MTE-01 Curve Tracing

Dec 2017

2(a) Show that the set X = [1,5]  [6,9]  [15,30] is a compact subset of  .

Ans:- Given set X = [1,5] [6,9]  [15,30].

Compact Set :- A set S of real numbers is compact if it is closed & bounded.

Here [1,5], [6,9] & [15,80] all are closed set and we know that union of finite closed sets is always closed.

So [1,5]  [6,9]  [15,30] is closed.  ……….. (1)

Let,  x S

X  [1,5]  [6,9]  [15,30]

Here       1  X 30

i.e. S is bounded below & bounded above by 1 & 30 respectively.

So, S is bounded ……….. (2)

Hence by the definite of compact set, (1) & (2)

S is compact subset of .

3(b) Prove that ( + ) is an algebraic number. Is every irrational number, an algebraic number? Justify your answer.

Ans :-

Algebraic Number :- A number is called an algebraic number if it satisfies a polynomial equation

a0xn + a1xn-1 + a2xn-2 + …………. an-1x + an = 0

where the coefficients a01 a11 a12 .......... a1n  are integers, n>1.

Given number         X =  +

X2 = ( + )2

X2 = ()2 + ()2  + 2()()

Second part :-

Every irrational number is not an algebraic number.

Ex:- ,e  etc.

Number , e are known as transcendental number

 

X2  = 2 + 5 + 2

X2  = 7 + 2

X2 – 7 = 2

(X2 – 7)2 = (22

X2 + 49 – 14X2 = 40

X4 – 14X2 + 9 = 0

\[ Ram\quad is\quad going\quad to\quad school.\\ x=\frac { -b+\sqrt { b^{ 2 }-4ac } }{ 2a } \\ 4x^{ 2 }-8x+9=0\\ \int _{ 4 }^{ 5 }{ 5x } dx=K \]

\[ \begin{array}{l}4x^2-8x+9=0\\\end{array} \]

\[ \left. \begin{array} { l } { \text { If } u = e ^ { x y z } \text { , find the value of } \frac { \partial ^ { 3 } u } { \partial x \partial y \partial z } } \\ { \qquad \left. \begin{array}{l}{ u = e ^ { x y z } }\\{ \frac { \partial u } { \partial z } = e ^ { x y z } ( x y ) }\\{ \frac { \partial ^ { 2 } u } { \partial y \partial z } = e ^ { x y z } ( x ) + e ^ { x y z } ( x z ) ( x y ) = e ^ { x y z } ( x + x ^ { 2 } y z ) }\\{ \frac { \partial ^ { 3 } u } { \partial x \partial y \partial z } = e ^ { x y z } ( 1 + 2 x y z ) + e ^ { x y z } ( y z ) \cdot ( x + x ^ { 2 } y z ) }\\{ = e ^ { x y z } [ 1 + 2 x y z + x y z + x ^ { 2 } y ^ { 2 } z ^ { 2 } ] }\\{ = e ^ { x y z } [ 1 + 3 x y z + x ^ { 2 } y ^ { 2 } z ^ { 2 } ] }\end{array} \right. } \end{array} \right. \]

\[ \left. \begin{array} { | c | c | c | c | c | c | c | c | c | c | } \hline \text { Iteration-1 } & { } & { C _ { j } } & { 5 } & { 10 } & { 8 } & { 0 } & { 0 } & { 0 } & { } \\ \hline B & { C _ { B } } & { X _ { B } } & { x _ { 1 } } & { x _ { 2 } } & { x _ { 3 } } & { S _ { 1 } } & { S _ { 2 } } & { S _ { 3 } } & { } \\ \hline \end{array} \right. \]

Iteration-1   Cj 5 10 8 0 0 0  
B CB XB x1 x2 x3 S1 S2 S3 MinRatio
XBx2
S1 0 60 3 (5) 2 1 0 0 605=12
S2 0 72 4 4 4 0 1 0 724=18
S3 0 100 2 4 5 0 0 1 1004=25
Z=0   Zj 0 0 0 0 0 0  
    Zj-Cj -5 -10 -8 0 0 0

Problem is

Max Z =   5 x1 + 10 x2 + 8 x3
subject to
  3 x1 + 5 x2 + 2 x3 60
  4 x1 + 4 x2 + 4 x3 72
  2 x1 + 4 x2 + 5 x3 100
and x1,x2,x30;

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint-1 is of type '' we should add slack variable S1

2. As the constraint-2 is of type '' we should add slack variable S2

3. As the constraint-3 is of type '' we should add slack variable S3

\[ \int _{\:0}^{\pi }\sin \left(x\right)dx \]

\[ \begin{pmatrix}5&4&5&r\\ g&t&y&5\\ 6&6&6&5\\ 4&3&5&t\\ v&t&t&5\end{pmatrix} \]

\[ \left. \begin{array} { l } { \text { Taylor series of function } f ( x ) \text { at a is defined as: } } \\ { \qquad f ( x ) = f ( a ) + \frac { f ( a ) } { 1 ! } ( x - a ) + \frac { f ^ { \prime \prime } ( a ) } { 2 ! } ( x - a ) ^ { 2 } + \frac { f ^ { \prime \prime } ( a ) } { 3 ! } ( x - a ) ^ { 3 } + \ldots } \\ { \text { Maclaurin series of function } f ( x ) \text { is a taylor series of function } f ( x ) \text { at: } a = 0 } \\ { \qquad f ( x ) = f ( 0 ) + \frac { f ( 0 ) } { 1 ! } ( x ) + \frac { f ( 0 ) } { 2 ! } ( x ) ^ { 2 } + \frac { f ^ { \prime \prime } ( 0 ) } { 3 ! } ( x ) ^ { 3 } + \ldots } \end{array} \right. \]

\[ \left. \begin{array} { | c | c | c | c | c | c | c | c | c | c | } \hline \text { Iteration-1 } & { } & { C _ { j } } & { 5 } & { 10 } & { 8 } & { 0 } & { 0 } & { 0 } & { } \\ \hline B & { C _ { B } } & { X _ { B } } & { x _ { 1 } } & { x _ { 2 } } & { x _ { 3 } } & { S _ { 1 } } & { S _ { 2 } } & { S _ { 3 } } & { } \\ \hline \end{array} \right. \]

\[ \left. \begin{array} { l } { 3 x _ { 1 } + 5 x _ { 2 } + 2 x _ { 3 } \leq 60 } \\ { 4 x _ { 1 } + 4 x _ { 2 } + 4 x _ { 3 } \leq 72 } \\ { 2 x _ { 1 } + 4 x _ { 2 } + 5 x _ { 3 } \leq 100 } \\ { \text { and } x _ { 1 } , x _ { 2 } , x _ { 3 } \geq 0 } \end{array} \right. \]

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