Every trading desk, portfolio manager, and financial engineer faces the same mathematical skeleton: choose a variable — a quantity to produce, a price to set, a hedge ratio to adopt — to maximise revenue or minimise cost, subject to a resource constraint. The calculus machinery developed in the previous lessons (critical points, the closed-interval method, the First Derivative Test for absolute extrema) is precisely what solves these problems. What was missing was the word problem skill: extracting a single-variable function from a verbal or geometric description before applying derivatives.
This note teaches the six-step optimisation procedure through worked examples across geometry, economics, and finance, from the classic minimum-cost container to the revenue-maximising price schedule.
Learning objectives. By the end of this note you should be able to:
- translate a word problem into an objective function of one variable with a stated domain;
- apply the closed-interval method or the First Derivative Test for absolute extrema to find the optimum;
- verify the nature of a critical point (max or min) by the sign of on either side;
- interpret marginal revenue, marginal cost, and the condition as a derivative condition;
- solve profit-maximisation and price-optimisation problems.
In this lesson
- The six-step optimisation procedure
- Geometric optimisation
- Economics and finance applications
- Distance and geometry optimisation
- Common problem types and their mathematical forms
Before you start
- Maximum and Minimum Values and the Closed-Interval Method
- Curve Sketching: Concavity, Inflection Points, and the Derivative Tests
The full lesson, with worked examples and exercise solutions, is for members.
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