11DIFFERENTIATION - Department of Mathematics

Using Technology
FIGURE T1
702
EXAMPLE 1
F INDING THE R ATE OF C HANGE OF A F UNCTION
We can use the numerical derivative operation of a graphing utility to obtain
the value of the derivative at a given value of x. Since the derivative of a
function f(x) measures the rate of change of the function with respect to x,
the numerical derivative operation can be used to answer questions pertaining
to the rate of change of one quantity y with respect to another quantity x,
where y f(x), for a specific value of x.
Let y 3t 3 2t.
a. Use the numerical derivative operation of a graphing utility to find how
fast y is changing with respect to t when t 1.
b. Verify the result of part (a), using the rules of differentiation of this section.
SOLUTION ✔ a. Write f(t) 3t3 2t. Using the numerical derivative operation of a
graphing utility, we find that the rate of change of y with respect to t when
t 1 is given by f(1) 10.
b. Here, f(t) 3t3 2t1/2 and
f(t) 9t 2 21 2 t1/2 9t 2 1
t
Using this result, we see that when t 1, y is changing at the rate of
f(1) 9(12 ) 1
10
1
units per unit change in t, as obtained earlier.
EXAMPLE 2
SOLUTION ✔
According to the U.S. Department of Energy and the Shell Development
Company, a typical car’s fuel economy depends on the speed it is driven and
is approximated by the function
f(x) 0.00000310315x 4 0.000455174x 3
0.00287869x 2 1.25986x (0 x 75)
where x is measured in mph and f(x) is measured in miles per gallon (mpg).
a. Use a graphing utility to graph the function f on the interval [0, 75].
b. Find the rate of change of f when x 20 and x 50.
c. Interpret your results.
Source: U.S. Department of Energy and the Shell Development Company
a. The result is shown in Figure T1.
b. Using the numerical derivative operation of a graphing utility, we see that
f(20) 0.9280996. The rate of change of f when x 50 is given by
f(50) 0.314501.
c. The results of part (b) tell us that when a typical car is being driven at
20 mph, its fuel economy increases at the rate of approximately 0.9 mpg
per 1 mph increase in its speed. At a speed of 50 mph, its fuel economy
decreases at the rate of approximately 0.3 mpg per 1 mph increase in
its speed.