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Calculus 1

Unit 1: Limits and continuity

Limits intro: Limits and continuityEstimating limits from graphs: Limits and continuityEstimating limits from tables: Limits and continuityFormal definition of limits (epsilon-delta): Limits and continuityProperties of limits: Limits and continuityLimits by direct substitution: Limits and continuityLimits using algebraic manipulation: Limits and continuityStrategy in finding limits: Limits and continuity
Squeeze theorem: Limits and continuityTypes of discontinuities: Limits and continuityContinuity at a point: Limits and continuityContinuity over an interval: Limits and continuityRemoving discontinuities: Limits and continuityInfinite limits: Limits and continuityLimits at infinity: Limits and continuityIntermediate value theorem: Limits and continuity

Unit 2: Derivatives: definition and basic rules

Average vs. instantaneous rate of change: Derivatives: definition and basic rulesSecant lines: Derivatives: definition and basic rulesDerivative definition: Derivatives: definition and basic rulesEstimating derivatives: Derivatives: definition and basic rulesDifferentiability: Derivatives: definition and basic rulesPower rule: Derivatives: definition and basic rules
Derivative rules: constant, sum, difference, and constant multiple: Derivatives: definition and basic rulesCombining the power rule with other derivative rules: Derivatives: definition and basic rulesDerivatives of cos(x), sin(x), 𝑒ˣ, and ln(x): Derivatives: definition and basic rulesProduct rule: Derivatives: definition and basic rulesQuotient rule: Derivatives: definition and basic rulesDerivatives of tan(x), cot(x), sec(x), and csc(x): Derivatives: definition and basic rulesProof videos: Derivatives: definition and basic rules

Unit 3: Derivatives: chain rule and other advanced topics

Chain rule: Derivatives: chain rule and other advanced topicsMore chain rule practice: Derivatives: chain rule and other advanced topicsImplicit differentiation: Derivatives: chain rule and other advanced topicsImplicit differentiation (advanced examples): Derivatives: chain rule and other advanced topicsDifferentiating inverse functions: Derivatives: chain rule and other advanced topicsDerivatives of inverse trigonometric functions: Derivatives: chain rule and other advanced topics
Strategy in differentiating functions: Derivatives: chain rule and other advanced topicsDifferentiation using multiple rules: Derivatives: chain rule and other advanced topicsSecond derivatives: Derivatives: chain rule and other advanced topicsDisguised derivatives: Derivatives: chain rule and other advanced topicsLogarithmic differentiation: Derivatives: chain rule and other advanced topicsProof videos: Derivatives: chain rule and other advanced topics

Unit 4: Applications of derivatives

Meaning of the derivative in context: Applications of derivativesStraight-line motion: Applications of derivativesNon-motion applications of derivatives: Applications of derivativesIntroduction to related rates: Applications of derivatives
Solving related rates problems: Applications of derivativesApproximation with local linearity: Applications of derivativesL’Hôpital’s rule: Applications of derivativesL’Hôpital’s rule: composite exponential functions: Applications of derivatives

Unit 5: Analyzing functions

Mean value theorem: Analyzing functions Extreme value theorem and critical points: Analyzing functions Intervals on which a function is increasing or decreasing: Analyzing functions Relative (local) extrema: Analyzing functions Absolute (global) extrema: Analyzing functions Concavity and inflection points intro: Analyzing functions
Analyzing concavity and inflection points: Analyzing functions Second derivative test: Analyzing functions Sketching curves: Analyzing functions Connecting f, f', and f'': Analyzing functions Solving optimization problems: Analyzing functions Analyzing implicit relations: Analyzing functions Calculator-active practice: Analyzing functions

Unit 6: Integrals

Accumulations of change introduction: IntegralsApproximation with Riemann sums: IntegralsSummation notation review: IntegralsRiemann sums in summation notation: IntegralsDefining integrals with Riemann sums: IntegralsFundamental theorem of calculus and accumulation functions: IntegralsInterpreting the behavior of accumulation functions: IntegralsProperties of definite integrals: Integrals
Fundamental theorem of calculus and definite integrals: IntegralsReverse power rule: IntegralsIndefinite integrals of common functions: IntegralsDefinite integrals of common functions: IntegralsIntegrating with u-substitution: IntegralsIntegrating using long division and completing the square: IntegralsIntegrating using trigonometric identities: IntegralsProof videos: Integrals

Unit 7: Differential equations

Differential equations introduction: Differential equationsVerifying solutions for differential equations: Differential equationsSketching slope fields: Differential equations
Reasoning using slope fields: Differential equationsSeparation of variables: Differential equationsParticular solutions to differential equations: Differential equationsExponential models: Differential equations

Unit 8: Applications of integrals

Average value of a function: Applications of integralsStraight-line motion: Applications of integralsNon-motion applications of integrals: Applications of integralsArea: vertical area between curves: Applications of integralsArea: horizontal area between curves: Applications of integralsArea: curves that intersect at more than two points: Applications of integrals
Volume: squares and rectangles cross sections: Applications of integralsVolume: triangles and semicircles cross sections: Applications of integralsVolume: disc method (revolving around x- and y-axes): Applications of integralsVolume: disc method (revolving around other axes): Applications of integralsVolume: washer method (revolving around x- and y-axes): Applications of integralsVolume: washer method (revolving around other axes): Applications of integralsCalculator active practice: Applications of integrals

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