Bad News Bears: Mr.K as - The Rock 'n Rolla

Today, we have received some news. Some serious news. As those who were in class, already know, but Mr. Kuropatwa will be gone next week for about 9 days. He will be gone for reasons I will not say, for they are very personal. We are all praying and wishing him the best.

Due to these events, Dr. Eviatar will return to substitute for 3 lessons of Chapter 8. Mr. K also gave us tinyurl's to assist us during that week should we have any concerns or confusions. I will post the tinyurl's now.

http://tinyurl.com/b94cgo - Chapter 8.1
http://tinyurl.com/c5c2nm - Chapter 8.2 & 8.3

All of them are interactive lessons. When he returns, we should be finishing up Chapter 8.3 or starting Chapter 8.4

Oh yeah, we have a test on Friday, be there or be cubed!

Starting off our 'Spare Lesson', we reviewed the Left and Right hand sums. After that we reviewed the midpoint and trapezoid sums. (I find it that I shouldn't have to really explain it here as it is explained in detail in the scribe post below)

Now after doing the homework that Mr. K assigned us, we found that the Midpoint errors were approximately half of that of a Trapezoid sum, therefore Midpoint sums are more accurate by approximately double that of a Trapezoid sum.

Error Estimates for Trapezoid & Midpoint Sums

The Equation for Trapezoidal Sums

The Equation for Midpoint Sums

Where...

• a & b are endpoints of an Interval
• n is the number of sub divisions
• M2 is the maximum value of the second derivative on the given interval.
  

For the Midpoint sum equation, it's 24n^2 because it is more accurate.

So to use this, we 'choose' the accuracy then find n, so we know how many rectangles there are.

The Simpson Sum


No, Not Homer Simpson. The next sum was made by a mathematician who found a way to combine the two sums. He tried it in a different way, he tried to calculate using little parabolas.
So for example, we'll use M4 and T4, because if you have both, you can find a Simpson Sum.
M4 = 0.2422
T4 = 0.2656
So just plug it in and...

((2(0.2422)) + (0.2656)) / 3 = 0.25

0.25 is the exact point of the integral we were using, but even though it was exact in this case, we'll call it an exact approximation. This tells us that the Simpson Sum is even more accurate than both sums.

Approximate

to within an error of 0.001 using a:
a)trapezoidal sum - How many intervals, n, are required?
b)How many intervals are required using a midpoint sum?

A)

f(x) = x^-2 f'(x) = -2x^-3

f"(x) = 6x^-4, so max of f" on [1,2] occurs at 1.
f"(1) = 6, therefore M2 = 6

0.001 = 1/(2n^2)
500 = n^2
22.3607 = n

From here, we round up because it can't be less than the error that we wanted. Less Intervals = Less Accuracy. So, n = 23.

B)

0.001 = 6(2-2)^3/24n^2
0.001 = 1/4n^2
250 = n^2
15/822 = n --> 16 = n

Wow, did that honestly take me two hours? Geeez ¬__¬" Anyways, I still have loads of homework to do, so I'm gonna go do that and stay up for a few hours. Expect me to be either grumpy or tired so yeah :D. Oh yeah, next scribe will be Francis.

Also I thought I'd bring this to everyone's attention.

Take the Vow

Over one billion children live in poverty. Hunger is only one of their challenges; exploitation, abuse and discrimination haunt them on a daily basis. Many live in remote areas of the world and have little, or no, education.

They have no rights. They have no voice.

On March 5, 2009, people around the world will remain silent for 24 hours in support of those who are unable to speak up themselves.

Join the quiet revolution.

Take the Vow of Silence.

Go to I am Silent for more information.