Can I become really good at calculus studying MIT OpenCourses?

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Is it good to use the site as a primary source or more as a complementary source?

Should I buy books to follow what is taught in the course?

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Best source for learning are books. There are tons of books in the market, just select one (Tom M. Apostol is good ), just get deep inside and you will find yourself learning much more.

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I would recommend learning from books. Of course if you get stuck at some point then you can watch those videos or find help here on this site. Best is the latter option in my opinion. However you must remember to study your stuff in a rather formal/serious way. I'm saying this because we do best when we have some motivation which at universities, in most of the cases, is the final exam. Like it's mentioned in the comments you can use problems to check if you are serious enough.

The following books may be of some use:

  1. How to think about analysis: Lara Alcock.

  2. Calculus: James Stewart.

  3. Understanding Analysis: Stephen Abbott
  4. Elementary Analysis- The theory of calculus: Kenneth Ross.

There is an online resource called Paul's Online Math Notes which should help you get an idea about the stuff you must study given that you are about to self-study the subject.