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Putting Differentials Back Into Calculus
Putting Differentials Back into Calculus Tevian Dray Corinne A. Manogue Department of Mathematics Department of Physics Oregon State University Oregon State University Corvallis, OR 97331 Corvallis, OR 97331 [email protected] [email protected] September 14, 2009 ... many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children. Bill McCallum [16] Calculus is rightly viewed as one of the intellectual triumphs of modern civilization. Math- ematicians are also justly proud of the work of Cauchy and his contemporaries in the early 19th century, who provided rigorous justification of the methods introduced by Newton and Leibniz 150 years previously. Leibniz introduced the language of differentials to describe the calculus of infinitesimals, which were later ridiculed by Berkeley as “ghosts of departed quantities” [4]. Modern calculus texts mention differentials only in passing, if at all. Nonetheless, it is worth remembering that calculus was used successfully during those 150 years. In practice, many scientists and engineers continue to this day to apply calculus by manipulating differentials, and for good reason. It works. The differentials of Leibniz [15] — not the weak imitation found in modern texts — capture the essence of calculus, and should form the core of introductory calculus courses. 1 Practice The preliminary terror ... can be abolished once for all by simply stating what is the meaning — in common-sense terms — of the two principal symbols that are used in calculating. These dreadful symbols are: (1) d, which merely means “a little bit of”. -
Calculus Without Limits—Almost
Calculus Without Limits—Almost By John C. Sparks 1663 LIBERTY DRIVE, SUITE 200 BLOOMINGTON, INDIANA 47403 (800) 839-8640 www.authorhouse.com Calculus Without Limits Copyright © 2004, 2005 John C. Sparks All rights reserved. No part of this book may be reproduced in any form—except for the inclusion of brief quotations in a review— without permission in writing from the author or publisher. The exceptions are all cited quotes, the poem “The Road Not Taken” by Robert Frost (appearing herein in its entirety), and the four geometric playing pieces that comprise Paul Curry’s famous missing-area paradox. Back cover photo by Curtis Sparks ISBN: 1-4184-4124-4 First Published by Author House 02/07/05 2nd Printing with Minor Additions and Corrections Library of Congress Control Number: 2004106681 Published by AuthorHouse 1663 Liberty Drive, Suite 200 Bloomington, Indiana 47403 (800)839-8640 www.authorhouse.com Printed in the United States of America Dedication I would like to dedicate Calculus Without Limits To Carolyn Sparks, my wife, lover, and partner for 35 years; And to Robert Sparks, American warrior, and elder son of geek; And to Curtis Sparks, reviewer, critic, and younger son of geek; And to Roscoe C. Sparks, deceased, father of geek. From Earth with Love Do you remember, as do I, When Neil walked, as so did we, On a calm and sun-lit sea One July, Tranquillity, Filled with dreams and futures? For in that month of long ago, Lofty visions raptured all Moonstruck with that starry call From life beyond this earthen ball.. -
Aaboe, Asger Episodes from the Early History of Mathematics QA22 .A13 Abbott, Edwin Abbott Flatland: a Romance of Many Dimensions QA699 .A13 1953 Abbott, J
James J. Gehrig Memorial Library _________Table of Contents_______________________________________________ Section I. Cover Page..............................................i Table of Contents......................................ii Biography of James Gehrig.............................iii Section II. - Library Author’s Last Name beginning with ‘A’...................1 Author’s Last Name beginning with ‘B’...................3 Author’s Last Name beginning with ‘C’...................7 Author’s Last Name beginning with ‘D’..................10 Author’s Last Name beginning with ‘E’..................13 Author’s Last Name beginning with ‘F’..................14 Author’s Last Name beginning with ‘G’..................16 Author’s Last Name beginning with ‘H’..................18 Author’s Last Name beginning with ‘I’..................22 Author’s Last Name beginning with ‘J’..................23 Author’s Last Name beginning with ‘K’..................24 Author’s Last Name beginning with ‘L’..................27 Author’s Last Name beginning with ‘M’..................29 Author’s Last Name beginning with ‘N’..................33 Author’s Last Name beginning with ‘O’..................34 Author’s Last Name beginning with ‘P’..................35 Author’s Last Name beginning with ‘Q’..................38 Author’s Last Name beginning with ‘R’..................39 Author’s Last Name beginning with ‘S’..................41 Author’s Last Name beginning with ‘T’..................45 Author’s Last Name beginning with ‘U’..................47 Author’s Last Name beginning -
Calculus Made Easy Books by Martin Gardner
CALCULUS MADE EASY BOOKS BY MARTIN GARDNER Fads and Fallacies in the Name Order and Surprise of Science The Whys of a Philosophical Mathematics, Magic, and Mystery Scrivener Great Essays in Science (ed.) Puzzles from Other Worlds Logic Machines and Diagrams The Magic Numbers of Dr. The Scientific American Book of Matrix Mathematical Puzzles and Knotted Doughnuts and Other Diversions Mathematical Entertainments The Annotated Alice The Wreck of the Titanic Foretold The Second Scientific American Riddles of the Sphinx Book of Mathematical Puzzles The Annotated Innocence of and Diversions Father Brown Relativity for the Million The No-Sided Professor (short The Annotated Snark stories) The Ambidextrous Universe Time Travel and Other The Annotated Ancient Mariner Mathematical Bewilderments New Mathematical Diversions The New Age: Notes of a Fringe from Scientific American Watcher The Annotated Casey at the Bat Gardner's Whys and Wherefores Perplexing Puzzles and Penrose Tiles to Trapdoor Ciphers Tantalizing Teasers How Not to Test a Psychic The Unexpected Hanging and The New Ambidextrous Universe Other Mathematical Diversions More Annotated Alice Never Make Fun of a Turtle, My The Annotated Night Before Son (verse) Christmas The Sixth Book of Mathematical Best Remembered Poems (ed.) Games from Scientific American Fractal Music, Hypercards, and Codes, Ciphers, and Secret Writing More Space Puzzles The Healing Revelations of Mary The Snark Puzzle Book Baker Eddy The Flight of Peter Fromm (novel) Martin Gardner Presents Mathematical Magic Show -
Calcconf2019 Papers 190910.Pdf
Calculus in upper secondary and beginning university mathematics Conference held at the University of Agder, Kristiansand, Norway August 6-9, 2019 This conference was supported by the Centre for Research, Innovation and Coordination of Mathematics Teaching (MatRIC) Edited by John Monaghan, Elena Nardi and Tommy Dreyfus Suggested reference: J. Monaghan, E. Nardi and T. Dreyfus (Eds.) (2019), Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC. Retrieved on [date] from https://matric-calculus.sciencesconf.org/ Table of contents Introduction Calculus in upper secondary and beginning university mathematics 1 Plenary lectures Biehler, Rolf The transition from Calculus and to Analysis – Conceptual analyses and supporting steps for students 4 Tall, David The Evolution of Calculus: A Personal Experience 1956–2019 18 Thompson, Patrick W. Making the Fundamental Theorem of Calculus fundamental to students’ calculus 38 Plenary panel González-Martín, Alejandro S.; Mesa, Vilma; Monaghan, John; Nardi, Elena (Moderator) From Newton’s first to second law: How can curriculum, pedagogy and assessment celebrate a more dynamic experience of calculus? 66 Contributed papers Bang, Henrik; Grønbæk, Niels; Larsen Claus Teachers’ choices of digital approaches to upper secondary calculus 67 Biza, Irene Calculus as a discursive bridge for Algebra, Geometry and Analysis: The case of tangent line 71 Bos, Rogier; Doorman, Michiel; Drijvers, Paul Supporting the reinvention of slope 75 Clark, Kathleen -
Computational Geology 5 If Geology, Then Calculus
Journal of Geoscience Education, v. 47, n. 2, p. 166-175, March1999 (edits, June 2005) Computational Geology 5 If Geology, Then Calculus H.L. Vacher, Department of Geology, University of South Florida, 4202 E. Fowler Ave., Tampa FL 33620 Introduction This column is the fifth in the series, and so it closes the first year of "Computational Geology." I would be remiss if I let a year go by without discussing or giving an example of how calculus informs the practice of geology, or vice versa. Therefore, I will devote this column to the intersection of geology and calculus. Rather than discuss a computational problem illustrating the intersection, I shall try to prove the following assertion: "If you understand geology, you understand calculus." Examining the Proposition If you have had a course such as Critical Thinking or freshman logic, you will recognize the statement "If you understand geology, you understand calculus" as a compound proposition of the form, p v q (read: "If p, then q"). The p and the q each stand for an elementary (i.e., noncompound) proposition, where a proposition is defined as a statement that can assume either of only two values (T or F, called truth values; i.e., a proposition cannot have the truth value of "maybe"). In the case here, the two elementary propositions are: p = "you understand geology" q = "you understand calculus" (1) The "if-then" statement is called a conditional. The p-statement is the antecedent; the q- statement is the consequent. Just as the p and q can be only T or F, a conditional (p v q) can be only T or F (i.e., again there is no "maybe"). -
A Mind at Play an Interview with Martin Gardner
A Mind at Play An Interview with Martin Gardner KENDRICK FRAZIER is mind is highly philosophical, at home with the most abstract concepts, yet his thinking and writ Hing crackle with clarity—lively, crisp, vivid. He achieved worldwide fame and respect for the three decades of his highly popular mathematical games column for Scientific American, yet he is not a mathematician. He is by every standard an eminent intellectual, yet he has no Ph.D. or academic position. He has a deep love of science and has written memorable science books (The Ambidextrous Universe and The Relativity Explosion, for instance), and yet he has devoted probably more time and effort to—and has been more effective than any thinker of the twentieth cen tury in—exposing pseudoscience and bogus science. He is considered a hard-nosed, blunt-speaking scourge of paranormalists and all who would deceive themselves or the public in the name of science, yet in person he is a gentle, soft-spoken, even shy man who likes nothing better than to stay in his home with his beloved wife Charlotte in 34 March/April 1998 SKEPTICAL INQUIRER Hendersonville, North Carolina, and write on his electric typewriter. His critics see him as serious, yet he has a playful mind, is often more amused than outraged by nonsense, and believes with Mencken that "one horselaugh is worth ten thousand syl logisms." He is deeply knowledgeable about conjuring and delights in learning new magic tricks. He retired from Scientific American more than fifteen years ago, but his output of books, articles, and reviews has, if anything, accelerated since then. -
Putting Differentials Back Into Calculus
Putting Differentials Back into Calculus Tevian Dray Corinne A. Manogue Department of Mathematics Department of Physics Oregon State University Oregon State University Corvallis, OR 97331 Corvallis, OR 97331 [email protected] [email protected] June 18, 2009 Abstract We argue that the use of differentials in introductory calculus courses provides a unifying theme which leads to a coherent view of calculus. Along the way, we will meet several interpretations of differentials, some better than others. 1 Introduction ... many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children. Bill McCallum [16] Calculus is rightly viewed as one of the intellectual triumphs of modern civilization. Math- ematicians are also justly proud of the work of Cauchy and his contemporaries in the early 19th century, who provided rigorous justification of the methods introduced by Newton and Leibniz 150 years previously. Leibniz introduced the language of differentials to describe the calculus of infinitesimals, which were later ridiculed by Berkeley as “ghosts of departed quantities” [4]. Modern calculus texts mention differentials only in passing, if at all. Nonetheless, it is worth remembering that calculus was used successfully during those 150 years. In practice, many scientists and engineers continue to this day to apply calculus by manipulating differentials, and for good reason. It works. The differentials of Leibniz [15] — not the weak imitation found in modern texts — capture the essence of calculus, and should form the core of introductory calculus courses. 1 2 Practice The preliminary terror .. -
The Geometries of Situation and Emotion and the Calculus of Change in Negotiation and Mediation
Valparaiso University Law Review Volume 29 Number 1 Fall 1994 pp.1-120 Fall 1994 The Geometries of Situation and Emotion and the Calculus of Change in Negotiation and Mediation John W. Cooley Follow this and additional works at: https://scholar.valpo.edu/vulr Part of the Law Commons Recommended Citation John W. Cooley, The Geometries of Situation and Emotion and the Calculus of Change in Negotiation and Mediation, 29 Val. U. L. Rev. 1 (1994). Available at: https://scholar.valpo.edu/vulr/vol29/iss1/1 This Article is brought to you for free and open access by the Valparaiso University Law School at ValpoScholar. It has been accepted for inclusion in Valparaiso University Law Review by an authorized administrator of ValpoScholar. For more information, please contact a ValpoScholar staff member at [email protected]. Cooley: The Geometries of Situation and Emotion and the Calculus of Chang VALPARAISO UNIVERSITY LAW REVIEW Volume 29 Fall 1994 Number 1 Articles THE GEOMETRIES OF SITUATION AND EMOTIONS AND THE CALCULUS OF CHANGE IN NEGOTIATION AND MEDIATION JOHN W. COOLEY* I. Introduction .................................. 3 II. Leibniz-The Person; The Lawyer; The Problem Solver ...... 5 A. Leibniz-The Person .......................... 5 B. Leibniz-The Lawyer ......................... 12 C. Leibniz-The Problem Solver .................... 15 1. Solver of Mathematical Problems .................16 2. Solver of People Problems ................... 18 III. Leibniz's Geometry of Situation .................... 19 A. Problem Solving-General ...................... 19 B. The Origin of Geometry of Situation .. ..............20 John W. Cooley is a former United States Magistrate, Assistant United States Attorney, Senior Staff Attorney for the United States Court of Appeals for the Seventh Circuit, and a partner in a Chicago law firm. -
Simplifying and Refactoring Introductory Calculus
Simplifying and Refactoring Introductory Calculus Jonathan Bartlett November 9, 2018 Abstract First year calculus is often taught in a way that is very burdensome to the student. Students have to memorize a diversity of processes for essentially performing the same task. However, many calculus processes can be simplified and streamlined so that fewer concepts can provide more flexibility and capability for first-year students. 1 Introduction While the exact set of topics in any particular calculus book or course may vary, the general method of calculus training has been essentially set in stone for the last hundred years. Nonetheless, there are many issues with this methodology that have been insufficiently addressed over the years. These issues range from the sequencing of topics to the content of the topics themselves. A well-recognized phenomena in computer programming is known as “code debt.” Code debt occurs when new ideas and changes get incorporated into a program, but the rest of the program doesn’t change sufficiently to take these new ideas and changes into account. Because of this, there winds up being a lot of duplication and confusion over the right ways of doing things. It is known as code debt because eventually, to relieve the tension, future work will have to be done to the rest of the system to bring it back into alignment. To relieve code debt, computer programmers engage in in a process known as “refactoring.” The idea behind refactoring is to reconceptualize the whole of the computer program in order to discover which facilities are the core, distinguishable pieces and which ones are merely a variant or permutation of those pieces.