IB Maths AI 2.7 Notes

This page contains our IB Maths AI notes for 2.7. By reading each one of these notes, you will fully cover the content for IB Maths AI 'Advanced modelling functions'.

All Topic 2 Sub-topics 2.1: Forms Of Lines 2.2: Introduction To Functions 2.3: Introduction To Modelling Functions 2.4: Modelling Applications (HL)2.5: Further functions (HL)2.6: Graph transformations (HL)2.7: Advanced modelling functions (HL)2.8: Logarithmic graphs

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Chapters

1. Exponential models & half-life

2. Natural logarithmic models

3. Sinusoidal models

4. Logistic models

5. Piecewise models

6. Choosing the right model

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Exponential models & half-life

Exponential decay models are used when a quantity decreases by a constant percentage over equal time intervals.

A common form is

$f(t)=Ae^{-kt}$

where $A$ is the initial amount and $k\gt 0$ .

A very important application is half-life. The half-life is the time taken for a quantity to reduce to half of its original value.

Suppose a substance is modelled by $f(t)=Ae^{-kt}$ .

To find the half-life, let $f(t)=\frac{A}{2} =Ae^{-kt}$

Divide by $A$ : $\frac{1}{2}=e^{-kt}$

Take natural logarithms: $\ln\left(\frac{1}{2}\right)=-kt$

So $t=\frac{\ln2}{k}$ .

This gives the half-life formula for an exponential decay model.

$t=\frac{\ln2}{k}$

Suppose a radioactive sample is modelled by $f(t)=500e^{-0.08t}$ , find the half-life.
Its half-life is $t=\frac{\ln2}{0.08}\approx8.66$ .
So the quantity halves every $8.66$ time units.

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