Average Rate of Change Revision Notes for HSC SSCE Mathematics Advanced

Average Rate of Change

What is average rate of change?

The average rate of change is a fundamental concept in calculus that helps us understand how a function behaves over a specific interval. It tells us how quickly the output of a function changes as the input values change between two points.

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Think of it as measuring the overall rate at which something is changing. For example, if you're tracking the distance a car travels over time, the average rate of change would tell you the average speed over that time period.

Key definitions

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Average rate of change: The change in one quantity divided by the corresponding change in another quantity.

Secant line: A straight line that passes through two points on the graph of a function.

The formula

For a function $y = f(x)$ over the interval $[a, b]$ , the average rate of change is calculated using:

$\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$

Where:

$\Delta y$ represents the change in the function's output: $f(b) - f(a)$
$\Delta x$ represents the change in the input: $b - a$
$a$ is the starting point of the interval
$b$ is the ending point of the interval

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This formula essentially calculates "rise over run" between two points on the function. The Greek letter delta (Δ) is used to represent "change in" a quantity.

Geometric interpretation

On a graph, the average rate of change corresponds to the gradient (slope) of the secant line connecting two points on the curve. The secant line joins the points $(a, f(a))$ and $(b, f(b))$ .

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In the diagram above, you can see:

The curve of the function $f(x)$
Two points: $(a, f(a))$ and $(b, f(b))$
The secant line connecting these points
The horizontal change: $b - a$
The vertical change: $f(b) - f(a)$

The gradient of this secant line equals the average rate of change over the interval.

Worked examples

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Worked Example 1: Quadratic Function

Question: For the function $f(x) = x^2$ , find the average rate of change over the interval $[1, 3]$ .

Solution:

Step 1: Identify the values

$a = 1$ and $b = 3$

Step 2: Evaluate $f(x)$ at $x = 1$

f(x) = x^2 f(1) = 1^2 f(1) = 1

Step 3: Evaluate $f(x)$ at $x = 3$

f(x) = x^2 f(3) = 3^2 f(3) = 9

Step 4: Apply the formula

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4

Answer: The average rate of change is 4.

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This result represents the gradient of the secant line between the points $(1, 1)$ and $(3, 9)$ on the graph of $y = x^2$ , as shown in the diagram above.

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Worked Example 2: Real-World Application

Question: A water tank's volume is modelled by $V(t) = 50t + 200$ , where $V$ is in litres and $t$ is in hours. Find the average rate of change of volume from $t = 2$ to $t = 5$ .

Solution:

Step 1: Identify the values

$t_1 = 2$ and $t_2 = 5$

Step 2: Evaluate $V(t)$ at $t = 2$

V(t) = 50t + 200 V(2) = 50(2) + 200 V(2) = 100 + 200 V(2) = 300

Step 3: Evaluate $V(t)$ at $t = 5$

V(t) = 50t + 200 V(5) = 50(5) + 200 V(5) = 250 + 200 V(5) = 450

Step 4: Apply the formula

\frac{\Delta V}{\Delta t} = \frac{V(t_2) - V(t_1)}{t_2 - t_1} = \frac{V(5) - V(2)}{5 - 2} = \frac{450 - 300}{5 - 2} = \frac{150}{3} = 50

Answer: The average rate of change is 50 litres per hour.

Interpretation: This result tells us that the volume increases at an average rate of 50 litres per hour over the interval $[2, 5]$ . This also represents the gradient of the secant line on the graph of $V(t)$ .

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Key Points to Remember:

The average rate of change measures how much a function's output changes per unit change in input over a given interval.
The formula is: $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$
Geometrically, the average rate of change equals the gradient of the secant line connecting two points on the function's graph.
To calculate it: evaluate the function at both endpoints, find the difference in outputs, divide by the difference in inputs.
The average rate of change has practical applications in many real-world contexts, such as calculating average speed, rate of volume change, or rate of population growth.

Average Rate of Change (HSC SSCE Mathematics Advanced): Revision Notes