Problem 19 - Precalculus with Limits

Question

In Exercises 17-22, find a formula for the slope of the graph of \(f\) at the point \((x, f(x))\). Then use it to find the slope at the two given points. \(f(x) = \dfrac{1}{x+4}\) (a) \((0, \frac{1}{4})\) (b) \((-2, \frac{1}{2})\)

Step-by-Step Solution

Answer

The slopes at the points \((0, \frac{1}{4})\) and \((-2, \frac{1}{2})\) are -1/16 and -1/4, respectively.

Step 1: Find the Derivative of Function \(f(x)\)

The function \(f(x) = \dfrac{1}{x+4}\) is a rational function. It can be written as \(f(x) = (x+4)^{-1}\). The derivative of this function can be found using the power rule for derivatives, which states: if \(f(x) = x^n\), then \(f'(x) = n \cdot x^{n-1}\). \n\nFollowing this rule, the derivative of \(f(x)\) is: \n\n\f'(x) = -1 \cdot (x+4)^{-2}\). This can be rewritten as \(- \dfrac{1} {(x + 4)^2}\)

Step 2: Find the Slope at the Point (0, 1/4)

To find the slope at the given point \((0, \frac{1}{4})\), substitute \(x = 0\) into the derivative \(- \dfrac{1}{(x+4)^2}\). \n\nSo, at \((0, \frac{1}{4})\), \(f'(x) = -\frac{1}{16}\)

Step 3: Find the Slope at the Point (-2, 1/2)

Similarly, to find the slope at the given point \((-2, \frac{1}{2})\), substitute \(x = -2\) into the derivative \(- \dfrac{1}{(x+4)^2}\).\n\nSo, at \((-2, \frac{1}{2})\), \(f'(x) = -\frac{1}{4}\)

Key Concepts

Power Rule in Derivatives

The power rule is an essential tool in calculus for finding derivatives quickly and effectively. It simplifies the process of differentiation, which involves finding the rate at which a function changes. The power rule states:

• If a function is expressed in the form \(f(x) = x^n\), its derivative is \(f'(x) = n \cdot x^{n-1}\).

This rule is extremely useful when dealing with polynomial functions where each term can be differentiated separately. For instance, to differentiate \(f(x) = (x+4)^{-1}\), you treat \((x+4)\) as a single variable raised to the power of \(-1\). Apply the rule by bringing down the exponent as a coefficient and subtracting one from the exponent. This yields:

• Derivative: \(f'(x) = -1 \cdot (x + 4)^{-2}\)

Breaking these steps down using the power rule makes the differentiation process smoother and less complex.

Understanding Rational Functions

In mathematics, a rational function is a type of function where the formula involves a ratio of two polynomials. The function \(f(x) = \frac{1}{x+4}\) is expressed as a rational function because it can be rewritten in the form \(f(x) = (x+4)^{-1}\). The denominator \((x+4)\) indicates that there is a restriction placed on the function, as division by zero leads to undefined values.
The domain of rational functions excludes any value that makes the denominator zero, meaning for \(f(x) = \frac{1}{x+4}\), \(x = -4\) is not included in its domain. Rational functions often pose unique challenges when finding derivatives or performing other operations, but using strategies like rewriting the function in a more manageable form helps simplify these tasks. Function behavior and limits are particularly important in understanding how these functions operate over different intervals.

Interpreting the Slope of a Tangent

The slope of a tangent line to a curve at a given point is significant as it provides information about the function's rate of change at that exact spot. When we say we are finding the slope of the tangent, we are essentially identifying the derivative value at a specific point. For example, with the function \(f(x) = \frac{1}{x+4}\), its derivative was found as \(f'(x) = -\frac{1}{(x+4)^2}\).

• At point \(x = 0\), slope = \(-\frac{1}{16}\)
• At point \(x = -2\), slope = \(-\frac{1}{4}\)

Thus, this tells us how steep the graph is at those points. A negative slope indicates the function is decreasing, meaning the curve moves downward as you move right, which is noticeable at both points in this exercise. Understanding the slope is crucial in analyzing the behavior and tendency of rational functions near specific points.

Basics of Differentiation

Differentiation is the process used in calculus to compute a derivative. A derivative provides the rate at which a function's value changes as its input changes. Differentiation is fundamental for understanding motion, growth, decay, and other changes in various contexts. For any function \(f(x)\), its derivative \(f'(x)\) is found through differentiation.
Basic techniques for differentiation include:

• Using derivative rules like the power rule, product rule, quotient rule, and chain rule.
• Identifying and applying these rules based on the function form.

In this exercise, the power rule was applied as part of the differentiation technique for the function \(f(x) = \frac{1}{x+4}\) by rewriting it as \((x+4)^{-1}\). This approach allows easy handling of more complex functions by breaking them into manageable parts, thereby providing clear insights into how changes in input affect the output.