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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Tuesday, February 28, 2006

Scribe for Monday

Given f(x)=0.5x^2-2x+4
And g(x) = 2 + rt 4+4x-x^2
S is the region between f and g, write the interval that gives:
a) The volume generated by rotating s around the line y=5.
b) The Volume generated by rotating s around the line y=1.
c) The volume generated by rotating s around the y-axis.
Rotating two lines, It surface will be looks like a washer.
And the Volume of a cylinder V = pai (R^2-r^2)
a)---> pai S(rt (4+4x-X^2) +2 -5) ^2 - (0.5X^2-2x+4-5)^2 dx
paiS(rt(4+4x-x^2-3)^2-(0.5x^2-2x-1)^2 dx
b)---> pai S(rt (4+4x-x^2)+3)^2-(0.5x^2-2x+5)^2 dx
c)---> PaiS(3^2-y^(1/2)) dy

A roll of paper towels has dimesions
Find the volume of the papper
Volume of Pai r^2 h a cylinder
V = Pai h (R^2-r^2)
= pai h (R+h) (R-h)
S (2pai x) f(x) dx
(circle) (height) (width)

Nest scribe: steven :P



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Monday, February 27, 2006

Volumes by slicing

This is definitely the hardest topic I've encountered, so far. It's a good thing that we started the class with a bit of a couple of easy questions that somewhat headed to the real deal.

First of, getting the derivative and the second derivative. Well, we all know how to get them at this point, we're probably good at it, by now, considering myself! (can you believe that? :D) Then the difference between the net and total distance was tackled. The second question involved integrating the area of a region "S" enclosed by graphs f and g, which is by the way an easy thing to do because this whole stuff is not new to us, right? But (a BIG BUT) when we came to the part where we had to find the volume generated by rotating region "S" around the x-axis... Okay, where am I now?

it wasn't very easy for me to understand this chapter because first and foremost, I couldn't picture out the image we were visualizing. So that's probably one note to greatly put everyone's attention to, VISUALIZE. ;) It will absolutely be hard to go through all these questions if merely knowing what it looks like is not met.

Two things I've learned that I thought were new:
- the slicing task results to a WASHER (well, not always)
- the concept of BIG R and small r. (its not new but subtracting R - r is pretty unfamiliar to me, but then again, I know everyone understands it)

IMPORTANT:
> memorize the volume formulae (it helps, big time!)



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What should I do?

Mr. K you'll probably wondering where am I and howcome I'm not in your class for couple of days now. Well this is what happen last friday February 17, when I was in my gym class an unexpected incident happened. I didn't exactly remember what happened because all I heard after I jumped was a crack and at that moment I know I broke my leg. Mr. K I need help I don't know what to do? and I really don't want to drop this class because eventhough I don't have a good grade in this extremely hard class I learned a lot of neat things and I really enjoy being in your class. Well the problem is that I just got surgery on my leg and it will take 3 to 4 months to recover. Now, I'm sitting here with pain don't know what to do? and my hopes of graduating this year is minimal. Now I got 26 credits just need 2 more and I don't know how to get that. Mr. K I need some of your advise on what should I do in order to graduate. And the blog is the only way I can think of to communicate with you to tell you why I'm not in your class for this couple of days. Well thank you in advance Mr. K.....



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Sunday, February 26, 2006

Let It Grow Sunday!

This week's game is called Grow.



Draw each item to the center of the ball to Grow it. If you drag them in the right order you will reach the maximum growth level for each object -- that's the challenge and it's not easy. ;-) Lots of trial and error. The number of different ways to play this game is 479 001 600. Can you find the winning strategy?

The Applied Math class will learn how to figure this out this week. Pre-Cal will learn it in about two more months and the AP Calculus students should remember from the Pre-Cal class. Do you?

Have Fun!



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Tuesday, February 21, 2006

Tell Your Parents the Blog is Multilingual!

You'll notice that all posts on our blog now have a series of flags automatically added to the bottom. Click on a contry flag to have the blog translated into that country's language. You can choose from:

French, German, Italian, Portugese, Spanish, Japanese, Korean, Chinese and Filipino

If you speak any of these languages, let me know if they work well enough to be understood. And tell your parents all about it! ;-) Encourage them to leave comments on the blog as well.



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Sunday, February 19, 2006

Chinese? Checkers? Chess?



Chinese Checkers it is called in England. Kinasjakk (Chinese Chess) in Norway. The truth is that it has nothing to do with neither checkers, chess, nor China.

'The Chinese Checkers game board is in the shape of a six pointed star and is playable with two up to six people at the same time. Each player uses pegs or markers of a different color placed within one of the points of the star. The object is to move all your ten pegs across the board (move one step at the time or jump over adjacent pegs) to occupy the star point directly opposite. The player getting all pegs across first wins.' - More.

You can play it here.

(Thanks again to Think Again!)



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Monday, February 13, 2006

Life's not easy

Never in a million times will I ever find ANTIDIFFERENTIATING easy. So to make the long story short, here's what I prepared for, to "somehow" pass the test. It's practically a list of formulae and other important notes for me to remember.
The INVERSE CHAIN RULE:
$\displaystyle \int F'(g(x))\cdot g'(x)\, dx=F(g(x))+C
$
The method of SUBSTITUTION:
  1. SUBSTITUTE by choosing u = g (x) and write du = g' (x) dx = (du / dx) = dx. Then subsitute both u and du to the original integral producing a new integral in the form of f(u).
  2. ANTIDIFFERENTIATE in terms of u. F ' (u) = f (u)
  3. RESUBSTITUTE g(x) to obtain the antiderivative in terms of x.

The method of INTEGRATION BY PARTS:

\[ 
\int f(x)g^\prime(x)~dx = f(x)g(x) - \int f^\prime(x)g(x)~dx 
 \]
* L I A T E *
Antiderivatives of the inverse of Trigonometric functions:

d/(dx)sin^(-1)x=1/(sqrt(1-x^2))
d/(dx)cos^(-1)x=-1/(sqrt(1-x^2))
d/(dx)tan^(-1)x=

1/(1+x^2)

And the NUMERICAL INTEGRATION

  • the MIDPOINT RULE
  • the TRAPEZOID RULE
  • and the SIMPSON SUM (where twice the value of the MIDPOINT is added to the TRAPEZOID value all over 3)



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Sunday, February 12, 2006

A Prisoner's Sunday Dilemma

This week it's a logic puzzle ... we look in on a prisoner with a problem ....




For the last trial, the king used not two, nor three, but nine rooms! The prisoner was told that one room contained a Lady and the other eight were either empty or filled with a tiger. The sign on the Lady's door was true, the signs on room with tigers were false, and empty rooms had signs that were either true or false.

These were the signs:

  1. The lady is in an odd-numbered room.

  2. This room is empty.

  3. Either sign 5 is right or sign 7 is wrong.

  4. Sign 1 is wrong.

  5. Either sign 2 or sign 4 is right.

  6. Sign 3 is wrong.

  7. 7. The lady is not in room 1.

  8. This room contains a tiger and room 9 is empty.

  9. This room contains a tiger and 6 is wrong.


The prisoner studied the nine signs for a while and came to the conclusion that the problem was unsolvable. The king admitted his mistake and told the poor prisoner if room eight was empty or not.

The prisoner needed no more help. He deduced where the Lady was. What about you?

Problem source: The Lady or the Tiger and other Logic Puzzles by Raymond Smullyan. (With thanks to Think Again!)



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Friday, February 10, 2006

Antiderivatives Review

You can find a good review with a little practice on antiderivatives here. It covers substitution but not inegration by parts. You can also try your hand at a true or false quis over here and some review exercises there.

You can review and practice your Integration by Parts skills here. (The flash animations are really cool!)

You can review Numeric Integration here and there (More flash and animations!).

Lots of practice antidifferentiating using the arc trig functions can be found here. (You may find the solutions provided a little confusing as we used a slightly different technique to solve these problems. If you read it through carefully though you might learn something new!)

You can find LOTS of practice for using Substitution by taking this quiz. There are 50 questions but you can check your answer as you do each one.

There is a whole lot more you can review and learn with this COW (Calculus On the Web). The stuff you're looking for will be in the Calculus Book II link.

Study Hard! Remember, luck has nothing to do with it. It's all about how much effort you're willing to put in. ;-)

Cheers!



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Wednesday, February 08, 2006

Friday's Scribe post.

late once again...

Friday, the first day of second sem. Even though classes were shortened we had learned a lot. As usual, question were up on the board for us to answer.

The first question asked us to list all the pythagorean trigonometry identities.
- sin2x + cos2x = 1
- tan2x +1 = sec2x
- 1 + cot2x = cscx

The second question was a question similar to the questions we had for homework.

The third, told us to find the derivatives of the three inverse trig functions...

With Friday's lessons, we have added three new derivative rules and three antiderivative rules in our banks.

That's it from Friday.




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Tuesday, February 07, 2006

Scribe

Today in class we worked on a few antiderivatives. We learned the Simpson Sum which is the best way to find the integral with an estimate.
Simpson Sum = (Tn* + 2Mn*)/3
*(n is a subscript)

The Definite Integral from 0 to 4 for x² dx.
the antiderivative of x² is x^3/3
Evaluate for when x =4 ansubtractct when x=0.
= 1/3(4)^3 - (1/3)(0)^3
= 64/3
= 21.3333

Using Simpson Sum, you get 21.3333, when n = 2.
The Simpson Sum is a combination of the Trapezoid Sum and twice the Midpoint Sum, then divide by 3.
The Trapezoid Sum is a combination of Left Hand Sum and Right Hand Sum, divided by 2.
The Midpoint Sum can be found by using your calculator and one of the programs on the calculator.
The next scribe is Chris.



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