Unit 2 · Lesson 9 of 11

A proportional growth rate makes an exponential.

When growth is proportional to the amount already present, equal time intervals multiply rather than add the same quantity.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its population growth rate.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase time? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

0.729 · illustrative relative amount

Population

Time3 hGrowth constant0.2 1/h

population growth rate

P'(3)=0.2\left(2e^{0.6}\right)=0.728848

The instantaneous growth rate is proportional to current population, not a fixed amount added each hour.

Assumptions, units, and limits

Continuous exponential growth with fixed k and no carrying-capacity limitation.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Differentiating an exponential brings down its inner rate. Logarithms undo the exponential and have reciprocal derivatives. Hyperbolic sine and cosine are combinations of opposite exponentials, so their derivatives follow immediately from exponential rules.

P(t)=2e^{kt},\quad P'=kP,\quad\frac{d}{dx}\ln x=\frac1x

Keep the reasoning visible

(\sinh x)'=\cosh x,\quad(\cosh x)'=\sinh x,\quad(\ln x)'=1/x

The instantaneous growth rate is proportional to current population, not a fixed amount added each hour.

What makes this model useful?

Continuous exponential growth with fixed k and no carrying-capacity limitation.

Exponential growth is a model with a limited regime of validity. Resources, losses, or regulation can change the rate law.

A worked example

Calculate the starting scenario

Use time = 3 h; growth constant = 0.2 1/h.
$P'(3)=0.2\left(2e^{0.6}\right)=0.728848$
The instantaneous growth rate is proportional to current population, not a fixed amount added each hour. The checked quantity is population growth rate.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the population growth rate using time 3 h, growth constant 0.2 1/h. State the meaning of its sign and units.

Checked quantity: population growth rate · thousand/h. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

0.729 · illustrative relative amount

Population

Time3 hGrowth constant0.2 1/h

population growth rate

P'(3)=0.2\left(2e^{0.6}\right)=0.728848

The instantaneous growth rate is proportional to current population, not a fixed amount added each hour.

Assumptions, units, and limits

Continuous exponential growth with fixed k and no carrying-capacity limitation.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

P(t)=2e^{kt},\quad P'=kP,\quad\frac{d}{dx}\ln x=\frac1x

Calculate and explain

Apply the idea to a new scenario

At $t=5$ h and $k=0.1$ h⁻¹, find population growth rate from initial population 2 thousand.

Checked quantity: population growth rate · thousand/h. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

0.33 · illustrative relative amount

Population

Time5 hGrowth constant0.1 1/h

population growth rate

P'(5)=0.1\left(2e^{0.5}\right)=0.329744

The instantaneous growth rate is proportional to current population, not a fixed amount added each hour.

Assumptions, units, and limits

Continuous exponential growth with fixed k and no carrying-capacity limitation.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

P(t)=2e^{kt},\quad P'=kP,\quad\frac{d}{dx}\ln x=\frac1x

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

0.729 · illustrative relative amount

Population

Time3 hGrowth constant0.2 1/h

population growth rate

P'(3)=0.2\left(2e^{0.6}\right)=0.728848

The instantaneous growth rate is proportional to current population, not a fixed amount added each hour.

Assumptions, units, and limits

Continuous exponential growth with fixed k and no carrying-capacity limitation.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Differentiating an exponential brings down its inner rate. Logarithms undo the exponential and have reciprocal derivatives. Hyperbolic sine and cosine are combinations of opposite exponentials, so their derivatives follow immediately from exponential rules.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.