View source for AP Calculus

{{Short description|Two Advanced Placement courses and exams}}
{{Primary sources|date=March 2025}}
{{Advanced Placement}}
'''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 [[Limit (mathematics)|limits]], [[derivative|derivatives]], and [[integrals]]. '''AP Calculus BC''' covers all AP Calculus AB topics plus [[integration by parts]], infinite series, [[parametric equation]]s, [[vector calculus]], and [[polar coordinate]] functions, among other topics.

==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]]:{{blockquote |text=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.<ref name="collegeboard1">{{cite web|url=http://collegeboard.com/prod_downloads/ap/students/calculus/ap-cd-calc-0607.pdf |title=2006, 2007 AP Calculus Course Description |year=2005 |publisher=College Board |access-date=2008-11-29 |url-status=dead |archive-url=https://web.archive.org/web/20080720151001/http://www.collegeboard.com/prod_downloads/ap/students/calculus/ap-cd-calc-0607.pdf |archive-date=2008-07-20 }}</ref>}}

===Topic outline===
The material includes the study and application of [[derivative (calculus)|differentiation]] and [[integral|integration]], and graphical analysis including [[Limit (mathematics)|limits]], [[asymptotes]], and [[Continuous function|continuity]].<ref>{{cite web |url=http://collegeboard.com/student/testing/ap/calculus_ab/topic.html?calcab |title=Topic Outline |work=Calculus AB |publisher=College Board |access-date=2008-11-29 |archive-url=https://web.archive.org/web/20081219070626/http://www.collegeboard.com/student/testing/ap/calculus_ab/topic.html?calcab |archive-date=2008-12-19 |url-status=dead }}</ref> An AP Calculus AB course is typically equivalent to one semester of college calculus.<ref>{{cite web|url=http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf|title=Course Descriptions}}</ref>
*Analysis of graphs (predicting and explaining behavior)
*[[Limit of a function|Limits of functions]] (one and two sided)
*Asymptotic and unbounded behavior
*[[Continuous function|Continuity]]
*[[Derivative]]s
**Concept
**At a point
**[[Derivative#Derivative as a function|As a function]]
**[[Differential calculus#Applications of derivatives|Applications]]
**[[Derivative#Higher derivatives|Higher order derivatives]]
**Techniques
*[[Integrals]]
**[[Integral#Interpretations|Interpretations]]
**Properties
**[[Integral#Applications|Applications]]
**Techniques
**[[Numerical integration|Numerical approximations]]
*[[Fundamental theorem of calculus]]
*[[Antiderivative|Antidifferentiation]]
*[[L'Hôpital's rule]]
*[[Separation of variables|Separable differential equations]]

{| class="wikitable" border="1"
|-
!Score
!2017<ref name=":3">{{Cite web|url=https://secure-media.collegeboard.org/ap/pdf/ap17-calculus-ab-scoring-global.pdf|title=AP Calculus AB May 2017 Score Distributions|last=College Board|archive-url=https://web.archive.org/web/20180326032116/https://secure-media.collegeboard.org/ap/pdf/ap17-calculus-ab-scoring-global.pdf|archive-date=2018-03-26|url-status=dead|access-date=2017-06-22}}</ref>
!2018<ref name="secure-media.collegeboard.org">{{cite web|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2018/Student-Score-Distributions-2018.pdf | title=Student Score Distributions, AP Exams - May 2018 | website=The College Board | access-date=December 24, 2022}}</ref>
!2019<ref name=":1">{{Cite web|title=Student Score Distributions|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2019/Student-Score-Distributions-2019.pdf|access-date=July 9, 2020}}</ref>
!2020<ref>{{Cite web|title=Student Score Distributions|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2020/Student-Score-Distributions-2020.pdf|access-date=June 9, 2021}}</ref>
!2021<ref>"[https://reports.collegeboard.org/media/pdf/2021-ap-student-score-distributions_1.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2022<ref>"[https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2022.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2023<ref>"[https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2023.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2024<ref>"[https://apstudents.collegeboard.org/about-ap-scores/score-distributions 2024 AP Score Distributions]". Retrieved July 8, 2024.</ref>
|-
!5
|18.7%
|19.4%
|19.1%
|19.5%
|17.6%
|20.4%
|22.4%
|21.4%
|-
!4
|18%
|17.3%
|18.7%
|20.9%
|14.1%
|16.1%
|16.2%
|27.8%
|-
!3
|20.8%
|21%
|20.6%
|21.0%
|19.3%
|19.1%
|19.4%
|15.3%
|-
!2
|22%
|22.4%
|23.3%
|24.1%
|25.3%
|22.6%
|21.7%
|22.7%
|-
!1
|20.4%
|20%
|18.3%
|14.5%
|23.7%
|21.7%
|20.3%
|12.9%
|-
!% of Scores 3 or Higher
|57.5%
|57.7%
|58.4%
|61.4%
|51.0%
|55.7%
|58.0%
|64.4%
|-
!Mean
|2.93
|2.94
|2.97
|3.07
|2.77
|2.91
|2.99
|3.22
|-
!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
|278,657
|-
|}

==AP Calculus BC==
{{Calculus}}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, {{blockquote |text=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.<ref name="collegeboard1"/>|sign=|source=}}

===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 equation]]s
* [[Polar coordinate system|Polar functions]] (including arc length in polar coordinates and calculating area)
*[[Arc length]] calculations using integration
* [[Integration by parts]] 
* [[Improper integral]]s 
* [[logistic function|Differential equations for logistic growth]]
* Using [[partial fraction]]s to integrate [[rational function]]s<ref>{{cite web |url=http://collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc |title=Topic Outline |work=Calculus BC |publisher=College Board |access-date=2008-11-29 |archive-url=https://web.archive.org/web/20081219081215/http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc |archive-date=2008-12-19 |url-status=dead }}</ref>

{| class="wikitable" border="1"
|-
!Score
!2017<ref name=":0">{{Cite web|url=https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap17-chief-reader-report-calculus.pdf|title=AP Calculus AB and Calculus BC Chief Reader Report May 2017|archive-url=https://web.archive.org/web/20180725153322/https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap17-chief-reader-report-calculus.pdf|archive-date=2018-07-25|url-status=dead}}</ref>
!2018<ref name="secure-media.collegeboard.org"/>
!2019<ref name=":1" />
!2020<ref>{{Cite web|title=STUDENT SCORE DISTRIBUTIONS|url=https://www.https://secure-media.collegeboard.org/digitalServices/pdf/research/2020/Student-Score-Distributions-2020.pdf|access-date=June 9, 2021|archive-date=August 19, 2013|archive-url=https://web.archive.org/web/20130819141428/http://https/|url-status=dead}}</ref>
!2021<ref>"[https://reports.collegeboard.org/media/pdf/2021-ap-student-score-distributions_1.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2022<ref>{{cite web |url=https://twitter.com/AP_Trevor/status/1542598887558504450?s=20&t=a70qOYpBdGGcRdLzGQCZFQ |title=The 2022 AP Calculus BC scores |access-date=7 July 2022}}</ref>
!2023<ref>"[https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2023.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2024<ref>"[https://apstudents.collegeboard.org/about-ap-scores/score-distributions 2024 AP Score Distributions]". Retrieved July 8, 2024.</ref>
|-
!5
|42.6%
|40.4%
|43.0%
|44.6%
|38.3%
|41.2%
|43.5%
|47.7%
|-
!4
|18.1%
|18.6%
|18.5%
|17.6%
|16.5%
|15.6%
|15.9%
|21.1%
|-
!3
|19.9%
|20.7%
|19.5%
|19.4%
|20.4%
|20.1%
|19.0%
|12.1%
|-
!2
|14.3%
|14.6%
|13.9%
|14.1%
|18.2%
|16.4%
|15.2%
|13.9%
|-
!1
|5.3%
|5.6%
|5.2%
|4.3%
|6.6%
|6.8%
|6.3%
|5.2%
|-
!% of Scores 3 or Higher
|80.6%
|79.7%
|81.0%
|81.6%
|75.2%
|76.9%
|78.5%
|80.9%
|-
!Mean
|3.79
|3.73
|3.81
|3.84
|3.62
|3.68
|3.75
|3.92
|-
!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
|148,191
|-
|}
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. The 5-rate is consistently over 40% (much higher than almost all the other AP exams). {{Original research inline|date=March 2024}}
====AB sub-score distribution====
{|class="wikitable" border="1"
|-
!Score
!2017<ref name=":0" />
!2018<ref name="secure-media.collegeboard.org"/>
!2019<ref>{{Cite web|title=STUDENT SCORE DISTRIBUTIONS – AP Exams May 2019|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2019/Student-Score-Distributions-2019.pdf|access-date=9 July 2020 |publisher=[[College Board]]}}</ref>
!2020<ref>{{cite web|title= STUDENT SCORE DISTRIBUTIONS – AP Exams May 2020|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2020/Student-Score-Distributions-2020.pdf |access-date=20 July 2023 |publisher=[[College Board]] }}</ref>
!2021<ref>{{cite web|title= STUDENT SCORE DISTRIBUTIONS – AP Exams May 2021|url=https://secure-media.collegeboard.org/digitalServices/pdf/research/2021/2021-ap-student-score-distributions.pdf |access-date=20 July 2023 |publisher=[[College Board]]}}</ref>
!2022<ref>"[https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2022.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
!2023<ref>"[https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2023.pdf STUDENT SCORE DISTRIBUTIONS]" (PDF). Retrieved January 12, 2024.</ref>
|-
!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.<ref>{{Cite web|title=2018 AP Exam Dates {{!}} The Princeton Review|url=https://www.princetonreview.com/college/ap-exam-schedule|access-date=2018-08-20|website=www.princetonreview.com|language=en}}</ref>

===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.<ref>{{cite web|title=The Exam|url=http://collegeboard.com/student/testing/ap/calculus_ab/exam.html?calcab|url-status=dead|archive-url=https://web.archive.org/web/20081219070621/http://www.collegeboard.com/student/testing/ap/calculus_ab/exam.html?calcab|archive-date=2008-12-19|access-date=2008-11-29|work=Calculus AB|publisher=College Board}}</ref>

{| class="wikitable" border="1"
|-
!
! 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 [[Calculator|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.<ref>{{cite web|year=2005|title=2006, 2007 AP Calculus Course Description|url=http://www.collegeboard.com/prod_downloads/ap/students/calculus/ap-cd-calc-0607.pdf|url-status=dead|archive-url=https://web.archive.org/web/20080720151001/http://www.collegeboard.com/prod_downloads/ap/students/calculus/ap-cd-calc-0607.pdf|archive-date=2008-07-20|access-date=2008-11-29|publisher=College Board}}</ref>

The free response section is hand-graded by hundreds of AP teachers and professors each June.<ref>{{cite web|title=AP: The Grade-Setting Process|url=http://www.collegeboard.com/student/testing/ap/exgrd_set.html|url-status=dead|archive-url=https://web.archive.org/web/20081218181203/http://www.collegeboard.com/student/testing/ap/exgrd_set.html|archive-date=2008-12-18|access-date=2008-11-29|publisher=College Board}}</ref> 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.

== History ==
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.<ref name=":2">{{Cite web |last=Bressoud |first=David |date=July 1, 2022 |title=Thoughts on Advanced Placement Precalculus |url=https://www.mathvalues.org/masterblog/thoughts-on-advanced-placement-precalculus |access-date=September 13, 2022 |website=MAA Blog}}</ref> By the mid-2020s, AP Calculus AB is one of the top ten most popular AP exams.<ref>{{Cite news |last=Najarro |first=Ileana |date=February 28, 2025 |title=The 10 Most-Requested AP Exams of 2024 |url=https://www.edweek.org/teaching-learning/the-10-most-requested-ap-exams-of-2024/2025/02 |archive-url=https://archive.today/20250607171921/https://www.edweek.org/teaching-learning/the-10-most-requested-ap-exams-of-2024/2025/02 |archive-date=June 7, 2025 |access-date=June 7, 2025 |work=Education Week}}</ref>

In the early 21st century, there has been a demand for the creation of AP Multivariable Calculus and indeed, a number of American high schools have begun to offer this class, giving colleges trouble in placing incoming students.<ref name=":2" />

== See also ==
{{Portal|Education|Mathematics|United States}}
* [[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)

==References==
{{reflist}}

==External links==
*AP Calculus AB
**[https://web.archive.org/web/20081219070626/http://www.collegeboard.com/student/testing/ap/calculus_ab/topic.html?calcab College Board description of the AP Calculus AB course content]
**[https://web.archive.org/web/20081219070621/http://www.collegeboard.com/student/testing/ap/calculus_ab/exam.html?calcab College Board description of the AP Calculus AB examination]
*AP Calculus BC
**[https://web.archive.org/web/20081219081215/http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc College Board description of the AP Calculus BC course content]
**[https://web.archive.org/web/20130401114413/http://www.collegeboard.com/student/testing/ap/calculus_bc/exam.html?calcbc College Board description of the AP Calculus BC examination]

==Further reading==
*{{Cite book |last=Nahin |first=Paul |title=Inside Interesting Integrals |publisher=Springer |year=2014 |isbn=9781493912766}}

{{Glossaries of science and engineering}}

[[Category:AP courses in mathematics]]
[[Category:Calculus]]
[[Category:Advanced Placement]]

[[zh:大学先修课程#科目]]