Question

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.

Explanation:

Step1: Recall derivative relationships

Velocity is the derivative of position ($v = s'$), and acceleration is the derivative of velocity ($a=v'$). That is, acceleration is the second - derivative of position ($a = s''$).

Step2: Analyze slopes of curves

The slope of the position - function curve gives the velocity. The slope of the velocity - function curve gives the acceleration. Curve $c$ has a non - negative slope that starts positive, decreases to zero, and then becomes positive again. Curve $b$ has values that match the slope of curve $c$. The slope of curve $b$ is positive initially, then negative, and then positive again. Curve $a$ has values that match the slope of curve $b$.

Step3: Identify the functions

Since the velocity is the derivative of the position and the acceleration is the derivative of the velocity, the position function is $c$, the velocity function is $b$, and the acceleration function is $a$.

Answer:

position: $c$
velocity: $b$
acceleration: $a$