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I finished calculus books like Thomas and currently reading a book on advanced calculus and another one on real analysis.

I noticed recently that I don't solve enough "hard problems". I usually just solve the exercises on my books. so I figured that I need problem book(s) on calculus.

I want a book(s) that has many exercise on calculus topics like Limits(without l'hopital) ,derivatives, integrals , series and sequence , multiple integrals etc... I also want them to be challenging and interesting problems and not "too hard" I consider myself on an intermediate level as the advanced exercises on Thomas book seems very easy to me, I also don't want "forward substitute questions" which is plugging numbers in formulas or straight forward use of a theorem or equation as I want to increase my problem solving skills.

I will appreciate any suggestions ,thank you in advance .

pie
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    https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/ – Marius S.L. Jul 16 '23 at 23:43
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    Stewart and others have advanced exercises at the end of each chapter... have you tried those? I also suggest the book "Inside Interesting Integrals." – Sean Roberson Jul 16 '23 at 23:44
  • @SeanRoberson I solved pretty much all of them and I want more advanced exercises – pie Jul 16 '23 at 23:45
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    See Introduction to Calculus and Analysis by Courant/John -- Volume 1 (also here; solutions here) and Volume 2 (also here). See also Extremely "hard" books (or handouts) for undergrad studies. – Dave L. Renfro Jul 17 '23 at 00:01
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    Stewart's Calculus is a waste of paper. – NDB Jul 17 '23 at 01:02
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    If you just want challenging problems, you could visit the tags related to calculus here. – Accelerator Jul 17 '23 at 04:49
  • @Accelerator they are too difficult and I want something easier – pie Jul 17 '23 at 16:44
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    Dude... you should not be looking for easier problems if you want to get better. The only way you'd get better is if you struggle. Just jump into the challenging problems head-on and when you struggle, that's good and that's the whole point of math. Maybe it would show you would need to read a book about the topic to gain some fundamentals in your tool set. Not to say one learning trigonometry should jump straight into topology, but at least don't try to look for easier problems that lie in your comfort zone. – Accelerator Jul 17 '23 at 17:05
  • @Accelerator I understand that I must struggle to get out of my comfort zone but I think problems here are too difficult it is like saying if jump into the Riemann hypothesis and tried so hard that would make me better at math but I doubt that what would happen and that is R.H is too difficult problem.

    so I need some hard problem that currently above my level but not too much above

     – pie Jul 17 '23 at 19:37
  • @Accelerator also I need more "easy" source to use like a book as finding random problem that I can understand and solve will take time and this won't cover every type of problem on calculus – pie Jul 17 '23 at 19:40
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    I see... there's this book on Amazon if you're willing to spend the money. There are also the American Mathematical Monthly problems but speaking from personal experience, I find those problems extremely difficult. I wish I could provide a link but for some reason I can't open the page. – Accelerator Jul 17 '23 at 21:03
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    Try Spivak or Apostol’s calculus texts. Though I’m not convinced solving hard calculus problems is the best way to proceed. It may be better to move on to a new subject and learn it, you will invariably find the need for calculus in the future, and it will be more meaningful in this context. – Andrew Jul 25 '23 at 03:57

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My recommendation would be "Problems in Mathematical Analysis" by Demidovich. The book was written in Russian, but you can find the English translation online. In total there are over 3,000 problems with solutions in the book ranging from complete beginner problems to intermediate/advanced.

bb_823
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The following might be helpful:

Problems and Theorems in Analysis I and Problems and Theorems in Analysis II by G. Pólya and G. Szegö

This two volume book is a problem based classic (first edition from 1925). It is a highly inspiring source of many interesting problems in different fields of analysis together with detailed solutions. The AMS-review might be interesting.

Markus Scheuer
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  • are you sure these books aren't too hard for my level ? I heard about them as too difficult problem book on math – pie Jul 23 '23 at 18:33
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    @Pie: I recommend that you see for yourself. Just start right at the beginning with some problems from Volume I. – Markus Scheuer Jul 23 '23 at 18:44
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    @pie: These ARE too hard and too mathematically advanced for your current level. However, it might be mathematically inspiring for you to read the statements of some problems you have the background to understand -- also, any hints and/or solutions provided may indicate some interesting ways in which such problems can be approached. But I suggest not buying the books if you can't find them for free somewhere (internet, university library, etc.), unless you're rich, as there are more appropriate problem solving books for your level. – Dave L. Renfro Jul 23 '23 at 18:57
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GN Berman, A Problem Book in mathematical analysis is your best bet, no other book comes close for questions related to calculus but not analysis. https://www.scribd.com/document/557039546/A-Problem-Book-in-Mathematical-Analysis-G-N-Berman-1977#

The Grim Reaper
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I recommend you check out (Almost) Impossible Integrals, Sums, and Series written by Cornel Ioan Valean. Instead of paying for the book, you could click on the link and get it in PDF form. The way the problems are written is unique in that there is a chapter for questions, a chapter for hints, and a chapter for solutions.

Kamal Saleh
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I recommend you check out these books;

  1. "Calculus for JEE (Advanced) - Cengage" by G. Tewani, see here

The above book is for the specific exam known as the JEE advanced, however the book does come with intermediate level calculus questions.

  1. For Multiple Integrals and other calculus content, I recommend "Higher Engineering Mathematics" by B.S Grewal see here and "An introduction to Harmonic series and logarithmic integrals" by Ali Shadhar - see here.

  2. "Problems in calculus of one variable" by I.A Maron -see here

  3. "Black Book" by Vikas Gupta - see here

Also, to solve "Hard" problems, the best task would be to solve math contest papers, there are plenty exams and their archives you can find online.

You can find online pdfs for them. .

NadiKeUssPar
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  • The only google hit for the title "Cengage for Advanced Calculus" is your MSE answer, so either this book is virtually unknown or there is something amiss with the title you've given. Incidentally, if this is at the standard advanced calculus level (3rd to 4th undergraduate year in the U.S. -- several examples are given in this answer), then your comment "used by highschoolers in general" doesn't seem very reasonable. – Dave L. Renfro Jul 23 '23 at 19:05
  • @DaveL.Renfro thank you so much i missed a word in the title, i used the phrase "highschooler" because that book is more inclined to math competitions and hence has good set of problems in calculus as requested by the OP, also im not entirely sure of the US standards of "standard advanced calculus level" – NadiKeUssPar Jul 23 '23 at 19:08
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    For the usual (in the U.S.) topics and level of "advanced calculus", see the 2nd sentence in my previously cited MSE answer and this typical such text. (moments later) From your edit, I see now that "advanced" is used as a modifier for "JEE" and not as a modifier for "calculus" (confirmed with a google search for "JEE advanced"), so the title does not actually involve the phrase "advanced calculus". – Dave L. Renfro Jul 23 '23 at 19:29
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    Better is to use the exact title (helps with google searches such as freely available copies at archive.org and other places, could help with people seeing this many years from now, etc.), which seems to be "Calculus for JEE (Advanced)". Even better, give a link with the title -- Calculus for JEE (Advanced) (see here in case you don't know how). From what I know of the JEE advanced test, this could be a useful source of (not so real-analysis-theoretically-focused) problems for the OP. – Dave L. Renfro Jul 23 '23 at 19:47
  • @DaveL.Renfro although i do have the links, i have not given them as i am unsure whether they are safe to access(please suggest if i should edit in the links anyways). Also thank you very much for guiding me on how to write latex for a link. – NadiKeUssPar Jul 23 '23 at 19:53
  • If you're not sure the links are safe, then don't give them. But amazon.com, archive.org, university catalog links to items, Worldcat.org links, and other such links are safe. If you're not sure about a link (especially one that goes directly to a .pdf file), then for books it's probably best to use a link that gives correct information about the book (i.e. exact correct title, author(s), publisher, year, etc.) so that readers will know what to search for, such as library catalog links, amazon.com links, Worldcat.org links, etc. Note that amazon.com links (continued) – Dave L. Renfro Jul 23 '23 at 20:28
  • can be shortened by using only the "/dp/9353503787" part -- see what I did for the "Calculus for JEE (Advanced)" link I gave. Also, Stack Exchange question/answer links can be shortened by using only what you get when you click on "Share" (bottom left of question/answer). Similarly for other places (you'll have to experiment), such as google-books links -- see how I gave the URL's in this answer (note the use of "&pg=PA*" at the end of the "id=#" stuff, where 's are digits and # is google's identification number for the book). – Dave L. Renfro Jul 23 '23 at 20:29
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Sounds like you may be ready for Rudin's "Principles of Mathematical Analysis" -- it has a reputation for having challenging questions but within a limited domain in calculus -- may be good for grounding what you already know.

If that looks too intimidating then I'd recommend Ross' "Elementary Analysis" as a good intro to rigorous proofs in analysis.

One other book that may be helpful is Advanced Calculus by Buck -- it has a more problem-set feel but covers a lot of ground.

Annika
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  • I almost finished roberts real analysis and I will begin in rudin book in week or so

    but I need a problem book that focus on sharping my mathematical thinking by solving hard problems for my level

     – pie Jul 25 '23 at 07:58
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    @pie -- seeing your responses to others on this question, I think you are reaching a place in your mathematics understanding that there will no longer be simple line-liner proofs in the problems you need to solve. You don't need 100's of hard problems, you will be kept plenty busy by, say, 20 or so.

    Also, most books have a variety of problem difficulties, so pick the ones that work for you, mark others to come back later. Thomas' calculus is very very straightforward compared to even elementary real analysis texts, so I think yo are spoilt for choice by the answers here.

     – Annika Jul 25 '23 at 12:58