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AP Calculus BC
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Course content
547 Lessons
Limits and continuity
About the course
About the course
Khan Academy in the classroom
Khan Academy in the classroom
Sal interviews the AP Calculus Lead at College Board
Sal interviews the AP Calculus Lead at College Board
Defining limits and using limit notation
Defining limits and using limit notation
Limits intro
Limits intro
Estimating limit values from graphs
Estimating limit values from graphs
Estimating limit values from graphs
Estimating limit values from graphs
Unbounded limits
Unbounded limits
One-sided limits from graphs
One-sided limits from graphs
One-sided limits from graphs: asymptote
One-sided limits from graphs: asymptote
Connecting limits and graphical behavior
Connecting limits and graphical behavior
Estimating limit values from tables
Estimating limit values from tables
Approximating limits using tables
Approximating limits using tables
Estimating limits from tables
Estimating limits from tables
One-sided limits from tables
One-sided limits from tables
Determining limits using algebraic properties of limits: limit properties
Determining limits using algebraic properties of limits: limit properties
Limit properties
Limit properties
Limits of combined functions
Limits of combined functions
Limits of combined functions: piecewise functions
Limits of combined functions: piecewise functions
Limits of composite functions
Limits of composite functions
Determining limits using algebraic properties of limits: direct substitution
Determining limits using algebraic properties of limits: direct substitution
Limits by direct substitution
Limits by direct substitution
Undefined limits by direct substitution
Undefined limits by direct substitution
Limits of trigonometric functions
Limits of trigonometric functions
Limits of piecewise functions
Limits of piecewise functions
Limits of piecewise functions: absolute value
Limits of piecewise functions: absolute value
Determining limits using algebraic manipulation
Determining limits using algebraic manipulation
Limits by factoring
Limits by factoring
Limits by rationalizing
Limits by rationalizing
Trig limit using Pythagorean identity
Trig limit using Pythagorean identity
Trig limit using double angle identity
Trig limit using double angle identity
Selecting procedures for determining limits
Selecting procedures for determining limits
Strategy in finding limits
Strategy in finding limits
Determining limits using the squeeze theorem
Determining limits using the squeeze theorem
Squeeze theorem intro
Squeeze theorem intro
Limit of sin(x)/x as x approaches 0
Limit of sin(x)/x as x approaches 0
Limit of (1-cos(x))/x as x approaches 0
Limit of (1-cos(x))/x as x approaches 0
Exploring types of discontinuities
Exploring types of discontinuities
Types of discontinuities
Types of discontinuities
Defining continuity at a point
Defining continuity at a point
Continuity at a point
Continuity at a point
Worked example: Continuity at a point (graphical)
Worked example: Continuity at a point (graphical)
Worked example: point where a function is continuous
Worked example: point where a function is continuous
Worked example: point where a function isn’t continuous
Worked example: point where a function isn’t continuous
Confirming continuity over an interval
Confirming continuity over an interval
continuity over an interval
continuity over an interval
Functions continuous on all real numbers
Functions continuous on all real numbers
Functions continuous at specific x values
Functions continuous at specific x values
Removing discontinuities
Removing discontinuities
Removing discontinuities (factoring)
Removing discontinuities (factoring)
Removing discontinuities (rationalization) (Opens a modal)
Removing discontinuities (rationalization) (Opens a modal)
Connecting infinite limits and vertical asymptotes
Connecting infinite limits and vertical asymptotes
Introduction to infinite limits
Introduction to infinite limits
Infinite limits and asymptotes
Infinite limits and asymptotes
Analyzing unbounded limits: rational function
Analyzing unbounded limits: rational function
Analyzing unbounded limits: mixed function
Analyzing unbounded limits: mixed function
Connecting limits at infinity and horizontal asymptotes
Connecting limits at infinity and horizontal asymptotes
Introduction to limits at infinity
Introduction to limits at infinity
Functions with same limit at infinity
Functions with same limit at infinity
Limits at infinity of quotients (Part 1)
Limits at infinity of quotients (Part 1)
Limits at infinity of quotients (Part 2)
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 (odd power)
Limits at infinity of quotients with square roots (even power)
Limits at infinity of quotients with square roots (even power)
Working with the intermediate value theorem
Working with the intermediate value theorem
Intermediate value theorem
Intermediate value theorem
Worked example: using the intermediate value theorem
Worked example: using the intermediate value theorem
Justification with the intermediate calue theorem: table
Justification with the intermediate calue theorem: table
Justification with the intermediate value theorem: equation
Justification with the intermediate value theorem: equation
Optional videos
Optional videos
Formal definition of limits Part 1: intuition review
Formal definition of limits Part 1: intuition review
Formal definition of limits Part 2: building the idea
Formal definition of limits Part 2: building the idea
Formal definition of limits Part 3: the definition
Formal definition of limits Part 3: the definition
Formal definition of limits Part 4: using the definition
Formal definition of limits Part 4: using the definition
Differentiation: definition and basic derivative rules
Differentiation: composite, implicit, and inverse functions
Contextual applications of differentiation
Applying derivatives to analyze functions
Integration and accumulation of change
Differential equations
Applications of integration
Parametric equations, polar coordinates, and vector-valued functions
Infinite sequences and series
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