Problem 266 - Calculus Volume 1

Question

Use the functions \(y=f(x)\) to find a. \(\frac{d f}{d x}\) at \(x=a\) and b. \(\quad x=f^{-1}(y)\) c. Then use part b. to find \(\frac{d f^{-1}}{d y}\) at \(y=f(a)\). $$ f(x)=9-x^{2}, 0 \leq x \leq 3, x=2 $$

Step-by-Step Solution

Answer

a. \(-4\), b. \(x=\sqrt{9-y}\), c. \(-\frac{1}{4}\).

Step 1: Differentiate the function

To find \( \frac{d f}{d x} \) at \( x = a \), we first need to differentiate the function \( f(x) = 9 - x^2 \) with respect to \( x \). The derivative is: \[ \frac{d f}{d x} = -2x \]

Step 2: Evaluate the derivative at x = a

Now, substitute \( x = 2 \) into the derivative \[ \frac{d f}{d x} = -2(2) = -4 \] So, \( \frac{d f}{d x} \) at \( x = 2 \) is \( -4 \).

Step 3: Solve for x in terms of y using the inverse function

Given that \( f(x) = 9 - x^2 \), we want to express \( x \) in terms of \( y \). Solve the equation for \( x \): \[ y = 9 - x^2 \] Rearranging gives \[ x^2 = 9 - y \] \[ x = \sqrt{9 - y} \] Thus, \( f^{-1}(y) = \sqrt{9 - y} \).

Step 4: Differentiate the inverse function

To find \( \frac{d f^{-1}}{d y} \), differentiate \( f^{-1}(y) = \sqrt{9 - y} \) with respect to \( y \): \[ \frac{d f^{-1}}{d y} = \frac{d}{dy}(9 - y)^{1/2} = \frac{1}{2}(9 - y)^{-1/2} \cdot (-1) = -\frac{1}{2\sqrt{9 - y}} \]

Step 5: Evaluate the derivative of the inverse at y = f(a)

We need to find \( \frac{d f^{-1}}{d y} \) at \( y = f(2) \). First, calculate \( f(2) \): \[ f(2) = 9 - 2^2 = 5 \] Now substitute \( y = 5 \) into the derivative of the inverse: \[ \frac{d f^{-1}}{d y} = -\frac{1}{2\sqrt{9 - 5}} = -\frac{1}{2\sqrt{4}} = -\frac{1}{4} \]

Key Concepts

Inverse Functions

Inverse functions are essential tools in mathematics, allowing us to reverse a process or function. When you have a function like \( f(x) = 9 - x^2 \), its inverse function, \( f^{-1}(y) \), will work in the opposite direction.An inverse function essentially swaps the roles of input and output:

• For the function \( f \), if \( f(x) = y \), then for \( f^{-1} \), \( f^{-1}(y) = x \).
• The original function and its inverse "undo" each other.

To find an inverse function:1. Replace the function notation \( f(x) \) with \( y \).2. Swap \( x \) and \( y \). 3. Solve the resulting equation for \( x \). In our example, starting with the equation \( y = 9 - x^2 \), we solve for \( x \) in terms of \( y \): \( x = \sqrt{9 - y} \). This gives us the inverse function \( f^{-1}(y) = \sqrt{9 - y} \). Understanding inverse functions is crucial in algebra and calculus as they provide a way to switch perspectives on a problem.

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents the rate of change of a function with respect to one of its variables.For a given function like \( f(x) = 9 - x^2 \), we can find its derivative, \( \frac{d f}{d x} \):

• The derivative tells us how the function value changes as \( x \) changes.
• For the function \( f(x) = 9 - x^2 \), differentiating gives us \( \frac{d f}{d x} = -2x \).

Evaluating this derivative at a specific point, like \( x = 2 \), gives \( \frac{d f}{d x} = -4 \). This result means at \( x=2 \), the function \( f(x) \) is decreasing at a rate of \( 4 \) units per unit change in \( x \).Differentiation is a key concept in calculus, forming the foundation for more complex operations like integrals and multivariable calculus.

Mathematical Problem Solving

Mathematical problem solving often involves applying your understanding of concepts like inverse functions and differentiation in systematic and creative ways. Solving the exercise requires:

• Knowledge of how to take derivatives to find \( \frac{d f}{d x} \).
• Using inverse functions to switch between \( x \) and \( y \).
• Understanding derivatives of inverse functions to find \( \frac{d f^{-1}}{d y} \).

For example, in the problem:1. We first find the derivative \( \frac{d f}{d x} = -2x \) and evaluate it at \( x = 2 \).2. Then, use the inverse relationship of the function to solve \( x = \sqrt{9 - y} \).3. Finally, compute the derivative of the inverse function at specific values.These steps highlight the importance of logical thinking, manipulating algebraic expressions, and recognizing patterns within functions. By practicing these strategies, you enhance your ability to approach and solve a wide range of mathematical problems efficiently.