Average Rate of Change (VCE SSCE Mathematical Methods): Revision Notes

Average Rate of Change

Understanding rates of change

In real life, most objects don't travel at a steady speed. Think about driving a car through city traffic – your speedometer is constantly changing as you speed up, slow down, and stop at lights. This is very different from travelling at a constant speed on a highway.

Similarly, many functions have curved graphs rather than straight lines. This means the rate of change of the function varies at different points. For non-linear functions, we need a way to describe how quickly the function is changing over a particular interval.

Average speed

We can use distance-time graphs to understand the concept of average speed. Even when an object's speed is constantly changing, we can calculate an overall average speed for a journey.

Comparing constant and varying speeds

Consider two cars travelling away from point $O$, where $d$ represents distance travelled (in metres) and $t$ represents time (in seconds).

The straight red line shows a car travelling at a constant speed of $60$ km/h. The blue curved line shows a car travelling through the city with varying speeds. This car:

• Accelerates to reach $60$ km/h at point $A$
• Slows down for traffic lights at point $B$, where it stops for $10$ seconds
• Has another burst of speed
• Stops again at point $C$

Although we don't know the exact speed of the city car at every moment (except when stationary), we can calculate its average speed over the full $60$ seconds.

Calculating average speed

The average speed is found using:

$$\text{average speed} = \frac{\text{total distance travelled}}{\text{total time taken}}$$

For the city journey, the car travelled $300$ metres in $60$ seconds:

$$\text{average speed} = \frac{300}{60} = 5 \text{ metres per second}$$

Average speed over any interval

You can calculate average speed for any time interval, not just the whole journey. For the interval from $t = 15$ to $t = 30$, the average speed is given by the gradient of the line joining points $A$ and $B$:

$$\text{average speed} = \frac{100}{15} = 6\frac{2}{3} \text{ metres per second}$$

General formula

For an object moving between times $a$ and $b$, if we know the positions at these times (points $P$ and $Q$ on the graph), then:

$$\text{average speed for } a \leq t \leq b = \text{gradient of line } PQ$$

Average rate of change for a function

The concept of average speed can be extended to any function. Before we look at the general formula, we need to understand two important terms.

Secants and chords

Secant: A line that passes through two points on a curve is called a secant.

Chord: A line segment joining two points on a curve is called a chord.

The general formula

For any function $y = f(x)$, we can find how quickly $y$ changes with respect to $x$ over an interval $[a, b]$.

The average rate of change of $y$ with respect to $x$ over the interval $[a, b]$ is the gradient of the secant line passing through points $A(a, f(a))$ and $B(b, f(b))$.

For the function shown in the diagram, we can find the average rate of change over the interval $[1, 2]$ by calculating the gradient of secant $PQ$:

$$\text{gradient} = \frac{4 - 1}{2 - 1} = 3$$

So the average rate of change is $3$.