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Newton's calculus is just one example of a calculus. A calculus is any set of mechanical rules for manipulating mathematical symbols. Calculation is the process of applying those rules. Cf. the lambda calculus. The branch of mathematics involving limits, functions, derivatives, and integrals is analysis.

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>The branch of mathematics involving limits, functions, derivatives, and integrals is analysis.

Everywhere I've studied, it's been referred to as simply calculus. Wikipedia says the same. I did take a class called analysis of functions in high school as an advanced track pre-calculus, but your definition really does not apply in the States at any rate. Where did you hear Analysis used to specifically apply to what we call calculus in the States?

http://en.wikipedia.org/wiki/Analysis#Mathematics

My usage is that which is universally accepted in any university math department. As a freshman in a math/science/engineering major, you take (or finish after beginning in high school) three semesters of Calculus, always referred to with a capital "C" to mean specifically Newton's calculus. Here you learn to calculate integrals and derivatives so that you can use them as a tool. If you major in math, then as a junior or senior you will take analysis. Here is where you rigorously study integrals and derivatives. An introductory analysis course typically begins by defining Cauchy sequences, and then defining the real numbers as equivalence classes of Cauchy sequences of rationals. Building on this foundation, you then construct the basic theory of continuous functions, derivatives, and Riemann integrals. Finally, you use this theory to prove that the transformation rules of Newton's calculus are sound.

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