Page 211 - Engineering Mathematics Workbook_Final

Numerical Methods

1. The equation e − x 4x = 2 0 has a root 4. If  and  are the forward and the
between 4 and 5. Fixed point iteration backward difference operators

1 respectively, then  − is equal to
with iteration function e x /2 .
2 (a) − (b) 

(a) diverges 
(c)  + (d)
(b) converges 

(c) oscillates 5. One root of the equation e − x 3x = 2 0
lies in the interval (3, 4). The least
(d) converges monotonically
number of iterations of the bisection
−
3
2. The formula method so that error  10 are
 1   
1

A f −  + A f ( ) 0 + A f  
2   
0  
 

2
 2  2   (a) 10 (b) 8
which approximates the integral (c) 6 (d) 4
 1 f ( ) x dx is exact for polynomials
− 1 6. The least squares approximation of
of degree less than or equal to 2 if first degree to the function
f x =
( ) sin x over the interval
4 2
(a) A = A = , A =  
2
0
3 1 3  −  ,  
  2 2  is
(b) A = A = A = 1  
1
0
2
24x 24x
4 2 (a) (b)
(c) A = A = , A = −  3  2
0 2 1
3 3
24x
(d) none of the above (c) (d) 24x

3. If x = is a double root of the
( )
f x =
f
equation ( ) 0 and if ''' x is 7. The order of the numerical
differentiation formula
continuous in a neighbourhood of  1
x
then the iteration scheme for '' f ( ) = 12h 2 [−  ( f x − 2h ) + ( f x + 2h ) +
0
0
0
determining  : 16  ( f x − ) h + ( f x + h ) 30 f− ( )]x 0
0
0
( )
2 f x (a) 2 (b) 3
x n+ = x − n has the order
( )
n
1
' f x n (c) 4 (d) 1
(a) 2 (b) 1
(c) less than 2 (d) less than 1

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