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 integration by parts, infinite series, parametric equations, vector calculus, and polar coordinate functions, among other topics.^[1]

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.

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.^[2]

AP Calculus AB is approximately equivalent to the first semester of college calculus, providing students with an opportunity to earn college credit or advanced placement.^[3] Research indicates that achieving higher scores on the exam can positively impact college completion rates by facilitating credit toward graduation requirements.^[4]

The material includes the study and application of differentiation and integration, and graphical analysis including limits, asymptotes, and continuity.^[5] An AP Calculus AB course is typically equivalent to one semester of college calculus.^[6][7]

• 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[8]	2018[9]	2019[10]	2020[11]	2021[12]	2022[13]	2023[14]	2024[15]	2025[16]
5	18.7%	19.4%	19.1%	19.5%	17.6%	20.4%	22.4%	21.4%	20.3%
4	18%	17.3%	18.7%	20.9%	14.1%	16.1%	16.2%	27.8%	28.9%
3	20.8%	21%	20.6%	21.0%	19.3%	19.1%	19.4%	15.3%	15.0%
2	22%	22.4%	23.3%	24.1%	25.3%	22.6%	21.7%	22.7%	22.8%
1	20.4%	20%	18.3%	14.5%	23.7%	21.7%	20.3%	12.9%	13.0%
% of Scores ≥3	57.5%	57.7%	58.4%	61.4%	51.0%	55.7%	58.0%	64.4%	64.2%
Mean	2.93	2.94	2.97	3.07	2.77	2.91	2.99	3.22	3.21
Standard Deviation	1.40	1.40	1.38	1.36	1.41	1.44	1.44	1.35	1.34
Number of Students	316,099	308,538	300,659	266,430	251,639	268,352	273,987	278,657	286,722

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).^[17]

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.^[2]

The course is designed for students with strong mathematical ability, aiming to provide advanced placement or credit equivalent to two semesters of college calculus.^[18]

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^[19][20]

Score	2017[21]	2018[9]	2019[10]	2020[22]	2021[23]	2022[24]	2023[25]	2024[26]	2025[27][28]
5	42.6%	40.4%	43.0%	44.6%	38.3%	41.2%	43.5%	47.7%	44.0%
4	18.1%	18.6%	18.5%	17.6%	16.5%	15.6%	15.9%	21.1%	21.9%
3	19.9%	20.7%	19.5%	19.4%	20.4%	20.1%	19.0%	12.1%	12.8%
2	14.3%	14.6%	13.9%	14.1%	18.2%	16.4%	15.2%	13.9%	15.2%
1	5.3%	5.6%	5.2%	4.3%	6.6%	6.8%	6.3%	5.2%	6.2%
% of Scores 3 or Higher	80.6%	79.7%	81.0%	81.6%	75.2%	76.9%	78.5%	80.9%	78.6%
Mean	3.79	3.73	3.81	3.84	3.62	3.68	3.75	3.92	3.82
Standard Deviation	1.28	1.28	1.27	1.25	1.33	1.33	1.32	1.27	1.30
Number of Students	132,514	139,376	139,195	127,864	124,599	120,238	135,458	148,191	160,954

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 1/2 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, often self-selecting into the course due to greater intrinsic motivation and prior achievement.^[29] The 5-rate is consistently over 40% (much higher than almost all the other AP exams).^[30] In 2025, AP Calculus BC was rated as one of the most satisfying AP exams by students, with an average satisfaction score of 8 out of 10.^[31]

Score	2017[21]	2018[9]	2019[32]	2020[33]	2021[34]	2022[35]	2023[36]
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

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.^[37] AP exam scores, including for Calculus, serve as strong predictors of college performance and are valued in admissions processes at institutions like the University of California system.^[38]

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.^[39][40]

	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.

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.^[41]

The free response section is hand-graded by hundreds of AP teachers and professors each June.^[42] 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.

There was considerable debate about whether or not calculus should be included when the Advanced Placement Mathematics course was first proposed in the early 1950s. AP Mathematics was eventually developed into AP Calculus thanks to physicists and engineers, who convinced mathematicians of the need to expose students in these subjects to calculus early on in their collegiate programs.^[43] The AP program originated in the 1950s amid Cold War concerns over educational rigor, with the first AP exams piloted in 1954 and nationally rolled out in 1956, including Mathematics as one of the original ten subjects.^[44] AP Calculus was established in 1955 as a single exam covering a full year of college-level calculus; in 1969, the AB exam was introduced for the first semester, while the original became BC.^[45] The program grew rapidly from the 1980s, with exam takership increasing fivefold by the mid-1990s, and by 2019, over 450,000 students took AP Calculus annually, representing about 20% of U.S. high school graduates.^[46] By the mid-2020s, AP Calculus AB was one of the top ten most popular AP exams.^[47]

In the early 21st century, there has been demand for the creation of AP Multivariable Calculus and indeed, a number of American high schools have begun to offer "Calculus III", giving colleges trouble in placing incoming students.^[43][48]

• AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism
• AP Precalculus
• Glossary of calculus
• Mathematics education in the United States
• Stand and Deliver (1988 film)

• AP Calculus AB
  • College Board description of the AP Calculus AB course content
  • College Board description of the AP Calculus AB examination
• AP Calculus BC
  • College Board description of the AP Calculus BC course content
  • College Board description of the AP Calculus BC examination

• Nahin, Paul (2014). Inside Interesting Integrals. Springer. ISBN 9781493912766.