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

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