Problem 53 - 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. $$ \begin{array}{l} f(x)=m x+b\\\ (m \text { and } b \text { are constants) } \end{array} $$

Step-by-Step Solution

Answer

The derivative, \(f'(x) = m\), is constant because the function is linear.

Step 1: Understand the Function

The function given is a linear function in the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. We need to find its derivative using the definition of a derivative.

Step 2: Definition of the Derivative

The derivative of a function \(f(x)\) at a point \(x\) is defined as: \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \). We will apply this definition to our function \( f(x) = mx + b \).

Step 3: Substitute into the Derivative Definition

Substitute \(f(x) = mx + b\) into the definition to get: \( f'(x) = \lim_{{h \to 0}} \frac{{(m(x+h) + b) - (mx + b)}}{h} \).

Step 4: Simplify the Expression

Simplify the expression: \( f'(x) = \lim_{{h \to 0}} \frac{{mx + mh + b - mx - b}}{h} \). This reduces to: \( f'(x) = \lim_{{h \to 0}} \frac{{mh}}{h} \).

Step 5: Cancel \(h\) and Evaluate the Limit

Cancel \(h\) from the numerator and the denominator: \( f'(x) = \lim_{{h \to 0}} m = m \). The remaining expression is simply \(m\), which is a constant.

Step 6: Explanation of Why the Derivative is Constant

Since the original function \(f(x) = mx + b\) is a linear function, its graph is a straight line. The slope of this line (rate of change) is constant and given by \(m\), thus the derivative \(f'(x) = m\) is constant for all \(x\).

Key Concepts

Linear Functions

Linear functions are mathematical expressions of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. These functions create a straight line when graphed on a coordinate plane. The constant \(m\) represents the slope of the line, showing how much \(f(x)\) changes for each unit change in \(x\). Meanwhile, \(b\) is the y-intercept, indicating where the line crosses the y-axis. Understanding these components is crucial because they define the behavior of the function. Here's how you can think about it:

• **Slope \((m)\):** Describes the steepness of the line. A positive slope means the line rises as \(x\) increases, while a negative slope means it falls.
• **Y-intercept \((b)\):** Shows the starting point of the line on the y-axis when \(x = 0\).

Linear functions are among the simplest types of functions, making them a starting point for understanding more complex relationships in calculus.

Constant Derivative

A derivative of a function represents the rate at which the function's value changes as the input changes. For linear functions, like \(f(x) = mx + b\), the derivative is particularly intriguing because it is always constant. This means that the rate of change never varies, no matter the value of \(x\).In our case, after applying the definition of the derivative, the derivative \(f'(x)\) is found to be \(m\), where \(m\) is the slope of the original linear function. This constant derivative is a direct result of the function's linearity:

• The graph of the function is a straight line.
• The slope \(m\) represents this constant rate of change.

So, when dealing with a straight line, the derivative remains constant because each unit increase in \(x\) results in the same increase \(m\) in \(f(x)\). This feature is unique to linear functions, and highlights their predictability and simplicity in calculus.

Definition of the Derivative

The derivative is a fundamental concept in calculus that quantifies how a function's output changes as its input changes. It is formally defined as:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]This definition, also known as the "difference quotient," captures what happens to the average rate of change of \(f(x)\) over a small interval as the interval approaches zero. When applied to the linear function \(f(x) = mx + b\), this definition simplifies beautifully:- By substituting \(f(x) = mx + b\) into this limit process, the algebra shows that the difference quotient simplifies to just \(m\), the slope of the line.- Since \(m\) is constant, the derivative \(f'(x)\) is also a constant value equal to \(m\).This step-by-step simplification not only demonstrates how derivatives are calculated but also why certain types of functions, like linear ones, have particular derivative characteristics. Mastering the definition of the derivative is essential for understanding the behavior of not just linear functions but any function that can be differentiated.