Problem 32 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=\frac{x^{2}}{x^{2}+4} $$

Step-by-Step Solution

Answer

The critical number of the function f(x) = \(\frac{x^{2}}{x^{2}+4}\) is 0. The function decreases over the interval (-∞, 0) and increases over the interval (0, ∞).

Step 1: Differentiate the function

Differentiate function \(f(x) = \frac{x^{2}}{x^{2}+4}\) using the quotient rule. Using the quotient rule for derivatives \(\frac{d}{dx}\left[ \frac{u}{v} \right] = \frac{vu' - uv'}{v^2}\), where \(u = x^2\) and \(v = x^2 + 4\). The derivative \(f'(x)\) is then computed as \(f'(x) = \frac{2x(x^{2}+4) - x^2(2x)}{(x^{2}+4)^2}\). This simplifies to \(f'(x) = \frac{8x}{(x^{2}+4)^2}\).

Step 2: Find critical points

Critical numbers are those where the derivative is either zero or does not exist. Since our derivative function \(f'(x)\) does not have any values for which it is undefined, we will set \(f'(x)\) to zero to find the critical points. The equation \(f'(x) = 0\) leads to \(0 = \frac{8x}{(x^{2}+4)^2}\). The numerator must equal zero, so \(x = 0\) is the only critical number.

Step 3: Determine intervals of increase or decrease

Perform interval testing on the slope of \(f'(x)\). By checking numbers to the left and right of the critical point \(x = 0\), we find that the function is decreasing on the interval \(-(∞, 0)\), and increasing on the interval \( (0, ∞)\). This means the function \(f(x)\) has a minimum at the point \(x = 0\).

Step 4: Graph the function

Plotting the original function \(f(x) = \frac{x^{2}}{x^{2}+4}\) on a graphing calculator or software should show a decreasing slope until \(x = 0\), after which the slope starts to increase.

Key Concepts

Derivative

In calculus, the derivative of a function provides us with a powerful tool to understand how the function behaves at any given point. It's like having a magnifying glass to see how a function is changing. When you differentiate, you are finding the slope of the tangent line at any point of the function. This slope tells us whether the function is increasing or decreasing. For the function \( f(x) = \frac{x^2}{x^2 + 4} \), the derivative is computed using the quotient rule. This rule helps when you have a function that is one fraction over another, allowing you to differentiate it by using a set formula.

Quotient Rule

The quotient rule is essential when dealing with functions that are ratios. The rule states that if \( u \) and \( v \) are functions of \( x \), then the derivative of the quotient \( \frac{u}{v} \) is given by:

• \( \frac{d}{dx}\left[ \frac{u}{v} \right] = \frac{vu' - uv'}{v^2} \).

Using this, we find the derivative of the given function \( f(x) = \frac{x^2}{x^2 + 4} \) by assigning \( u = x^2 \) and \( v = x^2 + 4 \). Compute \( u' = 2x \) and \( v' = 2x \). Substituting into the formula gives:

• \( f'(x) = \frac{2x(x^2 + 4) - x^2(2x)}{(x^2 + 4)^2} \).

Simplifying, we find:

• \( f'(x) = \frac{8x}{(x^2 + 4)^2} \).

This calculation helps identify critical points and observe the behavior of the function.

Intervals of Increase and Decrease

Once we have the derivative, we can use it to determine intervals where the function is increasing or decreasing. These intervals tell us sections of the graph where the function rises or falls as \( x \) progresses.

• To find where a function increases, find where the derivative is positive.
• To find where it decreases, observe where the derivative is negative.

For \( f'(x) = \frac{8x}{(x^2 + 4)^2} \):

• The function decreases on \((−∞, 0)\) because the derivative is negative there.
• The function increases on \((0, ∞)\) since the derivative is positive.

This provides insight into where the function achieves minimum or maximum values.

Graphing Functions

Graphing functions is a critical skill for visualizing the behavior of a function. It allows you to see trends like where the function increases or decreases, and identify points of interest such as minima and maxima. To graph the function \( f(x) = \frac{x^2}{x^2 + 4} \), you can use graphing software or a calculator.

• Start by plotting a few points, especially near the critical point \( x = 0 \).
• Observe that the graph descends as \( x \) approaches zero from the left, and ascends for values greater than zero.
• This confirms the intervals of increase and decrease found earlier.

Using these visual tools helps reinforce your understanding of how the algebraic work translates into the real world.