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

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Problem 1

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Given

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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)