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:
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
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
- At a point
- As a function
- Higher Order derivatives
- Numerical approximations
- Fundamental theorem of calculus
- 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.
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.
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
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
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".