Optimization Problems
Set up optimization problems by identifying the objective function and constraints. Reduce to one variable, apply calculus to find the global maximum or minimum, and interpret the result in context at AP exam depth.
A lesson of AP Calculus AB, in Analytical Applications of Differentiation, about 10 minutes.
By the end of this lesson, you will be able to set up and solve optimization problems, identify when a critical point gives a maximum vs. minimum, and interpret solutions in real-world context.
What Is Optimization?
Optimization problems ask: what input maximizes or minimizes a given quantity?
- What dimensions of a box maximize its volume given a fixed amount of material?
- What price maximizes a company's revenue?
- What path minimizes travel distance?
These problems use derivatives to find the critical points where maxima and minima occur.
The General Setup
- Identify the objective function — the quantity you're maximizing or minimizing
- Identify the constraint — the condition that limits your variables
- Write the objective as a function of ONE variable using the constraint
- Differentiate and find critical points
- Verify which critical point gives the maximum/minimum (First or Second Derivative Test; or compare values on closed interval)
- Interpret the answer in context
Example 1: Area Maximization
Problem: A farmer wants to enclose a rectangular plot adjacent to a river. He has 300 feet of fencing and doesn't need to fence the river side. Maximize the enclosed area.
Setup:
- Variables: width = x (perpendicular to river), length = y (parallel to river)
- Constraint: 2x + y = 300 (only one length side needed, since river = other side)
- Objective: Maximize A = xy
Solve: From constraint: y = 300 - 2x
Substitute: A(x) = x(300 - 2x) = 300x - 2x²
Differentiate: A'(x) = 300 - 4x = 0 → x = 75
Verify maximum: A''(x) = -4 < 0 → concave down → x = 75 is a maximum.
Dimensions: x = 75 ft, y = 300 - 2(75) = 150 ft
Maximum area: A = 75 × 150 = 11,250 square feet
Example 2: Volume Optimization (Box)
Problem: An open box is to be made from a 12-inch square sheet of cardboard by cutting equal squares of side x from each corner, then folding up the sides. Maximize the volume.