Scribe Post May 16, 2007

May 16, 2007
As usual we started the class with the problem of the day.

POD 5/16
My son Julian was swinging on his swing and I was pushing him so that the arc lenght of his swing was 3 meters. After I stopped pushing him the arc lenght of each swing decreased 1/6 of the previous distance. How far does Julian travel during 10 swings after I stop pushing him?

Nobody in the class was able to answer the problem so Mr. A answered the problem.

There are two major things you have to understand in this problem.

1. Is the sequence geometric or arithmetic.
   
2. What to do with the 1/6.
   

Mr. A then gave us a tip, "Always try to list the few terms first, to have a btter view of the problem."
So since we know T1= 3, and that 1/6 of 3 is 0.5, we can assume that T2 = 2.5. Doing this step many times would be very long so there is an easier way.

Instead of multiplying by 1/6 and subtracting from the previous term, just multiply the term by 5/6.

Why?

By multiplying the # by 5/6 you are throwing awy the 1/6 we dont want and conserving the other 5/6 we need. Then by multiplying 2.5 x 5/6 = 2.0833, we get the first three terms.

3, 2.5, 2.0833
(Now its clearer to see that the sequence is not arithmetic, but geometric)

Now, we only have to plug in the Geometric series formula.
Sn = T1(1-r^n)/(1-r)
S10 = 3(1-(5/6)^10)/(1-(5/6))
S10= Approx 15.09 meters

Now that we were finally finished with the POD, we started with limits.

Remember that a limit is a boundary, and that this statement means "What happens to Tn as n gets really big?"

So we had an example to solve:
Question: Find the limit of Tn as n goes towards infinity if the explicit formula is Tn= (0.99)^n

Mr. A, then showed us that there are four was to approach this answer.

1. Plug in large values for n, this way we can analyze what is Tn getting closer to.


2. List terms, as Mr. A said this helps us get a better view.


3. Repeated multiplication of a number less than 1, and one bigger.


4. 99% or as Isabella said 99/100. Why? 99/100 x 99/100 x 99/100, shows that the denominator is always bigger, therefore the numerator will get smaller.

This are basic ideas to reach the limit of the equation. We got to the conlusion that the equation tries to get closer to 0, even thought it yet never reaches 0.

So, the limit of Tn as n approaches infinity is 0.

At last in class, Mr. A taught us a second perspective of how to analyze limits: GRAPHS.

As an example we used a complex sequence.

1/2, 3/4, 7/8, 15/16

Still, to graph this is the GC there are several steps.

• Under mode change to seguqence and to dot.


• Go to Y and assign the following values.


• nMin= 1


• u(n) or equation= (2^n-1)/2^n


• u(nMin)Which is the first term = 0.5


• Then go to ZStandard in Zoom.


• Go to Window: Ymin= o and Y max= 2


• Graph

This graph can also be made as a simple function:

F(x)= (2^n-1)/(2^n)

CLASS ENDS!

There is a new HW - Read pg. 493-496. pg 496 #1-25 odd

Remember there is a test next week.

The next scribes are DANIEL GILCHRIST and JORGE VERGARA!!!!!