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AP®︎/College Calculus BC
Course: AP®︎/College Calculus BC > Unit 12
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- Units:
- Lessons:
- Defining average and instantaneous rates of change at a point
- Defining the derivative of a function and using derivative notation
- Estimating derivatives of a function at a point
- Interpreting the meaning of the derivative in context
- Straight-line motion: connecting position, velocity, and acceleration
- Rates of change in other applied contexts (non-motion problems)
- Introduction to related rates
- Solving related rates problems
- Approximating values of a function using local linearity and linearization
- Exploring accumulations of change
- Finding the average value of a function on an interval
- Connecting position, velocity, and acceleration functions using integrals
- Using accumulation functions and definite integrals in applied contexts
- Finding the area between curves expressed as functions of x
- Finding the area between curves expressed as functions of y
- Finding the area between curves that intersect at more than two points
- Volumes with cross sections: squares and rectangles
- Volumes with cross sections: triangles and semicircles
- Volume with disc method: revolving around x- or y-axis
- Volume with disc method: revolving around other axes
- Volume with washer method: revolving around x- or y-axis
- Volume with washer method: revolving around other axes
- The arc length of a smooth, planar curve and distance traveled
- Calculator-active practice
- Defining and differentiating parametric equations
- Second derivatives of parametric equations
- Finding arc lengths of curves given by parametric equations
- Defining and differentiating vector-valued functions
- Finding the area of a polar region or the area bounded by a single polar curve
- Finding the area of the region bounded by two polar curves
- Calculator-active practice
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- Units:
- Lessons:
- Videos:
- 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
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- Newton, Leibniz, and Usain Bolt
- Derivative as a concept
- Derivative as slope of curve
- The derivative & tangent line equations
- 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
- Articles:
- Exercises:
- Lessons:
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- Exercises:
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- Derivative as slope of curve
- The derivative & tangent line equations
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- Derivative as slope of curve
- The derivative & tangent line equations
- 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
- Articles:
- Exercises:
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- Lessons:
- Interpreting the meaning of the derivative in context
- Straight-line motion: connecting position, velocity, and acceleration
- Rates of change in other applied contexts (non-motion problems)
- Introduction to related rates
- Solving related rates problems
- Approximating values of a function using local linearity and linearization
- Defining and differentiating parametric equations
- Second derivatives of parametric equations
- Defining and differentiating vector-valued functions
- Videos:
- Interpreting the meaning of the derivative in context
- 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
- Applied rate of change: forgetfulness
- 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
- Related rates: Approaching cars
- Related rates: Falling ladder
- Related rates: water pouring into a cone
- Related rates: shadow
- Related rates: balloon
- Local linearity
- Local linearity and differentiability
- Worked example: Approximation with local linearity
- Linear approximation of a rational function
- Parametric equations intro
- Parametric equations differentiation
- Second derivatives (parametric functions)
- Vector-valued functions intro
- Vector-valued functions differentiation
- Second derivatives (vector-valued functions)
- Articles:
- Exercises:
- Interpreting the meaning of the derivative in context
- Interpret motion graphs
- Motion problems (differential calc)
- Rates of change in other applied contexts (non-motion problems)
- Analyzing related rates problems: expressions
- Analyzing related rates problems: equations
- Differentiate related functions
- Related rates intro
- Related rates (multiple rates)
- Related rates (Pythagorean theorem)
- Related rates (advanced)
- Approximation with local linearity
- Parametric equations differentiation
- Second derivatives (parametric functions)
- Vector-valued functions differentiation
- Second derivatives (vector-valued functions)
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- Introduction to integral calculus
- Definite integrals intro
- Worked example: accumulation of change
- Average value over a closed interval
- Calculating average value of function over interval
- Mean value theorem for 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
- 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)
- Articles:
- Exercises:
- Accumulation of change
- Average value of a function
- Analyzing motion problems (integral calculus)
- Motion problems (with integrals)
- Interpreting definite integrals in context
- Analyzing problems involving definite integrals
- Problems involving definite integrals (algebraic)
- Contextual and analytical applications of integration (calculator-active)
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- Finding the area between curves expressed as functions of x
- Finding the area between curves expressed as functions of y
- Finding the area between curves that intersect at more than two points
- Volumes with cross sections: squares and rectangles
- Volumes with cross sections: triangles and semicircles
- Volume with disc method: revolving around x- or y-axis
- Volume with disc method: revolving around other axes
- Volume with washer method: revolving around x- or y-axis
- Volume with washer method: revolving around other axes
- Finding the area of a polar region or the area bounded by a single polar curve
- Finding the area of the region bounded by two polar curves
- Calculator-active practice
- Videos:
- 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
- Area between a curve and the 𝘺-axis
- Horizontal area between curves
- Volume with cross sections: intro
- Volume with cross sections: squares and rectangles (no graph)
- Volume with cross sections perpendicular to y-axis
- Volume with cross sections: semicircle
- Volume with cross sections: triangle
- Disc method around x-axis
- Generalizing disc method around x-axis
- Disc method around y-axis
- Disc method rotation around horizontal line
- Disc method rotating around vertical line
- Calculating integral disc around vertical line
- Solid of revolution between two functions (leading up to the washer method)
- Generalizing the washer method
- 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
- Area bounded by polar curves
- Worked example: Area enclosed by cardioid
- Worked example: Area between two polar graphs
- Exercises:
- Area between a curve and the x-axis
- Area between two curves given end points
- Area between two curves
- Horizontal areas between curves
- Area between curves that intersect at more than two points (calculator-active)
- Volumes with cross sections: squares and rectangles (intro)
- Volumes with cross sections: squares and rectangles
- Volumes with cross sections: triangles and semicircles
- Disc method: revolving around x- or y-axis
- Disc method: revolving around other axes
- Washer method: revolving around x- or y-axis
- Washer method: revolving around other axes
- Area bounded by polar curves intro
- Area bounded by polar curves
- Area between two polar curves
- Lessons:
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- Exercises:
- Lessons:
- Videos:
- Disc method around x-axis
- Generalizing disc method around x-axis
- Disc method around y-axis
- Disc method rotation around horizontal line
- Disc method rotating around vertical line
- Calculating integral disc around vertical line
- Solid of revolution between two functions (leading up to the washer method)
- Generalizing the washer method
- 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
- Exercises:
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- Lessons:
- Working with the intermediate value theorem
- Connecting differentiability and continuity: determining when derivatives do and do not exist
- Applying the power rule
- Derivative rules: constant, sum, difference, and constant multiple: introduction
- Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule
- Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)
- The product rule
- The quotient rule
- Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions
- The chain rule: introduction
- The chain rule: further practice
- Implicit differentiation
- Differentiating inverse functions
- Differentiating inverse trigonometric functions
- Selecting procedures for calculating derivatives: strategy
- Selecting procedures for calculating derivatives: multiple rules
- Calculating higher-order derivatives
- Using the mean value theorem
- Extreme value theorem, global versus local extrema, and critical points
- Determining intervals on which a function is increasing or decreasing
- Using the first derivative test to find relative (local) extrema
- Using the candidates test to find absolute (global) extrema
- Determining concavity of intervals and finding points of inflection: graphical
- Determining concavity of intervals and finding points of inflection: algebraic
- Using the second derivative test to find extrema
- Sketching curves of functions and their derivatives
- Connecting a function, its first derivative, and its second derivative
- Solving optimization problems
- Exploring behaviors of implicit relations
- Calculator-active practice
- The fundamental theorem of calculus and accumulation functions
- Interpreting the behavior of accumulation functions involving area
- Applying properties of definite integrals
- The fundamental theorem of calculus and definite integrals
- Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule
- Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals
- Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals
- Integrating using substitution
- Integrating functions using long division and completing the square
- Using integration by parts
- Integrating using linear partial fractions
- Modeling situations with differential equations
- Verifying solutions for differential equations
- Sketching slope fields
- Reasoning using slope fields
- Approximating solutions using Euler’s method
- Finding general solutions using separation of variables
- Finding particular solutions using initial conditions and separation of variables
- Exponential models with differential equations
- Logistic models with differential equations
- Solving motion problems using parametric and vector-valued functions
- Defining polar coordinates and differentiating in polar form
- Units:
- Lessons:
- Videos:
- Intermediate value theorem
- Worked example: using the intermediate value theorem
- Justification with the intermediate value theorem: table
- Justification with the intermediate value theorem: equation
- 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
- Critical points introduction
- Finding critical points
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- Units:
- Lessons:
- Applying the power rule
- Derivative rules: constant, sum, difference, and constant multiple: introduction
- Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule
- Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)
- The product rule
- The quotient rule
- Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions
- The chain rule: introduction
- The chain rule: further practice
- Implicit differentiation
- Differentiating inverse functions
- Differentiating inverse trigonometric functions
- Selecting procedures for calculating derivatives: strategy
- Selecting procedures for calculating derivatives: multiple rules
- Calculating higher-order derivatives
- Defining polar coordinates and differentiating in polar form
- Videos:
- Power rule
- Power rule (with rewriting the expression)
- Basic derivative rules
- Basic derivative rules: find the error
- Basic derivative rules: table
- Differentiating polynomials
- Differentiating integer powers (mixed positive and negative)
- Tangents of polynomials
- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivative of 𝑒ˣ
- Derivative of ln(x)
- Product rule
- Differentiating products
- Worked example: Product rule with table
- Worked example: Product rule with mixed implicit & explicit
- Quotient rule
- Worked example: Quotient rule with table
- Differentiating rational functions
- Derivatives of tan(x) and cot(x)
- Derivatives of sec(x) and csc(x)
- 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
- 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
- Implicit differentiation
- Worked example: Implicit differentiation
- Worked example: Evaluating derivative with implicit differentiation
- Showing explicit and implicit differentiation give same result
- Derivatives of inverse functions
- Derivatives of inverse functions: from equation
- Derivatives of inverse functions: from table
- Derivative of inverse sine
- Derivative of inverse cosine
- Derivative of inverse tangent
- Differentiating functions: Find the error
- Manipulating functions before differentiation
- 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))
- Second derivatives
- Second derivatives (implicit equations): find expression
- Second derivatives (implicit equations): evaluate derivative
- Polar functions derivatives
- Worked example: differentiating polar functions
- Articles:
- Justifying the power rule
- Justifying the basic derivative rules
- Proving the derivatives of sin(x) and cos(x)
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Proof: the derivative of ln(x) is 1/x
- Proving the product rule
- Product rule review
- Quotient rule review
- Chain rule
- Proving the chain rule
- Derivative rules review
- Implicit differentiation review
- Differentiating inverse trig functions review
- Strategy in differentiating functions
- Second derivatives review
- Exercises:
- Power rule (positive integer powers)
- Power rule (negative & fractional powers)
- Power rule (with rewriting the expression)
- Basic derivative rules: find the error
- Basic derivative rules: table
- Differentiate polynomials
- Differentiate integer powers (mixed positive and negative)
- Tangents of polynomials
- Derivatives of sin(x) and cos(x)
- Derivatives of 𝑒ˣ and ln(x)
- Differentiate products
- Product rule with tables
- Differentiate quotients
- Quotient rule with tables
- Differentiate rational functions
- Derivatives of tan(x), cot(x), sec(x), and csc(x)
- Identify composite functions
- Chain rule intro
- Chain rule with tables
- Derivatives of aˣ and logₐx
- Chain rule capstone
- Implicit differentiation
- Derivatives of inverse functions
- Derivatives of inverse trigonometric functions
- Differentiating functions: Find the error
- Manipulating functions before differentiation
- Differentiating using multiple rules: strategy
- Differentiating using multiple rules
- Second derivatives
- Second derivatives (implicit equations)
- Differentiate polar functions
- Tangents to polar curves
- Lessons:
- Videos:
- Power rule
- Power rule (with rewriting the expression)
- Basic derivative rules
- Basic derivative rules: find the error
- Basic derivative rules: table
- Differentiating polynomials
- Differentiating integer powers (mixed positive and negative)
- Tangents of polynomials
- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivative of 𝑒ˣ
- Derivative of ln(x)
- Articles:
- Exercises:
- Power rule (positive integer powers)
- Power rule (negative & fractional powers)
- Power rule (with rewriting the expression)
- Basic derivative rules: find the error
- Basic derivative rules: table
- Differentiate polynomials
- Differentiate integer powers (mixed positive and negative)
- Tangents of polynomials
- Derivatives of sin(x) and cos(x)
- Derivatives of 𝑒ˣ and ln(x)
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- 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
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- 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
- 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
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- Determining intervals on which a function is increasing or decreasing
- Using the first derivative test to find relative (local) extrema
- Using the candidates test to find absolute (global) extrema
- Determining concavity of intervals and finding points of inflection: graphical
- Determining concavity of intervals and finding points of inflection: algebraic
- Using the second derivative test to find extrema
- Sketching curves of functions and their derivatives
- Connecting a function, its first derivative, and its second derivative
- Solving optimization problems
- Exploring behaviors of implicit relations
- Calculator-active practice
- Videos:
- Finding decreasing interval given the function
- Finding increasing interval given the derivative
- 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)
- Finding absolute extrema on a closed interval
- Absolute minima & maxima (entire domain)
- Concavity introduction
- Analyzing concavity (graphical)
- Inflection points introduction
- Inflection points (graphical)
- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
- Second derivative test
- Curve sketching with calculus: polynomial
- Curve sketching with calculus: logarithm
- Analyzing a function with its 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)
- 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
- Horizontal tangent to implicit curve
- Articles:
- Increasing & decreasing intervals review
- Finding relative extrema (first derivative test)
- Relative minima & maxima review
- Absolute minima & maxima review
- Analyzing the second derivative to find inflection points
- Concavity review
- Inflection points review
- Justification using first derivative
- Justification using second derivative
- Exercises:
- Increasing & decreasing intervals
- Relative minima & maxima
- Absolute minima & maxima (closed intervals)
- Absolute minima & maxima (entire domain)
- Concavity intro
- Inflection points intro
- Analyze concavity
- Find inflection points
- Second derivative test
- Justification using first derivative
- Justification using second derivative
- Connecting f, f', and f'' graphically
- Optimization
- Tangents to graphs of implicit relations
- Analyze functions (calculator-active)
- Lessons:
- Determining intervals on which a function is increasing or decreasing
- Using the first derivative test to find relative (local) extrema
- Using the candidates test to find absolute (global) extrema
- Determining concavity of intervals and finding points of inflection: graphical
- Determining concavity of intervals and finding points of inflection: algebraic
- Using the second derivative test to find extrema
- Sketching curves of functions and their derivatives
- Connecting a function, its first derivative, and its second derivative
- Calculator-active practice
- Videos:
- Finding decreasing interval given the function
- Finding increasing interval given the derivative
- 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)
- Finding absolute extrema on a closed interval
- Absolute minima & maxima (entire domain)
- Concavity introduction
- Analyzing concavity (graphical)
- Inflection points introduction
- Inflection points (graphical)
- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
- Second derivative test
- Curve sketching with calculus: polynomial
- Curve sketching with calculus: logarithm
- Analyzing a function with its 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)
- Articles:
- Increasing & decreasing intervals review
- Finding relative extrema (first derivative test)
- Relative minima & maxima review
- Absolute minima & maxima review
- Analyzing the second derivative to find inflection points
- Concavity review
- Inflection points review
- Justification using first derivative
- Justification using second derivative
- Exercises:
- Increasing & decreasing intervals
- Relative minima & maxima
- Absolute minima & maxima (closed intervals)
- Absolute minima & maxima (entire domain)
- Concavity intro
- Inflection points intro
- Analyze concavity
- Find inflection points
- Second derivative test
- Justification using first derivative
- Justification using second derivative
- Connecting f, f', and f'' graphically
- Analyze functions (calculator-active)
- Lessons:
- Videos:
- Curve sketching with calculus: polynomial
- Curve sketching with calculus: logarithm
- Analyzing a function with its 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)
- Articles:
- Exercises:
- Lessons:
- Videos:
- 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)
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- Lessons:
- Videos:
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- Exercises:
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- Videos:
- Concavity introduction
- Analyzing concavity (graphical)
- Inflection points introduction
- Inflection points (graphical)
- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
- Articles:
- Exercises:
- Lessons:
- Videos:
- Concavity introduction
- Analyzing concavity (graphical)
- Inflection points introduction
- Inflection points (graphical)
- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
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- Applying properties of definite integrals
- The fundamental theorem of calculus and definite integrals
- Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule
- Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals
- Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals
- Integrating using substitution
- Integrating functions using long division and completing the square
- Using integration by parts
- Integrating using linear partial fractions
- Videos:
- 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
- Antiderivatives and indefinite integrals
- Reverse power rule
- Indefinite integrals: sums & multiples
- Rewriting before integrating
- Indefinite integral of 1/x
- Indefinite integrals of sin(x), cos(x), and eˣ
- 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
- 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
- Integration using long division
- Integration using completing the square and the derivative of arctan(x)
- 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
- Integration with partial fractions
- Articles:
- Exercises:
- Finding definite integrals using area formulas
- Finding definite integrals using algebraic properties
- Definite integrals over adjacent intervals
- The fundamental theorem of calculus and definite integrals
- Antiderivatives and indefinite integrals
- Reverse power rule
- Reverse power rule: negative and fractional powers
- Reverse power rule: sums & multiples
- Reverse power rule: rewriting before integrating
- Indefinite integrals: eˣ & 1/x
- Indefinite integrals: sin & cos
- Definite integrals: reverse power rule
- Definite integrals: common functions
- Definite integrals of piecewise functions
- 𝘶-substitution: defining 𝘶
- 𝘶-substitution: indefinite integrals
- 𝘶-substitution: definite integrals
- Integration using long division
- Integration using completing the square
- Integration by parts
- Integration by parts: definite integrals
- Integration with partial fractions
- Lessons:
- Videos:
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- 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
- Articles:
- Exercises:
- Lessons:
- Videos:
- The fundamental theorem of calculus and definite integrals
- Antiderivatives and indefinite 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
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- Reverse power rule
- Indefinite integrals: sums & multiples
- Rewriting before integrating
- Indefinite integral of 1/x
- Indefinite integrals of sin(x), cos(x), and eˣ
- 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
- Articles:
- Exercises:
- Reverse power rule
- Reverse power rule: negative and fractional powers
- Reverse power rule: sums & multiples
- Reverse power rule: rewriting before integrating
- Indefinite integrals: eˣ & 1/x
- Indefinite integrals: sin & cos
- Definite integrals: reverse power rule
- Definite integrals: common functions
- Definite integrals of piecewise functions
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- 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
- Integration using long division
- Integration using completing the square and the derivative of arctan(x)
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- Modeling situations with differential equations
- Verifying solutions for differential equations
- Sketching slope fields
- Reasoning using slope fields
- Approximating solutions using Euler’s method
- Finding general solutions using separation of variables
- Finding particular solutions using initial conditions and separation of variables
- Exponential models with differential equations
- Logistic models with differential equations
- Videos:
- Differential equations introduction
- Writing a differential equation
- Verifying solutions to differential equations
- Slope fields introduction
- Worked example: equation from slope field
- Worked example: slope field from equation
- Worked example: forming a slope field
- Approximating solution curves in slope fields
- Worked example: range of solution curve from slope field
- Euler's method
- Worked example: Euler's method
- Separable equations introduction
- Addressing treating differentials algebraically
- Worked example: separable differential equations
- Worked example: identifying separable equations
- 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 & differential equations (Part 1)
- Exponential models & differential equations (Part 2)
- Worked example: exponential solution to differential equation
- Growth models: introduction
- The logistic growth model
- Worked example: Logistic model word problem
- Logistic equations (Part 1)
- Logistic equations (Part 2)
- Articles:
- Exercises:
- Write differential equations
- Verify solutions to differential equations
- Slope fields & equations
- Reasoning using slope fields
- Euler's method
- Separable differential equations: find the error
- Separable differential equations
- Identify separable equations
- Particular solutions to differential equations
- Particular solutions to separable differential equations
- Differential equations: exponential model equations
- Differential equations: exponential model word problems
- Differential equations: logistic model word problems
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- Defining limits and using limit notation
- Estimating limit values from graphs
- Estimating limit values from tables
- Determining limits using algebraic properties of limits: limit properties
- Determining limits using algebraic properties of limits: direct substitution
- Determining limits using algebraic manipulation
- Selecting procedures for determining limits
- Determining limits using the squeeze theorem
- Exploring types of discontinuities
- Defining continuity at a point
- Confirming continuity over an interval
- Removing discontinuities
- Connecting infinite limits and vertical asymptotes
- Connecting limits at infinity and horizontal asymptotes
- Further practice connecting derivatives and limits
- Using L’Hôpital’s rule for finding limits of indeterminate forms
- Approximating areas with Riemann sums
- Riemann sums, summation notation, and definite integral notation
- Evaluating improper integrals
- Defining convergent and divergent infinite series
- Working with geometric series
- The nth-term test for divergence
- Integral test for convergence
- Harmonic series and p-series
- Comparison tests for convergence
- Alternating series test for convergence
- Ratio test for convergence
- Determining absolute or conditional convergence
- Alternating series error bound
- Finding Taylor polynomial approximations of functions
- Lagrange error bound
- Radius and interval of convergence of power series
- Finding Taylor or Maclaurin series for a function
- Representing functions as power series
- Units:
- Lessons:
- Defining limits and using limit notation
- Estimating limit values from graphs
- Estimating limit values from tables
- Determining limits using algebraic properties of limits: limit properties
- Determining limits using algebraic properties of limits: direct substitution
- Determining limits using algebraic manipulation
- Selecting procedures for determining limits
- Determining limits using the squeeze theorem
- Videos:
- Limits intro
- Estimating limit values from graphs
- Unbounded limits
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- Connecting limits and graphical behavior
- Approximating limits using tables
- Estimating limits from tables
- One-sided limits from tables
- Limit properties
- Limits of combined functions
- Limits of combined functions: piecewise functions
- Limits of composite functions
- Limits by direct substitution
- Undefined limits by direct substitution
- Limits of trigonometric functions
- Limits of piecewise functions
- Limits of piecewise functions: absolute value
- Limits by factoring
- Limits by rationalizing
- Trig limit using Pythagorean identity
- Trig limit using double angle identity
- Strategy in finding limits
- Squeeze theorem intro
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(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
- Articles:
- Exercises:
- Limits intro
- Estimating limit values from graphs
- One-sided limits from graphs
- Connecting limits and graphical behavior
- Creating tables for approximating limits
- Estimating limits from tables
- One-sided limits from tables
- Limits of combined functions: sums and differences
- Limits of combined functions: products and quotients
- Limits of composite functions
- Limits by direct substitution
- Direct substitution with limits that don't exist
- Limits of trigonometric functions
- Limits of piecewise functions
- Limits by factoring
- Limits using conjugates
- Limits using trig identities
- Conclusions from direct substitution (finding limits)
- Next steps after indeterminate form (finding limits)
- Strategy in finding limits
- Squeeze theorem
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- Limits by factoring
- Limits by rationalizing
- Trig limit using Pythagorean identity
- Trig limit using double angle identity
- Strategy in finding limits
- Squeeze theorem intro
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(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
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- Types of discontinuities
- 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
- Continuity over an interval
- Functions continuous on all real numbers
- Functions continuous at specific x-values
- Removing discontinuities (factoring)
- Removing discontinuities (rationalization)
- Introduction to infinite limits
- Infinite limits and asymptotes
- Analyzing unbounded limits: rational function
- Analyzing unbounded limits: mixed function
- 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)
- Exercises:
- Classify discontinuities
- Continuity at a point (graphical)
- Continuity at a point (algebraic)
- Continuity over an interval
- Continuity and common functions
- Removable discontinuities
- Infinite limits: graphical
- Infinite limits: algebraic
- Limits at infinity: graphical
- Limits at infinity of quotients
- Limits at infinity of quotients with square roots
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- Introduction to infinite limits
- Infinite limits and asymptotes
- Analyzing unbounded limits: rational function
- Analyzing unbounded limits: mixed function
- 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)
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- 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 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
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- Defining convergent and divergent infinite series
- Working with geometric series
- The nth-term test for divergence
- Integral test for convergence
- Harmonic series and p-series
- Comparison tests for convergence
- Alternating series test for convergence
- Ratio test for convergence
- Determining absolute or conditional convergence
- Alternating series error bound
- Videos:
- Convergent and divergent sequences
- Worked example: sequence convergence/divergence
- Partial sums intro
- Partial sums: formula for nth term from partial sum
- Partial sums: term value from partial sum
- Infinite series as limit of partial sums
- Worked example: convergent geometric series
- Worked example: divergent geometric series
- Infinite geometric series word problem: bouncing ball
- Infinite geometric series word problem: repeating decimal
- nth term divergence test
- Integral test
- Worked example: Integral test
- Worked example: p-series
- Direct comparison test
- Worked example: direct comparison test
- Limit comparison test
- Worked example: limit comparison test
- Proof: harmonic series diverges
- Alternating series test
- Worked example: alternating series
- Ratio test
- Conditional & absolute convergence
- Alternating series remainder
- Worked example: alternating series remainder
- Articles:
- Exercises:
- Lessons:
- Defining convergent and divergent infinite series
- Working with geometric series
- The nth-term test for divergence
- Integral test for convergence
- Harmonic series and p-series
- Comparison tests for convergence
- Alternating series test for convergence
- Ratio test for convergence
- Determining absolute or conditional convergence
- Videos:
- Convergent and divergent sequences
- Worked example: sequence convergence/divergence
- Partial sums intro
- Partial sums: formula for nth term from partial sum
- Partial sums: term value from partial sum
- Infinite series as limit of partial sums
- Worked example: convergent geometric series
- Worked example: divergent geometric series
- Infinite geometric series word problem: bouncing ball
- Infinite geometric series word problem: repeating decimal
- nth term divergence test
- Integral test
- Worked example: Integral test
- Worked example: p-series
- Direct comparison test
- Worked example: direct comparison test
- Limit comparison test
- Worked example: limit comparison test
- Proof: harmonic series diverges
- Alternating series test
- Worked example: alternating series
- Ratio test
- Conditional & absolute convergence
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- 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
- 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
- Power series intro
- Worked example: interval of convergence
- Function as a geometric series
- Geometric series as a function
- Power series of arctan(2x)
- Power series of ln(1+x³)
- 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
- Integrating power series
- Differentiating power series
- Finding function from power series by integrating
- Interval of convergence for derivative and integral
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- Function as a geometric series
- Geometric series as a function
- Power series of arctan(2x)
- Power series of ln(1+x³)
- 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
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- Videos:
- Function as a geometric series
- Geometric series as a function
- Power series of arctan(2x)
- Power series of ln(1+x³)
- 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
- Exercises:
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- Videos:
- Function as a geometric series
- Geometric series as a function
- Power series of arctan(2x)
- Power series of ln(1+x³)
- 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
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