# AP Calculus BC exam guide > All ten AP Calculus BC units with published exam weightings, including parametric and polar functions and infinite series. - Canonical page: https://www.arcadesacademy.com/exams/ap-calculus-bc - Markdown source: https://www.arcadesacademy.com/exams/ap-calculus-bc/llms.txt - Exam family: AP exams - Governing body: College Board - Official exam information: https://apstudents.collegeboard.org/courses/ap-calculus-bc - Who takes it: High school students taking a full-year calculus course covering first- and second-semester college content, typically headed toward a quantitative major. - Follows the fall 2020 course and exam description, from 3 May 2021 - Guide structure last checked against official sources: August 30, 2026 ## Overview AP Calculus BC covers everything in AP Calculus AB and adds two further units: parametric equations, polar coordinates, and vector-valued functions, and infinite sequences and series. It also extends several AB units with additional techniques such as integration by parts, partial fractions, improper integrals, Euler's method, and the logistic model. The published weightings shift substantially from AB. The eight shared units are individually lighter because the two additional units together carry close to thirty percent of the multiple-choice section, with infinite series alone weighted comparably to integration. Students who treat series as an afterthought consistently underperform relative to their calculus ability. Arcade Academy is an independent study workspace. Use the official Course and Exam Description as the authoritative content list and the official exam page for current format details, and use the unit pages below to organize problem practice and timed free-response work. ## Exam format - **Total time:** 3 hours 10 minutes - **Section I:** 42 multiple-choice questions in 1 hour 40 minutes, 50% of the exam score - **Section I detail:** Part A is 29 questions in 62 minutes with no calculator; Part B is 13 questions in 38 minutes with a graphing calculator required - **Section II:** 6 free-response questions in 1 hour 30 minutes, 50% of the exam score - **Section II detail:** Part A is 2 questions in 30 minutes with a graphing calculator required; Part B is 4 questions in 60 minutes with no calculator - **Format update:** The multiple-choice count and section timing changed beginning with the May 2027 exam; confirm the current numbers on the official exam page - **Delivery:** Hybrid digital: multiple-choice questions are answered and free-response prompts are read in College Board's Bluebook app, and free-response answers are handwritten in a paper booklet - **AB subscore:** College Board reports an AB subscore alongside the BC score, based on the portion of the exam covering AB content - **Score scale:** 1 to 5 ## Scoring The multiple-choice and free-response sections each contribute half of a composite score, which College Board converts to the reported 1 to 5 scale using cut points set for that year's exam. BC students also receive an AB subscore, and colleges decide independently how to apply either score for credit or placement. ## Units AP Calculus BC is organized into 10 units. Each one below lists its published weighting where the governing body publishes one, the topics it covers, and study tips specific to it. ### 1. Unit 1: Limits and Continuity - Position: 1 of 10 - Published weighting: 4–7% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-1-limits-and-continuity Unit 1 develops the limit and defines continuity in terms of it, working algebraically, graphically, and numerically. The squeeze theorem, asymptotic behavior, and the Intermediate Value Theorem complete the unit. In BC this material is weighted lightly on the exam, but it underlies the convergence reasoning in Unit 10. #### Topics - **Defining and estimating limits** — Reading limits from graphs and tables, one-sided limits, and when a limit fails to exist. - **Determining limits algebraically** — Substitution, factoring, rationalizing, and manipulating indeterminate forms. - **Squeeze theorem and special limits** — Bounding arguments and the standard trigonometric limits reused throughout the course. - **Continuity** — The three-part definition, classifying discontinuities, and continuity on an interval. - **Asymptotes and limits at infinity** — Relating vertical and horizontal asymptotes to infinite limits and end behavior. - **Intermediate Value Theorem** — Stating hypotheses and using the theorem to justify that a value is attained. #### Study tips - Get limit notation exactly right now; it is required in Unit 10 convergence arguments where sloppiness costs points. - Practice limits at infinity for rational and exponential comparisons, since growth-rate reasoning recurs in series. - Rehearse stating theorem hypotheses before conclusions, which is how justification points are awarded. ### 2. Unit 2: Differentiation: Definition and Fundamental Properties - Position: 2 of 10 - Published weighting: 4–7% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-2-differentiation-definition-and-fundamental-properties Unit 2 defines the derivative through difference quotients and establishes the basic differentiation rules. Differentiability and its relationship to continuity, along with derivatives of the standard elementary functions, are covered. The rules built here are used without comment in every later unit. #### Topics - **Defining the derivative** — Average rate of change, the difference quotient, and the derivative as its limit. - **Derivative as a function** — Estimating derivative values from graphs and tables and sketching derivative graphs. - **Differentiability and continuity** — Why differentiability implies continuity and where differentiability fails. - **Basic derivative rules** — Power, constant multiple, sum, and difference rules on polynomial and radical functions. - **Derivatives of transcendental functions** — Sine, cosine, exponential, and natural logarithm derivatives. - **Product and quotient rules** — Applying both rules and recognizing when simplification first is faster. #### Study tips - Keep the limit definition fresh; it appears in multiple-choice questions phrased as a limit to be recognized. - Drill the elementary derivatives to automaticity so attention can go to setup in later units. - Practice moving between a function graph and its derivative graph in both directions. ### 3. Unit 3: Differentiation: Composite, Implicit, and Inverse Functions - Position: 3 of 10 - Published weighting: 4–7% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-3-differentiation-composite-implicit-and-inverse-functions Unit 3 covers the chain rule, implicit differentiation, and derivatives of inverse functions including the inverse trigonometric functions. Higher-order derivatives and the judgment of which technique to apply round out the unit. The chain rule in particular is a prerequisite for parametric and polar differentiation in Unit 9. #### Topics - **The chain rule** — Differentiating composite functions, including nested compositions and mixed rules. - **Implicit differentiation** — Differentiating relations and solving for the derivative, including tangent lines to implicit curves. - **Derivatives of inverse functions** — Using the inverse relationship to find derivative values without an explicit inverse formula. - **Inverse trigonometric derivatives** — Arcsine, arccosine, and arctangent derivatives in composite expressions. - **Higher-order derivatives** — Second and higher derivatives and their use in concavity and motion analysis. - **Selecting differentiation procedures** — Choosing the correct rule or combination for an unfamiliar expression. #### Study tips - Do mixed drills where the required rule is not announced, since procedure selection is the tested skill. - Practice implicit second derivatives, which appear in BC more often than students expect. - Connect the chain rule to Unit 9 explicitly; parametric derivatives are chain-rule applications. ### 4. Unit 4: Contextual Applications of Differentiation - Position: 4 of 10 - Published weighting: 6–9% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-4-contextual-applications-of-differentiation Unit 4 applies derivatives in context: motion along a line, related rates, linear approximation, and L'Hospital's rule. BC extends the indeterminate-form work to additional cases. Interpreting a derivative in words, with units, is required on the free-response section. #### Topics - **Interpreting the derivative in context** — Explaining a rate of change in a described situation with correct units. - **Straight-line motion** — Position, velocity, acceleration, speed, and direction changes. - **Related rates** — Relating quantities through an equation and differentiating with respect to time. - **Linear approximation** — Tangent-line estimates and determining whether they overestimate or underestimate. - **L'Hospital's rule** — Evaluating indeterminate forms after confirming the hypotheses hold. #### Study tips - Include units in every interpretation; rubrics award points for them specifically. - Set up related rates geometry before differentiating, and label the given and wanted rates. - Verify the indeterminate form before applying L'Hospital's rule; applying it otherwise loses the point. ### 5. Unit 5: Analytical Applications of Differentiation - Position: 5 of 10 - Published weighting: 8–11% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-5-analytical-applications-of-differentiation Unit 5 covers the Mean Value and Extreme Value Theorems, the first and second derivative tests, concavity, curve sketching, and optimization. The justification language required here is the same language rewarded across the whole free-response section. #### Topics - **Mean Value and Extreme Value Theorems** — Stating hypotheses correctly and applying each theorem on a closed interval. - **Increasing and decreasing behavior** — Using the first derivative sign to describe intervals of increase and decrease. - **First and second derivative tests** — Classifying critical points and knowing when each test applies or is inconclusive. - **Concavity and inflection points** — Second derivative sign analysis and locating changes in concavity. - **Curve sketching** — Combining derivative information into a complete graph with asymptotes and endpoints. - **Optimization** — Building a one-variable model and justifying that the candidate is the required extremum. #### Study tips - Write justifications in rubric language: name the derivative, state its sign change, and draw the conclusion. - Finish optimization problems with the justification, not just the candidate value. - Practice reading a derivative graph and describing the original function's behavior from it. ### 6. Unit 6: Integration and Accumulation of Change - Position: 6 of 10 - Published weighting: 17–20% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-6-integration-and-accumulation-of-change Unit 6 carries the largest published weighting among the shared units. Riemann sums lead to the definite integral, the Fundamental Theorem of Calculus links integration and differentiation, and BC adds integration by parts, partial fractions, and improper integrals to the technique set. #### Topics - **Riemann sums and definite integrals** — Left, right, midpoint, and trapezoidal approximations and the limit defining the integral. - **Fundamental Theorem of Calculus** — Evaluating definite integrals and differentiating accumulation functions. - **Substitution** — Changing variables for composite integrands, including adjusting limits of integration. - **Integration by parts** — Choosing the parts sensibly, and handling repeated applications and cyclic cases. - **Partial fractions** — Decomposing rational integrands with distinct linear factors before integrating. - **Improper integrals** — Rewriting an infinite limit or an unbounded integrand as a limit and testing for convergence. #### Study tips - Build a decision routine for choosing a technique: try substitution, then parts, then partial fractions. - Always convert an improper integral to a limit explicitly; the notation itself earns credit. - Give this unit heavy practice; it carries the largest weighting of the eight shared units. ### 7. Unit 7: Differential Equations - Position: 7 of 10 - Published weighting: 6–9% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-7-differential-equations Unit 7 covers slope fields, separable equations with initial conditions, and exponential models, and BC adds Euler's method for numerical approximation and the logistic model for constrained growth. Interpreting the carrying capacity and the point of fastest growth in a logistic model is a recurring exam task. #### Topics - **Modeling with differential equations** — Translating a described rate of change into an equation and interpreting it in context. - **Slope fields** — Sketching and reading slope fields and matching them to equations and solution curves. - **Separation of variables** — Separating, integrating, and applying an initial condition to find the particular solution. - **Euler's method** — Stepping forward numerically from an initial condition and judging whether the estimate is high or low. - **Exponential and logistic models** — Growth proportional to amount versus growth limited by carrying capacity, and the fastest-growth point. #### Study tips - Tabulate Euler's method steps rather than doing them in your head; the exam rewards the visible work. - Memorize what the carrying capacity is and where the logistic curve grows fastest, since those are asked directly. - Apply the initial condition immediately after integrating to avoid constant-of-integration errors. ### 8. Unit 8: Applications of Integration - Position: 8 of 10 - Published weighting: 6–9% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-8-applications-of-integration Unit 8 uses definite integrals to compute average value, accumulated change, area between curves, volumes by cross sections and revolution, and, in BC, arc length. As in AB, most of the credit lives in the setup, so a correct integrand and correct limits matter more than the final arithmetic. #### Topics - **Average value of a function** — The integral definition of average value and its interpretation in applied contexts. - **Accumulated change and motion** — Position from velocity, total distance from speed, and net change from a rate function. - **Area between curves** — Setting up integrals with respect to either variable, including multi-piece regions. - **Volumes with known cross sections** — Integrating cross-sectional area for square, triangular, and semicircular slices. - **Volumes of revolution** — Disk and washer setups about horizontal and vertical lines, including non-axis lines. - **Arc length** — Computing the length of a curve given in function form, including setting up the integral only. #### Study tips - Sketch the region and a representative slice before writing the integral, every time. - Separate displacement from total distance explicitly; the distinction is a standard trap. - Practice problems that ask only for the setup, which is common and rewards careful notation. ### 9. Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions - Position: 9 of 10 - Published weighting: 11–12% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-9-parametric-equations-polar-coordinates-and-vector-valued-functions Unit 9 is the first of the two BC-only units. It extends calculus to curves described parametrically, to vector-valued functions describing planar motion, and to polar coordinates. Derivatives become quotients of parameter derivatives, and areas in polar coordinates use a different integral entirely. #### Topics - **Parametric equations and derivatives** — Finding the first and second derivatives with respect to the independent variable from parameter derivatives. - **Arc length of parametric curves** — Setting up the arc length integral for a curve given in parametric form. - **Vector-valued functions** — Position, velocity, and acceleration vectors, and finding speed and total distance in the plane. - **Motion in the plane** — Integrating a velocity vector with an initial position to recover the path. - **Polar coordinates and derivatives** — Converting between polar and rectangular form and differentiating polar curves. - **Area in polar coordinates** — Setting up area integrals for a polar region, including regions between two curves. #### Study tips - Memorize the polar area integral separately from the rectangular area formula; conflating them is the common error. - Practice finding the second derivative for parametric curves, which requires differentiating a quotient and dividing again. - Work vector motion problems as two independent one-dimensional problems joined by shared time. ### 10. Unit 10: Infinite Sequences and Series - Position: 10 of 10 - Published weighting: 17–18% - Page: https://www.arcadesacademy.com/exams/ap-calculus-bc/unit-10-infinite-sequences-and-series Unit 10 carries a weighting comparable to integration and is the material most unique to BC. It covers convergence tests for series, power series and radius of interval of convergence, Taylor and Maclaurin polynomials, and error bounds including the alternating series and Lagrange error bounds. Choosing the right test quickly is the defining skill. #### Topics - **Sequences and series convergence** — The nth-term test, geometric series, and the distinction between a sequence and its series. - **Convergence tests** — Integral, comparison, limit comparison, ratio, and alternating series tests and when each applies. - **Absolute and conditional convergence** — Distinguishing the two and understanding what each implies about rearrangement and behavior. - **Power series** — Radius and interval of convergence, including checking the endpoints separately. - **Taylor and Maclaurin series** — Constructing series from derivatives, and the standard series for the exponential, sine, cosine, and geometric forms. - **Error bounds** — The alternating series error bound and the Lagrange error bound for a Taylor polynomial remainder. #### Study tips - Build a decision tree for convergence tests and practice choosing a test in under fifteen seconds. - Memorize the standard Maclaurin series and practice deriving new ones by substitution, differentiation, and integration. - Practice both error bounds explicitly, since error questions appear regularly and are often skipped in review. - Start series early rather than in the final weeks; this unit carries close to a fifth of the exam. ## Study strategies - Front-load the two BC-only units, since parametric and polar functions plus infinite series together carry close to thirty percent of the multiple-choice section. - Practice justification in rubric language, because on the free-response section reasoning earns points independently of the final answer. - Split practice between calculator and no-calculator conditions to match the four separately timed parts of the exam. - Maintain a convergence-test decision tree and rehearse it until test selection is fast and confident. - Rotate through the recurring free-response formats: accumulation from a rate, a table problem, particle motion in the plane, an area or volume setup, and a series problem. - Take at least one full timed exam to build stamina, since the exam runs over three hours across four timed parts. ## Frequently asked questions ### How is the AP Calculus BC Exam structured? The exam runs 3 hours 10 minutes. Section I is 42 multiple-choice questions in 1 hour 40 minutes for 50% of the score, split into a 29-question no-calculator part and a 13-question calculator part. Section II is 6 free-response questions in 1 hour 30 minutes for the other 50%, split into a 2-question calculator part and a 4-question no-calculator part. Confirm current numbers on the official exam page, since the format changed beginning with the May 2027 exam. ### Which BC topics do students find most challenging? College Board does not rank difficulty. Students most often report that infinite series demands the most practice, particularly test selection and error bounds, and that polar area setups are a frequent source of lost points. Your own practice data should guide your plan. ### How is BC scored, and what is the AB subscore? The two sections contribute equally to a composite score converted to the reported 1 to 5 scale using cut points set each year. BC students also receive an AB subscore based on the portion of the exam covering AB content, which some colleges use for placement when the BC score alone does not qualify. ### How much extra material does BC cover beyond AB? BC adds two full units, parametric, polar, and vector-valued functions and infinite sequences and series, and extends several shared units with techniques such as integration by parts, partial fractions, improper integrals, Euler's method, and the logistic model. ### When do students typically start series review? A common approach is to begin series practice as soon as the unit starts and keep a small amount of series work in every study session afterward, rather than saving it for a final review block. Its published weighting is comparable to integration, so late starts are hard to recover from. ### Is Arcade Academy affiliated with College Board? No. AP® is a registered trademark of College Board, which was not involved in the production of, and does not endorse, this site. Arcade Academy is an independent educational resource. ## Related exam guides - [AP Calculus AB](https://www.arcadesacademy.com/exams/ap-calculus-ab): All eight AP Calculus AB units with published exam weightings, from limits through applications of integration. - [AP Physics C: Mechanics](https://www.arcadesacademy.com/exams/ap-physics-c-mechanics): Calculus-based mechanics across the seven units of the revised framework, from kinematics to oscillations. - [AP Precalculus](https://www.arcadesacademy.com/exams/ap-precalculus): Three assessed units plus a fourth studied but not tested, covering polynomial, exponential, trigonometric, and polar functions. - [SAT](https://www.arcadesacademy.com/exams/sat): Study the digital, adaptive SAT module by module, with the College Board's published content domains and their share of every section. ## Independence and trademark notice AP® is a registered trademark of College Board, which was not involved in the production of, and does not endorse, this site. Arcade Academy is an independent educational resource. Confirm current exam format, dates, and policies with College Board directly. AP® is a registered trademark of College Board, which was not involved in the production of, and does not endorse, this site. Arcade Academy is an independent educational resource.