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Suggest some books to cover the topics of calculus taught in first year of under graduation so that I can improve my problem solving skills in it ( I've knowledge of calculus and pre calculus of high school level). Also recommend books for real analysis and algebra of the same level (Mainly to improve problem solving skills but also to build a strong foundation of theory and applications and stuff).

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You might try to have a look at similar questions - you can fine several such questions when you search for calculus+problems and calculus+exercises among the question tagged (reference-request). And also related links which MSE tried to generate should be useful. – Martin Sleziak Jul 13 '12 at 6:22
I recommend to activly participate here! I wonder if this will not improve your problem solving skills... – draks ... Jul 13 '12 at 7:39

If you are looking a book which has plenty of exercises on basic calculus, you could try B.P. Demidovich: Problems in mathematical analysis (originally Demidovich: Zbornik zadac i uprazneni po matematicheskomu analizu, there are translations to several languages). Even Russian edition might be good for you - for computing problems, like calculating a limit, a derivative, an integral you do not need understand the text of exercise. You can probably find it online, e.g. try searching for: demidovich zbornik djvu or Демидович Сборник задач djvu or demidovich problems djvu and similar searches (you can try pdf instead of djvu).

I very much like the selection of exercises in this book.

Kaczor, W. J.; Nowak, M. T. Problems in mathematical analysis. I. Real numbers, sequences and series. Google Books MR1751334

This book has two more volumes (II. Continuity and differentiation and III. Integration). There exists English, French and Polish edition.

Many interesting problems on real analysis are in

Rădulescu, Teodora-Liliana T.; Rădulescu, Vicenţiu D.; Andreescu, Titu. Problems in real analysis. Advanced calculus on the real axis. Google Books, MR2514007.

Although I am not sure whether this is at the level you're aiming for. (But you asked about real analysis in your question, too.)

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+1 for the great classic calculus problem book in Demodovich. Worth hunting down and photocopying-it's rare and insanely expensive now to buy,sadly. Here's hoping for a low priced Dover edition someday. – Mathemagician1234 Jan 24 '14 at 9:02

You might try Bluman, "Problem Book for First Year Calculus", Springer-Verlag 1984.

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