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AP Calculus
AP Calculus
AP Calculus
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Advanced Placement (AP) Calculus (also known as AP Calc, Calc AB , AB Calc or simply AB) 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 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

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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

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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]

Topic outline

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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]

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 or Higher 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 —
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

[edit]
Part of a series of articles about
Calculus
Differential
Definitions
Concepts
Rules and identities
Integral
Definitions
Integration by
Series
Convergence tests
Vector
Theorems
Multivariable
Formalisms
Definitions
Advanced
Specialized
Miscellanea

AP Calculus 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]

Purpose

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According to the College Board,

Calculus 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]

Topic outline

[edit]

AP Calculus includes all of the topics covered in AP Calculus AB, as well as the following:

Score 2017[21] 2018[9] 2019[10] 2020[22] 2021[23] 2022[24] 2023[25] 2024[26] 2025[27]
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 —
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, often self-selecting into the course due to greater intrinsic motivation and prior achievement.[28] The 5-rate is consistently over 40% (much higher than almost all the other AP exams).[29] 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.[30]

AB sub-score distribution

[edit]
Score 2017[21] 2018[9] 2019[31] 2020[32] 2021[33] 2022[34] 2023[35]
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

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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.[36] 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.[37]

Format

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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.[38][39]

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

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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.[40]

The free response section is hand-graded by hundreds of AP teachers and professors each June.[41] 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

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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.[42] 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.[43] 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.[44] 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.[45] By the mid-2020s, AP Calculus AB is one of the top ten most popular AP exams.[46]

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.[42][47]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

AP Calculus

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AP Calculus encompasses two Advanced Placement courses—AP Calculus AB and AP Calculus BC—offered by the to high school students, providing rigorous, college-level instruction in the core principles of . These courses emphasize the study of limits, derivatives, integrals, and their applications to model change in real-world contexts, preparing students for and related fields such as , , and . Successful performance on the associated exams can earn college credit or at participating institutions. AP Calculus AB is equivalent to a first-semester course, covering foundational topics including limits and continuity (10–12% of the ), differentiation and its applications (44–54%), integration and accumulation of change (17–20%), differential equations (6–12%), and applications of integration (10–15%). It requires prerequisites such as four years of secondary , including , , , , and elementary functions. The course develops skills in conceptual understanding, procedural fluency, and problem-solving through multiple representations (graphical, numerical, analytical, and verbal), with an consisting of 45 multiple-choice questions (50% of the score) and 6 free-response questions (50%), lasting 3 hours and 15 minutes. In contrast, AP Calculus BC builds on AB by incorporating all AB topics plus advanced material, equivalent to two semesters of calculus and representing about 50% more content overall. Exclusive BC units include parametric equations, polar coordinates, and vector-valued functions (11–12% of the exam) as well as infinite sequences and series (17–18%), along with deeper explorations of integration techniques like and improper integrals. Like AB, it shares the same prerequisites but assumes greater for its expanded scope. The BC exam follows the same format and duration as AB but weights units differently to reflect the additional topics, fostering advanced reasoning, justification, and communication in . Minor clarifications to the course framework, effective August 2025, include updates to emphasize career preparation alongside college readiness and a typographical correction in the series topic.

Overview

Introduction

AP Calculus is a program developed and administered by the , providing high school students with the opportunity to enroll in college-level courses through two distinct offerings: AP Calculus AB and AP Calculus BC. These courses enable motivated students to engage with rigorous mathematical content typically encountered in introductory university settings, fostering analytical skills applicable to various disciplines. The program's primary objectives include cultivating a deep understanding of differential and calculus equivalent to first-semester or full-year coursework, facilitating the potential for credit or based on exam performance, and equipping students for success in STEM fields such as engineering, physics, and . By emphasizing conceptual comprehension, problem-solving, and real-world applications, AP prepares participants for higher education and professional pursuits that rely on mathematical modeling of change. In 2024, 278,657 students took the AP Calculus AB exam, with 64.4% achieving a score of 3 or higher, while 148,191 students sat for the AP Calculus BC exam, resulting in an 80.9% pass rate at that threshold. In 2025, the 3+ pass rate for AB was 64% and for BC was 81%. AP Calculus AB aligns with a single semester of college-level calculus, covering foundational topics like limits, derivatives, and integrals, whereas AP Calculus BC encompasses the AB curriculum plus advanced subjects such as parametric equations and series, equivalent to a complete academic year.

Prerequisites

Students preparing for AP Calculus AB or BC must have a solid foundation in secondary mathematics equivalent to four years of college-preparatory coursework. This typically includes successful completion of Algebra I and II, Geometry, Trigonometry, and Precalculus (or an integrated equivalent program). These courses ensure proficiency in the analytical skills necessary for calculus concepts. Key prerequisite concepts encompass a range of functions and their properties, including linear, , rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. Students should be adept at graphing these functions, interpreting domain and range, identifying , zeros, intercepts, and such as increasing or decreasing intervals. Additionally, solving equations and inequalities algebraically, along with basic trigonometric identities and the unit circle for sine and cosine values at key angles (e.g., 0, π/6, π/4, π/3, π/2 and multiples), forms essential groundwork. Recommended skills include strong algebraic manipulation, such as simplifying expressions, factoring, and working with rational functions, as well as an intuitive grasp of rates of change through concepts like and average velocity. These build toward the notion of instantaneous rates in limits. To prepare effectively, students should self-assess by reviewing topics and addressing common gaps, such as mastery of the unit circle or trigonometric identities, through practice problems or diagnostic tests provided by educational resources.

AP Calculus AB

Description

AP Calculus AB is an course designed to provide high school students with a rigorous introduction to , equivalent to a first-semester college-level course in scope and depth. Its primary purpose is to develop students' knowledge conceptually, computationally, and creatively, fostering independent thinkers capable of using definitions and theorems to build arguments and justify conclusions. By emphasizing big ideas such as modeling change and approximate change, the course prepares learners for further studies in , , , and related fields, enabling them to earn college credit or advanced placement upon successful exam performance. The course typically spans a full in high school, comprising approximately 140–150 class periods of 45 minutes each on a five-day-per-week schedule. It requires prerequisites such as a strong foundation in secondary , including , , , , and elementary functions, with strongly recommended to ensure understanding of function properties and trigonometry. AP Calculus AB targets motivated, college-bound high school students who aspire to pursue studies in fields requiring calculus. The instructional approach emphasizes conceptual understanding through a multi-representational framework, integrating graphical, numerical, analytical, and verbal methods to express concepts, results, and problems, thereby deepening understanding and promoting clear communication of mathematical reasoning. Minor clarifications to the course framework, effective August 2025, emphasize career preparation alongside college readiness.

Curriculum Topics

The AP Calculus AB curriculum covers foundational topics in differential and integral calculus, organized into eight units with the following approximate exam weightings: Limits and Continuity (10–12%), Differentiation: Definition and Fundamental Properties (10–12%), Differentiation: Composite, Implicit, and Inverse Functions (9–13%), Contextual Applications of Differentiation (10–15%), Analytical Applications of Differentiation (15–18%), Integration and Accumulation of Change (17–20%), Differential Equations (6–12%), and Applications of Integration (10–15%). These units emphasize conceptual understanding of rates of change, accumulation, and function analysis, using multiple representations to model real-world change and prepare students for college-level calculus. Limits and continuity form the starting point, where students explore the concept of limits as values functions approach, including one-sided and infinite limits. The limit definition of the derivative is f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, with continuity requiring lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a). Students estimate limits graphically and numerically, analyze discontinuities, and apply the Intermediate Value Theorem for existence of roots. These tools establish the foundation for differentiation and behavior of functions. Differentiation begins with basic rules: the power rule ddx[xn]=nxn−1\frac{d}{dx} [x^n] = n x^{n-1}, product rule (fg)′=f′g+fg′(fg)' = f'g + fg', quotient rule (fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}, and chain rule for composites (f(g(x)))′=f′(g(x))g′(x)(f(g(x)))' = f'(g(x)) g'(x). Implicit differentiation solves for derivatives in equations like x2+y2=1x^2 + y^2 = 1, yielding dydx=−xy\frac{dy}{dx} = -\frac{x}{y}, while inverse function derivatives use ddx[f−1(x)]=1f′(f−1(x))\frac{d}{dx} [f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}. These enable analysis of rates of change for polynomials, rationals, exponentials, logarithms, and trigonometric functions. Contextual applications interpret derivatives as instantaneous rates, such as from position or in . Related rates problems, like inflating spheres where dVdt=4πr2drdt\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}, use implicit differentiation and . Linearization approximates functions near a point: f(x)≈f(a)+f′(a)(x−a)f(x) \approx f(a) + f'(a)(x - a), supporting line uses for . Analytical applications justify function behavior using : increasing/decreasing intervals via sign charts, concavity with s (f′′>0f'' > 0 for concave up), and inflection points where f′′f'' changes sign. The states f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a} for some c∈(a,b)c \in (a,b), linking average and instantaneous rates. Optimization identifies maxima/minima through critical points (f′=0f' = 0 or undefined) and first/ tests, applied to enclosed areas or resource allocation. resolves indeterminate forms like 00\frac{0}{0} by lim⁡fg=lim⁡f′g′\lim \frac{f}{g} = \lim \frac{f'}{g'}. These confirm extrema, symmetry, and end behavior. Integration reverses differentiation, with antiderivatives satisfying F′=fF' = f. Definite integrals ∫abf(x) dx\int_a^b f(x) \, dx represent net accumulation, approximated by Riemann sums (left, right, midpoint). The connects differentiation and integration: if F′(x)=f(x)F'(x) = f(x), then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a), and ddx∫axf(t) dt=f(x)\frac{d}{dx} \int_a^x f(t) \, dt = f(x). Basic integration rules mirror derivatives, including ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C for n≠−1n \neq -1. Numerical approximations like enhance computation when antiderivatives are unavailable. Differential equations introduce modeling with separable equations, solved by ∫dyg(y)=∫f(x) dx\int \frac{dy}{g(y)} = \int f(x) \, dx. Exponential models dydt=ky\frac{dy}{dt} = ky yield y=y0ekty = y_0 e^{kt}, representing growth/decay. Slope fields visualize solutions graphically, verifying particular solutions against initial conditions. These address or cooling rates. Applications of integration compute accumulated change: areas between curves ∫ab[f(x)−g(x)] dx\int_a^b [f(x) - g(x)] \, dx (for f≥gf \geq g), volumes of solids via disks/washers ∫abπ[R(x)]2 dx\int_a^b \pi [R(x)]^2 \, dx, or average value 1b−a∫abf(x) dx\frac{1}{b-a} \int_a^b f(x) \, dx. These extend to particle motion (displacement as of ) and real-world modeling like work or fluid accumulation.

AP Calculus BC

Description

AP Calculus BC is an course designed to provide high school students with a rigorous introduction to , equivalent to both the first and second semesters of -level in scope and depth. Its primary purpose is to develop students' mathematical knowledge conceptually, computationally, and creatively, fostering independent thinkers capable of using definitions and theorems to build arguments and justify conclusions. By emphasizing big ideas such as modeling change and the analysis of functions, the course prepares learners for comprehensive preparation, enabling them to earn or advanced placement upon successful completion. The course typically spans a full in high school, comprising approximately 180–190 class periods of 45 minutes each on a five-day-per-week schedule. It builds directly on the foundation of AP Calculus AB, incorporating all AB topics while extending them with additional content for greater breadth and depth, and is often taken sequentially or concurrently after AB. This structure allows students to progress from introductory concepts to more advanced applications, with the BC providing a subscore based on AB material to recognize partial achievement. AP Calculus BC targets motivated, college-bound high school students who have completed four years of secondary mathematics, including algebra, geometry, trigonometry, and functions, and who aspire to pursue deeper studies in mathematics, engineering, or related fields. The instructional approach emphasizes theoretical depth through a multi-representational framework, integrating graphical, numerical, analytical, and verbal methods to express concepts, results, and problems, thereby deepening understanding and promoting clear communication of mathematical reasoning.

Curriculum Topics

The AP Calculus BC curriculum encompasses all topics covered in AP Calculus AB, providing a foundation in limits and continuity, differentiation (including definitions, fundamental properties, composite, implicit, and inverse functions), contextual and analytical applications of differentiation, integration and accumulation of change, introductory differential equations, and applications of integration. These shared elements emphasize conceptual understanding of rates of change, accumulation, and function , preparing students for college-level . Building on this base, BC extends into more advanced areas, equivalent to a full two-semester college sequence, with a focus on parametric, polar, and vector representations; infinite series; sophisticated integration techniques; expanded models; and additional applications such as and numerical methods. A key extension involves parametric equations, polar coordinates, and vector-valued functions, where students learn to define and differentiate these forms to model curves and motion. For parametric equations given by x=f(t)x = f(t) and y=g(t)y = g(t), the first is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, and the second follows via rule as d2ydx2=d/dt(dy/dx)dx/dt\frac{d^2 y}{dx^2} = \frac{d/dt (dy/dx)}{dx/dt}. of a parametric curve from t=at = a to t=bt = b is calculated using the ∫ab(dx/dt)2+(dy/dt)2 dt\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt
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