AP Calculus

AP Calculus, or Advanced Placement Calculus, refers to the two Advanced Placement Calculus courses run by the College Board. The two courses are AP Calculus AB and AP Calculus BC. AP Calculus AB is supposed to be roughly equal to the first semester and a half of a typical year-long introductory, single-variable college calculus course, while AP Calculus BC is allegedly equal to the full year. Students may take the AP Calculus AB or BC exam administered every year in May for potential college credit. Like all other AP courses, students need not to actually take the class; they may take just the exam for possible college credit.

Course Content

The AP Calculus AB covers the following topics:

• Limits
  • Rational functions
  • Trigonometric, logarithmic, and exponential limits
  • Properties of limits
  • Limits at infinity
  • L'Hôpital's rule (will be added to AB in 2017)
• Continuity
  • Showing a function is continuous at a point
• Differentiation
  • Definition of derivative
  • Linear differentiation rules
  • Product, quotient, and chain rule
  • Trigonometric, exponential, and logarithmic functions
  • Logarithmic differentiation
  • Second and higher order differentiation
• Applications of the derivative
  • Mean Value Theorem, Intermediate Value Theorem, and Extreme Value Theorem (MVT, IVT, EVT)
  • Optimization and Related Rates
  • Linear approximation
  • Concavity
  • Relationship to position, velocity, and acceleration
• Integration
  • Properties of integration
  • Fundamental Theorem of Calculus
  • Indefinite Integrals
    • substitution (reverse chain rule)
    • Partial fraction decomposition
    • Integration by parts
• First order differential equations
  • Modeling exponential growth/decay with differential equations
  • Separable differential equations
  • Euler's method
• Applications of the definite integral
  • Net change
  • Relationship to position, velocity, and acceleration
  • Areas in the plane
  • Volumes of cross sections
  • Solids of revolution

AP Calculus BC covers everything found in AB, plus the following topics:

• Limits
  • Limits of sequences/series at infinity
  • L'Hôpital's rule (will also be covered in AB in the 2017 exam)
• Integration
  • Improper integrals
• Integration applications
  • Arc lengths of polar and "ordinary" functions
• Sequences
  • Arithmetic sequences defined recursively and explicitly
  • Geometric sequences defined recursively and explicitly
  • Oscillating sequences
• Series
  • Finite arithmetic and geometric series
  • Infinite and power series
  • Maclaurin and Taylor series
• Parameterization
• Polar coordinates
• Vectors in the 2-dimensional place

Class format

This course is taught like other math courses in school: a formula is presented, and students are to repeatedly apply them to routine exercises. However, not all exercises have an obvious approach and does demand a certain degree of creativity and problem solving skills that is not seen in previous math courses. As opposed to other math classes at school, little time is provided to review past content. Thus students planning to take either of the classes (especially BC) should have a firm understanding of precalculus concepts.

Prerequisites

The prerequisite for the courses varies depending on the school. Typically for BC one must successfully complete honors precalculus, while for AB they may complete either on-level or honors precalculus.

Difficulty

The difficulty of this course widely varies, depending heavily on the level of understanding the students have. Those who do not understand trigonometry, geometry, and/or algebra very well often have a tough time, especially since each topic in calculus frequently depends on past content, including previous courses. On the other hand, those that do understand the prerequisites (especially geometry) tend to describe the class to be very easy. Furthermore, those with past elementary-level problem solving skills (e.g. MathCounts and AMC 8) almost always find the class extremely easy.