Problem 148 - Algebra and Trigonometry

Question

If \(f(x)=m x+b,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) (Section \(2.2,\) Example 8 )

Step-by-Step Solution

Answer

The value is \(m\).

Step 1: Substitution

Substitute \(f(x)\) and \(f(x+h)\) into the equation. This will give \(\frac{m(x+h) + b - (mx + b)}{h}\).

Step 2: Simplification

Simplify the expression in the numerator, while keeping track of the terms. You will get \(\frac{mh + mx + b - mx - b}{h}\). Note that the terms \(mx\), \(b\), \(-mx\) and \(-b\) will cancel out to give \(\frac{mh}{h}\).

Step 3: Cancel Common Factors

Notice that \(h\) is a common factor in the numerator and the denominator. Cancel out these common factors. The expression simplifies to \(m\).

Key Concepts

Linear Function

A linear function is one of the simplest forms of function in mathematics, often represented as \(f(x) = mx + b\). This equation represents a line when plotted on a graph. Here, "\(m\)" is the slope of the line, and "\(b\)" is the y-intercept, which is where the line crosses the y-axis.
Linear functions have a constant rate of change which makes them very predictable and easy to analyze.

• When you increase \(x\) by 1, the value of \(f(x)\) will increase by \(m\).
• The graph of a linear function is always a straight line.
• This function is also called an affine function when \(b eq 0\).

Slope of a Line

The slope of a line in a linear function is a measure of how steep the line is. In the context of our equation \(f(x) = mx + b\), the slope is represented by the coefficient \(m\).
The slope can be understood as the rise over run, describing how much \(f(x)\) changes as \(x\) changes. If you imagine walking along a line:

• If the line goes up as you move from left to right, \(m\) is positive.
• If the line goes down as you move from left to right, \(m\) is negative.
• If the line is horizontal, \(m\) is zero, indicating no change.

Understanding the slope helps us interpret how two quantities are related. For example, an incline in a road is similar to a positive slope of a line.

Derivative Concept

The derivative is a fundamental concept in calculus that represents an instantaneous rate of change of a function concerning its variable. It is often seen as the slope of a function at a given point. The difference quotient you calculated \[\frac{f(x+h)-f(x)}{h}\]reflects the average rate of change of the function over an interval \(h\).
Consider it like calculating the slope over an infinitesimally small section of the curve, as \(h\) becomes very small.
For linear functions specifically, the derivative provides the exact slope of the line:

• Since linear functions have constant slopes, the derivative for \(f(x) = mx + b\) is \(m\).
• In non-linear functions, the derivative can vary depending on \(x\).
• Calculating derivatives gives insight into how quickly changes occur within the system modeled by the function.