Problem 54 - Precalculus with Limits

Question

In Exercises 51-54, use a graphing utility to graph \(f\) over the interval \([-2, 2]\) and complete the table. Compare the value of the first derivative with a visual approximation of the slope of the graph. $$ f(x) = \dfrac{x^2 - 4}{x+4} $$

Step-by-Step Solution

Answer

The first derivative \(f'(x) = \frac{x^2 + 8x + 4}{(x+4)^2}\) represents the slope of the function at a specific point. By graphing the function and its derivative and comparing the slopes, we can observe how the derivative function accurately depicts the slopes at each point.

Step 1: Function Analysis

At first, consider the function \(f(x) = \frac{x^2 - 4}{x+4}\).

Step 2: Find the First Derivative

The derivative of the function can be found by using the rules of differentiation. Here, we use the quotient rule and simplify: \(f'(x) = \frac{(x+4)*2x - (x^2-4)*1}{(x+4)^2} = \frac{2x^2+8x - x^2 +4}{(x+4)^2} = \frac{x^2 + 8x + 4}{(x+4)^2}\).

Step 3: Graphing the function

Then, Graph the function and its derivative over the interval [-2, 2]. Note that the value of the first derivative at a certain point x corresponds to the slope of the function at the same point.

Step 4: Completing the Table and Comparing

Complete the table with the function values, slopes from the first derivative and the visual approximation of the slopes from the graph for various x-values between -2 and 2. Then, compare the values from the derivative function (exact slopes) and the visual approximation ones.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps us visualize functions and their derivatives. These utilities can include calculators or software programs that plot graphs over a specified interval. In this exercise, we are tasked with graphing the function \( f(x) = \frac{x^2 - 4}{x+4} \) over the interval \([-2, 2]\).Using a graphing utility involves entering the function equation and setting the interval for which you want to see the graph. This allows you to see how the function behaves across the chosen range of x-values.
Graphing utilities are particularly useful for:

• Visualizing the overall shape of a function.
• Identifying key characteristics like peaks, troughs, and inflection points.
• Comparing the function and its derivative visually.

By graphing both the original function and its derivative, you can easily compare visual estimates of the slope to the exact mathematical derivatives at various points.

Quotient Rule

The quotient rule is a fundamental tool in calculus used to differentiate functions that are expressed as a division of two other functions. When dealing with a function like \(f(x) = \frac{x^2 - 4}{x+4}\), we apply the quotient rule to find its derivative. The quotient rule states:For two differentiable functions \(u(x)\) and \(v(x)\), if \(f(x) = \frac{u(x)}{v(x)}\), then its derivative \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}.\)In our example:

• \(u(x) = x^2 - 4\), and \(v(x) = x + 4\).
• The derivative \(u'(x) = 2x\).
• The derivative \(v'(x) = 1\).

Applying the quotient rule gives you:\(f'(x) = \frac{(2x)(x+4) - (x^2-4)(1)}{(x+4)^2}\). After simplifying, we get:\(f'(x) = \frac{x^2 + 8x + 4}{(x + 4)^2}\).This result represents the slope of the tangent line to the graph of the function at any point \(x\).

Slope

The slope of a function at a particular point indicates the steepness of the tangent to the graph at that point. It is a measure of how fast the function is changing at that point. For the function \(f(x) = \frac{x^2 - 4}{x+4}\), the slope at any point \(x\) is given by the derivative \(f'(x)\).
Understanding slope is crucial because:

• The slope provides insight into the behavior of a function - whether it's increasing, decreasing, or constant.
• A positive slope implies the function is increasing, while a negative slope implies it's decreasing.
• When the slope is zero, the function may have a local maximum, minimum, or a point of inflection.

During the exercise, you compare the analytically computed slope from \(f'(x)\) with the visual estimation from the graph. Observing the graph confirms whether the calculated slopes align with the visual impressions at various points along the interval \([-2, 2]\).

Function Analysis

Function analysis involves exploring various properties and characteristics of a function to understand how it behaves over a certain range. For \(f(x) = \frac{x^2 - 4}{x+4}\), it's essential to perform detailed analysis before interpreting or graphing.Some critical steps in function analysis include:

• Determining the domain of the function. For instance, \(f(x)\) is undefined where \(x+4 = 0\), so \(x eq -4\).
• Identifying intercepts, which are points where the graph crosses the axes.
• Finding horizontal or vertical asymptotes, indicating how the function behaves as \(x\) approaches specific values.
• Evaluating the derivative \(f'(x)\) to find slopes and critical points where the function's behavior changes.

Through function analysis, you gain a comprehensive understanding, enabling a deeper comparison between the visual graph and calculated details from derivatives. This process not only helps in understanding the precise behavior of the function over the interval \([-2, 2]\), but also highlights key attributes that inform about its global structure.