Problem 50 - Brief Applied Calculus

Question

For each function: a. Find \(f^{\prime}(x)\) using the definition of the derivative. b. Explain, by considering the original function, why the derivative is a constant. $$ f(x)=2 x-9 $$

Step-by-Step Solution

Answer

The derivative \(f'(x) = 2\) is constant because \(f(x) = 2x - 9\) is a linear function with a constant slope.

Step 1: Understand the Problem

We need to find the derivative of the function \(f(x) = 2x - 9\) using the definition of the derivative and then explain why this derivative is a constant.

Step 2: Use the Definition of the Derivative

The derivative of a function \(f(x)\) at a point \(x\) can be found using the limit definition: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]Let's identify the expressions for \(f(x+h)\) and \(f(x)\).

Step 3: Find f(x+h)

Substitute \(x+h\) into the function: \[ f(x+h) = 2(x+h) - 9 = 2x + 2h - 9 \]

Step 4: Substitute into the Derivative Formula

Now use the expressions for \(f(x+h)\) and \(f(x)\) into the definition of the derivative:\[ f'(x) = \lim_{h \to 0} \frac{(2x + 2h - 9) - (2x - 9)}{h} \]

Step 5: Simplify the Expression

Simplify the expression in the numerator:\[ f'(x) = \lim_{h \to 0} \frac{2x + 2h - 9 - 2x + 9}{h} = \lim_{h \to 0} \frac{2h}{h} \]

Step 6: Evaluate the Limit

Cancel \(h\) in the numerator and denominator:\[ f'(x) = \lim_{h \to 0} 2 = 2 \]Thus, the derivative of \(f(x) = 2x - 9\) is 2.

Step 7: Explain Why the Derivative is a Constant

The original function \(f(x) = 2x - 9\) is a linear function with a constant slope of 2. The derivative of a linear function \(ax + b\) is always the coefficient \(a\), which in this case is 2. Therefore, the derivative is constant because the slope of the line represented by \(f(x)\) does not change.

Key Concepts

Definition of the derivative

The concept of a derivative is essential in calculus and plays a crucial role in understanding how functions behave. In simple terms, the derivative of a function at a certain point tells us the rate at which the function is changing at that point. It is a way of quantifying how a small change in the input of the function produces a change in the output. The derivative can be thought of as a slope of the tangent line to the curve at a specific point.

The most common way to define the derivative is through a process called the "limit process." This involves calculating the derivative using the formula:

• \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

This expression involves taking the difference between the function values at two points, dividing by the distance between these points (denoted as \(h\)), and finding the limit as this distance approaches zero.

By applying this definition to a function, like \(f(x) = 2x - 9\), one can determine its derivative, which indicates how steep the line is or how quickly it is rising or falling.

Linear functions

Linear functions are the simplest type of functions you will come across in algebra and calculus. They are represented in the form of \(f(x) = ax + b\), where \(a\) and \(b\) are constants:

• The graph of a linear function is always a straight line.
• The constant \(a\) represents the slope of the line, which tells you how tilted or steep the line is.
• The constant \(b\) is known as the y-intercept, indicating where the line crosses the y-axis.

Understanding the characteristics of a linear function helps in interpreting its derivative. Since the slope \(a\) is constant throughout the function, the derivative of a linear function is the same constant value no matter where you differentiate it. For example, in the function \(f(x) = 2x - 9\), the slope \(a = 2\). Thus, the derivative is consistently 2, reflecting the unchanged slope of the line.

Limit process

The limit process is a fundamental technique in calculus used to find the derivatives and integrals of functions. It involves evaluating the behavior of functions as they approach a particular point.

When finding the derivative of a function via the limit process, you essentially determine what happens as the distance between two points on the function approaches zero. This process helps capture the 'instantaneous' rate of change, revealing the function's behavior on a very small scale.

Here’s a quick run through this with the earlier example, \(f(x) = 2x - 9\):

• Calculate \(f(x+h)\) for a small h, giving \(2(x+h) - 9\).
• Subtract \(f(x)\), which is \(2x - 9\), from \(f(x+h)\).
• Simplify the resulting expression.
• Take the limit as \(h\) approaches 0.

The difference quotient becomes \(\frac{2h}{h}\), which simplifies to 2 as \(h\) tends to zero. This concludes that the derivative \(f'(x) = 2\), demonstrating the uniform slope of the linear function.