Today we continued to work on limits,by using graph and logic
-We began by solving the daily POD:
The east section of a stadium has 30 rows of seats the 10th row has 100 seats and every row has 3 more seats than the row above it. How many seats are in the east section of the stadium?
Analisis:
-First we identify what the problem is asking us to do: It is asking for the total amount of seats, so it is a series.
-It also sais that each row has 3 more seats than the row above it,so the difference is 3 seats , so that makes it an arithmetic sequence.
-We are given T10that is 100and the r (ratio)is -3 because we are decreasing 3 seats from the of the term before.
- so first we find what T1 equals by using the explicit definition of an arithmetic sequence.
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- Now that we know what T1 is we can solve for T30:
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-Knowing what T1 and T30 are, we can solve for the sum of all rows using the formula for arithmetic series
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-There are 2,205 seats in the east side of the stadium
-Continuing with the lesson on limits; We began studying how to find LIM without a calculator.
-There are two acceptable ways
A) list the first few terms
b)think
-the first problem we solved was:
Tn=(-n)^2 Lim Tn?
-Just by looking at the problem we can induce that with any N we plug in the value on Tn will be positive ( n is rise to an even power) then if we make a table of values for the first few terms, we realize that the outputs are increasing.
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Therefore the limit of Tn as n approaches infiniti, because Tn continues getting bigger.
The second problem was:
Tn=(-1)^n-1· n/10n Lim Tn?
-We made a table of values to find the tendency of Tn as n increases:
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-We can now state that the limit of Tn as n approaches to infiniti is 0, because the values of Tn keep getting closer to 0 as n increases.
The 3rd problem we solved was:
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-As we did in the last problem, we started with a tabla of values which is the following:
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- If we analizethis way we can see that the value of Tn is not really approaching to one real number,it approaches both 1 and -1.This makes the limits DNA ,does not exist.
-The 4th problem we solevd was a little bit different because it asks for the limit of cosine.
Tn=cos(n) Lit tn=?
- The graph of cos is :
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And just like in problem #3, Tn is not approching 1 single real number, it approaches 1 and -1. the limit DNE .
-The last problem was:
Tn= Ln (n) Lim Tn= ?
-to solve this problem we have to know what the graph of LnX look like. We new how the graph of e^x was, and since LnX and e^x are apposites LnX is the reflection of e^x in the y=x axis
-Graph of e^x and y=x axis with it`s reflection
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- By seeing the graph, we can know that the limit of Tn as n approches infinity is infinity, because as n gets, Tn continues increasing as well
- We hope this was helpfull for you and that it cleared the doubts that you had. If you have any questions feel free to post them or add comments and we will be happy to answer them or explain.
!!!!!!!!!!!There is no¨NEW¨ homework.
- Next scribe Fiore and Caro I
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