A-Level Maths: Modelling Growth and Decay

Using Exponential Functions and Differential Equations

One of the most powerful ideas students meet in A-Level Mathematics is that very different real-world situations can be described by the same mathematics. Whether we are modelling population growth, radioactive decay, charging a capacitor, or the spread of a virus, the same exponential structure keeps appearing.

This makes growth and decay a perfect topic for mathematical modelling — and a favourite with examiners.

---

1. The Core Idea: Rate Proportional to Size

At the heart of exponential models is a simple assumption:

The rate of change of a quantity is proportional to the amount present.

In mathematical form:

$$\frac{dN}{dt} = kN$$

• $N$ = quantity (population, mass, charge, number of bacteria…)

• $t$ = time

• $k$ = constant of proportionality

• $k > 0$ → growth

• $k < 0$ → decay

This single differential equation underpins the whole topic.

---

2. Solving the Differential Equation

Separating variables:

$$\frac{1}{N} dN = k \, dt$$

Integrating:

$$\ln N = kt + C$$

Exponentiating:

$$N = Ae^{kt}$$

where $A = e^C$ is the initial value when $t = 0$.

👉 This is the exponential model used throughout A-Level Maths.

---

3. Exponential Growth Models

Used when quantities increase over time:

• Population growth (with unlimited resources)

• Bacterial cultures

• Compound interest

• Early stages of epidemics

General form:

$$N = N_0 e^{kt}$$

Key features students should recognise:

• Constant percentage increase

• Doubling time is constant

• Graph gets steeper with time

---

4. Exponential Decay Models

Used when quantities decrease over time:

• Radioactive decay

• Cooling (simplified models)

• Discharging capacitors

• Drug concentration in the bloodstream

Model:

$$N = N_0 e^{-kt}$$

Important exam ideas:

• Half-life is constant

• The quantity never quite reaches zero

• Logarithms are often used to find $k$

---

5. Connecting to Data and Modelling Assumptions

In modelling questions, marks are often earned (or lost!) on interpretation.

Typical assumptions:

• No limiting factors (no carrying capacity)

• Constant rate of growth or decay

• Continuous change (not step-by-step)

Common exam tasks:

• Find $k$ from given data

• Predict future values

• Interpret what $k$ means in context

• Comment on the validity of the model

---

6. Why Differential Equations Matter

Differential equations:

• Explain why exponential models arise

• Link calculus with real-world behaviour

• Prepare students for A-Level Physics, Chemistry, Biology and university STEM

For many students, this is the moment maths stops being abstract and starts to describe reality.

---

7. Teaching Tip (From the Lab)

At Hemel Private Tuition, we often:

• Plot real experimental data

• Fit exponential curves

• Linearise models using $\ln N$

• Compare theory with real-world limitations

Seeing the maths emerge from data makes it far more memorable — and exam-proof.