Every AP Calculus BC FRQ Sorted By Unit
UPDATED for the 2026-27 School Year. Below is every single AP Calculus BC FRQ from 2015-2026 sorted by topic.
At the end is the predicted FRQs for the upcoming 2027 exam. Good luck!
PRO TIP: Links open in FRQ Atlas — You find the FRQ and upload your working for free, instant AI grading based on scoring guidelines.
FRQ Types
Breakdown of common question formats and strategies.
Free-Response
Standard multi-part problems. Tip: Show every step of your algebraic manipulation; bald answers get zero points.
Frequency: 55 questions
Calculator Active
Requires a graphing calculator for integration or finding roots. Tip: Write the setup integral before plugging it into your calculator.
Frequency: 3 questions
Calculator Inactive
Pure analytical work. Tip: Leave answers in unsimplified exact forms (e.g., ln(3) + 4) to save time and avoid arithmetic errors.
Frequency: 4 questions
Table and Related Rates
Data interpretation from tables. Tip: Watch units carefully and use the specific values given in the table for approximations.
Frequency: 2 questions
Graph Analysis
Extracting data from function or derivative graphs. Tip: Clearly label coordinates and reference the specific graph (e.g., “graph of f'”) in your justification.
Frequency: 4 questions
Skills
Key mathematical practices tested.
Implementing Processes
Frequency: 88 questions
1.E: Apply Rules
Execute procedures like integration or differentiation accurately.
Count: 60
1.D: Identify Rules
Recognize which theorem or rule applies to a relationship.
Count: 16
1.C: Classify Rules
Select procedures based on expression type.
Count: 4
Connecting Representations
Frequency: 45 questions
2.B: Identify Info
Pull data correctly from graphs, tables, or equations.
Count: 19
2.D: Relate Properties
Link function features across different forms.
Count: 14
2.E: Describe Relationships
Explain connections between functions and derivatives.
Count: 6
Justification
Frequency: 100 questions
3.E: Provide Rationale
Explain “why” clearly using mathematical reasoning.
Count: 36
3.D: Apply Theorems
Use definitions like MVT or IVT appropriately.
Count: 32
3.F: Explain Meaning
Interpret mathematical results in the problem’s context.
Count: 9
Units
Frequency of topics by unit.
Unit 6: Integration & Accumulation
The heavyweight champion. Mastering accumulation functions and the Fundamental Theorem is non-negotiable.
Frequency: 29 questions
Unit 5: Analytical Differentiation
Connecting analytical derivatives to graph behavior (MVT, extrema) is a staple of almost every exam.
Frequency: 28 questions
Unit 4: Contextual Differentiation
Expect real-world rate problems (tanks, particles) that require translating word problems into calculus.
Frequency: 21 questions
Unit 8: Applications of Integration
Area and volume are consistent, but look out for average value and motion problems here too.
Frequency: 20 questions
Unit 2: Differentiation Definition
Fundamental properties often appear as parts of larger problems rather than standalone questions.
Frequency: 14 questions
Unit 10: Infinite Sequences & Series
The dreaded BC exclusive. Taylor series and error bounds are guaranteed to show up in Q6.
Frequency: 14 questions
Unit 9: Parametric, Polar, Vector
Another BC exclusive. Usually a dedicated FRQ involving particle motion or polar area.
Frequency: 12 questions
Unit 1: Limits and Continuity
Often tested via L’Hospital’s Rule or IVT within other contexts rather than isolation.
Frequency: 7 questions
Unit 7: Differential Equations
Slope fields and separation of variables are classic, reliable points if you know the procedure.
Frequency: 6 questions
Unit 3: Composite/Implicit/Inverse
Implicit differentiation often hides inside related rates or differential equation problems.
Frequency: 5 questions
Unit 1: Limits and Continuity
- 2026 Q1 (Free-Response) — Bird arrival rates, accumulation, average value.
- 2026 Q5 (Free-Response) — Cube root region, arc length perimeter, improper integral.
- 2025 Q1 (Free-Response) — Invasive species, average value, rate of change, limits at infinity.
- 2025 Q3 (Free-Response) — Reading rates, estimation, IVT, trapezoidal sums.
- 2022 Q4 (Table and Related Rates) — Melting cone, tabular data, approximation, related rates.
- 2022 Q5 (Area and Volume) — Rational functions, improper integral, bounded region area.
- 2019 Q5 (Analytical / Improper Integral) — Parameter family, tangent slopes, partial fractions, improper integral.
- 2017 Q5 (Function Analysis and Series) — Rational function, slope extrema, improper integral, series convergence.
- 2016 Q1 (free-response) — Water tank, rate function, removal table, net change.
Unit 2: Differentiation: Definition and Fundamental Properties
- 2026 Q1 (Free-Response) — Bird arrival rates, accumulation, average value.
- 2026 Q4 (Free-Response) — Graph of f′, extrema and concavity analysis.
- 2025 Q1 (Free-Response) — Invasive species, average value, rate of change, limits at infinity.
- 2025 Q3 (Free-Response) — Reading rates, estimation, IVT, trapezoidal sums.
- 2024 Q1 (Calculator Active) — Coffee temp table, derivative approximation, Riemann sum, rate change.
- 2022 Q4 (Table and Related Rates) — Melting cone, tabular data, approximation, related rates.
- 2021 Q1 (free-response) — Bacterial density table, derivative estimation, Riemann sums.
- 2021 Q4 (free-response) — Integral-defined function, concavity, piecewise graph, MVT.
- 2019 Q3 (Graph Analysis) — Piecewise graph, accumulation function extrema, limit evaluation.
- 2019 Q5 (Analytical / Improper Integral) — Parameter family, tangent slopes, partial fractions, improper integral.
- 2018 Q2 (free-response) — Plankton density, boat parametric velocity, total distance.
- 2018 Q4 (free-response) — Tree growth table, derivative estimation, MVT, related rates.
- 2017 Q3 (Graph Analysis) — Derivative graph, function values, extrema, second derivative.
- 2017 Q5 (Function Analysis and Series) — Rational function, slope extrema, improper integral, series convergence.
- 2016 Q1 (free-response) — Water tank, rate function, removal table, net change.
- 2015 Q3 (free-response) — Velocity table, acceleration, average velocity, distance.
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
- 2025 Q5 (Free-Response) — Implicit higher-order derivatives, Taylor polynomials, Lagrange error.
- 2023 Q1 (free-response) — Gasoline pumping rate, Riemann sum, MVT, flow average.
- 2023 Q3 (free-response) — Milk cooling, slope field, tangent approximation, concavity error.
- 2016 Q4 (free-response) — Differential equation, second derivative, extrema categorization, limit.
- 2015 Q4 (free-response) — Slope fields, second derivatives, concavity, extrema tests.
Unit 4: Contextual Applications of Differentiation
- 2026 Q3 (Free-Response) — Cooling pie differential equation, separation of variables.
- 2025 Q2 (Free-Response) — Polar curves, radius rate, area, coordinate optimization.
- 2023 Q1 (free-response) — Gasoline pumping rate, Riemann sum, MVT, flow average.
- 2023 Q3 (free-response) — Milk cooling, slope field, tangent approximation, concavity error.
- 2023 Q4 (free-response) — Derivative graph, extrema, concavity, L’Hospital’s Rule.
- 2022 Q4 (Table and Related Rates) — Melting cone, tabular data, approximation, related rates.
- 2021 Q1 (free-response) — Bacterial density table, derivative estimation, Riemann sums.
- 2021 Q4 (free-response) — Integral-defined function, concavity, piecewise graph, MVT.
- 2019 Q1 (Rate In/Rate Out) — Fish entering/leaving lake, net change, rate extrema.
- 2019 Q3 (Graph Analysis) — Piecewise graph, accumulation function extrema, limit evaluation.
- 2019 Q4 (Related Rates / Differential Equations) — Draining barrel, related rates, height differential equation.
- 2018 Q1 (free-response) — Escalator line rates, accumulation, time solution, optimization.
- 2018 Q2 (free-response) — Plankton density, boat parametric velocity, total distance.
- 2018 Q4 (free-response) — Tree growth table, derivative estimation, MVT, related rates.
- 2018 Q5 (free-response) — Polar curves, area, tangent slope, particle rate.
- 2017 Q1 (Rate and Accumulation) — Tank volume, Riemann sums, cross-section model, related rates.
- 2017 Q4 (Differential Equations) — Potato cooling, linear approx, concavity, separation of variables.
- 2016 Q4 (free-response) — Differential equation, second derivative, extrema categorization, limit.
- 2016 Q5 (free-response) — Funnel dimensions, average value, volume, related rates.
- 2015 Q1 (free-response) — Drainpipe rainwater flow, total flow, extrema, overflow.
- 2015 Q3 (free-response) — Velocity table, acceleration, average velocity, distance.
- 2015 Q5 (free-response) — Rational function, tangent line, critical points, partial fractions.
Unit 5: Analytical Applications of Differentiation
- 2026 Q2 (Free-Response) — Polar curve region, area, average distance.
- 2026 Q3 (Free-Response) — Cooling pie differential equation, separation of variables.
- 2026 Q4 (Free-Response) — Graph of f′, extrema and concavity analysis.
- 2025 Q1 (Free-Response) — Invasive species, average value, rate of change, limits at infinity.
- 2025 Q4 (Free-Response) — Integral-defined function, inflection points, absolute extrema.
- 2024 Q1 (Calculator Active) — Coffee temp table, derivative approximation, Riemann sum, rate change.
- 2024 Q3 (Calculator Inactive) — Seawater depth, slope field, separation of variables.
- 2024 Q4 (Calculator Inactive) — Integral-defined function g, graph area, critical points.
- 2023 Q1 (free-response) — Gasoline pumping rate, Riemann sum, MVT, flow average.
- 2023 Q3 (free-response) — Milk cooling, slope field, tangent approximation, concavity error.
- 2023 Q4 (free-response) — Derivative graph, extrema, concavity, L’Hospital’s Rule.
- 2022 Q1 (Rate and Accumulation) — Toll plaza vehicle rate, integrals, optimization.
- 2022 Q3 (Graph Analysis) — Derivative graph, inflection points, extrema, function values.
- 2021 Q2 (free-response) — Planar particle motion, velocity vector, speed, distance.
- 2021 Q3 (free-response) — Function family, area, radius maximization, volume.
- 2021 Q4 (free-response) — Integral-defined function, concavity, piecewise graph, MVT.
- 2019 Q1 (Rate In/Rate Out) — Fish entering/leaving lake, net change, rate extrema.
- 2019 Q3 (Graph Analysis) — Piecewise graph, accumulation function extrema, limit evaluation.
- 2019 Q4 (Related Rates / Differential Equations) — Draining barrel, related rates, height differential equation.
- 2018 Q1 (free-response) — Escalator line rates, accumulation, time solution, optimization.
- 2018 Q3 (free-response) — Derivative graph, function values, concavity, inflection points.
- 2018 Q4 (free-response) — Tree growth table, derivative estimation, MVT, related rates.
- 2017 Q2 (Polar Coordinates) — Polar curves, area, dividing ray equation, distance.
- 2017 Q3 (Graph Analysis) — Derivative graph, function values, extrema, second derivative.
- 2017 Q4 (Differential Equations) — Potato cooling, linear approx, concavity, separation of variables.
- 2017 Q5 (Function Analysis and Series) — Rational function, slope extrema, improper integral, series convergence.
- 2016 Q3 (free-response) — Integral-defined function g, piecewise linear f, properties.
- 2016 Q4 (free-response) — Differential equation, second derivative, extrema categorization, limit.
- 2015 Q1 (free-response) — Drainpipe rainwater flow, total flow, extrema, overflow.
- 2015 Q4 (free-response) — Slope fields, second derivatives, concavity, extrema tests.
- 2015 Q5 (free-response) — Rational function, tangent line, critical points, partial fractions.
Unit 6: Integration and Accumulation of Change
- 2026 Q1 (Free-Response) — Bird arrival rates, accumulation, average value.
- 2026 Q5 (Free-Response) — Cube root region, arc length perimeter, improper integral.
- 2025 Q1 (Free-Response) — Invasive species, average value, rate of change, limits at infinity.
- 2025 Q3 (Free-Response) — Reading rates, estimation, IVT, trapezoidal sums.
- 2025 Q4 (Free-Response) — Integral-defined function, inflection points, absolute extrema.
- 2024 Q1 (Calculator Active) — Coffee temp table, derivative approximation, Riemann sum, rate change.
- 2024 Q4 (Calculator Inactive) — Integral-defined function g, graph area, critical points.
- 2024 Q5 (Calculator Inactive) — Function properties, integral derivative, arc length, Euler’s method.
- 2023 Q1 (free-response) — Gasoline pumping rate, Riemann sum, MVT, flow average.
- 2023 Q4 (free-response) — Derivative graph, extrema, concavity, L’Hospital’s Rule.
- 2023 Q5 (free-response) — Area between curves, improper integral, integration by parts.
- 2022 Q1 (Rate and Accumulation) — Toll plaza vehicle rate, integrals, optimization.
- 2022 Q2 (Parametric Motion) — Parametric motion, slope, speed, acceleration, distance.
- 2022 Q3 (Graph Analysis) — Derivative graph, inflection points, extrema, function values.
- 2022 Q4 (Table and Related Rates) — Melting cone, tabular data, approximation, related rates.
- 2022 Q5 (Area and Volume) — Rational functions, improper integral, bounded region area.
- 2021 Q1 (free-response) — Bacterial density table, derivative estimation, Riemann sums.
- 2021 Q3 (free-response) — Function family, area, radius maximization, volume.
- 2021 Q4 (free-response) — Integral-defined function, concavity, piecewise graph, MVT.
- 2019 Q3 (Graph Analysis) — Piecewise graph, accumulation function extrema, limit evaluation.
- 2019 Q5 (Analytical / Improper Integral) — Parameter family, tangent slopes, partial fractions, improper integral.
- 2018 Q3 (free-response) — Derivative graph, function values, concavity, inflection points.
- 2018 Q4 (free-response) — Tree growth table, derivative estimation, MVT, related rates.
- 2017 Q1 (Rate and Accumulation) — Tank volume, Riemann sums, cross-section model, related rates.
- 2017 Q3 (Graph Analysis) — Derivative graph, function values, extrema, second derivative.
- 2017 Q5 (Function Analysis and Series) — Rational function, slope extrema, improper integral, series convergence.
- 2016 Q1 (free-response) — Water tank, rate function, removal table, net change.
- 2016 Q2 (free-response) — Particle motion, parametric equations, component graph.
- 2016 Q3 (free-response) — Integral-defined function g, piecewise linear f, properties.
- 2015 Q3 (free-response) — Velocity table, acceleration, average velocity, distance.
- 2015 Q5 (free-response) — Rational function, tangent line, critical points, partial fractions.
Unit 7: Differential Equations
- 2026 Q3 (Free-Response) — Cooling pie differential equation, separation of variables.
- 2024 Q3 (Calculator Inactive) — Seawater depth, slope field, separation of variables.
- 2023 Q3 (free-response) — Milk cooling, slope field, tangent approximation, concavity error.
- 2021 Q5 (free-response) — Differential equation, Taylor polynomial, Euler’s method.
- 2019 Q4 (Related Rates / Differential Equations) — Draining barrel, related rates, height differential equation.
- 2017 Q4 (Differential Equations) — Potato cooling, linear approx, concavity, separation of variables.
- 2015 Q4 (free-response) — Slope fields, second derivatives, concavity, extrema tests.
Unit 8: Applications of Integration
- 2026 Q1 (Free-Response) — Bird arrival rates, accumulation, average value.
- 2026 Q2 (Free-Response) — Polar curve region, area, average distance.
- 2026 Q5 (Free-Response) — Cube root region, arc length perimeter, improper integral.
- 2025 Q1 (Free-Response) — Invasive species, average value, rate of change, limits at infinity.
- 2025 Q3 (Free-Response) — Reading rates, estimation, IVT, trapezoidal sums.
- 2024 Q1 (Calculator Active) — Coffee temp table, derivative approximation, Riemann sum, rate change.
- 2023 Q1 (free-response) — Gasoline pumping rate, Riemann sum, MVT, flow average.
- 2023 Q5 (free-response) — Area between curves, improper integral, integration by parts.
- 2022 Q1 (Rate and Accumulation) — Toll plaza vehicle rate, integrals, optimization.
- 2022 Q5 (Area and Volume) — Rational functions, improper integral, bounded region area.
- 2021 Q1 (free-response) — Bacterial density table, derivative estimation, Riemann sums.
- 2021 Q3 (free-response) — Function family, area, radius maximization, volume.
- 2019 Q1 (Rate In/Rate Out) — Fish entering/leaving lake, net change, rate extrema.
- 2019 Q2 (Polar Coordinates) — Polar region, area, average distance, circle limits.
- 2018 Q1 (free-response) — Escalator line rates, accumulation, time solution, optimization.
- 2018 Q2 (free-response) — Plankton density, boat parametric velocity, total distance.
- 2018 Q4 (free-response) — Tree growth table, derivative estimation, MVT, related rates.
- 2017 Q1 (Rate and Accumulation) — Tank volume, Riemann sums, cross-section model, related rates.
- 2017 Q2 (Polar Coordinates) — Polar curves, area, dividing ray equation, distance.
- 2016 Q1 (free-response) — Water tank, rate function, removal table, net change.
- 2016 Q5 (free-response) — Funnel dimensions, average value, volume, related rates.
- 2015 Q1 (free-response) — Drainpipe rainwater flow, total flow, extrema, overflow.
- 2015 Q3 (free-response) — Velocity table, acceleration, average velocity, distance.
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 2026 Q2 (Free-Response) — Polar curve region, area, average distance.
- 2026 Q5 (Free-Response) — Cube root region, arc length perimeter, improper integral.
- 2025 Q2 (Free-Response) — Polar curves, radius rate, area, coordinate optimization.
- 2024 Q2 (Calculator Active) — Parametric motion, speed, distance, position vector.
- 2024 Q5 (Calculator Inactive) — Function properties, integral derivative, arc length, Euler’s method.
- 2023 Q2 (free-response) — Parametric motion, acceleration, speed, total distance.
- 2022 Q2 (Parametric Motion) — Parametric motion, slope, speed, acceleration, distance.
- 2021 Q2 (free-response) — Planar particle motion, velocity vector, speed, distance.
- 2019 Q2 (Polar Coordinates) — Polar region, area, average distance, circle limits.
- 2018 Q2 (free-response) — Plankton density, boat parametric velocity, total distance.
- 2018 Q5 (free-response) — Polar curves, area, tangent slope, particle rate.
- 2017 Q2 (Polar Coordinates) — Polar curves, area, dividing ray equation, distance.
- 2016 Q2 (free-response) — Particle motion, parametric equations, component graph.
- 2015 Q2 (free-response) — Velocity vector, position, tangent slope, speed.
Unit 10: Infinite Sequences and Series
- 2026 Q6 (Free-Response) — Maclaurin series for g, Taylor polynomial error bound.
- 2025 Q5 (Free-Response) — Implicit higher-order derivatives, Taylor polynomials, Lagrange error.
- 2025 Q6 (Free-Response) — Taylor series convergence, derivative series, geometric series.
- 2024 Q6 (Calculator Inactive) — Maclaurin series, alternating error, derivative series.
- 2023 Q5 (free-response) — Area between curves, improper integral, integration by parts.
- 2023 Q6 (free-response) — Higher-order derivative, Taylor polynomial, Lagrange error.
- 2022 Q6 (Power Series) — Power series, interval of convergence, error bound.
- 2021 Q5 (free-response) — Differential equation, Taylor polynomial, Euler’s method.
- 2021 Q6 (free-response) — Maclaurin series, Integral Test, Ratio Test, error.
- 2019 Q6 (Series) — Taylor polynomials, Maclaurin manipulation, error bounds.
- 2018 Q6 (free-response) — Logarithmic Maclaurin series, interval convergence, alternating error.
- 2017 Q5 (Function Analysis and Series) — Rational function, slope extrema, improper integral, series convergence.
- 2017 Q6 (Taylor Series) — Maclaurin construction, convergence analysis, alternating error.
- 2016 Q6 (free-response) — Taylor series construction, convergence interval, error estimation.
- 2015 Q6 (free-response) — Maclaurin radius, derivative series, related Taylor polynomial.
2027 FRQ Topics Prediction
We used Phy AI + the frequency of topics above to make an educated guess on what you might see on the upcoming 2027 AP Calculus BC Exam FRQ:
- Unit 6: Integration and Accumulation — 32 appearances. Hit twice in 2026 (bird-rate accumulation and the cube-root region). Expect a function defined as an integral with Fundamental Theorem reasoning.
- Unit 5: Analytical Differentiation — 31 appearances. The 2026 exam ran the classic f′-graph analysis. Look for “justify your answer” questions built on the Mean Value Theorem and sign charts.
- Unit 8: Applications of Integration — 24 appearances. 2026 asked for a region’s perimeter via arc length plus an improper integral — area/volume setups with a BC-only twist are trending.
- Unit 10: Infinite Sequences and Series — 15 appearances. The dedicated Series FRQ (usually Q6) is virtually guaranteed — 2026 delivered a Maclaurin series with derivative and error-bound parts.
- Unit 9: Parametric/Polar/Vector — 14 appearances. 2026 went polar (region area and average distance); a parametric or vector-valued particle motion FRQ is due next.
Use Phy to start solving Mathematics questions for free.
This article was written by Jason