# Introduction

## An Experimental Approach

A traditional approach to Calculus starts with the analytic foundations.
This time tested method introduces the critical concepts of limits,
continuity, and the completeness of the Real numbers as the foundation
for the development of the derivative and integral.
This development culminates in the proofs of the fundamental theorems
of Calculus linking the derivative and the integral.

With the fundamental theorem, each known derivative can be reinterpreted
as an integral. Various techniques of integration are presented and
the student is invited to practice these in the exercises.
The course then moves on. Perhaps to polynomial approximation, or
Calculus in higher dimensions.

Integration remains a challenge.

The experiment is to bring integration early in the development.
This is done by deferring the analytic foundations, and
presenting differentiation and integration, more precisely
antidifferentiation, in formal terms.
This allows the reader to get to integration quickly and to
focus on developing familiarity with the techniques of integration.

The analytic foundations are hinted at as the motivation for the
the formal derivative of a function is based on linear approximations
to the function. Likewise the motivation for the antiderivative of a
function is based on the area under that function.

## Connecting to Tradition

With the formal machiney developed and understood, the analytic
definitions can be presented. The formal rules are proved, and
with that all the techniques are on solid analytical ground.

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