Problem 80 - Precalculus with Limits

Question

Graphical Analysis Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. (a) \(f(x)=x^{2} e^{-x} \quad\) (b) \(g(x)=x 2^{3-x}\)

Step-by-Step Solution

Answer

For function \(f(x)=x^{2} e^{-x}\), it's generally decreasing in negative x values, increasing at around 0, and then again decreasing for positive x values. For \(g(x)=x 2^{3-x}\), it's decreasing until around x=1, increases between x=1 and x=3 (approx.), and then decreases for x more than 3.

Step 1: Graph Function

Graph the first function \(f(x)=x^{2} e^{-x}\) using a graphing utility. Analyze the graph to understand the function's behavior. The locations where the slope changes from positive to negative or vice versa can determine where the function is increasing or decreasing, these points are also potential local minima or maxima.

Step 2: Function Features - \(f(x)=x^{2} e^{-x}\)

Identify the intervals where the function is increasing and decreasing, and approximate any relative maximum and minimum points. Generally for this function, it is negatively sloped (decreasing) as x moves from negative infinity to 0, positively sloped at around 0 (increasing), and then back to negatively sloped (decreasing) for x greater than 0.

Step 3: Graph Function

Repeat the process for the second function \(g(x)=x 2^{3-x}\). Analyze the graph to determine the function's behavior. The locations where the slope changes from positive to negative or vice versa can indicate where the function is increasing and decreasing.

Step 4: Function Features - \(g(x)=x 2^{3-x}\)

Identify the intervals where the function increases and decreases, and approximate any relative maximum and minimum points by visually inspecting the graph. For the function \(g(x)=x 2^{3-x}\), it is negatively sloped (decreasing) as x moves from negative infinity to around 1 (approx.), positively sloped (increasing) for x values between 1 and 3 (approx.), and then decreasing for x greater than 3 (approx.)

Key Concepts

Increasing and Decreasing Intervals

Understanding increasing and decreasing intervals of a function is crucial for analyzing its behavior. This concept involves looking at the slopes of the function's graph.
When a function is increasing, the graph moves upwards as you move from left to right. Conversely, a function is decreasing when the graph moves downwards.

• A positive slope indicates that the function is increasing.
• A negative slope shows that the function is decreasing.

To find these intervals, we focus on critical points where the slope of the function changes. This occurs at points where the derivative is zero or undefined. In our examples, by using a graphing utility for functions like \(f(x)=x^{2} e^{-x}\) and \(g(x)=x 2^{3-x}\), we can visualize these intervals effectively.

Relative Maximum

A relative maximum is a point on the graph where the function reaches a peak within a certain neighborhood. This does not mean it's the highest point overall, but it's a local high point.
Identifying relative maxima helps in understanding the main peaks of the function which are important for many applications such as optimization.

• The function changes from increasing to decreasing at a relative maximum.
• Graphically, it is the top point of a hill in the function's curve.

For instance, when graphing \(f(x)=x^{2} e^{-x}\) or \(g(x)=x 2^{3-x}\), a graphing utility can help approximate these points visually, where the slope transitions from positive to negative.

Relative Minimum

Much like a relative maximum, a relative minimum is a point where the function dips to its lowest within a specific interval. It's a local low point and doesn't have to be the absolute lowest point of the entire function.
Understanding minima in a graph is essential, especially when analyzing patterns or trends.

• At a relative minimum, the function goes from decreasing to increasing.
• Graphically, it represents the bottom point of a valley.

When plotting \(f(x)=x^{2} e^{-x}\) and \(g(x)=x 2^{3-x}\), using a graphing utility can help identify these points where the slope goes from negative to positive.

Graphing Utility

A graphing utility is an invaluable tool for visualizing and analyzing functions. Whether you're using software, an app, or a calculator equipped with graphing capabilities, it helps to graph functions easily and efficiently.
Using a graphing utility, students can:

• Quickly generate graphs for complex functions.
• Identify increasing and decreasing intervals visually.
• Approximate positions of relative maxima and minima.

When dealing with functions with more intricate expressions, like \(f(x)=x^{2} e^{-x}\) and \(g(x)=x 2^{3-x}\), a graphing utility simplifies the otherwise challenging task of manually plotting these functions.

Function Graphing

Function graphing involves plotting a mathematical function on a coordinate plane to understand its behavior and characteristics visually.
By graphing, you can explore key features like intercepts, slopes, and curvatures that indicate where the function takes certain values or changes behavior.

• It provides a visual representation that is easier to understand than a list of values.
• Key features such as relative maxima and minima are readily apparent.
• Helpful in identifying patterns or anomalies in data.

In exercises where you work with functions like \(f(x)=x^{2} e^{-x}\) and \(g(x)=x 2^{3-x}\), graphing offers a vital, intuitive view of how changes in \(x\) affect \(f(x)\).