IB DP Maths: AA HL復習筆記5.5.1 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
- If other variables are initially involved,?constraints?or?assumptions?about them will need to be made; for example
- 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 as 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
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
- 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.
c)? ? ? ? Find the value of that minimises the perimeter.
e)? ? ? ?Justify that this is the minimum perimeter.
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