Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=-x^{2}+4 x$$

Step-by-Step Solution

Answer

The derivative of \( f(x) = -x^2 + 4x \) is \( f'(x) = -2x + 4 \).

Step 1: Understand the Alternative Definition of Derivative

The alternative definition of a derivative of a function \( f(x) \) at a point \( x \) is given by: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. We will use this definition to find the derivative of \( f(x) = -x^{2} + 4x \).

Step 2: Express the Function at \( x + h \)

Substitute \( x + h \) into the function \( f(x) \). So, \( f(x+h) = -(x+h)^2 + 4(x+h) \).

Step 3: Simplify \( f(x+h) \)

Expand the expression: \((x+h)^2 = x^2 + 2xh + h^2\). Thus, \( f(x+h) = -(x^2 + 2xh + h^2) + 4x + 4h = -x^2 - 2xh - h^2 + 4x + 4h \).

Step 4: Compute the Difference \( f(x+h) - f(x) \)

Substitute the original function \( f(x) = -x^{2} + 4x \) and calculate the difference: \( f(x+h) - f(x) = [-x^2 - 2xh - h^2 + 4x + 4h] - [-x^2 + 4x] \). Simplify by distributing and combining like terms: \( = -2xh - h^2 + 4h \).

Step 5: Set Up the Limit

Substitute the result from Step 4 into the derivative formula: \( f'(x) = \lim_{h \to 0} \frac{-2xh - h^2 + 4h}{h} \).

Step 6: Simplify the Expression

Factor out \( h \) in the numerator: \( = \lim_{h \to 0} \frac{h(-2x - h + 4)}{h} \). Cancel \( h \) in the numerator and denominator: \( = \lim_{h \to 0} (-2x - h + 4) \).

Step 7: Evaluate the Limit

Evaluate the limit as \( h \) approaches 0: \( = -2x + 4 \). Thus, \( f'(x) = -2x + 4 \).

Key Concepts

Definition of Derivative

Derivatives can feel complex at first, but they're simply a tool for understanding how a function changes. When we talk about the derivative of a function, we're talking about the rate at which the function's value changes as its input changes. Imagine driving a car: the derivative would be analogous to your speed at a precise moment, showing how fast your position is changing.

• The derivative tells us the slope of a function at a specific point. It tells us how steeply the function is inclined — or declining — in its graph.
• In mathematical terms, the derivative of a function \( f(x) \) at a point \( x \) can be defined using the limit process, touching on the core foundational aspects of calculus.

These concepts allow us to do all sorts of things, from finding maximum and minimum points on graphs to determining how quantities change over time.
You won't always "see" derivatives in everyday life, but they are everywhere, from physics problems to engineering marvels.

Limit Process

The limit process is fundamental to calculus and understanding derivatives. When calculating a derivative, we look at how a function behaves as we get closer and closer to a specific point. This is where limits come in.

• A limit helps us handle the situation when we can't substitute a value directly due to indeterminacy, such as dividing by zero.
• In our exercise context, we want to find the derivative at a point \( x \) by examining how the function \( f(x+h) \) changes as \( h \) approaches zero.

To simplify, if you imagine a number line, and you focus on a pinpoint along it, limits help you understand what happens as you get infinitively close to that point.
When deriving the derivative, the limit process transforms a mere approximation into an exact measurement of change. This makes calculus extremely powerful for predicting and understanding patterns.

Polynomial Differentiation

Polynomial differentiation is a specific and manageable case within the broader category of finding derivatives. When your function is a polynomial, you can use some straightforward rules to find its derivative.

• For our function \( f(x) = -x^2 + 4x \), a polynomial, each term can be differentiated separately according to a simple rule: bring down the power as a coefficient and subtract one from the power.
• This means \(-x^2\) becomes \(-2x\), and \(4x\) becomes \(4\) after differentiation.

This makes polynomial functions a breeze to work with when using derivatives. It allows us to find the slope at any point along the polynomial's curve by just applying and combining simple rules.
The result for our polynomial function showed that the derivative \(f'(x) = -2x + 4\) gives a line representing the rate of change, or slope, of the original function at any point \(x\).

Alternative Definition

The alternative definition of a derivative might seem like a lot to take in, but it's super useful. It's another way to approach finding how a function changes.

• This method uses the limit, \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\), to find the derivative. It essentially measures the average rate of change of a function over an interval and lets that interval shrink to zero.
• Our exercise used this method to carefully calculate the derivative of \( f(x) = -x^2 + 4x \), by expanding the function for \(x+h\), then finding the difference, and applying the limit as \(h\) approached zero.

Using this alternative definition lets us fully see how the derivative "works" behind the scenes by connecting algebra with calculus.
It’s a comprehensive approach that reinforces and complements the polynomial differentiation by letting us derive the same result: \(f'(x) = -2x + 4\). This is particularly helpful when dealing with non-polynomial functions or when verifying results.