Optimization problems are a classic AP Calculus topic — they test your ability to apply derivatives to real-world scenarios. You’ll see them in Free Response Questions (FRQs), often worth 6–9 points, and they require a clear strategy to earn full credit.
In this RevisionDojo guide, you’ll learn:
- How to recognize an optimization problem
- The 5-step method for solving them
- Common pitfalls that cost points
- A complete AP-style example
📚 What Are Optimization Problems?
In calculus, optimization means finding the maximum or minimum value of a function, often with real-life applications:
- Maximizing profit or area
- Minimizing cost or distance
- Finding optimal dimensions for a shape
🔍 Step-by-Step Optimization Strategy
1. Read the Problem Carefully
Identify:
- The quantity to optimize (maximize/minimize)
- The constraint (relationship between variables)
2. Draw a Diagram (If Applicable)
Sketch the situation — it helps translate words into math.
3. Write the Equation to Optimize
This is the function Q(x)Q(x) you want to maximize or minimize.
4. Reduce to One Variable
Use the constraint equation to eliminate extra variables.
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:
- Define variables: Let width = xx, length = yy (parallel to river)
- Constraint: 2x+y=6002x + y = 600
- Area equation: A=xyA = xy
- Substitute yy: A=x(600−2x)A = x(600 - 2x)
- Differentiate: A′(x)=600−4xA'(x) = 600 - 4x
- Set to zero: 600−4x=0600 - 4x = 0 → x=150x = 150
- Find yy: y=600−2(150)=300y = 600 - 2(150) = 300
- 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.
