Optimization Problems on the AP Calculus Exam: Explained | RevisionDojo

5. Differentiate and Solve

• Take derivative Q′(x)Q'(x)
• Set Q′(x)=0Q'(x) = 0 to find critical points
• Test each critical point (and endpoints, if given)

📝 Example AP-Style Problem

Problem:
A farmer wants to fence a rectangular field along a river. No fence is needed along the river. The farmer has 600 meters of fencing. What dimensions maximize the area?

Solution:

1. Define variables: Let width = xx, length = yy (parallel to river)
2. Constraint: 2x+y=6002x + y = 600
3. Area equation: A=xyA = xy
4. Substitute yy: A=x(600−2x)A = x(600 - 2x)
5. Differentiate: A′(x)=600−4xA'(x) = 600 - 4x
6. Set to zero: 600−4x=0600 - 4x = 0 → x=150x = 150
7. Find yy: y=600−2(150)=300y = 600 - 2(150) = 300
8. Conclusion: Max area with 150m width and 300m length

⚠️ Common Mistakes to Avoid

• Forgetting to check endpoints when given a range
• Mixing up which variable to eliminate
• Not stating the answer with units (the AP exam requires this for full credit)
• Skipping the justification for why your point is a max/min

📊 Practice Strategy from RevisionDojo

• Solve at least 2 optimization problems per week from past AP exams
• Always write equations before substituting numbers to avoid calculation errors
• Practice explaining your reasoning — AP graders love clear, justified solutions

🧭 Final Advice from RevisionDojo

Optimization problems aren’t just about algebra — they’re about translating a real-world scenario into a mathematical model.
Once you lock in the 5-step process, these problems become some of the easiest FRQ points you can earn.