Thursday, March 22, 2007

Friday's Test Topics

Here’s a list of topics that will be covered on this Friday’s Chapter 7-9 Test.

Chapter 7-9 Test Topics
Integration by parts (Sec. 7.1, #3,7,21)
Arc length (Sec. 8.1, #1,3,5,9,11)
Approximate Integration – Midpoint/Trapezoidal Rule (Sec. 7.7, #1,3,7,29)
Slope Fields/Differential Equations –Solutions (Sec. 9.2, #11,13)
Exponential Growth/Decay – Newton’s Law of Cooling (You knew it was coming!) (Sec. 9.4, 13,15)

As always, your homework is a good place to start reviewing, and the book has several other problems to give you more practice!

That’s it! I’ll be around after school on Thursday until 3:00 and back after 4:15 (faculty meeting) and in early on Friday. Donut holes and OJ!

I don’t know whether my life has been a success or a failure. But not having any anxiety about becoming one instead of the other, and just taking things as they came a long, I’ve had a lot of extra time to enjoy life.

Thursday, March 15, 2007

Friday's Quiz Topics

Here’s a list of topics that will be covered on this Friday’s Quiz.

Quiz – Sections 9.2-4
Solve a differential equation (Sec. 9.3, #1,5)
Solve a differential equation (IVP) (Sec. 9.3, #11,15)
Exponential Growth/Decay – Formulas, Rates, Values, Times and Graphs (Sec. 9.4, #1,3,9)
Slope/Direction Fields (Sec. 9.2, #11,13)

That’s it for now! I’ll be around after school on Thursday, online Thursday evening/night and in early on Friday – OJ and donut holes!

I like nonsense, it wakes up the brain cells. Fantasy is a necessary ingredient in living, It's a way of looking at life through the wrong end of a telescope. Which is what I do, And that enables you to laugh at life's realities.
- Dr. Seuss

And for those of you that didn’t see it, here’s a cute set of instructions for properly hugging a baby

9.4 Exponential Growth and Decay

Relative Rate of Growth: Things are growing based on what you already have.
  • Constant change based on what you start with

1. Separate the variables

2. Take the integral of both sides

3. Evaluate the integral (take the antiderivative).
Don't forget to add +C!

4. Come up with an equation for P (Isolate P).

This is a constant and the initial value:

P = population
P sub o = initial
K = constant
t= time

To find value of K:

Half-Life: Rate of decline or breakdown.

The (.5) is the rate at which the substance declines or breaks down.

I am going to use the carbon-14 (C-14) example even though we did it in class because it makes sense to most people after taking Bio and Chem.


Newton's Law of Cooling:
Difference between an object and its surroundings.
Ok now that I totally get this (and so does Genny!) here goes:

mac and cheese (that is what I am eating right now) at 110 F.
Room at 68 F.
30 minutes later, the mac and cheese is at 100 F.
Find the temperature after 1 hour.
1. Set up specific equation

The ratio inside of ln is (later data point)/(original data point).
The units for time do not matter as long as you are consistent.

2. Solve
Plug in 60 for t.

3. Add T sub o to the answer in part 2. (This is the typical mistake that people make, at least Genny and I did)
Add 68 to the answer in part 2.

Sorry I could not put this up sooner. I realize you may have needed it last night, but I did get it done and I really was not sure if I could. Thank goodness blogger was nice to me today.

Here is a good website. It is the one that Dartmouth uses for its books I think:


I have been a die-hard OC fan since the beginning, even when it went kinda crazy. It is strange that today is a Thursday and there is no OC two weeks in a row. In memory of what was once one of the best shows to ever hit television:

Tuesday, March 13, 2007

9.3 Separable Equations

Hello friends. I was thinking that we'd talk about Separable Equations today. Doesn't that sound nice? I think so.

Separable equations are differential equations of the first order (i.e. equations that contain a function and its derivative) that we can solve explicitly by separating (wow) the two variables involved into the two sides of the equation and by using integrals on both sides to get rid of the dx's and dy's. (Brainstorm: what if there were a three-sided equals sign? Wouldn't that be cool? Or would it just be pointless?)

There are certain steps we need to follow in order to get full credit on the AP's Free Response Questions. Don't worry, I'll cover them. How about an example? I think so.

Find the solution of the differential equation that satisfies the given condition.

(dy/dx) = y2 + 1, where y(1) = 0.

Step One: Separate the variables using algebra.
(dy/ (y2 + 1)) = dx

Steps Two and Three: Take the antiderivative of both sides. (⌠= integral sign)
⌠(dy/ (y2 + 1)) = ⌠(1dx)
tan-1y = x

NOTE - The integral of the left side of the equation is one of the rare occurrences of a derivative of an inverse trigonometric function. Yes, it's annoying. (sad face).

Step Four: Recognize the constant of integration.
tan-1y = x + C

Step Five: Solve for "C" using the given condition (y(1) = 0).
tan-1(0) = (1) + C
0 = 1 + C
= -1
Good job.

Step Six: We end up with tan-1y = x - 1. Our book seems to like solving for y at this point. I'm not sure if this is really necessary, but being the brilliant maths students we are, why not do it?

y = tan (x - 1). This is the answer you've all been waiting for, or, rather, for which you've all been waiting.

SOPHIE. Wake up and take notes; you're posting next.

And, if you couldn't handle my colloquial math tongue, this website presents separable equations in more textbooky way. It also has some sample problems you can try, if by chance you've done all of the ones in our own textbook.

Here is a video that my brother made for his Chinese class at Vassar College. It's about drugs and their effect on the psyche. Not really, but it is about drugs. And it's in Chinese, so it's funny. And it has a cute old Chinese man at the end with a cool voice. I hope you enjoy it, even if their accents sound like an electronic dictionary (Laurie).

More Drugs, More Chinese Problems

Bye bye.

Saturday, March 10, 2007

9.2 Direction/Slope Fields

Direction/Slope Fields!!!!

Direction fields are a way to approximate the solution of the differential equation graphically. (But it is extremely imprecise.)
(Remember, differential equations are equations with the derivative of the function and the function itself.)

It is easiest to learn through an example so....

Given the differential equation y’ = 2x + y, sketch the graph of the equation going through the origin.

1) It is important to remember that slope fields are an extremely imprecise way to find the solution. The first step of solving a problem is to make a chart.

X Y Y’
0 0 0
0 1 1
1 1 3
and so on and so forth. Etc etc.

At each point, draw a small line with the calculated slope until you have a graph something like this....
2) Since the problem is asking for the graph through the origin, so start at the origin and following the slope lines until a graph forms.

This is an extra link to help you!
:) It’s...cute.