Modelling with Differentiation (DP IB Applications & Interpretation (AI)): Revision Note

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

• Situations 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 or A 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

• 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, set the derivative equal to zero, and solve to find the “x"-coordinate(s) of any stationary points
   

• STEP 3
  If there is more than one stationary point, or if the question requires you 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