A. Rate of Change · Limits and Rates of Change

Rates of Change in the Real World

A1.1About 65 minutes

Learning goals

I can describe real-world rates of change in words, with a table, with a graph and with an equation.
I can calculate an average rate of change from each representation and state it with correct units.
I can tell whether a statement describes an average or an instantaneous rate of change.
I can interpret the sign and size of a rate of change in context.

Words to know

rate of change
How fast a dependent quantity changes compared with an independent quantity: Δy ÷ Δx, in units of y per unit of x.

average rate of change
The rate of change over an interval [a, b]: (f(b) − f(a)) ÷ (b − a).

instantaneous rate of change
The rate of change at one particular value of the independent variable, such as a speedometer reading at one moment.

interval
All the values between two endpoints. [a, b] means a ≤ x ≤ b.

calculus
The mathematics of change. Differential calculus studies rates of change; integral calculus studies accumulation.

representation
A way of showing a relationship: in words, numerically (a table), graphically or algebraically (an equation).

Welcome to calculus

Calculus is the mathematics of change. Engineers use it to design bridges that flex safely in the wind. Doctors use it to model how fast a medicine leaves the bloodstream. Economists use it to decide what price earns the most money. Every first-year university program in science, engineering, business and health sciences has a calculus course, and this course is the doorway.

It all starts with one question: how fast is something changing?

Remember from MHF4U: in the Advanced Functions unit Rates of Change, you found average rates of change as slopes of secants, estimated instantaneous rates of change as slopes of tangents by shrinking the interval (preceding, following and centred intervals), and, for polynomials, simplified the difference quotient (f(a + h) − f(a)) ÷ h to find tangent slopes such as 2a for y = x². In Combining Functions and Modelling you added, multiplied and composed functions to build models of real situations such as damped motion and profit. This unit takes those ideas and makes them exact with a new tool: the limit.

The rate of change, in one line

average rate of change of f over [a, b] = (f(b) − f(a)) ÷ (b − a) = Δy ÷ Δx

Three habits will save you many marks in this course:

Always write units. They are "units of y per unit of x": km/h, L/min, $ per item, people per year, °C per metre.
Read the sign. Positive means the quantity is increasing; negative means it is decreasing; zero means no overall change on that interval.
Say what kind of rate it is. An average rate belongs to an interval ("from 2 p.m. to 5 p.m.", "over the first 10 minutes"). An instantaneous rate belongs to a moment ("at 3 p.m.", "when the radius is 4 cm", "right now").

Four representations

A rate of change can be shown in words, numerically in a table, graphically or algebraically with an equation. Here is one situation in all four.

A 500 L water tank on a farm drains through a valve at the bottom. Water drains quickly when the tank is full (more pressure) and slowly when it is nearly empty. A model for the volume is V(t) = 500(1 − t/40)² litres, t minutes after the valve opens, for 0 ≤ t ≤ 40.

From the equation (or the table), the average rate over the first 10 minutes is (281.25 − 500) ÷ 10 = −21.875 L/min. Over the last 10 minutes it is (0 − 31.25) ÷ 10 = −3.125 L/min. On the graph, the first secant is much steeper than the last.

One situation, four representations. The table and equation give the numbers; the graph shows the same story as steep secants early and shallow secants late.

year	2016	2018	2020	2022	2024
population	12 400	12 950	13 380	14 100	14 900

Population of Pine Lake (a made-up town), recorded every two years.

Worked example

Example 1: A rate from a table

The population of the town of Pine Lake (a made-up town) was recorded every two years, as shown in the table above. Find the average rate of change of the population from 2016 to 2024, from 2016 to 2018 and from 2022 to 2024. Describe the trend.

2016 to 2024: (14 900 − 12 400) ÷ (2024 − 2016) = 2500 ÷ 8 = 312.5 people per year.
2016 to 2018: (12 950 − 12 400) ÷ 2 = 550 ÷ 2 = 275 people per year.
2022 to 2024: (14 900 − 14 100) ÷ 2 = 800 ÷ 2 = 400 people per year.
The two-year rates are not all equal, so the rate of change is not constant.

Overall the town grew by about 312.5 people per year, but growth sped up: 275 people/year at the start and 400 people/year at the end. (A fraction of a person is fine for an average rate.)

Worked example

Example 2: A rate from an equation

A stone falls from a bridge. Its height above the river is h(t) = 45 − 4.9t² metres after t seconds. Find its average velocity over [1, 3] and say what the sign means.

h(1) = 45 − 4.9(1) = 40.1 m.
h(3) = 45 − 4.9(9) = 45 − 44.1 = 0.9 m.
Average velocity = (0.9 − 40.1) ÷ (3 − 1) = −39.2 ÷ 2 = −19.6 m/s.

The average velocity is −19.6 m/s. The negative sign means the height is decreasing: the stone is moving down.

A cyclist's distance from home. Each straight segment has a constant rate (the speed on that part of the ride). The average rate for the whole ride is the slope from (0, 0) to (40, 9).

Worked example

Example 3: A rate from a graph

Use the cyclist's graph above. Find the average speed from t = 0 to t = 10 min, from t = 20 to t = 30 min, and for the whole ride, in km/h.

0 to 10 min: Δd = 3 km and Δt = 10 min = 1/6 h. Speed = 3 ÷ (1/6) = 18 km/h.
20 to 30 min: Δd = 5 − 5 = 0 km, so the speed is 0 km/h. The cyclist stopped (maybe at a red light, or for a snack).
Whole ride: Δd = 9 km and Δt = 40 min = 2/3 h. Speed = 9 ÷ (2/3) = 13.5 km/h.

18 km/h, 0 km/h, and 13.5 km/h for the whole ride. The average for the whole ride hides the stop and the fast final stretch.

Average or instantaneous?

Look for the clue words.

Average: "from … to …", "over", "during", "between", "in the first 3 hours", "last year".
Instantaneous: "at", "when", "at the moment", "right now", "at exactly 2 s", a speedometer or radar-gun reading.

The rest of this unit is about finding instantaneous rates exactly.

Worked example

Example 4: Interpreting statements

Classify each rate as average or instantaneous, and give its units.

(a) "At 9:00 a.m. the river was rising at 4 cm per hour." (b) "The car's value fell by $2800 per year over its first five years." (c) "When the radius is 10 cm, the balloon's volume is growing at 400 cm³/s." (d) "Between January and June, the store's sales grew by 6% per month."

(a) "At 9:00 a.m." names one moment: instantaneous, cm/h.
(b) "Over its first five years" is an interval: average, $/year (negative, because the value fell).
(c) "When the radius is 10 cm" names one moment: instantaneous, cm³/s.
(d) "Between January and June" is an interval: average, % per month.

(a) instantaneous, (b) average, (c) instantaneous, (d) average.

Just for fun

Isaac Newton called changing quantities fluents and their rates of change fluxions. So if calculus homework ever feels overwhelming, you can honestly say you are in a state of fluxion.

Hands-on activity

The draining bottle

You need: an empty plastic water bottle (500 mL or larger), masking tape, a ruler, a marker, a timer, a sink, and a grown-up with a thumbtack or small nail.

Grown-up step: make one small hole in the side of the bottle about 2 cm above the bottom. Be careful with the sharp point.
Stick a strip of tape up the side of the bottle and mark it every centimetre from the hole upward.
Cover the hole with your finger and fill the bottle with water over the sink.
Uncover the hole and start the timer. Every 15 s, call out the water height above the hole. Have someone write it down (or film it and read it later).
Make a table. Find the average rate of change of the height for each 15 s interval, in cm/s.
Sketch the graph of height against time. Describe it in words.

What to look for: the height falls quickly at first and more slowly later, so the average rates are negative and get closer to 0. The graph curves down and flattens out, just like the tank model V(t) = 500(1 − t/40)². That shape appears whenever the outflow depends on how much water is pushing on the hole.

Your turn

A sugar bush (a maple forest tapped for sap) collects 1500 L of sap in 6 hours. Find the average rate of collection.

For the tank V(t) = 500(1 − t/40)², find the average rate of change over [10, 20].

For Pine Lake (Example 1), find the average rate of change of the population from 2018 to 2022.

Average or instantaneous? "The radar gun read 96 km/h as the car passed the school."

Using the cyclist's graph, find the speed from t = 10 to t = 20 min in km/h.

Find the average velocity of the stone h(t) = 45 − 4.9t² over [0, 2].

Air temperature typically changes by about −0.65 °C per 100 m of height gained. What does this mean, and about how much would the temperature change on an 800 m climb?

More practice: warm-up

Find the rate of change from (2, 7) to (6, 19).

What are the units of the rate of change of volume (in litres) with respect to time (in seconds)?

A car travels 240 km in 3 h. Find its average speed.

The temperature is falling. Is its rate of change positive or negative?

Find the average rate of change of f(x) = x² from x = 1 to x = 3.

More practice: on your own

For f(x) = 2ˣ, find the average rate of change over [0, 3] and over [3, 6]. What do you notice?

A mug of hot chocolate cools according to T(t) = 20 + 70(0.95)ᵗ °C after t minutes. Find the average rate of change of temperature over the first 10 minutes.

Snow depth on a deck was 12 cm at 6 a.m., 20 cm at 9 a.m. and 23 cm at noon. Find the average rate of snowfall for each 3-hour period and describe what happened.

A savings account starts at $200 and grows by $25 per week. Write an equation, describe the graph and give the rate of change.

For the tank, find the average rate of change over [0, 40]. Why is it different from the rate over [0, 10]?

Challenge

Show that for f(x) = x², the average rate of change over [a, b] is a + b (for a ≠ b).

For the tank V(t) = 500(1 − t/40)², find k so that the average rate over [0, k] is −20 L/min.

A driver goes 60 km at 60 km/h, then 60 km at 120 km/h. Is the average speed for the trip 90 km/h? Explain.

Check your understanding

Quiz

Which statement describes an instantaneous rate of change?

Find the average rate of change of f(x) = x³ over [1, 3].

A town's population falls from 8400 to 7900 over 4 years. What is the average rate of change?

The cost C (in dollars) of making x hoodies is measured. What are the units of the rate of change of C with respect to x?

On a distance-time graph, a horizontal segment means:

For h(t) = 45 − 4.9t², what is the average velocity over [1, 3]?

For parents and teachers

Big idea (A1.1): a rate of change compares two changes, and it can be read from words, a table, a graph or an equation. This lesson reviews and extends the MHF4U unit Rates of Change and sets up the rest of calculus.

How to help: Point out rates in daily life (fuel economy in L/100 km, a phone's battery drain, prices per kilogram at the grocery store). Ask: "Is that an average or an instantaneous rate? What are its units? Is it positive or negative?"

Common difficulties:

Leaving off units, or writing them upside down (hours per km instead of km/h).
Sign errors when subtracting negative values, such as (−8) − 2.
Mixing minutes and hours. Convert before dividing (10 min = 1/6 h).
Thinking the average of two speeds is the average speed (Challenge 3).