Problem 278 - Calculus Volume 1

Question

For each of the given functions \(y=f(x),\) a. find the slope of the tangent line to its inverse function \(f^{-1}\) at the indicated point \(P,\) and b. find the equation of the tangent line to the graph of \(f^{-1}\) at the indicated point. $$f(x)=x^{5}+3 x^{3}-4 x-8, P(-8,1)$$

Step-by-Step Solution

Answer

Slope of tangent: \(\frac{1}{10}\). Equation: \(y = \frac{1}{10}x + \frac{9}{5}\).

Step 1: Verify Function and Inverse Points

The function is given as \( f(x) = x^5 + 3x^3 - 4x - 8 \). We need to find the point \( Q \) on the graph of the function such that \( Q = (x_0, -8) \) because \( P = (-8, 1) \) is on \( f^{-1} \). Thus, compute \( f(x_0) = -8 \). Testing \( x = 1 \), we find \( f(1) = 1^5 + 3*1^3 - 4*1 - 8 = -8 \), confirming that \( 1 \) is a solution. Therefore, \( Q = (1, -8) \).

Step 2: Calculate Derivative of the Original Function

Find the derivative of the function \( f(x) \) to use in the formula for the inverse's slope. The derivative is \( f'(x) = 5x^4 + 9x^2 - 4 \).

Step 3: Evaluate the Derivative at the Corresponding x-value

Evaluate \( f'(1) \) to find the slope of the original function at \( x = 1 \). So, \( f'(1) = 5*1^4 + 9*1^2 - 4 = 10 \).

Step 4: Find Slope of the Tangent to the Inverse

The slope of the tangent line to the inverse function at \( P \) is the reciprocal of the slope of the tangent to \( f(x) \). Thus, the slope at \( P(-8, 1) \) is \( \frac{1}{f'(1)} = \frac{1}{10} \).

Step 5: Equation of the Tangent Line

To find the equation of the tangent line at \( P(-8, 1) \), use the point-slope formula: \( y - y_1 = m(x - x_1) \), where \( m = \frac{1}{10} \), \( x_1 = -8 \), and \( y_1 = 1 \). So, the equation is \( y - 1 = \frac{1}{10}(x + 8) \).

Key Concepts

Inverse Functions

An inverse function essentially "reverses" the original function. If a function \( f \) pairs input \( x \) with output \( y \), then its inverse \( f^{-1} \) pairs \( y \) back with \( x \). In simpler terms, inverse functions allow us to "undo" a function.
For example, with the function \( f(x) = x^5 + 3x^3 - 4x - 8 \), evaluating \( f(1) \) gives \(-8\). Its inverse function \( f^{-1} \) will reverse this, mapping \(-8\) back to \( 1 \).
When dealing with inverse functions, verifying points is key to ensure they lie on the graph of the inverse. In our example, point \( (1, -8) \) on \( f \) corresponds with \( (-8, 1) \) on \( f^{-1} \), confirming their relationship.
Remember, not all functions have inverses. A function must be bijective (both injective and surjective) to possess an inverse. In some cases, you might need to restrict the domain to make it invertible.

Derivative

The derivative is a fundamental concept in calculus. It measures how one quantity changes with respect to another. In the context of a graph, the derivative at a point represents the slope of the tangent line at that point.
For the function \( f(x) = x^5 + 3x^3 - 4x - 8 \), we calculated the derivative as \( f'(x) = 5x^4 + 9x^2 - 4 \). This derivative tells us how the function changes at any specific \( x \)-value.
Calculating the derivative involves using rules such as the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \). Applying this rule correctly is necessary for finding accurate slopes of tangent lines.
In our exercise, adjusting the derivative to \( f'(1) = 10 \) showed us the slope of the function \( f(x) \) at \( x = 1 \), crucial for finding the slope of the corresponding tangent to the inverse.

Point-Slope Formula

When finding the equation of a tangent line, one incredibly useful tool is the point-slope formula: \( y - y_1 = m(x - x_1) \). This equation is perfect when you know:

• An exact point \((x_1, y_1)\) where the tangent line touches the curve
• The slope \( m \) of the tangent line at that point

For the inverse function \( f^{-1} \), with the given point \( P(-8, 1) \) and slope \( m = \frac{1}{10} \), we use the point-slope formula to write the equation for the tangent line as \( y - 1 = \frac{1}{10}(x + 8) \).
This formula not only simplifies finding the tangent line equation but also helps visualize how the curve behaves.
It's a straightforward method that applies widely across different functions, aiding in the study of curves and their approximations.