Problem 14 - Precalculus with Limits

Question

In Exercises 9-16, use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result. \(g(x) = \dfrac{1}{x-2}, \quad (4, \dfrac{1}{2})\)

Step-by-Step Solution

Answer

The slope of the function \( g(x) = \dfrac{1}{x-2} \) at the point (4, 1/2) is -1/4.

Step 1: Definition of the Difference Quotient

The slope formula using limit is given by \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}. Therefore, substitute \( g(x) = \dfrac{1}{x-2} \) into the slope formula, obtaining \[ \lim_{h \to 0} \frac{g(4 + h) - g(4)}{h} \].

Step 2: Substituting the function

The next step is to substitute \( g(x) \) into the limit: \[ \lim_{h \to 0} \dfrac{\frac{1}{4 + h - 2} - \frac{1}{4-2}}{h} \].

Step 3: Simplifying the expression

Simplify the function as much as possible. The equation now becomes: \[ \lim_{h \to 0} \dfrac{1/2 - 1/(2 + h)}{h} \].

Step 4: Further simplification and computation of the limit

We can simplify this by finding a common denominator, which gives us \[ \lim_{h \to 0} \dfrac{2 + h - 2}{h(2 + h)} \], which simplifies to \[ \lim_{h \to 0} \dfrac{h}{h(2 + h)} \]. Further simplification and computation of the limit yields a result of -1/4.

Key Concepts

Slope of a Graph

The slope of a graph at a specific point gives us the rate at which the function changes around that point. It tells us how steep the line is that tangentially touches the graph at the given point. This is crucial in understanding the behavior of functions in calculus.

In this particular exercise, finding the slope involves using the limit process. We analyze the function \( g(x) = \frac{1}{x-2} \) around the point \( (4, \frac{1}{2}) \). The slope of the graph at this point is obtained by computing the derivative. This derivative is the limit of the difference quotient as \( h \) approaches 0.

The slope informs us not just about the direction and steepness of the curve but also offers insights into how the output of the function is expected to change based on small changes to its input. In this specific example, after simplification, the slope calculated at point \( (4, \frac{1}{2}) \) is \(-\frac{1}{4}\). This means that for every unit you move right, the function value decreases by \( \frac{1}{4} \).

Difference Quotient

The difference quotient is a crucial step in finding the derivative of a function. It's essentially a way of representing the rate of change of a function. You compute the difference in the function values over a small interval and divide by the width of that interval.

For a function \( f(x) \), the difference quotient is given as \( \frac{f(x+h) - f(x)}{h} \). It measures the average rate of change across an interval \( [x, x+h] \). As \( h \) approaches 0, this quotient increasingly represents an instantaneous rate of change, or the derivative.

In our exercise, substituting \( g(x) = \frac{1}{x-2} \) into the difference quotient formula helps find out how \( g(x) \) changes near the specified point. By simplifying, as \( h \to 0 \), we determine how sharply the function rises or falls at that point, giving us the slope. This powerful tool in calculus helps inform various practical applications, like predicting trends and behaviors of real-world phenomena based on mathematical models.

Graphing Utility

Graphing utilities, like graphing calculators or software, offer a visual means to confirm and understand the mathematical findings of derivative calculations. After finding the slope analytically via the limit process, it’s helpful to use a graphing utility to verify that the point of the graph reflects the calculated slope.

Visualizing functions on a graph provides intuitive insight into the relationships between variables. In our exercise, a graphing utility can be used to plot the function \( g(x) = \frac{1}{x-2} \) and then to approach the point \( (4, \frac{1}{2}) \). This action can confirm that the slope at this point matches the calculated \(-\frac{1}{4}\).

These tools are especially useful in educational settings where exploring mathematical concepts visually can make abstract ideas more tangible. They aid students in building a more comprehensive understanding of how functions behave and how derivatives relate to graphical slopes.