University of Florida/Egm6341/s10.team2.niki/HW1

Problem 1

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Given

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Find

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Determine the limit of the given function   and plot it in the interval  

Solution

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Problem 9

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Problem Statement
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Pg. 7-1

 

1) Expand   in Taylor Series w/ remainder:

 

2) Find Taylor Series Expansion and Remainder of f(x). eq. 4 of p 6-3.--Egm6341.s10.team2.niki 02:26, 26 January 2010 (UTC)

Solution
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Given:

 

[equation 4 p 2-2]

 

[equation 1 p 2-3]

Part 1
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for the case that  , we get,

 

  = 

Using equation 1 p 2-3, we get the remainder as

 

for  , we get

 

finally,  

Part 2
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dividing both sides by x we get,

 

and remainder becomes

 

since  , we have

  where  

Finally,

 

problem 4

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Problem Statement

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Pg 5-1.

Prove the Integral Mean Value Theorem (IMVT) p. 2-3 for w(.) non-negative. i.e  

Solution

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We have the IMVT as

 


For a given function   Let m be the minimum of the function and M be the maximum of the same function

Then we know that,  

multiplying the inequality throughout by   and integrating between   we get

 

 

writing  , we get

 

It is seen that when w(x) = 0, the result is valid. Consider the case when w(x) > 0

dividing throughout by  

 

From the Intermediate Value Theorem, we know that there exists   such that

 

i.e

 

Hence Proved --Egm6341.s10.team2.niki 02:28, 27 January 2010 (UTC)