IB DP Maths: AI SL复习笔记5.1.3 Modelling with Differentiation
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
- If other variables are initially involved,?constraints?or?assumptions?about them will need to be made; for example
How do I solve optimisation problems?
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
Exam 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.
d)? ? ? ? Hence find the minimum perimeter.
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