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2020 | Buch

Examining Sets and Relations Kapitel lesen Erstes Kapitel lesen

Solving Problems in Mathematical Analysis, Part I

Sets, Functions, Limits, Derivatives, Integrals, Sequences and Series

verfasst von: Prof. Dr. Tomasz Radożycki

Verlag: Springer International Publishing

Buchreihe : Problem Books in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Examining Sets and Relations
Abstract
In this chapter, we deal with sets and perform various operations on them. We will also get acquainted with the notion of the “relation.” For the convenience of the reader, a number of basic properties of sets is first collected, which will be then used in the remainder of this chapter to help solve problems. The same applies to the subsequent chapters.
Tomasz Radożycki
Chapter 2. Investigating Basic Properties of Functions
Abstract
The present chapter is concerned with elementary properties of functions. It is the principal notion in mathematical analysis. A function called also a mapping defined on a certain set X (or D) called a domain with it values in another set Y called a codomain is an assignment that with every x ∈ X (the argument) associates a unique element y ∈ Y (the value), although not every y has to be associated with a certain x. This is denoted as y = f(x) or more formally:
$$\displaystyle f:X\,\mbox{\reflectbox {{}\(\in \)}}\, x\longmapsto y\in Y. $$
The subset of Y containing all and only values that correspond to at least one argument is denoted by f(X) and is called the image of the function f.
Tomasz Radożycki
Chapter 3. Defining Distance in Sets
Abstract
The present chapter is devoted to the notion of a metric. We will learn how to check whether a given function is a metric and draw the special sets called balls and segments.
Tomasz Radożycki
Chapter 4. Using Mathematical Induction
Abstract
In this chapter we learn how to use mathematical induction in various proofs. The method of proving different claims, identities, and inequalities, which is called the mathematical induction, can be formulated as follows. Assume a certain thesis is to be demonstrated for all \(n\in \mathbb {N}\). Then the inductive proof is composed of two steps:
1.
First, one must check that the thesis to be shown is true for n = 1. In general the verification of this fact is very simple.
 
2.
The second step is to demonstrate the veracity of the following claim: if one assumes that the thesis is true for certain \(k\in \mathbb {N}\) (the inductive hypothesis), then it is also true for k + 1 (the inductive thesis). This part of the proof is generally much more difficult.
 
The implementation of these two steps means that the inductive proof is accomplished and it allows for the conclusion that the thesis is true for every natural n.
Tomasz Radożycki
Chapter 5. Investigating Convergence of Sequences and Looking for Their Limits
Abstract
This chapter is devoted to the investigation of infinite sequences. We will be particularly concerned how to prove the convergence (or divergence) of a sequence and to calculate its limit.
Tomasz Radożycki
Chapter 6. Dealing with Open, Closed, and Compact Sets
Abstract
In this chapter we deal with basic topological properties of sets, which are necessary for the proper formulation of limit and continuity of real functions.
Tomasz Radożycki
Chapter 7. Finding Limits of Functions
Abstract
This chapter is devoted to the notion of the limit of a function at a given point. This notion is necessary for the formulation of the continuity and differentiability of functions dealt with in the following chapters. Here we will learn how to find the limits using some special tricks or substitutions.
Tomasz Radożycki
Chapter 8. Examining Continuity and Uniform Continuity of Functions
Abstract
One of the fundamental notions in topology is that of the continuity of functions, which constitutes our concern in this chapter.
Tomasz Radożycki
Chapter 9. Finding Derivatives of Functions
Abstract
The main subject of the present chapter constitutes the notion of the differentiability of functions. We will learn how to verify whether or not there exists a derivative of a given function and we will find derivatives from the definition. Also, some less trivial examples are considered.
Tomasz Radożycki
Chapter 10. Using Derivatives to Study Certain Properties of Functions
Abstract
The derivative of a function embodies the powerful tool for the investigation of function properties. Some ideas are dealt with in the present chapter, but this subject will continue in Chap. 12, where higher derivatives come into play.
Tomasz Radożycki
Chapter 11. Dealing with Higher Derivatives and Taylor’s Formula
Abstract
The present chapter is concerned with higher derivatives and Taylor’s formula. It is also shown how to use the latter to easily find some special limits of functions.
Tomasz Radożycki
Chapter 12. Looking for Extremes and Examine Functions
Abstract
The present chapter is devoted to the applications of the differential calculus to the comprehensive investigation of the behavior of functions. In particular, we will learn how to find monotonicity intervals, extremal points, points of inflection, etc. The monotonicity has already been touched in the theoretical summary at the beginning of Chap. 10.
Tomasz Radożycki
Chapter 13. Investigating the Convergence of Series
Abstract
In this chapter we deal with numerical series. We will learn how to check their convergence by the definition or by applying various tests.
Tomasz Radożycki
Chapter 14. Finding Indefinite Integrals
Abstract
The principal notion of the present chapter is that of the indefinite integral. We will become acquainted with the definition and we will learn to use several methods to calculate the integrals.
Tomasz Radożycki
Chapter 15. Investigating the Convergence of Sequences and Series of Functions
Abstract
In Chaps. 5 and 13, we were dealing with numerical sequences and series. This chapter is concerned with sequences and series whose terms are real functions and not ordinary numbers. We will be interested in their convergence and investigate the properties of limiting functions.
Tomasz Radożycki
Backmatter
Metadaten
Titel
Solving Problems in Mathematical Analysis, Part I
verfasst von
Prof. Dr. Tomasz Radożycki
Copyright-Jahr
2020
Electronic ISBN
978-3-030-35844-0
Print ISBN
978-3-030-35843-3
DOI
https://doi.org/10.1007/978-3-030-35844-0

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