## Friday, November 18, 2005

### Equation Dexterity

I'm liking how I'm introducing integrals in calculus. I figured we'd concentrate on 1 to 2 functions and work them in all sorts of ways to build up to an understanding of what's so cool about the (first?) fundamental theorem of calculus. So we approximated area, we brainstormed ways to make it more accurate, we found lower and upper bounds to exact area under the curve, and I then built up how to use limits and summation to get the exact area.

Sheesh, these things always take longer than you think they will. In my mind (hah!) I figured they could finish up simplifying this one equation in class and then I'd assign the 2nd problem for homework. Hah. Hah. What took most of them all period was to simplify an equation of this form (note that it's not a matter of calculus right now, it's basic algebra .... or so ... FOILing, distributing, adding fractions, simplifying):

[1/2 (3i/n - 2)(3i/n - 2) + 1](3/n).

Okay, ick (sort of), but doable ..... (apparently doable in 40 minutes and not the 5 or so minutes I envisioned in my fantasy teaching mind).

1. Anonymous6:24 AM

Well, I've just spent a little while trying to simplify this, and I've come up with:

(3i/2n)^3 - (6i/n)^2 + (9/n)

Am I miles off?

2. Anonymous12:21 AM

You are pretty darn close. In your first term, the i is squared, and the 2 is to the 1st power. The rest of the first term is correct.

In your 2nd term, only the n and 3 are squared.the 2 and the i are not.

Or:
(27i^2)/(2n^3) - 18i/(n^2) + (9/n)

3. Anonymous9:46 AM

Ah yes, I see where I went wrong before.

It's actually quite a simple simplification, if you know what steps to follow. :)

I can see why you thought it would only take five minutes.

I had much the same issue when I was giving a course on Sql lately. I asked them to do a puzzle which I thought would keep them busy for ten minutes, but forty minutes later they were still beavering away.

Bless em. :)

4. Anonymous12:03 PM
5. Anonymous12:27 AM

Good day !.
You re, I guess , perhaps curious to know how one can manage to receive high yields .
There is no initial capital needed You may commense earning with as small sum of money as 20-100 dollars.

AimTrust is what you thought of all the time
AimTrust incorporates an offshore structure with advanced asset management technologies in production and delivery of pipes for oil and gas.

It is based in Panama with affiliates everywhere: In USA, Canada, Cyprus.
Do you want to become a happy investor?
That`s your chance That`s what you wish in the long run!

I feel good, I began to take up income with the help of this company,
and I invite you to do the same. If it gets down to select a correct partner who uses your money in a right way - that`s the AimTrust!.
I earn US\$2,000 per day, and my first investment was 500 dollars only!
It`s easy to start , just click this link http://ruqobokux.1accesshost.com/sojuku.html
and go! Let`s take our chance together to get rid of nastiness of the life

6. Anonymous5:04 AM

Hello !.
You may , perhaps curious to know how one can collect a huge starting capital .
There is no initial capital needed You may start earning with as small sum of money as 20-100 dollars.

AimTrust is what you haven`t ever dreamt of such a chance to become rich
AimTrust incorporates an offshore structure with advanced asset management technologies in production and delivery of pipes for oil and gas.

Its head office is in Panama with structures everywhere: In USA, Canada, Cyprus.
Do you want to become an affluent person?
That`s your chance That`s what you desire!

I`m happy and lucky, I started to take up income with the help of this company,
and I invite you to do the same. It`s all about how to select a correct partner utilizes your funds in a right way - that`s the AimTrust!.
I make 2G daily, and my first investment was 500 dollars only!
It`s easy to start , just click this link http://umetocehig.fcpages.com/amiropex.html
and lucky you`re! Let`s take this option together to get rid of nastiness of the life