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Calculus I
Limits, derivatives, applications, and the first ideas of accumulation.
Start with Antiderivatives →39 complete · 39 total lessons
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01AntiderivativesA rigorous, example-driven guide to antiderivatives, including hypotheses, method choice, verification, and practice.Full lesson + sheet02Area Under a CurveA rigorous, example-driven guide to area under a curve, including hypotheses, method choice, verification, and practice.Full lesson + sheet03Average Value of a FunctionA rigorous, example-driven guide to average value of a function, including hypotheses, method choice, verification, and practice.Full lesson + sheet04Basic Differentiation RulesCore differentiation rules turn constants, powers, sums, and scalar multiples into instantaneous-rate formulas efficiently.Full lesson + sheet05Chain RuleThe chain rule differentiates a function inside another by multiplying the outer rate by the inner rate.Full lesson + sheet06ConcavityConcavity describes whether slopes are increasing or decreasing and is analyzed with the second derivative.Full lesson + sheet07ContinuityA rigorous, example-driven guide to continuity, including hypotheses, method choice, verification, and practice.Full lesson + sheet08Critical PointsCritical numbers are domain inputs where a derivative is zero or undefined and are candidates for extrema or other changes in behaviour.Full lesson + sheet09Curve SketchingCurve sketching combines algebra, limits, first derivatives, and second derivatives into a coherent graph.Full lesson + sheet10Definite IntegralA definite integral measures signed accumulation over an interval and can be evaluated using antiderivatives.Full lesson + sheet11DerivativeA derivative measures instantaneous rate of change and gives the slope of a function at a point.Full lesson + sheet12Derivative as a Rate of ChangeA rigorous, example-driven guide to derivative as a rate of change, including hypotheses, method choice, verification, and practice.Full lesson + sheet13Derivative from First PrinciplesThe derivative from first principles is the limit of secant slopes as the second point approaches the first.Full lesson + sheet14Derivatives of Exponential FunctionsA rigorous, example-driven guide to derivatives of exponential functions, including hypotheses, method choice, verification, and practice.Full lesson + sheet15Derivatives of Logarithmic FunctionsA rigorous, example-driven guide to derivatives of logarithmic functions, including hypotheses, method choice, verification, and practice.Full lesson + sheet16Derivatives of Trigonometric FunctionsA rigorous, example-driven guide to derivatives of trigonometric functions, including hypotheses, method choice, verification, and practice.Full lesson + sheet17DifferentialsA rigorous, example-driven guide to differentials, including hypotheses, method choice, verification, and practice.Full lesson + sheet18First Derivative TestThe first derivative test classifies critical points by tracking whether a function changes from increasing to decreasing or vice versa.Full lesson + sheet19Fundamental Theorem of CalculusA rigorous, example-driven guide to fundamental theorem of calculus, including hypotheses, method choice, verification, and practice.Full lesson + sheet20Implicit DifferentiationA rigorous, example-driven guide to implicit differentiation, including hypotheses, method choice, verification, and practice.Full lesson + sheet21Increasing and Decreasing FunctionsA rigorous, example-driven guide to increasing and decreasing functions, including hypotheses, method choice, verification, and practice.Full lesson + sheet22Infinite LimitsA rigorous, example-driven guide to infinite limits, including hypotheses, method choice, verification, and practice.Full lesson + sheet23Inflection PointsA rigorous, example-driven guide to inflection points, including hypotheses, method choice, verification, and practice.Full lesson + sheet24Intermediate Value TheoremA rigorous, example-driven guide to intermediate value theorem, including hypotheses, method choice, verification, and practice.Full lesson + sheet25LimitsA limit describes the value a function approaches as its input moves near a target, whether or not the function is defined there.Full lesson + sheet26Limits at InfinityA rigorous, example-driven guide to limits at infinity, including hypotheses, method choice, verification, and practice.Full lesson + sheet27Linear ApproximationA rigorous, example-driven guide to linear approximation, including hypotheses, method choice, verification, and practice.Full lesson + sheet28Mean Value TheoremA rigorous, example-driven guide to mean value theorem, including hypotheses, method choice, verification, and practice.Full lesson + sheet29One-Sided LimitsA rigorous, example-driven guide to one-sided limits, including hypotheses, method choice, verification, and practice.Full lesson + sheet30OptimizationOptimization uses constraints, derivatives, and endpoint comparisons to find the best feasible value of a model.Full lesson + sheet31Product RuleThe product rule differentiates two changing factors by adding the change from each factor while the other is held in place.Full lesson + sheet32Quotient RuleThe quotient rule differentiates one changing function divided by another while preserving the denominator's domain restriction.Full lesson + sheet33Related RatesA rigorous, example-driven guide to related rates, including hypotheses, method choice, verification, and practice.Full lesson + sheet34Review of FunctionsA rigorous, example-driven guide to review of functions, including hypotheses, method choice, verification, and practice.Full lesson + sheet35Riemann SumsA rigorous, example-driven guide to riemann sums, including hypotheses, method choice, verification, and practice.Full lesson + sheet36Second Derivative TestA rigorous, example-driven guide to second derivative test, including hypotheses, method choice, verification, and practice.Full lesson + sheet37Sigma NotationA rigorous, example-driven guide to sigma notation, including hypotheses, method choice, verification, and practice.Full lesson + sheet38Substitution RuleA rigorous, example-driven guide to substitution rule, including hypotheses, method choice, verification, and practice.Full lesson + sheet39What Is CalculusA rigorous, example-driven guide to what is calculus, including hypotheses, method choice, verification, and practice.Full lesson + sheet
