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Calculus Without Limits

Posted: Tue Nov 03, 2009 5:28 am by ndaru

Calculus Without Limits

Author : Karl Heinz Dovermann, Professor of Mathematics, University of Hawaii
Publication Date : July 5, 1999

License Reminder:

Karl Heinz Dovermann wrote:

(C)Copyright 1999 by the author. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. Printed in the United States of America.

Excerpts from the Book:

Karl Heinz Dovermann wrote:

These notes are written for a one-semester calculus course which meets three times a week and is, preferably, supported by a computer lab. The course is designed for life science majors who have a precalculus background, and whose primary interest lies in the applications of calculus. We try to focus on those topics which are of greatest importance to them and use life science examples to illustrate them. At the same time, we try of stay mathematically coherent without becoming technical. To make this feasible, we are willing to sacrifice generality. There is less of an emphasis on by hand calculations. Instead, more complex and demanding problems find their place in a computer lab. In this sense, we are trying to adopt several ideas from calculus reform. Among them is a more visual and less analytic approach. We typically explore new ideas in examples before we give formal definitions.

In one more way we depart radically from the traditional approach to calculus. We introduce differentiability as a local property without using limits. The philosophy behind this idea is that limits are the a big stumbling block for most students who see calculus for the first time, and they take up a substantial part of the first semester. Though mathematically rigorous, our approach to the derivative makes no use of limits, allowing the students to get quickly and without unresolved problems to this concept. It is true that our definition is more restrictive than the ordinary one, and fewer functions are differentiable in this manuscript than in a standard text. But the functions which we do not recognize as being differentiable are not particularly important for students who will take only one semester of calculus. In addition, in our opinion the underlying geometric idea of the derivative is at least as clear in our approach as it is in the one using limits.

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A New Calculus without the use of limits.

Posted: Wed Oct 12, 2011 10:45 am by john_gabriel

Dovermann's book is interesting, but unfortunately it is incomplete and still relies on limit theory. He explains the Lipschitz condition, which uses the methods of real analysis (i.e. limits) to establish the general method.

My new calculus does not use any of the theory of real analysis. I have also shown that Cauchy's definition of the derivative is a fake and explain why it works.
Limits are not the reason calculus works and are not required at all in any mathematics.

To read about the new calculus visit:

http://thenewcalculus.weebly.com