D. Characteristics of Functions · Rates of ChangeRates of Change All Around Us
D1.1 D1.2About 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 a rate of change as change in the dependent variable divided by change in the independent variable, with correct units.
- I can tell whether a rate of change is zero, constant or changing, and explain how I know.
- I can interpret the sign of a rate of change in context.
Words to know
rate of change
How fast one quantity changes compared with another: (change in the dependent variable) ÷ (change in the independent variable).
independent variable
The input quantity, usually time or x, that you change or that changes on its own.
dependent variable
The output quantity, usually y or f(x), that depends on the independent variable.
Δ (delta)
A Greek capital letter meaning "change in". Δy = y₂ − y₁ and Δx = x₂ − x₁.
first differences
In a table with equal steps in x, the differences between consecutive y-values.
constant rate of change
A rate that is the same over every interval. Its graph is a straight line.
Your phone is doing math right now
You get on the bus with your phone at 80% battery. After a 45-minute ride, it shows 62%. The battery dropped 18 percentage points in 45 minutes. That is about 0.4 percentage points per minute.
That number is a rate of change. It tells you how fast one quantity (battery level) changes compared with another (time). Rates of change are everywhere: the speed of a snowplow, the growth of a town, the price of groceries, the number of new flu cases each day.
This unit is about rates of change. It is also the bridge to calculus. In Grade 12 Calculus and Vectors (MCV4U), the unit Limits and Rates of Change starts exactly where this unit ends.
What is a rate of change?
A rate of change compares two changes:
rate of change = (change in the dependent variable) ÷ (change in the independent variable) = Δy ÷ Δx
The symbol Δ (delta) means "change in". If y goes from y₁ to y₂, then Δy = y₂ − y₁.
The units are always "units of y per unit of x": km/h, $/month, °C/h, people/year, L/min. Writing the units tells you what the number means.
The sign matters too.
- A positive rate means y goes up as x goes up.
- A negative rate means y goes down as x goes up.
- A rate of zero means y does not change.
For the phone: Δ(battery) = 62 − 80 = −18 percentage points and Δt = 45 min. The rate is −18 ÷ 45 = −0.4 percentage points per minute. The negative sign tells you the battery is draining.
Worked exampleExample 1: A rate from words
At 6 p.m. the temperature in Thunder Bay was 2 °C. By 11 p.m. it was −8 °C. Find the rate of change of temperature, and say what it means.
- The dependent variable is temperature, T. The independent variable is time, t.
- ΔT = (−8) − 2 = −10 °C. Watch the signs: subtracting 2 from −8 gives −10.
- Δt = 11 − 6 = 5 h.
- Rate = ΔT ÷ Δt = −10 ÷ 5 = −2 °C/h.
The rate of change is −2 °C/h. On average, the temperature fell 2 degrees every hour.
Four ways to show a rate of change
The same rate can be shown in words, in a table, on a graph or with an equation. Being able to move between them is a big part of this course.
Here is one situation in all four ways. A rain barrel already holds 50 L. A hose fills it at 8 L every minute.
- Words: "It starts with 50 L and gains 8 L each minute."
- Table: see below.
- Graph: a straight line that starts at 50 on the V-axis and rises 8 for every 1 to the right.
- Equation: V = 50 + 8t, where V is the volume in litres and t is the time in minutes.
| t (min) | 0 | 1 | 2 | 3 | 4 |
|---|
| V (L) | 50 | 58 | 66 | 74 | 82 |
| first difference | | +8 | +8 | +8 | +8 |
Zero, constant or changing?
Rates of change come in three kinds.
- Zero rate: the quantity does not change. A plane cruising at a steady altitude has a rate of change of altitude of 0 m/min. The graph is horizontal.
- Constant rate: the quantity changes by the same amount in every equal time step. A jet flying at a steady 840 km/h covers 840 km every hour. The graph is a straight line that is not horizontal.
- Changing rate: the amount of change is different from one step to the next. A cyclist climbing a hill slows down as the hill gets steeper, so her distance grows less in each minute. The graph is curved.
In a table with equal steps in x, look at the first differences. All zero means a zero rate. All equal (but not zero) means a constant rate. Different differences mean a changing rate.
Remember from Grade 11: in Functions (MCR3U) you used first and second differences to tell linear and quadratic functions apart. A linear function is exactly a function with a constant rate of change.
Worked exampleExample 2: Is the rate constant?
A circle's area is A = πr². Find the change in area when the radius grows from 1 cm to 2 cm, from 2 cm to 3 cm and from 3 cm to 4 cm. Is the rate of change of area constant?
- A(1) = π, A(2) = 4π, A(3) = 9π, A(4) = 16π (in cm²).
- From 1 to 2: ΔA = 4π − π = 3π ≈ 9.42 cm². Δr = 1 cm, so the rate is about 9.42 cm² per cm.
- From 2 to 3: ΔA = 9π − 4π = 5π ≈ 15.71 cm² per cm.
- From 3 to 4: ΔA = 16π − 9π = 7π ≈ 21.99 cm² per cm.
- The changes grow each time, even though the radius grows by the same 1 cm.
The rate is changing (increasing). A bigger circle gains more area for each extra centimetre of radius, because the new ring around the outside is longer.
Worked exampleExample 3: A trick with percentages
Prices rise by 2% per year (the inflation rate). A snack costs $5.00 today. Find its price for the next three years, and the change in dollars each year. Is the rate of change of the price, in dollars per year, constant?
- Each year, multiply by 1.02. Year 1: 5.00 × 1.02 = $5.10. Year 2: 5.10 × 1.02 = $5.202. Year 3: 5.202 × 1.02 ≈ $5.306.
- Changes: 5.10 − 5.00 = $0.10, then 5.202 − 5.10 = $0.102, then 5.306 − 5.202 ≈ $0.104.
- The percent rate stays at 2% per year, but the dollar change grows a little every year.
In dollars per year the rate is changing (slowly increasing), even though the percent rate is constant. Always ask: "rate of change of what, per what?"
Rates of change in many fields
Rates of change show up in every science and in daily life. A few examples:
- Physical sciences: speed (m/s), acceleration (m/s per s), the rate a hot drink cools (°C/min), the rate a battery drains.
- Natural sciences: the growth of a bacteria culture (cells per hour), the spread of an infection (new cases per day), the rate a glacier retreats (m/year).
- Social sciences and business: population growth (people per year), inflation (% per year), the change in a school's graduation rate over time (percentage points per year), the cost of making one more item ($ per item).
For example, educators and First Nations, Métis and Inuit communities in Ontario track how graduation rates change from year to year to see whether programs are working. That is a rate of change measured in percentage points per year.
Notice that the independent variable is not always time. The rate of change of a circle's area is measured per centimetre of radius.
Just for funWhy did the horizontal line refuse to go to the gym? Its rate of change was zero, and it was totally fine with that.
Hands-on activityTime your kitchen tap
You need: a 1 L measuring cup or a clean jug with millilitre marks, a timer (a phone is fine), a pencil and paper.
- Turn the cold tap on to a gentle, steady flow. Do not change it during the experiment.
- Hold the cup under the tap and start the timer. Call out the time each time the water reaches 250 mL, 500 mL, 750 mL and 1000 mL. Have someone write the times down (or film it and pause the video).
- Make a table of volume against time. Find the first differences in time for each 250 mL.
- Calculate the rate of change in mL/s for each 250 mL step, and for the whole litre.
- Now fill a round bowl instead, and use a ruler to measure the height of the water every 5 s. Is the rate of change of height constant?
What to look for: with a steady tap, the volume rate should be nearly constant (often somewhere between about 30 mL/s and 120 mL/s, depending on your tap). The height in a round bowl rises faster at the bottom and slower near the top, because the bowl gets wider. That is a changing rate even though the water flows at a constant rate. Turn the tap off when you finish, and use the water for plants.
Your turn
1A snowplow clears 18 km of road in 1.5 h. Find its average rate of clearing.
2The temperature drops from 4 °C at 6 p.m. to −8 °C at midnight. Find the rate of change of temperature.
ΔT = −8 − 4 = −12 °C over 6 h, so the rate is −12 ÷ 6 = −2 °C/h. The temperature falls 2 °C per hour on average.
3A table has x = 0, 2, 4, 6 and y = 3, 7, 11, 15. Is the rate of change zero, constant or changing? What is it?
The y-values go up by 4 each time x goes up by 2, so the rate is constant: 4 ÷ 2 = 2 (units of y per unit of x). Careful: the first difference is 4, but the step in x is 2.
4Classify the rate of change of each function as zero, constant or changing: (a) y = 5 (b) y = 3x − 2 (c) y = x².
(a) zero (horizontal line); (b) constant, 3; (c) changing (a parabola: first differences are not equal).
5A pool is draining. Its water level is measured in centimetres, and time in minutes. Name the independent and dependent variables and the units of the rate of change.
Independent: time (min). Dependent: water level (cm). Units: cm/min. The rate is negative while the pool drains.
6A candle is 20 cm tall and burns down at a constant 1.5 cm/h. Write an equation for its height h after t hours. When does it burn out?
h = 20 − 1.5t. It burns out when h = 0: 1.5t = 20, so t = 20 ÷ 1.5 ≈ 13.3 h (13 h 20 min).
7A square has side s cm and area A = s². Find the average rate of change of area from s = 2 to s = 3, and from s = 3 to s = 4. Is the rate constant?
From 2 to 3: (9 − 4) ÷ 1 = 5 cm² per cm. From 3 to 4: (16 − 9) ÷ 1 = 7 cm² per cm. The rate is changing (increasing).
More practice: warm-up
1A car travels 150 km in 2 h. What is its average speed?
2You earn $36 in 3 h. What is your rate of pay?
3What is the rate of change of y = 4?
0. The value never changes.
4Your heart beats 72 times in one minute. What are the units of this rate?
5Find the slope of the line through (0, 1) and (2, 7).
(7 − 1) ÷ (2 − 0) = 6 ÷ 2 = 3.
More practice: on your own
1A bank account has $500 at month 0, $470 at month 1, $440 at month 2 and $410 at month 3. Find the rate of change, write an equation, and say when the balance would first drop below $0 if the pattern continues.
Each month the balance falls $30, so the rate is a constant −$30/month and B = 500 − 30m. B = 0 when m = 500 ÷ 30 ≈ 16.7, so the balance first drops below $0 during month 17 (after month 16 it is $20).
2A town has 2000 people in year 0, 2100 in year 1, 2205 in year 2 and 2315 (rounded) in year 3. Is the rate of change constant? Explain.
First differences: 100, 105, about 110. They are not equal, so the rate is changing (increasing). Each year the town grows by 5% of a larger number.
3Find the average rate of change of the area of a circle as the radius grows from 5 cm to 6 cm.
A(6) − A(5) = 36π − 25π = 11π ≈ 34.6 cm² per cm. (Bigger than the 3π for radius 1 to 2, so the rate is increasing.)
4A line passes through (−2, 9) and (4, −3). Find its slope. If x is time in hours and y is litres of water in a tank, what does the slope mean?
Slope = (−3 − 9) ÷ (4 − (−2)) = −12 ÷ 6 = −2. The tank loses 2 L every hour.
5Give one real-world example each of a zero, a constant and a changing rate of change. Include units.
Model answer: zero: the height of a parked car's roof above the road (0 m/s); constant: the cost of gas at $1.50/L as you pump more litres ($1.50/L); changing: the height of a ball thrown upward (m/s, slowing as it rises). Any examples that fit each type, with correct units, are fine.
Challenge
1Show that the average rate of change of a circle's area as the radius grows from r to r + 1 is π(2r + 1). Use it to explain why the rate is not constant, and evaluate it for r = 10.
A(r + 1) − A(r) = π(r + 1)² − πr² = π(r² + 2r + 1 − r²) = π(2r + 1), and Δr = 1. The expression depends on r, so it changes as the circle grows: the rate is not constant. For r = 10: 21π ≈ 66.0 cm² per cm.
2Prices rise 3% per year. An item costs $40.00 now. Find its price after 1, 2 and 3 years and the dollar change each year. Is the rate of change in dollars per year constant?
Prices: 40 × 1.03 = $41.20; 41.20 × 1.03 = $42.436 ≈ $42.44; 42.436 × 1.03 ≈ $43.71. Dollar changes: $1.20, about $1.24, about $1.27. The percent rate is constant, but the dollar rate is changing (increasing).
3Runner A runs at a constant 4 m/s. Runner B starts at the same place and time, and her distance is d = 0.25t² metres after t seconds. When is B's average speed since the start equal to 4 m/s? What happens at that moment?
B's average speed over [0, t] is 0.25t² ÷ t = 0.25t. Set 0.25t = 4, so t = 16 s. At that moment B has run 0.25(256) = 64 m, and A has run 4(16) = 64 m. They are side by side: equal average speeds from the same start means equal distances.
Quiz
1. A rate of change compares which two things?
2. The snow on a deck stays 30 cm deep all day. What is the rate of change of the snow depth?
3. A table has x = 0, 1, 2, 3 and y = 5, 8, 11, 14. What is the rate of change?
4. The volume of water in a pool (in litres) is changing over time (in minutes). What are the units of the rate of change?
5. Which situation has a changing rate of change?
6. The temperature rises from −3 °C to 9 °C over 4 hours. What is the average rate of change?
Big ideas (D1.1, D1.2): a rate of change is Δy ÷ Δx, it always has units ("per"), and it can be zero, constant or changing. The same rate can be shown in words, a table, a graph or an equation.
How to help: When you see a rate in daily life (a gas price per litre, a speed limit, a data plan in GB per month, an interest rate), ask your teen: "What is the dependent variable? What is the independent variable? Is this rate constant, or does it change?"
Common difficulties:
- Subtracting in the wrong order, or mixing orders (y₂ − y₁ on top but x₁ − x₂ on the bottom). The order must match.
- Sign errors with negative numbers, such as −8 − 2.
- Forgetting units, or writing them upside down (min/L instead of L/min).
- Reading the first difference as the rate when the x-step is not 1.
- Thinking a constant percent rate means a constant rate in dollars or people. It does not.