B. Derivatives and Their Applications · Applications of Derivatives: Motion, Rates and Optimization

Derivatives as Real-World Rates: Inflation, Population and Flow

B2.2About 65 minutes

Learning goals

I can interpret a derivative in context, with correct units.
I can identify intervals of increase and decrease (such as inflation and deflation) from a graph or equation.
I can find when a quantity is changing fastest, using the second derivative or an inflection point.
I can calculate and interpret a relative rate of change.

Words to know

rate of change in context
A derivative described in words, with units (units of output per unit of input).

inflation
A period when prices are rising; the rate of inflation is how fast they rise.

deflation
A period when prices are falling.

relative rate of change
f′(t) ÷ f(t): the rate compared with the size of the quantity, often given as a percent.

flow rate
The rate at which a volume of fluid passes a point or enters or leaves a container, such as litres per minute.

Rates are everywhere

A news report says "prices rose 2.4% over the past year". A town planner says "the population is growing by about 300 people a year". A city engineer says "the reservoir is draining at 200 m³ per hour". Each statement is about a rate of change, and each one is really a derivative in disguise.

In this lesson you will translate derivative statements into everyday language (with the right units), and read real-world rates straight from graphs: when something is increasing or decreasing, and when it is changing fastest.

Reading a derivative in context

If y = f(x), then f′(a) is the instantaneous rate of change of y with respect to x at x = a.

Units: (units of y) per (unit of x).
Sign: f′(a) > 0 means y is increasing at that moment; f′(a) < 0 means it is decreasing.
Size: |f′(a)| tells you how fast.
Estimate: for a small change Δx, Δy ≈ f′(a) × Δx. So f′(a) is about how much y changes when x increases by one unit near a.

For example, if V(t) is the volume of water in a reservoir in cubic metres, t hours after noon, then V′(5) = −200 means: at 5 p.m., the water is draining at 200 m³ per hour. Over the next half hour, about 100 m³ will drain (if the rate stays about the same).

Relative rates

Sometimes the size of the rate compared with the size of the quantity matters more. The relative rate of change is f′(t) ÷ f(t), often written as a percent. A town growing by 300 people per year is growing fast if it has 2000 people (15% per year) and slowly if it has 2 000 000 (0.015% per year). Statistics Canada reports inflation as the percent change in the Consumer Price Index (CPI) over 12 months, which is a relative rate.

Worked example

Example 1: Words, units and estimates

T(h) is the air temperature in °C at a height of h metres above the ground on a winter morning. Interpret T′(500) = −0.006, and estimate the change in temperature from 500 m to 600 m.

Units: °C per metre.
Meaning: at a height of 500 m, the temperature is decreasing by about 0.006 °C for each metre you go up.
Estimate: ΔT ≈ T′(500) × Δh = −0.006 × 100 = −0.6.

The air is about 0.6 °C colder at 600 m than at 500 m (if the rate stays about the same).

Worked example

Example 2: Inflation, deflation and the fastest inflation

An economist models a price index as P(t) = 100 + 6t² − t³ for 0 ≤ t ≤ 6, where t is in years. (a) When is there inflation (prices rising) and when deflation (prices falling)? (b) When are prices rising fastest, and how fast? (c) Express that rate as a relative rate.

(a) P′(t) = 12t − 3t² = 3t(4 − t). For 0 < t < 4, P′ > 0: inflation. For 4 < t < 6, P′ < 0: deflation.
(b) The rate of inflation is P′(t). It is largest where its own derivative is zero: P″(t) = 12 − 6t = 0 at t = 2. Since P″ changes from + to − there, P′ has a maximum. P′(2) = 24 − 12 = 12 index points per year.
On the graph of P, t = 2 is the point of inflection: the price curve is steepest there.
(c) P(2) = 100 + 24 − 8 = 116. Relative rate = 12 ÷ 116 ≈ 0.103, about 10.3% per year.

Inflation for 0 < t < 4 years, deflation for 4 < t < 6. Prices rise fastest at t = 2 years, at 12 points per year (about 10.3% per year).

Left: the price index P(t) from Example 2. Prices rise (inflation, green band) until t = 4, then fall (deflation, yellow band). Right: the rate P′(t). Its highest point, at t = 2, matches the steepest point of P, where P changes from concave up to concave down.

Worked example

Example 3: Population growth from a graph

The graph below shows the population of a new suburb. Describe how the growth rate changes over time and say when the population grows fastest.

At first the curve is nearly flat: the population grows slowly (the derivative is small and positive).
Then the curve gets steeper: the growth rate increases (concave up, N″ > 0).
At the inflection point the curve is steepest: this is the maximum growth rate.
After that the curve is still rising but less steeply (concave down, N″ < 0): growth slows as the suburb fills up and the population levels off near 5000.

The population always increases, but the growth rate rises to a maximum at the inflection point (about t ≈ 5.5 years, when the population is about 2500) and then falls toward 0.

An S-shaped (logistic) growth curve. The dashed tangent at the inflection point is the steepest tangent anywhere on the curve, so that is when the population grows fastest.

Worked example

Example 4: Flow rate from a draining bathtub

A bathtub drains so that the volume of water left is V(t) = 200 − 40t + 2t² litres, for 0 ≤ t ≤ 10 minutes. Find the flow rate at the start and after 5 minutes, and check that the model makes sense at t = 10.

V′(t) = −40 + 4t litres per minute. The outflow rate is −V′(t), the rate at which water leaves.
At t = 0: V′(0) = −40, so water leaves at 40 L/min.
At t = 5: V′(5) = −40 + 20 = −20, so water leaves at 20 L/min.
At t = 10: V(10) = 200 − 400 + 200 = 0 and V′(10) = 0. The tub is empty and the flow has stopped, so the model fits its domain.

40 L/min at the start, 20 L/min after 5 minutes. The flow slows down as the water gets shallower, which matches experience: water drains fastest when the tub is full.

Just for fun

"The rate of increase of inflation is decreasing" sounds like great news. Count the derivatives: prices are P, inflation is P′, its rate of increase is P″, and "decreasing" means P‴ < 0. That is a third-derivative sentence, and prices are still going up.

Hands-on activity

Measure a flow rate

You need: a sink or a large bowl with a drain (or a plastic bottle with a small hole poked near the bottom by an adult, set in the sink), a ruler, a phone stopwatch, and a helper.

Fill the sink or bottle with water. Stand the ruler upright in the water.
Pull the plug (or uncover the hole) and start the stopwatch. Every 10 s, the helper reads the water depth in centimetres. Record until the water is nearly gone.
Make a table of t and depth d. Calculate Δd/Δt for each 10 s interval (cm/s).
Plot d against t. Sketch tangents at the start, the middle and near the end.

What to look for: the depth falls fastest at the start and more and more slowly as the water gets shallower, so the d-t graph is falling and concave up, like V(t) in Example 4. Your Δd/Δt values should be negative and shrink in size. (Physics explains this: the outflow speed depends on the depth of water above the hole.)

Your turn

C(x) is the cost, in dollars, of producing x hockey sticks. What are the units of C′(x), and what does C′(500) = 42 mean?

P(t) is the population of a town t years after 2020. Interpret P′(6) = −150.

A price index has P′(t) > 0 and P″(t) < 0 during a year. Describe what is happening to prices.

Using the index P(t) = 100 + 6t² − t³, find the rate of change of prices at t = 5. Is this inflation or deflation?

A town of 8000 people grows at 240 people per year. What is its relative growth rate?

The volume of water in a tank is V(t) = 1000 − 30t litres. What is the flow rate out of the tank?

For the bathtub V(t) = 200 − 40t + 2t², when is the outflow rate 10 L/min?

More practice: warm-up

If y is in kilograms and x is in days, what are the units of dy/dx?

f′(3) < 0. Is f increasing or decreasing at x = 3?

On a graph of prices over time, what does a falling section show?

Where on an S-shaped growth curve is the growth rate greatest?

f′(10) = 4. Estimate f(10.5) − f(10).

More practice: on your own

A maple tree's height is H(t) = 0.5t + 0.02t² metres, t years after planting (for 0 ≤ t ≤ 15). How fast is it growing at t = 10? Is its growth rate increasing?

A price index is P(t) = 100 + 9t² − t³ for 0 ≤ t ≤ 9. When is inflation fastest, and what is the rate?

For the same index, when does inflation turn into deflation?

A reservoir's volume is V(t) = 5000 + 300t − 20t² m³ (t in hours, 0 ≤ t ≤ 20). When does the volume stop increasing, and what is the maximum volume?

The number of bacteria in a culture is B(t) = 2000(1.5)^t, t in hours. Find the growth rate at t = 2 and the relative growth rate.

Challenge

A price index is P(t) = 100 + at² − t³ for t ≥ 0, where a > 0. Show that inflation is fastest at t = a/3 and find that maximum rate in terms of a.

Explain why the relative growth rate of any exponential model A(t) = A₀bᵗ is constant.

A water tank is filled and drained at once. Water comes in at 12 L/min and the volume is V(t) = 300 + 12t − 0.5t² L. At what time does the outflow rate equal the inflow rate, and what is the volume then?

Check your understanding

Quiz

V(t) is the volume of water in a pool, in litres, t minutes after a hose is turned on. What does V′(20) = 35 mean?

On a price-over-time graph, when is inflation fastest?

A population is 50 000 and growing at 1500 people per year. What is its relative growth rate?

For P(t) = 100 + 6t² − t³, what is P′(1)?

If f′(a) > 0 and f″(a) < 0, which description fits?

C(x) is the cost in dollars of x kilograms of maple syrup. What are the units of C′(x)?

For parents and teachers

Big idea (B2.2): a derivative is a rate with units. Reading it in words, finding where it is positive or negative, and finding where it is largest (often at an inflection point) are the skills behind economics, population studies and engineering.

How to help: pick a news item with a rate (gas prices, inflation, population) and ask your teen to say what quantity, what units, and whether the rate itself is going up or down.

Common difficulties:

Confusing a quantity with its rate ("35 L in the pool" versus "35 L per minute").
Thinking the fastest growth is at the highest point of a graph (it is where the graph is steepest).
Forgetting the units, or putting them upside down.
Mixing up a rate (people per year) and a relative rate (percent per year).