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Tuesday, February 3, 2009

The Curious Case of Substitution

I apologise for the lateness of this scribe post, work went well into the night.

So guys, im Not Paul, as opposed to Paul, who incidentally came into our class to tell us about scholarships. He directed us to this page:

Which gives you links to alot of good resources not just for university application but job hunting. Be sure to check it out and head to Mr. Ssembwere's office to pick up some scholarship applications!

Anyway, after that short interlude, we went back to working on understanding the substitution method.

We didnt go really in depth with it, mostly reviewing what we've already done and trying to understand the idea. The the figure on the second slide explains the process pretty clearly:

Take integral of original function and substitute a function within that integral with u. Make sure the integral is in terms of u, and now antidifferentiate the simplified function. Replace the substituted function into the new function and volia, you antidifferentiated an integral.

Slide three is the first example we did. This question introduced the idea that we would have to find x in terms of u in order to write the function in terms of u. However, once you encounter and overcome that small hurdle, the rest is rather easy.

For the fourth slide, we were tasked with solving the same question, except instead of being able to choose x + 7 = u, we were given that u = (x + 7)^(1/2). In this case, the solution is similar, only longer. But remember kids, longer does not nessecarily mean harder!


Anyway, the new thing we learned today was how to find the antiderivative of an integral at certain limits. The trick is quite simple, and there's two ways to do it:

Antidifferentiate using the substitution method, then input the limit values into that function. You will get two sets of values, and you can use the integral ?theorem/law/thingeh? that states integral = (f(b) - f(a)) to find your answer.

ORRRR, and this may be easier, harder or even the same depending on the situation, you can rearrange your function in terms of u where u is any function inside the integral. In this way, you can have u in terms of x, and you know x because of the limits. Solve for u, and input into newly simplified function. Use thingeh, find value, which will be the same as if you did it the other way! Cool amirite?

Now on the final slide, we are given a beast of an integral and tasked with antidifferentation. This is no easy task. The problem is the function doesnt really jive with what we're doing. Like when we find the derivative du, we cant seem to find a place to put it in. Thus we have to do some mathematical massage (its easy and you dont even need a license) to get it in there. The trick works like this, depicted since it's not clearly explained on the slides:

And urgh, need more practise with using my tablet. My writing is all wonky >:

Anyway, since we do that, we can put du in there, moving one step closer to our plans of integral domination. And by domination I mean simplification. Since we have u, we just rearrange to solve for x in terms of u, substitute u in so our integral is in terms of u, antidifferentiate and put the good stuff back in. Simple as pie. Even better that our simplified integral is a polynomial (because those radicals are nasty) since that makes it easier to antidifferentiate.

Anyway, that concludes todays lesson. Hope you guys did your homework.

As for the Team PSY video which is oddly missing, I decided to resync some audio and redo some parts I rushed through, so I hopefully will have it uploaded Friday - Saturday at the latest. Hopefully will not disappoint when all is said and done.

Anyway, Im going to bed, see you all tomorrow (today).

Oh and someone's the scribe. Duck, duck, might want to remove Shelly and cross out my name and J+ME's, duck, duck, Joyce. Also what is it with this whole

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