Calculus Without Limits (Almost)

342 Pages·2004·2.1 MB·English

by  John Sparks

Calculus
Without
Limits
—Almost

By John C. Sparks

1663 LIBERTY DRIVE, SUITE 200
BLOOMINGTON, INDIANA 47403
[phone]
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
[phone]
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...
Not wedded to its surface.

But marriage is of dust to dust
Where seasoned limbs reclaim the ground
Though passing thoughts still fly around
Supernal realms never found
On the planet of our birth.

And I, a man, love you true,
Love as God had made it so,
Not angel rust when then aglow,
But coupled here, now rib to soul,
Dear Carolyn of mine.

July 2002: 33rd Wedding Anniversary

v
Hippocrates’ Lune: Circa 440BC

This is the earliest known geometric figure
having two curvilinear boundaries for which a
planar area could be exactly determined.

vi
Forward

I first began to suspect there was something special about John
Sparks as a teacher back in 1994 when I assumed the role of department
chair and got a chance to see the outstanding evaluations he consistently
received from his students. Of course I knew that high student ratings
don’t always equate to good teaching. But as I got to know John better I
observed his unsurpassed enthusiasm, his unmitigated optimism and
sense of humor, and his freshness and sense of creativity, all important
qualities of good teaching. Then when I attended several seminars and
colloquia at which he spoke, on topics as diverse as Tornado Safety,
Attention Deficit Syndrome and Design of Experiments, I found that his
interests were wide-ranging and that he could present material in a clear,
organized and engaging manner. These are also important qualities of
good teaching. Next I encountered John Sparks the poet. From the
poems of faith and patriotism which he writes, and the emails he
periodically sends to friends, and the book of poems, Mixed Images,
which he published in 2000, I soon discovered that this engineer by trade
is a man with one foot planted firmly on each side of the intellectual divide
between the arts and the sciences. Such breadth of interest and ability is
most assuredly an invaluable component of good teaching. Now that he
has published Calculus without Limits, the rest (or at least more) of what
makes John Sparks special as a teacher has become clear. He has the
ability to break through those aspects of mathematics that some find
tedious and boring and reveal what is fascinating and interesting to
students and what engages them in the pursuit of mathematical
knowledge. By taking a fresh look at old ideas, he is able to expose the
motivating principles, the intriguing mysteries, the very guts of the matter
that are at the heart of mankind’s, and especially this author’s, abiding
love affair with mathematics. He manages to crack the often times
opaque shell of rules and formulas and algorithms to bring to light the
inner beauty of mathematics. Perhaps this completes my understanding
of what is special about John Sparks as a teacher. Or perhaps he still
has more surprises in store for me. Anyway, read this book and you will
begin to see what I mean.

Al Giambrone
Chairman
Department of Mathematics
Sinclair Community College
Dayton, Ohio October 2003

vii
(This page is blank)
viii
Table of Contents

“Significance” 1

List of Tables and Figures 2

1) Introduction 5

1.1 General 5
1.2 Formats, Symbols, and Book Use 7
1.3 Credits 13

“A Season for Calculus” 14

2) Barrow’s Diagram 15

3) The Two Fundamental Problems of Calculus 19

4) Foundations 25

4.1 Functions: Input to Output 25
4.2 Inverse Functions: Output to Input 33
4.3 Arrows, Targets, and Limits 37
4.4 Continuous Functions 44
4.5 The True Meaning of Slope 49
4.6 Instantaneous Change Ratios 54
4.7 ‘Wee’ Little Numbers Known as Differentials 62
4.8 A Fork in the Road 67

5) Solving the First Problem 70

5.1 Differential Change Ratios 70
5.2 Process and Products: Differentiation 77
5.3 Process Improvement: Derivative Formulas 81
5.4 Applications of the Derivative 98
5.5 Process Adaptation: Implicit Differentiation 131
5.6 Higher Order Derivatives 139
5.7 Further Applications of the Derivative 148
ix
Table of Contents
cont

6) Antiprocesses 158

6.1 Antiprocesses Prior to Calculus 158
6.2 Process and Products: Antidifferentiation 161
6.3 Process Improvement: Integral Formulas 168
6.4 Antidifferentiation Applied to Differential Equations 186

7) Solving the Second Problem 201

7.1 The Differential Equation of Planar Area 201
7.2 Process and Products: Continuous Sums 209
7.3 Process Improvement: The Definite Integral 212
7.4 Geometric Applications of the Definite Integral 218

8) Sampling the Power of Differential Equations 242

8.1 Differential Equalities 242
8.2 Applications in Physics 245
8.3 Applications in Finance 273

9) Conclusion: Magnificent Shoulders 288

“O Icarus…” 291

Appendices 292

A. Algebra, the Language of “X” 292
B. Formulas from Geometry 293
C. Formulas from Algebra 299
D. Formulas from Finance 309
E. Summary of Calculus Formulas 311

Answers to Problems 315

Short Bibliography 330

x
Significance

The wisp in my glass on a clear winter’s night
Is home for a billion wee glimmers of light,
Each crystal itself one faraway dream
With faraway worlds surrounding its gleam.

And locked in the realm of each tiny sphere
Is all that is met through an eye or an ear;
Too, all that is felt by a hand or our love,
For we are but whits in the sea seen above.

Such scales immense make wonder abound
And make a lone knee touch the cold ground.
For what is this man that he should be made
To sing to The One whose breath heavens laid?

July 1999

1
List of Tables and Figures

Tables

Number and Title Page

1.1: Calculus without Limits Syllabus 12
2.1: Guide to Barrow’s Diagram 16
6.1: Selected Processes and Antiprocesses 159
8.1: Elementary Differential Equations 243
8.2: Fixed Rate Mortgage Comparison 285

Figures

Number and Title Page

2.1: Barrow’s Diagram 15
2.2: Two Visual Proofs of the Pythagorean Theorem 17
3.1: Two Paths of Varying Complexity 19
3.2: Different Points, Same Slope 20
3.3: Slope Confusion 21
3.4: Curry’s Paradox 23
4.1: The General Function Process 25
4.2: Function Process for f(x)= x2 −3x−4 27
4.3: Graph of f(x)= x2 −3x−4 28
4.4: Graph of h(t)=3000−16.1t2 45
4.5: The Impossible Leap 46
4.6: Graph of f(x)=3000−16.1x2 46
4.7: Line Segment and Slope 49
4.8: Similar Walks—Dissimilar Coordinates 50
4.9: A Walk on the Curve 54
4.10: Failure to Match Exact Slope 55
4.11: Conceptual Setup for Exact Change Ratios 56
′
4.12: Better and Better Estimates for f (a) 57
4.13: Three Slopes for f(x)= x2 −3x−4 60
4.14: Differential Change Relationship for y = x2 63
4.15: And that has made all the Difference… 68
2