# Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises: 52

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

#### Work Step by Step

We've learned that the velocity function is the derivative of the position function, the acceleration function is the derivative of the velocity function, and the jerk function is the derivative of the acceleration function. Therefore, we can call the position function $f$, the velocity function $f'$, the acceleration function $f''$, and the jerk function $f'''$. Now look at curve a. Curve a has 2 local extrema and passes the $Ox$ 2 times. Now if curve a represents $f$ or $f'$ or $f''$, there must exist 1 other curve that passes the $Ox$ 2 times. Nevertheless, none of the other 3 curves pass the $Ox$ 2 times. Therefore, curve a represents $f'''$, or the jerk of the car. Then look at curve b. Curve b has 1 local extrema and passes the $Ox$ 1 times. Now if curve b represents $f$ or $f'$, there must exist another curve of the remaining ones that passes the $Ox$ 1 time. However, both curve c and d don't pass the $Ox$ at all. Therefore, curve b represents $f''$, or the acceleration of the car. Now look at the remaining 2 curves. Curve d is always increasing and goes up steepily at the end, while curve c increases before slowing down and slightly decreasing at the end. If curve c represents $f$ and curve d represents $f'$, then curve d must decrease at the end and even reaches negative values. In fact, it does not. Therefore, curve c represents $f'$, or the velocity of the car. Curve d represents $f$, or the position function of the car.

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