Exponential Growth and Decay (HSC SSCE Mathematics Advanced): Revision Notes

Exponential Growth and Decay

Introduction

This section explores exponential growth and decay, building on concepts from Year 11. These mathematical models help us understand how populations grow, how prices increase with inflation, and how radioactive materials decay over time.

The most important property of exponential functions is that the derivative of $e^x$ equals the same function $e^x$. This unique characteristic makes exponential functions ideal for modelling real-world situations where rates of change depend on current values.

Understanding exponential derivatives

The basic exponential function

When we have the function $Q = e^t$, its derivative is remarkably simple:

$\frac{dQ}{dt} = e^t$

This can be rewritten as:

$\frac{dQ}{dt} = Q$

The general exponential function

When we replace $t$ with $kt$, the function becomes $Q = e^{kt}$. The derivative now includes the constant $k$:

$\frac{dQ}{dt} = ke^{kt}$

This can be rewritten as:

$\frac{dQ}{dt} = kQ$

The gradient at any point on the graph equals $k$ times the height at that point. These functions $Q = e^{kt}$ are what we use throughout this topic to model growth and decay situations.

Exponential growth

What is exponential growth?

Consider a growing population (people in a country, rabbits on an island, or bacteria in a culture). The larger the population, the more new individuals are born in each time period. This means the rate of population growth is proportional to the current population size.

We write this mathematically as:

$\frac{dP}{dt} = kP$

where $k$ is a constant of proportionality.

This situation is called exponential growth, and a population growing this way obeys the law of exponential growth.

The exponential growth theorem

Proof of the theorem

We need to prove two things: that the function satisfies the differential equation, and that $Q_0$ is indeed the initial value.

Step 1: Verify the function satisfies the differential equation

Substitute $Q = Q_0 e^{kt}$ into the equation $\frac{dQ}{dt} = kQ$:

$$\text{LHS} = \frac{dQ}{dt} = \frac{d}{dt}(Q_0 e^{kt}) = kQ_0 e^{kt} = kQ = \text{RHS}$$

The function satisfies the differential equation.

Step 2: Verify the initial value

Substitute $t = 0$ into $Q = Q_0 e^{kt}$:

$$Q = Q_0 e^{k \times 0} = Q_0 \times 1 \quad \text{(because } e^0 = 1\text{)} = Q_0$$

Therefore, $Q_0$ is the initial value when $t = 0$.

Working with exponential growth problems

The constant $k$ can usually be calculated from data given in the problem. Once found, store the approximate value of $k$ in your calculator memory for later calculations.

Exponential decay

What is exponential decay?

Some quantities decrease at a rate proportional to their current amount. Radioactive substances decay in this manner. If $M$ is the mass of a substance at time $t$, and $M$ is decreasing, then the derivative $\frac{dM}{dt}$ is negative:

$\frac{dM}{dt} = -kM$

where $k$ is a positive constant.

Applying our theorem: $M = M_0 e^{-kt}$, where $M_0$ is the initial mass when $t = 0$.

The exponential decay theorem