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IBMaths: AI HLDPRevision Notes5. Calculus5.1 Differentiation5.1.3 Modelling with Differentiation

Modelling with Differentiation (DP IB Maths: AI HL)

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Paul

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Modelling with Differentiation

What can be modelled 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

What modelling assumptions are used in 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

How do I solve optimisation problems?

  • In optimisation problems, letters other thanspace x comma space y andspace f are often used including capital letters
    • space V is often used for volume,space S for surface area
    • space 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 usespace x andspace 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

Examiner Tip

  • 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

Worked example

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 pi space straight m squared.

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

P equals pi stretchy left parenthesis r plus 100 over r stretchy right parenthesis

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

b) Find fraction numerator straight d P over denominator straight d r end fraction.

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

c)
Find the value of r that minimises the perimeter.

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

d)
Hence find the minimum perimeter.

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

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    Previous:5.1.2 Applications of DifferentiationNext:5.2.1 Differentiating Special Functions
    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.