Problem 75
Question
Suppose \(y\) is a linear function of \(x\); that is, \(y=m x+b\). What will happen to the percentage rate of change of \(y\) with respect to \(x\) as \(x\) increases without bound? Explain.
Step-by-Step Solution
Verified Answer
The percentage rate of change of \(y\) with respect to \(x\) remains constant as \(x\) increases without bound.
1Step 1: Identify the Linear Function
Given: The function is linear, expressed as \(y = mx + b\). Where \(m\) is the slope and \(b\) is the y-intercept.
2Step 2: Define the Percentage Rate of Change
The percentage rate of change of \(y\) with respect to \(x\) is defined as \(\frac{\text{change in } y}{\text{change in } x} \times 100\%\).
3Step 3: Calculate the Derivative of y
The rate of change of a linear function \(y = mx + b\) is given by its derivative. The derivative \( \frac{dy}{dx} = m \). Thus, the rate of change is constant and equal to the slope \(m\).
4Step 4: Determine the Percentage Rate of Change
Since the rate of change \(m\) is constant, the percentage rate of change of \(y\) with respect to \(x\) remains \(m \times 100\%\) at all values of \(x\).
5Step 5: Interpret the Behavior as x Increases
As \(x\) increases without bound, the percentage rate of change of \(y\) with respect to \(x\) does not change and remains constant because \(m\) does not depend on \(x\).
Key Concepts
Rate of ChangePercentage Rate of ChangeDerivative
Rate of Change
The rate of change in a mathematical context tells us how one quantity changes in relation to another. For a linear function, described by the equation \( y = mx + b \), the rate of change is the slope \( m \). This essentially tells us how much \( y \) changes for a unit change in \( x \).
For example, if \( m = 2 \), then for every 1 unit increase in \( x \), \( y \) increases by 2 units. If the slope (\( m \)) is negative, \( y \) would decrease as \( x \) increases.
The rate of change is critical in understanding the behavior of a function and facilitates predictions about how changes in one variable will impact another. It is constant for linear functions, which makes them particularly straightforward to analyze.
For example, if \( m = 2 \), then for every 1 unit increase in \( x \), \( y \) increases by 2 units. If the slope (\( m \)) is negative, \( y \) would decrease as \( x \) increases.
The rate of change is critical in understanding the behavior of a function and facilitates predictions about how changes in one variable will impact another. It is constant for linear functions, which makes them particularly straightforward to analyze.
Percentage Rate of Change
The percentage rate of change provides a relative measure of change expressed as a percentage. It indicates how much \( y \) changes as a percentage relative to \( x \).
In a linear function \( y = mx + b \), because the rate of change is constant, the percentage rate of change is also constant. This is given by the formula: \( \frac{\text{change in } y}{\text{change in } x} \times 100\text{\textbackslash %} \times \frac{1}{x} \).
Since the change in \( y \) (which is \( m \)) and the change in \( x \) are both linear, as \( x \) increases, each unit change in \( x \) will produce the same percentage change in \( y \). This percentage does not change no matter how large \( x \) gets.
In a linear function \( y = mx + b \), because the rate of change is constant, the percentage rate of change is also constant. This is given by the formula: \( \frac{\text{change in } y}{\text{change in } x} \times 100\text{\textbackslash %} \times \frac{1}{x} \).
Since the change in \( y \) (which is \( m \)) and the change in \( x \) are both linear, as \( x \) increases, each unit change in \( x \) will produce the same percentage change in \( y \). This percentage does not change no matter how large \( x \) gets.
Derivative
The derivative of a function represents its instantaneous rate of change at any given point. For linear functions, the derivative is straightforward.
Given the function \( y = mx + b \), the derivative \( \frac{dy}{dx} \) is simply the slope \( m \). This derivative tells us that the rate of change of \( y \) with respect to \( x \) is constant and equals \( m \).
Since the derivative is a fundamental concept in calculus that provides detailed insights into the function's rate of change, understanding this for linear equations helps in grasping more complex functions. In simple terms, the slope \( m \) does not change, so the rate of change remains steady no matter how much \( x \) increases or decreases.
Given the function \( y = mx + b \), the derivative \( \frac{dy}{dx} \) is simply the slope \( m \). This derivative tells us that the rate of change of \( y \) with respect to \( x \) is constant and equals \( m \).
Since the derivative is a fundamental concept in calculus that provides detailed insights into the function's rate of change, understanding this for linear equations helps in grasping more complex functions. In simple terms, the slope \( m \) does not change, so the rate of change remains steady no matter how much \( x \) increases or decreases.
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