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Sunday, April 5, 2009

Uggg spring break scribe :[

Okay guys here's my much delayed scribe from last Friday. I apoligize if it sounds funny, as I actually typed it during the class its written for, so most of it will be in present tense, although tried to keep it in past tense. SO basically, bear with me, and we'll get this thing under way.

Alas this is technically the last scribe of the course; however, it is not the last scribe of our meetings together. Although we may be finished the course today, Mr. K mentioned how he would like to spend some more time on this last little bit. The reason Mr. K gave for as to why he would like to drag out this last little piece comes from the fact that Differential Equations are, “The doorway to the rest of calculus.” Or something along those lines. Basically, he said you take Cal 2, 3, and then differential equations, and after that. You can take differential equations for the rest of your life.

After that little intro, we began to talk about our plans for April, and therefore the rest of our time together, now that the course is basically finished. The result of these discussions revealed that April will be the month we kick it into overdrive manic preparation mode. So Mr. K decided to help us out, and post pretty much infinity AP Calculus Exams from previous years so that we can a.) Work on exam style questions to prepare for this exam and b.) Prepare for the exam by beginning to recognize the patterns emergent in its construction. These practice exams can be found on the blog, in a little box on the right hand side, available for download as pdf's

An important thing to note, is that the solution/scoring guidelines are extremely terse/to the point. They also happen to include the main, “Mark docking errors” (ie. Which mistakes will cause you to lose marks.)

We also got into a discussion about how intense the AP exam is. Some interesting topics of discussion included;

-The fact that the exam takes approximately 5 years to create (in essence, the exam we will write was completed around 2003, 2004ish)
-You have to sign a contract/written oath that you will not divulge certain aspects of the exam (namely the multiple choice questions). If you fail to maintain this you can (and likely will) be sued.

After all that, we began the good stuff.

Solving Differential Equations Symbolically and Newton’s Law of Cooling

There are 6 types of differential equations you can do, and within each type you can have numerous orders (first second third, etc.)

Separable differential equations are equations where it is possible to separate the variables. These are the type of differential equations we're concerned with in this course.

The reason we like the liebniz notation is because you can use it in separable equations, something you can’t do with f’(x) notation.

So now heres an example of this :]

- To start, your gonna rewrite dy/dx = ky as dy = ky dx, as seen in the slide.
- From there your gonna continue by getting all the y's on one side and the x's on the other by multiplying both sides by 1/y
- Next your going to take the integral of both sides which brings you to the next line on the slide.
- Now because we have Two C's on either side of the equation, we'll just combine them into one, since we know that they're constants.
- To continue we take the ln of both sides so that we might find y. This gives us y = e^(kx) * e^c
- The next thing we do is let e^c = A or something like that, which gives us our final solution of y = Ae^(kx)

After that problem, we got to the fabled, "Newtons Law of Cooling Problem"

Newtons Law of Cooling problem.

According to Newtons law of cooling a hot object cools at a rate proportional to the difference between its own temperature and that of its environment. If a roast at room temperature, 68 degrees F, is put into a 20 degree F freezer, and if after 2 hours the temperature of the roast is 40 degrees F

a.) What is the temperature of the roast after 5 hours
b.) How long will it take for the temperature of the roast to fall to 21 degrees F?

Looking at the question we find that we know there is a relationship between temperature and time, but we don’t know the parent function of this relationship. Instead we know the derivative for this relationship (temperature with respect to time.)

So just like with most problems we've solved, we begin by lying out what we know, and what we have to work with. Following the strategy we used before, we begin by getting all our x's on one side and our y's on the other (or in this case, the T's on one and the t's on the other.) We then take the integral of both sides, and fill in everything we know.

T(0) = 68 and T(2) = 40.

By plugging in the value of T at 0 we can find the value of A. This allows us to begin the process of finding the value of k, the point in the first place. Some more plugging in and algebraic massage, and we find the value of k to be


Part B. of the question is solved mostly by the tried and true plug and chug. Some of this process is started on the next slide, where the empty box, is used to represent the value of k.

Okay guys, I think that's everything, hopefully I didnt muck to much up. Sorry for the late post, I had lots of work to do this break. So much so it didn't really feel like a break. The scribe for monday I pick to be Lawrence, if lawrence already did it, kristina, and if she did it I'll pick on monday, cause the last scribe list is olddddd.

Alrighty, night guys, see everyone tomorrow :] :[

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