Good book on calculus
Mike N. Christoff
calculus books out there but only a few are good to actually *learn* from. Also, I'm not looking for a book on
applied calculus, but one that includes derivations for all of the laws/theorems, etc... Another big factor is
that it goes over limits in depth with a large number of epsilon/delta proof questions.
The book I had was 'Calculus' by Michael Spivak.
l8r, mike
Michael Varney
"Mike N. Christoff" <mchri...@sympatico.ca> wrote in message
news:1103_978280357@administrator...
Calculus the easy way. Cute book told in story format.
:-D
Mike N. Christoff
Isn't suggesting a book you published yourself a conflict of interest?
l8r, mike
>
> :-D
>
>
AIfred E Neuman
>
Hmm... suggesting a book by anyone else would be a conflict of self-interest,
wouldn't it? Aside from that, anything that said xxxx "the easy way" might not
be something for someone looking for a complete education.
~~Aleph
Clark
with Analytic Geometry". I've found his books fairly easy to work from.
I've certainly seen much worse.
Clark
"Mike N. Christoff" <mchri...@sympatico.ca> wrote in message
news:1103_978280357@administrator...
Bob Silverman
IMO, the two volumes by Tom Apostol are *superb*. Calculus Vol I
and Calculus Vol II.
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
Sent via Deja.com
http://www.deja.com/
Bob Silverman
IMO, the two volumes by Tom Apostol are *superb*. Calculus Vol I
MCKAY john
Motto: What one fool can do, so can another.
Came out in 19th century - likely still in print.
John McKay
In article <92tcu6$fjh$1...@nnrp1.deja.com>,
--
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.
crankyho
"Clark" <clan...@utmb.edu> wrote:
> Many years ago I managed to teach myself from Howard Anton's book, "Calculus
> with Analytic Geometry". I've found his books fairly easy to work from.
> I've certainly seen much worse.
Thanks. Does this go into epsilon/delta proofs, etc... or is it a more
applied book on calculus? I'm looking for a book more on the mathematical
vs. applied side of things.
l8r, mike
>
> Clark
>
> "Mike N. Christoff" <mchri...@sympatico.ca> wrote in message
> news:1103_978280357@administrator...
> > I'm looking for a text book on calculus that is easy to learn from via
> self-study. There are a lot of good
> > calculus books out there but only a few are good to actually *learn* from.
> Also, I'm not looking for a book on
> > applied calculus, but one that includes derivations for all of the
> laws/theorems, etc... Another big factor is
> > that it goes over limits in depth with a large number of epsilon/delta
> proof questions.
> >
> > The book I had was 'Calculus' by Michael Spivak.
> >
> >
> >
> > l8r, mike
> >
> >
> >
>
>
crankyho
mc...@cs.concordia.ca (MCKAY john) wrote:
> Try Silvanus P. Thompsom: Calculus made easy.
>
> Motto: What one fool can do, so can another.
>
> Came out in 19th century - likely still in print.
>
Thanks. I checked it out at Amazon (its only $17.50 US). Unfortunately
'Calc made easy' apparently suppresses the very info I'm interested in. Up
until 6 days ago I had (til I lost it in a taxi) Spivak's Calculus but have
always found it lacking. I can characterize my feeling about Spivak as
'uneven'. I found that he skimmed over the stuff I found difficult and
over-explained the easier stuff. I was very dissapointed with his section on
limits. In my opinion he should have introduced the actual defn of limit far
earlier and spent his time proving limits from the ACTUAL defn, not his
provisional def. This would have helped me greatly.
However, Spivak's book still remains a classic, but as more of a detailed
technical history of calculus. He does a great job of showing how all of the
underpinnings of calc follow from one to another within an interesting and
enthusiatic narrative.
Duane Jones
Cheers,
Duane Jones
"crankyho" <cran...@my-deja.com> wrote in message
news:92tj75$lfc$1...@nnrp1.deja.com...
crankyho
popular, but as I've mentioned, I had problems with it. I'm also aware the
Apostle book teaches integration before differentiation, but I don't see a
problem with this as long as everything follows logically.
I'm more interested in the coherence of the explanations in Apostol vs.
Spivak, number and quality of examples stepped through, etc...
Dave L. Renfro
[sci.math Sun, 31 Dec 2000 16:19:52 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote
Are you saying that you don't find Spivak's book suitable? If so,
you might want to say what it is about Spivak's book that you don't
like since it often gets mentioned whenever a thread along the
lines of "What is the best math book ever written?" comes up.
Among the standard level calculus texts (i.e. the kind that you'll
find used in virtually all HS/college calculus courses), I like
Thomas/Finney's book and Anton's book.
Here are some books that you might want to look at (besides Spivak's
book) if you want something that is bit different from any of the
several hundred standard calculus texts that have appeared in the
past 30 years.
Richard Courant and Fritz John, "Introduction to Calculus and
Analysis", Volume 1, Springer-Verlag, 1999. [Reprint of the 1965
edition (I think).]
Richard Courant and Herbert Robbins, "What Is Mathematics?,
2'nd edition (revised with the assistance of Ian Stewart), 1996.
[Not entirely calculus, but definitely worth a look.]
G. H. Hardy, A Course In Pure Mathematics", Cambridge Univ.
Press, 1947. [You can find this in almost any college library.]
Tom M. Apostol, "Calculus", 2'nd edition, Blaisdell, 1967 (volume 1)
and 1969 (volume 2). [Used at CalTech. Volume 2 was used for the
2'nd year honors calculus course where I was an undergraduate.]
David Bressoud, "Second Year Calculus: From Celestial Mechanics to
Special Relativity", Springer-Verlag, 1991.
<http://www.macalester.edu/~bressoud/books.html#2YC>
If you can read Spivak without much difficulty, then you might also
want to supplement your reading by looking at some real analysis
texts. Here are a couple that I think would be useful to look at
from time to time:
David Bressoud, "A Radical Approach to Real Analysis", MAA, 1994.
<http://www.macalester.edu/~bressoud/books.html#2YC>
Victor Bryant, "Yet Another Introduction to Analysis", Cambridge
University Press, 1990. [Bryant's book probably requires less
mathematical maturity than Spivak, Courant/John, Hardy, or Apostol.]
Dave L. Renfro
Kevin Foltinek
<cran...@my-deja.com> writes:
> [Spivak's Calculus]
> In my opinion he should have introduced the actual defn of limit far
> earlier and spent his time proving limits from the ACTUAL defn, not his
> provisional def. This would have helped me greatly.
A book you may (or may not) enjoy is Gaskill and Narayanaswami,
"Foundations of Analysis: The Theory of Limits". Unfortunately it
seems to be out of print. It is a very verbose book, with lots of
what might be described as intuitive motivation and discussion, yet is
very rigourous in that it presents axioms/definitions and carefully
proves things from them; and it is very clear to distinguish the
discussion from the rigour. It covers the definition of the real
numbers, limits, continuity, sequences, series, analyticity, etc.; all
the usual stuff.
The same authors also wrote "Elements of Real Analysis", but I have
not looked at this book so cannot make any informed comments about
it. The blurb on amazon.com suggests that it is similar to their
previous book.
Kevin.
Dave L. Renfro
[sci.math Tue, 02 Jan 2001 22:03:23 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote
> Thanks. Does this go into epsilon/delta proofs, etc... or is it
> a more applied book on calculus? I'm looking for a book more on
> the mathematical vs. applied side of things.
In addition to the other books I mentioned in another post in
this thread, you might want to look at this book:
Andrew M. Gleason, "Fundamentals of Abstract Analysis",
A. K. Peters Ltd., 1991.
<http://206.204.3.133/dir_akp/akp_funabs.html>
The book I remember was published in 1966 by Addison-Wesley, so I
assume this is a new edition or a reprint of the 1966 book.
[Neither amazon.com nor A. K. Peters mention the 1966 edition
that I can find.] I read bits and pieces of the 1966 edition
throughout my undergraduate years. This book is VERY carefully
written and EVERYTHING is developed from scratch. From what I
recall, the book begins with truth tables and propositional logic,
then it proceeds to predicate logic, then to set theory, then to
the Peano axioms for the natural numbers and a model of them in ZF
set theory, then to constructions of the integers, rational numbers,
real numbers, and complex numbers, ... Gleason gives a lot of
carefully written explanations but somehow still manages to get
all the way up to things like the Cauchy integral formula.
*****************************************************************
*****************************************************************
Review at amazon.com by Christopher C. Sze
Fundamentals of Abstract Analysis was our prescribed text in
my Advanced Calculus class. I got hold of this book from our
college library. The way Gleason presented topics form this
course was brilliant. Although this course is not a basic
calculus course, this is a must for mathematics majors who
wish to pursue masters and doctorate degrees in the future.
This book is recommended for students with a more mathematical
maturity than that afforded by the usual freshman-sophomore
courses in calculus. And one excellent way to gain such maturity
is through a beginning course in linear algebra, although it
is not a pre-requisite for the book, but of an advantage. This
book tackles the "structure" of calculus, wherein theorems are
presented in much greater depths and goes "behind the scenes"
of the theorems in calculus. I recommend this book for
undergraduate math majors and graduates as well.
*****************************************************************
*****************************************************************
Dave L. Renfro
Herman Rubin
> Mike N. Christoff <mchri...@sympatico.ca> wrote:
>> I'm looking for a text book on calculus that is easy to learn from
>via self-study. There are a lot of good
>> calculus books out there but only a few are good to actually *learn*
>from. Also, I'm not looking for a book on
>> applied calculus, but one that includes derivations for all of the
>laws/theorems, etc... Another big factor is
>> that it goes over limits in depth with a large number of
>epsilon/delta proof questions.
>> The book I had was 'Calculus' by Michael Spivak.
<IMO, the two volumes by Tom Apostol are *superb*. Calculus Vol I
<and Calculus Vol II.
They are, but most are unable to use them because of
lack of preparation and ability. The preparation is
to know how to think logically.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Rubin
Duane Jones <ga...@ziplink.net> wrote:
>Yes Anton's text has an optional chapter that treats limits rigorously.
It is my belief that the rigorous treatment of limits,
continuity, etc., can be more easily understood by
someone who has not been drilled in computations than
by someone who has. If fact, much of this should be
mandatory when decimals are introduced; except for
terminating decimals, the meaning is lost without it.
>> l8r, mike
>> > Clark
>> > > l8r, mike
Herman Rubin
crankyho <cran...@my-deja.com> wrote:
>Any opinions on Spivak vs. Apostol? I realize the Spivak book is far more
>popular, but as I've mentioned, I had problems with it. I'm also aware the
>Apostle book teaches integration before differentiation, but I don't see a
>problem with this as long as everything follows logically.
Why should teaching integration before differentiation cause
any problem whatever? Except that they did not know how to
define limits precisely, the Greeks understood the "Riemann"
integral, including the limit aspects.
>I'm more interested in the coherence of the explanations in Apostol vs.
>Spivak, number and quality of examples stepped through, etc...
Apostol is written for those who can understand the subject,
not for those who can learn to calculate.
D. J. Bernstein
> I'm not looking for a book on applied calculus, but one that includes
> derivations for all of the laws/theorems, etc...
You might try my 12-page booklet: http://cr.yp.to/papers.html#calculus
---Dan
crankyho
hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
> In article <92tqql$s7v$1...@nnrp1.deja.com>,
> crankyho <cran...@my-deja.com> wrote:
> >Any opinions on Spivak vs. Apostol? I realize the Spivak book is far more
> >popular, but as I've mentioned, I had problems with it. I'm also aware the
> >Apostle book teaches integration before differentiation, but I don't see a
> >problem with this as long as everything follows logically.
>
> Why should teaching integration before differentiation cause
> any problem whatever? Except that they did not know how to
> define limits precisely, the Greeks understood the "Riemann"
> integral, including the limit aspects.
Its only a 'problem' in the sense that it is the opposite of the order they
are usually taught.
>
> >I'm more interested in the coherence of the explanations in Apostol vs.
> >Spivak, number and quality of examples stepped through, etc...
>
> Apostol is written for those who can understand the subject,
> not for those who can learn to calculate.
>
Have you read Spivak? If so, are you saying you found his text to be one for
people who simply want to 'learn to calculate'?
Also, if you have the book, I was wondeing, for curiosity's sake, how many
pages (excluding the question section) the chapter on limits is. (Proving
limits has always been my weakest point when it comes to calculus).
crankyho
Thanks for the link. Now if only I could find a DVI viewer that works on NT
4.0.
Karl Hahn
recommend my website, Karl's Calculus Tutor, which has
step by step development of the delta-epsilon process as
well as having rigorous proofs of theorems like the
Intermediate Value Theorem, the Extreme Value Theorem,
Rolles Theorem, etc. Most of the proofs are listed as
optional material because first year calc students are
very unlikely to be tested on them. There are a number
of worked delta-epsilon problems as well.
URL = http://www.netsrq.com/~hahn/calculus.html
Throughout the text I have tried to make the concepts
concrete using the teaching method my dad used on me,
that is using parable and metaphor to illustrate the
abstract. BTW my dad was gifted at teaching math --
he taught me basic group theory when I was only 8.
And just to blunt the self-interest criticism, please
know that I get no money for writing and maintaining
this website. It is free for anybody who wants to
browse it and has no advertising except for links
to Amazon on the recommended books page.
Also note that the website is still a work in progress.
Karl Hahn
Dave L. Renfro
[sci.math Tue, 02 Jan 2001 22:25:26 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote (in part)
> I was very dissapointed with his section on limits. In my opinion
> he should have introduced the actual defn of limit far earlier
> and spent his time proving limits from the ACTUAL defn, not his
> provisional def. This would have helped me greatly.
If you're interested in limits, try these web pages:
1. A number of non-trivial epsilon/delta proofs can be found
at the Univ. of Calif.--Davis' calculus site at
<http://www.math.ucdavis.edu/~kouba/ProblemsList.html>.
[Click on "limit of a function using the precise
epsilon/delta definition of limit".]
2. I posted a lot of "elementary" ways to evaluate limits at
<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul>.
3. You might also want to look through the extensive list of
web pages I posted at
<http://forum.swarthmore.edu/epigone/sci.math/grersterdwee>.
4. My Sept. 24, 2000 post at
<http://forum.swarthmore.edu/epigone/ap-calc/grisworleld>
(which isn't included in #3 above) might be helpful.
Dave L. Renfro
Dave L. Renfro
[sci.math Wed, 03 Jan 2001 00:13:16 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/stoshexspex>
wrote
> Any opinions on Spivak vs. Apostol? I realize the Spivak book
> is far more popular, but as I've mentioned, I had problems with
> it. I'm also aware the Apostle book teaches integration before
> differentiation, but I don't see a problem with this as long as
> everything follows logically.
>
> I'm more interested in the coherence of the explanations in
> Apostol vs. Spivak, number and quality of examples stepped
> through, etc...
For what it's worth, the Richard Courant and Fritz John book
that I mentioned in my list of books at
<http://forum.swarthmore.edu/epigone/sci.math/regronble/y4g4um...@forum.mathforum.com>
also covers integration before differentiation.
Based on what you've said in your posts I think you should visit a
college library and spend a few hours looking through the books
in the QA 300's to QA 330's section. NOW would be a good time,
since due to the semester break there are likely to be more
books on the shelves than anytime in the next 5 months.
Unless you're in a very isolated area of the country (as I
happened to be in for three years, ending about 18 months ago)
[I'm assuming you live in the USA], there should be a decent
college or university library close enough to you to do this
without having to stay overnight. Even better, if it's possible,
arrange to drive through several college towns and visit every
library you can. For instance, while college A's copy of Apostol
might be checked out, you may find it on the shelves at
college B. Make sure that you take plenty of notes on what you
see in the various books you look at. You might also want to
photocopy portions of the books (e.g. the section on limits, the
table of contents, etc.) for later comparison. [With this in mind,
you'll want to be prepared, so bring along a roll of quarters
and some "fresh" one and five dollar bills.]
Naturally, before you make such a trip, you'll want to look through
the amazon.com reviews of various calculus books, the MAA reviews
(see <http://www.maa.org/reviews/topics-index.html#Calculus>),
math discussion posts on these books (do various topic and author
searches at
<http://forum.swarthmore.edu/discussions/epi-search/all.html>),
and other things easily available on the internet.
Dave L. Renfro
Dave L. Renfro
appeared with the URL's I provided omitted.
Dave L. Renfro
reason appeared without the URL's I had provided.
Herman Rubin
crankyho <cran...@my-deja.com> wrote:
>In article <930n9u$14...@odds.stat.purdue.edu>,
> hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>> In article <92tqql$s7v$1...@nnrp1.deja.com>,
>> crankyho <cran...@my-deja.com> wrote:
>> >Any opinions on Spivak vs. Apostol? I realize the Spivak book is far more
>> >popular, but as I've mentioned, I had problems with it. I'm also aware the
>> >Apostle book teaches integration before differentiation, but I don't see a
>> >problem with this as long as everything follows logically.
>> Why should teaching integration before differentiation cause
>> any problem whatever? Except that they did not know how to
>> define limits precisely, the Greeks understood the "Riemann"
>> integral, including the limit aspects.
>Its only a 'problem' in the sense that it is the opposite of the order they
>are usually taught.
Integration is sums of products, and limits of such. Discrete
integration is about 5000 years old. That it was this simple
was essentially lost after differentiation was introduced.
>> >I'm more interested in the coherence of the explanations in Apostol vs.
>> >Spivak, number and quality of examples stepped through, etc...
>> Apostol is written for those who can understand the subject,
>> not for those who can learn to calculate.
>Have you read Spivak? If so, are you saying you found his text to be one for
>people who simply want to 'learn to calculate'?
>Also, if you have the book, I was wondeing, for curiosity's sake, how many
>pages (excluding the question section) the chapter on limits is. (Proving
>limits has always been my weakest point when it comes to calculus).
Understanding limits is the important part of calculus. The
rest is computation. For the user of mathematics, being able
to formulate, and understand what it means, is the important
part. Being able to calculate is not; the machine, or the one
who knows how to calculate, can get the answer to a properly
formulated problem.
Herman Rubin
Dave L. Renfro <ren...@central.edu> wrote:
>crankyho <cran...@my-deja.com>
>[sci.math Wed, 03 Jan 2001 00:13:16 GMT]
><http://forum.swarthmore.edu/epigone/sci.math/stoshexspex>
............
>Based on what you've said in your posts I think you should visit a
>college library and spend a few hours looking through the books
>in the QA 300's to QA 330's section.
Not all libraries use the Library of Congress classification.
I believe it is 517 in the Dewey system, but I am not sure.
NOW would be a good time,
>since due to the semester break there are likely to be more
>books on the shelves than anytime in the next 5 months.
Dave L. Renfro
[sci.math 6 Jan 2001 14:44:28 -0500]
<http://forum.swarthmore.edu/epigone/sci.math/stoshexspex>
wrote (in reply to something I wrote):
>> Based on what you've said in your posts I think you should visit a
>> college library and spend a few hours looking through the books
>> in the QA 300's to QA 330's section.
>
> Not all libraries use the Library of Congress classification.
>
> I believe it is 517 in the Dewey system, but I am not sure.
Yeah, I should know--this year I'm teaching at a college whose
library uses the Dewey decimal system! Right now I'm nearly
900 miles away and I only brought 5 math books with me, but
one of them happens to be Victor Bryant's "Yet Another
Introduction to Real Analysis" (mentioned in one of my
Jan. 3 posts in the thread "Good book on calculus" at
<http://forum.swarthmore.edu/epigone/sci.math/regronble>), and
its call number is 515 Br 9 Y.
When I was working on my Ph.D. dissertation (early 1990's at
NC State Univ.) I often drove to the nearby (each within 35
miles) UNC-Chapel Hill and Duke Univ. math/physics libraries
for papers and books that NC State Univ.'s library didn't
have. Each of these other two universities has one of the
most complete mathematics library collections in the U.S.
(top two dozen, I'd bet). In fact, of the 30 to 40 university
libraries I've visited, only the mathematics collection at
Univ. of Michigan (Ann Arbor) is clearly superior. Michigan's
library is superior in another way, incidentally: Their library
has a lot of shelf space, and so you can browse their stacks for
even the obscure stuff, rather than having to fill out some
type of "off-site storage request form" and wait a day or two.
[This is expensive if you're only visiting and paying hotel
costs for each day, and it's awkward--regardless of whether
you're visiting or if you live there--when you're trying to
thoroughly research a topic rather than simply look up
references that someone else cited.] Anyway, Duke Univ.'s
libraries use the Dewey Decimal system, and I always felt
lost there since the mapping between these two classification
systems isn't order preserving.
Dave L. Renfro
crankyho
First off, thanks for the links, I've been reading through them.
For the proof that lim_{x-1}f(x) does not exist when f(x) = {x, x < 1; 3 - x,
x >= 1}, I came up with a proof that didn't use the triangle inequality.
However, I did use a very unmathematical looking '+/- d' which I'm not 100%
sure is valid. (Of course, one could also use left and right limits, but
anyways here's the 'proof'):
---------------
Let f(x) = {x, x < 1; 3 - x, x >= 1}
We want to show that lim_{x->1}f(x) does not exist.
Reworking things a bit, we create g(x) = x and h(x) = 3 - x. This means we
can renotate f(x) as f(x) = {g(x), x < 1; h(x), x >= 1}.
Assuming lim_{x->1}f(x) exists, we have that:
For all e > 0 there exists a d > 0 such that when |x-1| < d, then |f(x) - L|
< e.
This is equivalent to saying:
For all e > 0 there exists a d > 0 such that when |x-1| < d, then:
|g(x) - L| < e, and
|h(x) - L| < e
From looking at the graph*, we can see that close to x = 1, g(x) is close to
1 and h(x) is close to 2.
* - graph is at
<http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/Prec
LimSol.html#SOLUTION 15>
The important thing to note is that close to x=1, g(x) and h(x) are about a
value of 1 apart. By making e = 1/2 its obvious that there is no L such that
|g(x) - L| < 1/2 AND |h(x) - L| < 1/2.
However, we've assumed a limit L does exist, therefore, for e = 1/2 we assume
there exists a d such that:
|g(1 +- d) - L| < 1/2
which implies
|1 +/- d - L| < 1/2 => -1/2 < 2 +/- d - L < 1/2 => -3/2 + L < +-d < -1/2 + L,
AND
|h(1 +/- d) - L| < 1/2
which implies
|2 +/- d - L| < 1/2 => -1/2 < 2 +/- d - L < 1/2 => -5/2 + L < +/- d < -3/2 +
L.
However, for both of these to be true requires
-3/2 + L < +/- d < -3/2 + L,
an impossibility which contradicts our initial assumption. Therefore the
limit does not exist.
l8r, mike
Dave L. Renfro
[sci.math Mon, 08 Jan 2001 04:25:34 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote
> First off, thanks for the links, I've been reading through them.
>
> For the proof that lim_{x-1}f(x) does not exist when
> f(x) = {x, x < 1; 3 - x, x >= 1}, I came up with a proof that
> didn't use the triangle inequality. However, I did use a very
> unmathematical looking '+/- d' which I'm not 100% sure is valid.
> (Of course, one could also use left and right limits, but anyways
> here's the 'proof'):
>
> ---------------
>
> Let f(x) = {x, x < 1; 3 - x, x >= 1}
>
> We want to show that lim_{x->1}f(x) does not exist.
>
> Reworking things a bit, we create g(x) = x and h(x) = 3 - x.
> This means we can renotate f(x) as
> f(x) = {g(x), x < 1; h(x), x >= 1}.
>
> Assuming lim_{x->1}f(x) exists, we have that:
>
> For all e > 0 there exists a d > 0 such that when |x-1| < d,
> then |f(x) - L| < e.
Careful. You're assuming the limit exists, but what you wrote next
was that the limit is L. [Instead, write "Assuming ... = L,
we have that:"] By the way, there *are* ways of showing that a
limit exists without knowing what it is. Two examples that come to
mind (for sequences in the reals) are: (a) when you show the
sequence is a Cauchy sequence, and (b) when you show the sequence
is nondecreasing (nonincreasing) and is bounded above (is bounded
below).
>
> This is equivalent to saying:
>
> For all e > 0 there exists a d > 0 such that when |x-1| < d, then:
>
> |g(x) - L| < e, and
> |h(x) - L| < e
This is not equivalent, at least not in the sense of how you should
be working with a piecewise defined function. [Technically, the
statements *are* equivalent, though, since each is false, and
"False <==> False" is a true statement in logic.] More precisely,
what you wrote earlier doesn't imply what you just wrote (although
the reverse implication appears O-K). To fix this up, change the
word "and" to "or".
By the way, it's not necessary for something to actually be
logically equivalent for you to be able to use it. All you usually
need in a proof is that, at each step, what you just wrote follows
logically from stuff you've previously written. Whenever I see
"equivalent" in a proof, I wonder why the reader is telling me
more than I expected to have to know at that point. (Of course,
sometimes "equivalent" is needed in a proof.)]
>
> From looking at the graph*, we can see that close to x = 1,
> g(x) is close to 1 and h(x) is close to 2.
>
> * - graph is at
>
<http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html#SOLUTION
15>
>
> The important thing to note is that close to x=1, g(x) and h(x)
> are about value of 1 apart. By making e = 1/2 its obvious that
> there is no L such that |g(x) - L| < 1/2 AND |h(x) - L| < 1/2.
Since you didn't say what the domains of g(x) and h(x) are,
I just assumed that they were chosen so that f(x) is the union
of g(x) and h(x) (when identifying a function with its graph).
This means that g(x) has domain (-infinity, 1) and h(x) has
domain [1, infinity). Therefore, there **IS NO** L such that
|g(x) - L| < 1/2 AND |h(x) - L| < 1/2, since there isn't even
a value of x for which BOTH g(x) and h(x) are defined.
[snip rest]
In another post I'll give some more web page URL's that you
(and maybe some other sci.math readers) might find useful.
Dave L. Renfro
Dave L. Renfro
[sci.math Mon, 08 Jan 2001 04:25:34 GMT]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote (first sentence only):
> First off, thanks for the links, I've been reading through them.
Here are some more that you might find useful:
1. Manya Raman's (Univ. of Calif. Berkeley) Delta-Epsilon Tutorial
<http://socrates.berkeley.edu/~manya/deproject/>
2. Phill Schultz's (Univ. of Western Australia) historical essays at
<http://www.maths.uwa.edu.au/~schultz/3M3/Table_of_Contents.html>
Here are some essays that you might find interesting:
(a) "Zeno's Paradoxes"
<http://www.maths.uwa.edu.au/~schultz/3M3/L5Zeno.html>
(b) "Berkeley's criticism of the foundations of Calculus"
<http://www.maths.uwa.edu.au/~schultz/3M3/L21Berkeley.html>
(c) "D'Alembert's foundational work on limits"
<http://www.maths.uwa.edu.au/~schultz/3M3/L23DAlembert.html>
(d) "Euler's explanation of limits"
<http://www.maths.uwa.edu.au/~schultz/3M3/L24Euler.html>
(e) "Cauchy's definition of differentiability"
<http://www.maths.uwa.edu.au/~schultz/3M3/L27Cauchy.html>
(f) "Weierstrass, Dedekind and Cantor on the Foundations
of Mathematics"
<http://www.maths.uwa.edu.au/~schultz/3M3/L28Weierstr,Dede,Cantor.html>
3. Walter Felscher's (Univ. of Tuebingen) essay "Cauchy v. Bolzano"
<http://www.maths.uwa.edu.au/Staff/schultz/3M3/Bolzano_v_Cauchy.html>
4. CalTech's Ma 1a - Freshman Mathematics (Fall 2000)
<http://www.math.caltech.edu/ma_9/sols.html>
5. CalTech's Ma 1a - Probability and Calculus of One and Several
Variables (Fall 2000)
<http://www.math.caltech.edu/ma1a/notes.html>
<http://www.math.caltech.edu/ma1a/sols.html>
6. CalTech's Ma 1b - Probability and Calculus of One and Several
Variables [Practical Track] (2'nd term 2000-2001)
<http://www.math.caltech.edu/ma1b-pr/notes.html>
7. CalTech's Ma 1b - Probability and Calculus of One and Several
Variables [Analytical Track] (2'nd term 2000-2001)
<http://www.math.caltech.edu/ma1b-an/>
8. CalTech's Ma 108a - Classical Analysis (Fall 2000)
<http://www.math.caltech.edu/courses/00ma108a.html>
Dave L. Renfro
Dave L. Renfro
[sci.math 8 Jan 01 12:25:40 -0500 (EST)]
<http://forum.swarthmore.edu/epigone/sci.math/regronble>
wrote (in part)
> By the way, it's not necessary for something to actually be
> logically equivalent for you to be able to use it. All you usually
> need in a proof is that, at each step, what you just wrote follows
> logically from stuff you've previously written. Whenever I see
> "equivalent" in a proof, I wonder why the reader is telling me
^^^^^^^^^^
Ooops! I mean the writer.
> more than I expected to have to know at that point. (Of course,
> sometimes "equivalent" is needed in a proof.)]
Dave L. Renfro