Real-world optimization with derivatives
MathematicsCalculusAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Calculus
Pick a real-world problem – designing a can that uses the least material, folding an open-top box with the largest volume, or setting a price for the largest profit – then drag the variable to watch the picture change alongside the graph of the objective function f(x), its derivative f'(x) and the variation table. The optimum sits exactly where f'(x) changes sign, and the simulation compares it with real-world designs.
Lesson: Using inequalities and derivatives to solve optimization problems
What it shows
Three real-world problems show the full optimization process with derivatives: building an objective function, restricting it to a sensible domain, and locating its maximum or minimum. A cylindrical can of fixed volume uses the least material when its height equals its diameter. An open-top box folded from a square sheet of side a has its largest volume when squares of side a/6 are cut from the corners. A profit function L(g) = (g − c)(2000 − 10g) peaks at g = 100 + c/2. In each case the optimum lies where f'(x) changes sign.
How to use
Choose a Problem and move the slider of the variable (Base radius r, Cut-corner side x or Selling price g): the picture, the graphs of f(x) and f'(x) and the variation table change together. Students locate the optimum from the sign of the derivative before reading it. Then change the problem parameter (Can volume, Cardboard side a or Unit cost c) and see how the optimum moves; for the can, discuss why real cans differ.
Parameters you can change
- Problem Can using the least material, Open-top box with the largest volume, Selling price for the largest profit
- Variable value (fraction of the domain) 0.02–0.98
- Problem parameter (can volume mL / cardboard side cm / unit cost c) 10–500
- Currency (selling price problem) From the language, US dollar, Euro, Indian rupee, Indonesian rupiah, Saudi riyal
Questions to explore
- For the can that uses the least material, how does its height compare with the diameter of its base?
- Why must you find the domain of the variable before building the variation table?
- If f'(x) = 0 at a point, is that point always the optimum? What must happen to the sign of f'(x) there?
