Problem 20 - Calculus Early Transcendentals

Question

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x)=m x+b\)

Step-by-Step Solution

Answer

The derivative of \( f(x)=mx+b \) is \( m \), with domain \( \mathbb{R} \) for both \( f(x) \) and \( f'(x) \).

Step 1: Understand the Derivative Definition

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

Step 2: Substitute the Function into the Definition

Substitute \( f(x + h) = m(x+h) + b \) and \( f(x) = mx + b \) into the definition: \[ f'(x) = \lim_{{h \to 0}} \frac{{m(x + h) + b - (mx + b)}}{h} \]. Simplify inside the numerator.

Step 3: Simplify the Expression

Simplify the numerator: \[ m(x+h) + b - (mx + b) = mx + mh + b - mx - b = mh \]. The limit becomes \[ f'(x) = \lim_{{h \to 0}} \frac{{mh}}{h} \].

Step 4: Divide and Apply the Limit

Cancel \( h \) in the fraction: \[ \frac{{mh}}{h} = m \]. Therefore, the limit becomes \[ f'(x) = \lim_{{h \to 0}} m = m \]. Thus, the derivative of \( f(x) = mx + b \) is \( m \).

Step 5: State the Domains

The domain of \( f(x) = mx + b \) is all real numbers (\( \mathbb{R} \)) because it's a linear function. Similarly, the derivative \( f'(x) = m \) is also defined for all real numbers, so its domain is also \( \mathbb{R} \).

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus. It measures how a function changes as its input changes. In simple terms, it tells us the slope of the tangent line to the function at any given point. For a linear function like \( f(x) = mx + b \), the derivative represents the constant rate of change. When we apply the definition of derivative, we aim to find this rate.

• A derivative at a point refers to the instantaneous rate of change.
• It's often represented by \( f'(x) \) or \( \frac{df}{dx} \).
• For linear functions, this rate is constant, meaning it doesn’t depend on \( x \).

Understanding derivatives involves calculating limits, which helps us find how much a function's output changes when its input is "infinitesimally" increased.

Definition of Derivative

The definition of the derivative is crucial for identifying how functions behave. It is defined as the limit:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \]This expression represents the function's response to tiny changes and captures the concept of instantaneous rate of change. When we use this definition, the goal is to determine the rate at which \( f(x) \) changes as \( x \) approaches a specific point.

• Start with the expression for \( f(x + h) \).
• Subtract \( f(x) \) to find the change in function value.
• Divide by \( h \) to find the rate of change per unit change in x.

The concept behind this is vital when learning calculus because it links algebraic manipulation with geometric interpretation.

Linear Function

A linear function is probably the simplest type of function you'll encounter in calculus. It takes the form \( f(x) = mx + b \) where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Linear functions are special because their graphs are straight lines. This means the rate of change, or derivative, is the same everywhere on the line. Thus, when you use the derivative definition for a linear function, the result is simply the slope \( m \). This makes linear functions easy to work with when beginning to learn about differentiation.

Domains of Functions

Domains in mathematics are critical because they define all the input values (\( x \) values) for which a function produces valid outputs. For linear functions like \( f(x) = mx + b \), the domain includes all real numbers (\( \mathbb{R} \)). This is because you can plug any real number into a linear function and get a real number in return.

• The function \( f(x) \) is defined everywhere.
• The derivative \( f'(x) \), which we found to be \( m \), is also defined everywhere.

This wide applicability of linear functions and their derivatives makes them very useful in pure mathematics and practical applications. Understanding the domain helps ensure that your calculations and predictions about function behavior are valid across the range of interest.