Write a complete sentence to explain.
(b)
Consider the function \(g(v) = \frac{1}{f(v)}\text{.}\) What is the value of \(g(60)\text{?}\) What are the units on \(g\text{?}\) What does \(g\) measure?
(c)
Consider the function \(h(v) = v \cdot f(v)\text{.}\) What is the value of \(h(60)\text{?}\) What are the units on \(h\text{?}\) What does \(h\) measure?
(d)
Do \(f(60)\text{,}\) \(g(60)\text{,}\) and \(h(60)\) tell us fundamentally different information, or are they all essentially saying the same thing? Explain.
(e)
Suppose we also know that \(f(70) = 0.045\text{.}\) Find the average rate of change of \(f\) on the interval \([60,70]\text{.}\) What are the units on the average rate of change of \(f\text{?}\) What does this quantity measure? Write a complete sentence to explain.
