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Calculus

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In addition to devel­ oping 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. His aim is to present calculus as the first real encounter with it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote.

Such a procedure is feasible, but would soon cease to be if we considered collections of six, seven, or more numbers; it would be totally inadequate to prove the equality of all possible sums of an arbitrary finite collection of numbers ai,. Jane Bjorkgren performed prodigious feats of typing that compensated for my fitful production of the manuscript.If you like--or think you might like--the mathematician's ever-refining pursuit for rigour, for watertight argument, for abstract thought--then you'll love this book. It might seem reasonable to regard addition as an operation which can be performed on several numbers at once, and consider the sum a\ + • • • + an of n numbers a\,. Al principio puede hacerse algo espeso cuando no se está familiarizado con el tipo de razonamiento, pero creo que a la postre merece mucho la pena.

Eso está muy bien porque las matemáticas son teorías de formas puras en las que cada concepto está saturado de teoría. They range from straightforward, but not overly simple, exercises which develop basic techniques and test understanding of concepts, to problems of considerable diffi­ culty' and, I hope, of comparable interest.But seriously--if you're looking for practical application or pragmatic knowledge then this book isn't exactly for you. It is a good book for interesting problems, however this treatment will not make you any more successful in a Calculus course, perhaps the opposite since you are likely to be in a course that treats Calculus in a different way. The proof of this assertion involves nothing more than subtracting a from both sides of the equation, in other words, adding —a to both sides; as the following detailed proof shows, all three properties P1-P3 must be used to justify this operation. Although small changes have been made to some material, especially in Chap­ ters 5 and 20, this edition differs mainly in the introduction of additional problems, a complete update of the Suggested Reading, and the correction of numerous er­ rors. The most significant change in this third edition is the inclusion of a new (starred) Chapter 17 on planetary motion, in which calculus is employed for a substantial physics problem.

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