The exam at a glance
| Section | Weight | Timing | What you do |
|---|---|---|---|
| I · Multiple choice | 50% | 1 hr 40 min · 42 questions | Part A: 29 questions in 62 minutes, no calculator (35% of score). Part B: 13 questions in 38 minutes, graphing calculator required for some (15%). Functions appear as equations, graphs, tables and words. |
| II · Free response | 50% | 1 hr 30 min · 6 questions | Part A: 2 questions in 30 minutes, graphing calculator required (about 17%). Part B: 4 questions in 60 minutes, no calculator (about 33%). At least two questions use a real-world context. |
Click in Bluebook, write on paper
You answer multiple-choice questions and read the free-response questions in the Bluebook app, then handwrite free-response answers in a paper booklet.
Know your four calculator skills
On calculator parts you are expected to graph a function, solve an equation, find a numerical derivative and evaluate a definite integral. Check that your model is on College Board's approved list.
The multiple-choice section
Part A gives you 29 questions in 62 minutes without a calculator (about 2 minutes each). Part B gives 13 questions in 38 minutes with a calculator (about 3 minutes each). Wrong answers lose no points.
Strategies that work
- Read graphs of f′ carefully. Increasing f means f′ > 0, not f′ increasing.
- Know your limits toolkit: factoring, conjugates, the special trig limits and L'Hospital's Rule.
- Units tell you the answer type. A rate in gallons per hour, integrated over hours, gives gallons.
- In Part B, use the calculator for definite integrals and solving. Store intermediate values; don't round.
- Eliminate with estimates. A quick sketch often rules out two choices.
- Flag and move on after 3 minutes. Come back once the quick points are banked.
The 10 units, in plain English
Limits & Continuity
What a function approaches, and where it breaks.
Be ready to: Evaluate limits from graphs, tables and algebra; find asymptotes; apply continuity and the IVT.
Differentiation: Definition & Rules
The derivative as a limit and an instantaneous rate.
Be ready to: Use the limit definition and all basic derivative rules; connect differentiability and continuity.
Composite, Implicit & Inverse
Harder derivative rules.
Be ready to: Apply the chain rule, implicit differentiation and inverse-function derivatives.
Contextual Applications
Derivatives describing real change.
Be ready to: Interpret rates with units; solve motion and related-rates problems; linearize; use L'Hospital's Rule.
Analytical Applications
Using derivatives to analyze a function's shape.
Be ready to: Apply the MVT and EVT; justify extrema and concavity; solve optimization problems.
Integration & Accumulation
The integral as accumulated change.
Be ready to: Use Riemann sums and the FTC; integrate by substitution, parts and partial fractions; evaluate improper integrals.
Differential Equations
Equations that describe how something changes.
Be ready to: Read slope fields; use Euler's method; solve separable equations; analyze logistic growth.
Applications of Integration
Average value, area, volume and length.
Be ready to: Find average value, area, volume and the arc length of a curve.
Parametric, Polar & Vector
Curves described by t or θ, and motion in the plane.
Be ready to: Find derivatives, speed and arc length of parametric curves; analyze vector-valued motion; find polar slopes and areas.
Infinite Sequences & Series
Infinite sums and polynomial approximations.
Be ready to: Apply the convergence tests; find the radius and interval of convergence; build Taylor and Maclaurin series; bound errors with the alternating series and Lagrange bounds.
Series (Unit 10) is as heavy as integration and is the topic BC students most often under-prepare, so start it early and revisit it every week. Units 1–8 also appear on AB, and those topics produce your AB subscore.
The 6 free-response questions
90 minutes, 6 questions, 9 points each (54 raw points). Part A (2 questions, calculator) gets 30 minutes; Part B (4 questions, no calculator) gets 60 minutes, and you may return to Part A questions without a calculator.
Rates and accumulation
Rates in and out, amounts over time and maximums. Often in Part A with a calculator, and shared with AB.
Amount = start + ∫(in − out).
Parametric or vector motion
A particle moving in the plane: velocity vector, speed, total distance traveled and position at a time. Usually Part A.
Speed = √((dx/dt)² + (dy/dt)²).
Graph of f′ or a table
Analyze a function from its derivative's graph or from data in a table, with Riemann sums and full justifications.
Justify with the signs of f′ and f″.
Polar curves
Area inside polar curves or between them, and how r or position changes with θ.
Area = ½∫r² dθ. Find the bounds first.
Differential equations
Slope fields, Euler's method, separable equations and logistic models.
Logistic growth is fastest at half the carrying capacity.
Taylor series
Almost always Q6. Build a Taylor polynomial, find the interval of convergence and bound the error.
Check both endpoints separately with a named test.
Time plan: About 15 minutes per question. Write the setup before you compute: an integral written correctly with limits earns points even if you run out of time.
Habits that earn points
- Label every part: (a), (b), (c).
- Justify with the signs of f′ or f″, not "the graph".
- Include units in every answer in context.
- Give decimals to 3 places on calculator parts.
- Stuck? Write the setup integral or equation. It still scores.
10 traps that cost points
Skipping endpoint checks
The ratio test says nothing at the endpoints. Test each one separately and name the test.
Forgetting factorials
The nth Taylor coefficient is f⁽ⁿ⁾(a)/n!. Missing the n! is the most common series slip.
Polar bounds
For area, find where the curve starts and ends (or where curves intersect) before setting up ½∫r² dθ.
Speed vs velocity
Speed is the magnitude of the velocity vector. Total distance integrates speed, not dy/dt.
d²y/dx² for parametric curves
Differentiate dy/dx with respect to t, then divide by dx/dt again.
Justifying with "the graph"
Readers need a reason based on f′ or f″, such as "f′ changes from positive to negative".
Dropping +C or the initial condition
Add +C right after integrating and solve for it before isolating y.
Rounding too early
Store calculator values and round only the final answer to 3 decimal places.
Wrong error bound
Alternating series: the error is at most the first omitted term. Otherwise use the Lagrange bound with a maximum of the next derivative.
Leaving a part blank
Each part is scored separately. A correct setup can earn points even if the next step fails.
A study roadmap to May
A steady plan beats a last-minute rush. Shift the months to match your teacher's pacing.
- Sep – Oct
Units 1–4: fast foundations
Limits and every derivative rule, moving quickly. Drill derivatives daily until they are automatic.
- Nov – Dec
Units 5–6: analysis and integrals
MVT, extrema, the FTC and all integration techniques, including parts and partial fractions.
- Jan – Feb
Units 7–9: applications, parametric and polar
Euler, logistic growth, arc length, vector motion and polar area. One free-response question per week.
- Mar – Apr
Unit 10: series
Convergence tests, power series and Taylor polynomials with error bounds. Revisit series every week until May.
- Late April
Timed sections
Full multiple-choice and free-response sections under real timing, splitting calculator and no-calculator parts.
- May 2027
Exam day
Monday, May 10, 2027 (morning session). Sleep well and confirm your start time with your AP coordinator.
3 practice questions
Original questions written by ACT SAT GO in the style of the exam. The free PDF has 10, with full explanations.
- Unit 10 · Interval of convergence
What is the interval of convergence of Σ (x + 1)ⁿ / (n·2ⁿ), for n = 1 to ∞?
- A −3 ≤ x < 1
- B −3 < x < 1
- C −3 < x ≤ 1
- D −3 ≤ x ≤ 1
Show answer
Answer: A. The ratio test gives |x + 1| < 2. At x = 1 it is the harmonic series, which diverges; at x = −3 it is the alternating harmonic series, which converges.
- Unit 9 · Polar area
What is the area enclosed by the polar curve r = 2 + 2cos θ?
- A 3π
- B 4π
- C 6π
- D 8π
Show answer
Answer: C · 6π. Area = ½∫(2 + 2cos θ)² dθ from 0 to 2π = ½∫(4 + 8cos θ + 4cos²θ) dθ = ½(8π + 4π) = 6π.
- Unit 7 · Euler's method
dy/dx = x + y with y(0) = 1. Using Euler's method with two steps of size 0.5, what is the estimate for y(1)?
- A 2
- B 2.25
- C 2.5
- D 3
Show answer
Answer: C · 2.5. Step 1: slope 0 + 1 = 1, so y(0.5) ≈ 1.5. Step 2: slope 0.5 + 1.5 = 2, so y(1) ≈ 1.5 + 0.5(2) = 2.5.
How students score
Ranges reflect recent College Board score distributions and shift a little each year. Credit and placement policies vary by college.
Aiming for a 5 in AP Calculus BC?
Book a free diagnostic session. We'll find your weakest units, review your practice responses and give you a week-by-week plan to exam day.