AP Calculus BC
Course Content
Limits and continuity
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About the course
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Khan Academy in the classroom
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Sal interviews the AP Calculus Lead at College Board
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Defining limits and using limit notation
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Limits intro
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Estimating limit values from graphs
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Estimating limit values from graphs
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Unbounded limits
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One-sided limits from graphs
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One-sided limits from graphs: asymptote
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Connecting limits and graphical behavior
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Estimating limit values from tables
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Approximating limits using tables
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Estimating limits from tables
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One-sided limits from tables
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Determining limits using algebraic properties of limits: limit properties
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Limit properties
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Limits of combined functions
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Limits of combined functions: piecewise functions
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Limits of composite functions
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Determining limits using algebraic properties of limits: direct substitution
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Limits by direct substitution
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Undefined limits by direct substitution
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Limits of trigonometric functions
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Limits of piecewise functions
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Limits of piecewise functions: absolute value
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Determining limits using algebraic manipulation
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Limits by factoring
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Limits by rationalizing
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Trig limit using Pythagorean identity
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Trig limit using double angle identity
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Selecting procedures for determining limits
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Strategy in finding limits
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Determining limits using the squeeze theorem
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Squeeze theorem intro
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Limit of sin(x)/x as x approaches 0
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Limit of (1-cos(x))/x as x approaches 0
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Exploring types of discontinuities
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Types of discontinuities
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Defining continuity at a point
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Continuity at a point
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Worked example: Continuity at a point (graphical)
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Worked example: point where a function is continuous
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Worked example: point where a function isn’t continuous
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Confirming continuity over an interval
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continuity over an interval
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Functions continuous on all real numbers
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Functions continuous at specific x values
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Removing discontinuities
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Removing discontinuities (factoring)
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Removing discontinuities (rationalization) (Opens a modal)
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Connecting infinite limits and vertical asymptotes
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Introduction to infinite limits
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Infinite limits and asymptotes
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Analyzing unbounded limits: rational function
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Analyzing unbounded limits: mixed function
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Connecting limits at infinity and horizontal asymptotes
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Introduction to limits at infinity
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Functions with same limit at infinity
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Limits at infinity of quotients (Part 1)
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Limits at infinity of quotients (Part 2)
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Limits at infinity of quotients with square roots (odd power)
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Limits at infinity of quotients with square roots (even power)
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Working with the intermediate value theorem
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Intermediate value theorem
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Worked example: using the intermediate value theorem
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Justification with the intermediate calue theorem: table
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Justification with the intermediate value theorem: equation
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Optional videos
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Formal definition of limits Part 1: intuition review
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Formal definition of limits Part 2: building the idea
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Formal definition of limits Part 3: the definition
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Formal definition of limits Part 4: using the definition
Differentiation: definition and basic derivative rules
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Defining average and instantaneous rates of change at a point
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Newton, Leibniz, and Usain Bolt
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Derivative as a concept
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Secant lines & average rate of change
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Derivative as slope of curve
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The derivative & tangent line equations
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Defining the derivative of a function and using derivative notation
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Formal definition of the derivative as a limit
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Formal and alternate form of the derivative
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Worked example: Derivative as a limit
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Worked example: Derivative from limit expression
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The derivative of x² at x=3 using the formal definition
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The derivative of x² at any point using the formal definition
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Estimating derivatives of a function at a point
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Estimating derivatives
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Connecting differentiability and continuity: determining when derivatives do and do not exist
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Differentiability and continuity
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Differentiability at a point: graphical
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Differentiability at a point: algebraic (function is differentiable)
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Differentiability at a point: algebraic (function isn’t differentiable)
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Applying the power rule
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Power rule
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Power rule (with rewriting the expression)
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Derivative rules: constant, sum, difference, and constant multiple: introduction
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Basic derivative rules
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Basic derivative rules: find the error
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Basic derivative rules:table
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Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule Learn
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Differentiating polynomials
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Differentiating integer powers (mixed positive and negative)
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Tangents of polynomials
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Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)
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Derivatives of sin(x) and cos(x)
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Worked example: Derivatives of sin(x) and cos(x)
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Derivative of 𝑒ˣ
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Derivative of ln(x)
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The product rule
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Product rule
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Differentiating products
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Worked example: Product rule with table
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Worked example:Product rule with mixed implicit & explicit
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The quotient rule
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Quotient rule
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Worked example: Quotient rule with table
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Differentiating rational functions
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Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions
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Derivatives of tan(x) and cot(x)
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Derivatives of sec(x) and csc(x)
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Optional videos
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Proof: Differentiability implies continuity
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Justifying the power rule
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Proof of power rule for positive integer powers
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Proof of power rule for square root function
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Limit of sin(x)/x as x approaches 0
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Limit of (1-cos(x))/x as x approaches 0
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Proof of the derivative of sin(x
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Proof of the derivative of cos(x)
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Product rule proof
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Chain rule
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Common chain rule misunderstandings
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Identifying composite functions
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Worked example: Derivative of cos³(x) using the chain rule
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Worked example: Derivative of √(3x²-x) using the chain rule
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Worked example: Derivative of ln(√x) using the chain rule
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The chain rule: introduction
Differentiation: composite, implicit, and inverse functions
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Worked example: Chain rule with table
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Derivative of aˣ (for any positive base a)
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Derivative of logₐx (for any positive base a≠1)
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Worked example: Derivative of 7^(x²-x) using the chain rule
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Worked example: Derivative of log₄(x²+x) using the chain rule
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Worked example: Derivative of sec(3π/2-x) using the chain rule
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Worked example: Derivative of ∜(x³+4x²+7) using the chain rule
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The chain rule: further practice
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Implicit differentiation
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Worked example: Implicit differentiation
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Worked example: Evaluating derivative with implicit differentiation
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Showing explicit and implicit differentiation give same result
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Implicit differentiation
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Derivatives of inverse functions
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Derivatives of inverse functions: from equation
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Derivatives of inverse functions: from table
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Differentiating inverse functions
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Derivative of inverse sine
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Derivative of inverse cosine
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Derivative of inverse tangent
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Differentiating inverse trigonometric functions
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Differentiating functions: Find the error
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Manipulating functions before differentiation
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Selecting procedures for calculating derivatives: strategy
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Differentiating using multiple rules: strategy
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Applying the chain rule and product rule
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Applying the chain rule twice
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Derivative of eᶜᵒˢˣ⋅cos(eˣ)
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Derivative of sin(ln(x²))
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Selecting procedures for calculating derivatives: multiple rules
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Second derivatives
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Second derivatives (implicit equations): find expression
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Second derivatives (implicit equations): evaluate derivative
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Calculating higher-order derivatives
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Further practice connecting derivatives and limits
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Disguised derivatives
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Optional videos
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Proof: Differentiability implies continuity
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If function u is continuous at x, then Δu→0 as Δx→0(Opens a modal)
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Chain rule proof
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Quotient rule from product & chain rules
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Interpreting the meaning of the derivative in context
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Interpreting the meaning of the derivative in context
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Straight-line motion: connecting position, velocity, and acceleration
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Introduction to one-dimensional motion with calculus
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Interpreting direction of motion from position-time graph
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Interpreting direction of motion from velocity-time graph
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Interpreting change in speed from velocity-time graph
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Worked example: Motion problems with derivatives
Contextual applications of differentiation
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Rates of change in other applied contexts (non-motion problems)
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Applied rate of change: forgetfulness
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Introduction to related rates
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Related rates intro
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Analyzing related rates problems: expressions
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Analyzing related rates problems: equations (Pythagoras)
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Analyzing related rates problems: equations (trig)
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Differentiating related functions intro
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Worked example: Differentiating related functions
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Solving related rates problems
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Related rates: Approaching cars
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Related rates: Falling ladder
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Related rates: water pouring into a cone
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Related rates: shadow
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Related rates: balloon
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Approximating values of a function using local linearity and linearization
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Local linearity
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Local linearity and differentiability
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Worked example: Approximation with local linearity
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Linear approximation of a rational function
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Using L’Hôpital’s rule for finding limits of indeterminate forms
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L’Hôpital’s rule introduction
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L’Hôpital’s rule: limit at 0 example
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L’Hôpital’s rule: limit at infinity example
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Optional videos
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Proof of special case of l’Hôpital’s rule
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Using the mean value theorem
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Mean value theorem
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Mean value theorem example: polynomial
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Mean value theorem example: square root function
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Justification with the mean value theorem: table
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Justification with the mean value theorem: equation
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Mean value theorem application
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Extreme value theorem, global versus local extrema, and critical points
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Extreme value theorem
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Critical points introduction
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Finding critical points
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Determining intervals on which a function is increasing or decreasing
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Finding decreasing interval given the function
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Finding increasing interval given the derivative
Applying derivatives to analyze functions
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Using the first derivative test to find relative (local) extrema
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Introduction to minimum and maximum points
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Finding relative extrema (first derivative test)
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Worked example: finding relative extrema
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Analyzing mistakes when finding extrema (example 1)
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Analyzing mistakes when finding extrema (example 2)
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Using the candidates test to find absolute (global) extrema
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Finding absolute extrema on a closed interval
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Absolute minima & maxima (entire domain)
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Determining concavity of intervals and finding points of inflection: graphical
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Concavity introduction
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Analyzing concavity (graphical)
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Inflection points introduction
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Inflection points (graphical)
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Determining concavity of intervals and finding points of inflection: algebraic
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Analyzing concavity (algebraic)
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Inflection points (algebraic)
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Mistakes when finding inflection points: second derivative undefined
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Mistakes when finding inflection points: not checking candidates
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Using the second derivative test to find extrema
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Second derivative test
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Sketching curves of functions and their derivatives
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Curve sketching with calculus: polynomial
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Curve sketching with calculus: logarithm
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Analyzing a function with its derivative
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Connecting a function, its first derivative, and its second derivative
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Calculus based justification for function increasing
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Justification using first derivative
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Inflection points from graphs of function & derivatives
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Justification using second derivative: inflection point
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Justification using second derivative: maximum point
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Connecting f, f’, and f” graphically
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Connecting f, f’, and f” graphically (another example)
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Solving optimization problems
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Optimization: sum of squares
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Optimization: box volume (Part 1)
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Optimization: box volume (Part 2)
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Optimization: profit
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Optimization: cost of materials
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Optimization: area of triangle & square (Part 1)
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Optimization: area of triangle & square (Part 2)
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Motion problems: finding the maximum acceleration
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Exploring behaviors of implicit relations
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Horizontal tangent to implicit curve (Opens a modal)
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Exploring accumulations of change
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Introduction to integral calculus
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Definite integrals intro
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Worked example: accumulation of change
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Approximating areas with Riemann sums
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Riemann approximation introduction
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Over- and under-estimation of Riemann sums
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Worked example: finding a Riemann sum using a table
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Worked example: over- and under-estimation of Riemann sums
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Midpoint sums
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Trapezoidal sums
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Riemann sums, summation notation, and definite integral notation
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Summation notation
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Worked examples: Summation notation
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Riemann sums in summation notation
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Worked example: Riemann sums in summation notation
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Definite integral as the limit of a Riemann sum
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Worked example: Rewriting definite integral as limit of Riemann sum
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Worked example: Rewriting limit of Riemann sum as definite integral
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Functions defined by definite integrals (accumulation functions)
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Finding derivative with fundamental theorem of calculus
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Finding derivative with fundamental theorem of calculus: chain rule
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The fundamental theorem of calculus and accumulation functions
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The fundamental theorem of calculus and accumulation functions
Integration and accumulation of change
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Interpreting the behavior of accumulation functions involving area
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Interpreting the behavior of accumulation functions
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Applying properties of definite integrals
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Negative definite integrals
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Finding definite integrals using area formulas
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Definite integral over a single point
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Integrating scaled version of function
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Switching bounds of definite integral
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Integrating sums of functions
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Worked examples: Finding definite integrals using algebraic properties
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Definite integrals on adjacent intervals
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Worked example: Breaking up the integral’s interval
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Worked example: Merging definite integrals over adjacent intervals
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Functions defined by integrals: switched interval
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Finding derivative with fundamental theorem of calculus: x is on lower bound
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Finding derivative with fundamental theorem of calculus: x is on both bounds
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The fundamental theorem of calculus and definite integrals
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The fundamental theorem of calculus and definite integrals
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Antiderivatives and indefinite integrals
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Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule
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Reverse power rule
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Indefinite integrals : sum & multiples
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Rewriting before integrating
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Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals
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Indefinite integral of 1/x
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Indefinite integrals of sin(x), cos(x), and eˣ
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Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals
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Definite integrals: reverse power rule
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Definite integral of rational function
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Definite integral of radical function
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Definite integral of trig function
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Definite integral involving natural log
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Definite integral of piecewise function
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Definite integral of absolute value function
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Integrating using substitution
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𝘶-substitution intro
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𝘶-substitution: multiplying by a constant
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𝘶-substitution: defining 𝘶
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𝘶-substitution: defining 𝘶 (more examples)
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𝘶-substitution: rational function
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𝘶-substitution: logarithmic function
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𝘶-substitution: definite integrals
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𝘶-substitution: definite integral of exponential function
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Integrating functions using long division and completing the square
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Integration using long division
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Integration using completing the square and the derivative of arctan(x)
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Using integration by parts
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Integration by parts intro
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Integration by parts: ∫x⋅cos(x)dx
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Integration by parts: ∫ln(x)dx
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Integration by parts: ∫x²⋅𝑒ˣdx
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Integration by parts: ∫𝑒ˣ⋅cos(x)dx
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Integration by parts: definite integrals
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Integrating using linear partial fractions
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Integration with partial fractions
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Evaluating improper integrals
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Introduction to improper integrals
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Divergent improper integral
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Optional videos
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Proof of fundamental theorem of calculus
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Intuition for second part of fundamental theorem of calculus
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Modeling situations with differential equations
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Differential equations introduction
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Writing a differential equation
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Verifying solutions for differential equations
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Verifying solutions to differential equations
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Sketching slope fields
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Slope fields introduction
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Worked example: equation from slope field
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Worked example: slope field from equation
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Worked example: forming a slope field
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Reasoning using slope fields
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Approximating solution curves in slope fields
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Worked example: range of solution curve from slope field (Opens a modal)
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Approximating solutions using Euler’s method
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Euler’s method
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Worked example: Euler’s method
Differential equations
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Finding general solutions using separation of variables
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Separable equations introduction
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Addressing treating differentials algebraically
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Worked example: separable differential equations
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Worked example: identifying separable equations
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Finding particular solutions using initial conditions and separation of variables
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Particular solutions to differential equations: rational function
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Particular solutions to differential equations: exponential function
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Worked example: finding a specific solution to a separable equation
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Worked example: separable equation with an implicit solution
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Exponential models with differential equations
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Exponential models & differential equations (Part 1)
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Exponential models & differential equations (Part 2)
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Worked example: exponential solution to differential equation
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Logistic models with differential equations
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Growth models: introduction
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The logistic growth model
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Worked example: Logistic model word problem
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Logistic equations (Part 1)
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Logistic equations (Part 2)
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Finding the average value of a function on an interval
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Average value over a closed interval
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Calculating average value of function over interval
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Mean value theorem for integrals
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Connecting position, velocity, and acceleration functions using integrals
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Motion problems with integrals: displacement vs. distance
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Analyzing motion problems: position
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Analyzing motion problems: total distance traveled
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Worked example: motion problems (with definite integrals)
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Average acceleration over interval
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Using accumulation functions and definite integrals in applied contexts
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Area under rate function gives the net change
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Interpreting definite integral as net change
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Worked examples: interpreting definite integrals in context
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Analyzing problems involving definite integrals
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Worked example: problem involving definite integral (algebraic)
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Finding the area between curves expressed as functions of x
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Area between a curve and the x-axis
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Area between a curve and the x-axis: negative area
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Area between curves
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Worked example: area between curves
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Composite area between curves
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Finding the area between curves expressed as functions of y
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Area between a curve and the 𝘺-axis
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Horizontal area between curves
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Volumes with cross sections: squares and rectangles
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Volume with cross sections: intro
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Volume with cross sections: squares and rectangles (no graph)
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Volume with cross sections perpendicular to y-axis
Applications of integration
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Volume with cross sections: semicircle
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Volume with cross sections: triangle
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Volumes with cross sections: triangles and semicircles
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Disc method around x-axis
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Generalizing disc method around x-axis
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Disc method around y-axis
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Volume with disc method: revolving around x- or y-axis
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Disc method rotation around horizontal line
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Disc method rotating around vertical line
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Calculating integral disc around vertical line (Opens a modal)
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Volume with disc method: revolving around other axes
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Solid of revolution between two functions (leading up to the washer method)
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Generalizing the washer method
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Volume with washer method: revolving around x- or y-axis
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Arc length intro
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Worked example: arc length (Opens a modal)
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The arc length of a smooth, planar curve and distance traveled
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Volume with washer method: revolving around other axes
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Washer method rotating around horizontal line (not x-axis), part 1
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Washer method rotating around horizontal line (not x-axis), part 2
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Washer method rotating around vertical line (not y-axis), part 1
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Washer method rotating around vertical line (not y-axis), part 2
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Defining and differentiating parametric equations
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Parametric equations intro
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Parametric equations differentiation
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Second derivatives of parametric equations
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Second derivatives (parametric functions)
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Finding arc lengths of curves given by parametric equations
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Parametric curve arc length
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Worked example: Parametric arc length
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Defining and differentiating vector-valued functions
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Vector-valued functions intro
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Vector-valued functions differentiation
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Second derivatives (vector-valued functions)
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Solving motion problems using parametric and vector-valued functions
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Planar motion example: acceleration vector
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Motion along a curve: finding rate of change
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Motion along a curve: finding velocity magnitude
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Planar motion (with integrals)
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Defining polar coordinates and differentiating in polar form
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Polar functions derivatives
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Worked example: differentiating polar functions
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Finding the area of a polar region or the area bounded by a single polar curve
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Area bounded by polar curves
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Worked example: Area enclosed by cardioid
Parametric equations, polar coordinates, and vector-valued functions
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Finding the area of the region bounded by two polar curves
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Worked example: Area between two polar graphs
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Calculator-active practice
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Evaluating definite integral with calculator
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Partial sums: formula for nth term from partial sum
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Partial sums: term value from partial sum
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Infinite series as limit of partial sums
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Defining convergent and divergent infinite series
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Convergent and divergent sequences
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Worked example: sequence convergence/divergence
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Partial sums intro
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Worked example: convergent geometric series
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Worked example: divergent geometric series
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Infinite geometric series word problem: bouncing ball
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Infinite geometric series word problem: repeating decimal
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Working with geometric series
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nth term divergence test
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The nth-term test for divergence
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Integral test
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Worked example: Integral test
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Integral test for convergence
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Harmonic series and 𝑝-series
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Worked example: p-series
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Harmonic series and p-series
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Direct comparison test
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Worked example: direct comparison test
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Limit comparison test
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Worked example: limit comparison test
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Proof: harmonic series diverges
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Comparison tests for convergence
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Alternating series test
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Worked example: alternating series
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Alternating series test for convergence
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Ratio test
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Ratio test for convergence
Infinite sequences and series
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Conditional & absolute convergence (Opens a modal)
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Determining absolute or conditional convergence
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Alternating series remainder
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Worked example: alternating series remainder
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Alternating series error bound
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Taylor & Maclaurin polynomials intro (part 1)
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Taylor & Maclaurin polynomials intro (part 2)
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Worked example: Maclaurin polynomial
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Worked example: coefficient in Maclaurin polynomial
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Worked example: coefficient in Taylor polynomial
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Visualizing Taylor polynomial approximations
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Finding Taylor polynomial approximations of functions
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Taylor polynomial remainder (part 1)
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Taylor polynomial remainder (part 2)
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Worked example: estimating sin(0.4) using Lagrange error bound
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Worked example: estimating eˣ using Lagrange error bound
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Lagrange error bound
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Power series intro
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Worked example: interval of convergence
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Radius and interval of convergence of power series
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Maclaurin series of cos(x)
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Maclaurin series of sin(x)
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Maclaurin series of eˣ
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Worked example: power series from cos(x)
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Worked example: cosine function from power series
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Worked example: recognizing function from Taylor series
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Function as a geometric series
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Geometric series as a function
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Power series of arctan(2x)
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Power series of ln(1+x³)
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Visualizing Taylor series approximations
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Euler’s formula & Euler’s identity
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Geometric series interval of convergence
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Finding Taylor or Maclaurin series for a function
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Integrating power series
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Differentiating power series
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Finding function from power series by integrating
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Interval of convergence for derivative and integral
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Representing functions as power series
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Formal definition for limit of a sequence
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Proving a sequence converges using the formal definition
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Finite geometric series formula
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Infinite geometric series formula intuition
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Proof of infinite geometric series as a limit
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Proof of p-series convergence criteria
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Optional videos
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