Modelling with functionsAQA A-Level Maths: Revision notes

A mathematical model uses a function to describe a real situation. The process is a cycle:

1. State the assumptions that simplify the situation (for example, no air resistance).
2. Choose a function and fit it to the information given.
3. Use the function to make predictions.
4. Compare the predictions with real data or common sense.
5. Refine the model if it does not fit well enough, then repeat. No model is exactly right: each one is valid only within certain limits.

The shape of the data suggests the function:

• Linear, $y=mx+c$: constant rate of change, such as a fixed fee plus a cost per unit.
• Quadratic, $y=ax^2+bx+c$: one maximum or minimum, such as the height of a projectile.
• Exponential, $y=ab^x$: growth ($b>1$) or decay ($0<b<1$) at a rate proportional to the current size.
• Reciprocal, $y=\frac{k}{x}$: inverse proportion, with an asymptote.
• Trigonometric, $y=a+b\sin(cx)$: repeating patterns such as tides, with a regular period. Use the context (starting value, long-term behaviour, turning points) to pick between them.

Each constant in a model has a meaning. Explain it in words, with units.

• $C=450+0.2x$: $450$ is the fixed cost (the value when $x=0$) and $0.2$ is the cost per mile, the gradient.
• $h=1.5+12t-5t^2$: $1.5$ is the starting height; the maximum is found where $\frac{dh}{dt}=12-10t=0$, at $t=1.2$, giving $8.7$ m.
• $N=200\times1.5^t$: $200$ is the initial number and $1.5$ is the growth factor (a $50\%$ increase per hour).
• $\theta=20+70\times0.9^t$: $20$ is the long-term value; $70$ is how far above it the temperature starts; $0.9$ is the fraction left each minute.

A model only works for sensible inputs and outputs. The domain is the set of inputs for which the model is valid, and the range is the set of outputs it gives.

• The ball model is valid only from $t=0$ until the ball lands, when $h=0$: $5t^2-12t-1.5=0$ gives $t\approx2.52$ s.
• A cost model needs $x\geq0$.
• Counts of people or bacteria should be whole numbers, so the function is an approximation. A model that gives a negative height, a negative cost or a negative population is outside its valid domain.

A limitation is a way in which the model fails to match reality. A refinement changes the model to deal with it.

• Ball: ignores air resistance. A refinement is to add a resistance term, or use different values for the constants.
• Bacteria: $N=200\times1.5^t$ grows without limit. A refinement is a function that levels off at a maximum, because of limited food and space.
• Coffee: $\theta=20+70\times0.9^t$ assumes a room temperature of $20\,^\circ\text{C}$. If the room is at $18\,^\circ\text{C}$, use $\theta=18+72\times0.9^t$, which still gives $90\,^\circ\text{C}$ at $t=0$ but a lower temperature later: $43.1\,^\circ\text{C}$ after $10$ minutes, compared with $44.4\,^\circ\text{C}$. A good answer names the limitation and gives a specific change.

Models can be compared by equating them. Van 1 costs $C=450+0.2x$ and van 2 costs $D=300+0.35x$. Equating gives $x=1000$ miles. For $x<1000$ the second van is cheaper (at $700$ miles, £$545$ against £$590$) and for $x>1000$ the first is cheaper. To solve an exponential equation, take logarithms: for $20+70\times0.9^t=50$, $0.9^t=\frac37$ and $t=\frac{\ln(3/7)}{\ln0.9}=8.04$ minutes. Always conclude in the language of the context, not just with a number.