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

Look out for the word "initial" or similar, as 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 SL

2.3.3 Exponential Models