Single Variable Calculus

Volume 1: Single Variable Calculus

Galina Filipuk, Andrzej Kozłowski

• Publisher: De Gruyter
• Year: 2019
• eBook ISBN: 9783110590159
• Edition: 1
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

A computer algebra system such as Mathematica® is able to do much more than just numerics: This text shows how to tackle real mathematical problems from basic analysis. The reader learns how Mathematica® represents domains, qualifiers and limits to implement actual proofs – a requirement to unlock the huge potential of Mathematica® for a variety of applications.

1 Number systems

Usually, courses of analysis begin with number systems. Real analysis begins with the real numbers, which are either introduced axiomatically [12], [1] or constructed from something more “basic”, e. g., natural numbers or integers (which themselves are either introduced axiomatically [14] or defined in terms of sets [15]).

In this chapter we shall explain how Mathematica^® deals with sets, numbers and sequences of numbers and consider many other related topics.

1.1 Sets

One of the first notions one has to learn to study modern mathematics is the notion of a set, since the natural language of modern mathematics is provided by set theory. Mathematica^® uses the so called Wolfram Language, which does not have a built-in function corresponding to the mathematical notion of “set”. It is important to remark that the built-in function “Set” means something completely different. A finite ordered set exists in Mathematica^® as List. One can also study the properties of ordinary finite sets by using the fact that Mathematica^® has a natural way to sort (or order) any list of objects. This is done either by the function Sort or by the function Union:

In[·]:=	Sort[{b, 7, 3, "dog", "cat", a}]
Out[·]:=	{3, 7, cat, dog, a, b}

In the example above the list contains objects of different kinds (symbols, numbers and strings of texts), but Mathematica^® has a canonical way to arrange them. Because of this, any finite set of objects has a canonical representation as an ordered set so we can study sets by means of ...

Table of contents

• Title Page
• Copyright
• Contents
• Preface
• 1 Number systems
• 2 Recursive sequences, discrete dynamical systems and their limits
• 3 Series
• 4 Limits of functions and continuity
• 5 Differentiation
• 6 Sequences and series of functions
• 7 Integration
• Bibliography
• Subject Index