Problem 293 - Calculus Volume 1

Question

Use the given values to find \(\left(f^{-1}\right)^{\prime}(a)\). $$ f(1)=-3, f^{\prime}(1)=10, a=-3 $$

Step-by-Step Solution

Answer

\( (f^{-1})^{\prime}(-3) = \frac{1}{10} \).

Step 1: Identify the Need

To find the derivative of the inverse function at a given point, \( (f^{-1})^{ m\prime}(a) \), we will use the formula \[ (f^{-1})^{\prime}(a) = \frac{1}{f^{\prime}(b)} \] where \( f(b) = a \).

Step 2: Find Corresponding x-value

Since \( f(1) = -3 \), this implies that when \( x = 1 \), \( f(x) = -3 \). The value \( b = 1 \) is what we need for our calculation, because \( a = -3 \).

Step 3: Apply the Formula

Substitute \( b = 1 \) and \( f^{\prime}(1) = 10 \) into the formula: \[ (f^{-1})^{\prime}(-3) = \frac{1}{f^{\prime}(1)} = \frac{1}{10}\]

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change. It is the mathematical study of objects that change over time, such as motion and growth. Calculus is primarily divided into two branches: Differentiation and Integration. Differentiation focuses on analyzing rates of change, while Integration is about analyzing total quantities and areas.

• Differentiation helps us understand how a function changes at any given point.
• Integration provides the accumulation of quantities and the area under curves.

In the context of inverse function derivatives, calculus is essential for understanding how a function's slope behaves and how it inversely relates to its inverse function.

Derivative Calculation

Calculating the derivative involves finding how a function changes at a specific point. The derivative represents the slope of the function at that point, showing its rate of change.

The formula for calculating the derivative of a function's inverse, which is crucial in our exercise, is given by:

• \( (f^{-1})^{\prime}(a) = \frac{1}{f^{\prime}(b)} \)

This formula requires us to find the value of \( b \) such that \( f(b) = a \). Once we have this, we can easily substitute it to find the derivative of the inverse function.

To calculate the derivative, you should always:

• Identify the original function and its derivative.
• Find the corresponding x-value where the function's output matches the given y-value.
• Substitute these values into the formula for the inverse derivative.

Inverse Functions

Inverse functions reverse the input-output roles of the original function. In simple terms, if a function \( f(x) \) maps \( x \) to \( y \), then its inverse \( f^{-1}(y) \) maps \( y \) back to \( x \).

• An inverse function exists only if the original function is one-to-one, meaning each input maps to a unique output.
• Graphically, the inverse function is a reflection of the original function over the line \( y = x \).

For example, if \( f(1) = -3 \), then \( f^{-1}(-3) = 1 \), showing the reversal of roles. In the exercise, we used the inverse function to find the derivative at a specific value, which required careful application of the derivative rule for inverse functions.

Function Analysis

Function analysis is the detailed examination of functions and their behavior. This includes studying how functions perform across different domains and what their derivatives tell us about their nature.

In our example, function analysis involves:

• Identifying key points on the function, like where \( f(x) = a \).
• Understanding the behavior of the function around these points using derivatives.
• Applying concepts like the inverse derivative formula to determine how these key points affect the inverse function's behavior.

The exercise showed us how crucial it is to find where the function outputs a particular value, enabling us to understand its inverse dynamics and derive essential insights about the function itself.