توضیحات
‘This book provides a one-semester undergraduate introduction to counterexamples in calculus and analysis. It helps engineering, natural sciences, and mathematics students tackle commonly made erroneous conjectures. The book encourages students to think critically and analytically, and helps to reveal common errors in many examples.In this book, the authors present an overview of important concepts and results in calculus and real analysis by considering false statements, which may appear to be true at first glance. The book covers topics concerning the functions of real variables, starting with elementary properties, moving to limits and continuity, and then to differentiation and integration. The first part of the book describes single-variable functions, while the second part covers the functions of two variables.The many examples presented throughout the book typically start at a very basic level and become more complex during the development of exposition. At the end of each chapter, supplementary exercises of different levels of complexity are provided, the most difficult of them with a hint to the solution.This book is intended for students who are interested in developing a deeper understanding of the topics of calculus. The gathered counterexamples may also be used by calculus instructors in their classes. ‘–
‘In this manuscript we present counterexamples to different false statements, which frequently arise in the calculus and fundamentals of real analysis, and which may appear to be true at first glance. A counterexample is understood here in a broad sense as any example that is counter to some statement. The topics covered concern functions of real variables. The first part (chapters 1-6) is related to single-variable functions, starting with elementary properties of functions (partially studied even in college), passing through limits and continuity to differentiation and integration, and ending with numerical sequences and series. The second part (chapters 7-9) deals with function of two variables, involving limits and continuity, differentiation and integration. One of the goals of this book is to provide an outlook of important concepts and theorems in calculus and analysis by using counterexamples.We restricted our exposition to the main definitions and theorems of calculus in order to explore different versions (wrong and correct) of the fundamental concepts and to see what happens a few steps outside of the traditional formulations. Hence, many interesting (but more specific and applied) problems not related directly to the basic notions and results are left out of the scope of this manuscript. The selection and exposition of the material are directed, in the first place, to those calculus students who are interested in a deeper understanding and broader knowledge of the topics of calculus. We think the presented material may also be used by instructors that wish to go through the examples (or their variations) in class or assign them as homework or extra-curricular projects. In order to make the majority of the examples and solutions accessible to’– Read more…
Abstract: ‘This book provides a one-semester undergraduate introduction to counterexamples in calculus and analysis. It helps engineering, natural sciences, and mathematics students tackle commonly made erroneous conjectures. The book encourages students to think critically and analytically, and helps to reveal common errors in many examples.In this book, the authors present an overview of important concepts and results in calculus and real analysis by considering false statements, which may appear to be true at first glance. The book covers topics concerning the functions of real variables, starting with elementary properties, moving to limits and continuity, and then to differentiation and integration. The first part of the book describes single-variable functions, while the second part covers the functions of two variables.The many examples presented throughout the book typically start at a very basic level and become more complex during the development of exposition. At the end of each chapter, supplementary exercises of different levels of complexity are provided, the most difficult of them with a hint to the solution.This book is intended for students who are interested in developing a deeper understanding of the topics of calculus. The gathered counterexamples may also be used by calculus instructors in their classes. ‘–
‘In this manuscript we present counterexamples to different false statements, which frequently arise in the calculus and fundamentals of real analysis, and which may appear to be true at first glance. A counterexample is understood here in a broad sense as any example that is counter to some statement. The topics covered concern functions of real variables. The first part (chapters 1-6) is related to single-variable functions, starting with elementary properties of functions (partially studied even in college), passing through limits and continuity to differentiation and integration, and ending with numerical sequences and series. The second part (chapters 7-9) deals with function of two variables, involving limits and continuity, differentiation and integration. One of the goals of this book is to provide an outlook of important concepts and theorems in calculus and analysis by using counterexamples.We restricted our exposition to the main definitions and theorems of calculus in order to explore different versions (wrong and correct) of the fundamental concepts and to see what happens a few steps outside of the traditional formulations. Hence, many interesting (but more specific and applied) problems not related directly to the basic notions and results are left out of the scope of this manuscript. The selection and exposition of the material are directed, in the first place, to those calculus students who are interested in a deeper understanding and broader knowledge of the topics of calculus. We think the presented material may also be used by instructors that wish to go through the examples (or their variations) in class or assign them as homework or extra-curricular projects. In order to make the majority of the examples and solutions accessible to’
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ترجمه ماشینی :
این کتاب مقدمه ای یک ترم در مقطع کارشناسی برای مثال های متقابل در حساب دیفرانسیل و انتگرال و تجزیه و تحلیل ارائه می دهد. این به دانشجویان رشته های مهندسی، علوم طبیعی و ریاضی کمک می کند تا با حدس های اشتباه رایج مقابله کنند. این کتاب دانشآموزان را به تفکر انتقادی و تحلیلی تشویق میکند و به کشف خطاهای رایج در بسیاری از مثالها کمک میکند. در این کتاب، نویسندگان مروری بر مفاهیم و نتایج مهم در محاسبات و تحلیل واقعی با در نظر گرفتن عبارات نادرست، که ممکن است در نگاه اول درست به نظر برسد. این کتاب موضوعات مربوط به توابع متغیرهای واقعی را پوشش میدهد که از ویژگیهای ابتدایی شروع میشود، به سمت محدودیتها و تداوم حرکت میکند و سپس به تمایز و ادغام میپردازد. بخش اول کتاب توابع تک متغیری را توصیف میکند، در حالی که بخش دوم عملکرد دو متغیر را پوشش میدهد. مثالهای زیادی که در سراسر کتاب ارائه میشوند معمولاً از سطح بسیار ابتدایی شروع میشوند و در طول توسعه نمایش پیچیدهتر میشوند. در پایان هر فصل، تمرینهای تکمیلی با سطوح مختلف پیچیدگی، که سختترین آنها با اشاره به راهحل است، ارائه شده است. این کتاب برای دانشآموزانی است که علاقهمند به درک عمیقتر مباحث حساب دیفرانسیل و انتگرال هستند. نمونههای متقابل جمعآوریشده نیز ممکن است توسط مربیان حساب دیفرانسیل و انتگرال در کلاسهایشان استفاده شوند. ‘–
‘در این دستنوشته، نمونههایی متضاد برای گزارههای نادرست مختلف ارائه میکنیم، که اغلب در محاسبات و مبانی تحلیل واقعی مطرح میشوند، و ممکن است در نگاه اول درست به نظر برسند. مثال متضاد در اینجا به معنای وسیع به عنوان هر مثالی که مخالف برخی گزاره ها باشد درک می شود. موضوعات پوشش داده شده مربوط به توابع متغیرهای واقعی است. بخش اول (فص
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