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Code Scholars Tutoring · AP Calculus AB

Modeling and justifying optimal values

Unit 5 · Topics 5.10, 5.11. Original AP Calculus AB notes, worked example, and self-check.

Understand the idea

Define the objective and use the constraint to write it in a single variable. State the feasible domain from the context. Differentiate, locate candidates, and justify the global conclusion using endpoint comparison or a suitable derivative argument. Report the requested physical dimensions or value with units and interpret boundary behavior.

Optimize A(x)=x(P/2−x)for a rectangle of fixed perimeter P\text{Optimize }A(x)=x(P/2-x)\quad\text{for a rectangle of fixed perimeter }P

Worked example

A rectangle has perimeter 20\displaystyle 20 meters. Find the dimensions maximizing area.

Let one side be x\displaystyle x, so the other is 10−x\displaystyle 10-x, with 0<x<10\displaystyle 0<x<10. Area is A=x(10−x)\displaystyle A=x(10-x) and A′=10−2x\displaystyle A'=10-2x. The critical point is x=5\displaystyle x=5. The derivative changes from positive to negative, and area tends to zero at both boundaries, so the maximum is 25\displaystyle 25 square meters for a 5\displaystyle 5 by 5\displaystyle 5 rectangle.

Try it yourself

Find the positive number minimizing x+9/x\displaystyle x+9/x.

Show solution and reasoning

The domain is x>0\displaystyle x>0. The derivative 1−9/x2\displaystyle 1-9/x^2 changes from negative to positive at 3\displaystyle 3. The expression grows without bound toward either domain end, so x=3\displaystyle x=3 yields the absolute minimum 6\displaystyle 6.

Unit 5 MCQ