## Description

Table of Content

1 The laws of algebra

2 The Cartesian Plane

3 Trigonometry

4 Euclidean Geometry

5 Spherical Trigonometry

6 Solving Equations and Factorizing Polynomials

7 Functions

8 Limits

9 Algebra

10 Differential Calculus

Preface

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!)

It is the author’s opinion that mathematics can be appreciated and enjoyed more when

it is presented as a story. 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 the exercises as an essential part

of working through the book. The answers for many of the exercises can be found at the

author’s website http://www.tamug.edu/gacd/fs/math02.html

The range of topics (geometry, trigonometry, algebra and calculus) in this book is another

unique aspect of the book that educators should find refreshing. What’s more, there

are many attempts in the book to demonstrate the interplay and interconnectedness of

these topics. This is a most 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 Chapter 1, Chapter 2,

Chapter 6 and Chapter 9. They can be read independently of the other chapters. Chapter 3

(trigonometry) and Chapter 5 (spherical trigonometry) can also be read independently (after

the first two chapters have been read, if needed). Chapter 4 (Euclidean Geometry) does not

depend on any of the other chapters except for some symbolism from Chapter 1. Chapter 7,

which is an introduction to functions, can be regarded as the beginning of calculus because

the operations of calculus are applied to functions. The concepts and methods relating

to the calculation of limits, which underlie the operations of calculus, are introduced in

Chapter 8. Chapter 10 is an introduction to differential calculus. The definition of the

derivative, the rules for computing derivatives and the formulas for derivatives of all the

standard types of function (i.e. polynomial, rational, trigonometric, root, absolute value,

exponential and logarithmic functions) are introduced in this chapter.

Chapter 9, the final chapter on algebra, is placed after Chapter 7 and Chapter 8 because

the material in these two chapters can be learned without a high level of algebraic skill. 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 Chapter 7, Chapter 8, Chapter

9 and Chapter 10. It should also be mentioned that Chapter 9 includes an introduction

to partial fractions, a topic that is usually not taught in a course in algebra.

In the following two paragraphs, some comments are made regarding the presentation

of the material in Chapter 8 and Chapter 10.

It is usual, in most calculus textbooks, for the rules of limits to be taken as the starting

point for the evaluation of limits. In Chapter 8, 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

of limits are introduced, instead, at the end of the chapter, where they are used to evaluate

certain limits involving trigonometric functions.

In Chapter 10, 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 best approximating line to the graph near the origin.

Then, in the general case, the derivative can be defined at any point in the domain of a

function by means of an appropriate horizontal and vertical shift of the function so that

the derivative can be computed using the formula for the derivative in the special case

(at the origin). With this approach to the definition of the derivative, it is possible for

students to focus on the special case where the formula for the derivative is the simplest

possible. Furthermore, the concept of a tangent line as a “best approximating line” is a

much more flexible concept than the concept of the tangent line as a limit of secant lines,

as it is sometimes defined. Consequently, students should be able to identify a tangent

line more easily in certain situations. For example, they shouldn’t have too much difficulty

identifying the x-axis as the tangent line to the graph of x = y^3 at the origin. The definition

of the derivative in the special case (of a graph passing through the origin) also makes it

possible to derive the formula for the chain rule in the same special case, in which it is much

easier to understand why the slope of the tangent line (through the origin) to the graph of

a composition of functions is the product of the slopes of the tangent lines (through the

origin) to the graphs of each of the functions.

One of the innovative aspects of this book is the introduction of vectors at an early

stage (in Chapter 2). The concept of a vector is very natural and vectors have very practical

applications; however, the notation and terminology relating to vectors may be confusing

at first. The hope is that a gradual introduction to vectors will give students more time to

become comfortable with vectors. (Vectors are typically introduced for the first time in a

course in vector calculus, causing great alarm to the students!)

Another surprise in this book is the inclusion of Chapter 5 on spherical trigonometry.

Much of the material in this chapter was obtained from the book “Plane and Spherical

Trigonometry’’, by K. Nielsen and J. Vanlonkhuyzen, first published in 1944, which is one of

a few texts available on this topic. The purpose of including a chapter on spherical trigonometry

in this book is to give students the opportunity to acquire a deeper understanding of

trigonometry and also to give students some exposure to non–Euclidean geometry. Chapter

5 also has a section on vectors in three dimensions, including the definition of the cross

product of vectors, which students normally do not see until they take a course in vector

calculus. Spherical trigonometry is also interesting from a historical point of view: many

of the formulas relating to spherical trigonometry were discovered by Arabic and Iranian

mathematicians from the ninth to the thirteenth centuries.

A few comments regarding format, notation and mathematical language are in order:

new terminology is presented for the first time in italics, the most important definitions are

presented in highlighted definition boxes, statements that make important clarifications

are presented in highlighted remark boxes, and theorems, corollaries and lemmas are presented

in highlighted boxes. Examples are also presented in highlighted boxes in order

to distinguish them from the main text. A theorem is a general mathematical statement

which is proved on the basis of known mathematical truths. A corollary is a consequence

or a special case of a theorem that is important enough to be stated separately from the

theorem. Lemmas are statements that can be used as stepping stones toward the proofs of

more general statements (theorems). Throughout this book, the bi-conditional phrase “if,

and only if” is used when two statements are implied by each other.

This book can serve as a textbook, a reference book or a book that can be read for fun!

Please tell your friends about it, if you like it!

The author should be contacted regarding any errors in the book so that corrections can

be made for future editions. Suggestions for future editions will also be welcome.