Chapter 4.6, Problem 13E

Estimating speed Use the linear approximation given in Example 1 to answer the following questions.

7. If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?

EXAMPLE 1 Useful driving math Suppose you are driving along a highway at a nearly constant speed and you record the number of seconds it takes to travel between two consecutive mile markers. If it takes 60 seconds to travel one mile, then your average speed is 1 mi/60 s or 60 mi/hr. Now suppose that you travel one mile in 60 + x seconds; for example, if it lakes 62 seconds, then x = 2, and if it takes 57 seconds, then x = −3. In this case, your average speed over one mile is 1 mi/(60 + x) s. Because there are 3600 s in 1 hr, the function

$s(x)=\frac{3600}{60+x}=3600{(60+x)}^{-1}$

gives your average speed in mi/hr if you travel one mile in x seconds more or less than 60 seconds. For example, if you travel one mile in 62 seconds, then x = 2 and your average speed is s(2) ≈ 58.06 mi/hr. If you travel one mile in 57 seconds, then x = −3 and your average speed is s(−3) ≈ 63.16 mi/hr. Because you don’t want to use a calculator while driving, you need an easy approximation to this function. Use linear approximation to derive such a formula.