14.3PureStretch

Exponential Modelling

Populations, radioactive decay, cooling, compound interest — anything that grows or shrinks at a rate proportional to its size is modelled exponentially. This lesson interprets such models and their parameters.

25 min Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.3 Exponential ModellingWatch the full walkthrough before the notes below.

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What you'll be able to do

Interpret an exponential growth/decay model
Find values at a given time
Interpret the constants in the model
Understand the long-term behaviour

The model

Exponential models usually take the form $A = A_{0} e^{k t}$ . Here $A_{0}$ is the initial amount (at $t = 0$ ), and $k$ controls the rate: positive $k$ is growth, negative $k$ is decay.

A = A_{0} e^{k t}

A_{0}

= starting value;

k > 0

growth,

k < 0

decay.

Evaluating the model

Substitute the given time $t$ to find the amount, or the initial value by setting $t = 0$ (which makes $e^{0} = 1$ ).

1Set

t = 0

P = 200 e^{0} = 200 \times 1

Answer

200

Tip — The starting value is always the coefficient in front, because e⁰ = 1.

Interpreting the parameters

The constant in the front is the initial value; the sign of the exponent constant tells you growth vs decay; its size controls how fast. As $t \to \infty$ , a decay model tends to $0$ while a growth model increases without bound.

Formula recap

A = A_{0} e^{k t}

Standard exponential model.

t = 0 \Rightarrow A = A_{0}

Initial value.

k > 0 growth, k < 0 decay

Sign of k.

Common mistakes to avoid

Thinking the initial value needs calculation.

At t = 0, e⁰ = 1, so the initial value is just the leading coefficient.

Mixing up growth and decay.

Positive exponent constant ⟶ growth; negative ⟶ decay.

Key takeaways

Exponential models: A = A₀e^(kt).
A₀ is the initial value (t = 0).
k > 0 growth, k < 0 decay; |k| sets the speed.

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