Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=\frac{5}{x} ; x=2$$

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = -\frac{5}{4}x + 5\).

Step 1: Find the Derivative

To find the equation of the tangent line, we first need to calculate the derivative of the function \(f(x)\). The function is \(f(x) = \frac{5}{x}\). Using the power rule for differentiation, we can rewrite \(f(x) = 5x^{-1}\) and differentiate it. The derivative, \(f'(x)\), is \(-5x^{-2}\) or \(-\frac{5}{x^2}\).

Step 2: Evaluate the Derivative at x=2

Next, we substitute \(x = 2\) into the derivative to find the slope of the tangent line at this point. Thus, we have \(f'(2) = -\frac{5}{2^2} = -\frac{5}{4}\). So, the slope at \(x = 2\) is \(-\frac{5}{4}\).

Step 3: Find the Point on the Curve

We need a point on the curve to write the equation of the tangent line. Substitute \(x = 2\) into the original function to find the \(y\)-value. This gives \(f(2) = \frac{5}{2} = 2.5\). The point on the curve at \(x = 2\) is \((2, 2.5)\).

Step 4: Write the Equation of the Tangent Line

Using the point-slope formula for a line \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. Substituting \((2, 2.5)\) and the slope \(-\frac{5}{4}\), we get: \[ y - 2.5 = -\frac{5}{4}(x - 2) \].Simplifying, the equation becomes:\[ y = -\frac{5}{4}x + 5 \].

Step 5: Verify with Graph

Graph the function \(f(x) = \frac{5}{x}\) and the tangent line \(y = -\frac{5}{4}x + 5\) on a graphing calculator. Ensure both intersect at the point \( (2, 2.5) \). Confirm the slope at this point matches the derivative calculated.

Key Concepts

Derivative: Understanding the Slope of a Function

The derivative of a function gives us the slope at any point on its curve. For the given function,

• the function is \( f(x) = \frac{5}{x} \), which can be rewritten as \( f(x) = 5x^{-1} \) using the power rule for easier differentiation.
• The derivative, noted as \( f'(x) \), calculates to \( -5x^{-2} \) or equivalently \( -\frac{5}{x^2} \).

This expression represents how steep the line is at any given \( x \) on the curve. At \( x = 2 \), the slope identified is \( -\frac{5}{4} \). This negative slope indicates the tangent line is decreasing at that point.
Understanding derivatives is crucial as they reveal the behavior and direction of functions, akin to understanding the angle of a hill as you climb it.

Point-Slope Form: Constructing Equation of the Tangent Line

Creating an equation for the tangent line involves using the point-slope form. This form is expressed as:\[ y - y_1 = m(x - x_1) \]where:

• \( m \) is the slope, derived as \( -\frac{5}{4} \)
• \( (x_1, y_1) \) is a known point on the line, here \( (2, 2.5) \)

Substitute these values:\[ y - 2.5 = -\frac{5}{4}(x - 2) \]After simplifying, this results in the linear equation:\[ y = -\frac{5}{4}x + 5 \]

This form is an excellent tool for swiftly finding the equation of a line when you have a point and its slope. It helps bridge the connection between geometric intuition and algebraic representation.

Function Evaluation: Calculating the Curve's Specific Point

To determine the specific point needed for the equation of the tangent line, you evaluate the original function at the specific \( x \)-value.

• For \( x = 2 \), the function calculation is \( f(2) = \frac{5}{2} = 2.5 \).

This gives you the point \( (2, 2.5) \) that lies on the original curve. Calculating this point is vital since it serves as the anchor for the tangent line on the graph. This shows how function evaluation directly supports understanding how a line precisely kisses a curve at that point.

Graph Verification: Confirming Tangency through Visual Inspection

Once the equation of the tangent line is established, it's important to verify it graphically.

• Use a graphing calculator to plot both the original function \( f(x) = \frac{5}{x} \) and the tangent line \( y = -\frac{5}{4}x + 5 \).
• Ensure both graphs intersect exactly at the point \( (2, 2.5) \), confirming the accuracy of calculations.

Moreover, the slope at this intersection in the graph should visually reflect the calculated derivative slope.This step in the process of verifying graphically is indispensable, as it provides a visual confirmation of mathematical computations. It reassures that not only is the tangent line equation correct, but also that the understanding of how the curve and line interact is precise.