Problem 30 - Essential Calculus

Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=a x+b, \text { then } \Delta y / \Delta x=d y / d x $$

Step-by-Step Solution

Answer

The given statement is true. Both the average rate of change and the instantaneous rate of change for the linear function y = ax + b equal to the slope of the line, which is a.

Step 1: Reading the Statement

Understand the given statement clearly. \(If y = ax + b, then \Delta y / \Delta x = dy / dx\). This statement is stating that for the function y = ax + b, the average rate of change is equal to the instantaneous rate of change.

Step 2: Understanding the Definition of Average Rate of Change

The average rate of change of a function between two points is given by the difference in the y-coordinates of two points (denoted as Δy) divided by the difference in the x-coordinates of the same two points (denoted as Δx). In other words, for a function y = f(x), \(\Delta y / \Delta x = (f(x_2) - f(x_1))/(x_2 - x_1)\). So, for the linear function y = ax + b, \(\Delta y / \Delta x = (a*x_2 + b - (a*x_1 + b)) / (x_2 - x_1) = a\).

Step 3: Understanding the Definition of Instantaneous Rate of Change

The instantaneous rate of change at a certain x-coordinate (or point) in the function is given by the derivative of the function at that point. The derivative is represented as dy/dx. In this case, for the function y = ax + b, dy/dx = a.

Step 4: Comparing the Two Rates

Since 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 function y = ax + b for any two points on the function, the given statement is true.

Key Concepts

Average Rate of Change

The average rate of change provides a way to measure how a function behaves between two distinct points. It's akin to finding the slope of a straight line connecting these points on a graph. This is especially useful for non-linear functions, but it's incredibly straightforward for linear functions.

To calculate it for a function given by \(y = f(x)\), you subtract the initial value of the function from the final value, and then divide by the change in \(x\):

• Formula: \(\Delta y / \Delta x = (f(x_2) - f(x_1))/(x_2 - x_1)\)

For a linear function like \(y = ax + b\), this rate simplifies significantly. Since there's a constant slope, the calculation always yields the coefficient \(a\) of \(x\), showing that the function changes at a constant rate.

This concept allows us to see how predictable and uniform linear functions are. These calculations show that a straight line function doesn't speed up or slow down—it changes at a steady pace.

Instantaneous Rate of Change

The instantaneous rate of change represents how fast a function is changing at a specific point. Unlike the average rate, it is determined with great precision using calculus techniques. This is especially useful when dealing with curves or non-linear graph sections where the rate varies with every tiny movement along the \(x\)-axis.

The key tool for determining this rate is the derivative, often written as \(dy/dx\). For example, the derivative of \(y = ax + b\) at any point is simply \(a\), as it's a constant linear function.

• The derivative shows the function's steepness at any given point.
• For lines, this is uniformly \(a\) everywhere.

This means for a linear function, the instantaneous rate is identical to the average rate of any interval, confirming the consistency and unchanging nature of its slope.

Linear Function

Linear functions take the form \(y = ax + b\), known for having constant rates of change. They're defined by their straight-line graphs where \(a\) is the slope, indicating how steeply or gently the line ascends or descends.

This consistency is what makes them especially straightforward:

• The slope is constant and given by the coefficient \(a\).
• The intercept \(b\) indicates where the line crosses the \(y\)-axis.

Everything about this function revolves around its uniform change, exactly what students can count on when examining parts like average and instantaneous rates of change. Whether you're looking at large chunks of the graph or zooming into a single point, \(a\) remains unchanged, embodying simplicity and predictability.

Derivative

In calculus, the derivative is a fundamental tool used to measure how a function changes at an infinitesimally small point. Think of it as the function's rate of speed or velocity at a very precise spot. It extends the concept of slope from algebra to a wide array of curves and complex forms.

For example, in the case of a linear function \(y = ax + b\), the derivative \(dy/dx\) is equal to \(a\):

• Shows how much \(y\) increases for a small increase in \(x\).
• Remains the same across the entire length of the function.

This constancy highlights the linear function's predictability. In essence, whenever you compute \(dy/dx\) for linear functions, you confirm the uniform speed of change, like a car cruising on a straight highway with no turns or speed changes. This predictability allows students to confidently use the derivative to understand function behavior.