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).
Well, I've just spent a little while trying to simplify this, and I've come up with:
ReplyDelete(3i/2n)^3 - (6i/n)^2 + (9/n)
Am I miles off?
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.
ReplyDeleteIn your 2nd term, only the n and 3 are squared.the 2 and the i are not.
Your last term is dead on.
Or:
(27i^2)/(2n^3) - 18i/(n^2) + (9/n)
Ms. Cookie
Ah yes, I see where I went wrong before.
ReplyDeleteIt'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. :)
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