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AP Calculus BC

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Open to: Grades 9 - 12

Prerequisites: Qualifying math score and completion of Precalculus or the equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: This is an AP Calculus course, equivalent to a year-long Calculus BC sequence.

Course Code: CBC

Course Description


This AP Calculus BC course covers topics in single variable differential and integral calculus typically found in a first-year college Calculus I and Calculus II two semester course sequence. Students who have successfully completed AP Calculus AB should enroll in Calculus C.

While taking the Advanced Placement (AP) Calculus BC exam is not required, this course prepares students to succeed on the AP Calculus BC exam and subsequent courses that draw on material from this course.

Students will learn single variable calculus by actively engaging with the lectures, interacting with online resources, and by attempting many practice problems through homework and quizzes. Videos in the course are provided by Thinkwell.

Students progress through the course at their own pace, reviewing any materials with their own CTY Online Programs instructor as needed. The student’s knowledge will be assessed through weekly homework, chapter tests and cumulative exams. Exams contain Free Response questions, hand graded by your CTY Online Programs instructor, based on the actual AP Calculus exam.

This course has been reviewed and approved by the College Board to use the "AP" designation.

Topics include:

  • Precalculus review
  • Limits and continuity
  • Derivatives & applications
  • Curve sketching
  • Related rates
  • Techniques of integration & applications
  • Applications of integration
  • L’Hôpital’s rule and improper integrals
  • Applications of integral calculus
  • Parametric equations and polar coordinates
  • Differential equations
  • Sequences and series
  • Applications of series

For a detailed list of topics, click the List of Topics tab.

AP Calculus BC banner

Materials Needed

A textbook is not required for this course.
A graphing calculator is required, such as:

  • TI-83 PLUS
  • TI-84 PLUS
  • TI-85
  • TI-86
  • TI-89*


To see a full list of allowable calculators, visit the College Board website to view the AP Calculus calculator policy.

Full List of Topics

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following topics:

Precalculus Review

  • Overview
  • Functions and Graphing
  • Exponential Functions
  • Inverse Functions
  • Inverse Trigonometric Functions
  • Evaluating Logarithmic Functions


  • The Concept of the Limit
  • Calculating Limits
  • The Squeeze Theorem
  • Continuity and Discontinuity
  • Infinite Limits and Indeterminate Forms


  • Understanding the Derivative
  • Using the Derivative
  • Some Special Derivatives
  • The Power Rule
  • The Product and Quotient Rules
  • The Chain Rule
  • Derivatives of Trigonometric Functions
  • Derivatives of the Exponential Function and the Natural Logarithm
  • Implicit Differentiation
  • Differentiating Logarithms
  • Logarithmic Differentiation
  • Derivatives of Inverse Functions

Applications of the Derivative

  • Position and Velocity
  • Linear Approximation and Newton's Method
  • Optimization
  • Related Rates
  • An Introduction to Curve Sketching
  • Critical Points
  • Concavity and Inflection Points
  • Graphing Using the Derivative
  • Graphing Functions with Asymptotes
  • Indeterminate Quotients and L'Hospital's Rule
  • Other Indeterminate Forms


  • Antiderivatives
  • Integration by Substitution
  • Illustrating Integration by Substitution
  • The Fundamental Theorem of Calculus
  • Numerical Integration and Tables of Integrals
  • Trigonometric Substitution

Techniques of Integration

  • Integrals Involving Powers of Sine and Cosine
  • Integrals Involving Powers of Other Trigonometric Functions
  • An Introduction to Integration by Partial Fractions
  • Integration by Partial Fractions with Repeated Factors
  • Integration by Parts
  • An Introduction to Trigonometric Substitution
  • Trigonometric Substitution Strategy
  • Improper Integrals

Applications of Integration

  • Motion
  • Finding the Area between Two Curves
  • Integrating with Respect to y
  • The Average Value of a Function
  • Finding Volumes Using Cross-Sections
  • Disks and Washers
  • Shells
  • Work
  • Moments and Centers of Mass
  • Arc Lengths and Functions

Differential Equations

  • Separable Differential Equations
  • Direction Fields
  • Growth and Decay Problems
  • Euler's Method

Parametric Equations and Polar Coordinates

  •  Understanding Parametric Equations
  • Derivatives and Arc Length of Parametric Equations
  • Understanding Polar Coordinates
  • Polar Functions and Slope
  • Polar Functions and Area

Sequences and Series

  • Sequences
  • Infinite Series
  • Convergence and Divergence
  • The Integral Test and p-Series
  • The Comparison and Limit Comparison Test
  • The Alternating Series Test, Absolute and Conditional Convergence
  • The Ratio and Root Test
  • Polynomial Approximations of Elementary Functions
  • Taylor and Maclaurin Polynomials
  • Taylor and Maclaurin Series
  • Power Series
  • Power Series Representations of Functions

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Sample Video

Video Lecture

Click on the image below to view the online demo.

AP Calculus demo

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses an online virtual classroom for discussions with the instructor. The classroom works on standard computers with the Adobe Connect Add-in or Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile appStudents who are unable to attend live sessions will need a computer with the Adobe Connect Add-in or Adobe Flash plugin installed to watch recorded meetings. The Adobe Connect Add-in, Adobe Flash plugin, and Adobe Connect Mobile app are available for free download. Students who do not have the Flash plug-in installed or enabled on their browsers will be prompted to download and install the Adobe Connect add-in when accessing the virtual classroom.