# Mathematics

## Semester One

Calculus

Description :

COURSE GOALS AND COVERAGE

The main objective of Calculus I is for students to learn the basics of the calculus of functions of one variable. They will study transcendental functions, limits, differentiation and an introduction to the Riemann integral, culminating with the Fundamental Theorem of Calculus. They will also apply these ideas to a wide range of problems that include the equations of motion, related rates, curve sketching and optimization. The students should be able to interpret the concepts of Calculus algebraically, graphically and verbally. More generally, the students will improve their ability to think critically, to analyze a problem and solve it using a wide array of tools. These skills will be invaluable to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of a career in one of the other sciences.

STUDENT LEARNING OUTCOMES. By the end of the course you will be able to:

- Evaluate a variety of limits including limits at infinity, one-sided limits, and limits of indeterminate forms. Students should also be able to identify discontinuities in functions presented algebraically or graphically;

- Apply the definition of derivative to calculate and estimate derivatives from formulas, graphs, or data;

- Differentiate sums, product and quotients of composite polynomial, trigonometric, exponential, and logarithmic functions;

- Discuss the conceptual relations between derivatives, rates of change, and tangent lines in the context of an applied example;

- Use asymptotic, first and second derivatives to graph functions;

- Solve applied problems using calculus and justify answers;

- Estimate a definite integral with a Riemann sum, supply a sketch;

- Evaluate a simple definite integral using the FTC;

Topics

Chapter One Review of: FUNDAMENTALS/PRECALCULUS
1.1 Basic Algebra: Real Numbers and Inequalities
1.2 Intervals and Absolute Values
1.3 Laws of Exponents

Chapter Two: FUNCTIONS AND THEIR GRAPHS
2.1 Basic Properties of Functions, and Shifting and Scaling
2.2 A Catalogue of Essential Function
2.3 Composition of Functions.
2.3 Inverse of Functions

Chapter Three: LIMITS AND CONTINUITY

3.1 The Limit of a Function

3.2 Calculating Limits
3.3 Limits Involving Infinity
3.4 Continuity and, Discontinuity and Its Types

Chapter Four: DERIVATIVES
4.1 Derivatives and Rates of Change
4.2 The Derivative as a Function
4.3 Basic Differentiation Formulas
4.4 The Product and Quotient Rules
4.5 Derivatives of Some Special Functions and Their Inverses
4.6 The Chain Rule
4.7 Implicit Differentiation
4.8 Higher Derivatives
4.9 Related Rates

Chapter Five: APPLICATIONS OF DIFFERENTIATION
5.1 Maximum and Minimum Values
5.2 The Mean Value Theorem
5.3 Derivatives and the Shapes of Graphs
5.4 Curve Sketching
5.5 Optimization Problems
5.6 Antiderivatives

Chapter Six: INTEGRALS

6.1 Estimating with Finite Sums

6.2 Sigma Notation and Limits of Finite Sums

6.3 The Definite Integral

6.4 Evaluating Definite Integrals

6.5 The Fundamental Theorem of Calculus

6.6 The Substitution Rule

Chapter Seven: TECHNIQUES OF INTEGRATION
7.1 Trigonometric Integrals and Substitutions
7.2 Integration by Parts
7.3 Partial Fractions
7.4 Integration with Tables
7.5 Approximate Integration
7.6 Improper Integrals
7.7 Further Applications of Integration

Chapter Eight: AREAS AND VOLUME
8.1 Area Bounded by a Curve
8.2 Area Bounded by Two Curves
8.3 Volume

TEXT-BOOKS (REFERENCES)
- Thomas’s Calculus, International Edition, 11 E, 2005, (Primary).
- Essential Calculus by James Stewart, Thomson Learning, Inc, 2007, link: http://tinyurl.com/34py5te, (Secondary).
- Schaum's Outline Series Calculus I by Frank Ayres. McGraw-Hill Companies, 2009, (Secondary).
- The reader would benefit any Calculus Text Book.

Computer Skills

Description :

Course overview:

The wide spread usage of computer systems in our daily life forced us to inspect and explore the world of what is known as Information Technology, abbreviated by IT. As the academic life the students in colleges also need to be familiar with the term IT and its usages in a way that they need.
As the name of course denotes, applications are given to the students in their first stage. The course covers the learning of main applications. The application are used by students to manage their electronic-version of academic materials, like sheets, lectures and references on computers, by the means of Windows operating system and Microsoft office suite. The student also will be able to create presentations for lectures and their discussions.

Course objective:

By the course the students to be familiar with the IT topics needed in their study. Starting from basic principles of IT and main components of personal computer system hardware and software. Then going through applications starting with Windows operating system, as the most spread operating system, for managing and using files, folders and programs.
Then the students will get the programs from office suite, like Microsoft Word as a word processing program, by which they can create and edit documents such as letters, papers, and reports. Microsoft Excel also will be covered for working with numbers and accounting. After that, students will get the Microsoft Power Point for creating interactive and effective presentations they need in their study.

CONTENTS of course
Module 1 – Concepts of Information and Communication Technology (ICT)
Module 2 – Using the Computer and Managing Files
Module 3 – Word Processing
Module 4 – Spreadsheets
Module 5 – Presentation

Topics

• Introduction to Computer System
Information Technology IT
• Computer Software
• Microsoft Windows Overview
• Windows\Working With Windows
• Windows\Files and Folders
• Windows\Control Panel
• Windows\Text Editing
• Windows\Print Management
• Microsoft PowerPoint Overview
• PowerPoint\Working With Slides
• PowerPoint\Text and Images in Representation
• PowerPoint\Slide Show and Effects
• PowerPoint\Prepare Output and Printing Management
• Microsoft Word
• Excel \Preparing Output and Printing

1. International Computer Driving License (ICDL) Curriculum ver 6.

2. Perry G., (2010), “Teach Yourself Microsoft® Office 2010 All in One”, by Sams Publishing.

Foundations of Mathematics

Description :

Course overview:

In this subject, we will study some basic concepts of mathematics which are very important for a student of mathematics to feed himself with these basic concepts. The term "foundations of mathematics" is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory,. The search for foundations of mathematics is however also the central question of the philosophy of mathematics: on what ultimate basis can mathematical statements be called "true"? The current dominant mathematical paradigm is based on axiomatic set theory and formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic.

Course objective:
The goal of this course is to help students learn the language of rigorous mathematics, as structured by definitions, axioms, and theorems. Students will be trained in how to read, understand, devise and communicate proofs of mathematical statements.
This course book is designed to teach the university mathematics student the basics of mathematics. First semester we will study mathematical logic and algebra of sets and second semester we will study some basic concepts of number theory.

Student learning outcome:

- Student should be able Define and understand the mathematical of logic , and the properties and operations os statements.
- Student should be able to prove any true statement by technique of prove .
- Student should be able to formulate logic expressions for a variety of applications; convert a logic expression into a Boolean circuit, and vice versa; design relations.
-The student will be able to define and describe set , type of sets ( finite , infinite , countable , uncountable ) and the operations on sets .
- The student will be able to define factor, whole number, counting number, remainder, quotient and greatest common factor, multiple and least common multiple, prime, composite and inclusive and divisibility test.
- Students will demonstrate the ability to distinguish corresponding sets as representations of relations or functions by the analysis of graphical, numeric. Define type of relations and properties of functions.
- Students will demonstrate the ability to pove statements and solve problems involving divisibility, prime numbers and the Euclidean Algorithm.
- Students will able to solve of polynomial using techniques of polynomial factoring

The Topics:
1- Logic and Proofs
2 -Set Theory
3 -Relations, Functions + Cardinality of infinite sets
4-Elementary Number Theory
5- Polynomial
6-Elementary Group Theory

Course Reading List and References ::
1. Discrete Mathematics and Its Applications, Kenneth H. Rosen,2012
2. Fundamental approach to discrete mathematics , by D.P. Acharjya, Sreekumar, (2008)
3. Number system and foundations of analysis , by Elliott Mendelson , Academic press , New York. 1973.
4. Pre calculus: A Concise Course, by By Ron Larson, Robert P. Hostetler,U.S.A 2007.
5. Mathematical Induction: Mathematical Proof , Mathematical Logic, Frederic P. Miller, Agnes F. Vandome, John McBrewster.
6. Set theory and related topics, by S. Lipschutz. McGRAW-Hill New York,(1998).
7. Kenneth Rosen, Elementary Number Theory and its Applications