2.2.1 Rate of change and the slope of a tangent

What a rate of change measures

A rate of change says how fast one quantity changes as another one changes. It is the change in the output divided by the change in the input. Its units are output units per input unit.

A car that covers 150 km in 2 hours has an average speed of $150 \div 2 = 75$ km per hour. A tank that loses 40 litres in 8 minutes drains at $40 \div 8 = 5$ litres per minute.

The Greek letter $\Delta$ (delta) is short for "change in". If the input moves from $a$ to $b$ , then $\Delta x = b - a$ is the change in input and $\Delta y = f(b) - f(a)$ is the change in output.

Average rate of change

Between two inputs $a$ and $b$ , the average rate of change of $f$ is the change in output per unit of input. On the graph it is the slope of the straight line joining the two points. That line is called a chord.

Rule: Average rate of change

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

The rise over the run between two points of the graph, exactly as for a line in lesson 1.2.1. It describes the whole interval, not any single point in it.

The chord of $y = x^2$ from $x = 1$ to $x = 3$ . The run is 2, the rise is 8, so the average rate of change is 4.

For $f(x) = x^2$ from 1 to 3, the output goes from 1 to 9. The rise is 8 over a run of 2, so the average rate is 4. On average, the output grows by 4 for each step of 1 in the input.

Rising, falling and flat

The sign of the average rate tells you the direction of the change over the interval.

Positive: the output ends higher. For $x^2$ from 1 to 3 the rate is 4.
Negative: the output ends lower. For $g(x) = \frac{12}{x}$ from 1 to 4, the output falls from 12 to 3, so the rate is $\frac{3 - 12}{4 - 1} = -3$ .
Zero: the output ends where it started. For $x^2$ from $-2$ to 2 the rate is $\frac{4 - 4}{2 - (-2)} = 0$ .

A zero average does not mean nothing happened. Between $-2$ and 2, the curve $x^2$ falls to 0 and climbs back to 4.

The order of the two points does not matter, as long as top and bottom use the same order. From 3 back to 1, $x^2$ gives $\frac{1 - 9}{1 - 3} = 4$ again.

The interval matters

On a curve, the average rate changes with the interval. For $x^2$ it is 3 from 1 to 2, and 5 from 2 to 3. A straight line is the exception: its average rate equals its slope on every interval.

Try it average_rate is the formula above. The cell checks each example on this page. Every result is a decimal (4.0, not 4) because / in Python always gives a decimal.

def average_rate(f, a, b): return (f(b) - f(a)) / (b - a) def square(x): return x**2 def twelve_over(x): return 12 / x print(average_rate(square, 1, 3)) print(average_rate(square, 1, 2), average_rate(square, 2, 3)) print(average_rate(twelve_over, 1, 4)) print(average_rate(square, -2, 2))

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From a chord to a tangent

A curve is steeper in some places than others, so it has no single slope. To get the slope at one point $a$ , take the chord from $a$ to a nearby point $a + h$ . Then let the gap $h$ shrink.

The gap $h$ can be negative. Then the second point sits to the left of $a$ , and the chord comes in from that side.

For $f(x) = x^2$ at $a = 3$ , here are the chord slopes from both sides.

gap $h$	1	0.1	0.01	0.001
chord slope	7	6.1	6.01	6.001

gap $h$	$-1$	$-0.1$	$-0.01$	$-0.001$
chord slope	5	5.9	5.99	5.999

From the right the slopes fall towards 6. From the left they rise towards 6. Both one-sided limits (lesson 2.1.2) agree, so the slope at 3 is 6.

Explore

Move the gap h towards 0 from either side and watch the chord of y = x² turn into the tangent at 3.

Gap h = 1

Show the tangent

Second point: (4, 16)
Chord slope: (16 − 9) ÷ 1 = 7
Tangent slope at 3: 6, so the chord is off by 1

Algebra gives the same answer without a table. Expand the top, then cancel $h$ , which is allowed because $h \neq 0$ :

\frac{(3 + h)^2 - 9}{h} = \frac{6h + h^2}{h} = 6 + h

As $h \to 0$ , $6 + h \to 6$ . You cannot put $h = 0$ straight into the chord slope: it would read $\frac{0}{0}$ . The limit is how you get past that, as in lesson 2.1.1.

Rule: Slope at a point

\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

The limit of chord slopes as the second point slides into the first, from either side. It is the slope of the tangent line: the line that touches the curve at $a$ and points the same way. Lesson 2.2.2 turns this into a function, the derivative.

x^2

def f(x): return x**2 for h in [1, 0.1, 0.01, 0.001]: right = (f(3 + h) - f(3)) / h left = (f(3 - h) - f(3)) / -h print(h, round(right, 6), round(left, 6))

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A second example: $x^3$ at 2

Take $f(x) = x^3$ at $a = 2$ , where $f(2) = 8$ . With $h = 1$ the chord slope is $\frac{27 - 8}{1} = 19$ . With $h = 0.1$ it is $\frac{9.261 - 8}{0.1} = 12.61$ , and with $h = 0.01$ it is 12.0601.

From the left, $h = -0.1$ gives 11.41 and $h = -0.01$ gives 11.9401. Both sides settle on 12.

The algebra agrees. Expanding gives $(2 + h)^3 = 8 + 12h + 6h^2 + h^3$ , so the chord slope is $12 + 6h + h^2$ . As $h \to 0$ it tends to 12.

x^3

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The tangent line

Once you know the slope $m$ at $a$ , the tangent is the line through the point $(a, f(a))$ with that slope.

Rule: Tangent line at a

y = f(a) + m(x - a)

At $x = a$ this gives $f(a)$ , so the line passes through the point. Each step of 1 to the right adds $m$ , so its slope is $m$ . It is the same line as $y = mx + c$ from lesson 1.2.1, with $c = f(a) - ma$ .

For $x^2$ at 3 the tangent is $y = 9 + 6(x - 3)$ . Multiplying out gives $y = 6x - 9$ .

The tangent to $y = x^2$ at $(3, 9)$ , far away and close up. Far from 3 it drifts below the curve: at $x = 5$ the gap is 4. Inside the dashed box around $(3, 9)$ , enlarged, the two are hard to tell apart.

How closely the tangent copies the curve

Near $a$ the tangent is an excellent stand-in for the curve. Further away it drifts off. Here is $x^2$ against its tangent at 3.

$x$	3.1	3.5	4	5
curve $x^2$	9.61	12.25	16	25
tangent $6x - 9$	9.6	12	15	21
gap	0.01	0.25	1	4

The gap is the square of the distance from 3. Algebra shows why: $x^2 - (6x - 9) = (x - 3)^2$ . Move 10 times closer to 3 and the gap shrinks 100 times.

That is why the tangent is called the best straight-line copy of a curve near a point. It lets you estimate values quickly: $3.1^2$ is close to $6 \times 3.1 - 9 = 9.6$ . The same idea gives Newton's method (lessons 1.3.3 and 2.4.3).

A tangent does not have to stay on one side of the curve. For $x^3$ at 0, the chord slope is $\frac{h^3}{h} = h^2$ , which tends to 0. So the tangent is the line $y = 0$ , and the curve crosses it at 0.

x^2

def f(x): return x**2 def tangent(x): return 9 + 6 * (x - 3) for x in [3.1, 3.5, 4, 5]: print(x, round(f(x), 6), round(tangent(x), 6), round(f(x) - tangent(x), 6))

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Common mistakes

An average rate needs an interval; the slope at a point is a limit. "The rate of $x^2$ at 3" means the slope there, 6, not an average.
Divide by the change in input, $b - a$ , not by $b$ .
Keep the same order on top and bottom. $\frac{f(b) - f(a)}{a - b}$ has the wrong sign.
Do not put $h = 0$ straight in. It gives $\frac{0}{0}$ . Simplify first, then let $h$ shrink.
Check both sides. The slope at a point exists only when both one-sided limits agree. Lesson 2.2.2 shows a corner where they do not.
The tangent is only a good copy near the point. Far away it can miss badly, and it may cross the curve.

Rates of change and tangents

0 of 11 correct

1.
Find the average rate of change of $x^2$ from 1 to 2
2.
Find the average rate of change of $x^2 - 2x$ from 1 to 5
3.
Find the average rate of change of $\frac{12}{x}$ from 2 to 6
4.
Find the average rate of change of $5 - 2x$ from 1 to 4
5.
Find the average rate of change of $x^3$ from 0 to 2
6.
A car covers 150 km in 2 hours. What is its average speed in km per hour?
7.
Find the chord slope of $x^2$ at 3 with gap $h = 0.5$
8.
Find the chord slope of $x^3$ at 2 with gap $h = -0.1$
9.
The slope of $x^3$ at 2 is 12. Write the tangent line at 2 in the form $mx + c$
10.
Use the tangent $y = 9 + 6(x - 3)$ to estimate $3.1^2$
11.
The tangent to $x^2$ at 3 misses the curve by $(x - 3)^2$ . How far off is it at $x = 3.2$ ?

How to type answers

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Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

\frac{f(b) - f(a)}{b - a}

Rate of change and the slope of a tangent

What a rate of change measures

Average rate of change

Rule: Average rate of change

Rising, falling and flat

The interval matters

From a chord to a tangent

Rule: Slope at a point

A second example: $x^3$ at 2

The tangent line

Rule: Tangent line at a

How closely the tangent copies the curve

Common mistakes

Rates of change and tangents

Summary card

Free further reading

What a rate of change measures

Average rate of change

Rule: Average rate of change

Rising, falling and flat

The interval matters

From a chord to a tangent

Rule: Slope at a point

A second example: x3x^3x3 at 2

The tangent line

Rule: Tangent line at a

How closely the tangent copies the curve

Common mistakes

Summary card

Free further reading

A second example: $x^3$ at 2