Introduction to Derivatives · Grade 12

What is an average rate of change and how does it relate to the slope of a chord?

An average rate of change is the change in the function divided by the change in x over an interval. Geometrically it is the slope of the chord joining the two points on the graph — the straight line through them — which is why the same number answers both a question about speed and a question about slope.

Learning objectives

What exactly is being calculated

Take two values of x and the two function values that go with them. The average rate of change is the difference in the function values divided by the difference in the x values.

The order matters only in that it must be consistent. If you subtract the first from the second on top, do the same underneath — reversing one of them flips the sign of a rate that has not changed.

The word average is doing real work. Over the interval the function may rise, fall, and rise again; this number reports only the net effect from one end to the other.

Why it is the slope of the chord

The slope of a straight line is the vertical change divided by the horizontal change. That is exactly the expression just written, so the average rate of change of a function is the slope of the line joining its two endpoints.

Seeing it this way makes the next lesson almost inevitable. Bring the two points closer together and the chord turns into the tangent — and its slope becomes the derivative.

What it means in a word problem

If the function is distance against time, the average rate of change is average speed. If it is cost against quantity, it is the average cost per unit.

The units come from the division and are worth stating: metres per second, shekels per item. An answer without units in a word problem is only half an answer.

Worked examples

  1. Find the average rate of change of f(x) = x squared between x = 1 and x = 4

    1. Function values: f(1) is 1 and f(4) is 16
    2. Change in the function: 16 minus 1, that is 15
    3. Change in x: 4 minus 1, that is 3
    4. Divide: 15 over 3

    Answer: The average rate of change is 5

  2. A car travels 240 kilometres in 3 hours. What is its average speed?

    1. Distance against time, so the average rate of change is average speed
    2. Divide: 240 over 3
    3. The units come from the division

    Answer: 80 kilometres per hour

  3. Find the average rate of change of f(x) = x squared between x = −2 and x = 2

    1. Function values: f(−2) is 4 and f(2) is 4
    2. Change in the function: 4 minus 4, that is 0
    3. Divide by the change in x, which is 4

    Answer: Zero — the function fell and rose back, with no net change

Common mistakes

Subtracting in opposite orders above and below
If you subtract the first from the second on top, do the same underneath. Reversing one flips the sign, and a rate of change reported as negative when it is positive is a different claim about the situation.
Reading a zero rate as a constant function
A zero average rate means the function returned to where it started, not that it stayed there. It may have risen and fallen a great deal in between.
Leaving the answer without units
The units come from the division and carry the meaning. Eighty is not an answer to a question about speed; eighty kilometres per hour is.

What to remember

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