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Calculus I

Limits, derivatives, and integrals of single-variable functions. Every concept introduced with geometric intuition before algebraic formalism — understand what a derivative really means before computing one.

5 modules·82 concepts·52 practice problems·~40 hours
Practice Calc IBrowse Modules

Prerequisites

Math Foundations

Exam Relevance

AP Exams2 exams
AP Calculus AB100% of exam
AP Calculus BC60% of exam
University Exams1 exam
University Calculus I100% of exam

Module Breakdown

1.Limits & Continuity

13 concepts·12 problems

Build the limit concept from intuition to rigor, then use it to define continuity. The key theorems (IVT, EVT) show that continuous functions on closed intervals behave predictably.

13 concepts covered
Functions And Graphs ReviewPiecewise Functions And Domain ReviewIntuitive Notion Of A LimitOne Sided LimitsLimit Laws And Algebraic EvaluationLimits Involving Infinity And Horizontal AsymptotesVertical Asymptotes And Infinite LimitsIndeterminate FormsThe Squeeze TheoremDefinition Of ContinuityTypes Of DiscontinuitiesIntermediate Value TheoremExtreme Value Theorem

2.Derivatives

19 concepts·14 problems

From the limit definition to the full toolkit of differentiation rules. Each rule motivated geometrically — the product rule from area of a growing rectangle, the chain rule from composing rates.

19 concepts covered
Slopes And Rates Of Change ReviewTrigonometric Identities ReviewDerivative Definition Limit Of Difference QuotientDerivative As Rate Of ChangeDerivative As Slope Of Tangent LineDifferentiability And ContinuityPower RuleConstant Multiple And Sum Difference RulesProduct RuleQuotient RuleChain RuleImplicit DifferentiationDerivatives Of Inverse FunctionsDerivatives Of Sine And CosineDerivatives Of Tangent Secant Cosecant CotangentDerivatives Of Exponential FunctionsDerivatives Of Logarithmic FunctionsDerivatives Of Inverse Trigonometric FunctionsHigher Order Derivatives

3.Applications of Derivatives

16 concepts·10 problems

Where differentiation meets the real world: maxima and minima, curve sketching, related rates, linearization, and the mean value theorem.

16 concepts covered
Tangent Line Approximation And LinearizationDifferentialsRelated Rates Setup And StrategyRelated Rates Geometric ApplicationsRelated Rates Motion And Flow ApplicationsIncreasing And Decreasing FunctionsCritical PointsFirst Derivative Test For Relative ExtremaConcavity And The Second DerivativeInflection PointsSecond Derivative Test For Relative ExtremaCurve Sketching StrategyAbsolute Extrema On Closed IntervalsOptimization ProblemsRolles TheoremMean Value Theorem

4.Integration

18 concepts·8 problems

Build the integral from antiderivatives and Riemann sums. The Fundamental Theorem of Calculus bridges them — one of the most profound results in mathematics.

18 concepts covered
Summation Notation ReviewArea Approximation With Rectangles ReviewAntiderivative DefinitionPower Rule For IntegrationBasic Integration Formulas Trigonometric FunctionsBasic Integration Formulas Exponential And LogarithmicInitial Value ProblemsSigma Notation And Riemann SumsLeft Right And Midpoint ApproximationsDefinite Integral DefinitionProperties Of Definite IntegralsFtc Part 1 Derivative Of An IntegralFtc Part 2 Evaluation TheoremNet Change TheoremU Substitution Indefinite IntegralsU Substitution Definite IntegralsChoosing Substitution StrategyIntegrals Of Symmetric Functions

5.Applications of Integration

16 concepts·8 problems

The payoff: area between curves, kinematics via integration, work by variable force, and accumulation functions. Connects directly to physics and engineering courses.

16 concepts covered
Setting Up Definite Integrals From Word Problems ReviewCoordinate Geometry And Intersection Points ReviewArea Under A CurveArea Between Two Curves Vertical SlicesChoosing Axis Of Integration Horizontal SlicesArea With Multiple IntersectionsNet Displacement From VelocityTotal Distance From VelocityPosition Velocity And Acceleration Via IntegrationWork Done By A Constant ForceWork Done By A Variable ForceAverage Value Of A FunctionMean Value Theorem For IntegralsAccumulation Functions And Graphical InterpretationConnecting Derivatives And Integrals GraphicallyModeling With Integrals Rate In Minus Rate Out

Reference Textbooks

  • Stewart — Calculus: Early Transcendentals
  • Thomas — Calculus

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