The fastest 100m sprint ever was 9.58 seconds. But the runner was not running at 10.44 m/s for the entire race. His speed varied: slow at the start, explosive in the middle, then a slight fade at the end. In this lesson, you will learn how to measure average speed over an interval and glimpse how calculus will let us find his exact speed at any single instant.
Today's hook, A car travels 120 km in 2 hours, so its average speed is 60 km/h. Does this mean the speedometer showed exactly 60 km/h at every moment? What happens to the accuracy of the average speed as we measure it over shorter and shorter time intervals?
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A car travels 120 km in 2 hours, so its average speed is 60 km/h. Does this mean the speedometer showed exactly 60 km/h at every moment? What do you think happens to the accuracy of the average speed as we measure it over shorter and shorter time intervals?
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The two moves
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The two moves
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The average rate of change measures overall steepness between two points on a curve. It is the gradient of the secant line joining those points.
The instantaneous rate of change is what happens when those two points get infinitely close together. The secant becomes a tangent and we capture the exact rate at a single instant.
This is exactly the same calculation as finding the gradient of a straight line through the two points $(a, f(a))$ and $(b, f(b))$. On a curve, this line is called a secant it cuts through the curve at two points.
Average rate = slope of secant (grey dashed). Instantaneous rate = slope of tangent (red) at a single point.
From speed to any rate
The same idea applies to any changing quantity:
Speed: change in distance over change in time
Growth rate: change in population over change in time
Flow rate: change in volume over change in time
Cost per unit: change in cost over change in quantity
The geometric idea
Geometrically, as the second point on the curve moves closer to the first, the secant line pivots and approaches a limiting position: the tangent line at that point. The gradient of this tangent is the instantaneous rate of change.
Why this matters for sport. In elite athletics, coaches analyse split times over tiny intervals to understand exactly when an athlete accelerates, maintains speed, or decelerates. Usain Bolt's 9.58-second 100m world record involved an average speed of 10.44 m/s, but his peak speed was closer to 12.3 m/s around the 60-80 metre mark. Average speed hides this peak; instantaneous speed reveals it.
Average rate of change = $\dfrac{f(b)-f(a)}{b-a}$ = gradient of the secant through $(a, f(a))$ and $(b, f(b))$; Secant line: a straight line crossing the curve at two distinct points
Pause, copy the average rate of change formula $\dfrac{f(b)-f(a)}{b-a}$ and its geometric meaning (gradient of the secant line) into your book.
Quick check: True or false, the average rate of change over an interval equals the gradient of the tangent line at the midpoint of that interval.
Worked examples · 3 in a row, reveal as you go
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PROBLEM 1 · AVERAGE RATE FROM A TABLE
The temperature of a chemical solution is recorded over time:
Time (h)
0
2
4
6
8
Temp (°C)
15
22
28
24
18
Find the average rate of change of temperature over the first 4 hours.
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$T(0) = 15$ and $T(4) = 28$
Read the temperature values at $t = 0$ and $t = 4$ from the table.
The function value increased by 8 units while $x$ increased by 2 units.
PROBLEM 3 · ESTIMATING INSTANTANEOUS RATE
A runner's position after $t$ seconds is $s(t) = t^2$ metres. Estimate the instantaneous speed at $t = 2$ by calculating average speeds over $[2, 2.1]$ and $[2, 2.01]$.
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$s(2.1) = (2.1)^2 = 4.41$
$s(2) = 4$
$\displaystyle\text{Average speed} = \frac{4.41 - 4}{0.1} = 4.1$ m/s
Calculate the average speed over the wider interval $[2, 2.1]$.
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$s(2.01) = (2.01)^2 = 4.0401$
$\displaystyle\text{Average speed} = \frac{4.0401 - 4}{0.01} = 4.01$ m/s
Shrink the interval to $[2, 2.01]$ and recalculate.
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$4.1 \to 4.01$ as the interval shrinks
As the interval gets smaller, the average speed converges toward 4 m/s. We estimate the instantaneous speed at $t = 2$ is approximately 4 m/s.
Quick check: For $f(x) = x^2$, the average rate of change over $[2, 5]$ is:
Common errors · the 3 traps that cost marks
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Trap 01
Confusing average rate with function value
Students sometimes give $f(b)$ as the answer instead of $\frac{f(b)-f(a)}{b-a}$. The rate of change measures how steeply the function is changing, not how high it is.
Trap 02
Using wrong order in the difference quotient
Mixing up the numerator and denominator order gives the wrong sign or magnitude. Always keep the same order: $\frac{f(b)-f(a)}{b-a}$. If you swap one side, swap the other too.
Trap 03
Thinking instantaneous rate needs only one point
You cannot calculate a rate from a single point, rate requires change, and change requires two values. Instantaneous rate is found by letting the interval shrink to zero, not by ignoring the second point entirely.
Odd one out: Three of the following are examples of a rate of change. Which one is NOT?
Quick-fire practice · 5 problems
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Find the average rate of change of $f(x) = 3x + 2$ from $x = 0$ to $x = 4$.
2
Find the average rate of change of $f(x) = x^2$ from $x = 2$ to $x = 5$.
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A runner's distance from the start is recorded:
Time (s)
0
1
2
3
Distance (m)
0
3
8
15
Find the average speed over the first 3 seconds.
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Estimate the instantaneous rate of change of $f(x) = x^2$ at $x = 2$ using $h = 0.1$.
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Explain the difference between average rate of change and instantaneous rate of change.
Fill the blanks: drag each token into the matching blank.
secanttangentaverageinstantaneous
The gradient of the ___ line gives the
___ rate of change.
As the interval shrinks, this approaches the ___ rate, which equals the gradient of the ___.
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Revisit your thinking
Earlier you were asked: Does an average speed of 60 km/h mean the speedometer showed exactly 60 km/h at every moment? No. Average speed is calculated over a whole interval and smooths out all the variation. The speedometer shows instantaneous speed, which can be faster or slower than the average at any given moment. As we measure over shorter and shorter intervals, the average speed gets closer and closer to the instantaneous speed.
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Multiple choice
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Multiple choice
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Short answer
ApplyBand 4
Q1. The temperature of a liquid is given by $T(t) = 20 + 5t - t^2$, where $t$ is in minutes. Find the average rate of change of temperature from $t = 1$ to $t = 4$. Show all working. 3 MARKS
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Q2. A particle's position is $s(t) = t^3$ metres after $t$ seconds. Estimate the instantaneous velocity at $t = 2$ by using intervals of width $0.1$ and $0.01$. Show all working. 4 MARKS
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AnalyseBand 5
Q3. A car's speedometer shows instantaneous speed. Explain why the average speed over a trip can be different from the instantaneous speed at any moment. 3 MARKS
Drill 5: Average rate is calculated over an interval and smooths out all variation. Instantaneous rate is the exact rate at a single point, found by shrinking the interval to zero.
Q2 (4 marks): Over $[2, 2.1]$: $s(2.1) = 9.261$, $s(2) = 8$, rate $= \frac{9.261-8}{0.1} = 12.61$ m/s [1]. Over $[2, 2.01]$: $s(2.01) = 8.120601$, rate $= \frac{8.120601-8}{0.01} = 12.0601$ m/s [1]. As the interval shrinks, the values approach 12 [1]. Estimated instantaneous velocity at $t = 2$ is approximately $12$ m/s [1].
Q3 (3 marks): Average speed is total distance divided by total time over the whole trip [1]. Instantaneous speed is the exact speed at a single moment in time [1]. They differ because a car speeds up, slows down, and stops during a trip, so the average smooths out these changes while the instantaneous reading captures only one moment [1].
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