Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=x^{3} e^{-x+2}$$

Step-by-Step Solution

Answer

The filled graph from the graphing utility will show where the function \(f(x) = x^{3}e^{-x+2}\) increases and decreases. Approximate relative maximum and minimum values can be determined from these points.

Step 1: Draw the Graph of the Function

Utilize a graphing utility to plot \(f(x) = x^{3}e^{-x+2}\). Pay attention to how the graph is shaped, this will give you clues to where the function is increasing and decreasing, and where potential maximum/minimum points are located.

Step 2: Identify Intervals of Increasing and Decreasing

Observe the graph to determine on which intervals the function \( f(x) \) is increasing and decreasing. The function increases when the slope of the tangent line is positive, and decreases when the slope is negative. This will be visually visible on the graph.

Step 3: Approximate Relative Maximum and Minimum Values

Using the graph, approximate any local (relative) maximum or minimum values. These are the highest and lowest points of the function in a particular interval, respectively. Recall that relative maxima occur where the function changes from increasing to decreasing, while relative minima occur where the function changes from decreasing to increasing.

Key Concepts

Increasing and Decreasing Intervals

When you're analyzing the behavior of a function, it's crucial to identify where the function is increasing or decreasing. This tells you how the function moves across the xy-plane. An increasing interval means the function is rising as it moves from left to right. Conversely, during a decreasing interval, it falls.

To find these intervals, we often look at the graph of the derivative of the function. However, for visualization, you can directly examine the plotted graph. Simply see where the graph moves upwards or downwards.

• **Increasing Intervals:** The function's slope is positive here. The graph goes uphill as it moves right.
• **Decreasing Intervals:** The function’s slope is negative. The graph goes downhill as it moves right.

Understanding these intervals helps predict the behavior of real-world applications, like finding when an investment is growing or shrinking over time.
By carefully examining the graph, you can pick out increasing and decreasing trends even without precise calculations.

Relative Extrema

Relative extrema refer to the relative maximum and minimum points of a function. These points are crucial as they help identify the peaks and valleys in a graph—which correspond to optimal points a particular problem might be interested in.

These extrema occur at points where the function changes its direction:

• **Relative Maximum:** The peak of a particular interval where the function shifts from increasing to decreasing.
• **Relative Minimum:** The valley of a location where the function switches from decreasing to increasing.

To find these points, observe where the slope changes sign. On a graph, it means looking for places where peaks or troughs occur.
This concept is pivotal in optimization problems, helping determine the most or least favorable conditions in a scenario. Approximation using a graph is a great starting point when finding precise values analytically is not required.

Graphing Utilities

Graphing utilities are powerful tools that make complex graph analysis much easier. They allow you to visualize functions quickly and accurately. Using them can save time and provide insights that might be obscured by manual computations alone.

Here’s why graphing utilities are essential:

• **Ease of Visualization:** They offer clear visual illustrations of the function, making it simpler to identify important characteristics like increasing/decreasing intervals or relative extrema.
• **Interactivity:** Many graphing utilities let you zoom in, adjust parameters, or change the perspective of the graph, making subtle changes more noticeable.
• **Accuracy:** They provide more precise depictions of graphs than hand-drawn versions, reducing the risk of missed details.

Popular graphing tools range from graphing calculators, like the TI-84, to software applications such as Desmos and online resources.
These utilities serve not just in academic settings but are real-world assets, aiding professionals in fields like engineering and economics to analyze data trends effectively.