Optimization - Complete Interactive Lesson
Part 1: Setting Up Optimization Problems
Optimization
Part 1 of 7 โ Setting Up Optimization Problems
The Strategy
- Identify the quantity to maximize or minimize (the objective function)
- Write an equation for it in terms of your variables
- Use a constraint to eliminate a variable (reduce to one variable)
- Find critical points of the objective function
- Verify it's actually a max or min (use endpoints or Second Derivative Test)
Worked Example: Fencing Problem
A rancher has 200 m of fencing. She wants to enclose a rectangular area along a river (no fence needed on the river side). Find the maximum area.
Let = width (perpendicular to river), = length (parallel to river).
Objective: Maximize
Constraint: โ
Substitute:
โ
โ concave down โ maximum
. Maximum area = m.
Setting Up Optimization ๐ฏ
Key Takeaways โ Part 1
- Always define your variables clearly
- Write the objective function (what to optimize)
- Use the constraint to reduce to one variable
- Verify using the Second Derivative Test or endpoint analysis
Part 2: Geometric Optimization
Optimization
Part 2 of 7 โ Geometric Optimization
Box Problem (Classic AP Question)
An open-top box is made by cutting squares of side from corners of a 12 ร 8 sheet and folding up.
Objective: Maximize
Part 3: Cost & Revenue Optimization
Optimization
Part 3 of 7 โ Cost & Revenue Optimization
Business Applications
- Revenue: where is the price-demand function
- Profit: (revenue minus cost)
Part 4: 3D Optimization (Cylinders & Cones)
Optimization
Part 4 of 7 โ 3D Optimization (Cylinders & Cones)
Cylinder with Fixed Surface Area
Minimize the surface area of a cylinder with volume cm.
โ
Part 5: Distance & Angle Optimization
Optimization
Part 5 of 7 โ Distance & Angle Optimization
Minimizing Travel Distance
A lifeguard at point on the beach must reach a swimmer at point in the water. She runs on sand at 8 m/s and swims at 2 m/s. Where should she enter the water?
This uses Snell's Law: the optimal path has .
Part 6: AP-Style Workshop
Optimization
Part 6 of 7 โ AP-Style Workshop
Mixed optimization problems similar to AP free-response questions.
AP-Style Problems ๐ฏ
Workshop Complete!
Optimization is all about translating word problems into calculus.
Part 7: Comprehensive Assessment
Optimization โ Review
Part 7 of 7 โ Comprehensive Assessment
Optimization Checklist
- \u2705 Define variables and draw a picture
- \u2705 Write the objective function
- \u2705 Use constraint to reduce to one variable
- \u2705 Find critical points
- \u2705 Verify max/min (Second Derivative Test or domain check)
Final Assessment ๐ฏ
Optimization โ Complete! \u2705
You have mastered:
- \u2705 Setting up objective functions and constraints
- \u2705 Geometric, business, and distance optimization
- \u2705 Verifying solutions using calculus tests