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AP Calculus BC

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AP Calculus BC contains all of AB and adds parametric, polar and vector-valued functions, advanced integration techniques, improper integrals, logistic growth, and — the largest addition — infinite sequences and series, including Taylor and Maclaurin series with error bounds.

Series is where BC is won or lost. The convergence tests must be chosen, not tried in turn: given a series, you should know within seconds which test the form points to, and be able to name the test and verify its conditions.

BC also reports an AB subscore, so the AB material stays fully examinable and must not be left behind.

The notes cover the AB and BC material in CED order, giving series the space it needs. Released past papers are in the library. Since the AB subscore comes from the same paper, the AB units stay in the notes at full depth rather than being summarised away.

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Past Papers
  • 1

    Limits and Continuity

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    • 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)
    • 1.16 Exam tips
  • 2

    Differentiation: Definition and Fundamental Properties

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    • 2.1 Average and Instantaneous Rates of Change at a Point
    • 2.2 Defining the Derivative and Reading Its Notation
    • 2.3 Estimating a Derivative at a Point
    • 2.4 Differentiability and Continuity: When a Derivative Exists
    • 2.5 The Power Rule
    • 2.6 Constant, Sum, Difference, and Constant Multiple Rules
    • 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 Derivatives of Tangent, Cotangent, Secant, and Cosecant
    • 2.10 Exam tips
  • 3

    Differentiation: Composite, Implicit, and Inverse Functions

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    • 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
    • 3.6 Exam tips
  • 4

    Contextual Applications of Differentiation

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    • 4.1 Interpreting the Meaning of the Derivative in Context
    • 4.2 Straight-Line Motion: 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 Using Local Linearity and Linearization
    • 4.7 Using L'Hospital's Rule for Indeterminate Forms
    • 4.7 Exam tips
  • 5

    Analytical Applications of Differentiation

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    • 5.1 Using the Mean Value Theorem
    • 5.2 Extreme Values, Global vs Local Extrema, and Critical Points
    • 5.3 Where a Function Increases or Decreases
    • 5.4 The First Derivative Test for Local Extrema
    • 5.5 The Candidates Test for Absolute Extrema
    • 5.6 Concavity and Points of Inflection
    • 5.7 The Second Derivative Test for Extrema
    • 5.8 Sketching a Function and Its Derivative
    • 5.9 Connecting $f$, $f'$, and $f''$
    • 5.10 Introduction to Optimization Problems
    • 5.11 Solving Optimization Problems
    • 5.12 Exploring Behaviors of Implicit Relations
    • 5.12 Exam tips
  • 6

    Integration and Accumulation of Change

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    • 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
    • 6.6 Applying Properties of Definite Integrals
    • 6.7 The Fundamental Theorem of Calculus and Definite Integrals
    • 6.8 Finding Antiderivatives and Indefinite Integrals
    • 6.9 Integrating Using Substitution
    • 6.10 Integrating 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
    • 6.14 Exam tips
  • 7

    Differential Equations

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    • 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
    • 7.8 Exponential Models with Differential Equations
    • 7.9 Logistic Models with Differential Equations
    • 7.9 Exam tips
  • 8

    Applications of Integration

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    • 8.1 Finding the Average Value of a Function on an Interval
    • 8.2 Connecting Position, Velocity, and Acceleration 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 More Than Twice
    • 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 Curve and Distance Traveled
    • 8.13 Exam tips
  • 9

    Parametric Equations, Polar Coordinates, and Vector-Valued Functions

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    • 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 Finding the Area of a Polar Region
    • 9.9 Finding the Area of the Region Bounded by Two Polar Curves
    • 9.9 Exam tips
  • 10

    Infinite Sequences and Series

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    • 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
    • 10.15 Exam tips

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