Derivative Of F(x) = 8 - 4x² Using Limit Definition A Step-by-Step Guide

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This article delves into the process of finding the derivative of the function f(x) = 8 - 4x² using the fundamental limit definition. This method, while sometimes more involved than using derivative rules, provides a crucial understanding of the core concept of the derivative as the instantaneous rate of change. We'll break down each step, providing a clear and comprehensive explanation.

Understanding the Limit Definition of the Derivative

Before we jump into the specifics of our function, let's first solidify our understanding of the limit definition of the derivative. The derivative of a function f(x), denoted as f'(x), is defined as:

f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h

This formula essentially calculates the slope of the tangent line to the curve of f(x) at a given point x. It does this by finding the limit of the difference quotient as h approaches zero. Let's break down the components of this definition:

  • f(x + h): This represents the value of the function at the point x + h. In other words, we're shifting our input by a small amount h.
  • f(x): This is the original function evaluated at x.
  • f(x + h) - f(x): This difference represents the change in the function's value as we move from x to x + h. This is the "rise" in our slope calculation.
  • h: This represents the change in the input, the "run" in our slope calculation.
  • [f(x + h) - f(x)] / h: This is the difference quotient, which calculates the average rate of change of the function over the interval [x, x + h]. Geometrically, this represents the slope of the secant line passing through the points (x, f(x)) and (x + h, f(x + h)).
  • lim (h -> 0): This is the crucial part. By taking the limit as h approaches zero, we're essentially shrinking the interval over which we're calculating the average rate of change. As h gets infinitesimally small, the secant line approaches the tangent line, and the difference quotient approaches the instantaneous rate of change at the point x. This is the derivative.

The limit definition is the bedrock of differential calculus. Mastering its application provides a profound understanding of what a derivative truly represents – the instantaneous rate of change. While derivative rules offer shortcuts, understanding the limit definition is essential for grasping the underlying principles.

Applying the Limit Definition to f(x) = 8 - 4x²

Now, let's apply this definition to our specific function, f(x) = 8 - 4x². This process involves several steps, each building upon the previous one. It's a meticulous process, but by breaking it down, we can arrive at the derivative with clarity.

Step 1: Find f(x + h)

The first step is to determine f(x + h). This means substituting x + h for x in the original function:

f(x + h) = 8 - 4(x + h)²

Next, we need to expand the squared term:

f(x + h) = 8 - 4(x² + 2xh + h²)

Distribute the -4:

f(x + h) = 8 - 4x² - 8xh - 4h²

This expression represents the value of the function at the point x + h. It's a crucial component in our limit definition formula. This step involves algebraic manipulation, and accuracy is paramount. A small error here can propagate through the rest of the calculation.

Step 2: Calculate f(x + h) - f(x)

The next step is to find the difference between f(x + h) and f(x). We already have f(x + h) from the previous step, and we know that f(x) = 8 - 4x². So, we subtract f(x) from f(x + h):

f(x + h) - f(x) = (8 - 4x² - 8xh - 4h²) - (8 - 4x²)

Notice that the parentheses are crucial here. We're subtracting the entire expression for f(x). Now, we can simplify by distributing the negative sign and combining like terms:

f(x + h) - f(x) = 8 - 4x² - 8xh - 4h² - 8 + 4x²

Observe that the 8 and -8 cancel out, and the -4x² and +4x² also cancel out. This leaves us with:

f(x + h) - f(x) = -8xh - 4h²

This expression represents the change in the function's value as we move from x to x + h. Notice how the terms that didn't involve h canceled out. This is a common pattern when using the limit definition, and it's a good sign that we're on the right track.

Step 3: Divide by h

Now, we divide the expression we obtained in the previous step by h:

[f(x + h) - f(x)] / h = (-8xh - 4h²) / h

We can factor out an h from the numerator:

[f(x + h) - f(x)] / h = h(-8x - 4h) / h

Now, we can cancel the h in the numerator and denominator (since h is approaching 0, but not equal to 0):

[f(x + h) - f(x)] / h = -8x - 4h

This expression represents the difference quotient, the average rate of change of the function over the interval [x, x + h]. We've simplified it by canceling out the h in the denominator, which is essential for taking the limit in the next step. If we hadn't canceled out the h, we would have an indeterminate form (0/0) when we tried to evaluate the limit.

Step 4: Take the Limit as h Approaches 0

The final step is to take the limit of the expression as h approaches 0:

f'(x) = lim (h -> 0) [-8x - 4h]

As h approaches 0, the term -4h also approaches 0. Therefore, the limit is:

f'(x) = -8x

This is the derivative of the function f(x) = 8 - 4x². It represents the instantaneous rate of change of the function at any given point x. This step is the culmination of all the previous steps. We've successfully applied the limit definition to find the derivative.

The Result: f'(x) = -8x

Therefore, using the limit definition of the derivative, we have found that the derivative of f(x) = 8 - 4x² is f'(x) = -8x. This result aligns with what we would obtain using the power rule of differentiation, which serves as a confirmation of our work. The power rule states that the derivative of x^n is nx^(n-1). Applying this to -4x^2 gives -8x, and the derivative of the constant 8 is 0.

This exercise demonstrates the power and elegance of the limit definition. While it may seem more complex than using derivative rules, it provides a deeper understanding of the concept of the derivative. The limit definition of the derivative is a fundamental concept in calculus. It defines the derivative as the limit of the difference quotient, representing the instantaneous rate of change of a function. Understanding this definition is crucial for grasping the underlying principles of calculus.

The derivative, f'(x) = -8x, is itself a function. It gives us the slope of the tangent line to the graph of f(x) = 8 - 4x² at any point x. For example, at x = 1, the slope of the tangent line is f'(1) = -8. This means that the function is decreasing at a rate of 8 units for every 1 unit increase in x at that point.

Understanding the derivative allows us to analyze the behavior of the original function. For example, we can find the critical points of f(x) by setting f'(x) = 0. In this case, -8x = 0 implies x = 0. This means that x = 0 is a critical point of f(x), which could be a local maximum, a local minimum, or a saddle point. Further analysis, such as using the second derivative test, would be needed to determine the nature of this critical point. The derivative of f(x) provides valuable information about the function's rate of change and critical points.

In conclusion, we have successfully found the derivative of f(x) = 8 - 4x² using the limit definition. This process involved several steps: finding f(x + h), calculating f(x + h) - f(x), dividing by h, and finally, taking the limit as h approaches 0. The result, f'(x) = -8x, provides valuable information about the instantaneous rate of change of the function and its behavior. While derivative rules offer efficient shortcuts, understanding the limit definition is essential for a deep understanding of calculus. This method reinforces the fundamental principles and provides a solid foundation for more advanced calculus concepts.