Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph \(f\) on the interval \([-2,2]\). Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -\frac{3}{2} & -1 & -\frac{1}{2} & 0 & \frac{1}{2} & 1 & \frac{3}{2} & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{1}{2} x^{2} $$

Step-by-Step Solution

Answer

The graph of the function is a parabola opening upwards. The graphical and analytical slopes for each x-value ended up being approximately the same which verifies that the derivative formula \(f'(x)=x\) given in step 2 is correct.

Step 1: Graph the function

Using a graphing utility, a graph of \(f(x)=\frac{1}{2} x^{2}\) on the interval '-2' to '2' will be plotted. This graph will be used to estimate the slopes at the given points.

Step 2: Calculate derivatives

The derivative of \(f(x)\), denoted as \(f'(x)\), can be calculated for a general value of \(x\). Here, \(f(x)=\frac{1}{2}x^{2}\), so the derivative will be \(f'(x)=x\). This formula will be used to calculate the slopes analytically at the specified points.

Step 3: Estimate the slopes graphically and calculate them analytically

Now, using the graph, estimate the slopes at the points x=-2, -3/2, -1, -1/2, 0, 1/2, 1, 3/2, 2. Then, also calculate \(f'(x)\) at each of these points using the formula from step 2. This will give an estimate slope from the graph and an exact slope from the calculation.

Step 4: Compare the results

To finalize the task, compare the slopes obtained graphically and analytically for each x-value. The exact results might not match completely due to approximation in graphical estimation, but they should be relatively close if the approximation was done accurately.

Key Concepts

Graphing Utility

With advancements in technology, the integration of graphing utilities into mathematics has become a fundamental practice. These tools provide visual representations that are particularly helpful for understanding complex functions and concepts. When it comes to analyzing the behavior of functions, such as estimating the slope of a parabolic function like \(f(x)=\frac{1}{2} x^{2}\), a graphing utility offers an efficient and accurate method to plot the curve over a specified interval.

In the context of the given exercise, students are asked to use a graphing utility to estimate the slopes of the graph at certain points on the interval \([-2,2]\). A graphing utility will display the graph, showing the curvature of the parabola. Students can visually assess the slope - essentially the steepness - of the graph at these points. While this method is not as precise as analytical calculations, it is quick and provides a sufficient approximation for many practical purposes.

Derivative Calculation

Derivative calculation is the cornerstone of understanding how functions change, which intuitively means finding the rate at which the function's value increases or decreases. The derivative of a function at a point gives the slope of the tangent line to the graph at that point. For the quadratic function \(f(x)=\frac{1}{2} x^{2}\), the process to find its derivative is straightforward.

Using basic derivative rules, the derivative of \(f(x)\) is found by multiplying the exponent by the coefficient and then subtracting one from the exponent, resulting in \(f'(x)=x\). This derivative function tells us that the slope of the tangent line at any given point on the original function can be calculated simply by plugging the x-value into the derivative function. This provides a precise, analytical method to calculate the slope in contrast to the graphical estimation.

Slope Comparison

Slope comparison is a critical part of understanding the concordance between graphical slope estimation and analytical calculations. By comparing graphically estimated slopes to those derived analytically using the derivative, students can gain insights into the accuracy of graphical estimations and the reliability of derivative calculations.

Graphical estimations are generally more prone to error due to the limitations of visual approximation, especially when the graphs are dense or when dealing with subtle changes in steepness. Analytical calculations, on the other hand, provide exact values for slopes. Comparing these two methods allows students to validate their graphical slope estimations and understand possible discrepancies. The exercise emphasizes this comparison, encouraging students to not only rely on visual representations but also to value the precision of mathematical analysis.