Problem 42 - Essential Calculus

Question

(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). $$ f(x)=(x-4) \arcsin \frac{x}{4} $$

Step-by-Step Solution

Answer

Compute the function's graph and establish its absolute extremes. Then derive the critical numbers using calculus. Evaluate the original function at each critical number and the endpoints to find any absolute extremal points. Then compare the results from part (a) graphing and part (b) calculus.

Step 1: Graph the function

Using graphing software, plot the given function \(f(x) = (x-4) \arcsin (\frac{x}{4})\) on the real number line.

Step 2: Approximate absolute extrema

Analyze the plotted graph to approximately identify any absolute maximum and minimum points. These are the highest and lowest points on the graph over the entire domain.

Step 3: Find the function's derivative

Differentiate the function to find its derivative. The derivative of a function can help identify critical points on the graph. The derivative of the given function could be found using the product rule and the chain rule.

Step 4: Find the critical numbers

The critical numbers of a function are where its derivative equals zero or is undefined. After finding the derivative in the previous step, set the derivative equal to zero and solve for \(x\) to get the critical numbers.

Step 5: Find the absolute extrema using critical numbers

Evaluate the original function at the critical numbers and endpoints. The greatest of these values is the absolute maximum, and the least of these values is the absolute minimum. Identify any absolute extrema not located at the endpoints.

Step 6: Compare the results

Compare the results from part (a) and part (b) to verify that they match. They should fall in the same locations on the graph.

Key Concepts

Exploring Computer Algebra Systems for Graphing Functions

Computer Algebra Systems (CAS) like WolframAlpha, GeoGebra, or Desmos are powerful tools for visualizing mathematical functions. They allow you to quickly and accurately plot functions, offering insights that might be less obvious through manual sketching. When working with complicated functions, such as the one given here, these systems can show you the overall shape and also highlight any turning points or unique behaviors.

By using a CAS, you can graph the function \( f(x) = (x-4) \arcsin \left( \frac{x}{4} \right) \) over the desired interval. The visual output allows you to approxime the absolute extrema, showing what values the function reaches and where it peaks or dips.

• Graphing makes it easy to identify global maximum and minimum points visually.
• It complements algebraic calculations by verifying sketch accuracy.
• Aids in understanding how changes in function's parameters affect the graph's shape.

Using these systems optimizes time and enhances understanding, making them indispensable for modern mathematical analysis.

Understanding Derivative Calculation for Functions

Differentiating a function gives you a new function, known as the derivative, which tells you the rate of change of the original function. When applying this to \( f(x) = (x-4) \arcsin \left( \frac{x}{4} \right) \), you need to utilize rules such as the product rule and chain rule.

The product rule is used here because our function is a product of two separate terms: \( x-4 \) and \( \arcsin \left( \frac{x}{4} \right) \). The general product rule states that if you have a function in the form of \( u(x) \cdot v(x) \), its derivative is \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \).

• Apply the product rule first to derive each component.
• Use the chain rule for the derivative of \( \arcsin \left( \frac{x}{4} \right)\), involving an inner function \( \frac{x}{4} \).

Precisely calculating the derivative will offer you a function that reveals where the slope of the original graph is zero or undefined—these points are critical for finding critical numbers.

Identifying Critical Numbers in Functions

Critical numbers are values in the domain of a function where its derivative is zero or undefined. These values are crucial when determining potential extrema, given they indicate where a graph's slope changes.

For our function, once the derivative has been calculated, you'll set this derivative equal to zero and solve for \( x \) to find critical points. Remember:

• Where the derivative equals zero, the function has a horizontal tangent—possible extrema points.
• Where the derivative is undefined might also be points of interest.

Once you find these critical numbers, you assess each one by evaluating the original function at these points. Doing so reveals their significance in terms of maximum or minimum values on the graph.

Analyzing Graphs for Extrema Using Critical Points

Graph analysis is the visual exploration of a function's graph to find specific points where the function reaches maximum or minimum values, known as extrema. After determining the critical numbers as outlined, you locate where these points lie on your function's graph.

Evaluate the function at the critical numbers along with the interval's endpoints. This evaluation tells you the specific values and helps verify which among them are the highest and lowest within the interval.

• Evaluate the original function at the critical numbers.
• Check these values and the values at the endpoints to confirm the absolute maximum and minimum.
• Compare these findings with the plotted points from the graph to ensure accuracy.

This analytical procedure combines with graphical data ensures a comprehensive understanding of where the function achieves its peak and lowest values.