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I see a lot of places where "Calculus 1" is referred to as "Introduction to Calculus", or "Single-variable Calculus." "Calculus 3" is referred to as "Multiple-variable calculus."

Is there an alternate name for "Calculus 2"? Am I mistaken in thinking single variable calculus is Calculus 1?

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In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and exponential functions, and an introduction to anti-differentiation. "Calculus 2" typically starts with FTC, and works through techniques of integration and typically includes sequences, series, and Taylor series.

If I see a course titled "Introduction to Calculus," my first instinct is that it will be a course that rushes through limits, differentiation, and integration all in one semester with reduced attention to theory, and probably does not include trigonometric functions. Such a course would not be part of the calculus sequence taken by students majoring in STEM fields.

Of course there is variation on these observations. One source of confusion in US universities can result from institutions with an academic calendar based on a quarter system, as opposed to a semester system.

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  • $\begingroup$ Does the Calculus clep fully cover what is generally considered to be calculus 1? $\endgroup$ – Burt Aug 9 at 3:01
  • $\begingroup$ @burt--yes. CLEP calculus has good alignment with "calculus 1." $\endgroup$ – user52817 Aug 9 at 14:10
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The course titles "Calculus 1", "Calculus 2", etc. are not meaningful terms outside of the specific institutions where there are courses with these titles. These are names of classes, and not some internationally decided-upon list of topics or curriculum. The actual content of a class called "Calculus 1" might vary quite a lot from one institution to another, thus the best way to decide on a better course title would be to read the course catalog for your institution or track down a syllabus and determine what is actually taught.

A few examples from my own experience:

UNR

At the University of Nevada Reno (where I did my undergraduate and masters work), there is a three semester sequence of courses taught. The courses are titled Calculus I, II, and III. From the course catalog:

Calculus I (Math 181) Fundamental concepts of analytic geometry and calculus; functions, graphs, limits, derivatives and integrals.

Calculus II (Math 182) Methods of integration. Sequences and series, power series.

Calculus III (Math 283) Continuation of MATH 182 ; infinite series, three-dimensional calculus.

Given these course descriptions, reasonable titles for these might be:

  • Calculus I: Introduction to Calculus in One Variable
  • Calculus II: Techniques of Integrations; Sequences and Series
  • Calculus III: Three Dimensional Calculus

AP Calculus

In the US, high school students are often given the opportunity to to AP (or "Advanced Placement") courses, which prepare students for exams which may be in place of college courses. There are two AP calculus exams:

Calculus AB The material includes the study and application of differentiation and integration, and graphical analysis including limits, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.

  • Analysis of graphs (predicting and explaining behavior)
  • Limits of functions (one and two sided)
  • Asymptotic and unbounded behavior
  • Continuity
  • Derivatives
    • Concept
    • At a point
    • As a function
    • Applications
    • Higher Order derivatives
    • Techniques
  • Integrals
    • Interpretations
    • Properties
    • Applications
    • Techniques
    • Numerical approximations
  • Fundamental theorem of calculus
  • Antidifferentiation
  • L'Hôpital's rule, starting in the 2016-17 school year

Calculus BC Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics...Students who take an AP Calculus course should do so with the intention of placing out of a comparable college calculus course.[5]

AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as the following:

  • Convergence tests for series
  • Taylor series
  • The use of parametric equations
  • Polar functions (including arc length in polar coordinates)
  • Calculating curve length in parametric and function equations
  • Integration by parts
  • Improper integrals
  • Differential equations for logistic growth
  • Using partial fractions to integrate rational functions

Calculus AB is typically used in lieu of the first semester of college calculus in the US, i.e. it is equivalent to Calculus 1. The curriculum here is basically the same as the first semester of calculus at UNR, though it is usually taught with any eye towards computation, rather than theory. Calculus AB might reasonably be given the same title as UNR's Calculus I.

Calculus BC covers all of the same material, and also includes techniques of integration. The curriculum is approximately equivalent to about a year of UNR's calculus, so this course is, perhaps, the equivalent of Calculus I and II.

UCR

At my current institution (University of California Riverside), we are on a quarter system. Calculus is taught over five quarters:

First Year Calculus (Math 9A) Introduction to the differential calculus of functions of one variable.

First Year Calculus (Math 9B) Introduction to the integral calculus of functions of one variable.

First Year Calculus (Math 9C) Further topics from integral calculus, improper integrals, infinite series, Taylor’s series, and Taylor’s theorem.

Calculus of Several Variables (Math 10A) Topics include Euclidean geometry, matrices and linear functions, determinants, partial derivatives, directional derivatives, Jacobians, gradients, chain rule, and Taylor’s theorem for several variables.

Calculus of Several Variables (Math 10B) Covers vectors; differential calculus, including implicit differentiation and extreme values; multiple integration; line integrals; vector field theory; and theorems of Gauss, Green, and Stokes.

Here, there are really only two introductory calculus classes: the first year course, and the multivariable course. Hence Calculus 1 and 2 are, respectively, "Single Variable Calculus" and "Multivariable Calculus".

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  • Calculus 1 is Differential Calculus. You start off by learning how to find limits of Algebraic functions, then you learn how to derive every function you learned in High School Algebra.
  • Calculus 2 is Integral Calculus. You learn how to find the area under a curve and between two curves, which are solved using integrals. You will also learn the various techniques to solving integrals. Calculus 2 also covers sequences and series, as well as polar coordinates.
  • Calculus 3 is Multivariable Calculus. I never took this course, but I have heard that it involves vectors and three-dimensional space.
  • Calculus 4 is usually known as Differential Equations. This part of Calculus is a broad field with many classes specialized in Differential Equations. It is common for Engineering and Applied Math majors to take these classes.
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    $\begingroup$ The description of Calculus 1 and 2 should be clarified. Calculus 1 in my experience usually has an introduction to the Riemann integral and the Fundamental Theorem of calculus at the end (so students who take that course see the link between differentiation and integration) while Calculus 2 starts off with methods of integration that were not covered at the end of Calculus 1. $\endgroup$ – KCd Aug 15 at 18:24
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When I was an undergrad, I feel like our Calc 2 was “Multivariable Calculus”, and Calc 3 was “Vector Calculus”.

In Calc 1, we covered limits and the limit definitions for differentiation and integration, and the fundamental theorem of calculus.

In Calc 2, we covered multivariable calculus, such as implicit differentiation, and integrals in $\mathbb{R}^2$ and $\mathbb{R}^3$, to include change of variables with the Jacobian.

In Calc 3, we learned about divergence, gradients, and curl, plus line/surface/volume integrals, and covered Stokes’ theorem and Green’s theorem.

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If Calculus 2 means second-semester calculus to you, then call it second-semester calculus.

If Calculus 2 means multivariable calculus to you, then call it multivariable calculus.

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  • $\begingroup$ If I knew, then I would not be asking the question. I wanted what the general terminology meant. $\endgroup$ – Burt Aug 16 at 15:22
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Here in Chile, at UTFSM in the '70 we had:

  • Cálculo I: Sequences, limits, derivatives (the whole $\epsilon - \delta$ dance)
  • Cálculo II: Integrals (Riemann), integration techniques. Taylor series.
  • Cálculo III: Multiple variables. Taylor theorem, line integrals, Green's theorem, vector calculus (rudiments)

We also had a first semester class called Álgebra, which was more some of analytic geometry and a smattering of trigonometry.

Today's curriculum is different (the topics stretch out over four courses now), but the first ones are more or less the same.

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