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What is known to us simply as "calculus" is the "infinitesimal calculus", known
to mathematicians as "analysis", a branch of mathematics hinted at by ancient
and medieval figures like Eudoxus, Archimedes and Alhazen, synthesized by
Newton and Liebniz, and put onto rigorous and recognizable footing by the likes
of Euler, Cauchy, and Weirstrass. It concerns itself with four related

concepts:

(1) The infinite sequence, including the infinite sum.
(2) The limit
(3) The derivative, a linear approximation to a mathematical function
(4) The integral, or antiderivative.

A very rich theory has been built up around both fundamental and applied
questions in calculus, and concepts applicable to simple functions mapping single variables
from the real or natural numbers to the real or natural numbers have been
generalized to higher dimensions and abstract spaces whose utility is not
readily apparent to even a mathematically literate lay reader. The basics,
however, are understandable to anyone with a grasp of basic arithmetic/algebra.

(0) What is a function?

A discussion of the basics of the calculus would be dead in the water without an
understanding of what is meant by a "function" in mathematics.

A function is a relationship between two sets X and Y associating each element
of X with exactly one element of Y. The sets need not be different: the
function y=x^3 is a mapping from the whole real line to the whole real line.

An older usage allows functions to be multivalued; this faded nearly a century
ago but can still be found in some old books.

(1) The infinite sequence

Sequences are functions from the set of natural numbers (sometimes including
zero) to some other set, which we'll take to be the real numbers for
simplicity's sake.

For example, F(n)=(g^n-(1-g)^n)/sqrt(5), where g is the golden ratio, is a
sequence giving the Fibonacci numbers familiar from recreational mathematics.

We sometimes are concerned with the behavior of a sequence as n tends to
infinity, the limiting behavior or limit of the sequence. A sequence tends to
infinity if there exists a number N for any M, no matter how large, F(N)>M. A
sequence is said to have a limit L if for every (real number) epsilon, no
matter how small, there exists an N such that n>N implies |F(n)-L|< epsilon.

Some sequences of interest are sums, and we are often faced with sums of
infinite numbers of terms. Let's call the sum of the first n elements of a
sequence F(n) "sum(F(n))"=F(1)+F(2)+...+F(n) . The sum of an infinite number of
terms is just the limit of sum(F(n)),

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