Differential Equations BOB

When we first started this unit, it was quite easy. We learned a lot of stuff in this unit from our previous units. The Euler's Method when done by hand was tedious, but not hard. Slope fields were pretty cool to draw.

We went into the velocity and acceleration problems next. It's much easier to follow the question by writing down everything that's given in the question, and from the initial information, finding hidden information by doing a bit of work.

Newton's Law of Cooling was not bad in class, but looking back I can't remember much about it. I still have trouble on when to use certain equations and using the calculator to solve certain problems, so I'm still sort of worried. I just wish we had a class today for review or maybe a formula sheet, I suppose this would help by refreshing my memory on the different equations. Good luck on the test everyone.

BOB

So this last chapter was a nice closing to the course. What I had trouble with this unit was separating the variables. It wasn't the calculus that was messing me up but the algebra. After doing a lot of questions with separating, I'm used to doing them now without making mistakes. Some other things that have been messing me up on a lot of questions is antidifferentiating. I know it should be a breeze but I just can't remember that darn + c. It looks like it just doesn't belong there XD.

K so studying for the next 2 hours. I really wanna do good for this last chapter :(

Chapter 9 Pretest Mania...

Today's class was dedicated to the pretest. It was pretty straightforward and we are capable of getting nearly 100% on questions like these. I mean, I can say for at least myself that I seem to be over thinking things. Or if not, my mistakes seem to come from errors... that shouldn't really be errors. Most of my other mistakes are a result from not continuing my work. I'd have the right idea or I'd have the right concept, but I wouldn't think it through. The steps of these questions are like putting ideas from different units together. I'd get stuck somewhere.....but hey.. this isn't a BOB, or maybe I should just merge it in. Meh...I'll probably do one later.

Anyhow, I guess I'll explain the pretest. If you can explain something, your understanding of it can be deeper.

In this question, an equation is given for acceleration. As well as that, initial values are given to help determine the equations for velocity and position.

The objective of the question is to find an equation that follow the conditions of the initial values and to help find the correct equation, it's important to remember, that acceleration is the second derivative of position. Ergo, the method of finding the position under these circumstances is antidifferentiating; once to get the velocity and another time, to get the position. C is the correct answer.

Here, a derivative is given and it's asking for the AVERAGE rate of change. As soon as this is indicated, so is the pattern. The general formula of the average rate of change is 1/b-a multiplied by the integral of the function from a to b, with respect to x. From there, it's as easy as "plug and chug" to get the answer which happens to be A.
This is the multiple choice question that confused me. Here, it really helps to write down the information that the question is giving you and think of other ways to skin a cat. One of the things the class learned was why Leibniz notation is very much appreciated. The question tells us that the slope of the tangent line [derivative] of y is equal to e^y.

From there, we separate the variables and antidifferentiate, of course, not forgetting the "plus C" for the constant. In order to determine what value the constant is, we plug in the initial value to find the constant and it results in finding a general differentiable equation-- the answer for this question; letter c.

This is the last of the multiple choice questions. Another question that is easier to solve in Leibniz notation. This may look like a mini monster of a question, but one of the first things to do is to algebraically massage the equation to separate the x's on one side and the y's on the other.

Then, anti-differentiate both sides and find y. Again, to determine the constant, plug in the initial values given in the question and plug in that constant in order to determine a specific function, rather than settling with a family of functions caused by that "plus C".

Finally, there was the three part open response question.

For the first part, I was ashamed of myself... I spent ten more minutes on this part of the question than I should have. All of the needed information was already given in the question. All I needed was to plug the values in.

In part B, I got the right idea but I didn't get the right variables for Leibniz notation and I felt like i was going in circles. But I HAD IT!!! This question is following a very similar pattern to those questions from the multiple choice. Isolate variables, integrate each side to get the parent function... or the family of parent functions and solve for the constant to get the explicit equation.
Another way of checking if the equation is correct is to plug in the initial values and see if they are equal to each other.
Finally, since we've found the parent function in part B, all we have to do for this question is plug in the time it takes and determine the velocity.

There's the pretest for you. A quick reminder that Mr. K mentioned that he will not be here tomorrow and on our test day.. which is the next day after tomorrow. There isn't going to be a class tomorrow so.... enjoy.. your hour? haha.

The rest of the slides are question that are probably homework/review questions in preparation for the test.

Also.. our assignment.. Claim your wiki-questions soon!

Next scribe is....Kristina.

ZOMG>>>!! I almost forgot about our youtube tradition [that was broken...]



This one is really.. just.. incompetent.. but I love this movie.. it's been so long.. since I've seen this.

BOB for Differential Equations

This unit was a breeze after what we've done in the past units. The thing that I liked about this unit was that it combined almost everything we learned in the past units including the derivatives, integrals, and all of their applications. The slope fields part of the unit was something that I was looking forward to since the beginning of the course. Some of the students that took the AP Calc Course last year showed me the kinds of things that I will be learning and I saw the slope field. I was confused at how something like that could be drawn on the Cartesian Plain. Well now I know.

This is our last unit guys and it will be time to do some more crazy studying for the remaining of the month before the exam. With enough studying we will be ready to take on the exam.

Good luck on the test on Thursday (?) and on the exam next month.

BOB: Differential Equations

Looking back, this unit wasn't that bad. Most of the content in this chapter were things that we should already know, such as the first and second order equations. Those parts weren't so bad to deal with. I found myself becoming quite fond of the Newton's cooling law questions as well. They were fun to do once I actually started getting past my pre cal problems with them.

The slope field and Euler's method things were so so for me. I admit having some trouble seeing a certain function within some slope fields, it takes me a while for the more difficult ones. As for the Euler one, doing that manually was a real pain in the butt. There was so much more room for errors that I had to restart the assignment that was given around that time about five or six times. Yes, I messed up that much.

And of course there was the differentiating by separating the variables using the Leibniz notation thing. I really didn't like the Leibniz notation at first, but I can see why we should learn to love it. Admittedly, I did fail hard at recognizing where to do the separation of the variables on the pre-test. Maybe I'm just too used to the other notation better, or I was thinking too hard but I think I should be able to get it on the test! Other than my utter lack of ability to spot when to separate, I'm confident in actually differentiating using this method when I actually do spot it.

Emm..I think that's it. Yes, good luck on the test on Thursday my fellow peers. I shall now go back to cry at the fact that I can't edit my Question 9 on the wiki page.

BOB v.9: Differential Equations

The intro exercises were simple at first since it's a review of chapter four, including those questions where we had to antidifferentiate the acceleration due to gravity twice to get displacement, until they started asking questions that I haven't learned until later on in the chapter. Luckily, I caught on, but, unfortunately, I still forget the +C when antidifferentiating...and it becomes more of a problem when there are more than two +C's for those questions where we antidifferentiate both sides after separating the variables. And antidifferentiating e^-t...I'll be prepared the next time I'm asked to antidifferentiate e^-t.

Solving differential equations graphically via slope fields, as I remember, was the easiest out of the three ways to solve a differential equation, since all there is to do is draw lines at each (lattice) point on the graph OR use prgmSLOPEFLD to draw the lines for me.

Solving differential equations numerically via Euler's method was a bit too much to absorb at first, but reviewing Newton's linear approximation technique was helpful since those two methods are similar.

Solving differential equations symbolically via separation of variables was a doozy, although I personally prefer this method over the numerical method, just because. Comparing the separation of variables questions in our book to the questions on slides, to me the slides were harder (and more helpful) since the slides had just more.

I'll admit the pretest wasn't my best pretest though, but I did keep track of where I went wrong! The main thing I need to watch out for are those +C's!

Today's Slides: April 14

Here they are ...

AP Calculus AB April 14, 2009

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