Interpreting a differential-equation model | CAIE 9709 Pure Mathematics 3 Notes

Chapters

Pure Mathematics 3

Mathematics9709

Check the start, direction and long-term value

A solved function still needs to be read

increasing

tends to 0

approaches limit

Interpret a solution by checking its start, direction and long-term behaviour.

At the start: substitute the initial time and check the initial value.
During the motion: use the sign of the derivative to decide whether the quantity rises or falls.
In the long term: take the limit or identify an equilibrium value.

A model can solve correctly and still fit poorly

Worked example

Compare a prediction with observed data

The population model $P=\sqrt[3]{6t+8}$ predicts at $t=20$ :

P=\sqrt[3]{128}\approx5.04

If the observed value is 7, the model underestimates by about $1.96$ . The gap has become large, so the model is not a good long-term fit.

Key idea

“The prediction is 5.04” is calculation. “It underestimates the observed 7, so the long-term fit is poor” is interpretation.

(Pure Mathematics 2 & 3 Coursebook, Worked Example 10.5)

State a complete contextual conclusion

Keep the units used by the model.
State whether a limiting value is reached or only approached.
Do not extend a model beyond a domain where its variables make sense.

Your turnIndependent interpretation check [5 marks]

A population density is modelled by $P=\dfrac{12}{1+2e^{-0.3t}}$ , where t is in years. Find the initial value, state the long-term value and compare the model with an observed value of 10.0 at $t=5$ .

Show worked answer

P(0)=\frac{12}{1+2}=4

P(5)=\frac{12}{1+2e^{-1.5}}

P(5)=8.30\text{ (3 s.f.)}

Answer

P(0)=4

P\to12\text{ as }t\to\infty

10.0-8.30=1.70

At $t=5$ , the model underestimates the observed value by 1.70.

One comparison does not prove the whole model is poor, but it does show the direction and size of this error.

Common mistake

“It increases” does not answer a long-term question fully. State the value approached, whether it is reached in finite time and how the prediction compares with the data given.