I've recently begun Calculus 1 and if I'm honest, so far it's been challenging aswell as fun. The biggest problem for me so far has been understanding the content. Instead of attempting dozens of questions with intent on drilling methods into my head, I've been reading many different books and watching many different videos. However, I can still tell I don't understand the content fully. For example, I still can't construct Epsilon Delta proofs confidently and there are a few proofs I still don't understand.

I started Calculus three days ago and have spent about 18 hours in total. The topics I have covered so far is:

Derivative as Rate of Change, Limits and Continuity, Discontinuity, Calculating Derivatives, Derivatives of Sine and Cosine, Limits of Sine and Cosine, Product Rule, Quotient Rule, Chain Rule and Higher Derivatives

I would like to ask how can I acquire a deeper understanding of the topics I've learnt?

Also, should I continue watching videos in an attempt to fully understand the material or should I start attempting to solve problems?

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    $\begingroup$ Start working on problems! There are a lot of pretty self contained calc books that will have their own answers. And if you didn't know, Khan academy has a lot of great content. $\endgroup$ – Elliot G Oct 24 '16 at 6:30
  • $\begingroup$ Someone may provide an answer to this later; I would, but I'm about to sleep. Don't worry about $\epsilon - \delta$ proofs for now. Proofs are tricky to get a feel for until you take a course in writing mathematics. I'll provide an answer in the morning. $\endgroup$ – Sean Roberson Oct 24 '16 at 6:30
  • $\begingroup$ My advice is do a lot of problems. You likely won't be able to fully understand all of the concepts until you work out problems. They will make you more comfortable with the material, which will in turn help you understand the underlying concepts and proofs as you read about them. Also, that is a lot of stuff to learn in three days and I wouldn't expect it to all click in such a short time. $\endgroup$ – wgrenard Oct 24 '16 at 6:44
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    $\begingroup$ I started Calculus three days ago That's way too short to worry, allow it some more time to sink in. As it's been said already, work out some exercises in the meantime, since that's where the more abstract concepts meet concrete use-cases. $\endgroup$ – dxiv Oct 24 '16 at 6:50
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    $\begingroup$ If you are doing self study of calculus then I must say two things: First 3 days is too short to get an understanding of calculus, you probably need at least a few months and that too only if you study from a truly good book, which brings me to the second point: do get a copy of "A Course of Pure Mathematics" by G H Hardy, it is simply the best book for self-study. $\endgroup$ – Paramanand Singh Oct 24 '16 at 13:13

Maths is for most of us mortals a sequence of getting some info (through books, lectures, videos etc), then trying problems which shows what we have understood and also (always) what we haven't understood. Then there is a period of struggle/confusion which usually involves more problems/questions/answers and them comes the best bit, finally getting on top of that bit. Then it is onto the next new bit of maths and the cycle starts all over again.

The key is to do a mixture of getting information, trying problems and then (crucially) getting help when you don't understand. Mathematics is a "doing" subject. Like any sport you can only gain so much by watching other people. To improve you have to do it.

Most of the time it can feel like three steps forward and two steps back. If it reassures you I can tell you all professional mathematicians struggle trying to sort out whatever new bit of mathematics they are trying to understand and really get to grips with or develop.

Finally remember when you are learning mistakes are all part of it. That is how you improve your understanding.

Well, at least that is what I have seen over quite a number of years.

However my view is no more important that anybody elses.


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