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AP Calculus BC
Study AP Calculus BC
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- 1Limits and Continuity
- 2Differentiation: Definition and Fundamental Properties
- 3Differentiation: Composite, Implicit, and Inverse Functions
- 4Contextual Applications of Differentiation
- 5Analytical Applications of Differentiation
- 6Integration and Accumulation of Change
- 7Differential Equations
- 8Applications of Integration
- 9Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10Infinite Sequences and Series
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- Revision Sheet
- Syllabus Checklist
- Scheme of Work
- Exam Technique
- Command Words — A3 poster
- Lesson Plans — one page per topic
- Class Mark Book
1. Limits and Continuity 39 files
Handouts
Presentation slides
Exercise walk-through slides
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
Vocabulary
Topic Quizzes
Exercise sheets
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
Past Paper Questions
Unit Tests
2. Differentiation: Definition and Fundamental Properties 32 files
Presentation slides
Exercise walk-through slides
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Exercise sheets
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Past Paper Questions
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
3. Differentiation: Composite, Implicit, and Inverse Functions 20 files
Presentation slides
Exercise walk-through slides
Exercise sheets
Past Paper Questions
4. Contextual Applications of Differentiation 25 files
Presentation slides
Exercise walk-through slides
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
Topic Quizzes
Exercise sheets
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
Past Paper Questions
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
5. Analytical Applications of Differentiation 39 files
Presentation slides
Exercise walk-through slides
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.8 Sketching Graphs of Functions and Their Derivatives
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.11 Solving Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
Topic Quizzes
Exercise sheets
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.8 Sketching Graphs of Functions and Their Derivatives
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.11 Solving Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
Past Paper Questions
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
6. Integration and Accumulation of Change 44 files
Presentation slides
Exercise walk-through slides
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 6.14 Selecting Techniques for Antidifferentiation
Vocabulary
Topic Quizzes
Exercise sheets
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 6.14 Selecting Techniques for Antidifferentiation
Past Paper Questions
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
Unit Tests
7. Differential Equations 29 files
Handouts
Presentation slides
Exercise walk-through slides
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations
Vocabulary
Topic Quizzes
Exercise sheets
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations
Past Paper Questions
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
Unit Tests
8. Applications of Integration 40 files
Handouts
Presentation slides
Exercise walk-through slides
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Vocabulary
Topic Quizzes
Exercise sheets
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Past Paper Questions
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Unit Tests
9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions 31 files
Presentation slides
Exercise walk-through slides
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
Exercise sheets
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
Past Paper Questions
- 9.1 Defining and Differentiating Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
10. Infinite Sequences and Series 47 files
Handouts
Presentation slides
Exercise walk-through slides
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Vocabulary
Topic Quizzes
Exercise sheets
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Past Paper Questions
- 10.2 Working with Geometric Series
- 10.4 Integral Test for Convergence
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Unit Tests
- AP Calculus BC — Question index
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 3.1 The Chain Rule
- 3.2 Implicit Differentiation
- 3.6 Calculating Higher-Order Derivatives
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
- 9.1 Defining and Differentiating Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
- 10.2 Working with Geometric Series
- 10.4 Integral Test for Convergence
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Topic quizzes
- 1. Limits and Continuity
- 2. Differentiation: Definition and Fundamental Properties
- 3. Differentiation: Composite, Implicit, and Inverse Functions
- 4. Contextual Applications of Differentiation
- 5. Analytical Applications of Differentiation
- 6. Integration and Accumulation of Change
- 7. Differential Equations
- 8. Applications of Integration
- 9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10. Infinite Sequences and Series
Unit tests
- 1. Limits and Continuity
- 2. Differentiation: Definition and Fundamental Properties
- 3. Differentiation: Composite, Implicit, and Inverse Functions
- 4. Contextual Applications of Differentiation
- 5. Analytical Applications of Differentiation
- 6. Integration and Accumulation of Change
- 7. Differential Equations
- 8. Applications of Integration
- 9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10. Infinite Sequences and Series
Mock papers
Course companion
Glossary
Topic handouts
- 1. Limits and Continuity
- 2. Differentiation: Definition and Fundamental Properties
- 3. Differentiation: Composite, Implicit, and Inverse Functions
- 4. Contextual Applications of Differentiation
- 5. Analytical Applications of Differentiation
- 6. Integration and Accumulation of Change
- 7. Differential Equations
- 8. Applications of Integration
- 9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10. Infinite Sequences and Series
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
- 3.1 The Chain Rule
- 3.2 Implicit Differentiation
- 3.3 Differentiating Inverse Functions
- 3.4 Differentiating Inverse Trigonometric Functions
- 3.5 Selecting Procedures for Calculating Derivatives
- 3.6 Calculating Higher-Order Derivatives
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.8 Sketching Graphs of Functions and Their Derivatives
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.11 Solving Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 6.14 Selecting Techniques for Antidifferentiation
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Lesson slides
- 1. Limits and Continuity
- 2. Differentiation: Definition and Fundamental Properties
- 3. Differentiation: Composite, Implicit, and Inverse Functions
- 4. Contextual Applications of Differentiation
- 5. Analytical Applications of Differentiation
- 6. Integration and Accumulation of Change
- 7. Differential Equations
- 8. Applications of Integration
- 9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10. Infinite Sequences and Series
Lesson slides
- 1.1 Introducing Calculus: Can Change Occur at an Instant?
- 1.2 Defining Limits and Using Limit Notation
- 1.3 Estimating Limit Values from Graphs
- 1.4 Estimating Limit Values from Tables
- 1.5 Determining Limits Using Algebraic Properties of Limits
- 1.6 Determining Limits Using Algebraic Manipulation
- 1.7 Selecting Procedures for Determining Limits
- 1.8 Determining Limits Using the Squeeze Theorem
- 1.9 Connecting Multiple Representations of Limits
- 1.10 Exploring Types of Discontinuities
- 1.11 Defining Continuity at a Point
- 1.12 Confirming Continuity over an Interval
- 1.13 Removing Discontinuities
- 1.14 Connecting Infinite Limits and Vertical Asymptotes
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.2 Defining the Derivative of a Function and Using Derivative Notation
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5 Applying the Power Rule
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
- 3.1 The Chain Rule
- 3.2 Implicit Differentiation
- 3.3 Differentiating Inverse Functions
- 3.4 Differentiating Inverse Trigonometric Functions
- 3.5 Selecting Procedures for Calculating Derivatives
- 3.6 Calculating Higher-Order Derivatives
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3 Rates of Change in Applied Contexts Other Than Motion
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.8 Sketching Graphs of Functions and Their Derivatives
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.11 Solving Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9 Integrating Using Substitution
- 6.10 Integrating Functions Using Long Division and Completing the Square
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 6.14 Selecting Techniques for Antidifferentiation
- 7.1 Modeling Situations with Differential Equations
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8 Exponential Models with Differential Equations
- 7.9 Logistic Models with Differential Equations
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.5 Finding the Area Between Curves Expressed as Functions of y
- 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.8 Volumes with Cross Sections: Triangles and Semicircles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10 Volume with Disc Method: Revolving Around Other Axes
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
- 9.1 Defining and Differentiating Parametric Equations
- 9.2 Second Derivatives of Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
- 10.1 Defining Convergent and Divergent Infinite Series
- 10.2 Working with Geometric Series
- 10.3 The nth Term Test for Divergence
- 10.4 Integral Test for Convergence
- 10.5 Harmonic Series and p-Series
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Vocabulary lists
- 1. Limits and Continuity
- 2. Differentiation: Definition and Fundamental Properties
- 3. Differentiation: Composite, Implicit, and Inverse Functions
- 4. Contextual Applications of Differentiation
- 5. Analytical Applications of Differentiation
- 6. Integration and Accumulation of Change
- 7. Differential Equations
- 8. Applications of Integration
- 9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 10. Infinite Sequences and Series
Flashcards
Weekly vocab tests
- Week 1 · Topic 1 · Limits and Continuity
- Week 2 · Topic 1 · Limits and Continuity
- Week 3 · Topics 1–2 · Limits and Continuity, Differentiation: Definition and Fundamental Properties
- Week 4 · Topics 2–4 · Differentiation: Definition and Fundamental Properties, Differentiation: Composite, Implicit, and Inverse Functions, Contextual Applications of Differentiation
- Week 5 · Topics 4–6 · Contextual Applications of Differentiation, Analytical Applications of Differentiation, Integration and Accumulation of Change
- Week 6 · Topics 6–8 · Integration and Accumulation of Change, Differential Equations, Applications of Integration
- Week 7 · Topics 8–10 · Applications of Integration, Parametric Equations, Polar Coordinates, and Vector-Valued Functions, Infinite Sequences and Series
Weekly homework
Exercise sheets
- Complete pack — exercise sheets + past papers
- 1 Limits and Continuity — Part 1
- 1 Limits and Continuity — Part 2
- 2 Differentiation: Definition and Fundamental Properties — Part 1
- 2 Differentiation: Definition and Fundamental Properties — Part 2
- 3 Differentiation: Composite, Implicit, and Inverse Functions — Part 1
- 3 Differentiation: Composite, Implicit, and Inverse Functions — Part 2
- 4 Contextual Applications of Differentiation — Part 1
- 4 Contextual Applications of Differentiation — Part 2
- 5 Analytical Applications of Differentiation — Part 1
- 5 Analytical Applications of Differentiation — Part 2
- 6 Integration and Accumulation of Change — Part 1
- 6 Integration and Accumulation of Change — Part 2
- 7 Differential Equations — Part 1
- 7 Differential Equations — Part 2
- 8 Applications of Integration — Part 1
- 8 Applications of Integration — Part 2
- 9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions — Part 1
- 9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions — Part 2
- 10 Infinite Sequences and Series — Part 1
- 10 Infinite Sequences and Series — Part 2
Past papers
- 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16 Working with the Intermediate Value Theorem (IVT)
- 2.1 Defining Average and Instantaneous Rates of Change at a Point
- 2.3 Estimating Derivatives of a Function at a Point
- 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7 Derivatives of cos x, sin x, e^x, and ln x
- 2.8 The Product Rule
- 2.9 The Quotient Rule
- 3.1 The Chain Rule
- 3.2 Implicit Differentiation
- 3.6 Calculating Higher-Order Derivatives
- 4.1 Interpreting the Meaning of the Derivative in Context
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.4 Introduction to Related Rates
- 4.5 Solving Related Rates Problems
- 4.6 Approximating Values of a Function Using Local Linearity and Linearization
- 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
- 5.1 Using the Mean Value Theorem
- 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
- 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10 Introduction to Optimization Problems
- 5.12 Exploring Behaviors of Implicit Relations
- 6.1 Exploring Accumulations of Change
- 6.2 Approximating Areas with Riemann Sums
- 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6 Applying Properties of Definite Integrals
- 6.7 The Fundamental Theorem of Calculus and Definite Integrals
- 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.11 Integrating Using Integration by Parts
- 6.12 Integrating Using Linear Partial Fractions
- 6.13 Evaluating Improper Integrals
- 7.2 Verifying Solutions for Differential Equations
- 7.3 Sketching Slope Fields
- 7.4 Reasoning Using Slope Fields
- 7.5 Approximating Solutions Using Euler's Method
- 7.6 Finding General Solutions Using Separation of Variables
- 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 8.1 Finding the Average Value of a Function on an Interval
- 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4 Finding the Area Between Curves Expressed as Functions of x
- 8.7 Volumes with Cross Sections: Squares and Rectangles
- 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12 Volume with Washer Method: Revolving Around Other Axes
- 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
- 9.1 Defining and Differentiating Parametric Equations
- 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4 Defining and Differentiating Vector-Valued Functions
- 9.5 Integrating Vector-Valued Functions
- 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7 Defining Polar Coordinates and Differentiating in Polar Form
- 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9 Finding the Area of the Region Bounded by Two Polar Curves
- 10.2 Working with Geometric Series
- 10.4 Integral Test for Convergence
- 10.6 Comparison Tests for Convergence
- 10.7 Alternating Series Test for Convergence
- 10.8 Ratio Test for Convergence
- 10.9 Determining Absolute or Conditional Convergence
- 10.10 Alternating Series Error Bound
- 10.11 Finding Taylor Polynomial Approximations of Functions
- 10.12 Lagrange Error Bound
- 10.13 Radius and Interval of Convergence of Power Series
- 10.14 Finding Taylor or Maclaurin Series for a Function
- 10.15 Representing Functions as Power Series
Tips
Sort the released questions by topic, and start with series. Working papers in order spreads the hardest material thinly across months; grouping it makes the choice-of-test pattern obvious.
Name the test and check its conditions in writing. The rubric awards the justification separately from the conclusion, so "converges by the ratio test since lim = 1/2 < 1" scores where a bare "converges" does not.
Keep AB material in rotation — it is a large share of the paper and of the AB subscore, and it is the part most often neglected once series begins.