2.3.3 Exponential Models

Exponential growth or decay
- Exponential growth is represented by
  - $a^{x}$ where $a > 1$
  - $a^{- x}$ where $0 < a < 1$
  - $e^{r x}$ where $r > 0$
- Exponential decay is represented by
  - $a^{x}$ where $0 < a < 1$
  - $a^{- x}$ where $a > 1$
  - $e^{r x}$ where $r < 0$
They can be used when there a constant percentage increase or decrease
- Such as functions generated by geometric sequences
Examples include:
- V(t) is the value of car after t years
- S(t) is the amount in a savings account after t years
- B(t) is the amount of bacteria on a surface after t seconds
- T(t) is the temperature of a kettle t minutes after being boiled

An exponential growth model does not have a maximum
- In real-life this might not be the case
- The function might reach a maximum and stay at this value
Exponential models are monotonic
- In real-life this might not be the case
- The function might fluctuate

You may need to find the half-life of a substance
- This is the time taken for the mass of a substance to halve
Given an exponential model $f (t) = k a^{- t}$ or $f (t) = k e^{- r t}$ the half-life is the value of t such that:
- $f (t) = \frac{k}{2}$
For $f (t) = k a^{- t}$ the half-life is given by $t = \frac{\ln 2}{\ln a}$
- $\frac{k}{2} = k a^{- t}$
- $a^{t} = 2$
- $t \ln a = \ln 2$
For $f (t) = k e^{- r t}$ the half-life is given by $t = \frac{\ln 2}{r}$
- $\frac{k}{2} = k e^{- r t}$
- $e^{r t} = 2$
- $r t = \ln 2$

Look out for the word "initial" or similar, as a way of asking you to make the power equal to zero to simplify the equation
Questions regarding the boundary of the exponential model are also frequently asked

The value of a car, $V$ (NZD), can be modelled by the function

$V (t) = 25125 \times {0.8}^{t} + 8500, t \geq 0$

where $t$ is the age of the car in years.

State the initial value of the car.

2-3-3-ib-ai-sl-exponential-models-a-we-solution

Find the age of the car when its value is 17500 NZD.

2-3-3-ib-ai-sl-exponential-models-b-we-solution

DP IB Maths: AI HL

2.3.3 Exponential Models