# MTH603_3

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```					                       Assignment: # 3 (Spring 2010)
Mth603 (Numerical Analysis)
Lecture: 23 – 28
Total Marks = 25

Q. No. 1. (Marks 10).
Find the first and second order derivatives of f(x) at x = 1.5 if:

x        1.5            2.0      2.5        3.0           3.5            4.0

f(x)       3.37          7.00    13.62       24.00         38.87          59.00

Since x=1.5 appear at beginning of the table, it is appropriat to use formulae based on

forward differences to find the derivatives. The difference table for the given data is

depicted

X            Y               f ( x)        2 f ( x)     3 f ( x)           4 f ( x)       5 f ( x)

1.5          3.37

2.0          7.00            3.63

2.5          13.62           6.62          2.99

3.0          24.00           10.38         3.76          0.77

3.5          38.87           14.87         4.49          0.73            -0.04
4.0             59.00            20.13            5.26             0.77     0.04    0.08

H=0.5

Using forward difference formua for

1             2 f ( x)  3 f ( x)  4 f ( x)  5 f ( x)
Df ( x)  [f ( x)                                              ]
h                2            3         4          5
1          2.99 0.77 0.04 0.08
Df ( x)        [3.63                               ]
0.5            2         3       4        5
Df ( x)  4.8354
1                          11            5
D 2 f ( x)  2 [ 2 f ( x)   3 f ( x)   4 f ( x)   5 f ( x)]
h                           12            6
1                      11            5
D 2 f ( x)        2
[2.99  0.77  ( 0.04)  (0.08)]
(0.5)                     12            6
1
D 2 f ( x)        [2.99  0.77  0.0367  0.0667]
0.25
1
D 2 f ( x)        [2.22  0.0367  0.0667]  8.467
0.25
Q. No. 2.        (Marks 8).

Let P3(x) be the interpolating polynomial for the data (0, 0), (0.5, y), (1, 3) and (2, 2).
Find y if the coefficient of x3 in P3(x) is 6.

Solution:-
Solving for P3(x) gives
d

P3  x  
 x  0  x  1 x  2  y   x  0  x  0.5 x  2  3
 0.5 0.5 1.5                1 0.5 1

 x  0  x  0.5 x  1 2
 2 1.51
P3  x  
 x  0  x  1 x  2  y   x  0  x  0.5 x  2  3
 0.375                          0.5

 x  0  x  0.5 x  1 2
 3
  x  1 x  2      x  0.5 x  2  3
                   y                      
P3  x    x  0    0.375                    0.5        
  x  0.5  x  1                       
                     2                    

            3                            

  x  1 x  2           x  0.5 x  2  30
                   y1000                        
       375                        5          
P3  x    x  0 
  x  0.5  x  1                             
                     2                          

            3                                  

  x 2  3x  2           x 2  2.5x  1 30
                  y1000                      
  375                            5        
P3  x    x  0                                                
  x  1.5 x  0.5 
2

            3
2                       
                                              
 1              30      2  3
P3  x           y1000             x ..........
  375          5   3 
6   2.67 y  6  0.67 
6  6  0.67   2.67 y 
11.33  2.67 y
11.33
y
2.67
y  4.24

Q. No. 3.       (Marks 7).

The following data are given for the polynomial P(x) of unknown degree.

x                   0                   1               2

P(x)                  2                  -1               4
Determine the coefficient of x2 in P(x) if all the third order differences are 1.

x                   P(x)          1st differences   2nd differences

0                      2

1                     -1               -3                    8/2=4

2                      4               5

P2  x  
 x  1 x  2  2   x  0  x  2  (1)   x  0  x  1 (4)
 0  1 0  2  1  0 1  2                2  0  2  1
P2  x    
x   2
 3x  2 
2   x2  2x   2  x2  x 
2
P2  x   4 x  7 x  2
2

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