Calculus finds the best
Optimization problems ask for the biggest or smallest value of something: maximum volume, minimum cost, shortest time. Derivatives find these exactly.
The process
| Step | Do |
|---|---|
| 1 | Draw a diagram; define variables. |
| 2 | Write the objective function (what to optimize). |
| 3 | Write the constraint and use it to reduce to one variable. |
| 4 | Find the feasible domain. |
| 5 | Differentiate; find critical numbers. |
| 6 | Test critical numbers and endpoints. |
| 7 | Answer the question in context, with units. |
Example 1: maximum volume box
A 30 cm × 30 cm sheet of cardboard has equal squares (side x) cut from each corner, and the sides are folded up to make an open box. Find x for the maximum volume.
- V(x) = x(30 − 2x)² = 4x³ − 120x² + 900x
- Domain: 0 < x < 15
- V′(x) = 12x² − 240x + 900 = 12(x² − 20x + 75) = 12(x − 5)(x − 15)
- x = 5 (x = 15 is an endpoint where V = 0)
- V(5) = 5 × 20² = 2,000 cm³
Example 2: minimum material for a can
A closed cylinder must hold 500 cm³. Minimize the surface area.
- Constraint: πr²h = 500 → h = 500/(πr²)
- A = 2πr² + 2πrh = 2πr² + 1,000/r
- A′ = 4πr − 1,000/r² = 0 → r³ = 250/π → r ≈ 4.30 cm
- h = 500/(π × 4.30²) ≈ 8.60 cm → h = 2r
Example 3: maximum revenue
A theatre sells 400 tickets at $20; each $1 increase loses 10 sales. Price p = 20 + x.
- R(x) = (20 + x)(400 − 10x) = −10x² + 200x + 8,000
- R′(x) = −20x + 200 = 0 → x = 10
- Price $30; revenue $9,000
Example 4: shortest distance
Find the point on y = √x closest to (3, 0).
- Minimize D² = (x − 3)² + x = x² − 5x + 9 (squaring avoids roots)
- d/dx = 2x − 5 = 0 → x = 2.5
- Point (2.5, √2.5) ≈ (2.5, 1.58)
Checking it’s a max or min
Use the second derivative test, or compare values at critical numbers and endpoints.
Worked examples
Example 1: Two numbers add to 20. Maximize their product. P = x(20 − x) → x = 10 → 100.
Example 2: A rectangle with perimeter 60. Max area? 15 × 15 → 225.
Example 3: A farmer fences three sides of a field along a river with 400 m. Max area? A = x(400 − 2x) → x = 100 → 20,000 m².
Common mistakes
- Not reducing to one variable.
- Forgetting endpoints.
- Answering with x when the question asks for the area, cost, etc.
Try it yourself
- Two positive numbers have a product of 64. Minimize their sum.
- An open box from a 12 × 12 sheet: best corner square size?
- Revenue R(p) = 120p − 2p². Price for maximum revenue?
- A rectangle is inscribed under y = 12 − x² (above the x-axis, symmetric). Max area?
Answers: 1) 8 + 8 = 16 2) 2 (volume 128) 3) p = 30 4) x = 2 → width 4, height 8, area 32
Quick recap
- Optimization: write an objective function and a constraint.
- Differentiate, find critical numbers, and check endpoints.
- Answer in context.