Part of the Advanced Placement Program. Meant to simulate a first-year introductory college calculus course. It comes in two flavors: Calculus AB, and Calculus BC. Here are the curriculum outlines, taken from the College Board (www.collegeboard.org) website. See the site for a more detailed curriculum.

Calculus BC
(AB is a subset of BC. BC-only topics marked with *)

  1. Functions, Graphs, and Limits
  2. Derivatives
    • Concept of the derivative.
    • Derivative at a point.
    • Derivative as a function.
    • Second derivatives.
    • Applications of derivatives.
    • Computation of derivatives.
  3. Integrals
  4. * Polynomial Approximations and Series
The (AB) test itself, given at the end of the year, consists of the following:

Section I :
Multiple-choice questions on a wide variety of topics. There are two parts, one 50 minutes, one 55. The first, with 28 questions, is without calculator, and the second, with 17 questions, is with a calculator.

As a correction for guessing, one-fourth of the number of questions answered incorrectly in this section will be subtracted from the number answered correctly. This basically means that you are better off guessing if you can eliminate 2 or more answers.

Like all other AP exams I am familiar with, the questions are always of the same types: In the multiple choice section, there are about 15 types of questions, and each one has between 1 and 3 different representative questions.

These include questions like

  • Find the limit of F(x) at the point A on the pictured graph above.
  • Find the derivative of the composite of F(g(x)).
  • A problem about the definition of the derivative, such as a problem where you are given the unsimplified definition of the derivative of x^2 at 1, and told to find the limit, so that you either recognize the problem, or spend 25 minutes working out the limit.
  • Find the maximum of the function f(x).
  • What is the area under f(x) between 2 and 7.
  • And so on, ad nauseum

Section II:
6 free-response questions that require students to demonstrate their ability to solve problems involving a more complicated, multi-step calculations. The first three questions are with calculator, and the second three are without. Each section is 45 minutes, and one is allowed to work on the first section, which was done with calculators, during the second 45 minutes as well, however, a calculator may no longer be used.

The answers are written in booklets, which are provided to the students. All answers are graded based on all work shown and correctness of answer. Partial credit is possible.

Within each section, all questions are given equal weight. As the examinations are designed for full coverage of the subject matter, it is not expected that all students will be able to answer all the questions.

In the "essay" section, there are also some standard types, such as

  • A position, velocity, acceleration problem.
  • Finding the area under the curve, then the volume of the rotational solid formed by the following function, or between the following two functions.
  • A maximization problem for various shapes, such as finding the best type of box that can be created, given certain constraints.
  • A minimization problem with functions.
  • A related rates problem.
  • A question about whether a function is decreasing, increasing, concave up or down, usually based on a picture of a graph of the derivative of the function. A large percentage of students, even after told that the graph is of the derivative, then go on and answer problems as if it were a picture of the function.
  • There is usually at least one problem on the test where there is a misleading graph, for instance, where the intersection of two functions, which are given, looks like it is at (1,1), but is in reality at (1,cos(e))

The various types are subject to a significant amount of speculation among teachers and students about a particular year's essay test makeup, since there are only six questions, and there are clearly going to be some catergories that are left out, and others that will have double representation.

Teachers frequently speculate on whether a problem type will occur based on the fact that there has been a trend towards emphasizing that problem type in recent years, or because that type of problem has not shown up in a number of years. Many teachers, based on this and national discussion boards, know almost exactly what will be on a given year's test, which is a huge boon to that teacher's students. The test itself is graded on a scale from 1-5, with most colleges giving credit for a 4 or 5, and possibly also a 3. A 5 requires something like 65% of the possible points on the test, and a 3 requires about 40%. This is variable from year to year.

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