Problem 37 - Essential Calculus

Question

In Exercises \(37-42,\) (a) use a computer algebra system to graph the function and approximate any absolute extrema on the indicated interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a). $$ f(x)=3.2 x^{5}+5 x^{3}-3.5 x $$

Step-by-Step Solution

Answer

By following the above steps, the approximate extrema on the function \(f(x)=3.2 x^{5}+5 x^{3}-3.5 x\) can be identified visually from the graph and analytically by differentiating the function, finding the critical numbers and comparing their function values with the endpoints.

Step 1: Graph the function

Using a computer algebra system, plot the given function \(f(x)=3.2 x^{5}+5 x^{3}-3.5 x\). Observing the plot, identify any potential local maxima and minima by simply looking at the turns in the function.

Step 2: Find the derivative of the function

The derivative of the function \(f(x)=3.2 x^{5}+5 x^{3}-3.5 x\) is found using the power rule for differentiation. This gives \(f'(x)=16 x^{4}+15 x^{2}-3.5\). This will be used for finding the critical numbers.

Step 3: Find the critical numbers

The critical numbers of a function are those that make the derivative zero or undefined. Set the derivative equal to zero and solve for \(x: 16 x^{4}+15 x^{2}-3.5 = 0\). A computer algebra system can be useful for solving this polynomial.

Step 4: Find extrema using critical numbers

Substitute the critical numbers (solutions from previous step) into the original function to get the corresponding \(y\) values. Check the values of the function at these points. Compare these with the values at the endpoints of the interval to determine if they are absolute extrema.

Step 5: Compare results

Compare the results in Steps 1 and 4. They should match in terms of location of the extrema.

Key Concepts

Understanding Absolute Extrema

Absolute extrema, also known as absolute maximum and minimum, refer to the highest or lowest points on the entire graph of a function over a specified interval. To find these extrema, you must evaluate the function at critical numbers and at the endpoints of the interval. Here are a few key points about absolute extrema:

• Absolute maximum is the largest value a function achieves over its domain.
• Absolute minimum is the smallest value a function achieves over its domain.
• They are vital for determining the range and behavior of the graph.

When you have a function, graphing it with tools like a computer algebra system helps by visually showcasing where these extrema may lie. After plotting the function, you should look both at the peaks and troughs as well as assess the boundary points to ensure all potential locations for absolute extrema are covered.

Identifying Critical Numbers

Critical numbers play a crucial role in finding the extrema. A critical number is a specific value within the domain of a function where the derivative is zero or undefined. It is at these points that a function can have local maximums, minimums, or inflection points.

• Find the derivative of the function using the power rule.
• Set the derivative equal to zero, as this pinpoint local extrema candidates.
• Examine these x-values, and don't forget to solve for intervals where the derivative does not exist, too.

By solving the equation derived from setting the first derivative to zero, you identify these critical numbers. It's essential because they help you find local extrema on the interval, potentially leading to absolute ones. These points signal places where the graph either flattens out or changes direction.

Power Rule for Differentiation

The power rule for differentiation is a simple and widely-used technique in calculus. It allows you to find the derivative of powers of x quickly and efficiently. Here's the basic formula: if you have a function of the form \( f(x) = ax^n \), then its derivative is \( f'(x) = anx^{n-1} \).
Key steps with the power rule include:

• Identify the exponent \(n\) and the coefficient \(a\) of each term with the variable \(x\).
• Apply the power rule to find the derivative of each term.
• Combine these results to obtain the full derivative \(f'(x)\).

This method was applied when finding the derivative in the original exercise as well. It led us to the expression \( f'(x) = 16x^4 + 15x^2 - 3.5 \), which was critical for determining critical points.

Using Computer Algebra Systems

A computer algebra system (CAS) is an invaluable tool for solving complex algebraic problems, offering capabilities ranging from symbolic manipulation to graphing. When handling calculus problems like finding extrema, a CAS can be particularly helpful.

• They allow for precise graph plotting, making it easier to visually ascertain potential extrema.
• The system can solve complex polynomial equations derived while searching for critical numbers.
• Provides functionalities for checking and comparing results effectively.

By automating these tasks, a CAS not only saves time but also ensures accuracy in solutions—especially useful for more intricate functions or large calculations where manual errors could occur. It beautifully illustrates the theoretical concepts you learn in calculus through practical, instant results.