Solving Problems in Mathematical Analysis, Part I: Sets, Functions, Limits, Derivatives, Integrals, Sequences and Series (Problem Books in Mathematics)

Description

This textbook offers an extensive list of completely solved problems in mathematical analysis. This first of three volumes covers sets, functions, limits, derivatives, integrals, sequences and series, to name a few. The series contains the material corresponding to the first three or four semesters of a course in Mathematical Analysis. Based on the author’s years of teaching experience, this work stands out by providing detailed solutions (often several pages long) to the problems. The basic premise of the book is that no topic should be left unexplained, and no question that could realistically arise while studying the solutions should remain unanswered. The style and format are straightforward and accessible. In addition, each chapter includes exercises for students to work on independently. Answers are provided to all problems, allowing students to check their work. Though chiefly intended for early undergraduate students of Mathematics, Physics and Engineering, the book will also appeal to students from other areas with an interest in Mathematical Analysis, either as supplementary reading or for independent study. Preface Contents Definitions and Notation 1 Examining Sets and Relations 1.1 Demonstrating Simple Identities Problem 1 Solution Problem 2 Solution Identity (a) Identity (b) Identity (c) Problem 3 Solution 1.2 Finding Sets on a Plane Problem 1 Solution Problem 2 Solution 1.3 Finding Lower and Upper Bounds of Numerical Sets Problem 1 Solution Problem 2 Solution 1.4 Verifying Whether R Is an Equivalence Relation, Looking for Equivalence Classes and Drawing a Graph of the Relation Problem 1 Solution Problem 2 Solution Problem 3 Solution 1.5 Exercises for Independent Work 2 Investigating Basic Properties of Functions 2.1 Looking for Ranges (Images) and Level Sets Problem 1 Solution Problem 2 Solution 2.2 Verifying Whether a Function Is an Injection, Surjection, or Bijection and Looking for the Inverse Function Problem 1 Solution Problem 2 Solution Problem 3 Solution 2.3 Finding Images and Inverse Images of Sets Problem 1 Solution Problem 2 Solution 2.4 Exercises for Independent Work 3 Defining Distance in Sets 3.1 Examining Whether a Given Function Is a Metric Problem 1 Solution Problem 2 Solution 3.2 Drawing Balls and Segments Problem 1 Solution Problem 2 Solution 3.3 Exercises for Independent Work 4 Using Mathematical Induction 4.1 Proving Divisibility of Numbers and Polynomials Problem 1 Solution Problem 2 Solution 4.2 Proving Equalities and Inequalities Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 4.3 Demonstrating Some Important Formulas Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 4.4 Exercises for Independent Work 5 Investigating Convergence of Sequences and Looking for Their Limits 5.1 Some Common Tricks Useful for Calculating Limitsof Sequences Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 5.2 Using Various Criteria Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 5.3 Examining Recursive Sequences Problem 1 Solution Problem 2 Solution 5.4 When a Sequence Oscillates Problem 1 Solution Problem 2 Solution 5.5 Demonstrating Divergence of Sequences Problem 1 Solution Problem 2 Solution 5.6 Exercises for Independent Work 6 Dealing with Open, Closed, and Compact Sets 6.1 Examining Openness and Closeness of Sets Problem 1 Solution Problem 2 Solution Problem 3 Solution 6.2 Examining Compactness Problem 1 Solution Problem 2 Solution 6.3 Exercises for Independent Work 7 Finding Limits of Functions 7.1 Some Common Tricks Useful for Calculating Limitsof Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution 7.2 Using Substitutions Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 7.3 Exercises for Independent Work 8 Examining Continuity and Uniform Continuity of Functions 8.1 Demonstrating the Continuity of Functions with Heine's and Cauchy's Methods Problem 1 Solution Problem 2 Solution 8.2 Examining Functions in Their ``Gluing'' Points Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 8.3 Investigating Whether a Function Is Uniformly Continuous Problem 1 Solution Problem 2 Solution Problem 3 Solution 8.4 Exercises for Independent Work 9 Finding Derivatives of Functions 9.1 Calculating Derivatives of Functions by Definition Problem 1 Solution Problem 2 Solution 9.2 Examining the Differentiability of a Function Problem 1 Solution Problem 2 Solution 9.3 Finding Derivatives of Inverse Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution 9.4 Solving Several Intricate Problems Problem 1 Solution Problem 2 Solution Problem 3 Solution 9.5 Exercises for Independent Work 10 Using Derivatives to Study Certain Properties of Functions 10.1 Proving Identities and Inequalities Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution 10.2 Using Rolle's and Lagrange's Theorems Problem 1 Solution Problem 2 Solution Problem 3 Solution 10.3 Examining Curves on a Plane: Tangency and Angles of Intersection Problem 1 Solution Problem 2 Solution 10.4 Calculating Limits Using l'Hospital's Rule Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution Problem 6 Solution 10.5 Exercises for Independent Work 11 Dealing with Higher Derivatives and Taylor's Formula 11.1 Demonstrating by Induction Formulas for High OrderDerivatives Problem 1 Solution Problem 2 Solution 11.2 Expanding Functions Problem 1 Solution Problem 2 Solution Problem 3 Solution 11.3 Using Taylor's Formula to Calculate Limits of Functions Problem 1 Solution Problem 2 Solution 11.4 Exercises for Independent Work 12 Looking for Extremes and Examine Functions 12.1 Finding the Smallest and the Largest Values of a Function on a Given Set Problem 1 Solution Problem 2 Solution 12.2 Examining the Behavior of Functions from A to Z Problem 1 Solution Problem 2 Solution 12.3 Exercises for Independent Work 13 Investigating the Convergence of Series 13.1 Using Estimates Problem 1 Solution Problem 2 Solution 13.2 Using Various Tests Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution Problem 6 Solution Problem 7 Solution 13.3 Solving Several Interesting Problems Problem 1 Solution Problem 2 Solution Problem 3 Solution 13.4 Exercises for Independent Work 14 Finding Indefinite Integrals 14.1 Integrating by Parts and by Substitution Problem 1 Solution Problem 2 Solution Problem 3 Solution Problem 4 Solution Problem 5 Solution Problem 6 Solution 14.2 Using the Method of Recursive Formulas Problem 1 Solution Problem 2 Solution Problem 3 Solution 14.3 Integrating Rational Functions Problem 1 Solution Problem 2 Solution 14.4 Integrating Rational Functions of Trigonometric Functions Problem 1 Solution Problem 2 Solution 14.5 Using Euler's Substitutions Problem 1 Solution Problem 2 Solution 14.6 Making Use of Hyperbolic and Trigonometric Substitutions Problem 1 Solution Problem 2 Solution 14.7 Exercises for Independent Work 15 Investigating the Convergence of Sequences and Seriesof Functions 15.1 Finding Limits of Sequences of Functions Problem 1 Solution Problem 2 Solution 15.2 Examining Uniform Convergence of Functional Sequences Problem 1 Solution Problem 2 Solution 15.3 Examining Uniform Convergence of Functional Series Problem 1 Solution Problem 2 Solution 15.4 Calculating Sums of Series Problem 1 Solution Problem 2 Solution 15.5 Exercises for Independent Work Index