Problem 23 - Calculus Early Transcendentals

Question

Use a graphing utility to estimate the value of \(f^{\prime}(1)\) by zooming in on the graph of \(f,\) and then compare your estimate to the exact value obtained by differentiating. \(f(x)=\frac{x}{x^{2}+1}\)

Step-by-Step Solution

Answer

The estimated value of \(f'(1)\) is \(0\), which matches the exact value obtained analytically.

Step 1: Understand the Function

The function given is \(f(x) = \frac{x}{x^2 + 1}\). To solve the problem, we need to find its derivative, \(f'(x)\), and estimate \(f'(1)\).

Step 2: Use Graphing Utility

Graph the function \(f(x) = \frac{x}{x^2 + 1}\) using a graphing utility. Zoom in on the point where \(x = 1\). The slope of the tangent line to the graph at this point approximates \(f'(1)\). As you zoom in, note the slope of the curve at \(x = 1\).

Step 3: Differentiate Analytically

Differentiate the function \(f(x)\) using the quotient rule: If \(f(x) = \frac{u}{v}\), then \(f'(x) = \frac{u'v - uv'}{v^2}\). Here, \(u = x\) and \(v = x^2 + 1\), so \(u' = 1\) and \(v' = 2x\).

Step 4: Apply the Quotient Rule

Apply the quotient rule: \(f'(x) = \frac{(1)(x^2 + 1) - (x)(2x)}{(x^2 + 1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}\).

Step 5: Evaluate the Derivative at x=1

Substitute \(x = 1\) into the derivative \(f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}\): \(f'(1) = \frac{1 - 1^2}{(1^2 + 1)^2} = \frac{0}{4} = 0\). This is the exact value.

Step 6: Compare Results

Compare the estimated slope from the graphing utility with the exact value \(f'(1) = 0\). The closer you zoomed in, the closer your estimate should be to \(0\).

Key Concepts

Derivative Estimation

In calculus, estimating the derivative of a function at a given point is a common task. The derivative, denoted as \( f'(x) \), represents the slope of the tangent line to the curve at that point. To estimate \( f'(1) \), we can use graphing technology. By zooming into the graph of \( f(x) = \frac{x}{x^2 + 1} \) at \( x = 1 \), the slope of the tangent line becomes visible. As you continue to zoom in, the curve should appear more linear, allowing for a more accurate estimation of the slope which represents the derivative at that point. If the tangent line becomes horizontal, this suggests that \( f'(1) \) is close to zero.This graphical method is particularly useful when a calculator or computer is handy. However, remember that it's an estimation technique, which means it's most effective when accompanied by analytical methods for verification.

Graphing Utilities

Graphing utilities, like calculators and software programs, are valuable tools for visualizing functions and their behaviors. They allow you to plot the function \( f(x) \), and then interactively explore its properties through features like zooming, tracing, and analyzing.

• Plotting: Begin by plotting the function \( f(x) = \frac{x}{x^2 + 1} \) to get an overall picture of its shape and behavior.
• Zoom: Focus on areas of interest, such as \( x = 1 \), to observe how the graph behaves in small neighborhoods.
• Tracing: Use the trace feature to mark specific points and see the coordinates, which assists in estimating the slope at those points.

Using these utilities provides a deeper understanding of how the function behaves locally, and can help verify derivative calculations by showing a similar result graphically.

Quotient Rule

The quotient rule is a powerful technique in calculus that simplifies finding derivatives of quotients of functions. When a function is presented as a ratio, like \( f(x) = \frac{u}{v} \), the derivative \( f'(x) \) requires the following formula:\[f'(x) = \frac{u'v - uv'}{v^2}\]This means you differentiate the numerator \( u \) and the denominator \( v \) separately. In our case, for \( f(x) = \frac{x}{x^2 + 1} \):

• Let \( u = x \) and \( v = x^2 + 1 \).
• Then, \( u' = 1 \) and \( v' = 2x \).
• Substitute these into the formula to find \( f'(x) \).

After applying the quotient rule, we find:\[f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}\]For \( x = 1 \), this gives \( f'(1) = 0 \), confirming the horizontal tangent line estimation. The quotient rule is essential when dealing with such rational functions, providing exact derivatives that help verify graphical estimations.