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. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -2 & -1.5 & -1 & -0.5 & 0 & 0.5 & 1 & 1.5 & 2 \\ \hline f(x) & & & & & & & & & \\ \hline f^{\prime}(x) & & & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{1}{2} x^{2}\)

Short answer

The graphically estimated slopes at the given points should be generally close to the analytically calculated slopes, which are exactly equal to the x values of -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, and 2 due to the simple derivative of the function.

Step-by-step solution

Plot the Function

First, use a graphing utility to plot the function \(f(x) = \frac{1}{2} x^{2}\) between -2 and 2.

Estimate the Slope

Using the plotted graph, estimate the slope of the function at the points x = -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, and 2 by drawing a tangent line to the function at each of these points and calculating the rise over run.

Calculate the Derivative

The derivative of a function gives the slope of the tangent line to the function at any point. Thus, the derivative \(f'(x) = x\) of the function \(f(x)\) can be used to analytically calculate the slope of the function at the points x = -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, and 2.

Compare the Results

Finally, compare the graphically estimated slopes with the analytically calculated slopes by filling in the values in the table and observing whether they are close or identical.

Key concepts

Graphing Utilities

Graphing utilities, such as calculators or software programs, provide a visual representation of mathematical functions. They're particularly helpful in understanding complex concepts in calculus, such as the behavior of functions and their tangent lines at specific points.

Using a graphing utility to plot the function
\( f(x) = \frac{1}{2} x^{2} \) on the interval [-2,2], provides a visual curve that one can analyze to estimate slopes. This is especially useful when trying to grasp how a function changes at each point, which is fundamental in the study of calculus. Visually estimating the slope at each given point can serve as a preliminary step before delving into analytical calculations.

Furthermore, by comparing the graphical estimation to analytic results, students gain valuable insights into the accuracy and limitations of visual approximations. Always make sure the scales on the axes are correct to ensure accurate slope estimation.

Derivative Calculations

In calculus, the derivative represents the rate at which a function is changing at any given point. It is a fundamental concept as it describes the slope of the tangent line to the function's curve.

The derivative of the function
\( f(x) = \frac{1}{2} x^{2} \) is simply
\( f'(x) = x \), since the power of x decreases by one and is multiplied by the original power, in accordance with the power rule for derivatives.

Calculating Slopes Analytically

Once the derivative is known, you can calculate the slope at any point on the function by substituting the x-value into the derivative function. This gives an exact mathematical expression for what might have been first roughly estimated with the graphing utility. It's crucial for students to understand how to derive and apply these calculations, as they provide a more precise and reliable measure of the function's behavior.

Tangent Lines

One of the most practical applications of derivatives in calculus is finding the equation of tangent lines. A tangent line is a straight line that just touches the curve at a given point and has the same slope as the curve at that point. It's essential for approximating the value of functions near the points of tangency.

The slope of the tangent line at any given point on our function \( f(x) = \frac{1}{2} x^{2} \) can be found using the derivative \( f'(x) = x \). Graphically, this tangent line can be sketched by drawing a line that only touches the curve at the point of interest without crossing it.

Understanding tangent lines is pivotal for solving various application problems in physics, engineering, and economics, where rates of change are critical. They are a perfect tool for visualizing the local linear approximation of the function near the point of contact.

Analytic Analysis

Analytic analysis in calculus refers to solving problems using algebraic manipulations and calculations, rather than relying on visual or numerical approximations. This method is crucial for deriving precise results and proofs in calculus.

For our quadratic function \( f(x) = \frac{1}{2} x^{2} \), the analytic approach allows us to calculate the exact slope at any selected point by using the derivative. When completing the table for the given exercise, you ensure that every slope at each point like -2, -1, -0.5, and so forth is determined by substituting the x-value into \( f'(x) = x \). As a result, this method confirms or refines the estimates made using graphing utilities.

Comparing Graphical and Analytical Results

After calculating the slopes analytically, you compare them against the graphically estimated ones. This comparison solidifies the understanding of both graphical and analytic approaches in calculus and allows students to appreciate the reliability and necessity of analytic methods in solving complex mathematical problems.