D. Characteristics of Functions · Rates of Change

Average Rates of Change in Applications

D1.1 D1.4About 70 minutes

Learning goals

I can calculate average rates of change for linear, quadratic, polynomial, exponential and sinusoidal models.
I can work from an equation, a table (including unequal intervals) or a graph.
I can interpret average rates of change in the natural, physical and social sciences, with units.
I can compare average rates over different intervals to describe how a quantity is changing.

Words to know

model
An equation that describes a real situation, such as N(t) = 500(2)^(t/3) for a bacteria culture.

exponential growth
A quantity multiplied by the same factor in each equal time step.

sinusoidal function
A function whose graph is a sine-type wave, such as d(t) = 3 cos(πt/6) + 5.

radian mode
The calculator setting for angles in radians. Needed for models like cos(πt/6).

Same formula, any function

In the last lesson you met the formula for the average rate of change:

(f(b) − f(a)) ÷ (b − a)

It works for every kind of function you have studied in this course: polynomial (from Polynomial Functions and Equations), exponential and logarithmic (from Exponential and Logarithmic Functions), sinusoidal (from Radian Measure and Trigonometric Functions) and rational (from Rational Functions and Equations). The only things that change are how you evaluate f and what the answer means.

In this lesson you will work through real models from biology, finance, oceans and design.

Four steps for any average rate

Identify the interval [a, b] and the units of each variable.
Evaluate f(a) and f(b) carefully (keep extra digits; round only at the end).
Divide the change in f by the change in the input.
Interpret: number, units and a sentence with "on average".

Worked example

Example 1: Bacteria (natural science)

A bacteria culture has N(t) = 500(2)^(t/3) cells after t hours. Find the average rate of growth over [0, 3], [3, 6] and [6, 9].

N(0) = 500, N(3) = 500(2) = 1000, N(6) = 500(4) = 2000, N(9) = 500(8) = 4000.
[0, 3]: (1000 − 500) ÷ 3 ≈ 166.7 cells/h.
[3, 6]: (2000 − 1000) ÷ 3 ≈ 333.3 cells/h.
[6, 9]: (4000 − 2000) ÷ 3 ≈ 666.7 cells/h.

About 166.7, 333.3 and 666.7 cells per hour. The average rate doubles every 3 hours, just like the population does.

Average growth rates of the bacteria culture over three equal intervals. For an exponential model, the average rates over equal intervals grow by the same factor (here 2).

Worked example

Example 2: An investment (social science and finance)

$1000 is invested at 4% per year, compounded annually: A(t) = 1000(1.04)^t. Find the average rate of change of the value over the first 5 years and over years 5 to 10.

A(0) = 1000. A(5) = 1000(1.04)^5 ≈ 1216.65. A(10) = 1000(1.04)^10 ≈ 1480.24.
[0, 5]: (1216.65 − 1000) ÷ 5 ≈ 43.33 $/year.
[5, 10]: (1480.24 − 1216.65) ÷ 5 ≈ 52.72 $/year.

About $43.33 per year, then about $52.72 per year. The money grows faster later, because interest is earned on interest.

Worked example

Example 3: Tides (physical science, radians)

The water depth at a harbour is modelled by d(t) = 3 cos(πt/6) + 5, in metres, t hours after high tide. Find the average rate of change of depth over [0, 2], [2, 4] and [4, 6]. (Set your calculator to radian mode.)

d(0) = 3 cos 0 + 5 = 8. d(2) = 3 cos(π/3) + 5 = 3(0.5) + 5 = 6.5.
d(4) = 3 cos(2π/3) + 5 = 3(−0.5) + 5 = 3.5. d(6) = 3 cos π + 5 = −3 + 5 = 2.
[0, 2]: (6.5 − 8) ÷ 2 = −0.75 m/h. [2, 4]: (3.5 − 6.5) ÷ 2 = −1.5 m/h. [4, 6]: (2 − 3.5) ÷ 2 = −0.75 m/h.

−0.75, −1.5 and −0.75 m/h. The tide falls slowly just after high tide, fastest in the middle, and slowly again near low tide.

Secants on the tide curve over three equal 2-hour intervals. The middle secant is steepest, so the water drops fastest around mid-tide.

Worked example

Example 4: An open box (design, polynomial)

An open box is made from a 30 cm by 20 cm card by cutting x cm squares from each corner and folding up the sides. Its volume is V(x) = x(30 − 2x)(20 − 2x). Find the average rate of change of volume over [2, 4] and [4, 6].

V(2) = 2(26)(16) = 832 cm³. V(4) = 4(22)(12) = 1056 cm³. V(6) = 6(18)(8) = 864 cm³.
[2, 4]: (1056 − 832) ÷ 2 = 112 cm³ per cm.
[4, 6]: (864 − 1056) ÷ 2 = −96 cm³ per cm.

112 cm³/cm, then −96 cm³/cm. Bigger cuts first increase the volume, then decrease it, so the largest volume is somewhere between x = 2 and x = 6.

Worked example

Example 5: A table with unequal intervals (health)

A school nurse tracks the total number of students who have caught a cold since a cold started going around.

day: 0, 3, 5, 9
total cases: 2, 9, 20, 41

(Sample data.) Find the average rate of new cases per day on each interval. When was the cold spreading fastest?

[0, 3]: (9 − 2) ÷ 3 ≈ 2.33 cases/day.
[3, 5]: (20 − 9) ÷ 2 = 5.5 cases/day.
[5, 9]: (41 − 20) ÷ 4 = 5.25 cases/day.
The intervals have different lengths, so compare the rates, not the raw changes (21 cases in [5, 9] looks big, but it took 4 days).

Fastest over [3, 5], at 5.5 new cases per day on average.

Just for fun

Under ideal lab conditions, some bacteria, such as E. coli, can divide about every 20 minutes. If that went on for two days with no limits, one cell would outweigh the Earth. Luckily, real bacteria run out of food (and room) long before that.

Hands-on activity

Real data from your community

You need: a device with internet access, paper and a calculator.

Go to the Statistics Canada Census Profile (search for "Census Profile 2021 Statistics Canada") and look up your town or city. Find its population in 2016 and in 2021 (both are shown on the profile).
Calculate the average rate of change of population in people per year.
Do the same for a nearby town and for Ontario as a whole.
Which grew fastest in people per year? Which grew fastest as a percent of its 2016 population? Are the answers the same? Why not?

What to look for: a big city can have the largest rate in people per year but a smaller percent growth than a small fast-growing town. Each rate answers a different question, so always state which one you mean.

Your turn

For N(t) = 500(2)^(t/3), find the average rate of change over [0, 6].

For A(t) = 1000(1.04)^t, find the average rate of change over [0, 10].

For the tide d(t) = 3 cos(πt/6) + 5, find the average rate of change over [0, 6].

For the same tide, find the average rate of change over [0, 12] and interpret it.

For the box, V(x) = x(30 − 2x)(20 − 2x), find the average rate of change over [0, 2].

For the cold data, find the average rate of new cases per day over the whole period [0, 9].

Explain why the average rates of the bacteria culture in Example 1 double from one 3-hour interval to the next.

More practice: warm-up

Find the average rate of change of f(x) = 2x + 7 on [3, 8].

Find the average rate of change of f(x) = x² on [2, 5].

Find the average rate of change of f(x) = 3ˣ on [0, 2].

Find the average rate of change of f(x) = sin x on [0, π/2] (radians).

A population is measured in people and time in years. What are the units of its average rate of change?

More practice: on your own

A car bought for $28 000 loses value: V(t) = 28 000(0.85)^t after t years. Find the average rate of change over the first 3 years and interpret it.

A Ferris wheel seat's height is h(t) = 21 − 20 cos(πt/30) metres after t seconds. Find the average rate of change over [0, 10] and over [10, 20]. When is the seat rising faster?

A cup of tea cools: T(t) = 20 + 70(0.9)^t in °C after t minutes. Find the average rate of change over [0, 5] and [5, 10].

A theatre's revenue is R(p) = −50p² + 1000p dollars when tickets cost p dollars. Find the average rate of change of revenue over [5, 8] and over [8, 12]. Interpret.

For the cold data, a classmate says "the cold spread fastest from day 5 to 9 because the most new cases (21) happened then". Is this correct?

Challenge

For f(x) = a·bˣ, show that the average rate of change over [x, x + 1] is (b − 1)·f(x). What does this say about exponential growth?

For the tide d(t) = 3 cos(πt/6) + 5, find an interval of length 3 h with an average rate of change of 0. Explain.

For A(t) = 1000(1.04)^t, find the average rate of change over [10, 20], and show that it equals the rate over [0, 10] multiplied by 1.04^10.

Check your understanding

Quiz

For N(t) = 500(2)^(t/3), what is the average rate of change over [3, 6] hours?

For the tide d(t) = 3 cos(πt/6) + 5, the average rate of change over [0, 12] is 0 m/h. What does this mean?

What is the average rate of change of f(x) = sin x over [0, π] (radians)?

A car's value has an average rate of change of −$3600 per year over [0, 3]. Which interpretation is best?

For an exponential growth model, how do the average rates of change over consecutive equal intervals behave?

A table gives total cases: 9 on day 3, 20 on day 5 and 41 on day 9. Which interval had the greater average rate of new cases?

For parents and teachers

Big ideas (D1.1, D1.4): the average-rate formula works for every function family. Comparing average rates over equal intervals tells you how a quantity is changing: doubling rates for exponential growth, a symmetric pattern for a sinusoidal model, a sign change where a polynomial reaches a maximum.

How to help: Look at a real number together: your household's electricity or water use in two months, the price of a litre of milk this year and last year, or your town's population on the Statistics Canada website. Ask your teen to calculate an average rate and say what it means in a sentence.

Common difficulties:

Calculator in degree mode for models like cos(πt/6). Check: cos(π) should give −1.
Rounding A(5) to 1217 and then getting slightly wrong rates. Keep two decimals for money, more for intermediate steps.
Comparing raw changes instead of rates when intervals have different lengths.
Writing exponents incorrectly on the calculator: 500 × 2^(6/3) needs brackets around 6/3.