Free Response Strategies

Part 1 of 7 — The AP FRQ Format

Structure of the FRQ Section

• 6 questions in 90 minutes
• Questions 1-2: Calculator allowed (30 min)
• Questions 3-6: No calculator (60 min)

Common FRQ Types

1. Rate/Accumulation (graph or table of rate, find accumulation)
2. Particle Motion (position, velocity, acceleration)
3. Area/Volume (between curves, disk/washer)
4. Differential Equations (slope field, separation of variables)
5. Table Analysis (Riemann sums, MVT, IVT)
6. Connected Rates/Related Rates

FRQ Format 🎯

Key Takeaways — Part 1

1. Know the 6 common FRQ types
2. Show all work for full credit
3. Practice pacing: ~15 min per question

Free Response Strategies

Part 2 of 7 — Rate/Graph Problems

The Rate Problem Template

Given a graph of rate $R(t)$:

1. "Total amount" → $int_a^b R(t),dt$ (area under curve)
2. "Rate of change at $t = c$" → read $R(c)$ from graph
3. "Is amount increasing or decreasing?" → sign of rate
4. "When is amount maximum?" → where rate changes from $+$ to $-$

Rate Problem Strategies 🎯

$R(t)$ is the rate of water flow (gal/hr). $R(t) > 0$ for $0 < t < 3$, $R(t) < 0$ for $3 < t < 7$, $R(3) = 0$.

Key Takeaways — Part 2

1. Accumulation max where rate changes sign
2. Total = integral of rate

Free Response Strategies

Part 3 of 7 — Justification Language

Writing Justifications

The AP exam requires specific language. Here are templates:

For IVT: "Since $f$ is continuous on $[a, b]$ and $f(a) < N < f(b)$, by the Intermediate Value Theorem, there exists $c in (a, b)$ such that $f(c) = N$."

For MVT: "Since $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, by the Mean Value Theorem, there exists $c in (a, b)$ such that f'(c) = rac{f(b)-f(a)}{b-a}."

For increasing: "Since $f'(x) > 0$ on $(a, b)$, $f$ is increasing on $(a, b)$."

Justification 🎯

Key Takeaways — Part 3

1. Cite theorem names explicitly
2. State conditions (continuous, differentiable) before applying
3. Use precise mathematical language

Free Response Strategies

Part 4 of 7 — Calculator Strategies

When Calculators Are Allowed (Q1-Q2)

You may be asked to:

1. Evaluate a definite integral — use numerical integration
2. Find zeros of $f'$ — graph and find zeros
3. Find intersections — graph both functions
4. Evaluate $f(a)$ — direct computation

Important Rules

• Store intermediate values (don't round too early)
• Show the mathematical setup even if using calculator
• Write at least 3 decimal places

Calculator Skills 🎯

Key Takeaways — Part 4

1. Show setup + numerical answer
2. Use 3+ decimal places
3. Don't round intermediate results

Free Response Strategies

Part 5 of 7 — Common Mistakes to Avoid

Top 10 AP Calculus Mistakes

1. Forgetting $+C$ on indefinite integrals
2. Not stating hypotheses when using IVT/MVT
3. Using the wrong variable ($x$ vs $t$)
4. Not including units in contextual answers
5. Premature rounding on calculator problems
6. Confusing displacement and total distance
7. Forgetting absolute values for total distance
8. Not checking endpoints for absolute extrema
9. Missing a factor from the chain rule
10. Not answering the question asked

Avoid Common Mistakes 🎯

Key Takeaways — Part 5

1. Read the question carefully
2. Include units and justify theorems
3. Total distance and displacement are different!

Free Response Strategies

Part 6 of 7 — Scoring & Partial Credit

How FRQs Are Scored

Each question is worth 9 points, typically split:

• Part (a): 2 points
• Part (b): 2-3 points
• Part (c): 2-3 points
• Part (d): 2-3 points

Partial Credit Tips

• Each part is graded independently — always attempt every part
• You can use results from earlier parts even if wrong
• Show your setup for credit even if computation has errors
• A correct answer with no work may get 0 points

Scoring Strategy 🎯

Key Takeaways — Part 6

1. Never leave a part blank
2. Show work for every part
3. Parts are scored independently

Free Response Strategies — Review

Part 7 of 7 — Final Practice

FRQ Review 🎯

FRQ Strategies — Complete! ✅

You are ready to tackle AP FRQs with confidence!