M251_Quiz2_Spring20072008

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Unformatted text preview: AMERICAN
UNIVERSITY
OF
BEIRUT
 Faculty
of
Arts
and
Sciences
‐
Mathematics
Department.
 
 
 Test
II
­
MATH
251
 Closed
Book
–
75
mn
 SPRING
2007
–
2008
 
 
 
 
 
 
 STUDENT
NAME

























































































































.
 
 ID
NUMBER


































































































































.
 
 
 
 
 Problem
1





































/


14





.
 Problem
2






































/


14



.
 Problem
3






































/


22



.
 
 TOTAL











































/
50
 
 
 
 1
 TABLE
1:
Problems
1
and
2
 
 Consider
 the
 following
 table
 of
 data
 for
 the
 function
 f(x).
 All
 computations
 shall
 be
 carried
 out
with
8
significant
figures.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2
 #
 1
 ­
 a
 –
 Use
 the
 Central
 Difference
 formula
 to
 approximate
 f
 ‘(0.5).
 Improve
 this
 result
 using
 Richardson’s
 extrapolation
 of
 the
 1st
 and
 2nd
 orders.
 For
 that
 purpose,
 give
 the
 formulae
that
provide
 ,
then
fill
out
the
table
that
follows.






















(10
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3
 b
 –Using
 table
 1,
 calculate
 the
 third
 derivative
 f
 “’(1.000)
 ,
 using
 the
 Backward
 Difference
 Approximation.

 


















(4
points)
 f
“’
(x)
 
 
 f
‘”
(1.000)
 
 
 
 
 
 #
 2
 –
 a‐
 Using
 table
 1,
 write
 Newton’s
 interpolating
 polynomial
 of
 degree
 3,
 that
 would
 best
approximate
f(0.4).
Find

f(0.4).


 






















































































(4
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4
 b
–
Find
the
equation
of
the
quadratic
spline
s(x),
on
the
interval
[0.125,
0.5],
then
draw
 its
graph.
Draw
clearly
the
tangent
to
the
curve
at
each
node.
(Let
z1=
s’
(x1)
=
0)




 (10
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5
 #
3
­
Consider
the
following
4
by
4
square
matrix:
 
 a­ Apply
 on
 this
 matrix
 Gauss
 Elimination
 with
 the
 scaled
 partial
 pivoting
 strategy,
 showing
 the
 status
 of
 the
 4
 by
 4
 matrix
 after
 each
 reduction.
 (Each
 pivot
 row
 and
 the
 corresponding
 multipliers
 should
 be
 identified
 and
 circled).
 
 Specify
 the
 index
 vector
IV
after
each
reduction.
















































































(10
points)
 









































































































 Reduction
1:
 Modified
matrix
A
including
multipliers








scales
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7
 b­ Specify
the
elements
of
the
matrices
P,
L
and
U
that
satisfy
the
LU
decomposition
of
the
 matrix
A.









 
























































































































(4
points)
 
 
 
 
 
 
 
 
 c­ Use
the
matrices
P,
L
and
U,
to
find
the
determinant
of
A.

 













































































































































(2
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 8
 d­ Use
the
matrices
P,
L
and
U,
to
find
the

2nd

column
(ONLY)
of
the
inverse
of
A.



































 (6
points)
 
 
 
 
 
 
 
 9
 ...
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This note was uploaded on 02/26/2010 for the course MATH 251 taught by Professor Notrelevant during the Fall '07 term at American University of Beirut.

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