Optimisation
Turn a real problem into a function of one variable, find its critical points, and pick the best one. Check the ends of the allowed range too.
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Analogy
Finding the best seat in a theatre: define what 'best' means, then search for the peak.
Example
Sides x and y: 2x + 2y = 100, so y = 50 - x.Area A = x(50 - x) = 50x - x^2.A' = 50 - 2x = 0 gives x = 25.Square 25 by 25, area 625 m^2.
In practice
Used to maximise area, profit or strength and to minimise cost, time or material.
Practice
Flashcards
- Largest area of a rectangle with perimeter 100 m?625 m^2, from a 25 m by 25 m square.
Interview questions
- A rectangle has perimeter 100 m. Find its dimensions for maximum area and justify that it is a maximum.
- Let sides be x and 50 - x
- A = 50x - x^2
- A' = 50 - 2x = 0 gives x = 25
- A'' = -2 < 0 so a maximum
- Area is 625 m^2, a square
Questions people ask
What are the steps of an optimisation problem?
First, draw a picture and name the variables. Second, write the quantity to optimise as a function of one variable, using any constraint. Third, differentiate and solve for zero. Finally, check that it is a maximum or minimum and state the answer with units.
Why must I check the endpoints?
The best value may sit at the edge of the allowed range, where the derivative is not zero. A function restricted to x between 0 and 10 might be largest at x = 10. Compare the critical points and the endpoints and choose the best.