Any good calculus places to start?

[TylerH]

The harder part is to show that no number smaller than 1 is as large as all those finite decimals. Note that those finite decimals of form .9999. differ from 1 by a finite decimal of form .0001. So if there were a number lying strictly between 1 and all those finite decimals, it would differ from 1 by a positive number which is less than every number of form .000000...0001. I quit there in an elementary class saying that cannot happen. But for you, here is a sketch of the slightly tedious argument:

It amounts to showing there is no positive number smaller than all those numbers of form
.000000...0001. Now any non zero finite decimal has a first non zero digit in some position, and if we put a 0 in that position and follow it by a 1, we have a smaller number of the form .000000...0001. Thus no finite decimal can be smaller than all those.

As for an infinite decimal, it is at least as large as all its finite truncations by definition, and we can find a number of form .000000...0001. that is strictly smaller than one of those truncations. Thus also every infinite decimal is larger than some number of form .000000...0001.

Would this not be the proof, rendering "sup{.9, .99, .999, ...} = 1" extraneous?

 

[mathwonk]

Lazernugget, if you follow the notes in post 23, you are well on your way to understanding integrals.

 

[mathwonk]

Lazernugget, I will post one more set of notes on finding areas and volumes by antiderivatives. I hope some of this gives you a start. This is an actual lecture from my second semester honors college calculus class.

 

[Lazernugget]

Wow! I'll look at the pdf things later, but the other posts you made were amazing! Thanks for the help! I got to go read my trigonometry book...bye!

 

[mathwonk]

You are extremely welcome. The pleasure is mine. Some of those pdf files are really elementary, and you might glance at one or two of them.

the ones in post 34 were written for honors high school students, and the ones in 23 start out really elementary, with area of a circle. These are easier to read than a trig book.

 

[FeDeX_LaTeX]

Hello;

I started learning calculus around your age too. While I was a bit younger I remember "edugratis" helped me a bit. Unfortunately the website is no longer active, but you can find some of his videos on YouTube. Search "edugratis".

KhanAcademy is invaluable, too.

 

[mathwonk]

forgive me for this post but i have some honors calc notes that may interest someone who always wondered why his prof did not prove the basic theorems on continuity. Here is the fundamental boundedness result. This is the tedious technical result, not the more fun ones, but it is late and I am tired now of making it at least mildly compatible with current technology.