The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
$c$ is the position function
$b$ is the velocity
$a$ is the acceleration
So in this problem were given a graph That has three lines on it, a BNC. And we're told that one of these is positioned function of a car. One is the velocity of the car and one is the acceleration and were asked identify each curve and explain your choices. All right. Well, first of all we know that the acceleration is the derivative of the velocity and the velocity is a derivative of the position. And so if we think about this for mitt the derivative meaning the slope of that curve. Well, if we look at it for a second as C starts to climb directly, Starting out here at zero and see starts to climb linearly across here, you notice that be shows that linear acceleration happening right then. See turns upward across this interval here he turns up and then starts to decrease on this last interval. So it really looks like that. Be B is the slope or derivative of C. Then if we look closely again as B is climbing A is climbing even faster. So it looks like the slope of the derivative of be going on. And then as B starts to approach the top of the curve in this section here we see that we hit a maximum and start to decrease then right. Which is what the slope does until it starts to go negative right? And then becomes less and less negative out here toward the end. So it looks like the A. Hey, is the slope of B. So therefore see is the position of the car, B is the velocity and C is the acceleration of the car.