Chairperson : Professor Neeru Department Coordinator : Professor Vinay Kanwar Course Leader : Professor Vinay Kanwar B.A. SEMESTER – I, MATHEMATICS PAPER : II, CALCULUS CONTENTS Introductory Letter i Syllabus ii L. No. Title Author Page No. 1. Properties of Real Numbers Bhushan Kumar 1 - 26 2. Limit and Continuity Bhushan Kumar 27 - 56 3. Derivative of Hyperbolic and Bhushan Kumar 57 – 71 Inverse Hyperbolic Functions 4. Successive Differentiation Bhushan Kumar 72 - 89 5. Mean Value Theorems Bhushan Kumar 90 - 117 6. Indeterminate Forms Bhushan Kumar 118 - 138 Vetter & Submitted: Professor Vinay Kanwar E-mail of Department -
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i Introductory Letter Dear Student, We congratulate you on joining USOL and hope you will enjoy association with our Institute. It is our endeavour to make this association meaningful and a great learning experience. As you know you are now in the first semester of B.A. Mathematics, you are encouraged to solve exercises of this book as well as from other standard text books at your own instead of finding solutions from the help books. This block of lessons deals with the elementary calculus of single real variable. You will find in this block of lessons a detailed discussion on all the topics and their applications. The exercises given at the end of each lesson deal with the topics discussed in the text and are sufficient for practice and for examination point of view. Please go through them very carefully. If you have any difficulties do not hesitate to write to us or to visit us personally. Yours sincerely Deptt. of Mathematics
1 Lesson - 1 PROPERTIES OF REAL NUMBERS Structure 1.0 Objectives 1.1 Introduction 1.2 Real Number System 1.2.1 Field of Real Numbers 1.2.2 Field Properties of Real Number System 1.2.3 Order Properties in Real Number 1.3 Absolute (or Numerical or Modulus) Value 1.4 Bounds 1.4.1 Upper and Lower Bound of a Set 1.4.2 Bounded and Unbounded Set 1.4.3 Least Upper Bound and Greatest Lower Bound 1.5 Least Upper Bound Property [Order Completeness Property of Real Numbers] 1.6 Archimedean Property of Real Numbers 1.7 Summary 1.8 References 1.9 Further Readings 1.10 Model Questions 1.0 OBJECTIVES After going through this lesson, you will be able to: define real number and their properties. explain absolute or modulus value of a real number. discuss bounded and unbounded set. have the knowledge of order completeness property of real numbers. study Archimedean property of real numbers. 1.1 INTRODUCTION In this lesson, we will study the real number system along with their field and order properties. Further, we are going to learn the absolute or modulus value of a real number. In
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