Which Calculus Book Will Give Me the True Understanding I'm Looking For?

[Fisicks]

Last year I took ap calculus ab, and I am taking ap calculus bc next year. So i decided to refresh my memory this summer and went through The Calculus Lifesaver by Adrian Banner, and thoroughly enjoyed it since i didn't get the depth that it went into in class. Now i know there are more "in depth" books out there, specifically Apostol and Spivak.

I read what I could of them online, which was only like a couple of pages, and liked Apostol's approach a little better, but could someone lead me towards which one will give me a true understanding of calculus? I love a challenge, there would defiantly have to be problems-solutions, good explanations of proofs, theorems, defiantly have to cover calculus AB and BC, and really overall just give me that true mathematical understanding I'm looking for.

Thanks in advance.

 

[thrill3rnit3]

Tom Apostol's generally is the tougher to read among those two. But you can't go wrong with either one. I'm going through Spivak's right now, and I'm loving it.

Spivak's IMO is more enjoyable to read, Apostol's is a bit dry but I guess it's alright for serious students.

In short, they're pretty much interchangeable.

 

[n!kofeyn]

I agree with thrill3rnit3 about the two books being interchangeable. If you go through either book, you will be getting the best education in calculus you can get through a textbook in my opinion. I wish I had known about either of these books when I was first learning calculus. If you liked Apostol, don't be afraid to go with it, and you will definitely learn calculus thoroughly. I think he covers quite a bit more material than Spivak, although I believe there is an answer book for Spivak, which could be helpful for self-study. In answer to your question, they definitely cover calculus for AB, BC, and beyond.

These books can be expensive, especially Apostol, so be sure to see if your library has them or if you could get them through inter-library loan.

I thought I should mention that there is a third calculus book to complete the tribunal, and that is Differential and Integral Calculus by Richard Courant. It is a more intuitive and physical approach than the above two books. There is a newer version called Introduction to Calculus and Analysis that was supposed to be developed to fit the American calculus course better. Spivak and Apostol have much more rigor though.

 

[thrill3rnit3]

The most important thing in going through either book (especially Spivak's) is to do ALL or MOST of the exercises.

 

[Fisicks]

Thanks to both of you, but i still have one more question...
Im assuming then that apostol's book doesn't have answers to his exercises then?

 

[thrill3rnit3]

Spivak's doesn't

edit: actually he has answers to SELECTED problems

 

[n!kofeyn]

Yes, Spivak has answers only to selected exercises in the back of his book. But, there is a totally separate https://www.amazon.com/dp/0914098926/?tag=pfamazon01-20 that has actual solutions or hints to all the exercises.

I looked at Apostol again, and he does have answers (not solutions) to almost all of his exercises. The only ones he doesn't include in the back are the proof-like questions.

Apostol is very expensive, so you might want to look at getting Spivak and his answer book, which looks like it would cost you less than if you got Apostol's book. I've never really read through Spivak, but looking at it some more, it looks very, very good. It's unfortunate that Apostol is so expensive, as it is very good as well. It's a no lose situation.

 

[thrill3rnit3]

There are international versions available at Alibris.com that sells for about $22 each.

http://www.alibris.com/booksearch.detail?invid=9934742937&cart=1

I think these versions are paperback and are made of cheaper quality. Do you think it's a good bargain (because I'm planning to get one)??

 

[Feldoh]

There are international versions available at Alibris.com that sells for about $22 each.

http://www.alibris.com/booksearch.detail?invid=9934742937&cart=1

I think these versions are paperback and are made of cheaper quality. Do you think it's a good bargain (because I'm planning to get one)??

Considering last time I checked Apostol was closer to $100 so it seems like a fairly good deal.

 

[thrill3rnit3]

yeah but it's paperback and the printing would probably be a little bad :tongue:

 

[n!kofeyn]

There are international versions available at Alibris.com that sells for about $22 each.

http://www.alibris.com/booksearch.detail?invid=9934742937&cart=1

I think these versions are paperback and are made of cheaper quality. Do you think it's a good bargain (because I'm planning to get one)??

Yea, they are definitely paperback, but I'm not for sure of the quality. I know a lot of people swear by ordering the international editions, so they can't be that bad. I think they're more like nice photocopies than actual printings though. I myself strongly prefer hardback, and I must admit that I've seen a hardback Apostol in the library, which was an extremely nice book. The book is very well bound and is heavy because of the quality paper. The book is rather thick though, so the paperback may not hold up as long. I guess you could get the paperback, and then if you really do enjoy the book, then splurge for the nicer hardback, as it is definitely a nice looking book on the desk or bookshelf.

 

[rasmhop]

yeah but it's paperback and the printing would probably be a little bad :tongue:

I have an International edition of Apostol (I'm assuming it's the same as the publishers are identical) so I thought I would shed some light on the issue of the quality of the international edition of Apostol since some people seem to be worried about it. It's indeed paperback and the cover is uniformly colored, but the coloring is pretty bad so you will probably get these white lines all over it after just a little use. If you want something that looks impressive on the desk, then this isn't it. As with all international edition the pages are pretty thin, but they seem to be sufficiently thick to keep them from breaking (I haven't had a single tear in either of the volumes and I wouldn't consider myself especially careful). Personally I prefer these slightly thinner pages as they are easier to handle than the photo-like paper in some modern textbooks (especially science books). The binding also seems quite solid. As for the actual printing it seems fine. Every 20 or so pages a single letter is a little blurry, but easily readable, and sometimes the text becomes a bit bolder in what seems to be arbitrary places, but in my opinion it isn't a problem.

As for which book is actually the best I believe both can give you an excellent handle of calculus. Personally I find Apostol hard to learn from and his reasoning often seem unsupported in my opinion, but it contains all the material and if you are willing to spend a little extra time deciphering proofs and why they are structured as they are then Apostol can be a good book, and it may actually force you to think a bit more deeply about the material. Spivak is pretty talkative which can be an obstacle to some students who have a hard time extracting rigorous arguments or mathematical heuristics from such text, but personally if I were to re-learn calculus I would go with Spivak over Apostol.

I don't know how much alibris charges for shipping, but if you want to save 10-20$ you may want to take a look at AbeBooks. There they sell for about 3$, but the shipping seems awfully expensive so I don't know if it really is cheaper in total. AbeBooks is usually pretty reliable (just remember to order a new book and not a used one), though I have once received the wrong book (ordered An Introduction to Homological Algebra, but got An Introduction to Harmonic Analysis).

 

[n!kofeyn]

AbeBooks is usually pretty reliable.

I have also had good experience with AbeBooks. Thanks for the extra advice. I would really like to have time to go through either of these books for enjoyment and picking up a few things. I'm impressed by Fisicks being aware of these books and willing to go through them, while only in high school.

 

[thrill3rnit3]

abebooks is wonderful site, thanks for sharing

They sell volume 1 (international version) for about 3 dollars. Should I trust that? :tongue:

EDIT: nevermind, shipping for 2 (international) books is a whopping 57 dollars!

 

[qspeechc]

Sorry to confuse you more on the issue, but I thought I'd throw another book into the pot (aweful metaphor I admit!). The book by Courant and John Introduction to Calculus and Analysis I found was very good, at the same level as Spivak, but a very nice read, well explained. Spivak, however, is something of an institution, or rather, a religion these days. So there you are, not sure I helped any.

 

[thrill3rnit3]

The books by Spivak, Apostol, and Courant are pretty much the "big three". You can't go wrong with either one of them.

 

[n!kofeyn]

Sorry to confuse you more on the issue, but I thought I'd throw another book into the pot (aweful metaphor I admit!). The book by Courant and John Introduction to Calculus and Analysis I found was very good, at the same level as Spivak, but a very nice read, well explained. Spivak, however, is something of an institution, or rather, a religion these days. So there you are, not sure I helped any.

You're right, as Courant is very good. This also includes his original book, Differential and Integral Calculus. Although, I had already mentioned Courant back in the 3rd post. :)

 

[thrill3rnit3]

I haven't had the chance to go through Courant's books (both of them - Intro to Calculus and Analysis, and his original D & I Calculus). But I've been reading good reviews on them, and some people are even recommending them over Spivak's.

 

[n!kofeyn]

I haven't had the chance to go through Courant's books (both of them - Intro to Calculus and Analysis, and his original D & I Calculus). But I've been reading good reviews on them, and some people are even recommending them over Spivak's.

I have browsed Courant, but as I mentioned above, the biggest difference is that he seems to have discussion with more physical intuition than both Spivak and Apostol. From just glancing, it seems to be looser, but that gives it a different quality than the others. It was a warm read.

 

[snipez90]

Going over both might not be the best idea. His intro to calculus and analysis text is in many ways a more formal presentation of the concepts in his differential and integral calculus text. If you're already working on Spivak, go through that first. Courant is very good at tying everything together. For instance, when he introduces the limit of a sequence, he will talk about limits of functions, sequential criterion, continuity, Cauchy sequences, point of accumulation, bolzano-weierstrass theorem, limsup, liminf, etc. all in the same chapter.

Many would consider this good pedagogy, but Spivak forces you to do maybe half the work on your own. For instance, Spivak will show you the proof of the sequential criterion, the bolzano-weierstrass theorem in terms of sequences, and most of the proof of the Cauchy criterion. He will let you figure out how to generalize the proof by contradiction of one direction of the sequential criterion, how the Bolzano-Weierstrass theorem is proved using the notion of limit points, and leave a somewhat tedious but important part of the proof of the Cauchy criterion to you.

Seriously, I don't think Spivak is any better necessarily due to the readings, but I know that the problems are good, and worth doing. Besides, if you want a presentation like that of Courant, read chapters 5, 22, 6, 7, and 8 (including appendix).

 

[snipez90]

I have browsed Courant, but as I mentioned above, the biggest difference is that he seems to have discussion with more physical intuition than both Spivak and Apostol. From just glancing, it seems to be looser, but that gives it a different quality than the others. It was a warm read.

Yeah, it's definitely looser in a sense. But at the very least, he usually gives a precise definition iirc. For instance, he'll say that a sequence has a limit L if for every interval, no matter how small, marked off about point L, includes all but a finitely many terms of the sequence. Then he formalizes the concept using epsilon's and N's. I think he is a bit looser in the presentation in the appendix, but I forget. Either way, when he does give a proof, it is usually fairly rigorous.

 

[n!kofeyn]

Going over both might not be the best idea. His intro to calculus and analysis text is in many ways a more formal presentation of the concepts in his differential and integral calculus text. If you're already working on Spivak, go through that first.

I'm not for sure who this is aimed at. I don't see where anybody is working through both or suggesting to work through both. The original poster has already mentioned that he looked through Apostol and is going to get it. Only thrill3rnit3 mentioned he's going through Spivak, but didn't mention he was going through both or wanted to.

 

[snipez90]

I haven't had the chance to go through Courant's books (both of them - Intro to Calculus and Analysis, and his original D & I Calculus)... and some people are even recommending them over Spivak's.

Yeah, I guess I misinterpreted that with my first sentence, but thanks for pointing out the essential point of my post.

 

[thrill3rnit3]

Yeah I probably worded it wrong.

I haven't gone through any of Courant's books :tongue:

 

[JG89]

Quick question guys - I'm self studying Courant's Introduction to Calculus and Analysis right now. I'm almost done the text. I have to take an analysis course that is notorious for being very hard starting September that teaches from Spivak. Do you think the questions in Courant prepared me adequately for Spivak?

 

[snipez90]

Which course is it, if you don't mind me asking? I think you are going to be fairly bored. Reading Courant alone is more than enough preparation for Spivak. Courant's readings will likely cover everything Spivak covers and more.

 

[JG89]

It is U of T`s (University of Toronto) Analysis 1 class - MAT157. Here`s a course description for it:

MAT157Y1
Analysis I [72L, 48T]

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor’s theorem; sequences and series; uniform convergence and power series.

 

[n!kofeyn]

Quick question guys - I'm self studying Courant's Introduction to Calculus and Analysis right now. I'm almost done the text. I have to take an analysis course that is notorious for being very hard starting September that teaches from Spivak. Do you think the questions in Courant prepared me adequately for Spivak?

As I mentioned earlier in this thread, Spivak has an answer book that gives good hints and solutions if you are ever flat out stuck. If you have worked through Courant and have actually done a few of the exercises, then you should be more than prepared for your course.

 

[JG89]

Thanks for the responses guys. I feel less nervous about taking the course now.

 

[snipez90]

Ah, okay I actually studied from this guy's webpage:

http://www.math.toronto.edu/~drorbn/classes/0405/157AnalysisI/index.html

when I took a similar course this past year. There are multiple years of that course, but that is the most recent. Anyways, the term exams didn't seem particularly difficult, but that varies from course to course I'm sure. As long as you work out the problems that seem challenging to you in Spivak (in particular, the single-starred or double-starred problems), you'll probably gain a lot from the course.

 

[Fisicks]

anyone have any underground places where i can get a hardback copy of apostol cheap? lol

 

[thrill3rnit3]

Hardback? Not a chance. They're at least 100 bucks each.

You can get a softback international copy for 1/4th the price. the link is at the previous page, I believe

 

[Sankaku]

I got a hardback copy of Spivak's 3rd edition for about $38 Canadian from amazon.com but I suspect most places will try to sell you the more expensive 4th edition now.

I have seen pdf versions of both Spivak and Apostol floating around, but they are definitely not publisher-approved...

 

[thrill3rnit3]

I got a hardback copy of Spivak's 3rd edition for about $38 Canadian from amazon.com but I suspect most places will try to sell you the more expensive 4th edition now.

I have seen pdf versions of both Spivak and Apostol floating around, but they are definitely not publisher-approved...

I have PDFs of a lot of books. However, I am not a big fan of reading ebooks on your computer. I prefer having the actual book so I can take it wherever I want to go. I also get distracted easily when I'm in front of the computer :tongue:

 

[Landau]

You know, computers nowadays are often connected with a printer, so you can convert digital text into paper with ink :tongue: