# AP Calculus

Advanced Placement (AP) Calculus (also known as AP Calc, Calc AB / BC, AB / BC Calc or simply AB / BC) is a set of two distinct Advanced Placement calculus courses and exams offered by the American nonprofit organization College Board. AP Calculus AB covers basic introductions to limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus additional topics (including integration by parts, Taylor series, parametric equations, vector calculus, and polar coordinate functions).

## AP Calculus AB

AP Calculus AB is an Advanced Placement calculus course. It is traditionally taken after precalculus and is the first calculus course offered at most schools except for possibly a regular or honors calculus class. The Pre-Advanced Placement pathway for math helps prepare students for further Advanced Placement classes and exams.

### Purpose

According to the College Board:
   An AP course in calculus consists of a full high school academic year of work that is comparable to calculus courses in colleges and universities. It is expected that students who take an AP course in calculus will seek college credit, college placement, or both, from institutions of higher learning.
   The AP Program includes specifications for two calculus courses and the exam for each course. The two courses and the two corresponding exams are designated as Calculus AB and Calculus BC. Calculus AB can be offered as an AP course by any school that can organize a curriculum for students with advanced mathematical ability.

### Topic outline

The material includes the study and application of differentiation and integration, and graphical analysis including limits, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.
* Analysis of graphs (predicting and explaining behavior)
* Limits of functions (one and two sided)
* Asymptotic and unbounded behavior
* Continuity
* Derivatives
** Concept
** At a point
** As a function
** Applications
** Higher order derivatives
** Techniques
* Integrals
** Interpretations
** Properties
** Applications
** Techniques
** Numerical approximations
* Fundamental theorem of calculus
* Antidifferentiation
* L'Hôpital's rule
* Separable differential equations
| Score | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| 5 | 18.7% | 19.4% | 19.1% | 19.5% | 17.6% | 20.4% | 22.4% | 21% |
| 4 | 18% | 17.3% | 18.7% | 20.9% | 14.1% | 16.1% | 16.2% | 28% |
| 3 | 20.8% | 21% | 20.6% | 21.0% | 19.3% | 19.1% | 19.4% | 15% |
| 2 | 22% | 22.4% | 23.3% | 24.1% | 25.3% | 22.6% | 21.7% | 23% |
| 1 | 20.4% | 20% | 18.3% | 14.5% | 23.7% | 21.7% | 20.3% | 13% |
| % of Scores 3 or Higher | 57.5% | 57.7% | 58.4% | 61.4% | 51.0% | 55.7% | 58.0% | 64% |
| Mean | 2.93 | 2.94 | 2.97 | 3.07 | 2.77 | 2.91 | 2.99 | 3.21 |
| Standard Deviation | 1.40 | 1.40 | 1.38 | 1.36 | 1.41 | 1.44 | 1.44 | 1.35 |
| Number of Students | 316,099 | 308,538 | 300,659 | 266,430 | 251,639 | 268,352 | 273,987 | 251,000 |

## AP Calculus BC

Part of a series of articles about :
Calculus :
$\int _{a}^{b}f'(t)\,dt=f(b)-f(a)$ :
Fundamental theorem
Limits
Continuity
Rolle's theorem
Mean value theorem
Inverse function theorem :
Differential
Definitions
Derivative (generalizations)
Differential
infinitesimal
of a function
total
Concepts
Differentiation notation
Second derivative
Implicit differentiation
Logarithmic differentiation
Related rates
Taylor's theorem
Rules and identities
Sum
Product
Chain
Power
Quotient
L'Hôpital's rule
Inverse
General Leibniz
Faà di Bruno's formula
Reynolds :
Integral
Lists of integrals
Integral transform
Leibniz integral rule
Definitions
Antiderivative
Integral (improper)
Riemann integral
Lebesgue integration
Contour integration
Integral of inverse functions
Integration by
Parts
Discs
Cylindrical shells
Substitution (trigonometric, tangent half-angle, Euler)
Euler's formula
Partial fractions (Heaviside's method)
Changing order
Reduction formulae
Differentiating under the integral sign
Risch algorithm :
Series
Geometric (arithmetico-geometric)
Harmonic
Alternating
Power
Binomial
Taylor
Convergence tests
Summand limit (term test)
Ratio
Root
Integral
Direct comparison
Limit comparison
Alternating series
Cauchy condensation
Dirichlet
Abel :
Vector
Gradient
Divergence
Curl
Laplacian
Directional derivative
Identities
Theorems
Gradient
Green's
Stokes'
Divergence
generalized Stokes
Helmholtz decomposition :
Multivariable
Formalisms
Matrix
Tensor
Exterior
Geometric
Definitions
Partial derivative
Multiple integral
Line integral
Surface integral
Volume integral
Jacobian
Hessian :
Advanced
Calculus on Euclidean space
Generalized functions
Limit of distributions :
Specialized
Fractional
Malliavin
Stochastic
Variations :
Miscellanea
Precalculus
History
Glossary
List of topics
Integration Bee
Mathematical analysis
Nonstandard analysis :
vte :
AP Calculus BC is equivalent to a full year regular college course, covering both Calculus I and II. After passing the exam, students may move on to Calculus III (Multivariable Calculus).

### Purpose

According to the College Board,
   Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics... Students who take an AP Calculus course should do so with the intention of placing out of a comparable college calculus course.

### Topic outline

AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as the following:
* Convergence tests for series
* Taylor series
* Parametric equations
* Polar functions (including arc length in polar coordinates and calculating area)
* Arc length calculations using integration
* Integration by parts
* Improper integrals
* Differential equations for logistic growth
* Using partial fractions to integrate rational functions
| Score | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| 5 | 42.6% | 40.4% | 43.0% | 44.6% | 38.3% | 41.2% | 43.5% | 45% |
| 4 | 18.1% | 18.6% | 18.5% | 17.6% | 16.5% | 15.6% | 15.9% | 22% |
| 3 | 19.9% | 20.7% | 19.5% | 19.4% | 20.4% | 20.1% | 19.0% | 13% |
| 2 | 14.3% | 14.6% | 13.9% | 14.1% | 18.2% | 16.4% | 15.2% | 15% |
| 1 | 5.3% | 5.6% | 5.2% | 4.3% | 6.6% | 6.8% | 6.3% | 5% |
| % of Scores 3 or Higher | 80.6% | 79.7% | 81.0% | 81.6% | 75.2% | 76.9% | 78.5% | 80% |
| Mean | 3.79 | 3.73 | 3.81 | 3.84 | 3.62 | 3.68 | 3.75 | 3.87 |
| Standard Deviation | 1.28 | 1.28 | 1.27 | 1.25 | 1.33 | 1.33 | 1.32 | 1.27 |
| Number of Students | 132,514 | 139,376 | 139,195 | 127,864 | 124,599 | 120,238 | 135,458 | 119,000 |
It can be seen from the tables that the pass rate (score of 3 or higher) of AP Calculus BC is higher than AP Calculus AB. It can also be noted that about 1/3 as many take the BC exam as take the AB exam. A possible explanation for the higher scores on BC is that students who take AP Calculus BC are more prepared and advanced in math. The 5-rate is consistently over 40% (much higher than almost all the other AP exams).

#### AB sub-score distribution

| Score | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
| 5 | 48.4% | 48.7% | 49.5% | N/A | 46.7% | 48.5% | 46.4% |
| 4 | 22.5% | 20.2% | 23.5% | N/A | 17.7% | 20.9% | 19.6% |
| 3 | 14.1% | 15.9% | 13.2% | N/A | 16.0% | 11.8% | 18.8% |
| 2 | 10.0% | 9.9% | 9.7% | N/A | 13.0% | 12.2% | 8.6% |
| 1 | 4.9% | 5.3% | 4.2% | N/A | 6.6% | 6.7% | 6.6% |
| % of Scores 3 or Higher | 85.0% | 84.8% | 86.2% | N/A | 80.4% | 81.2% | 84.8% |
| Mean | 4.00 | 3.97 | 4.05 | N/A | 3.85 | 3.92 | 3.91 |
| Standard Deviation | 1.21 | 1.23 | 1.18 | N/A | 1.31 | 1.30 | 1.26 |
| Number of Students | 132,505 | 139,376 | 139,195 | N/A | 124,607 | 120,276 | 135,458 |

## AP Exam

The College Board intentionally schedules the AP Calculus AB exam at the same time as the AP Calculus BC exam to make it impossible for a student to take both tests in the same academic year, though the College Board does not make Calculus AB a prerequisite class for Calculus BC. Some schools do this, though many others only require precalculus as a prerequisite for Calculus BC. The AP awards given by College Board count both exams. However, they do not count the AB sub-score piece of the BC exam.

### Format

The structures of the AB and BC exams are identical. Both exams are three hours and fifteen minutes long, comprising a total of 45 multiple choice questions and six free response questions. They are usually administered on a Monday or Tuesday morning in May.
| | Multiple-Choice, Section I Part A | Multiple-Choice, Section I Part B | Free-Response, Section II Part A | Free-Response, Section II Part B |
| # of Questions | 30 | 15 | 2 | 4 |
| Time Allowed | 60 minutes | 45 minutes | 30 minutes | 60 minutes |
| Calculator Use | No | Yes | Yes | No |
The two parts of the multiple choice section are timed and taken independently.
Students are required to put away their calculators after 30 minutes have passed during the Free-Response section, and only at that point may begin Section II Part B. However, students may continue to work on Section II Part A during the entire Free-Response time, although without a calculator during the later two thirds.

### Scoring

The multiple choice section is scored by computer, with a correct answer receiving 1 point, with omitted and incorrect answers not affecting the raw score. This total is multiplied by 1.2 to calculate the adjusted multiple-choice score.
The free response section is hand-graded by hundreds of AP teachers and professors each June. The raw score is then added to the adjusted multiple choice score to receive a composite score. This total is compared to a composite-score scale for that year's exam and converted into an AP score of 1 to 5.
For the Calculus BC exam, an AB sub-score is included in the score report to reflect their proficiency in the fundamental topics of introductory calculus. The AB sub-score is based on the correct number of answers for questions pertaining to AB-material only.