User:Egm6341.s11.team5.oh/HW1

=Problem 1 - L'Hôpital's rule= From (meeting 3 page 1)

Given

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f(x) = \frac{e^{x}-1}{x} $$ $$
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 * $$\displaystyle
 * $$\displaystyle (1-1)
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Find
1. Find $$\displaystyle \lim \ f(x) $$ as $$\displaystyle \ x -> 0$$

2. Plot $$\displaystyle \ f(x) \ ; \ x \in [ 0,1 ]$$

1. Find $$\displaystyle \lim \ f(x) $$ as $$\displaystyle \ x -> 0$$
 We solved this on our own

[Background Knowledge] L'Hôpital's rule

According to L'Hôpital's rule,


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$$
 * $$\displaystyle \lim_{x\rightarrow 0} \frac{e^{x}-1}{x} \ = \ \lim_{x\rightarrow 0} \frac{e^{x}}{1} \ = \ e^{0} \ = \ 1$$
 * $$\displaystyle (1-2)
 * $$\displaystyle (1-2)
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Therefore,  $$\displaystyle \ \lim_{x\rightarrow 0} \frac{e^{x}-1}{x} \ = \ 1 $$
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2. Plot
 We refered to s10.team3 matlab code but we improved obtaining range from 0 to 10

[Matlab code]

$$\displaystyle 1. \ x \ \in \ [0,1]$$ Plot.jpg

$$\displaystyle 2. \ x \ \in \ [0,10]$$ Plot2.jpg

=Problem 2 - Taylor Series Expansion= From (meeting 3 page 4)

Solution
 

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=Problem 3 - Prove IMVT= From (meeting 4 page 3)

Solution
 

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=Problem 4 - Plot and find max norm of Functions= From (meeting 5 page 3)

Solution
 

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=References=