Problem 294 - Calculus Volume 1

Question

Use the given values to find \(\left(f^{-1}\right)^{\prime}(a)\). $$ f(1)=0, f^{\prime}(1)=-2, a=0 $$

Step-by-Step Solution

Answer

\((f^{-1})'(0) = -\frac{1}{2}\).

Step 1: Understanding the Problem

We need to find the derivative of the inverse function, \(f^{-1}(x)\), evaluated at \(x = a = 0\). The given values are \(f(1)=0\), \(f'(1) = -2\), and \(a=0\).

Step 2: Using the Inverse Function Derivative Formula

The formula to find the derivative of the inverse function is \((f^{-1})'(a) = \frac{1}{f'(b)}\)\,, where \(f(b) = a\). Since \(f(1) = 0\), we have \(b = 1\).

Step 3: Finding the Derivative of the Inverse

Substitute \(b=1\) and \(f'(1) = -2\) into the formula: \((f^{-1})'(a) = \frac{1}{f'(1)} = \frac{1}{-2}\).

Step 4: Calculating the Result

Simplify the expression to find that \((f^{-1})'(0) = -\frac{1}{2}\).

Key Concepts

Derivative of Inverse Functions

When working with inverse functions in calculus, understanding how to find their derivatives is essential. To do this, we utilize a special formula for the derivative of an inverse function. The formula is expressed as:

• \((f^{-1})'(a) = \frac{1}{f'(b)}\)

where \(f(b) = a\). This formula helps us determine the behavior of the inverse function at a specific point.

To use this formula effectively, it's crucial to identify both \(b\) and \(f'(b)\). Once you know \(b\), the formula allows you to calculate \((f^{-1})'(a)\) by taking the reciprocal of the derivative of the original function evaluated at \(b\).

This method elegantly connects the derivative of a function with its inverse, showcasing the symmetry inherent in inverse relationships. Each value in the original function has a corresponding value in the inverse, and this relationship is mirrored in their derivatives as well.

Inverse Functions

Inverse functions are an important component of calculus, helping us reverse or 'undo' a function's operation. An inverse function, denoted as \(f^{-1}(x)\), pairs each output of the original function with its corresponding input.

To determine if a function has an inverse, it must be one-to-one. This means each input produces a unique output. Elements such as horizontal line tests can visually confirm this property. If a function passes the horizontal line test, it is invertible.

Working with inverse functions requires careful attention to how values flip between inputs and outputs. For example, if \(f(1) = 0\), then \(f^{-1}(0) = 1\). This switch in roles is foundational in solving calculus problems involving inverse functions, particularly when determining derivatives.

Calculus Problems Involving Inverse Functions

Solving calculus problems with inverse functions often involves calculating their derivatives, especially when given specific values like in our exercise.

To successfully solve these problems:

• Identify the necessary values for \(a\) and \(b\) from the given function details.
• Use the inverse function derivative formula \((f^{-1})'(a) = \frac{1}{f'(b)}\).
• Ensure you have the correct derivative \(f'(b)\) of the original function.

In our example, the task was to find \((f^{-1})'(0)\) using \(f(1) = 0\) and \(f'(1) = -2\). By substituting the values into the formula, we found \((f^{-1})'(0) = -\frac{1}{2}\).

These steps highlight how applying known formulas and details can break down complex calculus problems into manageable pieces, offering insights into the structure and behavior of inverse functions.