AP Calculus BC
Course Content
Limits and continuity

About the course

Khan Academy in the classroom

Sal interviews the AP Calculus Lead at College Board

Defining limits and using limit notation

Limits intro

Estimating limit values from graphs

Estimating limit values from graphs

Unbounded limits

Onesided limits from graphs

Onesided limits from graphs: asymptote

Connecting limits and graphical behavior

Estimating limit values from tables

Approximating limits using tables

Estimating limits from tables

Onesided limits from tables

Determining limits using algebraic properties of limits: limit properties

Limit properties

Limits of combined functions

Limits of combined functions: piecewise functions

Limits of composite functions

Determining limits using algebraic properties of limits: direct substitution

Limits by direct substitution

Undefined limits by direct substitution

Limits of trigonometric functions

Limits of piecewise functions

Limits of piecewise functions: absolute value

Determining limits using algebraic manipulation

Limits by factoring

Limits by rationalizing

Trig limit using Pythagorean identity

Trig limit using double angle identity

Selecting procedures for determining limits

Strategy in finding limits

Determining limits using the squeeze theorem

Squeeze theorem intro

Limit of sin(x)/x as x approaches 0

Limit of (1cos(x))/x as x approaches 0

Exploring types of discontinuities

Types of discontinuities

Defining continuity at a point

Continuity at a point

Worked example: Continuity at a point (graphical)

Worked example: point where a function is continuous

Worked example: point where a function isn’t continuous

Confirming continuity over an interval

continuity over an interval

Functions continuous on all real numbers

Functions continuous at specific x values

Removing discontinuities

Removing discontinuities (factoring)

Removing discontinuities (rationalization) (Opens a modal)

Connecting infinite limits and vertical asymptotes

Introduction to infinite limits

Infinite limits and asymptotes

Analyzing unbounded limits: rational function

Analyzing unbounded limits: mixed function

Connecting limits at infinity and horizontal asymptotes

Introduction to limits at infinity

Functions with same limit at infinity

Limits at infinity of quotients (Part 1)

Limits at infinity of quotients (Part 2)

Limits at infinity of quotients with square roots (odd power)

Limits at infinity of quotients with square roots (even power)

Working with the intermediate value theorem

Intermediate value theorem

Worked example: using the intermediate value theorem

Justification with the intermediate calue theorem: table

Justification with the intermediate value theorem: equation

Optional videos

Formal definition of limits Part 1: intuition review

Formal definition of limits Part 2: building the idea

Formal definition of limits Part 3: the definition

Formal definition of limits Part 4: using the definition
Differentiation: definition and basic derivative rules

Defining average and instantaneous rates of change at a point

Newton, Leibniz, and Usain Bolt

Derivative as a concept

Secant lines & average rate of change

Derivative as slope of curve

The derivative & tangent line equations

Defining the derivative of a function and using derivative notation

Formal definition of the derivative as a limit

Formal and alternate form of the derivative

Worked example: Derivative as a limit

Worked example: Derivative from limit expression

The derivative of x² at x=3 using the formal definition

The derivative of x² at any point using the formal definition

Estimating derivatives of a function at a point

Estimating derivatives

Connecting differentiability and continuity: determining when derivatives do and do not exist

Differentiability and continuity

Differentiability at a point: graphical

Differentiability at a point: algebraic (function is differentiable)

Differentiability at a point: algebraic (function isn’t differentiable)

Applying the power rule

Power rule

Power rule (with rewriting the expression)

Derivative rules: constant, sum, difference, and constant multiple: introduction

Basic derivative rules

Basic derivative rules: find the error

Basic derivative rules:table

Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule Learn

Differentiating polynomials

Differentiating integer powers (mixed positive and negative)

Tangents of polynomials

Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)

Derivatives of sin(x) and cos(x)

Worked example: Derivatives of sin(x) and cos(x)

Derivative of 𝑒ˣ

Derivative of ln(x)

The product rule

Product rule

Differentiating products

Worked example: Product rule with table

Worked example:Product rule with mixed implicit & explicit

The quotient rule

Quotient rule

Worked example: Quotient rule with table

Differentiating rational functions

Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions

Derivatives of tan(x) and cot(x)

Derivatives of sec(x) and csc(x)

Optional videos

Proof: Differentiability implies continuity

Justifying the power rule

Proof of power rule for positive integer powers

Proof of power rule for square root function

Limit of sin(x)/x as x approaches 0

Limit of (1cos(x))/x as x approaches 0

Proof of the derivative of sin(x

Proof of the derivative of cos(x)

Product rule proof

Chain rule

Common chain rule misunderstandings

Identifying composite functions

Worked example: Derivative of cos³(x) using the chain rule

Worked example: Derivative of √(3x²x) using the chain rule

Worked example: Derivative of ln(√x) using the chain rule

The chain rule: introduction
Differentiation: composite, implicit, and inverse functions

Worked example: Chain rule with table

Derivative of aˣ (for any positive base a)

Derivative of logₐx (for any positive base a≠1)

Worked example: Derivative of 7^(x²x) using the chain rule

Worked example: Derivative of log₄(x²+x) using the chain rule

Worked example: Derivative of sec(3π/2x) using the chain rule

Worked example: Derivative of ∜(x³+4x²+7) using the chain rule

The chain rule: further practice

Implicit differentiation

Worked example: Implicit differentiation

Worked example: Evaluating derivative with implicit differentiation

Showing explicit and implicit differentiation give same result

Implicit differentiation

Derivatives of inverse functions

Derivatives of inverse functions: from equation

Derivatives of inverse functions: from table

Differentiating inverse functions

Derivative of inverse sine

Derivative of inverse cosine

Derivative of inverse tangent

Differentiating inverse trigonometric functions

Differentiating functions: Find the error

Manipulating functions before differentiation

Selecting procedures for calculating derivatives: strategy

Differentiating using multiple rules: strategy

Applying the chain rule and product rule

Applying the chain rule twice

Derivative of eᶜᵒˢˣ⋅cos(eˣ)

Derivative of sin(ln(x²))

Selecting procedures for calculating derivatives: multiple rules

Second derivatives

Second derivatives (implicit equations): find expression

Second derivatives (implicit equations): evaluate derivative

Calculating higherorder derivatives

Further practice connecting derivatives and limits

Disguised derivatives

Optional videos

Proof: Differentiability implies continuity

If function u is continuous at x, then Δu→0 as Δx→0(Opens a modal)

Chain rule proof

Quotient rule from product & chain rules

Interpreting the meaning of the derivative in context

Interpreting the meaning of the derivative in context

Straightline motion: connecting position, velocity, and acceleration

Introduction to onedimensional motion with calculus

Interpreting direction of motion from positiontime graph

Interpreting direction of motion from velocitytime graph

Interpreting change in speed from velocitytime graph

Worked example: Motion problems with derivatives
Contextual applications of differentiation

Rates of change in other applied contexts (nonmotion problems)

Applied rate of change: forgetfulness

Introduction to related rates

Related rates intro

Analyzing related rates problems: expressions

Analyzing related rates problems: equations (Pythagoras)

Analyzing related rates problems: equations (trig)

Differentiating related functions intro

Worked example: Differentiating related functions

Solving related rates problems

Related rates: Approaching cars

Related rates: Falling ladder

Related rates: water pouring into a cone

Related rates: shadow

Related rates: balloon

Approximating values of a function using local linearity and linearization

Local linearity

Local linearity and differentiability

Worked example: Approximation with local linearity

Linear approximation of a rational function

Using L’Hôpital’s rule for finding limits of indeterminate forms

L’Hôpital’s rule introduction

L’Hôpital’s rule: limit at 0 example

L’Hôpital’s rule: limit at infinity example

Optional videos

Proof of special case of l’Hôpital’s rule

Using the mean value theorem

Mean value theorem

Mean value theorem example: polynomial

Mean value theorem example: square root function

Justification with the mean value theorem: table

Justification with the mean value theorem: equation

Mean value theorem application

Extreme value theorem, global versus local extrema, and critical points

Extreme value theorem

Critical points introduction

Finding critical points

Determining intervals on which a function is increasing or decreasing

Finding decreasing interval given the function

Finding increasing interval given the derivative
Applying derivatives to analyze functions

Using the first derivative test to find relative (local) extrema

Introduction to minimum and maximum points

Finding relative extrema (first derivative test)

Worked example: finding relative extrema

Analyzing mistakes when finding extrema (example 1)

Analyzing mistakes when finding extrema (example 2)

Using the candidates test to find absolute (global) extrema

Finding absolute extrema on a closed interval

Absolute minima & maxima (entire domain)

Determining concavity of intervals and finding points of inflection: graphical

Concavity introduction

Analyzing concavity (graphical)

Inflection points introduction

Inflection points (graphical)

Determining concavity of intervals and finding points of inflection: algebraic

Analyzing concavity (algebraic)

Inflection points (algebraic)

Mistakes when finding inflection points: second derivative undefined

Mistakes when finding inflection points: not checking candidates

Using the second derivative test to find extrema

Second derivative test

Sketching curves of functions and their derivatives

Curve sketching with calculus: polynomial

Curve sketching with calculus: logarithm

Analyzing a function with its derivative

Connecting a function, its first derivative, and its second derivative

Calculus based justification for function increasing

Justification using first derivative

Inflection points from graphs of function & derivatives

Justification using second derivative: inflection point

Justification using second derivative: maximum point

Connecting f, f’, and f” graphically

Connecting f, f’, and f” graphically (another example)

Solving optimization problems

Optimization: sum of squares

Optimization: box volume (Part 1)

Optimization: box volume (Part 2)

Optimization: profit

Optimization: cost of materials

Optimization: area of triangle & square (Part 1)

Optimization: area of triangle & square (Part 2)

Motion problems: finding the maximum acceleration

Exploring behaviors of implicit relations

Horizontal tangent to implicit curve (Opens a modal)

Exploring accumulations of change

Introduction to integral calculus

Definite integrals intro

Worked example: accumulation of change

Approximating areas with Riemann sums

Riemann approximation introduction

Over and underestimation of Riemann sums

Worked example: finding a Riemann sum using a table

Worked example: over and underestimation of Riemann sums

Midpoint sums

Trapezoidal sums

Riemann sums, summation notation, and definite integral notation

Summation notation

Worked examples: Summation notation

Riemann sums in summation notation

Worked example: Riemann sums in summation notation

Definite integral as the limit of a Riemann sum

Worked example: Rewriting definite integral as limit of Riemann sum

Worked example: Rewriting limit of Riemann sum as definite integral

Functions defined by definite integrals (accumulation functions)

Finding derivative with fundamental theorem of calculus

Finding derivative with fundamental theorem of calculus: chain rule

The fundamental theorem of calculus and accumulation functions

The fundamental theorem of calculus and accumulation functions
Integration and accumulation of change

Interpreting the behavior of accumulation functions involving area

Interpreting the behavior of accumulation functions

Applying properties of definite integrals

Negative definite integrals

Finding definite integrals using area formulas

Definite integral over a single point

Integrating scaled version of function

Switching bounds of definite integral

Integrating sums of functions

Worked examples: Finding definite integrals using algebraic properties

Definite integrals on adjacent intervals

Worked example: Breaking up the integral’s interval

Worked example: Merging definite integrals over adjacent intervals

Functions defined by integrals: switched interval

Finding derivative with fundamental theorem of calculus: x is on lower bound

Finding derivative with fundamental theorem of calculus: x is on both bounds

The fundamental theorem of calculus and definite integrals

The fundamental theorem of calculus and definite integrals

Antiderivatives and indefinite integrals

Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule

Reverse power rule

Indefinite integrals : sum & multiples

Rewriting before integrating

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals

Indefinite integral of 1/x

Indefinite integrals of sin(x), cos(x), and eˣ

Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals

Definite integrals: reverse power rule

Definite integral of rational function

Definite integral of radical function

Definite integral of trig function

Definite integral involving natural log

Definite integral of piecewise function

Definite integral of absolute value function

Integrating using substitution

𝘶substitution intro

𝘶substitution: multiplying by a constant

𝘶substitution: defining 𝘶

𝘶substitution: defining 𝘶 (more examples)

𝘶substitution: rational function

𝘶substitution: logarithmic function

𝘶substitution: definite integrals

𝘶substitution: definite integral of exponential function

Integrating functions using long division and completing the square

Integration using long division

Integration using completing the square and the derivative of arctan(x)

Using integration by parts

Integration by parts intro

Integration by parts: ∫x⋅cos(x)dx

Integration by parts: ∫ln(x)dx

Integration by parts: ∫x²⋅𝑒ˣdx

Integration by parts: ∫𝑒ˣ⋅cos(x)dx

Integration by parts: definite integrals

Integrating using linear partial fractions

Integration with partial fractions

Evaluating improper integrals

Introduction to improper integrals

Divergent improper integral

Optional videos

Proof of fundamental theorem of calculus

Intuition for second part of fundamental theorem of calculus

Modeling situations with differential equations

Differential equations introduction

Writing a differential equation

Verifying solutions for differential equations

Verifying solutions to differential equations

Sketching slope fields

Slope fields introduction

Worked example: equation from slope field

Worked example: slope field from equation

Worked example: forming a slope field

Reasoning using slope fields

Approximating solution curves in slope fields

Worked example: range of solution curve from slope field (Opens a modal)

Approximating solutions using Euler’s method

Euler’s method

Worked example: Euler’s method
Differential equations

Finding general solutions using separation of variables

Separable equations introduction

Addressing treating differentials algebraically

Worked example: separable differential equations

Worked example: identifying separable equations

Finding particular solutions using initial conditions and separation of variables

Particular solutions to differential equations: rational function

Particular solutions to differential equations: exponential function

Worked example: finding a specific solution to a separable equation

Worked example: separable equation with an implicit solution

Exponential models with differential equations

Exponential models & differential equations (Part 1)

Exponential models & differential equations (Part 2)

Worked example: exponential solution to differential equation

Logistic models with differential equations

Growth models: introduction

The logistic growth model

Worked example: Logistic model word problem

Logistic equations (Part 1)

Logistic equations (Part 2)

Finding the average value of a function on an interval

Average value over a closed interval

Calculating average value of function over interval

Mean value theorem for integrals

Connecting position, velocity, and acceleration functions using integrals

Motion problems with integrals: displacement vs. distance

Analyzing motion problems: position

Analyzing motion problems: total distance traveled

Worked example: motion problems (with definite integrals)

Average acceleration over interval

Using accumulation functions and definite integrals in applied contexts

Area under rate function gives the net change

Interpreting definite integral as net change

Worked examples: interpreting definite integrals in context

Analyzing problems involving definite integrals

Worked example: problem involving definite integral (algebraic)

Finding the area between curves expressed as functions of x

Area between a curve and the xaxis

Area between a curve and the xaxis: negative area

Area between curves

Worked example: area between curves

Composite area between curves

Finding the area between curves expressed as functions of y

Area between a curve and the 𝘺axis

Horizontal area between curves

Volumes with cross sections: squares and rectangles

Volume with cross sections: intro

Volume with cross sections: squares and rectangles (no graph)

Volume with cross sections perpendicular to yaxis
Applications of integration

Volume with cross sections: semicircle

Volume with cross sections: triangle

Volumes with cross sections: triangles and semicircles

Disc method around xaxis

Generalizing disc method around xaxis

Disc method around yaxis

Volume with disc method: revolving around x or yaxis

Disc method rotation around horizontal line

Disc method rotating around vertical line

Calculating integral disc around vertical line (Opens a modal)

Volume with disc method: revolving around other axes

Solid of revolution between two functions (leading up to the washer method)

Generalizing the washer method

Volume with washer method: revolving around x or yaxis

Arc length intro

Worked example: arc length (Opens a modal)

The arc length of a smooth, planar curve and distance traveled

Volume with washer method: revolving around other axes

Washer method rotating around horizontal line (not xaxis), part 1

Washer method rotating around horizontal line (not xaxis), part 2

Washer method rotating around vertical line (not yaxis), part 1

Washer method rotating around vertical line (not yaxis), part 2

Defining and differentiating parametric equations

Parametric equations intro

Parametric equations differentiation

Second derivatives of parametric equations

Second derivatives (parametric functions)

Finding arc lengths of curves given by parametric equations

Parametric curve arc length

Worked example: Parametric arc length

Defining and differentiating vectorvalued functions

Vectorvalued functions intro

Vectorvalued functions differentiation

Second derivatives (vectorvalued functions)

Solving motion problems using parametric and vectorvalued functions

Planar motion example: acceleration vector

Motion along a curve: finding rate of change

Motion along a curve: finding velocity magnitude

Planar motion (with integrals)

Defining polar coordinates and differentiating in polar form

Polar functions derivatives

Worked example: differentiating polar functions

Finding the area of a polar region or the area bounded by a single polar curve

Area bounded by polar curves

Worked example: Area enclosed by cardioid
Parametric equations, polar coordinates, and vectorvalued functions

Finding the area of the region bounded by two polar curves

Worked example: Area between two polar graphs

Calculatoractive practice

Evaluating definite integral with calculator

Partial sums: formula for nth term from partial sum

Partial sums: term value from partial sum

Infinite series as limit of partial sums

Defining convergent and divergent infinite series

Convergent and divergent sequences

Worked example: sequence convergence/divergence

Partial sums intro

Worked example: convergent geometric series

Worked example: divergent geometric series

Infinite geometric series word problem: bouncing ball

Infinite geometric series word problem: repeating decimal

Working with geometric series

nth term divergence test

The nthterm test for divergence

Integral test

Worked example: Integral test

Integral test for convergence

Harmonic series and 𝑝series

Worked example: pseries

Harmonic series and pseries

Direct comparison test

Worked example: direct comparison test

Limit comparison test

Worked example: limit comparison test

Proof: harmonic series diverges

Comparison tests for convergence

Alternating series test

Worked example: alternating series

Alternating series test for convergence

Ratio test

Ratio test for convergence
Infinite sequences and series

Conditional & absolute convergence (Opens a modal)

Determining absolute or conditional convergence

Alternating series remainder

Worked example: alternating series remainder

Alternating series error bound

Taylor & Maclaurin polynomials intro (part 1)

Taylor & Maclaurin polynomials intro (part 2)

Worked example: Maclaurin polynomial

Worked example: coefficient in Maclaurin polynomial

Worked example: coefficient in Taylor polynomial

Visualizing Taylor polynomial approximations

Finding Taylor polynomial approximations of functions

Taylor polynomial remainder (part 1)

Taylor polynomial remainder (part 2)

Worked example: estimating sin(0.4) using Lagrange error bound

Worked example: estimating eˣ using Lagrange error bound

Lagrange error bound

Power series intro

Worked example: interval of convergence

Radius and interval of convergence of power series

Maclaurin series of cos(x)

Maclaurin series of sin(x)

Maclaurin series of eˣ

Worked example: power series from cos(x)

Worked example: cosine function from power series

Worked example: recognizing function from Taylor series

Function as a geometric series

Geometric series as a function

Power series of arctan(2x)

Power series of ln(1+x³)

Visualizing Taylor series approximations

Euler’s formula & Euler’s identity

Geometric series interval of convergence

Finding Taylor or Maclaurin series for a function

Integrating power series

Differentiating power series

Finding function from power series by integrating

Interval of convergence for derivative and integral

Representing functions as power series

Formal definition for limit of a sequence

Proving a sequence converges using the formal definition

Finite geometric series formula

Infinite geometric series formula intuition

Proof of infinite geometric series as a limit

Proof of pseries convergence criteria

Optional videos
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