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Math and Computer ScienceCalculus

AP Calculus BC

Study the full single-variable calculus sequence, including AB foundations plus advanced integration, differential equations, parametric and polar analysis, and infinite series.

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Unit 1Limits and Continuity16 topics

Build the language of limits, connect graphical, numerical, and algebraic representations, and use continuity theorems to justify conclusions about function behavior.

  1. 1.1Introducing Calculus: Can Change Occur at an Instant?
  2. 1.2Defining Limits and Using Limit Notation
  3. 1.3Estimating Limit Values from Graphs
  4. 1.4Estimating Limit Values from Tables
  5. 1.5Determining Limits Using Algebraic Properties of Limits
  6. 1.6Determining Limits Using Algebraic Manipulation
  7. 1.7Selecting Procedures for Determining Limits
  8. 1.8Determining Limits Using the Squeeze Theorem
  9. 1.9Connecting Multiple Representations of Limits
  10. 1.10Exploring Types of Discontinuities
  11. 1.11Defining Continuity at a Point
  12. 1.12Confirming Continuity over an Interval
  13. 1.13Removing Discontinuities
  14. 1.14Connecting Infinite Limits and Vertical Asymptotes
  15. 1.15Connecting Limits at Infinity and Horizontal Asymptotes
  16. 1.16Working with the Intermediate Value Theorem (IVT)

Unit 2Differentiation: Definition and Fundamental Properties10 topics

Develop the derivative from rates of change and limits, then apply foundational derivative rules while connecting differentiability with continuity.

  1. 2.1Defining Average and Instantaneous Rates of Change at a Point
  2. 2.2Defining the Derivative of a Function and Using Derivative Notation
  3. 2.3Estimating Derivatives of a Function at a Point
  4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
  5. 2.5Applying the Power Rule
  6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
  7. 2.7Derivatives of cos x, sin x, eˣ, and ln x
  8. 2.8The Product Rule
  9. 2.9The Quotient Rule
  10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Unit 3Differentiation: Composite, Implicit, and Inverse Functions6 topics

Extend differentiation to composite, implicit, inverse, and inverse-trigonometric functions and choose efficient procedures for higher-order derivatives.

  1. 3.1The Chain Rule
  2. 3.2Implicit Differentiation
  3. 3.3Differentiating Inverse Functions
  4. 3.4Differentiating Inverse Trigonometric Functions
  5. 3.5Selecting Procedures for Calculating Derivatives
  6. 3.6Calculating Higher-Order Derivatives

Unit 4Contextual Applications of Differentiation6 topics

Interpret derivatives as contextual rates, model motion and related rates, and use local linearity to make and assess approximations.

  1. 4.1Interpreting the Meaning of the Derivative in Context
  2. 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
  3. 4.3Rates of Change in Applied Contexts Other Than Motion
  4. 4.4Introduction to Related Rates
  5. 4.5Solving Related Rates Problems
  6. 4.6Approximating Values of a Function Using Local Linearity and Linearization

Unit 5Analytical Applications of Differentiation12 topics

Use derivative tests and major theorems to analyze extrema, monotonicity, concavity, graph behavior, optimization, and implicit relations.

  1. 5.1Using the Mean Value Theorem
  2. 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
  3. 5.3Determining Intervals on Which a Function Is Increasing or Decreasing
  4. 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
  5. 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
  6. 5.6Determining Concavity of Functions over Their Domains
  7. 5.7Using the Second Derivative Test to Determine Extrema
  8. 5.8Sketching Graphs of Functions and Their Derivatives
  9. 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
  10. 5.10Introduction to Optimization Problems
  11. 5.11Solving Optimization Problems
  12. 5.12Exploring Behaviors of Implicit Relations

Unit 6Integration and Accumulation of Change14 topics

Connect accumulation, Riemann sums, definite integrals, and the Fundamental Theorem of Calculus, then select effective antiderivative techniques.

  1. 6.1Exploring Accumulations of Change
  2. 6.2Approximating Areas with Riemann Sums
  3. 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
  4. 6.4The Fundamental Theorem of Calculus and Accumulation Functions
  5. 6.5Interpreting the Behavior of Accumulation Functions Involving Area
  6. 6.6Applying Properties of Definite Integrals
  7. 6.7The Fundamental Theorem of Calculus and Definite Integrals
  8. 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
  9. 6.9Integrating Using Substitution
  10. 6.10Integrating Functions Using Long Division and Completing the Square
  11. 6.11Integrating Using Integration by Parts
  12. 6.12Integrating Using Linear Partial Fractions
  13. 6.13Evaluating Improper Integrals
  14. 6.14Selecting Techniques for Antidifferentiation

Unit 7Differential Equations9 topics

Model change with differential equations, interpret slope fields, verify and approximate solutions, and solve contextual growth and decay problems.

  1. 7.1Modeling Situations with Differential Equations
  2. 7.2Verifying Solutions for Differential Equations
  3. 7.3Sketching Slope Fields
  4. 7.4Reasoning Using Slope Fields
  5. 7.5Approximating Solutions Using Euler’s Method
  6. 7.6Finding General Solutions Using Separation of Variables
  7. 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
  8. 7.8Exponential Models with Differential Equations
  9. 7.9Logistic Models with Differential Equations

Unit 8Applications of Integration13 topics

Apply definite integrals to average value, motion, net accumulation, planar area, cross-sectional volume, and solids of revolution.

  1. 8.1Finding the Average Value of a Function on an Interval
  2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
  4. 8.4Finding the Area Between Curves Expressed as Functions of x
  5. 8.5Finding the Area Between Curves Expressed as Functions of y
  6. 8.6Finding the Area Between Curves That Intersect at More Than Two Points
  7. 8.7Volumes with Cross Sections: Squares and Rectangles
  8. 8.8Volumes with Cross Sections: Triangles and Semicircles
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
  10. 8.10Volume with Disc Method: Revolving Around Other Axes
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
  12. 8.12Volume with Washer Method: Revolving Around Other Axes
  13. 8.13The Arc Length of a Smooth, Planar Curve and Distance Traveled

Unit 9Parametric Equations, Polar Coordinates, and Vector-Valued Functions9 topics

Differentiate and integrate parametric, vector-valued, and polar representations to analyze planar motion, arc length, and area.

  1. 9.1Defining and Differentiating Parametric Equations
  2. 9.2Second Derivatives of Parametric Equations
  3. 9.3Finding Arc Lengths of Curves Given by Parametric Equations
  4. 9.4Defining and Differentiating Vector-Valued Functions
  5. 9.5Integrating Vector-Valued Functions
  6. 9.6Solving Motion Problems Using Parametric and Vector-Valued Functions
  7. 9.7Defining Polar Coordinates and Differentiating in Polar Form
  8. 9.8Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
  9. 9.9Finding the Area of the Region Bounded by Two Polar Curves

Unit 10Infinite Sequences and Series15 topics

Determine convergence with appropriate tests, bound approximation error, and represent functions with Taylor, Maclaurin, and power series.

  1. 10.1Defining Convergent and Divergent Infinite Series
  2. 10.2Working with Geometric Series
  3. 10.3The nth Term Test for Divergence
  4. 10.4Integral Test for Convergence
  5. 10.5Harmonic Series and p-Series
  6. 10.6Comparison Tests for Convergence
  7. 10.7Alternating Series Test for Convergence
  8. 10.8Ratio Test for Convergence
  9. 10.9Determining Absolute or Conditional Convergence
  10. 10.10Alternating Series Error Bound
  11. 10.11Finding Taylor Polynomial Approximations of Functions
  12. 10.12Lagrange Error Bound
  13. 10.13Radius and Interval of Convergence of Power Series
  14. 10.14Finding Taylor or Maclaurin Series for a Function
  15. 10.15Representing Functions as Power Series