Book Title:
Introductory Mathematical Analysis
Author:
Said Taan EL Hajjar
Publisher:
Xlibris Corporation, 2011
1-888-795-4274
www.Xlibris.com : Orders@Xlibris.com
ISBN:
softcover:
978-1-4628-8789-7
EBook:
978-1-4628-8790-3
Overview: Introductory Mathematical Analysis includes topics from differential and integral
calculus that are of interest to students of business, economics, finance and the social sciences. It
begins with non-calculus topics such as equations, inequalities, functions, and mathematics of
finance. This book contains the theoretical development of the real number system, the continuity,
the differentiability, the integration of functions, and the convergence of sequences and series of
real numbers. It also includes the development of sequences and series of functions and an analysis
of the properties a limit function may inherit from its approximants. It is designed for students who
have an intuitive understanding and basic competency in the standard procedures of the calculus.
Some proofs are sufficiently described but are not overdone. Our guiding philosophy led us to
build on this foundation in such a way that pupils achieve the elementary results and acquire
fundamental skills in higher business and higher calculus. Partially fulfills Core Mathematics
requirement.
Contents
Preface…………………………………………………………………………………………2
Introduction …………………………………………………………………………………..3
Chapter 1 Equations and Inequalities
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Evaluating an expression ……………………………………………………………7
Linear equations……………………………………………………………………….8
Rational equations……………………………………………………………………10
Simple irrational equations…………………………………………………………..12
Absolute value equations…………………………………………………………….13
Quadratic equations…………………………………………………………………..16
Inequalities…………………………………………………………………………...22
Chapter 2 Sequences
2.1
2.2
2.3
2.4
2.5
Proof by Mathematical Induction …………………………………………………...36
Numerical Sequence…………………………………………………………………37
Arithmetic Sequence ………………………………………………………………...38
Geometric Sequence ………………………………………………………………...41
Recurrent sequence ………………………………………………………………….45
2.6
2.7
Monotonic Sequence…………………………………………………………………47
Bounded Sequence …………………………………………………………………..52
Chapter 3 Functions
3.1
3.2
3.3
3.4
3.5
Remarkable functions ………………………………………………………………..62
Polar coordinates and trigonometric functions….. …………………………………..78
Range and Sketch of functions……………………………………………………….86
Solving logarithm, exponential and trigonometric equations………………………...96
Combination and Inverse of functions……………………………………………….99
Chapter 4 Limits and Continuities
4.1 Limit of a function at a given point….........................................................................123
4.2 Limit of a function at  …………………………………………………………….127
4.3 Operations on the limits……………………………………………………………...132
4.4 Limits and inequalities ………………………………………………………………134
4.5 Continuity of functions ……………………………………………………………….135
Chapter 5 Differentiation
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
What is the Derivative?................................................................................................151
Rules of derivations………………………………………………………................. 156
Differentiability of trigonometric functions ……………...………………………….161
Geometric applications…………………………………………………………….....162
Physical applications ………………...……………………………………………... 165
Derivative of composite functions (Chain Rule)……………………………..............168
Second order differentiation ………………………………………………………....169
Implicit differentiation…………………………….………………………………….171
Chapter 6 Application on Derivatives
6.1 Minimum and Maximum values……………………………………………………..181
6.2 Increasing and decreasing functions…………………………………………………182
6.3 Critical number ……………………………………………………………………...186
6.4 Mean value theorem…………………………………………………………………187
6.5 Concavity…………………………………………………………………………….190
6.6 Variation and graph of polynomial functions………………………………………..196
6.7 Applications problems on maximum and minimum values…………………………200
6.8 Finance…………………………………………...…………………………………..202
Chapter 7 Integration
7.1 Antiderivative (primitive) of a function…………………………………………….222
7.2 Indefinite integral……………………………………………………………………224
7.3 Definite integral……………………………………………………………………...227
7.4
7.5
7.6
7.7
Graphical interpretation of the definite integral……………………………………..229
The fundamental theorem of integral calculus………………………………………231
Area between two curves……………………………………………………………233
Volume of a solid of revolution…………………………………………………..…234
7.8 Finance……………………………………………………………………… 245