Modelling with Differentiation

Recall that differentiation is about the rate of change of a function and provides a way of finding minimum and maximum values of a function
Anything that involves maximising or minimising a quantity can be modelled using differentiation; for example
- minimising the cost of raw materials in manufacturing a product
- the maximum height a football could reach when kicked
These are called optimisation problems

The quantity being optimised needs to be dependent on a single variable
- If other variables are initially involved, constraints or assumptions about them will need to be made; for example
  - minimising the cost of the main raw material – timber in manufacturing furniture say – the cost of screws, glue, varnish, etc can be fixed or considered negligible
- Other modelling assumptions may have to be made too; for example
  - ignoring air resistance and wind when modelling the path of a kicked football

In optimisation problems, letters other than $x, y$ and $f$ are often used including capital letters
- $V$ is often used for volume, $S$ for surface area
- $r$ for radius if a circle, cylinder or sphere is involved
Derivatives can still be found but be clear about which variable is independent ( $x$ ) and which is dependent ( $y$ )
- a GDC may always use $x$ and $y$ but ensure you use the correct variable throughout your working and final answer
Problems often start by linking two connected quantities together – for example volume and surface area
- where more than one variable is involved, constraints will be given such that the quantity of interest can be rewritten in terms of a single variable
Once the quantity of interest is written as a function of a single variable, differentiation can be used to maximise or minimise the quantity as required

STEP 1

Rewrite the quantity to be optimised as a single variable, using any constraints given in the question

STEP 2

Use your GDC to find the (local) maximum or minimum points as required

Plot the graph of the function and use the graphing features of the GDC to “solve for minimum/maximum” as required

STEP 3

Note down the solution from your GDC and interpret the answer(s) in the context of the question

The first part of rewriting a quantity as a single variable is often a “show that” question – this means you may still be able to access later parts of the question even if you can’t do this bit

A large allotment bed is being designed as a rectangle with a semicircle on each end, as shown in the diagram below.

5-1-3-ib-sl-ai-only-we-soltn-quest

The total area of the bed is to be $100 π m^{2}$ .

Show that the perimeter of the bed is given by the formula

P = π (r + \frac{100}{r})

5-1-3-ib-sl-ai-only-we-soltn-a

b) Find

\frac{d P}{d r}

5-1-3-ib-sl-ai-only-we-soltn-b

Find the value of

r

that minimises the perimeter.

5-1-3-ib-sl-ai-only-we-soltn-c

Hence find the minimum perimeter.

5-1-3-ib-sl-ai-only-we-soltn-d

Modelling with Differentiation (DP IB Maths: AI HL)