Problem 45 - A Graphical Approach to Precalculus with Limits

Question

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=4 x-x^{2}$$

Step-by-Step Solution

Answer

The derivative formula is \(f'(x) = 4 - 2x\).

Step 1: Identify the function

We start with the function given to us, which is \(f(x) = 4x - x^2\).

Step 2: Recall the definition of the derivative

The derivative \(f^{ ext{'}}(a)\) is defined as the rate of change of the function at a particular point \(a\). It is given by the limit \(f^{ ext{'}}(a) = \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h}\).

Step 3: Substitute in the definition

Using the definition of the derivative, we substitute \(f(x) = 4x - x^2\) into the limit definition: \[ f^{ ext{'}}(a) = \lim_{{h \to 0}} \frac{{(4(a+h) - (a+h)^2) - (4a - a^2)}}{h} \]

Step 4: Simplify the expression inside the limit

Expand and simplify the expression: \( f(a+h) = 4a + 4h - (a^2 + 2ah + h^2) = 4a + 4h - a^2 - 2ah - h^2 \). Subtract \(f(a) = 4a - a^2\): \[ f(a+h) - f(a) = (4a + 4h - a^2 - 2ah - h^2) - (4a - a^2) = 4h - 2ah - h^2 \]

Step 5: Simplify the fraction

Substitute back into the limit: \[ f^{ ext{'}}(a) = \lim_{{h \to 0}} \frac{{4h - 2ah - h^2}}{h} \]Factor \(h\) out of the numerator:\[ = \lim_{{h \to 0}} \frac{h(4 - 2a - h)}{h} \] Cancel \(h\) from the numerator and the denominator:

Step 6: Evaluate the limit

After cancelling \(h\), evaluate the limit as \(h\) approaches 0:\[ \lim_{{h \to 0}} (4 - 2a - h) = 4 - 2a \]

Step 7: Write the derivative formula

Therefore, the derivative \(f'(x)\) or \(f'(a)\) for any \(a\) is given by:\[ f'(x) = 4 - 2x \]

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus. It represents how a function changes at any given point and is often described as the "slope" of the function at a particular point. More precisely, the derivative of a function at a specific input value measures the rate at which the function's output changes as its input changes.

• For the function \( f(x) = 4x - x^2 \), the derivative tells us how this particular quadratic function is changing at any point \( a \).
• The derivative gives us a precise mathematical tool to predict small changes in the function's value in response to small increments in \( x \).
• A positive derivative means the function is increasing, while a negative derivative indicates it's decreasing at that point.

In our exercise, we calculated the derivative at point \( a \), using the limit definition, and found that the derivative, \( f'(x) \), equals \( 4 - 2x \). This formula helps us understand how the function behaves for any value of \( x \), where the negative term \(-2x \) shows that the rate of increase is reduced as \( x \) becomes larger.

Function Analysis

Function analysis involves understanding the behavior of functions. It includes identifying intervals where the function increases or decreases, finding maximum and minimum values, and determining points of inflection. When analyzing \( f(x) = 4x - x^2 \), we use its derivative \( f'(x) = 4 - 2x \) to determine where the function changes.

• If \( f'(x) > 0 \), the function is increasing. Solving \( 4 - 2x > 0 \) gives \( x < 2 \).
• If \( f'(x) < 0 \), the function is decreasing. Solving \( 4 - 2x < 0 \) yields \( x > 2 \).
• Finding where \( f'(x) = 0 \) helps locate critical points. Setting \( 4 - 2x = 0 \) gives \( x = 2 \), a point where the function may change from increasing to decreasing, indicating a potential maximum or minimum.

With these calculations, we know \( x = 2 \) is a maximum for the function because it transitions from increasing to decreasing behavior, characterizing a peak at that point.

Limit Definition

The limit definition of the derivative is a formal method for finding the derivative of a function at a specific point. Introduced by Isaac Newton and Gottfried Wilhelm Leibniz, it’s essential for understanding how functions behave.

• The limit definition states that the derivative \( f'(a) \) of a function \( f \) at a point \( a \) is given by the limit: \[ f'(a) = \lim_{{h \to 0}} \frac{{f(a+h) - f(a)}}{h} \]
• This expression calculates the slope of the tangent line to the graph of \( f \) at point \( a \), demonstrating how the function changes close to \( a \).
• We use this concept to derive the exact formula for the derivative, which is very significant in not only solving calculus problems but also in real-world applications like physics, engineering, and more.

For the function in our exercise, the limit process simplified to providing a derivative formula, \( f'(x) = 4 - 2x \), directly using the principles of the limit definition to observe changes around each point \( x \).