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Course I · Open courseware

AP Calculus AB

From the paradox of instant speed to volumes of revolution — the complete AB curriculum told as one connected story, in lessons of ten to sixteen minutes.

8

Units

46

Chapters

10–16

Min per lesson

$0

Always

Start with Chapter 1Watch a free lesson
01

Limits and Continuity

8 chapters
  1. 01Can Change Happen in an Instant?A speedometer reads 60 at a single instant — but speed needs two moments to measure. The paradox that forces a genuinely new idea: the limit.
  2. 02Reading Limits from Graphs and TablesBefore any algebra, eyes and numbers: what a graph shows about a limit, what a table of values whispers — and how both can lie.
  3. 03Computing Limits Exactly0/0 is not an answer — it is a message that the expression is hiding its limit behind a common factor. Algebra decodes it.
  4. 04The Squeeze TheoremWhen a function is too wild to touch directly, trap it between two functions you can — and settle what sin(x)/x does at 0.
  5. 05What It Means to Be ContinuousThere are exactly three ways a curve can break — and a three-part test that certifies it doesn't.
  6. 06Continuity on Intervals — and Repairing BreaksSome breaks are illusions: a single missing point that one line of definition can heal.
  7. 07Limits at the Edges: AsymptotesWhat a function does next to a forbidden point, and what it does out toward forever, are both questions limits answer.
  8. 08The Intermediate Value TheoremProve an equation has a solution without ever solving it.
02

Differentiation: Definition and Fundamental Properties

5 chapters
  1. 09Defining the DerivativeThe limit machinery, aimed back at the instant-speed paradox that opened the course — and closing it.
  2. 10When Derivatives Exist — and When They Don'tContinuity was about not breaking; differentiability is about not bending sharply. Corners, cusps, and vertical tangents.
  3. 11The First Derivative RulesRunning the limit definition on every function is unbearable. The power rule and linearity turn differentiation into arithmetic.
  4. 12The Derivatives Nature Chose: sin, cos, eˣ, lnSine's derivative is cosine; eˣ is its own derivative. These are not coincidences — they are why these functions matter.
  5. 13Products, Quotients, and the Rest of Trig(fg)′ is not f′g′ — a growing rectangle shows what really happens when two changing quantities multiply.
03

Differentiation: Composite, Implicit, and Inverse Functions

4 chapters
  1. 14The Chain RuleMost functions in the wild are nested. Rates multiply through a chain — like gears passing rotation along.
  2. 15Implicit DifferentiationA circle is not a function, yet it has a tangent line at every point. Differentiating a relationship instead of a formula.
  3. 16Derivatives of Inverse FunctionsReflecting a graph across y = x flips every slope into its reciprocal — and arcsin's derivative comes out, strangely, algebraic.
  4. 17The Complete Toolkit — and Higher DerivativesEvery differentiation rule is now in hand. The remaining skill is choosing — and then asking what the derivative of the derivative says.
04

Contextual Applications of Differentiation

5 chapters
  1. 18The Derivative in MotionThe derivative finally speaks in units — meters per second, then meters per second squared. Position, velocity, and acceleration as one story.
  2. 19Rates of Change Beyond MotionMarginal cost, a draining tank, a spreading rumor — the same derivative wearing different units.
  3. 20Related RatesThe ladder slides, the shadow stretches: when quantities are linked by geometry, their rates are linked by the chain rule.
  4. 21Linear ApproximationZoom in far enough and every differentiable curve is a line — so borrow the line when the curve is too hard.
  5. 22L'Hospital's Rule0/0 returns from Unit 1 — but now, with derivatives in hand, indeterminate forms resolve by comparing rates.
05

Analytical Applications of Differentiation

7 chapters
  1. 23The Mean Value TheoremIf you averaged 60 mph over an hour, at some instant you were doing exactly 60. Obvious-sounding — and one of the most load-bearing facts in calculus.
  2. 24Where Extremes Live: Critical PointsA maximum can only hide in a short list of places. The Extreme Value Theorem guarantees it exists; critical points say where to look.
  3. 25Finding Extrema with the First DerivativeThe sign of f' narrates f — rising, falling, turning. The candidates test then closes every case on a closed interval.
  4. 26Concavity and the Second Derivative Testf' says whether the graph rises; f'' says how the rise bends. Inflection points are where the bend flips.
  5. 27The Complete Portrait: f, f′, f″ TogetherFrom a function's derivative alone you can draw its shape blind. The three graphs are one object seen three ways.
  6. 28OptimizationLargest area, cheapest can, shortest path — calculus's oldest job: finding the best.
  7. 29How Implicit Curves BehaveBack to curves that aren't functions — now with the full toolkit: where they rise, where they bend, where they turn.
06

Integration and Accumulation of Change

7 chapters
  1. 30Accumulation: The Other Half of CalculusThe odometer from the speedometer: adding up change recovers the total. The question that will mirror everything since Unit 2.
  2. 31Riemann Sums and the Definite IntegralRectangles under a curve, more and thinner, until the sum stops changing: a limit again — and this time it builds a new object.
  3. 32The Fundamental Theorem, Part 1: Accumulation FunctionsThe area-so-far is itself a function. Differentiate it, and Unit 2 comes back — the bridge between the two halves of calculus.
  4. 33The Fundamental Theorem, Part 2: Evaluating IntegralsThe theorem flips into a computing tool: any antiderivative evaluates any definite integral in two substitutions.
  5. 34AntiderivativesDifferentiation run backwards — every rule you know, reversed, plus the quietly important + C.
  6. 35SubstitutionThe chain rule in reverse: spot the inner function, and the integral untangles.
  7. 36Harder Antiderivatives and Choosing a TechniqueWhen the integrand resists: long division, completing the square — and the meta-skill of recognizing which tool a form is asking for.
07

Differential Equations

4 chapters
  1. 37Differential Equations: Laws of ChangeNature rarely hands you the function; it hands you how the function changes. Equations whose unknowns are functions.
  2. 38Slope FieldsSee every solution at once without solving anything — a field of tiny slopes that any solution must comb through.
  3. 39Separation of VariablesThe one solving technique this course owns: split the variables to opposite sides, then integrate both.
  4. 40Exponential Growth and Decay'The rate is proportional to the amount' — the sentence behind populations, radioactivity, and cooling coffee — has exactly one shape of solution.
08

Applications of Integration

6 chapters
  1. 41The Average Value of a FunctionAveraging infinitely many values sounds impossible — the integral makes it one division.
  2. 42Integrals in Context: Motion and AccumulationUnit 4 asked how fast; the integral answers how far — displacement versus distance, and any quantity that accumulates.
  3. 43Area Between CurvesArea under a curve lifts to area between two — and sometimes the smarter move is to slice sideways.
  4. 44Volumes from Cross SectionsA loaf of bread is the sum of its slices: stack known cross-sections on a base region and integrate their areas.
  5. 45Volumes of Revolution: DiscsSpin a region around an axis and its slices become coins.
  6. 46Volumes of Revolution: WashersWhen the region doesn't touch the axis, the coins get holes.

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