Problem 40 - Calculus: Early Transcendentals

Question

Each limit represents the derivative of some function \( f \) at some number \( a \). State such an \( f \) and \( a \) in each case. \( \displaystyle \lim_{x \to 1/4} \frac{\frac{1}{x} - 4}{x - \frac{1}{4}} \)

Step-by-Step Solution

Answer

Function: \( f(x) = \frac{1}{x} \), Point: \( a = \frac{1}{4} \).

Step 1: Identify the Derivative Definition

The limit \( \lim_{x \to a} \frac{f(x) - f(a)}{x-a} \) represents the derivative \( f'(a) \). In our exercise, we need to identify \( f(x) \) and \( f(a) \) based on the given expression.

Step 2: Identify Function and Point

To match the expression \[ \lim_{x \to 1/4} \frac{\frac{1}{x} - 4}{x - \frac{1}{4}} \] to the form of the derivative, we recognize that \( f(x) = \frac{1}{x} \) and \( a = \frac{1}{4} \), so \( f(a) = \frac{1}{1/4} = 4 \).

Step 3: Express Function and Point

Through comparison, conclude that \( f(x) = \frac{1}{x} \) and \( a = \frac{1}{4} \). This matches the standard derivative form \( f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a} \).

Key Concepts

Limit Definition

To understand derivatives, it's vital to first grasp the concept of limits. A limit helps us find the value that a function approaches as the input gets closer to a specific point. In the context of the derivative, we often use the limit to explore how a function behaves near a certain point, which allows us to calculate the instantaneous rate of change.

In the expression \[\lim_{x \to a} \frac{f(x) - f(a)}{x-a},\]we observe the performance of the function as \( x \) approaches \( a \). This specific limit is the foundation of calculus as it leads to the definition of the derivative, a major concept that tells us how a function changes at any given point.

To simplify, think of it as peeking at where the function is headed. If you can predict its path close to a point \( a \), you've harnessed the power of limits in calculus.

Function Identification

The next step in solving our calculus problem involves identifying the function from a limit expression, which is often cleverly disguised. In our exercise, we're given the limit expression: \[\lim_{x \to 1/4} \frac{\frac{1}{x} - 4}{x - \frac{1}{4}}.\]

Notice how this needs to match the generic derivative form\[\frac{f(x) - f(a)}{x-a}.\]By peeling apart the given expression, we find that the function inside is \( f(x) = \frac{1}{x} \). The point \( a \) at which we're finding the derivative is \( x = \frac{1}{4} \).

One can verify this by substituting \( a \) into the function, giving \( f\left(\frac{1}{4}\right) = 4 \), confirming that our identification matches the form. This step is crucial as it sets the stage for further calculations, and ensures that you're measuring the rate of change at the correct point.

Derivative Calculation

Once you have your function \( f(x) = \frac{1}{x} \) and point \( a = \frac{1}{4} \), calculating the derivative at this point becomes straightforward. Understanding the derivative here means recognizing it as the slope of the tangent line to the curve at \( x = \frac{1}{4} \). This gives insight into how quickly the function's value is changing right there.

To perform this calculation, you use the derivative definition:\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}.\]For this exercise, it transforms into:\[ f'\left(\frac{1}{4}\right) = \lim_{x \to \frac{1}{4}} \frac{\frac{1}{x} - 4}{x - \frac{1}{4}}.\]

Evaluating this limit step-by-step shows the rate at which \( \frac{1}{x} \) changes as \( x \) approaches \( \frac{1}{4} \). After simplifying, this calculation reveals the derivative (rate of change) at that precise point. Understanding and being able to calculate this is key for deeper insights into how functions behave, which is essential for advanced calculus studies.