Calculus I · Unit 2B · review
Optimization Review
Optimization Review
Optimization requires an objective, a constraint, a one-variable model, a feasible domain, critical numbers, endpoints, and an interpretation. A derivative locates candidates; comparison and context determine the answer.
Find the rectangle of maximum area with perimeter .
Exercise 1 answer
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A farmer has m of fencing for three sides of a rectangle beside a river. Maximize area.
Exercise 2 answer
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Find the dimensions of a closed cylinder of fixed volume using minimum material.
Exercise 3 answer
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Find the point on closest to .
Exercise 4 answer
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Maximize on .
Exercise 5 answer
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Design a gutter by folding equal edges of a strip and maximize cross-sectional area.
Exercise 6 answer
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Explain why minimizing distance squared gives the same location as minimizing distance.
Exercise 7 answer
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Write a full model for the Norman window problem and identify its feasible domain.
Exercise 8 answer
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After the explanation
Use the section idea
Separate the objective from the constraint, reduce to one feasible variable, and interpret the winning candidate in the original design.
Optimization is a modeling problem first: the derivative only compares candidates after the geometry, units, and feasible domain are correct.
Write variables and units, identify the objective, use the constraint to eliminate a variable, then test critical and boundary candidates.
Optimizing the constraint, ignoring the feasible domain, or keeping an algebraic critical point that cannot exist in the real design.
Have you compared every feasible candidate and explained why the result is physically and economically reasonable?
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