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ISBN13: 978-0914098898

ISBN10: 0914098896 Edition: 3RD 94

Copyright: 1994

Publisher: Publish or Perish, Inc.

Published: 1994

International: No

ISBN10: 0914098896 Edition: 3RD 94

Copyright: 1994

Publisher: Publish or Perish, Inc.

Published: 1994

International: No

Written as a textbook, Calculus is used both for theoretical calculus courses and for ''Introduction to Analysis'' courses in several U.S. universities. It is also used (in rather larger numbers) in quite a few Canadian universities, as well as in a few universities overseas. Because of this, sales of the Answer Book, which are intended for instructors of the courses, are normally restricted.

Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of mathematical thinking developed, calculus ought to be the place in which to expect, rather than avoid, the strengthening of insight with logic. In addition to developing the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions.

Part I Prologue

1 Basic Properties of Numbers

2 Numbers of Various Sorts

Part II Foundations

3 Functions

Appendix. Ordered Pairs

4 Graphs

Appendix 1. Vectors

Appendix 2. The Conic Sections

Appendix 3. Polar Coordinates

5 Limits

6 Continuous Functions

7 Three Hard Theorems

8 Least Upper Bounds

Appendix. Uniform Continuity

Part III Derivatives and Integrals

9 Derivatives

10 Differentiation

11 Significance of the Derivative

Appendix. Convexity and Concavity

12 Inverse Functions

Appendix. Parametric Representation of Curves

13 Integrals

Appendix. Riemann Sums

14 The Fundamental Theorem of Calculus

15 The Trigonometric Functions

*16 Pi is Irrational

*17 Planetary Motion

18 The Logarithm and Exponential Functions

19 Integration in Elementary Terms

Appendix. The Cosmopolitan Integral

Part IV Infinite Sequences and Infinite Series

20 Approximation by Polynomial Functions

*21 e is Transcendental

22 Infinite Sequences

23 Infinite Series

24 Uniform Convergence and Power Series

25 Complex Numbers

26 Complex Functions

27 Complex Power Series

Part V Epilogue

28 Fields

29 Construction of the Real Numbers

30 Uniqueness of the Real Numbers

Suggested Reading

ISBN10: 0914098896 Edition: 3RD 94

Copyright: 1994

Publisher: Publish or Perish, Inc.

Published: 1994

International: No

Written as a textbook, Calculus is used both for theoretical calculus courses and for ''Introduction to Analysis'' courses in several U.S. universities. It is also used (in rather larger numbers) in quite a few Canadian universities, as well as in a few universities overseas. Because of this, sales of the Answer Book, which are intended for instructors of the courses, are normally restricted.

Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of mathematical thinking developed, calculus ought to be the place in which to expect, rather than avoid, the strengthening of insight with logic. In addition to developing the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions.

Table of Contents

Part I Prologue

1 Basic Properties of Numbers

2 Numbers of Various Sorts

Part II Foundations

3 Functions

Appendix. Ordered Pairs

4 Graphs

Appendix 1. Vectors

Appendix 2. The Conic Sections

Appendix 3. Polar Coordinates

5 Limits

6 Continuous Functions

7 Three Hard Theorems

8 Least Upper Bounds

Appendix. Uniform Continuity

Part III Derivatives and Integrals

9 Derivatives

10 Differentiation

11 Significance of the Derivative

Appendix. Convexity and Concavity

12 Inverse Functions

Appendix. Parametric Representation of Curves

13 Integrals

Appendix. Riemann Sums

14 The Fundamental Theorem of Calculus

15 The Trigonometric Functions

*16 Pi is Irrational

*17 Planetary Motion

18 The Logarithm and Exponential Functions

19 Integration in Elementary Terms

Appendix. The Cosmopolitan Integral

Part IV Infinite Sequences and Infinite Series

20 Approximation by Polynomial Functions

*21 e is Transcendental

22 Infinite Sequences

23 Infinite Series

24 Uniform Convergence and Power Series

25 Complex Numbers

26 Complex Functions

27 Complex Power Series

Part V Epilogue

28 Fields

29 Construction of the Real Numbers

30 Uniqueness of the Real Numbers

Suggested Reading

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