AP Calculus AB is an advanced placement calculus exam taken by some United States high school students. It comes after Precalculus, which is known as Introduction to Analysis in some places, and is the first calculus course offered at most schools. The Advanced Placement Program, commonly known as Advanced Placement, or AP, is a United States and Canada-based program that offers high school students the opportunity to receive university credit for their work during high school. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics education, pre-calculus, in reality just advanced algebra, is a foundational mathematical discipline. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ...

An AP Calculus AB course is typically equivalent to one semester of college calculus. The material includes Limits, differentiation, integration, and other topics standardly covered in college calculus courses. This course usually is recommended for those students who are thinking about a math-related degree once in college. It differs from AP Calculus BC in that it does not typically cover improper integrals or Taylor series. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... AP Calculus BC includes all of the topics in AP Calculus AB, as well as convergence tests for series, Taylor or Maclaurin series, vector, polar, and parametric functions. ... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... As the degree of the Taylor series rises, it approaches the correct function. ...

Calculus AB topics are those traditionally offered in the first year of calculus in college, and are designed for students who wish to obtain a semester of advanced placement in college. The topics studied include limits, continuity, derivatives and integrals of algebraic and transcendental functions and their applications, and elementary differential equations.

AP Calculus AB Exam

Topics Covered on the AP Calculus AB Exam

I. Functions, Graphs, and Limits

• Analysis of Graphs
  • With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
• Limits of Functions (including one-sided limits)
  • An intuitive understanding of the limiting process.
  • Calculating limits using algebra.
  • Estimating limits from graphs or tables of data.
• Asymptotic and Unbounded Behavior
  • Understanding asymptotes in terms of graphical behavior.
  • Describing asymptotic behavior in terms of limits involving infinity.
  • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
• Continuity as a Property of Functions
  • An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)
  • Understanding continuity in terms of limits.
  • Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

II. Derivatives

• Concept of the Derivative
  • Derivative presented graphically, numerically, and analytically.
  • Derivative interpreted as an instantaneous rate of change.
  • Derivative defined as the limit of the difference quotient.
  • Relationship between differentiability and continuity.
• Derivative at a Point
  • Slope of a curve at a point. Examples are emphasized, including points at which there are
  • Tangent line to a curve at a point and local linear approximation.
  • Instantaneous rate of change as the limit of average rate of change.
  • Approximate rate of change from graphs and tables of values.
• Derivative as a Function
  • Corresponding characteristics of graphs of f and f '.
  • Relationship between the increasing and decreasing behavior of f and the sign of f '.
  • The Mean Value Theorem and its geometric consequences.
  • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
• Second Derivatives
  • Corresponding characteristics of the graphs of f, f ', and f ".
  • Relationship between the concavity of f and the sign of f ".
  • Points of inflection as places where concavity changes.
• Applications of Derivatives
  • Analysis of curves, including the notions of monotonicity and concavity.
  • Optimization, both absolute (global) and relative (local) extrema.
  • Modeling rates of change, including related rates problems.
  • Use of implicit differentiation to find the derivative of an inverse function.
  • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
  • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
• Computation of Derivatives
  • Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
  • Basic rules for the derivative of sums, products, and quotients of functions.
  • Chain rule and implicit differentiation.

III. Integrals

• Interpretations and Properties of Definite Integrals
  • Computation of Riemann sums using left, right, and midpoint evaluation points.
  • Definite integral as a limit of Riemann sums over equal subdivisions.
  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
  • Basic properties of definite integrals. (Examples include additivity and linearity.)
• Applications of Integrals
  • Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.
• Fundamental Theorem of Calculus
  • Use of the Fundamental Theorem to evaluate definite integrals.
  • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
• Techniques of Antidifferentiation
  • Antiderivatives following directly from derivatives of basic functions.
  • Antiderivatives by substitution of variables (including change of limits for definite integrals).
• Applications of Antidifferentiation
  • Finding specific antiderivatives using initial conditions, including applications to motion along a line.
  • Solving separable differential equations and using them in modeling. In particular, studying the equation y' = ky and exponential growth.
• Numerical Approximations to Definite Integrals
  • Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

External links

• College Board description of the course content
• College Board description of the examination