Table Of ContentA #troll through calculus
A guide for the merely curious
Anthony Barcellos
American River College
Sacramento, California
Bassim Hamadeh, CEO and Publisher
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Copyright © 2015 by Anthony Barcellos. All rights reserved. No part of this publication may
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First published in the United States of America in 2015 by Cognella, Inc.
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are used only for identification and explanation without intent to infringe.
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Printed in the United States of America
ISBN: 978-1-63189-801-3 (pbk) / 978-1-63189-802-0 (br) / 978-1-63487-179-2 (sb)
0
Preface
This book is definitely for you. After all, you’re browsing the first pages roots that go back even as far as Archimedes, the greatest of its early
of a book whose title contains the word “calculus,” so you’re obviously pioneers.
curious. The pages that follow will help to satisfy your curiosity. Most
people think calculus is super-difficult math for geniuses. It’s not. A Stroll through Calculus is the kind of book I would have been
Calculus is about measuring things and how fast they change. The math delighted to find when I first began to learn about calculus. Since it
is often very simple and readily accessible to curious, intelligent people. didn’t exist, I finally had to sit down and write it.
I will be your tour guide during this stroll through calculus—not your
How to read this book
lecturer. I have deliberately kept the discussions in this book at an
The best way to read this book is with a pen or pencil in your hand and a
elementary level. In fact, I’m going to start with something as simple as
tablet of paper nearby. You’ll get the most out of the examples I present
the area of a rectangle. That should be a friendly entry level for almost
if you check the steps yourself. I try not to leave too much out, but the
everyone. It would be nice if you’re not too frightened of a little algebra,
author always has to perform a balancing act between brevity and detail.
and later on there will be places where some knowledge of trigonometry
A short explanation can be sweet, but terse discussions are often difficult
might help, but in every case I will lead you through step by step. If I
to follow, while a detailed explanation can be easy to follow while also
start explaining things you already know, then you can skim those parts
being boring and clogged with every possible tiny step. My plan is to
until you get to material that is more engaging and less obvious. My
always provide enough details and steps so that you can follow the
main concern is to avoid assuming too much mathematical knowledge
action, although you may occasionally find it useful to pause to verify
on the part of the reader.
what I’m saying with a line or two of calculation on your notepad.
Calculus is such an important part of math and science that it’s a shame
Participatory reading is the best way to understand what we encounter
more people aren’t acquainted with it. That’s what I intend to provide
on our stroll through calculus.
here: an acquaintance. This introduction won’t make you into a calculus
expert, but it will give you some appreciation for what calculus is, how
it’s used, and why it’s important. You’ll run into some famous names as
we examine the concepts of calculus because I occasionally try to put
things in historical context by giving due credit to the clever people who
initially explained or discovered the key ideas. Calculus has ancient
Dedication
This book is in memory of two men of mathematics:
Clyde M. Wilcoxon (1929–2004), late professor emeritus of Edmund Silverbrand (1912–2005), glider pilot, raconteur, teacher,
mathematics at Porterville College. Every day in the classroom I principal, education lobbyist, and adult school advocate. In all respects,
consciously and unconsciously draw on his lessons. He taught me math he was the quintessential gentleman scholar.
and he taught me teaching.
Table of Contents
Preface ............................................................................................................................................................................................v
Dedication ......................................................................................................................................................................................vi
1 The Area of a Rectangle .........................................................................................................................................................1
2 A Loved Triangle ....................................................................................................................................................................7
3 Sometimes a Great Notation ................................................................................................................................................13
4 Simplicity Patterns ...............................................................................................................................................................19
5 Integration from a to b .........................................................................................................................................................23
6 Numerical Interlude ............................................................................................................................................................27
7 Putting Pieces Together .......................................................................................................................................................33
8 It Slices, Dices, and Spins .....................................................................................................................................................39
9 In the General Area ..............................................................................................................................................................45
10 Hitting the Slopes ................................................................................................................................................................51
11 Off on a Tangent ..................................................................................................................................................................57
12 A Derivative Work ................................................................................................................................................................63
13 Everything You Always Wanted to Know About Polynomials .................................................................................................71
14 The Fundamentals of Newton and Leibniz ............................................................................................................................77
15 Down with Derivatives .........................................................................................................................................................83
16 Going Around in Circles .......................................................................................................................................................89
17 Go Forth and Multiply ..........................................................................................................................................................99
18 Only as Strong as the Weakest Link ....................................................................................................................................107
19 Transcendental Mediations on Functions ...........................................................................................................................115
20 As Easy as Falling Off a Logarithm ......................................................................................................................................123
21 An Integral of Many Parts ...................................................................................................................................................127
22 The Anti-Chain Rule ...........................................................................................................................................................131
23 The Best (or Worst) That You Can Be ..................................................................................................................................139
24 A Second-Derivative Look at Ups and Downs .....................................................................................................................145
25 We Conquer Gravity ..........................................................................................................................................................153
26 Population Boom ...............................................................................................................................................................161
27 Polynomials Forever! .........................................................................................................................................................167
28 So Very Closely Related ......................................................................................................................................................173
29 What Are the Chances? ......................................................................................................................................................179
30 Once Is Not Enough ..........................................................................................................................................................185
31 The Gravity of the Situation ................................................................................................................................................191
Afterword ....................................................................................................................................................................................199
Index ..........................................................................................................................................................................................201
Credits
Copyright © Chiswick Chap (CC BY-SA 3.0) at http://commons.
wikimedia.org/wiki/File:Cavalieri%27s_Principle_in_Coins.JPG
“Wood lathe class,” Missouri School for the Deaf. Copyright © by
Missouri School of the Deaf. Reprinted with permission.
“Ferris Wheel,” http://commons.wikimedia.org/wiki/File:Ferris-wheel.
jpg. Copyright in the Public Domain.
Copyright © 2010 Depositphotos/newlight.
NASA, “Robert Goddard with Early Rocket,” http://commons.
wikimedia.org/wiki/File:Goddard_and_Rocket.jpg. Copyright in the
Public Domain.
“Robert Goddard with Later Rocket,” http://www.flickr.com/
photos/49487266@N07/4729110188. Copyright in the Public Domain.
1
The Area of a Rectangle
Thinking inside the box
The area of a rectangle is length times width. What could be simpler?
Calculus can be used to find the area of more complicated regions, so
we’re going to work our way up to calculus by starting with the simplest
case we can. The first surprise is that the area of a rectangle, as simple as
it is, can represent a lot of different things—not just area.
Let’s take a rectangle whose length is 4 feet and whose width is 3 feet. Figure 1-1.
According to the length-times-width formula, its area is 4 × 3 = 12, 3
which has a nice illustration to go with it. Count the squares in Figure
1-1. There are 12 of them.
2
What does each square represent? Since I gave the units of measurement
in terms of feet, each square in Figure 1-1 is a square foot. When a 1
square is 1 unit long on each side, we call it a unit square. If we write the
area computation again, this time including the measurement units, we 1 2 3 4
have (4 feet)(3 feet) = 12 square feet. As you’ll recall from algebra, we
usually prefer to indicate a square with a superscript, like this: 12 feet2.
(Here you can see one of the reasons why I’m going to avoid footnotes
Figure 1-2.
in this book.) We’re therefore using feet to measure length and feet2 to
measure area.
3
The calculation of area works pretty well with fractions, too. Suppose
your rectangle has length 4.5 feet and width 3.5 feet. The area would be
(4.5 feet)(3.5 feet) = 15.75 feet2. Check out Figure 1-2 and try to count 2
the squares. The original 12 are still there, but now we have some half
squares (little rectangles) to count, as well as the small square in the
1
1
upper right-hand corner, which is of a regular square. Each pair of
4
1 2 3 4
The Area of a Rectangle | 11
little rectangles equals one of the unit squares, so we can count up our Figure 1-3.
1ft2 1ft2
results, as shown in Figure 1-3.
2
t
f
3 34
To save some space, we’re using the traditional abbreviation ft2 for
feet2. In addition to the original 12 ft2, we have two square feet across
2
the top, one square foot on the lower right edge, and the remaining 2 12 ft
2
3 3 ft
ft2 on the upper right edge. Therefore the total is 12 + 2 + 1 + = 1
4 4
1
15.75 ft2. Naturally we could have coaxed this result out of a calculator,
and some of us may recall the nearly lost art of performing decimal
1 2 3 4
multiplication by hand, but checking the results by counting squares in
Figure 1-3 makes the example nice and concrete. The picture matches
the calculation.
Thinking outside the box
I began by saying we were going to talk about area, and we have,
at least for the rectangle. We could now move on to other areas you
might recall—things like triangles, trapezoids, and circles—but we’re
going to stay with rectangles for a while longer. Before moving on to
other calculations of area, I need to make an important point about the
calculations we just made. They don’t have to be about area. While we
may compute something that seems to be the area of a geometric object,
it could actually represent something very different. Let me show you.
Consider our original rectangle again, but let’s change the measurement
units. We’ll keep the numbers 4 and 3, but we’re going to use
dimensions other than length. While this may sound strange, the result
will be quite familiar. Suppose that the “length” of the rectangle is
now given as 4 hours (that’s right, I said hours) and its “width” is
now 3 miles per hour. Using traditional abbreviations for the units of
measurement, this is what we get when we compute the “area” again:
(4 hr)(3 mi/hr) = 12 mi.
Notice how the “hr”—the hours—canceled and left us with miles
22 | The Area of a Rectangle
as the units of measurement. If that looks unfamiliar, this may help,
remembering that you can rearrange factors in a multiplication:
mi mi
4hr 3 = 4 3 hr = 12mi.
^ h` hr j ^ h^ hc hr m
We did the same computation as for area, but this time we ended up
with a distance for our answer. Galileo observed hundreds of years ago
that area calculations can be used to find distance, so we are simply
following in his footsteps. You probably recognize that our answer
seems reasonable enough: If you walk for 4 hours at 3 miles per hour,
you’ll travel 12 miles. This is a well-known formula: Distance equals
Rate times Time, often written as d = r × t.
If we already knew the formula for computing distance from rate
and time, why did we bother pretending it’s the same as an area
computation? Well, there’s no pretense about it. Math gets a lot of its
power by taking advantage of equivalent quantities. We often find that a
problem which seems difficult in a particular context can become quite
easy when approached in a different context. As long as we’re willing to
play with the units and get away from the idea that an area can represent
only an area, we begin to see the power of generalization.
The box itself
Let’s repeat our “area” calculation one more time, but instead of
obtaining an area or a distance, we’ll see that it can produce yet another
quantity. This time we’ll use a “length” of 4 feet (okay, that really is a
length) and a “width” of 3 feet2 (which you can see is already an area).
When we multiply our quantities together, we get
(4 ft)(3 ft2) = 12 ft3.
What does this mean? We multiplied a length by an area and got a
volume (cubic units, as they say). If you’re wondering how we could
represent this calculation physically, it’s simply the volume of a solid
The Area of a Rectangle | 33
