Question

Graphical, Numerical, and Analytic Analysis In Exercises 43 and \(44,\) use a graphing utility to graph \(f\) on the interval [-2,2] . Complete the table by graphically estimating the slope of the graph at the indicated points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -1.5 & -1 & -0.5 & 0 & 0.5 & 1 & 1.5 & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{1}{4} x^{3}\)

Short answer

The slopes of \(f(x)=\frac{1}{4} x^{3}\) at various points on the interval [-2,2] can be obtained graphically as well as analytically. Graphically, the slopes have to be estimated by visual assessment of the tangent lines to the curve. Analytically, the derivative of \(f(x)\), that is \(f'(x)=3/4*x^2\), must be evaluated at these points. Any changes between the graphical and analytical results are expected to be due to estimation errors in the graphical method.

Step-by-step solution

Graph the Function

Using a graphing utility, plot the function \(f(x)=\frac{1}{4} x^{3}\) on the interval [-2,2]. This graph displays the visual representation of the function and provides an initial understanding of its behavior within the given interval.

Estimate the Slopes Graphically

For each x-value in the given table, determine the slope of the tangent to the curve at that point. Do this visual estimation by assessing the change in y-value over the change in x-value near each point on the graph. Populate the second row of the table (row \(f'(x)\)) with these estimates.

Analytical Evaluation of the Slopes

First, calculate the derivative of \(f(x)\), that is \(f'(x) = 3/4*x^2\). Then substitute each x-value from the table into \(f'(x)\) to get the analytical value of the slope. Compare these results with the ones estimated graphically and note any differences or similarities.