AP Calculus
Encyclopedia
Advanced Placement Calculus (also known as AP Calculus, AP Calc AB/BC, or simply AP Calc) is used to indicate one of two distinct Advanced Placement courses and examinations offered by the College Board
, AP Calculus AB and AP Calculus BC.
course taken by high school students around the world, most of whom study at schools with an American curriculum. The course is traditionally taken after precalculus
and is the first calculus course offered at most schools except for the regular calculus class. The Pre-Advanced Placement pathway for math will help prepare students for further Advanced Placement classes and exams.
,
, and graphical analysis including limits
, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.
More specifically, the topics are
for series, Taylor
and/or Maclaurin series, the use of parametric equation
s, polar functions
, including arc length in polar coordinates, calculating curve length in parametric and function (y = f(x)) equations, L'Hôpital's rule
, integration by parts
, improper integral
s, Euler's method, differential equations for logistic growth
, and using partial fraction
s to integrate rational function
s. Students in AP Calculus BC generally receive two semesters of Advanced Placement in mathematics.
intentionally schedules the AP Calculus AB exam at the same time as the AP Calculus BC exam in order to make it impossible for a student to take both tests and receive college credit for both in the same academic year. They do not, however, bar a student from taking the two classes simultaneously; it is usually at the discretion of the individual school to label Calculus AB as a pre-requisite class for Calculus BC. However, the majority of schools only require precalculus as a prerequisite to Calculus BC.
questions and six free response
questions. They are further subdivided as follows:
The two parts of the Multiple-Choice section are timed and taken independently; students may work on the Section II Part A during the time for Section II Part B but are NOT allowed to resume using a calculator. The Free-Response section, however, is one hour-and-a-half administration. New to the exam in 2011, the calculator-required section will only contain 2 questions, while the non-calculator section will contain 4 questions, putting emphasis on the test-taker's knowledge of concepts and theorems. 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, albeit without a calculator during the latter half.
The free-response section is hand-graded by hundreds of educators 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 1-5.
Students generally receive this score report by mail in mid-July of the year they took the test. Alternately, they can receive their scores by phone as early as June 27 for a fee of $8 (although the College Board only officially recognizes July 1 as the first available date to receive grades by phone). 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.
The grade distribution for 2010 was:
The grade distribution for the 2010 BC scores was:
The mean AB subscore was a 3.96; the grade distribution was:
College Board
The College Board is a membership association in the United States that was formed in 1900 as the College Entrance Examination Board . It is composed of more than 5,900 schools, colleges, universities and other educational organizations. It sells standardized tests used by academically oriented...
, AP Calculus AB and AP Calculus BC.
AP Calculus AB
AP Calculus AB is an Advanced Placement calculusCalculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
course taken by high school students around the world, most of whom study at schools with an American curriculum. The course is traditionally taken after precalculus
Precalculus
In American mathematics education, precalculus , an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is actually two separate courses: Algebra and Trigonometry...
and is the first calculus course offered at most schools except for the regular calculus class. The Pre-Advanced Placement pathway for math will help prepare students for further Advanced Placement classes and exams.
Purpose
According to the College BoardCollege Board
The College Board is a membership association in the United States that was formed in 1900 as the College Entrance Examination Board . It is composed of more than 5,900 schools, colleges, universities and other educational organizations. It sells standardized tests used by academically oriented...
,
Course content
The material includes the study and application of differentiation and integrationIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, and graphical analysis including limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.
More specifically, the topics are
- Analysis of graphs (predicting and explaining behavior)
- Limits of functionsLimit of a functionIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
(one and two sided) - Asymptotic and unbounded behavior
- ContinuityContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
- DerivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s- Concept
- At a point
- As a function
- Applications
- Second derivatives
- Integrals
- Interpretations
- Properties
- Applications
- Techniques
- Numerical approximations
- Fundamental theorem of calculusFundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
- AntidifferentationAntiderivativeIn calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
Purpose
According to the College Board,Course content
AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as convergence testsConvergence tests
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series.- List of tests :...
for series, Taylor
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
and/or Maclaurin series, the use of parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
s, polar functions
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
, including arc length in polar coordinates, calculating curve length in parametric and function (y = f(x)) equations, L'Hôpital's rule
L'Hôpital's rule
In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
, integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
, improper integral
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
s, Euler's method, differential equations for logistic growth
Logistic function
A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...
, and using partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
s to integrate rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s. Students in AP Calculus BC generally receive two semesters of Advanced Placement in mathematics.
Growth
Between 1990 and 2004, the number of students taking the AP Calculus exams has increased more than threefold. The exams are now taken by more than 250,000 students each year. The College BoardCollege Board
The College Board is a membership association in the United States that was formed in 1900 as the College Entrance Examination Board . It is composed of more than 5,900 schools, colleges, universities and other educational organizations. It sells standardized tests used by academically oriented...
intentionally schedules the AP Calculus AB exam at the same time as the AP Calculus BC exam in order to make it impossible for a student to take both tests and receive college credit for both in the same academic year. They do not, however, bar a student from taking the two classes simultaneously; it is usually at the discretion of the individual school to label Calculus AB as a pre-requisite class for Calculus BC. However, the majority of schools only require precalculus as a prerequisite to Calculus BC.
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 choiceMultiple choice
Multiple choice is a form of assessment in which respondents are asked to select the best possible answer out of the choices from a list. The multiple choice format is most frequently used in educational testing, in market research, and in elections-- when a person chooses between multiple...
questions and six free response
Free response
Free response is a type of question used in tests in education, workplace, and government. Most free response questions ask or require the test-taker to state a belief, opinion, or write a short essay and support it with facts, examples, or other evidence...
questions. They are further subdivided as follows:
| 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 | 28 | 17 | 2 | 4 |
| Time Allowed | 55 minutes | 50 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 may work on the Section II Part A during the time for Section II Part B but are NOT allowed to resume using a calculator. The Free-Response section, however, is one hour-and-a-half administration. New to the exam in 2011, the calculator-required section will only contain 2 questions, while the non-calculator section will contain 4 questions, putting emphasis on the test-taker's knowledge of concepts and theorems. 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, albeit without a calculator during the latter half.
Scoring
The multiple-choice section is scored by computer, with a correct answer receiving 1 point, a blank answer receiving 0 points and an incorrect answer costing no points as a new change done by AP Central. This total is multiplied by 1.2 to calculate the adjusted multiple-choice score.The free-response section is hand-graded by hundreds of educators 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 1-5.
Students generally receive this score report by mail in mid-July of the year they took the test. Alternately, they can receive their scores by phone as early as June 27 for a fee of $8 (although the College Board only officially recognizes July 1 as the first available date to receive grades by phone). 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.
Grade distributions for AP Calculus AB
In the 2010 administration, 245,867 students took the exam. The mean score was a 2.81.The grade distribution for 2010 was:
| Score | Percent |
|---|---|
| 5 | 21.2% |
| 4 | 16.4% |
| 3 | 18.0% |
| 2 | 11.2% |
| 1 | 33.1% |
Grade distributions for AP Calculus BC
In the 2010 administration, 78,998 students took the exam. The mean score was a 3.86.The grade distribution for the 2010 BC scores was:
| Score | Percent |
|---|---|
| 5 | 49.4% |
| 4 | 15.4% |
| 3 | 18.0% |
| 2 | 5.8% |
| 1 | 11.4% |
The mean AB subscore was a 3.96; the grade distribution was:
| Score | Percent |
|---|---|
| 5 | 51.0% |
| 4 | 19.4% |
| 3 | 14.2% |
| 2 | 5.7% |
| 1 | 9.7% |