Radford Mathematics IB Mathematics resources • AI SL

IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus

Optimisation

Turn a real context into a one-variable objective function and find the required maximum or minimum.

AI SL · SL 5.7

Learning goal

Solve optimisation problems in context, using technology where appropriate.

Syllabus link

AI SL: SL 5.7 · Topic 5 Calculus.

Big idea

Optimisation is a modelling process: define the quantity to optimise, use the constraints, choose the correct domain, then interpret the optimum.

Key relationship

\(f\'(x)=0\quad\text{for interior candidates}\)

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Core idea

Optimisation is a modelling process: define the quantity to optimise, use the constraints, choose the correct domain, then interpret the optimum.

Solve optimisation problems in context, using technology where appropriate.

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Key results and method

\[f\'(x)=0\quad\text{for interior candidates}\]
\[\text{compare candidates and endpoints when the domain is closed}\]
  1. Define variables and the quantity to maximise or minimise.
  2. Use the constraint to write the objective as a function of one variable.
  3. State the feasible domain.
  4. Find candidate extrema with calculus/GDC.
  5. Interpret the value and units in context.
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Worked example

Question

A rectangle has perimeter 40 m. Find the dimensions of maximum area.

Solution
  1. Let the width be \(x\), so length is \(20-x\).
  2. \(A(x)=x(20-x)=20x-x^2\).
  3. \(A\'(x)=20-2x=0\Rightarrow x=10\).
Answer: The maximum-area rectangle is \(10\text{ m}\times10\text{ m}\).
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Practice

Try these questions before moving on.

  1. Maximise the volume of an open box formed from a rectangular sheet.
  2. Minimise cost when one material is more expensive than another.
  3. Explain why a feasible domain is part of the mathematical model.

Search the worksheet library for more practice →

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