Problem 74 - Calculus Early Transcendentals

Question

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\frac{\tan ^{-1}\left(x^{2}-x\right)}{x^{2}+4} $$

Step-by-Step Solution

Answer

Use a CAS to find derivatives and estimate key points by graphing them.

Step 1: Find the Derivatives

First, we'll find the first derivative \(f'(x)\) and the second derivative \(f''(x)\) of the function \(f(x) = \frac{\tan^{-1}(x^2-x)}{x^2+4}\). Use the quotient rule and chain rule for these calculations. The resulting derivatives can become complex, so using a CAS (Computer Algebra System) will simplify this step.

Step 2: Graph the Derivatives

Use a CAS tool to graph \(f'(x)\) and \(f''(x)\). These graphs will help you identify where the first derivative changes sign, indicating potential relative extrema, and where the second derivative changes sign, indicating possible points of inflection.

Step 3: Estimate the Relative Extrema

Analyze the graph of \(f'(x)\). Estimate the \(x\)-coordinates where \(f'(x)\) changes from positive to negative or vice versa. These points are the relative extrema of \(f(x)\).

Step 4: Verify with \(f(x)\)

Graph the original function \(f(x)\) and check if the estimated \(x\)-coordinates of the extrema from the graph of \(f'(x)\) match the peaks or valleys on the graph of \(f(x)\). This step serves as a verification of your estimates.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus. They represent the rate at which a function is changing at any given point. For a function like \[ f(x) = \frac{\tan^{-1}(x^2-x)}{x^2+4} \], finding the first and second derivatives involves applying the chain rule and quotient rule simultaneously.

• The chain rule is used when you have a composition of functions, like \( \tan^{-1}(x^2-x) \).
• The quotient rule is helpful when dividing two functions, as seen in the form \( \frac{u(v)}{w(v)} \), where \( u(v) \) and \( w(v) \) are differentiable functions.

These rules combine to provide \( f'(x) \) and \( f''(x) \). Calculations can become intricate, which is why using technology like CAS can be beneficial for ensuring precision and simplifying complex expressions.

Graphing

Graphing is an essential technique in calculus for visualizing the behavior of functions and their derivatives. By plotting \( f'(x) \) and \( f''(x) \), we can gain insights into where the function is increasing or decreasing as well as the curvature of the graph.

• A graph of the first derivative, \( f'(x) \), will show where the function has relative maxima and minima, as these occur when \( f'(x) = 0 \) or changes sign.
• The second derivative, \( f''(x) \), reveals points of inflection, where the concavity of \( f(x) \) changes.

By leveraging graphs, we can achieve a deeper understanding of how a function behaves over intervals and become more adept at predicting its behavior.

Relative Extrema

Relative extrema refer to the points on a graph where the function reaches a local maximum or minimum. These points can be found by analyzing the first derivative of the function, \( f'(x) \).

• At points where \( f'(x) = 0 \), the function may have a local max or min, known as critical points.
• If \( f'(x) \) changes from positive to negative, a local maximum is indicated.
• Conversely, if \( f'(x) \) changes from negative to positive, a local minimum is present.

It's crucial to evaluate these changes carefully, as relative extrema provide critical insights into the function's overall behavior and characteristics.

Computer Algebra Systems (CAS)

Computer Algebra Systems (CAS) are invaluable tools for solving and analyzing complex mathematical problems. They perform symbolic and numeric computations, which are especially useful for calculus tasks.

• CAS can find derivatives and integrals quickly and accurately, minimizing human errors.
• They offer graphical representations of functions and their derivatives, aiding in the comprehension of relationships and changes.
• CAS provides capabilities for algebraic manipulation, helping to simplify expressions that would be laborious to solve by hand.

Using CAS for visualizing graphs of \( f'(x) \) and \( f''(x) \) helps students estimate relative extrema and verify results efficiently. They bridge the gap between theoretical understanding and practical application, enhancing the learning experience.