Problem 264 - Calculus Volume 1

Question

For the following exercises, use the functions \(y=f(x)\) to find a. \(\frac{d f}{d x}\) at \(x=a\) and b. \(x=f^{-1}(y).\) c. Then use part b. to find \(\frac{d f^{-1}}{d y}\) at \(y=f(a).\) $$f(x)=6 x-1, x=-2$$

Step-by-Step Solution

Answer

a. \(6\); b. \(f^{-1}(y) = \frac{y + 1}{6}\); c. \(\frac{1}{6}\).

Step 1: Find the Derivative of the Function

To find \(\frac{d f}{d x}\), we need to differentiate the given function \(f(x) = 6x - 1\). The derivative of a linear function \(ax + b\) is simply the coefficient \(a\). Here, the coefficient of \(x\) is 6, so \(\frac{d f}{d x} = 6\).

Step 2: Evaluate the Derivative at the Given x-value

Now we need to evaluate the derivative at \(x = -2\). Since \(\frac{d f}{d x} = 6\) is constant, it does not depend on \(x\). Therefore, \(\frac{d f}{d x}\bigg|_{x=-2} = 6\).

Step 3: Solve for the Inverse Function

To find the inverse function \(f^{-1}(y)\), we need to swap \(x\) and \(y\) in the equation \(y = 6x - 1\) and solve for \(x\). This gives:\[ y = 6x - 1 \]\[ y + 1 = 6x \]\[ x = \frac{y + 1}{6} \]Thus, the inverse function is \(f^{-1}(y) = \frac{y + 1}{6}\).

Step 4: Find the Inverse Function's Derivative

Differentiate the inverse function \(f^{-1}(y) = \frac{y + 1}{6}\) with respect to \(y\). The derivative is:\[ \frac{d f^{-1}}{d y} = \frac{1}{6} \].

Step 5: Evaluate the Inverse Function's Derivative at y=f(a)

Evaluate the inverse function's derivative at \(y = f(-2)\). First, find \(f(-2)\):\[ f(-2) = 6(-2) - 1 = -12 - 1 = -13 \]Now, evaluate \(\frac{d f^{-1}}{d y}\) at \(y = -13\). Since \(\frac{d f^{-1}}{d y} = \frac{1}{6}\) is constant, it does not depend on \(y\). Thus, \(\frac{d f^{-1}}{d y}\bigg|_{y=-13} = \frac{1}{6}\).

Key Concepts

Derivatives

Derivatives lie at the heart of calculus. They represent how a function changes at any given point, essentially describing the "rate of change" or "slope" of the function. When working with a linear function like

• \( f(x) = 6x - 1 \), the derivative is simply the coefficient of x.
• The derivative \( \frac{d f}{d x} \) tells us how the function f(x) changes as x changes.
• Since the derivative of \( f(x) = 6x - 1 \) is constant at 6, this tells us that the rate of change is uniform, meaning that for every unit increase in x, y increases by 6.

This constancy in rate is a hallmark of linear functions, making them straightforward to analyze and predict.

Inverse Functions

Inverse functions reverse the operation of the original function. If you start with a value, apply a function, and get a result, applying the inverse function to that result returns you to the original value.

• For the function \( f(x) = 6x - 1 \), the inverse \( f^{-1}(y) \) can be found by solving the equation \( y = 6x - 1 \) for x.
• This involves swapping x and y and rearranging to isolate x, giving us \( f^{-1}(y) = \frac{y + 1}{6} \).
• Inverse functions have the property that if you input y to \( f^{-1}(y) \), you retrieve the original x used in \( f(x) \).

Understanding inverse functions is crucial, as they provide a way to "undo" operations performed by the original function.

Linear Functions

Linear functions are a fundamental concept in both algebra and calculus, characterized by their constant rate of change. They form straight lines when graphed.

• A general form of a linear function is \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
• In the specific case of \( f(x) = 6x - 1 \), \( a = 6 \) and \( b = -1 \).
• These functions are easy to work with due to their simple structure. Their derivatives are constant, which means the slope of the line is the same at every point.

This uniformity is what makes linear functions easy to predict and analyze, providing a reliable model in various applications.

Differentiation

Differentiation is the process of finding a derivative, which is a key tool in calculus.

• It involves calculating how a function's output changes as its input changes.
• For any function expressed as \( f(x) \), the derivative \( \frac{d f}{d x} \) is found through differentiation.
• For linear functions like \( f(x) = 6x - 1 \), differentiation is straightforward because the derivative is constant and equals the coefficient of x. Here, differentiation shows that \( \frac{d f}{d x} = 6 \).

Understanding differentiation is crucial for analyzing how various functions behave, and it's the stepping stone to many advanced concepts in calculus.