This wiki is associated with Cognito Mentoring, an advising service for learners run by Jonah Sinick and Vipul Naik. The wiki is very much in beta, so you're likely to find many broken links and incomplete pages. Please be patient with us as we continue to improve our offerings.
Please connect with us to offer feedback on the wiki content.
Single-variable calculus
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.
Contents
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.