Chapter 2.7, Problem 7E

Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.

Example 1

FIGURE 1

FIGURE 2

To determine

To sketch: The graph of $f^{\prime }$ below the graph of f.

Explanation of Solution

From the given graph, it is observed that the graph of f contains the horizontal tangent at one point. Let the point be A.

Note that, the value of the derivative will be zero at the point where the function has the horizontal tangent.

Thus, the graph of $f^{\prime }$ will be zero at the points A.

From the given graph, f has a vertical tangent line at the left end point. This implies that, the derivative function $f^{\prime }$ has a vertical asymptote at that point. Thus, f is not differentiable at that point.

From the point A to left, the slope of the graph f is positive which implies that the derivative graph $f^{\prime }$ must have a functional value in positive.

From the point A to right, the slope of the graph f is negative which implies that the derivative graph $f^{\prime }$ must have a functional value in negative.

From the given graph, f has a vertical tangent line at the right end point. This implies that, the derivative function $f^{\prime }$ has a vertical asymptote at that point. Thus, f is not differentiable at that point.

Graph:

Use the above information and obtain the graph of $f^{\prime }\left(x\right)$ as shown below in Figure 1.

Thus, $f^{\prime }$ is the required graph.