Philip Brown - Foundations of Mathematics: Algebra, Geometry, Trigonometry and Calculus

FOUNDATIONS
OF
MATHEMATICS

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FOUNDATIONS
OF
MATHEMATICS

Algebra, Geometry, Trigonometry, Calculus

Philip Brown

Texas A&M University at Galveston

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Philip Brown. Foundations of Mathematics: Algebra, Geometry, Trigonometry, Calculus
ISBN: 978-1-942270-75-1

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Library of Congress Control Number: 2015957711
161718321 This book is printed on acid-free paper.

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This book is intended primarily for university students. In particular, this book can be used as a textbook or an additional reference book by university students attending a course in algebra, trigonometry, geometry, or calculus. As a calculus textbook, this book is unique in that it contains all the mathematics (and more) that students will need to know in order to be successful in a calculus course. For this reason, the book will be invaluable for students who may need to fill in some gaps in their mathematical background, or review certain topics, while attending a university level calculus course. (Calculus professors will normally not have the time to do this!)

Mathematics can be appreciated and enjoyed more when it is presented as a development of ideas. Hence, there is a strong emphasis in this book on the unfolding and development of the concepts, and there are comments throughout the book to help the reader trace the historical development of mathematics. Of course, students also need to develop their skills. For this reason, there are many exercises included at the ends of the chapters, and students are encouraged to do all of them as an essential part of working through the book. The range of topics in this book, and examples that demonstrate their interplay and interconnectedness is another unique aspect of the book. This is a rewarding aspect of learning mathematics, which university students are typically not exposed to because of the way current-day university courses are organized.

The four chapters that make up the introduction to algebra are presents many of the theorems that are important in Euclidean Geometry, along with some guides to solving problems in geometry.

Because many students have difficulties with algebra when they enroll in a university calculus course, a first semester course in calculus may well consist of includes an introduction to partial fractions, a topic that is usually not taught to students until they reach their second semester of calculus.

In the following two paragraphs, some comments are made regarding the presentation of the material in .

It is usual, in most calculus textbooks, for the rules for limits to be taken as the starting point for the evaluation of limits. In , the slightly different approach to evaluating limits is to take as a fact the continuity of all of the standard functions and any algebraic combinations and compositions of the standard functions. (This fact can be proved using methods of real analysis, which is too advanced for this book.) This means that a limit to any point in the domain of any of these functions can be evaluated simply by making a substitution into the function (i.e., by an application of the equation of continuity). The rules for limits are introduced, instead, at the end of the chapter, where they are used to evaluate certain limits involving trigonometric functions.

In , the derivative of a function is first defined in the special case that the graph of the function passes through the origin, and the tangent line (through the origin) is defined in a precise way as the