download You Already Know Calculus by Scott Robertson

325 Pages·2016·11.47 MB·English

by  Scott Robertson

You Already Know Calculus
for inventors and other curious folk
Reformed calculus for the 23rd century.
by Scott Robertson
…a zeroeth draft! (2011)
IMPORTANT ADDITIONS AND FINAL APOLOGY:
This really is a ZEROETH draft. It is an unfinished labor of love by
someone who probably was not cut out to be either a mathematician
or an inventor. I believe this book is full of really awesome stuff. I
could not find a calculus book or even an "easy" calculus book that
was intelligible to me. This book is intelligible. It might contain
mistakes though, because it is unfinished. I don't think it contains
important mistakes, but there are some possible inconsistencies. It is
too full of rants, but so what. I stopped working on this book about
five years ago and will never finish it, so here it is, for what it's
worth. I like to say that I wrote a book about why calculus is hard to
learn, without learning any calculus. This is essentially correct, but I
am confident that if I were to re-read my book today, I would learn
something about calculus. An engineer I know told me that when he
was in college, he didn't get calculus. Then after summer vacation, he
went back to class and found that calculus had gotten him, while he
wasn't struggling with it. Kinda nice story but should it be that way?
Why shouldn't teachers and books be able to get their point across to
more people? Anyhow, enjoy this. I enjoyed writing it.
From my notes:
I have no idea what this means but it was to be a list of things to
change while trying to turn this zeroeth draft into a first draft...
which I never got around to doing.
No doubt this list is incomplete.
corrections and additions
1. point is not first dimension because dimension means measurement, so distance is first dimension
2. change some dx etc to Dx greek letter, look for unpaired differentials and get rid of them
3. footnote all occurrences of “indefinite integral” to reference chapter 13 for correct nomenclature
AUTHOR’S NOTE
When I say “calculus” in this book I am usually referring to basic
routines that can be learned in a few minutes, that is, the kind
of calculus actually used by engineers and inventors in everyday
design work. The other stuff is for mathematicians and it isn’t
practical calculus, it’s higher math. Higher math is for
mathematicians and theoreticalists, they pay their bills with it.
No one else needs it. It is on the curriculum because where do
most mathematicians have to seek jobs? In universities.
Prequiring more math than anyone wants or needs is a vicious
cycle, self-feeding, self-perpetuating, because it makes work for
that portion of the Curriculum Committee that everyone looks to
for the answer to the question, “Just how much math do people
really need to get their job done?” That’s like asking Ronald
McDonald to recommend a fine dining establishment.
This is not a calculus book.
It’s not even about calculus.
It’s about standing next to calculus until comfort sets in.
I’m not even going to end this book at a logical point. I’m going
to end it when comfort sets in.
Because, you see,
I wrote this book for me,
and I’ll let you read it,
but not for free!
In closing, let me assert that what this book lacks in
organization, it makes up for in something that mathematicians
wouldn’t understand.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing
interesting facts. That’s fine in context, and has its place just as learning a vocabulary
does—it helps you to create richer, more nuanced works of art. But it’s not the fact that
triangles take up half their box that matters. What matters is the beautiful idea of
chopping it with the line, and how that might inspire other beautiful ideas and lead to
creative breakthroughs in other problems—something a mere statement of fact can never
give you.
By removing the creative process and leaving only the results of that process, you
virtually guarantee that no one will have any real engagement with the subject. It is like
saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I
supposed to be inspired by that?
…By concentrating on what, and leaving out why, mathematics is reduced to an empty
shell. The art is not in the “truth” but in the explanation, the argument. It is the argument
itself which gives the truth its context, and determines what is really being said and
meant. Mathematics is the art of explanation. If you deny students the opportunity to
engage in this activity—to pose their own problems, make their own conjectures and
discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble
together their own explanations and proofs—you deny them mathematics itself. So no,
I’m not complaining about the presence of facts and formulas in our mathematics classes,
I’m complaining about the lack of mathematics in our mathematics classes.
If your art teacher were to tell you that painting is all about filling in numbered regions,
you would know that something was wrong…
—Paul Lockhart
“A Mathematician’s Lament”
published by Pneumatic Options Research Library
http://www.AirCarAccess.com
NOTE: If you don’t like Part One because you want just the facts, skip
directly to Part Two where no opinions about the Curriculum Committee, the
Discouragement Fraternity, Function Notation, etc. will be registered!*
INTRODUCTION: How fast did the chicken cross the
road and why? Or, the fear of thinking.
* Just kidding. Part 2 doesn’t exist. For exercises, just use the ones in any calculus book. I suggest the “easy calculus” books
like Calculus Made Easy, Calculus for Dummies, Idiot’s Guide to Calculus, etc. Because hard calculus books are stupid.
PART I: Diary of a Calculus Self-Teacher
CHAPTER ONE: Time fixes everything in place.
That’s why everything never stops moving.
CHAPTER TWO: Let’s get small. The fun never ends
no matter how small you think.
CHAPTER THREE: Watch me think. How I proved
that calculus should have been taught in
geometry class, without even knowing enough
calculus to throw a rubber chicken at.
CHAPTER FOUR: Limits? Limit This!
CHAPTER FIVE: What’s so Right About Triangles?
CHAPTER SIX: The Chain Rule, Tangent Line
Approximation, a Facelift for Dysfunctional
Notation, and an Identity that Encompasses All
That is Calculus
CHAPTER SEVEN: The Truth about Calculus.
The Linear Equation Meets the Tangent Line
Approximation and the Fundamental Theorem
of Calculus
CHAPTER EIGHT: The Power Rule Loses its Trick
Status, and, Quippling about the Powers that Be
CHAPTER NINE: More Rules for Doing Easy Calculus
CHAPTER TEN: Exponential Growth and its Opposite,
“Exponential Decay,” which was Apparently
Named in Honor of Nuclear Waste
CHAPTER ELEVEN: What in Quippledom is a
Logarithm?
CHAPTER TWELVE: Revisiting Integral Notation,
and Staying for the Numerical Approximations
CHAPTER THIRTEEN: What does Indefinite have to do
with Integrals; Integration Rules, and the Real
Power Rule!
CHAPTER FOURTEEN: Partial Differentials, Maxima
and Minima and other Curvaceous Phenomena
EPILOG: Why Save the Best for Last? Life’s Too
Short… How to Teach Calculus to Kindergarteners
PART II: The New Calculus, Rant-Free & Rent-Free…
calculus in a nutshell and taught right for the
first time in the history of western civilization
BIBLIOGRAPHY
INDEX
This book is guaranteed to teach you REAL PRACTICAL
(something about)
CALCULUS by showing you what calculus is, not by
(in my opinion)
shoving it down your throat. You will learn that calculus is the
natural language of reality, and you will learn why it seems so hard
the way it is usually taught. You’ll gain a beginner’s intuitive
understanding of easy basic calculus as a tool just by reading this
book. There are no exercises and little math, because calculus is so
easy. Suit yourself: waste 8 years of your life having superfluous
nonsense blown down your shirt, or get to the guts of the tool you
need to get your work done.
And if you don’t carefully follow the math herein, you won’t learn a
damn thing, like any other math book.
You Already Know Calculus
For Inventors and Other Curious Folk
by Scott Robertson
Not available at your college bookstore.
$21.00 by PayPal, credit card, or debit card
available only from http://AirCarAccess.com/onlinestore.htm
by the same author:
Air Car Hall of Fame
Compressed Air Power Secrets
This book is dedicated to the Death Lords of Academia…Just think
of this book as “tough love”…! (That was a joke…) The book is
REALLY dedicated to Steve the machinist and Tom the
engineer, for breathing new life into my research.

INTRODUCTION: How fast did the chicken cross the
road and why? Or, The Fear of Thinking.
I LIKE TO GET THE PUNCH LINE OUT OF THE WAY FIRST, SO HERE IT IS:
y = mx + b
A ROLLICKING ROMP THROUGH THE ROILING REELING RIOT OF RELATIONSHIPS
Re-imagine your first day of formal instruction in the fine art of adding 1 + 1 = 2 and
2 + 2 = 4. What if your teacher had started the lesson with these words:
“Now children, I know you are going to hate what we will do next, because it is really,
really hard. In fact that is why we have kept this information from you till you turned
6. Some of you still aren’t ready for it, and you will be the ones to fail. You might stay
in school till you’re 15, 18, or even all the way through graduate school and beyond,
but you will still be a failure. Because adding 1 + 1 = 2 and 2 + 2 = 4 is so very hard
that some of you will just have to fake your way through life, because you’re simply
not capable of learning this topic well enough to benefit from it.”
Then the teacher spends the next 6 hours going over the finer points of a formal proof
that 1 + 1 = 2, a topic that is beyond the scope of anyone except a professional
mathematician. When the day finally ends, several students have no intention of
returning the next day for the formal proof that 2 + 2 = 4.
In my high school, nothing past Plane Geometry With Proofs was required to graduate.
It was rumored that trigonometry and calculus were too hard for most people, and no
one had the vaguest idea, after 9 years of studying other math topics, what trig and
calc actually were or what they were good for. The “with proofs” part of plane
geometry certainly did nothing to encourage anyone to come back for more. The dour-
faced gentleman attempting to teach formal proofs to hormone-high 15-year-olds has
my sympathy today, but at the time I did my best to choke him with his own last
shred of sanity for taking away my intuitive respect for math. In spite of it all, I got an
easy A in the class, but swore off math forever. I had never been so bored in my life.
When I graduated high school a few years later, I was still an uneducated smart-ass
with an easy B average that I earned by doing the minimum, and when my parents
drove me home in my cap and gown, I still thought that an engineer is someone who
drives a train. Well a lot of American children thought so, up to the age of 7 or 8, but
in some more progressive decades than the mid-1970s, there has been more emphasis
on studying towards some goal other than “graduating and going to college.”
Anyone who actually remembers the mid-1970s has just now raised a mental
objection about the mid-1970s to the effect that this was a very progressive time. But