Applied Optimization

Reading

• Section 4.6

Practice problems

• Section 4.6: 9, 11, 13, 35, 61
• In class: 4.6 3, 10, 25, 14, 51, 59
• To turn in: 4.6 2, 4, 8, 20, 52
• Challenge: 4.6 44

Notes

Applied Optimization

Applied optimization problems are word problems that involve a physical design with some variable quantities and a target quantity that we want to minimize or maximize. We then turn this into a mathematical problem of finding the minimum or maximum of some function \(f(x)\) over some interval.

The main steps would be the following:

• Read the problem carefully, at least twice.
• Identify a proper sketch of the problem description.
• Identify known quantities in the sketch.
• Identify which quantity (or quantities) varies, assign a variable(s) to it.
• Identify what constraints your variables have, given the physical restrictions of the problem (for example lengths cannot be negative).
• Identify what quantity we want to maximize or minimize, and which of those two directions is our goal.
• Describe that quantity as a function of the variables in question.
• Apply the methods we have for finding the maximum or minimum of a function over an interval (or occasionally over an open range like \((0, \infty)\)), to find the value of the variable that achieves the desired objective.
• Make sure that value makes physical sense.