Problem 32 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} f(x)=0.01 x+0.05 $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 0.01 \).

Step 1: Write the definition of the derivative

The derivative of a function \( f(x) \) at a point \( x \) is given by:\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Step 2: Substitute the function into the limit expression

Given the function \( f(x) = 0.01x + 0.05 \), substitute \( f(x+h) \) and \( f(x) \) into the derivative formula:\[f(x+h) = 0.01(x+h) + 0.05 = 0.01x + 0.01h + 0.05\]\[f(x) = 0.01x + 0.05\]So the expression becomes:\[f'(x) = \lim_{h \to 0} \frac{(0.01x + 0.01h + 0.05) - (0.01x + 0.05)}{h}\]

Step 3: Simplify the expression inside the limit

Simplify the expression:\[f'(x) = \lim_{h \to 0} \frac{0.01x + 0.01h + 0.05 - 0.01x - 0.05}{h}\]Notice that \( 0.01x \) and \( 0.05 \) cancel out:\[f'(x) = \lim_{h \to 0} \frac{0.01h}{h}\]

Step 4: Simplify and evaluate the limit

You can simplify the fraction:\[f'(x) = \lim_{h \to 0} 0.01\]The limit as \( h \to 0 \) of a constant is the constant itself, thus:\[f'(x) = 0.01\]

Step 5: Conclusion: State the derivative

The derivative of the function \( f(x) = 0.01x + 0.05 \) is a constant value, which is \( f'(x) = 0.01 \).

Key Concepts

Definition of the Derivative

The derivative of a function is a fundamental concept in calculus, reflecting how a function behaves at any given point. It's like asking, "How quickly is my function changing here?" Imagine you're driving and checking your speedometer. Your speed at any moment is similar to what the derivative tells you about a function:

• The derivative provides the instantaneous rate of change of a function.
• It helps find slopes of tangent lines to curves.
• In mathematical terms, for a function \( f(x) \), the derivative \( f'(x) \) is defined as \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

This formula uses the concept of limits to capture the idea of "instantaneous" change. By computing the derivative, we essentially determine how a small change in \( x \) impacts the change in \( f(x) \). The definition of the derivative is essential for solving many real-world problems involving motion, growth, and change.

Limit Process

Central to understanding derivatives is the limit process. It's a method that lets us calculate the exact instantaneous rate of change. Let's break down what's happening:

• In the definition, \( h \) represents a tiny change in \( x \).
• We want to see what happens when \( h \) gets closer and closer to zero.
• The limit, \( \lim_{h \to 0} \), examines the behavior of the quotient \( \frac{f(x+h) - f(x)}{h} \) as \( h \) approaches zero.

By focusing on the limit, we capture the exact behavior of a function at a single point. This process of taking limits is not just confined to derivatives; it's a foundational idea throughout calculus, revealing the subtle underlying patterns in mathematics.

Linear Functions

A linear function is a simple yet crucial part of calculus and algebra. It's a function that depicts a straight line, and it's described by the formula:

• \( f(x) = mx + b \) where \( m \) and \( b \) are constants.
• \( m \) is the slope, which shows how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Linear functions have constant slopes, which means their derivatives are constants. If you recall the original function \( f(x) = 0.01x + 0.05 \), its derivative is \( f'(x) = 0.01 \). This outcome aligns with what we know about linear functions: their rate of change, or slope, doesn't vary. Understanding linear functions can simplify many calculus problems, making them a powerful tool when dealing with more complex situations.