Introductory Mathematical Analysis PDF

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Book Title:

Introductory Mathematical Analysis

Author:

Said Taan EL Hajjar

Publisher:

Xlibris Corporation, 2011 1-888-795-4274 www.Xlibris.com : [email protected] softcover:

ISBN:

EBook:

978-1-4628-8789-7 978-1-4628-8790-3

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 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Minimum and Maximum values……………………………………………………..181 Increasing and decreasing functions…………………………………………………182 Critical number ……………………………………………………………………...186 Mean value theorem…………………………………………………………………187 Concavity…………………………………………………………………………….190 Variation and graph of polynomial functions………………………………………..196 Applications problems on maximum and minimum values…………………………200 Finance…………………………………………...…………………………………..202

Chapter 7 Integration 7.1 Antiderivative (primitive) of a function…………………………………………….222 7.2 Indefinite integral……………………………………………………………………224

7.3 7.4 7.5 7.6 7.7

Definite integral……………………………………………………………………...227 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...