Honors Calculus

356 Pages·2014·2.639 MB·English

by  Pete L. Clark

Honors Calculus
Pete L. Clark
(cid:13)c Pete L. Clark, 2014.
Thanks to Ron Freiwald, Eilidh Geddes, Melody Hine, Nita Jain, Andrew Kane,
Bryan Oakley, Elliot Outland, Didier Piau, Jim Propp, Betul Tolgay, Troy Woo,
and Sharon Yao.
Contents
Foreword 7
Spivak and Me 7
What is Honors Calculus? 10
Some Features of the Text 10
Chapter 1. Introduction and Foundations 13
1. Introduction 13
2. Some Properties of Numbers 17
Chapter 2. Mathematical Induction 25
1. Introduction 25
2. The First Induction Proofs 27
3. Closed Form Identities 29
4. Inequalities 31
5. Extending Binary Properties to n-ary Properties 32
6. The Principle of Strong/Complete Induction 34
7. Solving Homogeneous Linear Recurrences 35
8. The Well-Ordering Principle 39
9. The Fundamental Theorem of Arithmetic 40
Chapter 3. Polynomial and Rational Functions 43
1. Polynomial Functions 43
2. Rational Functions 49
Chapter 4. Continuity and Limits 53
1. Remarks on the Early History of the Calculus 53
2. Derivatives Without a Careful Definition of Limits 54
3. Limits in Terms of Continuity 56
4. Continuity Done Right 58
5. Limits Done Right 63
Chapter 5. Differentiation 71
1. Differentiability Versus Continuity 71
2. Differentiation Rules 72
3. Optimization 77
4. The Mean Value Theorem 81
5. Monotone Functions 85
6. Inverse Functions I: Theory 92
7. Inverse Functions II: Examples and Applications 97
8. Some Complements 104
3
4 CONTENTS
Chapter 6. Completeness 105
1. Dedekind Completeness 105
2. Intervals and the Intermediate Value Theorem 113
3. The Monotone Jump Theorem 116
4. Real Induction 117
5. The Extreme Value Theorem 118
6. The Heine-Borel Theorem 119
7. Uniform Continuity 120
8. The Bolzano-Weierstrass Theorem For Subsets 122
9. Tarski’s Fixed Point Theorem 122
Chapter 7. Differential Miscellany 125
1. L’Hoˆpital’s Rule 125
2. Newton’s Method 128
3. Convex Functions 136
Chapter 8. Integration 149
1. The Fundamental Theorem of Calculus 149
2. Building the Definite Integral 152
3. Further Results on Integration 161
4. Riemann Sums, Dicing, and the Riemann Integral 170
5. Lesbesgue’s Theorem 176
6. Improper Integrals 179
7. Some Complements 182
Chapter 9. Integral Miscellany 185
1. The Mean Value Therem for Integrals 185
2. Some Antidifferentiation Techniques 185
3. Approximate Integration 187
4. Integral Inequalities 194
5. The Riemann-Lebesgue Lemma 196
Chapter 10. Infinite Sequences 199
1. Summation by Parts 200
2. Easy Facts 201
3. Characterizing Continuity 203
4. Monotone Sequences 203
5. Subsequences 206
6. The Bolzano-Weierstrass Theorem For Sequences 208
7. Partial Limits; Limits Superior and Inferior 211
8. Cauchy Sequences 214
9. Geometric Sequences and Series 216
10. Contraction Mappings Revisited 217
11. Extending Continuous Functions 224
Chapter 11. Infinite Series 229
1. Introduction 229
2. Basic Operations on Series 233
3. Series With Non-Negative Terms I: Comparison 236
4. Series With Non-Negative Terms II: Condensation and Integration 241
CONTENTS 5
5. Series With Non-Negative Terms III: Ratios and Roots 247
6. Absolute Convergence 250
7. Non-Absolute Convergence 254
8. Power Series I: Power Series as Series 260
Chapter 12. Taylor Taylor Taylor Taylor 265
1. Taylor Polynomials 265
2. Taylor’s Theorem Without Remainder 266
3. Taylor’s Theorem With Remainder 269
4. Taylor Series 271
5. Hermite Interpolation 277
Chapter 13. Sequences and Series of Functions 283
1. Pointwise Convergence 283
2. Uniform Convergence 285
3. Power Series II: Power Series as (Wonderful) Functions 290
Chapter 14. Serial Miscellany 293
1. ∞ 1 = π2 293
n=1 n2 6
2. Rearrangements and Unordered Summation 295
P
3. Abel’s Theorem 304
4. The Peano-Borel Theorem 308
5. The Weierstrass Approximation Theorem 311
6. A Continuous, Nowhere Differentiable Function 315
7. The Gamma Function 317
Chapter 15. Several Real Variables and Complex Numbers 323
1. A Crash Course in the Honors Calculus of Several Variables 323
2. Complex Numbers and Complex Series 325
3. Elementary Functions Over the Complex Numbers 327
4. The Fundamental Theorem of Algebra 329
Chapter 16. Foundations Revisited 335
1. Ordered Fields 336
2. The Sequential Completion 344
Bibliography 353

Foreword
Spivak and Me
Thedocumentyouarecurrentlyreadingbeganitslifeasthelecturenotesforayear
long undergraduate course in honors calculus. Specifically, during the 2011-2012
academic year I taught Math 2400(H) and Math 2410(H), Calculus With Theory,
atthe Universityof Georgia. This is acourse forunusuallytalented andmotivated
(mostly first year) undergraduate students. It has been offered for many years at
the University of Georgia, and so far as I know the course text has always been
Michael Spivak’s celebrated Calculus [S]. The Spivak text’s take on calculus is
sufficiently theoretical that, although it is much beloved by students and practi-
tioners of mathematics, it is seldomed used nowadays as a course text. In fact, the
UGAMath2400/2410courseistraditionallysomethingofaninterpolationbetween
standard freshman calculus and the fully theoretical approach of Spivak. My own
take on the course was different: I treated it as being a sequel to, rather than an
enriched revision of, freshman calculus.
I began the course with a substantial familiarity with Spivak’s text. The sum-
mer after my junior year of high school I interviewed at the University of Chicago
and visited Paul Sally, then (and still now, as I write this in early 2013, though
he is 80 years old) Director of Undergraduate Mathematics at the University of
Chicago. After hearing that I had taken AP calculus and recently completed a
summer course in multivariable calculus at Johns Hopkins, Sally led me to a sup-
plycloset, rummagedthroughit, andcameoutwithabeatupoldcopyofSpivak’s
text. “Thisishowwedocalculusaroundhere,”hesaid,presentingittome. During
my senior year at high school I took more advanced math courses at LaSalle Uni-
versity (which turned out, almost magically, to be located directly adjacent to my
high school) but read through Spivak’s text. And I must have learned something
from it, because by the time I went on to college – of course at the University of
Chicago – I placed not into their Spivak calculus course, but the following course,
“HonorsAnalysisinRn”. Thiscoursehasthereputationofsharingthehonorwith
Harvard’sMath55ofbeingthehardestundergraduatemathcoursethatAmerican
universities have to offer. I can’t speak to that, but it was certainly the hardest
mathcourseIevertook. Therewerethreetenweekquarters. Thefirstquarterwas
primarilytaughtoutofRudin’sclassictext[R],withanemphasisonmetricspaces.
The second quarter treated Lebesgue integration and some Fourier analysis, and
the third quarter treated analysis on manifolds and Stokes’s theorem.
However, that was not the end of my exposure to Spivak’s text. In my second
year of college I was a grader for the first quarter of Spivak calculus, at the (even
then)amazinglylowrateof$20perstudentfortheentire10weekquarter. Though
7
8 FOREWORD
the material in Spivak’s text was at least a level below the trial by fire that I had
successfully endured in the previous year, I found that there were many truly dif-
ficult problems in Spivak’s text, including a non-negligible percentage that I still
did not know how to solve. Grading for this course solidified my knowledge of this
elementary but important material. It was also my first experience with reading
proofs written by bright but very inexperienced authors: often I would stare at
an entire page of text, one long paragraph, and eventually circle a single sentence
whichcarriedtheentirecontentthatthewriterwastryingtoexpress. Ionlygraded
for one quarter after that, but I was a “drop in tutor” for my last three years of
college, meaning that I would field questions from any undergraduate math course
that a student was having trouble with, and I had many further interactions with
Spivak’s text. (By the end of my undergraduate career there were a small number
of double-starred problems of which I well knew to steer clear.)
Here is what I remembered about Spivak’s text in fall of 2011:
(i) It is an amazing trove of problems, some of which are truly difficult.
(ii) The text itself is lively and idiosyncratic.1
(iii) The organization is somewhat eccentric. In particular limits are not touched
until Chapter 5. The text begins with a chapter “basic properties of numbers”,
which are essentially the ordered field axioms, although not called such. Chapter
3 is on Functions, and Chapter 4 is on Graphs. These chapters are essentially con-
tentless. The text is broken up into five “parts” of which the third is Derivatives
and Integrals.
After teaching the 2400 course for a while, I lost no esteem for Spivak’s text,
but increasingly I realized it was not the ideal accompaniment for my course. For
one thing, I realized that my much more vivid memories of the problems than the
text itself had some basis in fact: although Spivak writes with a lively and distinct
voice and has many humorous turns of phrase, the text itself is rather spare. His
words are (very) well chosen, but few. When one takes into account the ample
margins (sometimes used for figures, but most often a white expanse) the chapters
themselves are very short and core-minded. When given the chance to introduce a
subtlety or ancillary topic, Spivak almost inevitably defers it to the problems.
I have had many years to reflect on Spivak’s text, and I now think its best use
is in fact the way I first encountered it myself: as a source of self study for bright,
motivated students with little prior background and exposure to university level
mathematics (in our day, this probably means high school students, but I could
imagine a student underwhelmed by an ordinary freshman calculus class for which
Spivak’s text would be a similarly mighty gift). Being “good for self study” is a
high compliment to pay a text, and any such text can a fortiori also be used in a
course...but not in a completely straightforward way. For my course, although the
1In between the edition of the book that Sally had given me and the following edition, all
instances of third person pronouns referring to mathematicians had been changed from “he” to
“she”. InitiallyIfoundthisamusingbutsilly. MorerecentlyIhavebegundoingsomyself.
SPIVAK AND ME 9
students who stuck with it were very motivated and hard-working, most (or all) of
themneededa lot ofhelpfrommeinallaspectsofthecourse. Ihadbeeninformed
in advance by my colleague Ted Shifrin that signing on to teach this course should
entail committing to about twice as many office hours as is typical for an under-
graduate course, and this did indeed come to pass: I ended up having office hours
fourdaysaweek,andIspentthemajorityofthemhelpingthestudentsmaketheir
way through the problems in Spivak’s text. Even the best students – who, by the
way,Ithoughtwereawfullygood–couldnotsolvesomeoftheproblemsunassisted.
There were many beautiful, multi-part problems introducing extra material that I
wanted my students to experience: and eventually I figured out that the best way
todothiswastoincorporatetheproblemsintomylecturesastheoremsandproofs.
All in all I ended up viewing Spivak’s book as being something of a “deconstruc-
tion”2ofthematerial,andmuchofmyteachingtimewasspent“reconstructing”it.
For me, the lightness of touch of Spivak’s approach was ultimately something I
appreciated aesthetically but could see was causing my students difficulty at vari-
ous key points. My experience rather convinced me that “more is more”: students
wantedtoseemoreargumentsincompletedetail,andalsosimplymoreproofsover-
all. Aside from the (substantial) content conveyed in the course, a major goal is to
give the students facility with reading, understanding, constructing and critiquing
proofs. Inthisregardbeinginaclassroomsettingwithothermotivatedstudentsis
certainlyinvaluable,andItriedtotakeadvantageofthisbyinsistingthatstudents
present their questions and arguments to each other as well as to me.
Butstudentsalsolearnabouthowtoreasonandexpressthemselvesmathemat-
ically by being repeatedly exposed to careful, complete proofs. I think experienced
mathematicians can forget the extent to which excellent, even brilliant, students
benefit from this exposure. Of course they also benefit immensely, probably more
so, by working things out for themselves. The value of this is certainly not in
question, and there is no competition here: the amount of “things todo” in honors
calculus is infinite – and the repository of problems in Spivak’s text is nearly so –
sobyworkingoutmoreforherself,theinstructorisnotleavingless forthestudents
to do, but only different things for them to do.
This brings me to the current text. As explained above, it is heavily indebted
to [S]. However, it is – or at least, I mean it to be – a new honors calculus text,
and not merely a gloss of [S]. Indeed:
The text is indebted to [S], but not uniquely so. It is also heavily indebted
•
toRudin’sclassictext[R], anditborrowsatkeypointsfromseveralothersources,
e.g. [A], [Go], [H], [L].
I do not view this borrowing as being in any way detrimental, and I certainly do
notattempttosuppressmyrelianceonothertexts. Themathematicsexposedhere
ishundredsofyearsoldandhasbeentreatedexcellentlyinmanyfamoustexts. An
undergraduate mathematics text which strove to be unlike its predecessors would
2My conception of the meaning of deconstruction in the sense of academic humanities is
painfullyvague. Iammorethinkingoftheterminthesensethatchefsuseit.
10 FOREWORD
almost certainly do so to the detriment of its audience.
Having said this, most undergraduate texts with “calculus” in the title offer
very little innovation indeed. It is distressing to me how many calculus books are
writtenwhichlooklikenearlyexactreplicasofsomeplatonic(butfarfromperfect)
calculus text written circa 1920. Apparently the issue of treating transcendental
functionsearlierratherthanlaterwasenoughtowarrantmanyneweditions, ifnot
wholly new texts.
This text does contain some novelties, up to and including a proof technique,
real induction,whichtomyknowledgehasnotappearedbeforeintexts. Spivak’s
text is a great innovation, and its distinctiveness – some might say eccentricity –
has not gone unnoticed by instructors. This text is also eccentric, and in its own
way: if you like Spivak’s text, you need not like this one.
What is Honors Calculus?
It was audacious of Spivak to call his text simply “calculus,” since it is light years
awayfromanyothertextofthatnamecurrentlyinprint. Ihavechosentocallthis
text: honors calculus. What is honors calculus?
By “honors calculus” I mean a certain take on undergraduate real analysis.
Some Features of the Text
Our approach is heavily theoretical but not heavily abstract.
•
The jumping off point for an honors calculus course is a theoretical understanding
oftheresultsseeninfreshmancalculus,especiallyrigorousdefinitionsofthevarious
limiting processes and proofs of the “hard theorems” of calculus, which inevitably
turnonthecompletenessofR. Wearenotinterestedincalculationforitsownsake,
and we do not give any halfway serious applications to any subject outside mathe-
matics. This theoretical perspective is of course a huge change from the practical,
problem-oriented approach of freshman calculus (with which we assume a prior
familiarity). It is certainly not as appealing to as broad an audience as freshman
calculus. At the University of Georgia, it happens that many freshman students
(especially, those in the honors program) are signed up for Math 2400 without any
realideaofwhattheyaregettinginto. Theattritionatthebeginningofthecourse
is therefore significant, and the students that remain are not necessarily the most
talentedorexperiencedbutratherthosethatremaininterestedinafullytheoretical
approach to calculus. This text assumes that interest on the part of the reader, in
fact probably even more so than Spivak’s text.
The idea of presenting a fully theoretical take on real analysis to a young un-
dergraduate audience is hardly new, of course. But most contemporary texts on
real analysis take a more sophisticated approach, making use of concepts from set
theory and topology. In this text we rarely speak explicitly about the connected-
ness and compactness of intervals on the real line. (More precisely, we never use
these concepts in the main text, but only in some digressions which briefly and
optionally present some advanced material: e.g. 10.10.5 introduces metric spaces
§
anddiscussedcompactnesstherein.) Insteadwemake(cid:15)-δ argumentswhichamount
to this. Let us freely admit that this concreteness makes certain arguments longer