550 AP Calculus AB & BC Practice Questions (College Test Preparation)

Princeton Review

Princeton Review | 2013 | 448 páginas | rar - epub | 29,8 Mb

link (password : matav)

THE PRINCETON REVIEW GETS RESULTS. Get extra preparation for an excellent AP Calculus AB & BC score with 550 extra practice questions and answers.

Practice makes perfect—and The Princeton Review’s 550 AP Calculus AB & BC Practice Questions gives you everything you need to work your way to the top. Inside, you’ll find tips and strategies for tackling and overcoming challenging questions, plus all the practice you need to get the score you want.
Inside The Book: All the Practice and Strategies You Need• 2 diagnostic exams (one each for AB and BC) to help you identify areas of improvement• 2 comprehensive practice tests (one each for AB and BC)• Over 300 additional practice questions• Step-by-step techniques for both multiple-choice and free-response questions• Practice drills for each tested topic: Limits, Functions and Graphs, Derivatives, Integration, Polynomial Approximations, and Series• Answer keys and detailed explanations for each drill and test question• Engaging guidance to help you critically assess your progress

Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research

Hongwei Chen

The Mathematical Association of America | 2010 | 316 páginas | pdf | 2,1 Mb

linklink1

Excursions in Classical Analysis introduces undergraduate students to advanced problem solving and undergraduate research in two ways. Firstly, it provides a colourful tour of classical analysis which places a wide variety of problems in their historical context. Secondly, it helps students gain an understanding of mathematical discovery and proof. In demonstrating a variety of possible solutions to the same sample exercise, the reader will come to see how the connections between apparently inapplicable areas of mathematics can be exploited in problem-solving. This book will serve as excellent preparation for participation in mathematics competitions, as a valuable resource for undergraduate mathematics reading courses and seminars and as a supplement text in a course on analysis. It can also be used in independent study, since the chapters are free-standing.

Contents
Two classical inequalities
A new approach for proving inequalities
Means generated by an integral
The L'Hôpital monotone rule
Trigonometric identities via complex numbers
Special numbers
On a sum of cosecants
The gamma products in simple closed forms
On the telescoping sums
Summation of subseries in closed form
Generating functions for powers of Fibonacci numbers
Identities for the Fibonacci powers
Bernoulli numbers via determinants
On some finite trigonometric power sums
Power series of (arcsin x)²
Six ways to sum [zeta] 2
Evaluations of some variant Euler sums
Interesting series involving binomial coefficients
Parametric differentiation and integration
Four ways to evaluate the Poisson integral
Some irresistible integrals.

Calculus Mysteries and Thrillers

(Classroom Resource Materials) 

R. Grant Woods 

The Mathematical Association of America | 1998 | 152 páginas | rar - pdf | 4,6 Mb

link (password: matav)

This book is a collection of a dozen mathematics projects. These are typically novel, interesting and several levels more complex than those usually found in textbooks. The nature of the projects makes them suitable for group working. The problems involve such diverse concepts as Newton's method for approximating roots, inverse trigonometric functions and surface area integrals. Although ideas from economics and physics are used in the problems no prior knowledge of these fields is required.

ContentsThe Purpose of This Book ix
An Overview of the Projects xiii
Detailed Mathematical Requirements xv
The Projects
1. The Case of the Parabolic Pool Table 3
2. Calculus for Climatologists 9
3. The Case of the Swiveling Spotlight 13
4. Finding the Salami Curve 19
5. Saving Lunar Station Alpha 23
6. An Income Policy for Mediocria 29
7. The Case of the Cooling Cadaver 35
8. Designing Dipsticks 41
9. The Case of the Gilded Goose-egg 47
10. Sunken Treasure 51
11. The Case of the Alien Agent 57
The Solutions
1. The Case of the Parabolic Pool Table—Solution 67
2. Calculus for Climatologists—Solution 71
3. The Case of the Swiveling Spotlight—Solution 75
4. Finding the Salami Curve—Solution 81
5. Saving Lunar Station Alpha—Solution 87
6. An Income Policy for Mediocria—Solution 95
7. The Case of the Cooling Cadaver—Solution 103
8. Designing Dipsticks—Solution 107
9. The Case of the Gilded Goose-egg—Solution 113
10. Sunken Treasure—Solution 121
11. The Case of the Alien Agent—Solution 127

A Tour of the Calculus

David Berlinski 

Vintage | 1997 | 352 páginas | rar - epub | 4 Mb

link (password: matav)

Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brio. Even as he initiates us into the mysteries of real numbers, functions, and limits, Berlinski explores the furthest implications of his subject, revealing how the calculus reconciles the precision of numbers with the fluidity of the changing universe.

Contents
Introduction
A Note to the Reader
The Frame of the Book
Chapter   1   Masters of the Symbols
Chapter   2   Symbols of the Masters
Chapter   3   The Black Blossoms of Geometry
Chapter   4   Cartesian Coordinates
Chapter   5   The Unbearable Smoothness of Motion
Chapter   6   Yo
Chapter   7   Thirteen Ways of Looking at a Line
Chapter   8   The Doctor of Discovery
Chapter   9   Real World Rising
Chapter 10   Forever Familiar, Forever Unknown
Chapter 11   Some Famous Functions
Chapter 12   Speed of Sorts
Chapter 13   Speed, Strange Speed
Chapter 14   Paris Days
Chapter 15   Prague Interlude
Chapter 16   Memory of Motion
Chapter 17   The Dimpled Shoulder
Chapter 18   Wrong Way Rolle
Chapter 19   The Mean Value Theorem
Chapter 20   The Song of Igor
Chapter 21   Area
Chapter 22   Those Legos Vanish
Chapter 23   The Integral Wishes to Compute an Area
Chapter 24   The Integral Wishes to Become a Function
Chapter 25   Between the Living and the Dead
Chapter 26   A Farewell to Continuity
Epilogue
Acknowledgments

Outros livros do mesmo autor:

The king of infinite space : Euclid and his Elements por David Berlinski Idioma: Inglês   Editora: New York : Basic Books, [2013]

One, two, three : absolutely elementary mathematics por David Berlinski Idioma: Inglês   Editora: New York : Pantheon Books, ©2011.

Infinite ascent : a short history of mathematics por David Berlinski Idioma: Inglês   Editora: New York : Modern Library, 2005.

Paradoxes and Sophisms in Calculus

Sergiy Klymchuk e Susan G. Staples

The Mathematical Association of America | 2013 | páginas | rar - pdf |965 kb

link (password : matav)

In the study of mathematics, surprising and counterintuitive examples can offer a fascinating insight into the development of the subject, and inspire a learner's passion for discovery. With a carefully chosen selection of so-called paradoxes and sophisms, this book offers a delightful supplementary resource to enhance the study of single variable calculus. By paradox, the authors mean an unexpected statement that looks invalid, but is in fact true. The word sophism describes intentionally invalid reasoning that looks formally correct, but, in fact, contains a subtle mistake or flaw. This collection of over fifty paradoxes and sophisms showcases the subtleties of calculus and leads students to contemplate the underlying concepts. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences and series are explored, with full explanations provided for each example. The book is an ideal resource for those studying or teaching calculus at high school and university level.

Contents
Introduction;
Part I. Paradoxes:
1. Functions and limits;
2. Derivatives and integrals;
Part II. Sophisms:
3. Functions and limits;
4. Derivatives and integrals;
Part III. Solutions to Paradoxes:
5. Functions and limits;
6. Derivatives and integrals;
Part IV. Solutions to Sophisms:
7. Functions and limits;
8. Derivatives and integrals;
References.

Pré-cálculo pra leigos

Deborah Rumsey , Krystle Rose Forseth , Christopher Burger , Michelle Rose Gilman

Alta books editora |2011| 400 páginas | rar - pdf | 9,3 Mb

link (password : matav)

Está preparado para cálculo, mas se sente confuso? Não tenha medo! Este nada intimidador guia prático leva você ao tópicos essenciais, desde valor absoluto e equações quadráticas a logaritmos e funções exponenciais para identidades trigonométricas e operações matriz. Você vai entender os conceitos – não apenas a mastigação de números – e ver como efetuar todos os testes, de gráficos até o confronto de provas.

Descubra como: • Aplicar os principais teoremas e fórmulas • Aplicar funções de gráficos de trigonometria como profissional • Encontrar os valores da trigonometria em um círculo unitário • Identificar funções limites e continuidade

Neste livro você encontrará: • Explicações em português de fácil entendimento. • Informações fáceis de localizar e passo a passo. • Ícones e outros recursos de identificação e memorização. • Folha de cola para destacar com informações práticas. • Listas dos 10 melhores itens relacionados ao assunto. • Um toque de humor e diversão.

A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus

Nancy Baxter Hastings

 Mathematical Association of America | 2006 | 409 páginas | rar - pdf |2,8 Mb

link (password: matav)

Each year, over 1,000,000 students take college-level courses below calculus such as precalculus, college algebra and others that fulfill general education requirements. Most college algebra courses, and certainly all precalculus courses, were originally intended to prepare students for calculus. Most are still offered in this spirit, even though only a small percentage of students have any intention of taking calculus. This volume examines how the courses below calculus might be refocused to provide better mathematical experiences for all students. This initiative involves a greater emphasis on conceptual understanding with a de-emphasizing on rote manipulation. It encourages the use of realistic applications, math modeling and data analysis that reflect the ways mathematics is used in other disciplines. It promotes the use of active learning approaches, including group work, exploratory activities and projects. It emphasizes communication skills: reading, writing, presenting and listening. It endorses the appropriate use of technology to enhance conceptual understanding, visualization, and to enable students to tackle real-world problems.The 49 papers in this volume seek to focus attention on the problems and needs of the courses and to provide guidance to the mathematics community. Major themes include: new visions for introductory collegiate mathematics, transition from high school to college, needs of other disciplines, research on student learning, implementation issues, and ideas and projects that work.

Contents
Preface  . vii
Introduction .  .1
1 The Conference: Rethinking the Preparation for Calculus, Jack Narayan and Darren Narayan . .  3
2 Twenty Questions about Precalculus, Lynn Arthur Steen . 8
Background  13
3 Who are the Students Who Take Precalculus?, Mercedes A. McGowen . . 15
4 Enrollment Flow to and from Courses Below Calculus, Steven R. Dunbar .  . 28
5 What Have We Learned from Calculus Reform? The Road to Conceptual Understanding, Deborah Hughes Hallett . . 43
6 Calculus and Introductory College Mathematics: Current Trends and Future Directions, Susan L. Ganter .46
Theme 1. New Visions for Introductory Collegiate Mathematics . . . 55
7 Refocusing Precalculus: Challenges and Questions, Nancy Baxter Hastings . . . 57
8 Preparing Students for Calculus in the Twenty-First Century, Sheldon P. Gordon . . 64
9 Preparing for Calculus and Preparing for Life, Bernard L. Madison .. 78
10 College Algebra: A Course in Crisis, Don Small . . 83
11 Changes in College Algebra, Scott R. Herriott .  . 90
12 One Approach to Quantitative Literacy: Understanding our Quantitative World, Janet Andersen .. 101
Theme 2. The Transition from High School to College . . . 109
13 High School Overview and the Transition to College, Zalman Usiskin  . 111
14 Precalculus Reform: A High School Perspective, Daniel J. Teague . . 121
15 The Influence of Current Efforts to Improve School Mathematics on the
Preparation for Calculus, Eric Robinson and John Maceli . . 129
Theme 3. The Needs of Other Disciplines . . 151
16 Fundamental Mathematics: Voices of the Partner Disciplines, William Barker and Susan L. Ganter .153
17 Skills versus Concepts at West Point, Rich West . 160
18 Integrating Data Analysis into Precalculus Courses, Allan J. Rossman . 169
Theme 4. Student Learning and Research. . .179
19 Assessing What Students Learn: Reform versus Traditional Precalculus and Follow-up Calculus, Florence S. Gordon . .. 181
20 Student Voices and the Transition from Reform High School Mathematics to College Mathematics, Rebecca Walker .  . . 193
Theme 5. Implementation . .. .211
21 Some Political and Practical Issues in Implementing Reform, Robert E. Megginson . .. 213
22 Implementing Curricular Change in Precalculus: A Dean’s Perspective, Judy E. Ackerman .. . 219
23 The Need to Rethink Placement in Mathematics, Sheldon P. Gordon  . 224
24 Changing Technology Implies Changing Pedagogy, Lawrence C. Moore and David A. Smith .. . 229
25 Preparing for Calculus and Beyond: Some Curriculum Design Issues, Al Cuoco . 235
26 Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course, Bonnie Gold 249
Theme 6. Influencing the Mathematics Community . . 255
27 Launching a Precalculus Reform Movement: Influencing the Mathematics Community, Bernard L. Madison . 257
28 Mathematics Programs for the Rest-of-Us, Naomi D. Fisher and Bonnie Saunders . . 265
29 Where Do We Go From Here? Creating a National Initiative to Refocus the Courses below Calculus, Sheldon P. Gordon . 274
Ideas and Projects that Work: Part 1  . . 283
30 College Precalculus Can Be a Barrier to Calculus: Integration of Precalculus with Calculus Can Achieve Success, Doris Schattschneider . 285
31 College Algebra Reform through Interdisciplinary Applications, William P. Fox . 295
32 Elementary Math Models: College Algebra Topics and a Liberal Arts Approach, Dan Kalman . . 304
33 The Case for Labs in Precalculus, Brigitte Lahme, Jerry Morris, and Elias Toubassi  310
34 The Fifth Rule: Direct Experience of Mathematics, Gary Simundza   320
Ideas and Projects that Work: Part 2 . . 329
35 Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics, Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick, and Sylvia Svitak . . 333
36 Precalculus: Concepts in Context, Marsha Davis .  . . 337
37 Rethinking College Algebra, Benny Evans . . . 341
38 From The Bottom Up, Sol Garfunkel .  . 345
39 The Functioning in the Real World Project, Florence S. Gordon and Sheldon P. Gordon 348
40 The Importance of a Story Line: Functions as Models of Change, Deborah Hughes Hallett 352
41 Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus, Nancy Baxter Hastings . . 355
42 Maricopa Mathematics, Alan Jacobs .  . 360
43 College Algebra/Quantitative Reasoning at the University of Massachusetts, Boston, Linda Almgren Kime . . 364
44 Developmental Algebra: The First Mathematics Course for Many College Students, Mercedes A. McGowen .. . 369
45 Workshop Precalculus: Functions, Data, and Models, Allan J. Rossman . 376
46 Contemporary College Algebra, Don Small  . 380
47 Precalculus: A Study of Functions and Their Applications, Todd Swanson . . . . 386
48 Success and Failures of a Precalculus Reform Project, David M. Wells and Lynn Tilson 390
49 The Earth Math Projects, Nancy Zumoff and Christopher Schaufele . .393

Mathematical Analysis Fundamentals

Agamirza Bashirov

Elsevier | 2014 | 363 páginas | rar - pdf | 2,77 Mb

link (password: matav)

The author's goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options.

• Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces 

• Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus.

• Friendly and well-rounded presentation of pre-analysis topics such as sets, proof techniques and systems of numbers.

Contents
1 Sets and Proofs 1
1.1 Sets, Elements, and Subsets 1
1.2 Operations on Sets 3
1.3 Language of Logic 4
1.4 Techniques of Proof 7
1.5 Relations 11
1.6 Functions 15
1.7* Axioms of Set Theory 18
Exercises 20
2 Numbers 25
2.1 System N 25
2.2 Systems Z and Q 29
2.3 Least Upper Bound Property and Q 32
2.4 System R 34
2.5 Least Upper Bound Property and R 37
2.6* Systems R, C, and *R 41
2.7 Cardinality 43
Exercises 47
3 Convergence 51
3.1 Convergence of Numerical Sequences 51
3.2 Cauchy Criterion for Convergence 53
3.3 Ordered Field Structure and Convergence 55
3.4 Subsequences 57
3.5 Numerical Series 59
3.6 Some Series of Particular Interest 62
3.7 Absolute Convergence 64
3.8 Number e 71
Exercises 74
4 Point Set Topology 79
4.1 Metric Spaces 79
4.2 Open and Closed Sets 83
4.3 Completeness 89
4.4 Separability 93
4.5 Total Boundedness 95
4.6 Compactness 96
4.7 Perfectness 100
4.8 Connectedness 103
4.9* Structure of Open and Closed Sets 104
Exercises 106
5 Continuity 113
5.1 Definition and Examples 113
5.2 Continuity and Limits 117
5.3 Continuity and Compactness 120
5.4 Continuity and Connectedness 121
5.5 Continuity and Oscillation 123
5.6 Continuity of Rk-valued Functions 124
Exercises 126
6 Space C(E, E) 131
6.1 Uniform Continuity 131
6.2 Uniform Convergence 134
6.3 Completeness of C(E, E) 137
6.4 Bernstein and Weierstrass Theorems 138
6.5* Stone and Weierstrass Theorems 142
6.6* Ascoli–Arzelà Theorem 144
Exercises 146
7 Differentiation 149
7.1 Derivative 149
7.2 Differentiation and Continuity 151
7.3 Rules of Differentiation 155
7.4 Mean-Value Theorems 158
7.5 Taylor’s Theorem 163
7.6* Differential Equations 165
7.7* Banach Spaces and the Space C1 (a, b) 169
7.8 A View to Differentiation in Rk 172
Exercises 174
8 Bounded Variation 177
8.1 Monotone Functions 177
8.2 Cantor Function 181
8.3 Functions of Bounded Variation 183
8.4 Space BV(a, b) 185
8.5 Continuous Functions of Bounded Variation 189
8.6 Rectifiable Curves 191
Exercises 192
9 Riemann Integration 195
9.1 Definition of the Riemann Integral 195
9.2 Existence of the Riemann Integral 199
9.3 Lebesgue Characterization 204
9.4 Properties of the Riemann Integral 207
9.5 Riemann Integral Depending on a Parameter 213
9.6 Improper Integrals 217
Exercises 220
10 Generalizations of Riemann Integration 225
10.1 Riemann–Stieltjes Integral 225
10.2* Helly’s Theorems 232
10.3* Reisz Representation 236
10.4* Definition of the Kurzweil–Henstock Integral 239
10.5* Differentiation of the Kurzweil–Henstock Integral 245
10.6* Lebesgue Integral 246
Exercises 250
11 Transcendental Functions 253
11.1 Logarithmic and Exponential Functions 253
11.2* Multiplicative Calculus 256
11.3 Power Series 262
11.4 Analytic Functions 268
11.5 Hyperbolic and Trigonometric Functions 274
11.6 Infinite Products 279
11.7* Improper Integrals Depending on a Parameter 287
11.8* Euler’s Integrals 295
Exercises 300
12 Fourier Series and Integrals 307
12.1 Trigonometric Series 307
12.2 Riemann-Lebesgue Lemma 310
12.3 Dirichlet Kernels and Riemann’s Localization Lemma 313
12.4 Pointwise Convergence of Fourier Series 315
12.5* Fourier Series in Inner Product Spaces 321
12.6* Cesàro Summability and Fejér’s Theorem 328
12.7 Uniform Convergence of Fourier Series 332
12.8* Gibbs Phenomenon 335
12.9* Fourier Integrals 338
Exercises 342
Bibliography 347

A Historian Looks Back: The Calculus as Algebra and Selected Writings

(Spectrum)

Judith V. Grabiner

 Mathematical Association of America |  2010 |304 páginas | rar - pdf | 1,7 Mb

link (password: matav)

Judith Grabiner, the author of A Historian Looks Back, has long been interested in investigating what mathematicians actually do, and how mathematics actually has developed. She addresses the results of her investigations not principally to other historians, but to mathematicians and teachers of mathematics. This book brings together much of what she has had to say to this audience.

The centerpiece of the book is The Calculus as Algebra: J.-L. Lagrange, 1736-1813. The book describes the achievements, setbacks, and influence of Lagrange s pioneering attempt to reduce the calculus to algebra. Nine additional articles round out the book describing the history of the derivative; the origin of delta-epsilon proofs; Descartes and problem solving; the contrast between the calculus of Newton and Maclaurin, and that of Lagrange; Maclaurin s way of doing mathematics and science and his surprisingly important influence; some widely held myths about the history of mathematics; Lagrange s attempt to prove Euclid s parallel postulate; and the central role that mathematics has played throughout the history of western civilization.

The development of mathematics cannot be programmed or predicted. Still, seeing how ideas have been formed over time and what the difficulties were can help teachers find new ways to explain mathematics. Appreciating its cultural background can humanize mathematics for students. And famous mathematicians struggles and successes should interest -- and perhaps inspire -- researchers. Readers will see not only what the mathematical past was like, but also how important parts of the mathematical present came to be.

Contents

Introduction .. . . xi

Part I. The Calculus as Algebra .. . . .1

Preface to the Garland Edition .. .3

Acknowledgement. . . .7

Introduction . . . . .9

1. The Development of Lagrange’s Ideas on the Calculus: 1754–1797 .. . 17

2. The Algebraic Background of the Theory of Analytic Functions . . . 37

3. The Contents of the Fonctions Analytiques . . . 63

4. From Proof-Technique to Definition: The Pre-History of Delta-Epsilon Methods . . 81

Conclusion .  . . .101

Appendix . .103

Bibliography .. . .105

Part II. Selected Writings  . .125

1. The Mathematician, the Historian, and the History of Mathematics . . . .127

2. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus∗ . . . 135

3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass† . . 147

4. The Centrality of Mathematics in the History of Western Thought† . . . 163

5. Descartes and Problem-Solving† . . . .175

6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy . .191

7. Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions∗ . . 209

8. Newton, Maclaurin, and the Authority of Mathematics∗ .. .229

9. Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while PromotinMaths . . .243

10. Why Did Lagrange “Prove” the Parallel Postulate?∗ .  . 257

Index .. . 275

About the Author .  . .287

Everyday Calculus: Discovering the Hidden Math All around Us

Oscar E. Fernandez 

Princeton University Press | 2014 | 165 páginas | rar - pdf | 1,5 Mb

link (password : matav)

Calculus. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun, accessible, and surrounds us everywhere we go. In Everyday Calculus, Oscar Fernandez shows us how to see the math in our coffee, on the highway, and even in the night sky.

Fernandez uses our everyday experiences to skillfully reveal the hidden calculus behind a typical day's events. He guides us through how math naturally emerges from simple observations--how hot coffee cools down, for example--and in discussions of over fifty familiar events and activities. Fernandez demonstrates that calculus can be used to explore practically any aspect of our lives, including the most effective number of hours to sleep and the fastest route to get to work. He also shows that calculus can be both useful--determining which seat at the theater leads to the best viewing experience, for instance--and fascinating--exploring topics such as time travel and the age of the universe. Throughout, Fernandez presents straightforward concepts, and no prior mathematical knowledge is required. For advanced math fans, the mathematical derivations are included in the appendixes.

Whether you're new to mathematics or already a curious math enthusiast, Everyday Calculus invites you to spend a day discovering the calculus all around you. The book will convince even die-hard skeptics to view this area of math in a whole new way.

CONTENTS

Preface ix

Calculus Topics Discussed by Chapter xi

CHAPTER 1 Wake Up and Smell the Functions 1

What’s Trig Got to Do with Your Morning? 2

How a Rational Function Defeated Thomas Edison, and Why Induction Powers the World 5

The Logarithms Hidden in the Air 10

The Frequency of Trig Functions 14

Galileo’s Parabolic Thinking 17

CHAPTER 2 Breakfast at Newton’s 21

Introducing Calculus, the CNBC Way 21

Coffee Has Its Limits 25

A Multivitamin a Day Keeps the Doctor Away 30

Derivatives Are about Change 34

CHAPTER 3 Driven by Derivatives 35

Why Do We Survive Rainy Days? 36

Politics in Derivatives, or Derivatives in Politics? 39

What the Unemployment Rate Teaches Us about the Curvature of Graphs 41

America’s Ballooning Population 44

Feeling Derivatives 46

The Calculus of Time Travel 47

CHAPTER 4 Connected by Calculus 51

E-Mails, Texts, Tweets, Ah! 51

The Calculus of Colds 53

What Does Sustainability Have to Do with Catching a Cold? 56

What Does Your Retirement Income Have to Do with Traffic? 58

The Calculus of the Sweet Tooth 61

CHAPTER 5 Take a Derivative and You’ll Feel Better 65

I “Heart” Differentials 65

How Life (and Nature) Uses Calculus 67

The Costly Downside of Calculus 73

The Optimal Drive Back Home 75

Catching Speeders Efficiently with Calculus 77

CHAPTER 6 Adding Things Up, the Calculus Way 81

The Little Engine That Could . . . Integrate 82

The Fundamental Theorem of Calculus 90

Using Integrals to Estimate Wait Times 93

CHAPTER 7 Derivatives Integrals: The Dream Team 97

Integration at Work—Tandoori Chicken 98

Finding the Best Seat in the House 101

Keeping the T Running with Calculus 104

Look Up to Look Back in Time 108

The Ultimate Fate of the Universe 109

The Age of the Universe 113

Epilogue 116

Appendix A Functions and Graphs 119

Appendices 1–7 125

Notes 147

Index 149

Forgotten Calculus

Barbara Lee Bleau

Barron's Educational Series | 2001 - 3ª edição | 480 páginas | rar - mobi | 10 Mb

link (password: matav)

Updated and expanded to include the optional use of graphing calculators, this combination textbook and workbook is a good teach-yourself refresher course for men and women who took a calculus course in school, have since forgotten most of what they learned, and now need some practical calculus for business purposes or advanced education. The book is also very useful as a supplementary text for students who are taking calculus and finding it a struggle. Each progressive work unit offers clear instruction and worked-out examples. Special emphasis has been placed on business and economic applications. Topics covered include functions and their graphs, derivatives, optimization problems, exponential and logarithmic functions, integration, and partial derivatives.

From Calculus to Computers Using the last 200 years of mathematics history in the classroom

(Mathematical Association of America Notes)

Amy Shell-Gellasch e Dick Jardine

The Mathematical Association of America | 2005 | 268 páginas | rar - pdf | 1,9 Mb

link (password: matav)

To date, much of the literature prepared on the topic of integrating mathematics history into undergraduate teaching contains, predominantly, ideas from the 18th century and earlier. This volume focuses on nineteenth- and twentieth-century mathematics, building on the earlier efforts but emphasizing recent history in the teaching of mathematics, computer science, and related disciplines. From Calculus to Computers is a resource for undergraduate teachers that provides ideas and materials for immediate adoption in the classroom and proven examples to motivate innovation by the reader. Contributions to this volume are from historians of mathematics and college mathematics instructors with years of experience and expertise in these subjects. Examples of topics covered are probability in undergraduate statistics courses, logic and programming for computer science, undergraduate geometry to include non-Euclidean geometries, numerical analysis, and abstract algebra.
Emphasizes mathematics history from the nineteenth and twentieth centuries
Provides ideas and material for immediate adoption in the classroom
Topics covered range from Galois theory to using the history of women and minorities in teaching

Table of Contents
Preface
Introduction
Part I. Algebra, Number Theory, Calculus, and Dynamical Systems:
1. Arthur Cayley and the first paper on group theory David J. Pengelley
2. Putting the differential back into differential calculus Robert Rogers
3. Using Galois' idea in the teaching of abstract algebra Matt D. Lunsford
4. Teaching elliptic curves using original sources Lawrence D'Antonio
5. Using the historical development of predator-prey models to teach mathematical modeling Holly P. Hirst
Part II. Geometry:
6. How to use history to clarify common confusions in geometry Daina Taimina and David W. Henderson
7. Euler on Cevians Eisso J. Atzema and Homer White
8. Modern geometry after the end of mathematics Jeff Johannes
Part III. Discrete Mathematics, Computer Science, Numerical Methods, Logic, and Statistics:
9. Using 20th century history in a combinatorics and graph theory class Linda E. MacGuire
10. Public key cryptography Shai Simonson
11. Introducing logic via Turing machines Jerry M. Lodder
12. From Hilbert's program to computer programming William Calhoun
13. From the tree method in modern logic to the beginning of automated theorem proving Francine F. Abeles
14. Numerical methods history projects Dick Jardine
15. Foundations of Statistics in American Textbooks: probability and pedagogy in historical context Patti Wilger Hunter
Part IV. History of Mathematics and Pedagogy:
16. Incorporating the mathematical achievements of women and minority mathematicians into classrooms Sarah J. Greenwald
17. Mathematical topics in an undergraduate history of science course David Lindsay Roberts
18. Building a history of mathematics course from a local perspective Amy Shell-Gellasch
19. Protractors in the classroom: an historical perspective Amy Ackerberg-Hastings
20. The metric system enters the American classroom:
1790-1890 Peggy Aldrich Kidwell
21. Some wrinkles for a history of mathematics course Peter Ross
22. Teaching history of mathematics through problems John R. Prather

Journey through Mathematics Creative Episodes in Its History

Enrique A. González-Velasco

Springer | 2011| 478 páginas | pdf | 4,4 Mb

link direto
link

This book offers an accessible and in-depth look at some of the most  important episodes of two thousand years of mathematical history. Beginning with trigonometry and moving on through logarithms, complex numbers, infinite series, and calculus, this book profiles some of the lesser known but crucial contributors to modern day mathematics. It is unique in its use of primary sources as well as its accessibility; a knowledge of first-year calculus is the only prerequisite. But undergraduate and graduate students alike will appreciate this glimpse into the fascinating process of mathematical creation.

The history of math is an intercontinental journey, and this book showcases brilliant mathematicians from Greece, Egypt, and India, as well as Europe and the Islamic world. Several of the primary sources have never before been translated into English. Their interpretation is thorough and readable, and offers an excellent background for teachers of high school mathematics as well as anyone interested in the history of math.

TABLE OF CONTENTS
Preface ix
1 TRIGONOMETRY 1
1.1 The Hellenic Period 1
1.2 Ptolemy’s Table of Chords 10
1.3 The Indian Contribution 25
1.4 Trigonometry in the Islamic World 34
1.5 Trigonometry in Europe 55
1.6 From Viète to Pitiscus 65
2 LOGARITHMS 78
2.1 Napier’s First Three Tables 78
2.2 Napier’s Logarithms 88
2.3 Briggs’ Logarithms 101
2.4 Hyperbolic Logarithms 117
2.5 Newton’s Binomial Series 122
2.6 The Logarithm According to Euler 136
3 COMPLEX NUMBERS 148
3.1 The Depressed Cubic 148
3.2 Cardano’s Contribution 150
3.3 The Birth of Complex Numbers 160
3.4 Higher-Order Roots of Complex Numbers 173
3.5 The Logarithms of Complex Numbers 181
3.6 Caspar Wessel’s Breakthrough 185
3.7 Gauss and Hamilton Have the Final Word 190
4 INFINITE SERIES 195
4.1 The Origins 195
4.2 The Summation of Series 203
4.3 The Expansion of Functions 212
4.4 The Taylor and Maclaurin Series 220
5 THE CALCULUS 230
5.1 The Origins 230
5.2 Fermat’s Method of Maxima and Minima 234
5.3 Fermat’s Treatise on Quadratures 248
5.4 Gregory’s Contributions 258
5.5 Barrow’s Geometric Calculus 275
5.6 From Tangents to Quadratures 283
5.7 Newton’s Method of Infinit Series 289
5.8 Newton’s Method of Fluxions 294
5.9 Was Newton’s Tangent Method Original? 302
5.10 Newton’s First and Last Ratios 306
5.11 Newton’s Last Version of the Calculus 312
5.12 Leibniz’ Calculus: 1673–1675 318
5.13 Leibniz’ Calculus: 1676–1680 329
5.14 The Arithmetical Quadrature 340
5.15 Leibniz’ Publications 349
5.16 The Aftermath 358
6 CONVERGENCE 368
6.1 To the Limit 368
6.2 The Vibrating String Makes Waves 369
6.3 Fourier Puts on the Heat 373
6.4 The Convergence of Series 380
6.5 The Difference Quotient 394
6.6 The Derivative 401
6.7 Cauchy’s Integral Calculus 405
6.8 Uniform Convergence 407
BIBLIOGRAPHY 412

Elliptic Tales: Curves, Counting, and Number Theory

Avner Ash e Robert Gross 

Princeton University Press | 2012 | 276 páginas | pdf | 3 Mb

link direto
link

Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.

Contents
Prologue 1
PART I. DEGREE
Chapter1. Degree of aCurve 13
1. Greek Mathematics 13
2. Degree 14
3. Parametric Equations 20
4. Our Two Definitions of Degree Clash 23
Chapter 2. AlgebraicClosures 26
1. Square Roots of Minus One 26
2. Complex Arithmetic 28
3. Rings and Fields 30
4. Complex Numbers and Solving Equations 32
5. Congruences 34
6. Arithmetic Modulo a Prime 38
7. Algebraic Closure 38
Chapter3. The Projective Plane 42
1. Points at Infinity 42
2. Projective Coordinates on a Line 46
3. Projective Coordinates on a Plane 50
4. Algebraic Curves and Points at Infinity 54
5. Homogenization of Projective Curves 56
6. Coordinate Patches 61
Chapter 4. Multiplicities and Degree 67
1. Curves as Varieties 67
2. Multiplicities 69
3. Intersection Multiplicities 72
4. Calculus for Dummies 76
Chapter5. Bezout’s ´ Theorem 82
1. A Sketch of the Proof 82
2. An Illuminating Example 88
PART II. ELLIPTIC CURVES AND ALGEBRA
Chapter6. Transition to EllipticCurves 95
Chapter7. Abelian Groups 100
1. How Big Is Infinity? 100
2. What Is an Abelian Group? 101
3. Generations 103
4. Torsion 106
5. Pulling Rank 108
Appendix: An Interesting Example of Rank and Torsion 110
Chapter8. Nonsingular Cubic Equations 116
1. The Group Law 116
2. Transformations 119
3. The Discriminant 121
4. Algebraic Details of the Group Law 122
5. Numerical Examples 125
6. Topology 127
7. Other Important Facts about Elliptic Curves 131
5. Two Numerical Examples 133
Chapter9. Singular Cubics 135
1. The Singular Point and the Group Law 135
2. The Coordinates of the Singular Point 136
3. Additive Reduction 137
4. Split Multiplicative Reduction 139
5. Nonsplit Multiplicative Reduction 141
6. Counting Points 145
7. Conclusion 146
Appendix A: Changing the Coordinates of the Singular Point 146
Appendix B: Additive Reduction in Detail 147
Appendix C: Split Multiplicative Reduction in Detail 149
Appendix D: Nonsplit Multiplicative Reduction in Detail 150
Chapter10. EllipticCurves over Q 152
1. The Basic Structure of the Group 152
2. Torsion Points 153
3. Points of Infinite Order 155
4. Examples 156
PART III. ELLIPTIC CURVES AND ANALYSIS
Chapter11. Building Functions 161
1. Generating Functions 161
2. Dirichlet Series 167
3. The Riemann Zeta-Function 169
4. Functional Equations 171
5. Euler Products 174
6. Build Your Own Zeta-Function 176
Chapter12. AnalyticContinuation 181
1. A Difference that Makes a Difference 181
2. Taylor Made 185
3. Analytic Functions 187
4. Analytic Continuation 192
5. Zeroes, Poles, and the Leading Coefficient 196
Chapter13.L-functions 199
1. A Fertile Idea 199
2. The Hasse-Weil Zeta-Function 200
3. The L-Function of a Curve 205
4. The L-Function of an Elliptic Curve 207
5. Other L-Functions 212
Chapter14. Surprising Properties of L-functions 215
1. Compare and Contrast 215
2. Analytic Continuation 220
3. Functional Equation 221
Chapter15. TheConjecture of Birch and Swinnerton-Dyer 225
1. How Big Is Big? 225
2. Influences of the Rank on the Np’s 228
3. How Small Is Zero? 232
4. The BSD Conjecture 236
5. Computational Evidence for BSD 238
6. The Congruent Number Problem 240
Epilogue 245
Retrospect 245
Where Do We Go from Here? 247

Sugestão de tibu

Distilling Ideas An Introduction to Mathematical Thinking

Brian P. Katz e Michael Starbird
Mathematical Association of America | 2013 | 188 páginas | pdf | 2,1 Mb

link direto
rar - pdf - 1,7 Mb - link

Mathematics is not a spectator sport: successful students of mathematics grapple with ideas for themselves. Distilling Ideaspresents a carefully designed sequence of exercises and theorem statements that challenge students to create proofs and concepts. As students meet these challenges, they discover strategies of proofs and strategies of thinking beyond mathematics. In order words, Distilling Ideas helps its users to develop the skills, attitudes, and habits of mind of a mathematician and to enjoy the process of distilling and exploring ideas.
Distilling Ideas is an ideal textbook for a first proof-based course. The text engages the range of students' preferences and aesthetics through a corresponding variety of interesting mathematical content from graphs, groups, and epsilon-delta calculus. Each topic is accessible to users without a background in abstract mathematics because the concepts arise from asking questions about everyday experience. All the common proof structures emerge as natural solutions to authentic needs.Distilling Ideas or any subset of its chapters is an ideal resource either for an organized Inquiry Based Learning course or for individual study.
A student response to Distilling Ideas: "I feel that I have grown more as a mathematician in this class than in all the other classes I've ever taken throughout my academic life."

Contents
Preface ix
1 Introduction 1
1.1 Proof and Mathematical Inquiry . . . . . . . . . . 1
2 Graphs 5
2.1 The Konigsberg Bridge Problem . . . . . . . . . . 5
2.2 Connections . . . . . . . . . . . . . . . . . . . . . 6
2.3 Taking a Walk . . . . . . . . . . . . . . . . . . . . 16
2.4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Planarity . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Euler Characteristic . . . . . . . . . . . . . . . . . 27
2.7 Symmetries . . . . . . . . . . . . . . . . . . . . . 32
2.8 Colorability . . . . . . . . . . . . . . . . . . . . . 34
2.9 Completing theWalk around Graph Theory . . . . 40
3 Groups 43
3.1 Examples Lead to Concepts . . . . . . . . . . . . . 43
3.2 Clock-Inspired Groups . . . . . . . . . . . . . . . 52
3.3 Symmetry Groups of Regular Polygons . . . . . . 55
3.4 Subgroups, Generators, and Cyclic Groups . . . . . 56
3.5 Sizes of Subgroups and Orders of Elements . . . . 62
3.6 Products of Groups . . . . . . . . . . . . . . . . . 64
3.7 Symmetric Groups . . . . . . . . . . . . . . . . . 65
3.8 Maps between Groups . . . . . . . . . . . . . . . . 68
3.9 Normal Subgroups and Quotient Groups . . . . . . 76
3.10 More Examples* . . . . . . . . . . . . . . . . . . 80
3.11 Groups in Action* . . . . . . . . . . . . . . . . . . 81
3.12 The Man Behind the Curtain . . . . . . . . . . . . 85
4 Calculus 89
4.1 Perfect Picture . . . . . . . . . . . . . . . . . . . . 89
4.2 Convergence . . . . . . . . . . . . . . . . . . . . . 91
4.3 Existence of Limits . . . . . . . . . . . . . . . . . 104
4.4 Continuity . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Zeno’s ParadoxTM . . . . . . . . . . . . . . . . . 123
4.6 Derivatives . . . . . . . . . . . . . . . . . . . . . 127
4.7 Speedometer Movie and Position . . . . . . . . . . 135
4.8 Applications of the Definite Integral . . . . . . . . 137
4.9 Fundamental Theorem of Calculus . . . . . . . . . 141
4.10 From Vague to Precise . . . . . . . . . . . . . . . 145
5 Conclusion 149
5.1 Distilling Ideas . . . . . . . . . . . . . . . . . . . 149
Annotated Index 153
List of Symbols 165

Sugestão de tibu

Calculus and its Origins

(Spectrum)
David Perkins

The Mathematical Association of America | 2012 | 180 páginas | pdf | 3,7 Mb

link direto
link

Calculus answers questions that had been explored for centuries before calculus was born. Calculus and Its Origins begins with these ancient questions and details the remarkable story of how subsequent scholars wove these inquiries into a unified theory. This book does not presuppose knowledge of calculus, it requires only a basic knowledge of geometry and algebra (similar triangles, polynomials, factoring). Inside you will find the accounts of how Archimedes discovered the area of a parabolic segment, ibn Al-Haytham calculated the volume of a revolved area, Jyesthadeva explained the infinite series for sine and cosine, Wallis deduced the link between hyperbolas and logarithms, Newton generalized the binomial theorem, Leibniz discovered integration by parts, and much more. Each chapter ends with further results, in the form of exercises, by such luminaries as Pascal, Maclaurin, Barrow, Cauchy and Euler.

Contents
Preface ix
1 The Ancients 1
1.1 Zeno holds a mirror to the infinite . . 2
1.2 The ‘infinitely small’  . . 4
1.3 Archimedes exhausts a parabolic segment . . 5
1.4 Patterns . . 7
1.5 The evolution of notation  . . 10
1.6 Furthermore  . . 10
2 East of Greece 15
2.1 Ibn al-Haytham sums the fourth powers . . 15
2.2 Ibn al-Haytham’s parabolic volume . . 17
2.3 Jyesthadeva expands 1/(1+x)  . . 20
2.4 Jyesthadeva expresses as a series . . 23
2.5 Furthermore. . 26
3 Curves 29
3.1 Oresme invents a precursor to a coordinate system . . 29
3.2 Fermat studies the maximums of curves. . 32
3.3 Fermat extends his method to tangent lines . . 33
3.4 Descartes proposes a geometric method . . 35
3.5 Furthermore . . 37
4 Indivisibles 43
4.1 Cavalieri’s quadrature of the parabola . . 43
4.2 Roberval’s quadrature of the cycloid  . . 47
4.3 Worry over indivisibles. . 50
4.4 Furthermore. . 51
5 Quadrature 57
5.1 Gregory studies hyperbolas . . 57
5.2 De Sarasa invokes logarithms . . 59
5.3 Brouncker finds a quadrature of a hyperbola. . 61
5.4 Mercator andWallis finish the task . . 63
5.5 Furthermore  . . 65
6 The Fundamental Theorem of Calculus 77
6.1 Newton links quadrature to rate of change  . . 77
6.2 Newton reverses the link  . . 79
6.3 Leibniz discovers the transmutation theorem  . . 81
6.4 Leibniz attains Jyesthadeva’s series for  pi. . 83
6.5 Furthermore  . . 85
7 Notation 95
7.1 Leibniz describes differentials. . 95
7.2 The fundamental theorem with new notation  . . 98
7.3 Leibniz integrates the cycloid . . 100
7.4 Furthermore . . 102
8 Chords 109
8.1 Preliminary results known to the Greeks. . 109
8.2 Jyesthadeva finds series for sine and cosine. . 110
8.3 Newton derives a series for arcsine . . 116
8.4 Furthermore . . 118
9 Zero over zero 123
9.1 D’Alembert and the convergence of series  . . 123
9.2 Lagrange defines the ‘derived function’. . 126
9.3 Taylor approximates functions. . 128
9.4 Bolzano and Cauchy define convergence  . . 130
9.5 Furthermore . . 133
10 Rigor 137
10.1 Cauchy defines continuity . . 137
10.2 Bolzano invents a peculiar function . . 140
10.3 Weierstrass investigates the convergence of functions. . 145
10.4 Dirichlet’s nowhere-continuous function . . 147
10.5 A few final words about the infinite. . 149
10.6 Furthermore . . 150

Sugestão de tibu

Explorations in Complex Analysis

Michael A. Brilleslyper, Michael J. Dorff, Jane M. McDougall, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, Kenneth Stephenson


Classroom Resource Materials

Mathematical Association of America | 2012 | 392 páginas | rar - pdf | 29,5 Mb

link
(password: matav)

This book is written for mathematics students who have encountered basic complex analysis and want to explore more advanced project and/or research topics. It could be used as (a) a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics, (b) a project resource for a senior capstone course for mathematics majors, (c) a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or (d) a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis. Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation. There are more than 15 Java applets that allow students to explore the research topics without the need for purchasing additional software.