IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
Optimisation
Build and analyse an objective function, including closed-domain endpoint checks.
Learning goal
Solve optimisation problems using differentiation and interpret the result in context.
Syllabus link
AA HL: SL 5.8 · Topic 5 Calculus.
Big idea
The calculus is usually not the hardest part: the essential step is reducing the context to a correct one-variable model and domain.
Key relationship
\(f\'(x)=0\quad\text{for interior candidates}\)
Core idea
The calculus is usually not the hardest part: the essential step is reducing the context to a correct one-variable model and domain.
Solve optimisation problems using differentiation and interpret the result in context.
Key results and method
- Draw/define the geometry.
- Use the constraint to eliminate one variable.
- State the feasible domain.
- Differentiate and find candidates.
- Check nature/endpoints and interpret units.
Worked example
A rectangle has area \(100\text{ cm}^2\). Minimise its perimeter.
- Let sides be \(x\) and \(100/x\).
- \(P(x)=2x+200/x\), \(x>0\).
- \(P\'(x)=2-200/x^2=0\Rightarrow x=10\).
Practice
Try these questions before moving on.
- Optimise a cylinder with fixed volume.
- Solve a problem where the maximum is at an endpoint.
- Explain the meaning of the derivative condition in a profit/cost context.
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