49. 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.
To identify: Each curves on the given graph and give proper explanation.
The curve c is position curve, b is velocity curve and a is acceleration curve.
The given graph is shown as in Figure 1,
Observe the graph of b and a carefully.
The point where a(t)=0 is the same point where graph of b(t) has horizontal tangent.
Recall that the derivative of a function is zero where the function has a horizontal tangent.
Therefore, a(t) is the derivative of the graph b(t). That is, b′(t)=a(t).
Observe the graph of a and c carefully.
The graph of a has both positive and negative values. Hence a can be either velocity or acceleration.
The points where the graph of a has horizontal tangent, the functional value of c is not zero at that point.
This implies that, a′(t)≠c(t).
The only possibility is that a is the acceleration curve. This implies that c″(t)=a(t).
Thus, c is position curve, b is velocity curve and a is acceleration curve.