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- Thread starter Bashyboy
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EDIT: if you want to help yourself, go back and make sure your foundation is strong with limits/derivatives/integrals. There are theorems that are important (MVT, FTOC) but they will have no immediate impact on learning Calculus 2

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If you're not a math major, then you can probably safely skip them.

What theorems are we talking about anyway??

Perhaps you could start a thread in the math forum asking for assistence with the proof?? I'll be happy to help you!!

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Usually Analysis is a junior/senior level math course.

I am not saying you shouldn't understand them. By all means, if you can follow the proofs then go right ahead. What I am trying to say is there is a whole branch of mathematics devoted to studying Calculus. Also most of the time the proofs in the textbooks are just terrible.

You should definitely know what the theorem says but proving the theorem at this point in your academic career has very little benefit (as compared to solidifying your differentiation/integration skills)

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A few of the suggestions I've seen and heeded are as follows:

For a general introduction to proofs, try 'How to Prove It: A Structured Approach' by Velleman. For a 'proofy' treatment of calculus (great for aspiring math majors, with REALLY hard problems) try Calculus by Michael Spivak. I'm working through these now, the proof book is easy enough to follow on my own but Spivak.. well, I can follow the chapters but most of the exercises are above my ability (so far).

I think Spivak would be great for you, you can 'review' the calculus you've already learned in a new light, acquire some skills at proving/analysis and perhaps dip into calc II. If you want intuition, well, you acquire that by understanding theory behind the math. At least that's how I've always seen it. There isn't much intuition to learning formulaic shortcuts if you don't understand what is really going on in the background.

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Fredrik

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It's hard for everyone to understand proofs in math books. I don't think the problem is that you haven't taken a full course on logic. It would however help to understand truth tables of logical operations, so that you understand things like that the statements

##A\Rightarrow B##

##\lnot B\Rightarrow\lnot A##

##\lnot(A\land\lnot B)##

are equivalent. The best way to see that is to write down their truth tables. If you do, you will see that they're all the same.

I don't think it would help much to study more logic than this. You just have to study more examples of proofs, and practice doing proofs on your own. If you want to be good at proofs, I would recommend that you do something like this: Study the proof in the book first. Then try to prove the theorem without looking at the book. Then do it over and over, while imagining yourself explaining it to someone else. Keep doing it until you don't see a way to improve your explanation. If it's a difficult proof, it can take many attempts to get to that point.

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In addition to this wonderful post, I would add that mathematics is not a spectator sport. You learn by doing and trying it yourself, not by reading things in a book.

It's hard for everyone to understand proofs in math books. I don't think the problem is that you haven't taken a full course on logic. It would however help to understand truth tables of logical operations, so that you understand things like that the statements

##A\Rightarrow B##

##\lnot B\Rightarrow\lnot A##

##\lnot(A\land\lnot B)##

are equivalent. The best way to see that is to write down their truth tables. If you do, you will see that they're all the same.

I don't think it would help much to study more logic than this. You just have to study more examples of proofs, and practice doing proofs on your own. If you want to be good at proofs, I would recommend that you do something like this: Study the proof in the book first. Then try to prove the theorem without looking at the book. Then do it over and over, while imagining yourself explaining it to someone else. Keep doing it until you don't see a way to improve your explanation. If it's a difficult proof, it can take many attempts to get to that point.

For example, proving something by induction is something you should have done quite a lot of times yourself before you really get what's going on. The same holds with proofs by contradiction, which seem quite exotic at first.

So in addition to trying to understand the proofs yourself, try to prove some easier things first. Then let somebody (perhaps on this forum) read your proof and indicate what's wrong about it. These are the only ways to really get it.

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