Question

Shown is the graph of the population function \(P\left( t \right)\) for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative \(P'\left( t \right)\). What does the graph of \(P'\) tell us about the yeast population?

Short answer

The graph of \(P'\) shows that the yeast culture increases more rapidly after 6 hours, and the rate of growth decreases.

Step-by-step solution

Graph the derivative \(P'\left( t \right)\)

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

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

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

The tangent at \(t = A\) and \(t = D\) is horizontal. Therefore the slope is zero and it follows that the value of \(P'\left( A \right) = P'\left( D \right) = 0\).

It is observed from the graph of \(y = P\left( t \right)\) that there is a steepest at \(t = B\). Hence, the value of \(P'\left( B \right)\) is 50.

The slope of the function \(P\left( t \right)\) is \(P'\left( C \right)\) can be determined as the ratio of the rise and run that appears to be equal. The function \(P\left( t \right)\) is increasing at \(x = C\). It follows that the value of \(P'\left( C \right) = 70\).

It is observed that the tangent of four points has a positive slope, so \(P'\left( t \right)\) is positive.

Use the above point to estimate the derivative of the function as shown below:

Explain the graph of \(P'\)shows about the yeast population

It is observed from the graph of \(y = P\left( t \right)\) that the slope of the tangent lines is always positive; therefore, the \(y - \)values of \(y = P'\left( t \right)\) are always positive.

These values begin at relatively small and continue increasing, reaching a maximum at about \(t = 6\).

Eventually, the \(y - \)values of \(y = P'\left( t \right)\) decrease and reach zero. The graph of \(P'\) shows that the yeast culture increases more rapidly after 6 hours and the rate of growth decreases.