AP Calculus BC | 1 Big School (By Blue Brain)

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
One-sided limits from graphs
One-sided limits from graphs: asymptote
Connecting limits and graphical behavior
Estimating limit values from tables
Approximating limits using tables
Estimating limits from tables
One-sided 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 (1-cos(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 (1-cos(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 √(3x²-x) 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^(x²-x) using the chain rule
Worked example: Derivative of log₄(x²+x) using the chain rule
Worked example: Derivative of sec(3π/2-x) 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 higher-order 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
Straight-line motion: connecting position, velocity, and acceleration
Introduction to one-dimensional motion with calculus
Interpreting direction of motion from position-time graph
Interpreting direction of motion from velocity-time graph
Interpreting change in speed from velocity-time graph
Worked example: Motion problems with derivatives

Contextual applications of differentiation

Rates of change in other applied contexts (non-motion 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 under-estimation of Riemann sums
Worked example: finding a Riemann sum using a table
Worked example: over- and under-estimation 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
𝘶-substitution intro
𝘶-substitution: multiplying by a constant
𝘶-substitution: defining 𝘶
𝘶-substitution: defining 𝘶 (more examples)
𝘶-substitution: rational function
𝘶-substitution: logarithmic function
𝘶-substitution: definite integrals
𝘶-substitution: 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 x-axis
Area between a curve and the x-axis: 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 𝘺-axis
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 y-axis

Applications of integration

Volume with cross sections: semicircle
Volume with cross sections: triangle
Volumes with cross sections: triangles and semicircles
Disc method around x-axis
Generalizing disc method around x-axis
Disc method around y-axis
Volume with disc method: revolving around x- or y-axis
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 y-axis
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 x-axis), part 1
Washer method rotating around horizontal line (not x-axis), part 2
Washer method rotating around vertical line (not y-axis), part 1
Washer method rotating around vertical line (not y-axis), 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 vector-valued functions
Vector-valued functions intro
Vector-valued functions differentiation
Second derivatives (vector-valued functions)
Solving motion problems using parametric and vector-valued 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 vector-valued functions

Finding the area of the region bounded by two polar curves
Worked example: Area between two polar graphs
Calculator-active 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 nth-term test for divergence
Integral test
Worked example: Integral test
Integral test for convergence
Harmonic series and 𝑝-series
Worked example: p-series
Harmonic series and p-series
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 p-series convergence criteria
Optional videos

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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

One-sided limits from graphs

One-sided limits from graphs: asymptote

Connecting limits and graphical behavior

Estimating limit values from tables

Approximating limits using tables

Estimating limits from tables

One-sided 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 (1-cos(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