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 derivative of the function can be obtained using the graph of the function by noticing the slope of the tangent line to the curve at each point. It can be done by drawing the tangent at the point \(\left( {x,f\left( x \right)} \right)\) and estimating its slope. The slope of the graph of \(f\) becomes the \(y\)-value on the graph of \(f'\).

Step-by-step solution

The slope of the tangent

The derivative of the function can be obtained using the graph of the function by noticing the slope of the tangent line to the curve at each point. It can be done by drawing the tangent at the point \(\left( {x,f\left( x \right)} \right)\) and estimating its slope. The slope of the graph of \(f\) becomes the \(y\)-value on the graph of \(f'\).

Draw the graph of the derivative

Take some points \(A,B,C,D,E,F,G,H\) on the graph of the given function \(f\).

There are corners at the point \(A,B,C,D,E,F,G,H\). Therefore, the graph of \(f'\) is discontinuous at points \(A,B,C,D,E,F,G,H\).

From point \(C\) to \(D\) and \(F\)to\(G\), the graph of the function is a horizontal straight line which is parallel to \(x\)-axis. So, the approximate value of the derivativefrom point \(C\) to \(D\) and \(F\)to\(G\)is \(0\).

From \(A\) to \(B\), \(D\) to \(E\), and \(G\) to \(H\), the graph of the function is a straight line and has a slope of \( - 1\). Therefore, the approximate value of the derivative from \(A\) to \(B\), \(D\) to \(E\), and \(G\) to \(H\) is \( - 1\).

From \(B\) to \(C\), \(E\) to \(F\), the graph of the function is a straight line and has a slope of \(1\). Therefore, the approximate value of the derivative from \(B\) to \(C\), \(E\) to \(F\) is \(1\).

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