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Single-variable calculus

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Single-variable calculus, also called calculus of one variable or calculus, is a topic within mathematics that has wide applications both within and outside mathematics. It builds on knowledge of algebra, trigonometry, and functions (sometimes collectively called precalculus). The material is typically covered in either late high school or early college. Typical follow-up courses to single-variable calculus are multivariable calculus and linear algebra.

Stages where it's covered

Part of the information in this section is specific to the United States.

Advanced Placement (AP) in school

Advanced Placement Calculus is typically taught in high school, either in senior year (grade 12) or junior year (grade 11). There are two versions:

  • Advanced Placement AB Calculus: This covers the basics of differentiation and integration but excludes Taylor series and excludes some of the trickier examples and methods.
  • Advanced Placement BC Calculus: This covers differentiation, integration, and Taylor series.

Many schools in the United States, particularly in large areas with academically proficient student bodies, offer AP Calculus. However, not all do. Among the schools that offer AP Calculus, not all of them offer BC. Schools in some other countries (such as South Korea), that have a tradition of sending students to the United States, also offer AP Calculus.

International Baccalaureate (IB)

To be filled in later

College calculus

Further information: Lower division undergraduate mathematics, lower division undergraduate mathematics course structure

Most colleges have calculus courses, and many colleges require statements to take at least one quarter or semester of calculus. Colleges have a range of policies regarding whether they accept AP or IB credit in lieu of the calculus courses at the college. At top colleges, calculus courses are generally more demanding than AP Calculus, either proofs-wise or computation-wise or both. For instance, Princeton University's placement guidelines say that an AP BC score of 5 is likely comparable to a C in Math 104, their second semester calculus course targeted at non-math majors.

Components

The following are the major components of single-variable calculus:

Component Part covered in AP AB Calculus Part covered in AP BC Calculus that is not covered in AB Calculus Part covered in a comprehensive college calculus course not covered in BC Calculus
Limits Basic conceptual definition and computational techniques for limits of functions L'Hopital's rule Limits and convergence of sequences, formal definition of limit ()
Differential calculus Basic conceptual definition and computational techniques for limits of functions Parametric differentiation More detailed treatment of piecewise functions, greater focus on conceptual definition
Integral calculus Conceptual definition of integral, basic integration techniques Integration by parts, rational functions with linear factors of denominator, and improper integrals Inverse trigonometric function integration, completing the square, more advanced partial fractions
Differential equations (a very preliminary peek) Concept, separable differential equations, autonomous case Numerical solution methods (Euler's method) First-order linear differential equations, special cases of higher-order linear differential equations
Sequences and series, including Taylor series and power series -- Definition, alternating series, geometric series, comparison tests, integral tests, determining convergence, Taylor series, interval of convergence of power series Rearrangement theorems, rigorous definitions of concepts, more on sequences in relation with series.

Learning calculus

Typical learning trajectories

Further information: High school mathematics course structure, Lower division undergraduate mathematics course structure

We recommend learning calculus in high school, taking Advanced Placement BC Calculus if possible, and then starting with the appropriate calculus placement in college based on placement tests and information about optimal starting points.

Recommendations

See our single-variable calculus learning recommendations for recommendations on learning calculus.