Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=-\frac{2}{x} ; x=4$$

Step-by-Step Solution

Answer

The slope of the tangent line at \( x = 4 \) is \( \frac{1}{8} \).

Step 1: Understand the problem

We need to find the slope of the tangent line to the curve defined by the function \( f(x) = -\frac{2}{x} \) at the point where \( x = 4 \). The slope of the tangent line is given by the derivative of the function evaluated at this specific point.

Step 2: Differentiate the function

Find the derivative of \( f(x) = -\frac{2}{x} \). To do this, we rewrite the function as \( f(x) = -2x^{-1} \). Use the power rule for differentiation, which is \( \frac{d}{dx} (x^n) = nx^{n-1} \). Therefore, the derivative \( f'(x) \) is calculated as follows:\[ f'(x) = -2(-1)x^{-1-1} = 2x^{-2} \].Thus, \( f'(x) = \frac{2}{x^2} \).

Step 3: Evaluate the derivative at the given point

Now that we have \( f'(x) = \frac{2}{x^2} \), substitute \( x = 4 \) into this expression to find the slope of the tangent line at this point:\[ f'(4) = \frac{2}{(4)^2} = \frac{2}{16} = \frac{1}{8} \].

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that represents the rate at which a function is changing at any given point. It is like a real-world measurement of change. Imagine you are driving a car, the derivative would be analogous to the speedometer which tells you how fast you are changing your position. When you differentiate a function, you find a new function known as the derivative. This derivative function gives you the slope of the tangent line at any point on the original function's curve. It is crucial in understanding how the function behaves. To compute the derivative, differentiation rules such as the power rule, chain rule, and product rule are applied. In this exercise, we applied the power rule to derive the derivative of the function, making it easier to predict the slope of the curve at given points.

Slope of tangent line

The slope of a tangent line at a particular point on a curve gives information about the curve's direction and steepness at that point. It helps us understand how the function behaves in a small interval around that point.A tangent line is a straight line that touches a curve at a single point without crossing it. This line represents the instantaneous rate of change of the function at that specific point. In this exercise, you were asked to find the slope of the tangent line for the function at a specific value of \(x\). By evaluating the derivative at \(x = 4\), the slope was determined to be \(\frac{1}{8}\). This means that at the point \(x = 4\), the line that best approximates the function has a gentle upwards slope.

Power rule

The power rule is one of the most straightforward and commonly used rules in differential calculus. It deals with finding the derivative of polynomials. The rule can be stated as follows:

• If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

It's vital to notice that this rule applies when you have a power of \(x\) with any real number \(n\), including negative integers, fractions, etc.In the exercise, the original function \(f(x) = -\frac{2}{x}\) was rewritten as \(f(x) = -2x^{-1}\) to easily apply the power rule. By doing this, finding the derivative became straightforward, allowing us to identify how the function changes as \(x\) changes, ultimately leading to the computation of the slope of the tangent line.