page 1 of 3

Suppose *s*(*t*) = 2*t*^{3} represents the position of a race car along a straight track, measured in feet
from the starting line at time *t* seconds. What is the average rate of change of
*s*(*t*) from *t* = 2 to *t* = 3?

The average rate of change is equal to the total change in position divided by the total change in time:

Avg Rate | = | ||

= | |||

= | |||

= 38 ft per second |

In physics, velocity is the rate of change of position. Thus, 38 feet per second is the
average velocity of the car between times *t* = 2 and *t* = 3.

What is the instantaneous rate of change of the same race car at time *t* = 2?

The instantaneous rate of change measures the rate of change, or slope, of a curve at a
certain instant. Thus, the instantaneous rate of change is given by the derivative. In this
case, the instantaneous rate is *s'*(2).

s'(t) | = | 6t^{2} | |

s'(2) | = | 6(2)^{2} = 24 feet per second |

Thus, the derivative shows that the racecar had an instantaneous velocity of 24 feet per
second at time *t* = 2.

Take a Study Break!