AP Calculus BC
A complete course for the AP Calculus BC exam. It uses the AB course's lessons for the topics the two exams share, and adds the BC-only ones: integration by parts, partial fractions and improper integrals, Euler's method and logistic growth, arc length, parametric, polar and vector functions, and infinite series. Every unit ends with a BC-level review and a timed unit test, and the course ends with a full practice exam.
- 10 units
- 111 topics
- 76 lessons
- 10 unit reviews
- 10 unit tests
- a full practice exam
$49 once for the whole course, and the whole AP Calculus AB course, no subscription. Unit 1 is free with an account.
How every unit works
- Lessons
- Unit review
- Unit test
Then a full practice exam, timed like the real one. Sign in and your progress, XP and topic mastery are saved as you go.
Units
Unit 1 is free. The full course unlocks every unit and the practice exam.
Unit 1: Limits and Continuity5–10% of the exam16 topicsFree previewFree preview
- What is a limit?See why we need limits to talk about change at an instant, then learn to read and write limits, one-sided limits and "does not exist".Lesson35 minFree previewFree preview
- Limits from graphs and tablesEstimate limits and one-sided limits by reading graphs (holes and jumps) and tables of values.Lesson35 minFree previewFree preview
- Limit laws and direct substitutionSplit a limit into pieces with the sum, product, quotient and composition rules, and use plain substitution when it works.Lesson35 minFree previewFree preview
- Rewriting to find limitsWhen substitution gives 0/0, factor, multiply by a conjugate or simplify, and learn to pick the right move.Lesson40 minFree previewFree preview
- The squeeze theorem and connecting representationsTrap a tricky limit between two easy ones, then match the same limit across a graph, a table and a formula.Lesson40 minFree previewFree preview
- Continuity at a pointTell removable, jump and infinite discontinuities apart and check the three conditions for continuity at a point.Lesson35 minFree previewFree preview
- Continuity on intervals and fixing gapsKnow where common functions are continuous, then choose constants that make a piecewise function connect or fill in a hole.Lesson35 minFree previewFree preview
- Infinite limits and asymptotesFind vertical asymptotes from infinite limits, and use limits at infinity to find horizontal asymptotes.Lesson40 minFree previewFree preview
- The intermediate value theoremCheck the conditions of the IVT and write a complete justification that a function hits a value, often from a table.Lesson30 minFree previewFree preview
- Unit 1 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 1: limits and continuity.Unit review40 minCalculator in partsFree previewFree preview
- Unit 1 testTimed: 8 multiple choice (6 without a calculator, 2 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFree previewFree preview
Topics in this unit
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- 1.1Change at an instant
- 1.2What a limit means and how to write it
- 1.3Reading limits from graphs
- 1.4Reading limits from tables
- 1.5Limit laws
- 1.6Limits by rewriting (factoring, conjugates)
- 1.7Choosing a method for a limit
- 1.8The squeeze theorem
- 1.9Limits across graphs, tables and formulas
- 1.10Kinds of discontinuities
- 1.11Continuity at a point
- 1.12Continuity on an interval
- 1.13Removing a discontinuity
- 1.14Infinite limits and vertical asymptotes
- 1.15Limits at infinity and horizontal asymptotes
- 1.16The intermediate value theorem
Unit 2: Differentiation: Definition and Fundamental Properties5–10% of the exam10 topicsFull courseFull course
- From average rate to the derivativeMove from average to instantaneous rate of change, meet the limit definition of the derivative and its notation.Lesson40 minFull courseFull course
- Estimating derivatives from dataEstimate a derivative from a table or graph, with units, and pick the best nearby interval.Lesson30 minFull courseFull course
- When does a derivative exist?Spot corners, cusps, vertical tangents and breaks, and see why differentiable means continuous but not the other way round.Lesson35 minFull courseFull course
- Power rule and basic rulesDifferentiate powers (including negative and fractional ones) and sums, differences and constant multiples, then write tangent lines.Lesson40 minFull courseFull course
- Derivatives of sin, cos, eˣ and ln xLearn the derivatives of four key functions and why they are what they are.Lesson30 minFull courseFull course
- The product ruleDifferentiate a product of two functions, from formulas, tables and graphs.Lesson35 minFull courseFull course
- Quotient rule and the other trig derivativesDifferentiate quotients, then use the quotient rule to get the derivatives of tan, cot, sec and csc.Lesson40 minFull courseFull course
- Unit 2 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 2: differentiation: definition and fundamental properties.Unit review35 minCalculator in partsFull courseFull course
- Unit 2 testTimed: 8 multiple choice (6 without a calculator, 2 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFull courseFull course
Topics in this unit
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- 2.1Average and instantaneous rates of change
- 2.2The derivative as a limit, and its notation
- 2.3Estimating a derivative at a point
- 2.4When a derivative exists (differentiability and continuity)
- 2.5The power rule
- 2.6Constant, sum, difference and constant-multiple rules
- 2.7Derivatives of sin x, cos x, eˣ and ln x
- 2.8The product rule
- 2.9The quotient rule
- 2.10Derivatives of tan, cot, sec and csc
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions5–10% of the exam6 topicsFull courseFull course
- The chain ruleDifferentiate a function inside another function, working one layer at a time, from formulas and tables.Lesson40 minFull courseFull course
- Implicit differentiationDifferentiate both sides of an equation, solve for dy/dx and find tangent lines to curves that are not functions.Lesson40 minFull courseFull course
- Derivatives of inverses and inverse trigFind the slope of an inverse function from a table or graph, and learn the derivatives of arcsin, arccos and arctan.Lesson40 minFull courseFull course
- Choosing rules and higher derivativesDecide which rules a messy expression needs, then take second and higher derivatives, including implicit ones.Lesson40 minFull courseFull course
- Unit 3 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 3: differentiation: composite, implicit, and inverse functions.Unit review30 minCalculator in partsFull courseFull course
- Unit 3 testTimed: 8 multiple choice (6 without a calculator, 2 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFull courseFull course
Topics in this unit
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- 3.1The chain rule
- 3.2Implicit differentiation
- 3.3Derivatives of inverse functions
- 3.4Derivatives of inverse trig functions
- 3.5Choosing a differentiation method
- 3.6Second and higher derivatives
Unit 4: Contextual Applications of Differentiation5–10% of the exam7 topicsFull courseFull course
- What a derivative means in contextExplain what f′(a) tells you in a real situation, with units and time, in sentences that earn the point.Lesson35 minFull courseFull course
- Motion and other rates of changeConnect position, velocity, acceleration and speed, and use derivatives for rates in science and business.Lesson45 minFull courseFull course
- Related ratesSet up and solve problems where several changing quantities are tied together, with the right signs and units.Lesson45 minFull courseFull course
- Linear approximationUse a tangent line to estimate nearby values and decide, from concavity, whether the estimate is too high or too low.Lesson30 minFull courseFull course
- L'Hospital's ruleUse derivatives to evaluate 0/0 and ∞/∞ limits, checking the conditions first.Lesson30 minFull courseFull course
- Unit 4 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 4: contextual applications of differentiation.Unit review35 minCalculator in partsFull courseFull course
- Unit 4 testTimed: 8 multiple choice (5 without a calculator, 3 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFull courseFull course
Topics in this unit
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- 4.1What a derivative means in context
- 4.2Motion on a line: position, velocity, acceleration
- 4.3Rates of change in other settings
- 4.4Getting started with related rates
- 4.5Solving related rates problems
- 4.6Linear approximation
- 4.7L'Hospital's rule
Unit 5: Analytical Applications of Differentiation10–15% of the exam12 topicsFull courseFull course
- The mean value theoremState the MVT, check its conditions, find the point it promises and use it to justify claims from tables.Lesson35 minFull courseFull course
- Extreme values and critical pointsKnow what the extreme value theorem guarantees, separate local from global extrema and find critical points.Lesson35 minFull courseFull course
- Increasing, decreasing and the first derivative testUse the sign of f′ to find where a function rises and falls and to classify relative extrema, with a written justification.Lesson40 minFull courseFull course
- Absolute extrema: the candidates testFind the absolute maximum and minimum by comparing critical points and endpoints, and argue it is global.Lesson35 minFull courseFull course
- Concavity and the second derivative testUse f″ to find concavity and inflection points, and to classify critical points.Lesson40 minFull courseFull course
- Reading f, f′ and f″ togetherSketch f from the graph of f′ (and back), and answer graph questions with reasons tied to the picture.Lesson45 minFull courseFull course
- OptimizationTurn a word problem into a function, find its maximum or minimum on the right domain and prove it is the best.Lesson45 minFull courseFull course
- Behavior of implicit curvesFind horizontal and vertical tangents and judge concavity on curves given by an equation in x and y.Lesson35 minFull courseFull course
- Unit 5 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 5: analytical applications of differentiation.Unit review50 minCalculator in partsFull courseFull course
- Unit 5 testTimed: 12 multiple choice (8 without a calculator, 4 with) + 1 free response, 45 minutes.Unit test45 minCalculator in partsFull courseFull course
Topics in this unit
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- 5.1The mean value theorem
- 5.2Extreme value theorem, local and global extrema, critical points
- 5.3Where a function increases or decreases
- 5.4The first derivative test
- 5.5The candidates test for absolute extrema
- 5.6Concavity
- 5.7The second derivative test
- 5.8Sketching a function and its derivatives
- 5.9How f, f′ and f″ fit together
- 5.10Setting up optimization problems
- 5.11Solving optimization problems
- 5.12Implicit relations: tangents and behavior
Unit 6: Integration and Accumulation of Change15–20% of the exam14 topicsFull courseFull course
- Accumulation and Riemann sumsSee area under a rate graph as accumulated change, then estimate it with left, right, midpoint and trapezoid sums.Lesson45 minFull courseFull course
- From sums to the definite integralRead and write sigma notation, and see the definite integral as the limit of Riemann sums.Lesson35 minFull courseFull course
- The fundamental theorem and accumulation functionsDifferentiate a function defined by an integral, including ones with variable limits and the chain rule.Lesson40 minFull courseFull course
- Reading accumulation functions from graphsUse the graph of f to find values of g(x), where it rises, bends and reaches its extremes.Lesson40 minFull courseFull course
- Properties of definite integralsReverse limits, split intervals, pull out constants and find integrals from geometry or piecewise graphs.Lesson35 minFull courseFull course
- Evaluating integrals with the fundamental theoremEvaluate definite integrals with antiderivatives, and use net change to turn a rate into a final value.Lesson35 minFull courseFull course
- Antiderivatives and indefinite integralsLearn the basic antiderivative rules and notation, why +C matters and the ln|x| rule.Lesson35 minFull courseFull course
- u-substitutionUndo the chain rule by substitution, for indefinite and definite integrals, including changing the bounds.Lesson45 minFull courseFull course
- Long division, completing the square, choosing a techniqueRewrite fractions so they can be integrated, spot arctan forms and decide which technique a problem needs.Lesson40 minFull courseFull course
- Integration by partsUndo the product rule to integrate products like x·eˣ: choose u and dv, repeat parts when you need to, and use it on definite integrals and tables of values.Lesson40 minFull courseFull course
- Partial fractionsSplit a fraction whose bottom factors into different linear pieces into simpler fractions, then integrate each one (doing long division first when needed).Lesson35 minFull courseFull course
- Improper integrals and choosing a techniqueIntegrate out to infinity or across a vertical asymptote with a limit, decide whether the answer is a finite number, and pick the right technique for any integral.Lesson45 minFull courseFull course
- Unit 6 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 6: integration and accumulation of change.Unit review55 minCalculator in partsFull courseFull course
- Unit 6 testTimed: 18 multiple choice (13 without a calculator, 5 with) + 2 free response, 75 minutes.Unit test75 minCalculator in partsFull courseFull course
Topics in this unit
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- 6.1Accumulating change
- 6.2Riemann sums
- 6.3Sigma notation and the definite integral
- 6.4Fundamental theorem and accumulation functions
- 6.5How accumulation functions behave
- 6.6Properties of definite integrals
- 6.7Evaluating definite integrals with the fundamental theorem
- 6.8Antiderivatives and basic rules
- 6.9u-substitution
- 6.10Long division and completing the square
- 6.11Integration by parts
- 6.12Partial fractions (distinct linear factors)
- 6.13Improper integrals
- 6.14Choosing an integration technique
Unit 7: Differential Equations5–10% of the exam9 topicsFull courseFull course
- Writing and checking differential equationsTurn "the rate is proportional to..." into a differential equation and test whether a function solves it.Lesson35 minFull courseFull course
- Slope fieldsDraw a slope field, sketch solution curves through it and read behavior and equilibria from the picture.Lesson40 minFull courseFull course
- Euler's methodFollow a differential equation's slopes one small step at a time to estimate a solution, and tell from concavity whether the estimate is too high or too low.Lesson35 minFull courseFull course
- Separation of variablesSolve a differential equation by separating variables, and use the initial condition right after the +C.Lesson45 minFull courseFull course
- Exponential growth and decaySolve dy/dt = ky, use half-life and doubling time, and handle cooling-style models.Lesson35 minFull courseFull course
- Logistic growthRead a logistic model dy/dt = ky(a − y) without solving it: its carrying capacity, where it grows fastest, and what happens from different starting values.Lesson35 minFull courseFull course
- Unit 7 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 7: differential equations.Unit review35 minCalculator in partsFull courseFull course
- Unit 7 testTimed: 8 multiple choice (6 without a calculator, 2 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFull courseFull course
Topics in this unit
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- 7.1Writing differential equations from a situation
- 7.2Checking a solution
- 7.3Drawing slope fields
- 7.4Reading slope fields
- 7.5Euler's method
- 7.6General solutions by separating variables
- 7.7Particular solutions from an initial condition
- 7.8Exponential growth and decay
- 7.9Logistic models
Unit 8: Applications of Integration5–10% of the exam13 topicsFull courseFull course
- Average value of a functionFind the average value of a function on an interval and tell it apart from the average rate of change.Lesson30 minFull courseFull course
- Motion with integralsGet position and displacement from velocity, separate displacement from total distance and find average velocity.Lesson40 minFull courseFull course
- Accumulation in real situationsHandle rate-in and rate-out problems, find the amount at a time and the maximum or minimum amount, with units.Lesson40 minFull courseFull course
- Area between curves (in x)Find the area between two curves as top minus bottom, including curves that cross more than twice.Lesson40 minFull courseFull course
- Area between curves (in y)Slice horizontally and integrate right minus left when that is easier.Lesson30 minFull courseFull course
- Volumes with known cross sectionsStack squares, rectangles, triangles and semicircles on a base region and add up the volume.Lesson40 minFull courseFull course
- Disc methodSpin a region around an axis or another line and find the volume with discs.Lesson40 minFull courseFull course
- Washer methodFind volumes of solids with a hole using outer and inner radii, and write a complete setup.Lesson45 minFull courseFull course
- Arc lengthFind the length of a curve y = f(x) with an integral, write arc-length setups without evaluating them, and use arc-length functions.Lesson35 minFull courseFull course
- Unit 8 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 8: applications of integration.Unit review40 minCalculator in partsFull courseFull course
- Unit 8 testTimed: 8 multiple choice (5 without a calculator, 3 with) + 1 free response, 35 minutes.Unit test35 minCalculator in partsFull courseFull course
Topics in this unit
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- 8.1Average value of a function
- 8.2Position, velocity and acceleration with integrals
- 8.3Accumulation in applied settings
- 8.4Area between curves (functions of x)
- 8.5Area between curves (functions of y)
- 8.6Area when curves cross more than twice
- 8.7Cross sections: squares and rectangles
- 8.8Cross sections: triangles and semicircles
- 8.9Disc method around the x- or y-axis
- 8.10Disc method around other lines
- 8.11Washer method around the x- or y-axis
- 8.12Washer method around other lines
- 8.13Arc length of a graph (and distance traveled)
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions10–15% of the exam9 topicsFull courseFull course
- Parametric curves and their derivativesDescribe a curve with x(t) and y(t), follow its direction, and find its slope dy/dx, its tangent lines and its second derivative d²y/dx².Lesson45 minFull courseFull course
- Vector-valued functionsWrite position, velocity and acceleration as vectors, then differentiate and integrate them one part at a time, starting from a known position.Lesson40 minFull courseFull course
- Motion in the plane: speed and distanceFind speed, position from velocity, displacement and total distance traveled (the same integral as a parametric curve's length) for something moving in the plane.Lesson45 minFull courseFull course
- Polar curves and their slopesReview polar graphs, switch between polar and rectangular coordinates, and find slopes dy/dx and rates like dr/dθ on polar curves.Lesson45 minFull courseFull course
- Area in polar coordinatesFind the area inside a polar curve and between two polar curves: the limits for θ, which curve is outside, and calculator setups.Lesson45 minFull courseFull course
- Unit 9 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 9: parametric equations, polar coordinates, and vector-valued functions.Unit review45 minCalculator in partsFull courseFull course
- Unit 9 testTimed: 12 multiple choice (7 without a calculator, 5 with) + 2 free response, 60 minutes.Unit test60 minCalculator in partsFull courseFull course
Topics in this unit
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- 9.1Parametric curves and dy/dx
- 9.2Second derivatives of parametric curves
- 9.3Arc length of a parametric curve
- 9.4Vector-valued functions and their derivatives
- 9.5Integrating vector-valued functions
- 9.6Motion in the plane (parametric or vector)
- 9.7Polar coordinates and polar derivatives
- 9.8Area inside one polar curve
- 9.9Area between two polar curves
Unit 10: Infinite Sequences and Series15–20% of the exam15 topicsFull courseFull course
- Infinite series and geometric seriesSee what it means for an endless sum to add up to a number through its partial sums, and find the sum of a geometric series.Lesson40 minFull courseFull course
- The nth-term and integral testsSpot series that can't converge with the nth-term test, and use the integral test, with its conditions, to decide others.Lesson40 minFull courseFull course
- p-series and comparison testsKnow which p-series converge, then compare a new series with one you know, directly or with a limit, and write the full conclusion.Lesson40 minFull courseFull course
- The alternating series testCheck the three conditions of the alternating series test in writing to show that a series whose signs take turns converges.Lesson30 minFull courseFull course
- The ratio test; absolute and conditional convergenceUse the ratio test and know when it can't decide, tell absolute from conditional convergence, and choose a test for a new series.Lesson40 minFull courseFull course
- The alternating series error boundStop an alternating series early and bound the error with the first term you left out, written as a clear inequality.Lesson30 minFull courseFull course
- Taylor polynomialsBuild polynomials from a function's derivatives at a point (even from a differential equation), centered anywhere, and use them to estimate values.Lesson40 minFull courseFull course
- The Lagrange error boundPut a guaranteed limit on a Taylor polynomial's error using the largest value of the next derivative, written as an inequality chain.Lesson35 minFull courseFull course
- Radius and interval of convergenceUse the ratio test to find where a power series converges, then test both endpoints with named tests and write the interval.Lesson40 minFull courseFull course
- Taylor and Maclaurin seriesFind Taylor and Maclaurin series from derivatives, know the series for eˣ, sin x, cos x and 1/(1 − x), and recognize what a series adds up to.Lesson40 minFull courseFull course
- Building series from known onesBuild new series from known ones by substituting, multiplying, and differentiating or integrating term by term, and find their general terms and sums.Lesson45 minFull courseFull course
- Unit 10 reviewUntimed mixed practice at BC exam level that pulls together everything in Unit 10: infinite sequences and series.Unit review55 minCalculator in partsFull courseFull course
- Unit 10 testTimed: 18 multiple choice (14 without a calculator, 4 with) + 2 free response, 75 minutes.Unit test75 minCalculator in partsFull courseFull course
Topics in this unit
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- 10.1Series as limits of partial sums
- 10.2Geometric series
- 10.3The nth-term test for divergence
- 10.4The integral test
- 10.5The harmonic series and p-series
- 10.6Direct and limit comparison tests
- 10.7The alternating series test
- 10.8The ratio test
- 10.9Absolute and conditional convergence
- 10.10The alternating series error bound
- 10.11Taylor polynomials
- 10.12The Lagrange error bound
- 10.13Radius and interval of convergence
- 10.14Taylor and Maclaurin series
- 10.15Building power series from known series
Full-length practice exam
Timed like the real exam, with multiple-choice and free-response parts, and feedback on every question. It's part of the full course.
- Section I, Part A: multiple choiceTimed: 29 multiple choice, 62 minutes, no calculator.Practice exam62 minNo calculatorFull courseFull course
- Section I, Part B: multiple choiceTimed: 13 multiple choice, 38 minutes, graphing calculator required.Practice exam38 minCalculatorFull courseFull course
- Section II, Part A: free responseTimed: 2 free-response questions, 30 minutes, graphing calculator required.Practice exam30 minCalculatorFull courseFull course
- Section II, Part B: free responseTimed: 4 free-response questions, 60 minutes, no calculator.Practice exam60 minNo calculatorFull courseFull course
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