Problem 77 - Essential Calculus

Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x}{x^{2}-4} $$

Step-by-Step Solution

Answer

After performing the steps, you should be able to identify that there are no extrema, vertical asymptotes are at x = -2 and x = 2, and the horizontal asymptote is at y = 0.

Step 1: Differentiate the function

Find the derivative of \(f(x) = \frac{x}{x^{2}-4}\) using the quotient rule. Set the derivative equal to zero and solve for x to find the stationary points, which will help identify the extrema.

Step 2: Find the critical points and analyze them

Plug these x-values from step 1 into the original function to get their corresponding y-values. Check the points to the left and right of these stationary points in the derivative to find out whether these points are maxima or minima.

Step 3: Find the asymptotes

Vertical asymptotes can be found by setting the denominator of the function equal to zero and solving for x. Horizontal asymptotes can be found by finding the limit of the function as x tends to plus and minus infinity.

Key Concepts

Quotient Rule

When analyzing graphs, especially those involving rational functions, understanding the quotient rule is essential. The quotient rule is a technique for finding the derivative of a function that is the quotient of two differentiable functions. The general form of the quotient rule states that if you have a function \( h(x) = \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable, then the derivative \( h'(x) \) is given by:

• \( h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \)

In our exercise, where \( f(x) = \frac{x}{x^2 - 4} \), \( u(x) = x \) and \( v(x) = x^2 - 4 \). First, differentiate \( u(x) \) to obtain \( u'(x) = 1 \), and \( v(x) \) to obtain \( v'(x) = 2x \). Then substitute these into the quotient rule formula to find \( f'(x) \).
Using the quotient rule helps us determine where the slope of the function is zero, which will aid in identifying critical points that are necessary for understanding the behavior and shape of the graph.

Critical Points

Critical points play a significant role in understanding the behavior of a function's graph. To find critical points, one must solve for where the derivative of the function equals zero or is undefined. These points often indicate where the function changes direction, or where it could have local maxima or minima.
In the context of the given exercise, after applying the quotient rule, set the derivative equal to zero to find any stationary points:

• Find where \( f'(x) = 0 \) to get potential extrema.
• Check where \( f'(x) \) is undefined, which may also indicate critical points.

Next, analyze these points by substituting back into the function \( f(x) \) to calculate their value and determine the nature (maximum, minimum, or saddle point) by evaluating the sign of \( f'(x) \) around these points. This process gives valuable insight into the peaks and valleys of the graph, which is crucial for comprehensive graph analysis.

Asymptotes

Asymptotes provide crucial information about the behavior of a function as it tends toward extreme values. For rational functions like \( f(x) = \frac{x}{x^2 - 4} \), there are typically two types of asymptotes to consider: vertical and horizontal.

• Vertical Asymptotes: These occur where the denominator of the rational function is zero, and the function is undefined. In this function, setting \( x^2 - 4 = 0 \) and solving for \( x \) identifies the vertical asymptotes at \( x = 2 \) and \( x = -2 \).
• Horizontal Asymptotes: These are determined by examining the end behavior of the function as \( x \) approaches infinity. For this function, since the degree of the polynomial in the denominator is higher than that of the numerator, the horizontal asymptote is at \( y = 0 \), as the value of \( f(x) \) approaches zero.

Understanding where and how asymptotes affect the function helps greatly in sketching the graph accurately and predicting its behavior in regions not immediately evident from just basic plotting.