Seems like in math, not physics really, whenever I'm doing the same thing for a while it seems to get too repetitive and I cant wait to get out of that section, not to mention the speed seems slow. Could be because I'm taking AP Calc and not at a college level. For whatever reason, I seem to learn better going at a fast pace. Is anyone else like that?
Yes, that level of maths is pretty much just calculations. Most people I know agree that calculations are boring. Pick yourself up a baby rudin and get reading. Or a copy of Introduction to Linear Algebra by Gilbert Strang Or any college level maths book!
Yeah, don't do that. Not now anyway! Use the Calculus books by Spivak or Apostol. I've heard that both are gems but that Spivak might be a little more accessible. I have no first-hand experience with either (filling in the gaps of my algebra knowledge) but I intend on starting them soon.
Yes, calculus exercises can be repetitive and boring. But it's very important to make the repetitive exercises. You will want to know the different techniques and recognize when you need to use which technique. You can only do this by making repetitive exercises. Only making 10 exercises on the chain rule in derivatives (for example) is not enough. You have to know the chain rule inside out.
If something is too easy then it becomes boring. Just doing lots of *similar* "chain rule" exercises is going to be boring. Don't just blindly go through, say, every exercise in the book. Ask yourself if you really need to do the exercise! If you are certain you can do the exercise, move on to the next one...
Like recently I've just been not paying attention in class just so it isn't so boring when I do the problems at home lol. Makes it a little better, but still.. Does it get better in college :S
You should try to pick up Spivak's Calculus. It's accessible but in depth and rigorous at the same time. He motivates most of what he does including why definitions are the way they are.
So you need to solve more than 10 exercises on the chain rule? I don't agree with this. By doing that you would only remember the chain rule very well, which is not the same as understanding it very well. Understanding comes when you know why it works, when you can clearly mentally see what is happening when you are applying it, and that is acquired by thinking and trying to conceive why is it true, not by mechanically solving repetitive problems. Generally, in my opinion, problem solving is overrated. Trying to conceive, visualize theorems, concepts is much more important than mechanically using various rules.
Understanding the chain rule very well is easy. Being able to apply it and recognizing when to apply it are very different things. You will need to be able to calculate derivatives in a lot of places. So practising a lot on it is no wasted effort. I see the difference with me. Learning something with solving problems is far better than learning something without doing problems. You have to be able to recognize situations when to use which technique. This can only be done by solving many exercises (repetitive or not).
Understanding is continuous, hence you can't really say that understanding the chain rule is easy. No matter how good you already understand it, you can understand it better. You can always find new ways to look at it, new things to relate it to. This is what gives the ability to be able to recognize when to apply it. And this, at least for me, is best achieved not by solving as many problems as possible, but by thinking about it, trying to imagine, compare, relate, etc.
Understanding the chain rule is not easy. Probably 99.9% of calculus students don't understand the chain rule, they just know how to use it, if they are a decent student. The difference between me and those calculus students is not that I'm better at calculus because I understand the chain rule. The difference is that because I understand the chain rule, I can prove more general versions of it, such as the complex version or maybe for Banach spaces. Doing drill is sufficient for mundane purposes, but those with more lofty goals must try to understand WHY the theorems are true, not just how to use them.