AP Calculus AB
A complete course for the AP Calculus AB exam, made of the same interactive lessons as the rest of the site: worked examples you reveal step by step, graphs you can move, and practice that checks your answers. Every unit ends with a review and a timed unit test, and the course ends with a full practice exam.
- 8 units
- 81 topics
- 54 lessons
- 8 unit reviews
- 8 unit tests
- a full practice exam
$39 once for the whole 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 Continuity10–15% 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 that pulls together everything in Unit 1: limits and continuity.Unit review40 minCalculator in partsFree previewFree preview
- Unit 1 testTimed: 12 multiple choice (8 without a calculator, 4 with) + 1 free response, 45 minutes.Unit test45 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 Properties10–15% 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 that pulls together everything in Unit 2: differentiation: definition and fundamental properties.Unit review35 minCalculator in partsFull courseFull course
- Unit 2 testTimed: 12 multiple choice (9 without a calculator, 3 with) + 1 free response, 45 minutes.Unit test45 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 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 Differentiation10–15% 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 that pulls together everything in Unit 4: contextual applications of differentiation.Unit review35 minCalculator in partsFull courseFull course
- Unit 4 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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- 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 Differentiation15–20% 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 that pulls together everything in Unit 5: analytical applications of differentiation.Unit review50 minCalculator in partsFull courseFull course
- Unit 5 testTimed: 18 multiple choice (12 without a calculator, 6 with) + 2 free response, 75 minutes.Unit test75 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 exam11 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
- Unit 6 reviewUntimed mixed practice that pulls together everything in Unit 6: integration and accumulation of change.Unit review50 minCalculator in partsFull courseFull course
- Unit 6 testTimed: 18 multiple choice (12 without a calculator, 6 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
- 6.14Choosing an integration technique
Unit 7: Differential Equations5–10% of the exam7 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
- 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
- Unit 7 reviewUntimed mixed practice that pulls together everything in Unit 7: differential equations.Unit review30 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
- 7.6General solutions by separating variables
- 7.7Particular solutions from an initial condition
- 7.8Exponential growth and decay
Unit 8: Applications of Integration10–15% of the exam12 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
- Unit 8 reviewUntimed mixed practice that pulls together everything in Unit 8: applications of integration.Unit review40 minCalculator in partsFull courseFull course
- Unit 8 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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- 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
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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