Problem 10 - Calculus an Applied Approach

Question

Find the derivative of the function. $$ g(x)=3 x-1 $$

Step-by-Step Solution

Answer

The derivative of the function \( g(x) = 3x - 1 \) is 3.

Step 1: Identify the function

The function given is \( g(x) = 3x - 1 \). This is a linear function, as its graph is a straight line.

Step 2: Define the derivative

The derivative of the function, often denoted as \( g'(x) \), measures the rate at which the output of the function is changing at any given point. For a linear function, this is just the slope of the line. The derivative of a constant is zero, and the derivative of \( y = ax \), where \( a \) is a constant and \( x \) is the variable, equals to \( a \).

Step 3: Apply the derivative rules

Applying the derivative rules, the derivative of \( g(x) = 3x - 1 \) is the derivative of \( 3x \) minus the derivative of 1. The derivative of \( 3x \) is 3 and the derivative of a constant number, such as 1, is 0. So \( g'(x) = 3 - 0 = 3 \).

Key Concepts

Understanding Linear Functions

A linear function is one of the simplest types of mathematical expressions you can encounter. It takes the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants and \( x \) is the variable. This function produces a straight line when graphed on a coordinate plane.

Some key characteristics of linear functions include:

• Constant slope: The slope is the coefficient of \( x \), which is \( a \) in \( ax + b \). This means no matter where you are on the graph, the steepness or tilt of the line does not change.
• Simple to calculate: Because they involve only multiplication and addition/subtraction, linear functions are less complicated than other functions like quadratics or polynomials.
• Predictable behavior: Linear functions have no curves, bumps, or swerves. They proceed in a single, straight direction throughout their range.

Linear functions are a cornerstone of algebra, helping us model relationships that have a constant rate of change, like speed over time or prices with respect to quantity. Learning this helps you make more complex connections in mathematics.

Rate of Change in Functions

The concept of rate of change refers to how much a quantity increases or decreases over a particular interval. For linear functions, the rate of change is constant and equals the slope of the function. It essentially tells you how fast one quantity changes in relation to another.

For a linear function, the rate of change is easy to compute because:

• It is equal to the slope \( a \) in the equation \( f(x) = ax + b \).
• This slope means that for every 1 unit increase in \( x \), \( y \) will increase by \( a \) units.

Therefore, if you have the linear function \( g(x) = 3x - 1 \), the rate of change is 3. This indicates that the function's output, or \( g(x) \), increases by 3 for every additional unit of \( x \).

Understanding the rate of change is crucial in determining how rapidly variables in a function behave relative to each other.

Why Constant Functions Matter

Constant functions are a special subset of linear functions. They have the form \( f(x) = b \), where \( b \) is a constant, and there is no \( x \) variable involved. Such a line is perfectly horizontal on a graph, indicating that the output does not change regardless of the input.

Key points to understand about constant functions include:

• Zero slope: The slope of a constant function is always zero, indicating no change in \( y \) regardless of changes in \( x \).
• Derivative is zero: The derivative of a constant function is always zero because the rate of change is nil.

In the context of derivatives, knowing that the derivative of a constant is zero helps simplify calculations. In our example with \( g(x) = 3x - 1 \), the derivative of the \(-1\) is zero, simplifying the differentiation process.

Analyzing the Slope of the Line

The slope of the line is a key component in understanding linear functions. It is a measure of the steepness or incline of a line, indicating the rate of change over a specific range in the context of an equation.

Several important aspects of slope to remember:

• Positive slope: A positive slope, like \( 3 \) in the function \( g(x) = 3x - 1 \), means the line is increasing as you move from left to right.
• Interpretation: In a real-world context, slope can represent speed, efficiency, or intensity. It's often used to show how one variable increases in proportion to another.

Therefore, considering the slope's value is crucial for predicting and evaluating a function's behavior. In our case, a slope of 3 means for every step to the right on the \( x \)-axis, the line steps up 3 units, reinforcing that our function has a constant positive increase.