Question

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.

Short answer

The graph of \(f'\left( x \right)\) is shown below:

Step-by-step solution

Sketch the graph of \(f'\)

Consider some points \(A,B,C,D,\) and \(E\)to plot on the graph of the given function \(f\left( x \right)\). We can estimate the graph of derivatives from these points.

Sketch the graph of the given function with some points as shown below:

To determine the value of the derivative at any value of \(x\), we can draw the tangent at the point \(\left( {x,f\left( x \right)} \right)\) and estimate the slope.

It is observed from the given graph that for \(x = B\) and \(x = D\), the slope of the function \(f\left( x \right)\) is \(f'\left( B \right),f'\left( D \right)\). It can be determined as the ratio of the rise and run that appears to be equal.

The function \(f\left( x \right)\) is decreasing at \(x = B,D\). Hence, the value of \(f'\left( B \right)\) and \(f'\left( D \right)\) is \( - 1\).

The tangent at \(x = A\) and \(x = E\) is horizontal, therefore the slope is zero and it follows that the value of \(f'\left( A \right) = f'\left( E \right) = 0\).

The tangent at \(x = C\) is vertical, thus the slope is not defined and there is a decreasing function \(x = c\). It follows that the value of \(f'\left( C \right) = - \infty \).

Use the above points to sketch the graph of \(f'\left( x \right)\) as shown below: