User:Egm6341.s11.team3.rakesh

http://en.wikiversity.org/wiki/User:Egm6341.s11.team3.rakesh/hw2

= HW 1.1 =

Problem
Find the limit of function: $$ \lim_{x\to 0} f(x) = \lim_{x\to 0} \frac {e^x - 1}{x} $$

Find
$$ \lim_{x\to 0} f(x) = \lim_{x\to 0} \frac {e^x - 1}{x} $$,

Solution
Take the limit of the function:

$$ \lim_{x\to 0} f(x) = \frac {e^0 - 1}{0} = \frac {0}{0}$$

Hence we can apply L-Hopital Rule:

$$ \lim_{x\to 0} f(n) = \frac {f'(n)}{g'(n)} \qquad (1)$$

By above rule:

$$ \lim_{x\to 0} f(x) = \lim_{x\to 0} \frac {e^x - 1}{x} =\lim_{x\to 0} \frac {f(x)}{g(x)} \Rightarrow

\begin{matrix} f'(x) = e^x -0 \\ g'(x) = 1 \end{matrix} $$

$$ \Rightarrow \lim_{x\to 0} f(0) = \frac {f'(0)}{g'(0)} = \frac {e^0 - 0}{1} = 1 \ $$

$$ Ans.: \lim_{x\to 0} f(x) = \frac {e^x - 1}{x} = 1 $$

= HW 2.1 =

Problem
Do the integration by parts to reveal three more terms  :  $$ {f(x)=f(x_0)+\frac {x-x_0}{1!} f(x_0)+\int\limits_{x}^{x_0}(x-t)f''(t)dt } $$

Problem 2.1.2
use IMVT theorem to express the remainder $$ f^5(\S)for (x,x_0) $$

Problem
use IMVT theorem to express the remainder $$ f^5(\S)for (x,x_0) $$

Solution
a) Given That 
 * {| style="width:100%" border="0"

$$ {f(x)=f(x_0)+\frac {x-x_0}{1!} f(x_0)+\int\limits_{x}^{x_0}(x-t)f''(t)dt } $$|| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}(x-t)f(t)dt = [-\frac {(x-t)^2}{2}]_{t=x_0}^{t=x} f(t)-\int\limits_{x}^{x_0}\frac{-(x-t)^2}{2}f'''(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}(x-t)f(t)dt = \frac {(x-x_0)^2}{2} f(x_0)+\int\limits_{x}^{x_0}\frac {(x-t)^2}{2}f'''(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}\frac {(x-t)^2}{2}f(t)dt = -[\frac {(x-t)^3}{2*3}]_{t=x_0}^{t=x} f(t)-\int\limits_{x}^{x_0}\frac {-(x-t)^3}{2*3}f(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}\frac {(x-t)^2}{2}f(t)dt = \frac {(x-x_0)^3}{3!} f(t)+\int\limits_{x}^{x_0}\frac {(x-t)^3}{3!}f(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}\frac {(x-t)^3}{3}f'(t)dt = -[\frac {(x-t)^4}{4*3!}]_{t=x_0}^{t=x} f'(t)-\int\limits_{x}^{x_0}-\frac {(x-t)^4}{4!}f'(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ \int\limits_{x}^{x_0}\frac {(x-t)^3}{3}f'(t)dt = \frac {(x-x_0)^4}{4!} f'(t)+\int\limits_{x}^{x_0}\frac {(x-t)^4}{4!}f(t)dt $$| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }


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$$ {f(x)=f(x_0)+\frac {x-x_0}{1!} f(x_0)+\frac {(x-x_0)^2}{2} f(x_0)+ \frac {(x-x_0)^3}{3!} f'(t)+\frac {(x-x_0)^4}{4!} f'(x_0)+\int\limits_{x}^{x_0}\frac {(x-t)^4}{4!}f'(t)dt} $$|| $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }

<span id="(1)">
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$$ \int\limits_{x}^{x_0}\frac {(x-t)^4}{4}f^(4)(t)dt =  f^5(\xi)\frac {(x-x_0)^5}{5!} $$, $$   \xi  \in (x,x_0) $$ By IMVT THEROEM <span id="(1)">
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 * }


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 * style="width:95%" |Integrate by parts of $$R_{n+1}\left ( x \right )$$
 * $$\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt=\frac{1}{n!}\left \{ \left [ -\frac{(x-t)^{n}}{(n+1)}f^{(n+1)}\left ( t \right ) \right ]_{x_{0}}^{x} - \int_{x_{0}}^{x}\frac{-(x-t)^{n+1}}{(n+1)}f^{(n+1)}\left ( t \right )dt\right \}$$|| <p style="text-align:right;">$$\displaystyle
 * $$\frac{1}{n!}\int_{x_{0}}^{x}\left ( x-t \right )^{n}f^{(n+1)}\left ( t \right )dt=\frac{1}{n!}\left \{ \left [ -\frac{(x-t)^{n}}{(n+1)}f^{(n+1)}\left ( t \right ) \right ]_{x_{0}}^{x} - \int_{x_{0}}^{x}\frac{-(x-t)^{n+1}}{(n+1)}f^{(n+1)}\left ( t \right )dt\right \}$$|| <p style="text-align:right;">$$\displaystyle

$$
 * }

<span id="(1)">
 * {| style="width:100%" border="0"


 * style="width:95%" |
 * $$ \frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{n+1}}{(n+1)!}f^{(n+2)}dt$$<p style="text-align:right;">$$\displaystyle
 * $$ \frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right )+\int_{x_{0}}^{x}\frac{(x-t)^{n+1}}{(n+1)!}f^{(n+2)}dt$$<p style="text-align:right;">$$\displaystyle

$$
 * }

<span id="(1)">
 * {| style="width:100%" border="0"


 * style="width:95%" |
 * $$f\left ( x \right )=P_{n+1}\left ( x \right )+R_{n+2}\left ( x \right )$$|| <p style="text-align:right;">$$\displaystyle
 * $$f\left ( x \right )=P_{n+1}\left ( x \right )+R_{n+2}\left ( x \right )$$|| <p style="text-align:right;">$$\displaystyle

$$
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ P_{n+1}\left ( x \right )=f\left ( x_{0} \right )+\frac{(x-x_{0})}{1!}f^{(1)}\left ( x_{0} \right )+...+\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{(n+1)}\left ( x_{0} \right ) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right;">$$\displaystyle
 * }

<span id="(1)">
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$$ R_{n+2}\left ( x \right )=\frac{1}{(n+1)!}\int_{x_{0}}^{x}(x-t)^{n+1}f^{(n+2)}dt $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right;">$$\displaystyle
 * }

APPLYING IMVT TO EQUATION

<span id="(1)">
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$$\int\limits_{x}^{x_0}\frac{(x-t)^n}{n!}f^{n+1}(t)dt = f^{n+1}(\xi)\frac {(x-x_0)^n}{n+1!} \xi \in (x,x_0)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

= HW 2.2 =

 REMARK: We did the solution on our own 

HW 2.2.a
1. Make sure to leave spaces in between the code like I have, b'fore finishing the stuff. 2. The way I do is I make extra space, then copy there the code and when I finished doing that, I copy/paste the formulas to the code that I put above them and then delete the plain formulas.


 * Example*

<span id="(1)">
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$$ F = force \quad (N) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x, x\in(0,\pi) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x \quad at \quad x_0=\frac{\pi}{4} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$ f(x) = cos(x) \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

As for font, it seems that wiki has it's own.. though process or whatever...

use "\quad" command at the end to up the font. Quad is nothing more then saying "skip space" but it does a job.


 * example*

//

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f'(x) = \cos(x) \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \cos(x) \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }


 * end of example

!!!! also if u can do cos(x) in place of \cos x and similarly for sin

<span id="(1)">
 * {| style="width:100%" border="0"

$$f(x) = \sin x, x\in(0,\pi) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$f(x) = \sin x, x\in(0,\pi) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x  \quad $$ $$ x\in(0,\pi)$$   |$$ x_0=\frac{\pi}{4}$$ $$
 * style="width:95%"  |
 * style="width:95%"  |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
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$$   $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$  f'(x) = \cos x \quad$$
 * <p style="text-align:right">$$ \displaystyle
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$  f''(x) = -\sin x \quad $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$ f'''(x) = -\cos x \quad $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * $$ f(x) =  \sin x \quad $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

we know from taylor series <span id="(1)">
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$$ f(x) = P_n + R_n+1 \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

where as

<span id="(1)">
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$$P_n =f(x_0)+\frac {x-x_0}{1!} f(x_0)+\frac {(x-x_0)^2}{2} f(x_0)+\frac {(x-x_0)^3}{3!} f'(t)+.....+\frac {(x-x_0)^n}{n!} f^n(t) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
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$$R_n+1 = \frac {(x-x_0)^n+1}{n+1!} f^{n+1}(\S) $$ $$ when N =0
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$  f(x) = P_0+ R_1 \quad $$
 * <p style="text-align:right">$$ \displaystyle
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
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$$P_0 = f(x_0) = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
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$$R_1= (x-\frac{\pi}{4})$$   $$ \,f(\S)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x_0) = \frac{1}{\sqrt{2}}+(x-\frac{\pi}{4}) \,f(\S) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

when n=1 <span id="(1)">
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$$ f(x) = P_1+ R_2 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$P_0 = f(x_0) = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$P_1 = f(x_1) = \frac{x-\frac{\pi}{4}}{1!}\cos\frac{\pi}{4}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_2= \frac{(x-\frac{\pi}{4})^2}{2!} \,f^2(\S)$$$$ \displaystyle $$ $$P_1 + R_2= \frac{1}{\sqrt{2}}+\frac{x-\frac{\pi}{4}}{1!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^2}{2!} \,f^2(\S) $$ $$ when n=2
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = P_2+ R_3 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ =\frac{1}{\sqrt{2}}+\frac{x-\frac{\pi}{4}}{1!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^3}{3!} \,f^3(\S)  $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

when n=3

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = P_3+ R_4 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ =\frac{1}{\sqrt{2}}+\frac{x-\frac{\pi}{4}}{1!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^3}{3!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^4}{4!} \,f^4(\S)  $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

when n=4

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = P_4+ R_5 \quad$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ =\frac{1}{\sqrt{2}}+\frac{x-\frac{\pi}{4}}{1!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^3}{3!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{\pi}{4})}^4}{4!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^5}{5!} \,f^5(\S)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

so on and when n=10 <span id="(1)">
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$$ f(x) = P_{10}+ R_{11}\quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ =\frac{1}{\sqrt{2}}+\frac{x-\frac{\pi}{4}}{1!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}-\frac{({x-\frac{\pi}{4})}^3}{3!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{\pi}{4})}^4}{4!}\frac{1}{\sqrt{2}}+.......-\frac{({x-\frac{\pi}{4})}^{10}}{10!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^{11}}{11!} \,f^{11}(\S)  $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

finally the maximum can be computed by

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_{n+1} = \frac {(x-x_0)^{n+1}}{n+1!} f^{n+1}(\S)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_{n+1}\le\frac {(x-x_0)^{n+1}}{n+1!} |maxf^{n+1}(\S)|\le\frac {(x-x_0)^{n+1}}{n+1!}=\frac {(\pi/4)^{n+1}}{n+1!}\le(\pi/4)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

HW 2.2.b
<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x, x\in(0,\pi) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = \sin x \quad at \quad x_0=\frac{3\pi}{4} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f'(x) = COS(X) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f'(x) = COS(X) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$  f''(x) = -\sin x  $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * style="width:95%" |$$ f'''(x) = -\cos x  $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$
 * $$ f(x) =  \sin x  $$
 * <p style="text-align:right"> $$ \displaystyle
 * <p style="text-align:right"> $$ \displaystyle
 * }

we know from taylor series <span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = P_n + R_n+1 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

where as

<span id="(1)">
 * {| style="width:100%" border="0"

$$P_n =f(x_0)+\frac {x-x_0}{1!} f(x_0)+\frac {(x-x_0)^2}{2} f(x_0)+\frac {(x-x_0)^3}{3!} f'(t)+.....+\frac {(x-x_0)^n}{n!} f^n(t) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_n+1 = \frac {(x-x_0)^n+1}{n+1!} f^{n+1}(\S) $$ $$ when N =0
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$P_0 = f(x_0) = \sin\frac{3\pi}{4} = \frac{1}{\sqrt{2}}$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_1= (x-\frac{3\pi}{4})$$   $$ \,f(\S)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x_0) = \frac{1}{\sqrt{2}}+(x-\frac{3\pi}{4}) \,f(\S) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

when n=1 <span id="(1)">
 * {| style="width:100%" border="0"

$$ f(x) = P_1+ R_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$R_2= \frac{(x-\frac{3\pi}{4})^2}{2!} \,f^2(\S)$$$$ \displaystyle $$ $$f(x)= \frac{1}{\sqrt{2}}-\frac{x-\frac{3\pi}{4}}{1!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{3\pi}{4})^2}{2!} \,f^2(\S) $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

when n=2

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

$$ f(x) = P_2+ R_3 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

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

$$ =\frac{1}{\sqrt{2}}-\frac{x-\frac{3\pi}{4}}{1!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{3\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{3\pi}{4})^3}{3!} \,f^3(\S)  $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

When n=3

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

$$ f(x) = P_3+ R_4 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }

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

$$ =\frac{1}{\sqrt{2}}-\frac{x-\frac{3\pi}{4}}{1!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{3\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{3\pi}{4})}^3}{3!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{3\pi}{4})^4}{4!} \,f^4(\S)  $$ $$ ...so on When n=10 <span id="(1)">
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">$$ \displaystyle
 * }
 * {| style="width:100%" border="0"

$$ f(x) = P_{10}+ R_{11} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

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

$$ =\frac{1}{\sqrt{2}}-\frac{x-\frac{3\pi}{4}}{1!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{3\pi}{4})}^2}{2!}\frac{1}{\sqrt{2}}+\frac{({x-\frac{3\pi}{4})}^3}{3!}\frac{1}{\sqrt{2}}++.......-\frac{({x-\frac{3\pi}{4})}^{10}}{10!}\frac{1}{\sqrt{2}}+\frac{(x-\frac{\pi}{4})^{11}}{11!} \,f^{11}(\S)  $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

finally the maximum can be computed by

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

$$R_{n+1} = \frac {(x-x_0)^{n+1}}{n+1!} f^{n+1}(\S)$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

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

$$R_{n+1}\le\frac {(x-x_0)^{n+1}}{n+1!} |maxf^{n+1}(\S)|\le\frac {(x-x_0)^{n+1}}{n+1!}=\frac {(\pi/2)^{n+1}}{n+1!}\le(\pi/2)$$ $$ = 3.5 = $$E_{n}^{n+1}\left ( x \right )=F^{n+1}\left ( x \right)-0$$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

Solution
<span id="(1)">
 * {| style="width:100%" border="0"

$$ E_{n}^{n+1}\left ( x \right )=F^{n+1} \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.1)
 * }

We start with an general equation for error estimation

From we learn that

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

$$E_{n} = I-I_{n} \quad $$ $$ the first derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1) <span id="(1)">
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.2)
 * }
 * {| style="width:100%" border="0"

$$E_n^{1}=\int_{0}^{1}f^{1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{1}\left ( x \right )dx\, \quad $$ $$ the second derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1) <span id="(1)">
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.3)
 * }
 * {| style="width:100%" border="0"

$$E_n^{2}=\int_{0}^{1}f^{2}\left ( x \right )dx-\int_{0}^{1}f_{n}^{2}\left ( x \right )dx\, \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.4)
 * }

the third derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1)

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

$$E_n^{3}=\int_{0}^{b}f^{3}\left ( x \right )dx-\int_{0}^{b}f_{n}^{3}\left ( x \right )dx\, \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.5)
 * }

the fourth derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1)

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

$$E_n^{4}=\int_{0}^{1}f^{4}\left ( x \right )dx-\int_{0}^{1}f_{n}^{4}\left ( x \right )dx\, \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.6)
 * }

the nth derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1)

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

$$E_n^{n}=\int_{0}^{1}f^{n}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n}\left ( x \right )dx\, \quad $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.7)
 * }

the n+1 th derivative of $$E_{n}\,$$ would yield us the following result for an interval (0,1)

<span id="(1)">
 * {| style="width:100%" border="0"

$$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n+1}\left ( x \right )dx\, \quad $$ $$ we have learnt that
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.8)
 * }

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


 * style="width:95%" |
 * style="width:95%" |

$$f_{n}\left ( x \right )=\sum _{i=0}^{n}l_{i,n}\left ( x \right ) f\left ( x_i \right)\,$$ $$
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.9)
 * }

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


 * style="width:95%" |
 * style="width:95%" |

(i) where $$l_{i,n}\left ( x \right)=l_{i}\left ( x \right)=\prod_{j=0,i \neq j}^{n}\frac{x-x_{j}}{x_{0}-x_{j}}\,$$ $$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n+1}\left ( x \right )dx\, \quad $$ $$
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.10)
 * }

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

$$ \begin{align} \\n=1, & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}}  l_{0}=l\left ( x^{1}\right)\;  equation \mbox{ }of\mbox{ } degree\mbox{ } 1 \\n=2, & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}}\frac{x-x_{2}}{x_{0}-x_{2}}  l_{0}=l\left ( x^{2}\right)\; equation \mbox{ }of \mbox{ }degree \mbox{ } 2 \\.\, \\.\, \\.\, \\n=n, & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}}\frac{x-x_{2}}{x_{0}-x_{2}}...\frac{x-x_{n-1}}{x_{0}-x_{n-1}}\frac{x-x_{n}}{x_{0}-x_{n}}  l_{0}=l\left ( x^{n}\right)\, equation \mbox{ }of\mbox{ } degree\mbox{ } n \end{align} $$||<p style="text-align:right"> $$ \displaystyle $$
 * style="width:95%" |
 * style="width:95%" |
 * }

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

$$f\left( x_{i} \right)\,$$ is constant, Hence forth  $$f_n\left( x \right)\,$$ is $$x^{n}\,$$'s function $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle
 * }

which is inferred from our professors lecture

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

$$f_n^{n+1}\left ( x \right )=0\,$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.11)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n+1}\left ( x \right )dx\,$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.12)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1} 0 dx\,$$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.13)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ E_n^{n+1}=\!F^{n+1}\left ( x \right )dx- 0 \,$$ $$ Therefore,
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. 3.5.14)
 * }

$$ E_n^{n+1}=\!F^{n+1}\left ( x \right )dx- 0 \,$$