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**Unformatted text preview: **APEX C Version 3.0 Gregory Hartman, Ph.D.
Department of Applied Mathema cs
Virginia Military Ins tute Contribu ng Authors
Troy Siemers, Ph.D.
Department of Applied Mathema cs
Virginia Military Ins tute Brian Heinold, Ph.D.
Department of Mathema cs and Computer Science
Mount Saint Mary’s University Dimplekumar Chalishajar, Ph.D.
Department of Applied Mathema cs
Virginia Military Ins tute Editor
Jennifer Bowen, Ph.D.
Department of Mathema cs and Computer Science
The College of Wooster Copyright © 2015 Gregory Hartman
Licensed to the public under Crea ve Commons
A ribu on-Noncommercial 4.0 Interna onal Public
License Contents
Table of Contents iii Preface vii 1 Limits
1.1 An Introduc on To Limits . . . . .
1.2 Epsilon-Delta Deﬁni on of a Limit
1.3 Finding Limits Analy cally . . . . .
1.4 One Sided Limits . . . . . . . . .
1.5 Con nuity . . . . . . . . . . . . .
1.6 Limits Involving Inﬁnity . . . . . . .
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. 2 Deriva ves
2.1 Instantaneous Rates of Change: The Deriva ve
2.2 Interpreta ons of the Deriva ve . . . . . . . .
2.3 Basic Diﬀeren a on Rules . . . . . . . . . . .
2.4 The Product and Quo ent Rules . . . . . . . .
2.5 The Chain Rule . . . . . . . . . . . . . . . . .
2.6 Implicit Diﬀeren a on . . . . . . . . . . . . .
2.7 Deriva ves of Inverse Func ons . . . . . . . . .
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. 117 3 The Graphical Behavior of Func ons
3.1 Extreme Values . . . . . . . . . . .
3.2 The Mean Value Theorem . . . . . .
3.3 Increasing and Decreasing Func ons
3.4 Concavity and the Second Deriva ve
3.5 Curve Sketching . . . . . . . . . . . .
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. 123
123
131
136
144
152 4 Applica ons of the Deriva ve
159
4.1 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.2 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.3
4.4 Op miza on . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Diﬀeren als . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5 Integra on
5.1 An deriva ves and Indeﬁnite Integra on
5.2 The Deﬁnite Integral . . . . . . . . . . .
5.3 Riemann Sums . . . . . . . . . . . . . .
5.4 The Fundamental Theorem of Calculus . .
5.5 Numerical Integra on . . . . . . . . . . . .
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. 189
189
199
210
228
240 6 Techniques of An diﬀeren a on
6.1 Subs tu on . . . . . . . . . .
6.2 Integra on by Parts . . . . . .
6.3 Trigonometric Integrals . . . .
6.4 Trigonometric Subs tu on . .
6.5 Par al Frac on Decomposi on
6.6 Hyperbolic Func ons . . . . .
6.7 L’Hôpital’s Rule . . . . . . . .
6.8 Improper Integra on . . . . . .
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. 255
255
275
286
296
305
313
324
333 7 Applica ons of Integra on
7.1 Area Between Curves . . . . . . . . . . . . . . . . . . . .
7.2 Volume by Cross-Sec onal Area; Disk and Washer Methods
7.3 The Shell Method . . . . . . . . . . . . . . . . . . . . . .
7.4 Arc Length and Surface Area . . . . . . . . . . . . . . . .
7.5 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Fluid Forces . . . . . . . . . . . . . . . . . . . . . . . . . .
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. 345
346
353
361
369
378
388 8 Sequences and Series
8.1 Sequences . . . . . . . . . . . . . . . . . . .
8.2 Inﬁnite Series . . . . . . . . . . . . . . . . .
8.3 Integral and Comparison Tests . . . . . . . .
8.4 Ra o and Root Tests . . . . . . . . . . . . . .
8.5 Alterna ng Series and Absolute Convergence
8.6 Power Series . . . . . . . . . . . . . . . . . .
8.7 Taylor Polynomials . . . . . . . . . . . . . .
8.8 Taylor Series . . . . . . . . . . . . . . . . . . .
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. 397
397
411
426
435
441
452
465
477 9 Curves in the Plane
9.1 Conic Sec ons . . . . . . . . . . .
9.2 Parametric Equa ons . . . . . . .
9.3 Calculus and Parametric Equa ons
9.4 Introduc on to Polar Coordinates .
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503
513
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. 9.5 Calculus and Polar Func ons . . . . . . . . . . . . . . . . . . . 538 10 Vectors
10.1 Introduc on to Cartesian Coordinates in Space
10.2 An Introduc on to Vectors . . . . . . . . . . .
10.3 The Dot Product . . . . . . . . . . . . . . . . .
10.4 The Cross Product . . . . . . . . . . . . . . . .
10.5 Lines . . . . . . . . . . . . . . . . . . . . . . .
10.6 Planes . . . . . . . . . . . . . . . . . . . . . . .
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. 551
551
566
580
593
604
615 11 Vector Valued Func ons
11.1 Vector–Valued Func ons . . . . . . . . .
11.2 Calculus and Vector–Valued Func ons . .
11.3 The Calculus of Mo on . . . . . . . . . .
11.4 Unit Tangent and Normal Vectors . . . . .
11.5 The Arc Length Parameter and Curvature .
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. 623
623
629
642
655
664 12 Func
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8 ons of Several Variables
Introduc on to Mul variable Func ons . . . . .
Limits and Con nuity of Mul variable Func ons .
Par al Deriva ves . . . . . . . . . . . . . . . . .
Diﬀeren ability and the Total Diﬀeren al . . . .
The Mul variable Chain Rule . . . . . . . . . . .
Direc onal Deriva ves . . . . . . . . . . . . . .
Tangent Lines, Normal Lines, and Tangent Planes
Extreme Values . . . . . . . . . . . . . . . . . . .
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. 675
675
682
692
704
713
720
730
740 13 Mul
13.1
13.2
13.3
13.4
13.5
13.6 ple Integra on
Iterated Integrals and Area . . . . . . . . .
Double Integra on and Volume . . . . . . .
Double Integra on with Polar Coordinates .
Center of Mass . . . . . . . . . . . . . . .
Surface Area . . . . . . . . . . . . . . . . .
Volume Between Surfaces and Triple Integra .
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. 751
751
761
772
779
791
798 A Solu ons To Selected Problems
Index .
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on . A.1
A.33 P
A Note on Using this Text Thank you for reading this short preface. Allow us to share a few key points
about the text so that you may be er understand what you will ﬁnd beyond this
page.
This text comprises a three–volume series on Calculus. The ﬁrst part covers
material taught in many “Calc 1” courses: limits, deriva ves, and the basics of
integra on, found in Chapters 1 through 6.1. The second text covers material
o en taught in “Calc 2:” integra on and its applica ons, along with an introducon to sequences, series and Taylor Polynomials, found in Chapters 5 through
8. The third text covers topics common in “Calc 3” or “mul variable calc:” parametric equa ons, polar coordinates, vector–valued func ons, and func ons of
more than one variable, found in Chapters 9 through 13. All three are available
separately for free at .
Prin ng the en re text as one volume makes for a large, heavy, cumbersome
book. One can certainly only print the pages they currently need, but some prefer to have a nice, bound copy of the text. Therefore this text has been split into
these three manageable parts, each of which can be purchased for under $15
at Amazon.com. For Students: How to Read this Text
Mathema cs textbooks have a reputa on for being hard to read. High–level
mathema cal wri ng o en seeks to say much with few words, and this style
o en seeps into texts of lower–level topics. This book was wri en with the goal
of being easier to read than many other calculus textbooks, without becoming
too verbose.
Each chapter and sec on starts with an introduc on of the coming material,
hopefully se ng the stage for “why you should care,” and ends with a look ahead
to see how the just–learned material helps address future problems.
Please read the text; it is wri en to explain the concepts of Calculus. There
are numerous examples to demonstrate the meaning of deﬁni ons, the truth
of theorems, and the applica on of mathema cal techniques. When you encounter a sentence you don’t understand, read it again. If it s ll doesn’t make
sense, read on anyway, as some mes confusing sentences are explained by later
sentences.
You don’t have to read every equa on. The examples generally show “all”
the steps needed to solve a problem. Some mes reading through each step is
helpful; some mes it is confusing. When the steps are illustra ng a new technique, one probably should follow each step closely to learn the new technique.
When the steps are showing the mathema cs needed to ﬁnd a number to be
used later, one can usually skip ahead and see how that number is being used,
instead of ge ng bogged down in reading how the number was found.
Most proofs have been omi ed. In mathema cs, proving something is always true is extremely important, and entails much more than tes ng to see if
it works twice. However, students o en are confused by the details of a proof,
or become concerned that they should have been able to construct this proof on their own. To alleviate this poten al problem, we do not include the proofs
to most theorems in the text. The interested reader is highly encouraged to ﬁnd
proofs online or from their instructor. In most cases, one is very capable of understanding what a theorem means and how to apply it without knowing fully
why it is true. Interac ve, 3D Graphics
New to Version 3.0 is the addi on of interac ve, 3D graphics in the .pdf version. Nearly all graphs of objects in space can be rotated, shi ed, and zoomed
in/out so the reader can be er understand the object illustrated.
As of this wri ng, the only pdf viewers that support these 3D graphics are
Adobe Reader & Acrobat (and only the versions for PC/Mac/Unix/Linux computers, not tablets or smartphones). To ac vate the interac ve mode, click on
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scroll wheel on a mouse to zoom in/out. (A great way to inves gate an image
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One can also revert the graph back to its default view. If you wish to deac ve
the interac vity, one can right-click and choose the “Disable Content” op on. Thanks
There are many people who deserve recogni on for the important role they
have played in the development of this text. First, I thank Michelle for her support and encouragement, even as this “project from work” occupied my me
and a en on at home. Many thanks to Troy Siemers, whose most important
contribu ons extend far beyond the sec ons he wrote or the 227 ﬁgures he
coded in Asymptote for 3D interac on. He provided incredible support, advice
and encouragement for which I am very grateful. My thanks to Brian Heinold
and Dimplekumar Chalishajar for their contribu ons and to Jennifer Bowen for
reading through so much material and providing great feedback early on. Thanks
to Troy, Lee Dewald, Dan Joseph, Meagan Herald, Bill Lowe, John David, Vonda
Walsh, Geoﬀ Cox, Jessica Liber ni and other faculty of VMI who have given me
numerous sugges ons and correc ons based on their experience with teaching
from the text. (Special thanks to Troy, Lee & Dan for their pa ence in teaching
Calc III while I was s ll wri ng the Calc III material.) Thanks to Randy Cone for
encouraging his tutors of VMI’s Open Math Lab to read through the text and
check the solu ons, and thanks to the tutors for spending their me doing so.
A very special thanks to Kris Brown and Paul Janiczek who took this opportunity far above & beyond what I expected, me culously checking every solu on
and carefully reading every example. Their comments have been extraordinarily
helpful. I am also thankful for the support provided by Wane Schneiter, who as
my Dean provided me with extra me to work on this project. I am blessed to
have so many people give of their me to make this book be er. APEX – Aﬀordable Print and Electronic teXts
APEX is a consor um of authors who collaborate to produce high–quality,
low–cost textbooks. The current textbook–wri ng paradigm is facing a potenal revolu on as desktop publishing and electronic formats increase in popularity. However, wri ng a good textbook is no easy task, as the me requirements alone are substan al. It takes countless hours of work to produce text, write
examples and exercises, edit and publish. Through collabora on, however, the
cost to any individual can be lessened, allowing us to create texts that we freely
distribute electronically and sell in printed form for an incredibly low cost. Having said that, nothing is en rely free; someone always bears some cost. This text
“cost” the authors of this book their me, and that was not enough. APEX Calculus would not exist had not the Virginia Military Ins tute, through a generous
Jackson–Hope grant, given the lead author signiﬁcant me away from teaching
so he could focus on this text.
Each text is available as a free .pdf, protected by a Crea ve Commons Attribu on - Noncommercial 4.0 copyright. That means you can give the .pdf to
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We encourage others to adapt this work to ﬁt their own needs. One might
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need. The source ﬁles can be found at github.com/APEXCalculus.
You can learn more at . 1: L
Calculus means “a method of calcula on or reasoning.” When one computes
the sales tax on a purchase, one employs a simple calculus. When one ﬁnds the
area of a polygonal shape by breaking it up into a set of triangles, one is using
another calculus. Proving a theorem in geometry employs yet another calculus.
Despite the wonderful advances in mathema cs that had taken place into
the ﬁrst half of the 17th century, mathema cians and scien sts were keenly
aware of what they could not do. (This is true even today.) In par cular, two
important concepts eluded mastery by the great thinkers of that me: area and
rates of change.
Area seems innocuous enough; areas of circles, rectangles, parallelograms,
etc., are standard topics of study for students today just as they were then. However, the areas of arbitrary shapes could not be computed, even if the boundary
of the shape could be described exactly.
Rates of change were also important. When an object moves at a constant
rate of change, then “distance = rate × me.” But what if the rate is not constant
– can distance s ll be computed? Or, if distance is known, can we discover the
rate of change?
It turns out that these two concepts were related. Two mathema cians, Sir
Isaac Newton and Go ried Leibniz, are credited with independently formula ng
a system of compu ng that solved the above problems and showed how they
were connected. Their system of reasoning was “a” calculus. However, as the
power and importance of their discovery took hold, it became known to many
as “the” calculus. Today, we generally shorten this to discuss “calculus.”
The founda on of “the calculus” is the limit. It is a tool to describe a parcular behavior of a func on. This chapter begins our study of the limit by approxima ng its value graphically and numerically. A er a formal deﬁni on of
the limit, proper es are established that make “ﬁnding limits” tractable. Once
the limit is understood, then the problems of area and rates of change can be
approached. 1.1 An Introduc on To Limits We begin our study of limits by considering examples that demonstrate key concepts that will be explained as we progress.
Consider the func on y = sinx x . When x is near the value 1, what value (if
any) is y near?
While our ques on is not precisely formed (what cons tutes “near the value Chapter 1 Limits
1”?), the answer does not seem diﬃcult to ﬁnd. One might think ﬁrst to look at a
graph of this func on to approximate the appropriate y values. Consider Figure
1.1, where y = sinx x is graphed. For values of x near 1, it seems that y takes on
values near 0.85. In fact, when x = 1, then y = sin1 1 ≈ 0.84, so it makes sense
that when x is “near” 1, y will be “near” 0.84.
Consider this again at a diﬀerent value for x. When x is near 0, what value (if
any) is y near? By considering Figure 1.2, one can see that it seems that y takes
on values near 1. But what happens when x = 0? We have y
1 0.8 0.6 . y→ x
1 0.5 1.5 “0”
sin 0
→
.
0
0 The expression “0/0” has no value; it is indeterminate. Such an expression gives
no informa on about what is going on with the func on nearby. We cannot ﬁnd
out how y behaves near x = 0 for this func on simply by le ng x = 0.
Finding a limit entails understanding how a func on behaves near a par cular value of x. Before con nuing, it will be useful to establish some nota on. Let
y = f(x); that is, let y be a func on of x for some func on f. The expression “the
limit of y as x approaches 1” describes a number, o en referred to as L, that y
nears as x nears 1. We write all this as Figure 1.1: sin(x)/x near x = 1. y 1 lim y = lim f(x) = L. x→1 x→1 0.9 This is not a complete deﬁni on (that will come in the next sec on); this is a
pseudo-deﬁni on that will allow us to explore the idea of a limit.
Above, where f(x) = sin(x)/x, we approximated 0.8 . x −1 lim 1 Figure 1.2: sin(x)/x near x = 0. x
0.9
0.99
0.999
1
1.001
1.01
1.1 sin(x)/x
0.870363
0.844471
0.841772
0.841471
0.84117
0.838447
0.810189 x→1 and lim x→0 sin x
≈ 1.
x (We approximated these limits, hence used the “≈” symbol, since we are working with the pseudo-deﬁni on of a limit, not the actual deﬁni on.)
Once we have the true deﬁni on of a limit, we will ﬁnd limits analy cally;
that is, exactly using a variety of mathema cal tools. For now, we will approximate limits both graphically and numerically. Graphing a func on can provide
a good approxima on, though o en not very precise. Numerical methods can
provide a more accurate approxima on. We have already approximated limits
graphically, so we now turn our a en on to numerical approxima ons.
Consider again limx→1 sin(x)/x. To approximate this limit numerically, we
can create a table of x and f(x) values where x is “near” 1. This is done in Figure
1.3.
No ce that for values of x near 1, we have sin(x)/x near 0.841. The x = 1 row
is in bold to highlight the fact that when considering limits, we are not concerned Figure 1.3: Values of sin(x)/x with x near
1. Notes: 2 sin x
≈ 0.84
x 1.1 An Introduc on To Limits
with the value of the func on at that par cular x value; we are only concerned
with the values of the func on when x is near 1.
Now approximate limx→0 sin(x)/x numerically. We already approximated
the value of this limit as 1 graphically in Figure 1.2. The table in Figure 1.4 shows
the value of sin(x)/x for values of x near 0. Ten places a er the decimal point
are shown to highlight ho...

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