Lesson 3 of 7Rates of Change & Combining Functions

Unit 6 · Lesson 3 of 7

Rates of Change and the Shape of a Graph

The sign of the instantaneous rate of change tells you where a graph rises, where it falls and where it turns; its size tells you where it is steepest. Sinusoids change fastest at their axis, and exponentials change at a rate proportional to their value.

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Key things to remember

1Positive instantaneous rate on an interval: the function is increasing there. Negative: decreasing.
2At a local maximum the rate changes from positive to negative; at a local minimum, from negative to positive. The tangent is horizontal at both.
3A zero rate without a sign change, as for $y = x^{3}$ at $x = 0$ , is not a turning point.
4The graph is steepest where the rate is largest in size, often at a point of inflection between turning points.
5For a sinusoid the rate is zero at maximums and minimums and greatest in size at the axis, where it equals $∣ a ∣∣ k ∣$ .
6For an exponential function the rate is proportional to the value: rate ÷ value is the same at every point.

Imagine hiking a trail with a slope meter strapped to your boot. While the reading is positive you are climbing; while it is negative you are descending; and at the exact top of a hill, or the bottom of a valley, the reading passes through zero, because for an instant the ground is level. Without ever seeing a map, the sequence of readings tells you the shape of the trail. The instantaneous rate of change is that slope meter for a function. This lesson connects the numbers you learned to estimate in the last topic with the features of a graph you already know how to describe — increasing and decreasing intervals, local maximums and minimums, the steepest part of a curve — and then looks at two families whose rates follow a striking pattern.

The sign of the rate and the direction of the graph

If the instantaneous rate of change is positive at every point of an interval, every tangent line there slopes upward, so the function is increasing on that interval. If the rate is negative throughout, the function is decreasing. At a local maximum the curve stops rising and starts falling, so the rate changes sign from positive to negative and passes through zero; the tangent there is horizontal. At a local minimum the rate changes from negative to positive. The sign CHANGE is what matters. For $y = x^{3}$ the rate is zero at $x = 0$ , but it is positive on both sides, so the curve only flattens for an instant and keeps rising: a zero rate is a candidate for a turning point, not a guarantee.

x	f(x)	Estimated rate (centred difference)	What the graph is doing
0	1	9	Rising steeply
0.5	4.125	3.75	Still rising, but less steeply
1	5	0	Level: rate changes + to −, so a local maximum
2	3	−3	Falling as steeply as it ever falls between the turns
3	1	0	Level: rate changes − to +, so a local minimum
4	5	9	Rising again

Rates for f(x) = x³ − 6x² + 9x + 1. f increases for x < 1 and x > 3 and decreases for 1 < x < 3.

Fig 1The tangent slope at a point is the instantaneous rate; its sign shows whether the curve is rising or falling there.

VideoReads off where the tangent slopes on a graph are positive, negative and zero, which shows where the function rises, falls and levels off.Watch on YouTube

Where the rate is greatest

The rate of change is itself a quantity that changes. In the table, the rate drops from $9$ at $x = 0$ , through $0$ at the maximum, down to $- 3$ at $x = 2$ , then climbs back through $0$ to $9$ . So the curve is falling fastest at $x = 2$ , halfway between its two turning points. That point, where the graph switches from bending downward to bending upward, is the point of inflection, and on many real curves it is the most interesting moment: the instant a disease outbreak is spreading fastest, the place a road is steepest, the time a car on a distance–time graph is moving fastest. To find it from data, estimate the rate at several points and look for where the rate itself is largest or smallest. With a formula, a table of centred differences on a fine grid, or graphing technology, does the same job.

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Move the point along the curve and watch the tangent slope: positive while rising, zero at the turns, steepest between them.

Sinusoids: fastest at the axis

A Ferris wheel rider has height $h (t) = - 20 cos (\frac{2 π}{3} t) + 22$ m. At the bottom and the top the rider is moving sideways, so the vertical rate of change is $0$ : these are the local minimum and maximum. The height changes fastest when the rider passes the level of the axle, moving straight up on one side and straight down on the other. Estimating the rate at $t = 0.75$ min, a quarter turn after boarding, gives about $41.9$ m/min; at $t = 2.25$ min it is about $- 41.9$ m/min. That pattern holds for every sinusoid: the rate is zero at maximums and minimums and largest in size where the curve crosses its axis. In fact, numerical estimates show that the greatest rate of $y = a sin (k (x - d)) + c$ is $∣ a ∣∣ k ∣$ — the amplitude times $∣ k ∣$ . Here $20 \times \frac{2 π}{3} \approx 41.9$ . The same reasoning tells you that tides rise fastest at mid-tide, and days lengthen fastest around the spring equinox.

Fig 2A periodic graph's maximums and minimums are where the rate is zero; the steepest stretches lie between them.

Exponentials: rate proportional to value

A bacteria culture modelled by $N (t) = 500 (2)^{t /3}$ doubles every 3 hours. Estimate its instantaneous growth rate at several times and something remarkable appears: each time the population doubles, so does the rate. Dividing the rate by the population gives the same number, about $0.231$ per hour, at every time. That is the defining property of exponential change — the rate is proportional to the current value — and it is why exponential growth feels slow at first and then explosive. A population of 4000 is growing eight times as fast as a population of 500, simply because there is eight times as much of it to grow. Decay works the same way with a negative constant: a cooling drink or a decaying isotope changes fastest at the start and ever more slowly as it approaches its asymptote.

t (h)	N(t)	Estimated rate (bacteria/h)	Rate ÷ N(t)
0	500	115.5	0.231
3	1000	231.0	0.231
6	2000	462.1	0.231
9	4000	924.2	0.231

The rate doubles whenever the population doubles, so their ratio never changes.

A zero rate is not a zero value

At the top of the Ferris wheel the height is

42

m and the rate is

0

m/min. Students often mix the two up when interpreting. The value tells you WHERE the quantity is; the rate tells you how fast it is changing and in which direction. Always give a rate with units of output per input, such as m/min or bacteria per hour, and describe its meaning in words.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

For

f (x) = x^{3} - 6 x^{2} + 9 x + 1

, centred-difference estimates give rates of

9

0

- 3

0

and

9

x = 0, 1, 2, 3, 4

. Describe where

f

increases and decreases, and identify any local maximum or minimum.

Example 2

A ball's height is

h (t) = - 4.9 t^{2} + 19.6 t + 2

m. Find the instantaneous rate of change at

t = 1

2

and

3

s, and explain what each tells you.

Example 3

For

h (t) = - 20 cos (\frac{2 π}{3} t) + 22

m, find where the rider's height changes fastest and estimate that rate.

Example 4

For

N (t) = 500 (2)^{t /3}

bacteria, estimate the growth rate at

t = 0

3

and

6

h, and show that the rate is proportional to the population.

Where marks get lost

Concluding there is a turning point wherever the rate is zero, without checking that the rate changes sign.
Saying a function is decreasing because its VALUES are negative, when increasing and decreasing depend only on the sign of the rate.
Placing a sinusoid's fastest change at its maximum, where the rate is actually zero.
Confusing a value with a rate: the height at the top of a Ferris wheel is 42 m, but its rate there is 0 m/min.
Assuming an exponential grows at a constant rate. Its rate grows in proportion to its value.

Flashcards 9

Rate positive on an interval

Rate changes from + to −

Rate changes from − to +

Why is

x = 0

not a turning point of

y = x^{3}

+ 5 more

Practice

The instantaneous rate of change of f is negative at every point of the interval 2 < x < 5. What must be true?

At x = a the rate of change of f is 0, positive just to the left, and negative just to the right. What is at x = a?

For

f (x) = x^{3} - 6 x^{2} + 9 x + 1

, estimate the instantaneous rate at

x = 2

using a centred difference with

h = 0.1

. Give 2 decimal places.

For

h (t) = - 20 cos (\frac{2 π}{3} t) + 22

m, what is the greatest rate of change of height, in m/min to 1 decimal place?

m/min

At which features of a sinusoidal graph is the instantaneous rate of change zero?

N (t) = 500 (2)^{t /3}

grows at about

0.231 N

bacteria per hour. Estimate the growth rate when

t = 6

h, to the nearest whole number.

bacteria/h

For

f (x) = 2^{x}

, the ratio (instantaneous rate) ÷ (value) is the same at every

x

. Estimate it to 3 decimal places.

For

y = sin x

, order these points by their instantaneous rate of change, from most negative to most positive.

1 $x = 0$
2 $x = \frac{π}{4}$
3 $x = \frac{π}{2}$
4 $x = π$
5 $x = \frac{3 π}{4}$

Match each feature of a graph to the behaviour of its rate of change.

Local maximum

Local minimum

Increasing interval

Steepest point between two turns

For

y = x^{3}

, the rate of change at

x = 0

0

. Which statement is correct?

For a ball with

h (t) = - 4.9 t^{2} + 19.6 t + 2

m, the rate of change at

t = 3

s is

- 9.8

m/s. What does this mean?

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