User:EGM6341.s11.TEAM1.HW1

=Problem 1: Taylor Series Problem =

Find
Solve for: and plot $$ f(x) \ $$, for $$ x \epsilon\ [0,1] \ $$

Solution
The Taylor Series expansion for $$ e^x \ $$ is: subtracting 1 and dividing thru by x gives: Apply the limit as x goes to 0:

Author EGM6341.s11.TEAM1.WILKS 06:56, 24 January 2011 (UTC)EGM6341.s11.team1.Chiu

=Problem 2: Theorem of Taylor Series =

Find
Solve for:

and

at $$ x_o=0 \ $$

Solution
Author

Egm6341.s11.team1.arm 21:25, 23 January 2011 (UTC)egm6341.s11.team1.arm

=Problem 3: Integral Mean Value Theorem =

Given
a function:

PART A) for all $$ x \epsilon\ [a,b] $$

PART B) for all $$ x \epsilon\ [a,b] $$

Find
PART A) Prove the integral mean value theorem given Equations 3.1 and 3.2 PART B) Prove the integral mean value theorem given Equations 3.1 and 3.3

Solution
Part (A) Define m,M as below,

Consider,

Integrating each side of the inequality,

where, m and M are constants by definition of (3.3 and 3.4) Multiplying (3.7) by $$\displaystyle \frac{1}{\int^b_a w(x)dx}$$ yields

Applying Intermediate Value Theorem,

Thus,


 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) \geqq 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | (3.11)
 * }
 * }

Part (B) Another version of IMVT Either 1) w(x)>0 (done in Part(A)) or 2) w(x)<0

Define m,M as below,

Consider,

Integrating each side of the inequality, where, m and M are constants by definition of (3.3 and 3.4) Multiplying (3.13) by $$\displaystyle \frac{1}{\int^b_a |w(x)|dx}$$ yields

Since $$\displaystyle w(x)<0 $$, $$\displaystyle |w(x)| $$ becomes $$\displaystyle - w(x) $$,

Applying Intermediate Value Theorem,

Thus,


 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) < 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | (3.18)


 * }
 * }

Author EGM6341.s11.TEAM1.Yoon 09:45, 18 January 2011 (UTC) - Primary Author

=Problem 4: Infinity Norms=

Given
for $$ x\epsilon\ [0,1] $$ and

for $$x \epsilon\ [0,1] $$

Find
1) Plot $$ f(x) \ $$ and $$ g(x) \ $$ 2) Solve for:

Solution
Author --Sujin 00:26, 19 January 2011 (UTC)EGM6341.s11.TEAM1.JANG

=Signatures=

Solved problem 1 -- EGM6341.s11.TEAM1.WILKS 07:00, 24 January 2011 (UTC)EGM6341.s11.team1.Chiu

Solved problem 2 --EGM6341.s11.TEAM1.ARM 21:27, 23 January 2011 (UTC)

Solved problem 3 --EGM6341.s11.TEAM1.Yoon 09:45, 18 January 2011 (UTC)

Solved problem 4 --EGM6341.s11.TEAM1.Sujin 00:27, 19 January 2011 (UTC)

=Contributing Team Members=

=References=