Question

(a) use a computer algebra system to differentiate the function, (b) sketch the graphs of \(f\) and \(f^{\prime}\) on the same set of coordinate axes over the indicated interval, (c) find the critical numbers of \(f\) in the open interval, and (d) find the interval(s) on which \(f^{\prime}\) is positive and the interval(s) on which it is negative. Compare the behavior of \(f\) and the sign of \(f^{\prime}\). $$ f(t)=t^{2} \sin t,[0,2 \pi] $$

Short answer

The derivative of the function is \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\). The critical numbers and intervals where the derivative is positive or negative should be determined from the solution of \(f^{\prime}(t) = 0\). The behavior of \(f\) increases when \(f^{\prime}\) is positive, and \(f\) decreases when \(f^{\prime}\) is negative. These observations can be noticed from the graphs.

Step-by-step solution

Derive the Function

To derive the given function \(f(t) = t^{2} \sin(t)\) within a computer algebra system, the product rule will be used. The derivative is \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\).

Sketch the Functions

Next, the graphs of \(f(t)\) and \(f^{\prime}(t)\) should be sketched over the same set of coordinate axes. Note that the critical numbers (when the derivative is zero) will help in identifying the maximum and minimum values on the graph, which helps in sketching the curve.

Find the Critical Numbers

To find the critical numbers of the function, set \(f^{\prime}(t)\) equal to zero and solve for \(t\). From \(f^{\prime}(t) = 2t\sin(t) + t^{2}\cos(t)\), the critical numbers are obtained by solving \(2t\sin(t) + t^{2}\cos(t) = 0\).

Determine Intervals where \(f^{\prime}\) is Positive or Negative

To determine where the derivative of the function is positive or negative, make a sign test chart. Start by marking the critical numbers on the line. Then test numbers from each interval on the derivative equation to see whether the derivative is positive or negative.

Analyze Comparison between \(f\) and \(f^{\prime}\)

Finally, compare the behavior of the function \(f\) and its derivative \(\fcrep\). When the derivative is positive, \(f\) should be increasing. When the derivative is negative, \(f\) should be decreasing. This can be analyzed from the graphs of \(f\) and \(f^{\prime}\) that were sketched in step 2.