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Math · AP

Calculus & Probability

Surfaces, tangents, Riemann sums and dice you can actually roll.

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The course

7 worlds · 60 lessons

World 1 · Functions & Limits

FREEFrom graphs to the edge of calculus
  1. 01What is a function?Inputs, outputs and rules · 6 min
  2. 02Families of functionsLines, parabolas, waves · 7 min
  3. 03Slope of a lineRise over run · 6 min
  4. 04What is a limit?Closing in on a value · 8 min
  5. 05One-sided limitsApproaching from left and right · 7 min
  6. 06ContinuityNo holes, jumps or breaks · 7 min
  7. 07Average rate of changeThe secant line · 7 min
  8. 08Instantaneous rateWhen the secant becomes the tangent · 8 min
  9. 09Exponentials & logsGrowth and its inverse · 8 min
  10. 10Shifting & scalingTransforming any graph · 6 min
  11. 11Piecewise functionsDifferent rules on different intervals · 6 min
  12. 12Checkpoint — limitsPull it together · 7 min

World 2 · Derivative Lab

PRORules, tangents and 3D surfaces
  1. 01The derivative as slopef'(x) at every point · 7 minPRO
  2. 02The power ruleDifferentiate polynomials fast · 7 minPRO
  3. 03Tangents on a curveSee the slope live · 6 minPRO
  4. 04The product ruleDifferentiating f·g · 7 minPRO
  5. 05The quotient ruleDifferentiating f / g · 7 minPRO
  6. 06The chain ruleFunctions inside functions · 8 minPRO
  7. 07Higher derivativesAcceleration & concavity · 7 minPRO
  8. 08Tangent planes in 3DPartial derivatives · 8 minPRO
  9. 09Critical pointsMaxima, minima, saddles · 7 minPRO
  10. 10Related ratesLinked changing quantities · 8 minPRO
  11. 11OptimizationBest possible value · 8 minPRO
  12. 12The Mean Value TheoremAverage meets instantaneous · 7 minPRO
  13. 13Checkpoint — derivativesThe full toolbox · 6 minPRO

World 3 · Integral Lab

PROArea, accumulation and the FTC
  1. 01Area under a curveA new kind of question · 7 minPRO
  2. 02Riemann sumsRectangles that converge · 8 minPRO
  3. 03The definite integralThe limit of the sum · 7 minPRO
  4. 04AntiderivativesUndoing the derivative · 7 minPRO
  5. 05The Fundamental TheoremDerivatives meet integrals · 8 minPRO
  6. 06Net changeIntegrating a rate · 7 minPRO
  7. 07Average valueThe mean height of a curve · 6 minPRO
  8. 08Area between curvesTop minus bottom · 7 minPRO
  9. 09Volumes of revolutionSpinning a region into a solid · 8 minPRO
  10. 10Improper integralsAreas that run to infinity · 7 minPRO
  11. 11Numerical integrationWhen there is no formula · 7 minPRO
  12. 12Checkpoint — integralsAccumulation mastered · 6 minPRO

World 4 · Probability & Distributions

PROChance, dice and the bell curve
  1. 01Sample space & eventsListing what can happen · 6 minPRO
  2. 02Combining probabilitiesAnd, or, complement · 7 minPRO
  3. 03The two-dice sumWhy 7 is special · 7 minPRO
  4. 04Random variablesNumbers from chance · 7 minPRO
  5. 05Expected valueThe long-run average · 7 minPRO
  6. 06Variance & spreadHow wide is the distribution? · 7 minPRO
  7. 07The binomial distributionCounting successes · 8 minPRO
  8. 08The Poisson distributionRare events over time · 7 minPRO
  9. 09The normal distributionThe bell curve · 8 minPRO
  10. 10Monte-Carlo simulationEstimating by random trials · 7 minPRO
  11. 11Law of large numbersAverages settle down · 6 minPRO
  12. 12The Central Limit TheoremWhy the bell is everywhere · 8 minPRO
  13. 13Checkpoint — probabilityChance, mastered · 6 minPRO

World 5 · The shape of change

PRORead a landscape, then find its steepest path
  1. 01Two directions, two slopesProbe a surface at one point · 6 minPRO
  2. 02Walk without climbingContour lines and their normals · 7 minPRO
  3. 03Find the steepest climbA direction is a unit vector · 7 minPRO
  4. 04The best view on a circular trailA constrained maximum · 8 minPRO

World 6 · From contour map to terrain

PROReshape a hill and read the map it leaves
  1. 01Lift a hill through its contoursPRO
  2. 02Read steepness from a mapPRO

World 7 · Unit circle to sine wave

PROTurn an angle, unroll it, and watch sine and cosine appear
  1. 01Radians: measure an angle by its arcThe point (cos θ, sin θ) · 7 minPRO
  2. 02Amplitude, frequency and phaseMatch y = A·sin(ωθ + φ) · 8 minPRO
  3. 03The helix: two graphs in one curveSide views of (cos θ, sin θ, θ) · 7 minPRO
  4. 04e^(iθ) and the quarter-turn velocityWhy the derivative of sin is cos · 8 minPRO
Read first

Guides for Calculus & Probability

All guides →
Explainer
Codes & Ciphers LabClassic Ciphers: Caesar, ROT13, Atbash and AffineHow the Caesar, ROT13, Atbash, affine and keyword ciphers work, how to crack them by counting letters, and why none of them is secure today.4 min read
Explainer
Shape LabPlace Value, Fractions, Nets and Euler's FormulaSee how blocks show place value, how fractions are fair shares, how nets fold into solids and why faces minus edges plus vertices always equals 2.5 min read
Explainer
Vector LabVectors Explained: Sum, Dot Product, Cross ProductLearn how vectors add tip to tail, how the dot product detects right angles, what the cross product points to, and how projection casts a shadow.4 min read