Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=a x+b\), then \(\Delta y / \Delta x=d y / d x\)

Short answer

The statement is true for the linear function \(y = ax + b\). Both the average rate of change \(\Delta y / \Delta x\) and the instantaneous rate of change \(dy/dx\) equal to the slope \(a\) of the line.

Step-by-step solution

Understanding the Meaning

Firstly, it's important to note what \(\Delta y / \Delta x\) and \(dy/dx\) represent. \(\Delta y / \Delta x\) usually represents the average rate of change of \(y\) with respect to \(x\) over an interval, whereas \(dy/dx\) represents the instantaneous rate of change, or the derivative of \(y\) with respect to \(x\). The given equation \(y = ax + b\) is a linear function.

Calculating the Average and Instantaneous Rate of Change

For the given linear function \(y = ax + b\), the average rate of change on any interval is \(\Delta y / \Delta x = a\), which is constant. For the instantaneous rate of change \(dy/dx\), since the rate at which \(y\) changes with \(x\) is a constant \(a\), \(dy/dx = a\).

Comparing the Two Rates of Change

Since the average rate of change and the instantaneous rate of change are both equal to the slope \(a\) in the linear function, it's true to say that for this specific case \(\Delta y / \Delta x = dy/dx\).

Key concepts

Linear Functions

In mathematics, linear functions are equations that form a straight line when plotted on a graph. These functions can be written in the standard form, \(y = ax + b\), where:

• \(a\) represents the slope or gradient of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

The simplicity of linear functions makes them a cornerstone topic in algebra and calculus, as they provide a straightforward way to explore concepts like slope and intercepts. These functions are characterized by a constant rate of change, which means the relationship between x and y is uniform throughout.

A key concept with linear functions is their slope, \(a\). It indicates how much y changes for a unit increase in x. If the slope is positive, the function increases; if it is negative, the function decreases. However, this slope remains constant across the line, offering a clear opportunity to study rates of change easily.

Rate of Change

The rate of change is an essential concept that describes how a quantity changes over a specific interval relative to another quantity. In the context of functions, it typically refers to how the dependent variable y changes as the independent variable x changes. For linear functions, this rate is consistent and equals the slope \(a\).

In mathematical terms, the average rate of change over an interval \([x_1, x_2]\) is given by the formula:\[\Delta y / \Delta x = \frac{y_2 - y_1}{x_2 - x_1}\]This expression calculates the change in y (\(\Delta y\)) over the change in x (\(\Delta x\)). For linear functions, this always simplifies to the constant slope \(a\).

This characteristic makes it easy to determine the rate of change at any part of the line: it’s always the same. Consequently, linear functions provide a simple way to comprehend the concept of rates, proving fundamental in understanding more complex functions in calculus.

Derivative

A derivative, denoted as \(dy/dx\), represents the instantaneous rate of change of a function with respect to a variable, or in simpler terms, the slope of the tangent line to a function at a specific point. It provides insights into how a function changes at any given point, unlike the average rate of change which considers the whole interval.

For linear functions like \(y = ax + b\), the derivative is extremely straightforward. The function’s rate of change is consistent, meaning the derivative is equal to the slope \(a\) at any point on the function. Thus:\[\frac{dy}{dx} = a\]This consistency reflects the function's linear nature, where the slope doesn't vary, which differs from more complex functions where the slope changes when x values change.

Derivatives form the backbone of differential calculus and are widely applicable in various fields, such as physics for motion analysis, economics for cost changes, and biology for modeling population growth. Understanding the derivative as a concept begins with linear functions and extends to exploring changes in more intricate models.