Problem 75 - Calculus of a Single Variable

Question

Analyzing a Graph Using Technology In Exercises \(75-82,\) use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=9-\frac{5}{x^{2}} $$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 9 - \frac{5}{x^2}\) forms a hyperbola shifted upwards by 9 units along the y-axis. Based on the plot, there are no extrema. A horizontal asymptote exists at \(y = 9\) and vertical asymptotes at \(x = 0\).

Step 1: Understanding the function

Look at the given function \(f(x) = 9 - \frac{5}{x^2}\). The function seems to be a hyperbola shifted upwards along the y-axis by 9 units.

Step 2: Plotting the function

Plot the function using a computer algebra system. The x and y-axis should be clearly labelled. Note the shape and position of the graph.

Step 3: Identifying Extrema

Based on the graph, the extrema can be identified. The extrema of a function are the points at which the function reaches its maximum or minimum values.

Step 4: Looking for Asymptotes

Asymptotes are lines that the function approaches but never reaches. For the function \(f(x) = 9 - \frac{5}{x^2}\), there appears to be a horizontal asymptote at \(y = 9\) and vertical asymptotes at \(x = 0\). Double check on your computer algebra system to be certain.

Key Concepts

Computer Algebra System

Using a computer algebra system (CAS) can greatly simplify the process of analyzing complex functions. A CAS is a software tool that allows you to perform a wide range of mathematical calculations and visualizations. Generally, it can handle symbolic mathematics, such as solving equations, performing algebraic operations, and creating precise graphs.

To analyze the given function, you can input it into a CAS to graph it accurately. This visual approach helps in identifying key features of the graph, such as intercepts, asymptotes, and extrema. Commonly used CAS include software like Wolfram Alpha, Maple, and TI-Nspire.

• You can obtain a clear graph quickly without manual plotting.
• Labels for the axes and units can be automatically generated.
• It's easier to explore different aspects of the function by adjusting its parameters or viewpoint.

Using a CAS can be a powerful ally in understanding mathematics thoroughly and deeply.

Function Extrema

Extrema refer to the points on a graph where the function reaches either a maximum or minimum value. Determining extrema is crucial for understanding the behavior of a function.

To find the extrema, whether a function has them, and where they lie, you can analyze the derivative of the function. The extrema occur when the derivative is zero or undefined.
For the function \(f(x) = 9 - \frac{5}{x^2}\), observe that it has no global maximum or minimum, but it trends towards a particular value, indicating a local characteristic.

• Plots can give a clear visual of peaks and troughs.
• Using a CAS can help to find these exact points by calculating or setting the derivative to zero.

This understanding is key to analyzing various real-world problems where such points represent critical values or states.

Asymptotes

In calculus, an asymptote is a line that a graph approaches but never actually touches or crosses. Asymptotes can be horizontal, vertical, or even oblique, and they provide significant insights into the behavior of a function.

In the function \(f(x) = 9 - \frac{5}{x^2}\), we find:

• Horizontal asymptote at \(y = 9\).
• Vertical asymptote at \(x = 0\).

These asymptotes show the limits of the function as \(x\) approaches infinity or zero. By using a computer algebra system, we can easily visualize these lines on the graph. This makes the function's behavior at extreme values more intuitive.
Asymptotes help in understanding the extent and limit of applied problems, such as predicting boundaries in physical or economic models.

Hyperbola

A hyperbola is a type of curve on a graph that takes on the form of an inverted U. It is one of the conic sections formed by intersecting a plane with a double cone. The function \(f(x) = 9 - \frac{5}{x^2}\) resembles a hyperbola since it includes a division by the square of \(x\).

Unlike circles and ellipses, hyperbolas have asymptotes, which they approach but never meet. They are characterized by two separate curves, each situated on opposite sides of the graph.

• The curve opens vertically or horizontally depending on the signs and squares present in the equation.
• Hyperbolas can be shifted or repositioned along the axis, as seen in this exercise where it is shifted up by 9 units.

Understanding hyperbolas is crucial in various applied sciences, such as physics and engineering, where they appear in the analysis of light paths, optimization problems, and more.