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IBMaths: AA SLDPRevision Notes5. Calculus5.5 Optimisation5.5.1 Modelling with Differentiation

Modelling with Differentiation (DP IB Maths: AA SL)

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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 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 letter is representing the independent (x) variable and which letter is representing the dependent (y) variable
    • A GDC may always use x and y but ensure you use the correct letters 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 one 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 in terms of a single variable, using any constraints given in the question
 
STEP 2
Differentiate and solve the derivative equal to zero to find the “x"-coordinate(s) of any stationary points
 
STEP 3
If there is more than one stationary point, or the requirement to justify the nature of the stationary point, differentiate again
 
STEP 4
Use the second derivative to determine the nature of each stationary point and select the maximum or minimum point as necessary
 
STEP 5
Interpret the answer 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
  • Even when an algebraic solution is required you can still use your GDC to check answers and help you get an idea of what you are aiming for

Worked example

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

7-2-6-model-diff-diagram-for-example

The total area of the bed is to bespace 100 straight pi space straight m squared.

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

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

5-5-1-ib-sl-aa-only-we-soltn-a-b-c-d-e-

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

5-5-1-ib-sl-aa-only-we-soltn-b

c)

Find the value of r that minimises the perimeter.

5-5-1-ib-sl-aa-only-we-soltn-c

d)
Hence find the minimum perimeter.
5-5-1-ib-sl-aa-only-we-soltn-d
e)
Justify that this is the minimum perimeter.

5-5-1-ib-sl-aa-only-we-soltn-e

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    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.