Course syllabus
AP® Calculus BC
A High Bluff Academy course, authorized through the College Board's AP Course Audit.
AP® Calculus AB and AP Calculus BC are two College Board courses built on a single framework. The framework asks students to grasp the central ideas of calculus and put its methods to use, to move between a function's graph, a table of its values, its formula, and a description in words, and to back each conclusion with an argument that rests on definitions and theorems.
The College Board describes AP Calculus AB as the equivalent of a first-semester college calculus course in differential and integral calculus, and AP Calculus BC as the equivalent of both the first and the second semester of college calculus.
AP Calculus BC includes everything in AP Calculus AB and goes further. Within Units 6 to 8 it adds integration by parts, linear partial fractions, improper integrals, Euler's method, logistic models, and the arc length of a planar curve, and its two further units take up parametric equations, polar coordinates, and vector-valued functions (Unit 9) and infinite sequences and series (Unit 10).
The College Board's framework has ten units. AP Calculus AB covers Units 1 to 8, and AP Calculus BC covers all ten: Units 9 and 10, and topics 6.11, 6.12, 6.13, 7.5, 7.9, and 8.13, are part of BC only. The College Board presents the framework as helping to prepare students for college credit or placement.
Course identification
- Course title
- AP Calculus BC
- Subject area
- Mathematics
- AP authorization
- Authorized through the College Board's AP Course Audit
- Framework
- College Board, AP Calculus AB and BC Course and Exam Description, effective fall 2020, with its clarifications and corrections for fall 2026
- Length
- Two semesters of credit; the calendar length depends on the format
- Prerequisites
- The recommended preparation below; High Bluff Academy may ask for placement testing
- Term
- Set at enrollment
Credit and accreditation
- School of record
- High Bluff Academy
- CEEB code
- 053036
- AP Exam
- Given at High Bluff Academy, an official College Board testing site
- Accreditation
- Accrediting Commission for Schools, Western Association of Schools and Colleges (ACS WASC)
- Credits
- 10 credits: 5 for each semester
- Grading
- A letter grade for each semester, recorded on the High Bluff Academy transcript
- Transcript title
- AP Calculus BC, with the AP designation
High Bluff Academy is a private, college-preparatory school in Rancho Santa Fe, California. It offers coursework leading to a high school diploma and grants that diploma. The course is authorized through the College Board's AP Course Audit, which permits High Bluff Academy to use the AP designation for the course on student transcripts. One enrollment covers both semesters: a student earns 5 credits for each semester passed and 10 credits for the full course. A semester is passed with a grade of D or better.
How the course is offered
High Bluff Academy offers AP Calculus BC in person as a group class, one-on-one with a teacher, and online, with a High Bluff Academy teacher as the teacher of record. The course framework, the credit, and the transcript record are the same in every format. Availability of each format may vary with enrollment and demand.
Recommended preparation
The College Board expects every calculus student to have completed the equivalent of four years of college-preparatory high school mathematics, including algebra, geometry, trigonometry, analytic geometry, and elementary functions, with a solid grounding in algebraic reasoning and algebraic structures.
For AP Calculus AB and BC, a working knowledge of
- Linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions
- The properties, composition, algebra, and graphs of functions
- The language of functions: domain and range, odd and even, periodic, symmetry, zeros, intercepts, and increasing and decreasing
- The sine and cosine functions as defined from the unit circle, and the values of the trigonometric functions at 0, π/6, π/4, π/3, and π/2 and at their multiples
For AP Calculus BC, also
- Basic familiarity with sequences and series
- Some exposure to parametric and polar equations
High Bluff Academy may ask for an unofficial transcript and placement testing before a student enrolls in a mathematics course.
Units and topics
Unit 1
Limits and Continuity
Assessed on the AP Calculus AB and BC Exams
How change at an instant leads to the idea of a limit, then limits estimated from graphs and tables and found with algebraic properties, algebraic manipulation, and the squeeze theorem; types of discontinuity, continuity at a point and over an interval, infinite limits and limits at infinity with their asymptotes, and the Intermediate Value Theorem.
- 1.1Introducing Calculus: Can Change Occur at an Instant?
- 1.2Defining Limits and Using Limit Notation
- 1.3Estimating Limit Values from Graphs
- 1.4Estimating Limit Values from Tables
- 1.5Determining Limits Using Algebraic Properties of Limits
- 1.6Determining Limits Using Algebraic Manipulation
- 1.7Selecting Procedures for Determining Limits
- 1.8Determining Limits Using the Squeeze Theorem
- 1.9Connecting Multiple Representations of Limits
- 1.10Exploring Types of Discontinuities
- 1.11Defining Continuity at a Point
- 1.12Confirming Continuity over an Interval
- 1.13Removing Discontinuities
- 1.14Connecting Infinite Limits and Vertical Asymptotes
- 1.15Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16Working with the Intermediate Value Theorem (IVT)
Unit 2
Differentiation: Definition and Fundamental Properties
Assessed on the AP Calculus AB and BC Exams
Average and instantaneous rates of change, the definition of the derivative and its notation, estimating derivatives at a point, and how differentiability relates to continuity; then the basic rules: the power rule, the constant, sum, difference, and constant multiple rules, the derivatives of cosine, sine, e^x, and ln x, the product and quotient rules, and the derivatives of the other four trigonometric functions.
- 2.1Defining Average and Instantaneous Rates of Change at a Point
- 2.2Defining the Derivative of a Function and Using Derivative Notation
- 2.3Estimating Derivatives of a Function at a Point
- 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5Applying the Power Rule
- 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7Derivatives of cos x, sin x, e^x, and ln x
- 2.8The Product Rule
- 2.9The Quotient Rule
- 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3
Differentiation: Composite, Implicit, and Inverse Functions
Assessed on the AP Calculus AB and BC Exams
The chain rule, implicit differentiation, the derivatives of inverse functions and of inverse trigonometric functions, choosing a procedure for calculating a derivative, and derivatives of higher order.
- 3.1The Chain Rule
- 3.2Implicit Differentiation
- 3.3Differentiating Inverse Functions
- 3.4Differentiating Inverse Trigonometric Functions
- 3.5Selecting Procedures for Calculating Derivatives
- 3.6Calculating Higher-Order Derivatives
Unit 4
Contextual Applications of Differentiation
Assessed on the AP Calculus AB and BC Exams
The derivative read in context: position, velocity, and acceleration in straight-line motion, rates of change in applied settings other than motion, related rates, approximation by local linearity and linearization, and L'Hospital's Rule for limits of indeterminate forms.
- 4.1Interpreting the Meaning of the Derivative in Context
- 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3Rates of Change in Applied Contexts Other Than Motion
- 4.4Introduction to Related Rates
- 4.5Solving Related Rates Problems
- 4.6Approximating Values of a Function Using Local Linearity and Linearization
- 4.7Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
Unit 5
Analytical Applications of Differentiation
Assessed on the AP Calculus AB and BC Exams
The Mean Value Theorem and the Extreme Value Theorem, critical points, where a function increases and decreases, the first derivative test, the candidates test, and the second derivative test for extrema, concavity, sketching and connecting a function with its first and second derivatives, optimization problems, and the behavior of implicit relations.
- 5.1Using the Mean Value Theorem
- 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6Determining Concavity of Functions over Their Domains
- 5.7Using the Second Derivative Test to Determine Extrema
- 5.8Sketching Graphs of Functions and Their Derivatives
- 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10Introduction to Optimization Problems
- 5.11Solving Optimization Problems
- 5.12Exploring Behaviors of Implicit Relations
Unit 6
Integration and Accumulation of Change
Assessed on the AP Calculus AB and BC Exams
Accumulation of change and its approximation by Riemann sums, definite integral notation, the Fundamental Theorem of Calculus and accumulation functions, properties of definite integrals, antiderivatives and indefinite integrals, integration by substitution, by long division, and by completing the square, and choosing a technique of antidifferentiation. Integration by parts, linear partial fractions, and improper integrals are part of AP Calculus BC only.
- 6.1Exploring Accumulations of Change
- 6.2Approximating Areas with Riemann Sums
- 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6Applying Properties of Definite Integrals
- 6.7The Fundamental Theorem of Calculus and Definite Integrals
- 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9Integrating Using Substitution
- 6.10Integrating Functions Using Long Division and Completing the Square
- 6.11Integrating Using Integration by Parts (BC only)
- 6.12Integrating Using Linear Partial Fractions (BC only)
- 6.13Evaluating Improper Integrals (BC only)
- 6.14Selecting Techniques for Antidifferentiation
Unit 7
Differential Equations
Assessed on the AP Calculus AB and BC Exams
Differential equations as models of situations, verifying their solutions, sketching and reasoning with slope fields, general and particular solutions by separation of variables, and exponential models. Euler's method and logistic models are part of AP Calculus BC only.
- 7.1Modeling Situations with Differential Equations
- 7.2Verifying Solutions for Differential Equations
- 7.3Sketching Slope Fields
- 7.4Reasoning Using Slope Fields
- 7.5Approximating Solutions Using Euler's Method (BC only)
- 7.6Finding General Solutions Using Separation of Variables
- 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8Exponential Models with Differential Equations
- 7.9Logistic Models with Differential Equations (BC only)
Unit 8
Applications of Integration
Assessed on the AP Calculus AB and BC Exams
Definite integrals applied: the average value of a function, position, velocity, and acceleration, including displacement and the total distance traveled, accumulation in applied contexts, areas between curves, and the volumes of solids with known cross sections and of solids of revolution by the disc and washer methods. The arc length of a smooth planar curve is part of AP Calculus BC only.
- 8.1Finding the Average Value of a Function on an Interval
- 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4Finding the Area Between Curves Expressed as Functions of x
- 8.5Finding the Area Between Curves Expressed as Functions of y
- 8.6Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7Volumes with Cross Sections: Squares and Rectangles
- 8.8Volumes with Cross Sections: Triangles and Semicircles
- 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10Volume with Disc Method: Revolving Around Other Axes
- 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12Volume with Washer Method: Revolving Around Other Axes
- 8.13The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC only)
Unit 9
Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Assessed on the AP Calculus BC Exam only
Differentiating parametric equations, their second derivatives, and the arc length of parametric curves; differentiating and integrating vector-valued functions and using them, with parametric functions, to solve motion problems; and polar coordinates, differentiation in polar form, and the areas of regions bounded by one or two polar curves.
- 9.1Defining and Differentiating Parametric Equations (BC only)
- 9.2Second Derivatives of Parametric Equations (BC only)
- 9.3Finding Arc Lengths of Curves Given by Parametric Equations (BC only)
- 9.4Defining and Differentiating Vector-Valued Functions (BC only)
- 9.5Integrating Vector-Valued Functions (BC only)
- 9.6Solving Motion Problems Using Parametric and Vector-Valued Functions (BC only)
- 9.7Defining Polar Coordinates and Differentiating in Polar Form (BC only)
- 9.8Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve (BC only)
- 9.9Finding the Area of the Region Bounded by Two Polar Curves (BC only)
Unit 10
Infinite Sequences and Series
Assessed on the AP Calculus BC Exam only
Convergent and divergent infinite series; geometric series, the harmonic series, and p-series; the nth term, integral, comparison, alternating series, and ratio tests; absolute and conditional convergence; the alternating series and Lagrange error bounds; Taylor polynomial approximations; the radius and interval of convergence of power series; Taylor and Maclaurin series; and functions represented as power series.
- 10.1Defining Convergent and Divergent Infinite Series (BC only)
- 10.2Working with Geometric Series (BC only)
- 10.3The nth Term Test for Divergence (BC only)
- 10.4Integral Test for Convergence (BC only)
- 10.5Harmonic Series and p-Series (BC only)
- 10.6Comparison Tests for Convergence (BC only)
- 10.7Alternating Series Test for Convergence (BC only)
- 10.8Ratio Test for Convergence (BC only)
- 10.9Determining Absolute or Conditional Convergence (BC only)
- 10.10Alternating Series Error Bound (BC only)
- 10.11Finding Taylor Polynomial Approximations of Functions (BC only)
- 10.12Lagrange Error Bound (BC only)
- 10.13Radius and Interval of Convergence of Power Series (BC only)
- 10.14Finding Taylor or Maclaurin Series for a Function (BC only)
- 10.15Representing Functions as Power Series (BC only)
Mathematical practices
The College Board organizes the skills of AP Calculus AB and BC into four mathematical practices, which it expects students to develop throughout the course. Every task on the AP Exams draws on these skills, and Practice 4 is assessed only in the free-response section.
Practice 1
Implementing Mathematical Processes
Finding expressions and values by carrying out mathematical procedures and rules.
- 1.ARecognize what question is being asked or what problem needs solving (not assessed).
- 1.BPick out the information that matters for answering the question or solving the problem (not assessed).
- 1.CChoose a suitable rule or procedure from the form of the expression at hand, such as the chain rule for a composite function.
- 1.DChoose a suitable rule or procedure from how concepts or processes are related, such as rate of change and accumulation, or differentiation and antidifferentiation.
- 1.ECarry out the appropriate rules or procedures, with and without technology.
- 1.FExplain how an approximation relates to the value it approximates.
Practice 2
Connecting Representations
Moving mathematical information within one representation or between several.
- 2.ARecognize the common structure beneath problems set in different contexts.
- 2.BRead mathematical information from graphical, numerical, analytical, and verbal representations.
- 2.CRecognize the same mathematical information when it is expressed in another form.
- 2.DRecognize how the characteristics or properties of a function appear in different representations.
- 2.EDescribe how representations of a function and of its derivatives relate to one another.
Practice 3
Justification
Supporting reasoning and solutions with justified arguments.
- 3.AUse technology to form claims and conjectures (not assessed).
- 3.BChoose the definition, theorem, or test that applies.
- 3.CCheck that the hypotheses or conditions of the chosen definition, theorem, or test are met.
- 3.DApply the chosen definition, theorem, or test.
- 3.EGive reasons for solutions and conclusions.
- 3.FExplain what a mathematical solution means in its context.
- 3.GCheck that solutions are accurate and appropriate.
Practice 4
Communication and Notation
Communicating results and solutions with correct notation, language, and mathematical conventions.
- 4.AExpress ideas in precise mathematical language.
- 4.BAttach the correct units of measure.
- 4.CWrite with correct mathematical symbols and notation, including the notations for a derivative.
- 4.DDraw graphs with appropriate technique.
- 4.ERound results appropriately.
Three big ideas run through the College Board's framework: change, limits, and analysis of functions. Derivatives describe rates of change and definite integrals describe net change, and the Fundamental Theorem of Calculus ties the two together.
Assessment and grading
Each semester grade is built from the categories below, which carry the weights High Bluff Academy usually uses. Every format includes a midterm examination and a final examination each semester.
| Category | Weight | What it covers |
|---|---|---|
| Homework | 40% | Homework assigned during the semester |
| Tests | 40% | Tests during the semester, including the midterm examination |
| Final examination | 20% | The examination at the end of the semester |
Academic integrity
Work submitted for a grade must be the student's own. Tests and exams are taken under the conditions the teacher sets, including whether a calculator is allowed, and the College Board's rules apply on the AP Exam.
Graphing calculator
The College Board treats a graphing calculator as an integral part of AP Calculus AB and BC, and one is required on two parts of each AP Exam. It also expects students to explore ideas with technology, a graphing calculator above all, across the whole course.
Every student has individual access to a graphing calculator for coursework, and the web or app version of the Desmos graphing calculator is one way to have it. A handheld calculator from the College Board's approved list is optional; a student who plans to use one on the AP Exam should practice with it during the year.
The College Board expects a calculator used on the exam to graph a function in any viewing window, find zeros of functions, and compute a derivative and a definite integral numerically. When a free-response answer uses one of those capabilities, the student must show the setup, such as the equation to be solved or the derivative or integral to be computed, together with the calculator's result; any other calculator feature needs the mathematical steps that produce the result.
The AP Exam
The AP Calculus AB and AP Calculus BC Exams share one format. Each lasts 3 hours and 10 minutes and has two sections, with 42 multiple-choice questions and 6 free-response questions. The AB Exam assesses Units 1 to 8, and the BC Exam assesses all ten units.
| Part | Questions | Time | Share of score | Calculator |
|---|---|---|---|---|
| Section I, Part A: multiple choice | 29 | 62 minutes | 35% | Not permitted |
| Section I, Part B: multiple choice | 13 | 38 minutes | 15% | Graphing calculator required |
| Section II, Part A: free response | 2 | 30 minutes | 16.7% | Graphing calculator required |
| Section II, Part B: free response | 4 | 60 minutes | 33.3% | Not permitted |
| Section I | 42 | 100 minutes | 50% | |
| Section II | 6 | 90 minutes | 50% | |
| Exam total | 48 | 190 minutes | 100% |
On the exam, a Desmos graphing calculator is built into the testing app, available only in the parts that require a calculator, and students may also bring up to two handheld graphing calculators from the College Board's approved list.
Three of the six free-response questions are common to the AB and BC Exams and assess AB content, and each exam includes at least two questions set in a real-world context. Students answer the multiple-choice questions and read the free-response questions in the College Board's digital testing app, and they handwrite their free-response answers in a paper booklet.
Scores are reported on a scale of 1 to 5. A student who takes the AP Calculus BC Exam also receives a Calculus AB subscore on the same scale, which reports how the student did on the AB material, about 60 percent of the exam. This is the exam format in effect from the May 2027 exams. The AP Exams are given in schools in May, and a student may not take both the AB and the BC Exam in the same year.
High Bluff Academy is an official College Board testing site, and students enrolled in the course can take the AP Exam there.
A student who takes the AP Exam at a school other than the one where they take the course, or who takes the exam without taking the course, gets a join code from the AP coordinator at the school where they will test and joins an exam-only section there.
Materials and technology
- Graphing calculator
- The Desmos graphing calculator on the web or as an app, or a handheld model from the College Board's approved list
- Course framework
- apcentral.collegeboard.org/courses/ap-calculus-bc