# Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises: 51

Curve a - acceleration of the car. Curve b - velocity of the car. Curve c - position function of the car.

#### Work Step by Step

We've learned that the velocity function is the derivative of the position function, and the acceleration function is the derivative of the velocity function. Therefore, we can call the position function $f$, the velocity function $f'$ and the acceleration function $f''$. Now look at curve a. Curve a has 2 local extrema and passes the $Ox$ 1 time. Now if curve a represents $f$ or $f'$, there must exist 1 other curve that passes the $Ox$ 2 times. Nevertheless, none of the other 2 curves pass the $Ox$ 2 times. Therefore, curve a represents $f''$, or the acceleration of the car. Now look at the remaining 2 curves. Curve b has 1 local extrema and curve c does not pass the $Ox$ at all. If curve b represents $f$, then curve c should pass $Ox$ 1 time. But in fact, it does not. Therefore, curve b represents $f'$, or the velocity of the car. Curve c represents $f$, or the position function of the car.

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