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# AP®︎ Calculus BC

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## Meet an AP®︎ teacher who uses AP®︎ Calculus in his classroom

Bill Scott uses Khan Academy to teach AP®︎ Calculus at Phillips Academy in Andover, Massachusetts, and he’s part of the teaching team that helped develop Khan Academy’s AP®︎ lessons. Phillips Academy was one of the first schools to teach AP®︎ nearly 60 years ago.

Learn more## Meet an AP®︎ teacher who uses AP®︎ Calculus in his classroom

Bill Scott uses Khan Academy to teach AP®︎ Calculus at Phillips Academy in Andover, Massachusetts, and he’s part of the teaching team that helped develop Khan Academy’s AP®︎ lessons. Phillips Academy was one of the first schools to teach AP®︎ nearly 60 years ago.

Learn moreDefining limits and using limit notation: Limits and continuityEstimating limit values from graphs: Limits and continuityEstimating limit values from tables: Limits and continuityDetermining limits using algebraic properties of limits: limit properties: Limits and continuityDetermining limits using algebraic properties of limits: direct substitution: Limits and continuityDetermining limits using algebraic manipulation: Limits and continuitySelecting procedures for determining limits: Limits and continuityDetermining limits using the squeeze theorem: Limits and continuity

Exploring types of discontinuities: Limits and continuityDefining continuity at a point: Limits and continuityConfirming continuity over an interval: Limits and continuityRemoving discontinuities: Limits and continuityConnecting infinite limits and vertical asymptotes: Limits and continuityConnecting limits at infinity and horizontal asymptotes: Limits and continuityWorking with the intermediate value theorem: Limits and continuityOptional videos: Limits and continuity

Defining average and instantaneous rates of change at a point: Differentiation: definition and basic derivative rulesDefining the derivative of a function and using derivative notation: Differentiation: definition and basic derivative rulesEstimating derivatives of a function at a point: Differentiation: definition and basic derivative rulesConnecting differentiability and continuity: determining when derivatives do and do not exist: Differentiation: definition and basic derivative rulesApplying the power rule: Differentiation: definition and basic derivative rulesDerivative rules: constant, sum, difference, and constant multiple: introduction: Differentiation: definition and basic derivative rules

Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule: Differentiation: definition and basic derivative rulesDerivatives of cos(x), sin(x), 𝑒ˣ, and ln(x): Differentiation: definition and basic derivative rulesThe product rule: Differentiation: definition and basic derivative rulesThe quotient rule: Differentiation: definition and basic derivative rulesFinding the derivatives of tangent, cotangent, secant, and/or cosecant functions: Differentiation: definition and basic derivative rulesOptional videos: Differentiation: definition and basic derivative rules

The chain rule: introduction: Differentiation: composite, implicit, and inverse functionsThe chain rule: further practice: Differentiation: composite, implicit, and inverse functionsImplicit differentiation: Differentiation: composite, implicit, and inverse functionsDifferentiating inverse functions: Differentiation: composite, implicit, and inverse functionsDifferentiating inverse trigonometric functions: Differentiation: composite, implicit, and inverse functions

Selecting procedures for calculating derivatives: strategy: Differentiation: composite, implicit, and inverse functionsSelecting procedures for calculating derivatives: multiple rules: Differentiation: composite, implicit, and inverse functionsCalculating higher-order derivatives: Differentiation: composite, implicit, and inverse functionsFurther practice connecting derivatives and limits: Differentiation: composite, implicit, and inverse functionsOptional videos: Differentiation: composite, implicit, and inverse functions

Interpreting the meaning of the derivative in context: Contextual applications of differentiationStraight-line motion: connecting position, velocity, and acceleration: Contextual applications of differentiationRates of change in other applied contexts (non-motion problems): Contextual applications of differentiationIntroduction to related rates: Contextual applications of differentiation

Solving related rates problems: Contextual applications of differentiationApproximating values of a function using local linearity and linearization: Contextual applications of differentiationUsing L’Hôpital’s rule for finding limits of indeterminate forms: Contextual applications of differentiationOptional videos: Contextual applications of differentiation

Using the mean value theorem: Applying derivatives to analyze functions Extreme value theorem, global versus local extrema, and critical points: Applying derivatives to analyze functions Determining intervals on which a function is increasing or decreasing: Applying derivatives to analyze functions Using the first derivative test to find relative (local) extrema: Applying derivatives to analyze functions Using the candidates test to find absolute (global) extrema: Applying derivatives to analyze functions Determining concavity of intervals and finding points of inflection: graphical: Applying derivatives to analyze functions

Determining concavity of intervals and finding points of inflection: algebraic: Applying derivatives to analyze functions Using the second derivative test to find extrema: Applying derivatives to analyze functions Sketching curves of functions and their derivatives: Applying derivatives to analyze functions Connecting a function, its first derivative, and its second derivative: Applying derivatives to analyze functions Solving optimization problems: Applying derivatives to analyze functions Exploring behaviors of implicit relations: Applying derivatives to analyze functions Calculator-active practice: Applying derivatives to analyze functions

Exploring accumulations of change: Integration and accumulation of changeApproximating areas with Riemann sums: Integration and accumulation of changeRiemann sums, summation notation, and definite integral notation: Integration and accumulation of changeThe fundamental theorem of calculus and accumulation functions: Integration and accumulation of changeInterpreting the behavior of accumulation functions involving area: Integration and accumulation of changeApplying properties of definite integrals: Integration and accumulation of changeThe fundamental theorem of calculus and definite integrals: Integration and accumulation of changeFinding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule: Integration and accumulation of change

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals: Integration and accumulation of changeFinding antiderivatives and indefinite integrals: basic rules and notation: definite integrals: Integration and accumulation of changeIntegrating using substitution: Integration and accumulation of changeIntegrating functions using long division and completing the square: Integration and accumulation of changeUsing integration by parts: Integration and accumulation of changeIntegrating using linear partial fractions: Integration and accumulation of changeEvaluating improper integrals: Integration and accumulation of changeOptional videos: Integration and accumulation of change

Modeling situations with differential equations: Differential equationsVerifying solutions for differential equations: Differential equationsSketching slope fields: Differential equationsReasoning using slope fields: Differential equations

Approximating solutions using Euler’s method: Differential equationsFinding general solutions using separation of variables: Differential equationsFinding particular solutions using initial conditions and separation of variables: Differential equationsExponential models with differential equations: Differential equationsLogistic models with differential equations: Differential equations

Finding the average value of a function on an interval: Applications of integrationConnecting position, velocity, and acceleration functions using integrals: Applications of integrationUsing accumulation functions and definite integrals in applied contexts: Applications of integrationFinding the area between curves expressed as functions of x: Applications of integrationFinding the area between curves expressed as functions of y: Applications of integrationFinding the area between curves that intersect at more than two points: Applications of integrationVolumes with cross sections: squares and rectangles: Applications of integration

Volumes with cross sections: triangles and semicircles: Applications of integrationVolume with disc method: revolving around x- or y-axis: Applications of integrationVolume with disc method: revolving around other axes: Applications of integrationVolume with washer method: revolving around x- or y-axis: Applications of integrationVolume with washer method: revolving around other axes: Applications of integrationThe arc length of a smooth, planar curve and distance traveled: Applications of integrationCalculator-active practice: Applications of integration

Defining and differentiating parametric equations: Parametric equations, polar coordinates, and vector-valued functionsSecond derivatives of parametric equations: Parametric equations, polar coordinates, and vector-valued functionsFinding arc lengths of curves given by parametric equations: Parametric equations, polar coordinates, and vector-valued functionsDefining and differentiating vector-valued functions: Parametric equations, polar coordinates, and vector-valued functions

Solving motion problems using parametric and vector-valued functions: Parametric equations, polar coordinates, and vector-valued functionsDefining polar coordinates and differentiating in polar form: Parametric equations, polar coordinates, and vector-valued functionsFinding the area of a polar region or the area bounded by a single polar curve: Parametric equations, polar coordinates, and vector-valued functionsFinding the area of the region bounded by two polar curves: Parametric equations, polar coordinates, and vector-valued functionsCalculator-active practice: Parametric equations, polar coordinates, and vector-valued functions

Defining convergent and divergent infinite series: Infinite sequences and seriesWorking with geometric series: Infinite sequences and seriesThe nth-term test for divergence: Infinite sequences and seriesIntegral test for convergence: Infinite sequences and seriesHarmonic series and p-series: Infinite sequences and seriesComparison tests for convergence: Infinite sequences and seriesAlternating series test for convergence: Infinite sequences and seriesRatio test for convergence: Infinite sequences and series

Determining absolute or conditional convergence: Infinite sequences and seriesAlternating series error bound: Infinite sequences and seriesFinding Taylor polynomial approximations of functions: Infinite sequences and seriesLagrange error bound: Infinite sequences and seriesRadius and interval of convergence of power series: Infinite sequences and seriesFinding Taylor or Maclaurin series for a function: Infinite sequences and seriesRepresenting functions as power series: Infinite sequences and seriesOptional videos: Infinite sequences and series