Two Equations In, One Function Out

You'll learnto turn a paragraph into something differentiable — name the objective, write the constraint, use the constraint to remove a variable, and read the domain off the situation rather than off the algebra.

Every optimization problem arrives as two relationships and a question. One relationship is the thing you want extreme, and the other is what stops you from making it infinite. Nothing can be differentiated until the second has been used to remove a variable from the first, and nothing can be answered until the domain is on the page. This topic stops exactly there — at the moment the problem has become one function of one variable on a stated interval — because everything after that is Unit 5 machinery you already have.

8 STEPS · 6 QUICK CHECKS · ONE LETTER AND A STATED DOMAIN BEFORE ANY DERIVATIVE · v1

A = xy · 2x + y = 1200 ⇒ A(x) = x(1200 − 2x) ON 0 < x < 600

Before you start

What you're looking at

A rectangular plot with a river along one side, so only three sides need fencing. There are 1200 feet of fence. The two sides running away from the river are each x, and the side parallel to the river is y.

The question

What has to be written down before a derivative can be taken at all? Two equations, one substitution, and a domain — in that order, every time.

Watch for

The rectangle changes shape and its area changes with it, while the total fence never moves off 1200. The thing that never moves is the constraint; the thing that changes is what you are optimizing.

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