Math251_Final_Spring2007-08

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Unformatted text preview: AMERICAN
UNIVERSITY
OF
BEIRUT
 Faculty
of
Arts
and
Sciences
‐
Mathematics
Department.
 
 
 MATH
251
 Final
Exam
(2hrs)
 SPRING
2007
–
2008
 
 
 
 
 
 
 STUDENT
NAME

























































































































.
 
 ID
NUMBER


































































































































.
 
 
 
 
 Problem
1





































/


19





.
 Problem
2






































/


22



.
 Problem
3






































/


15



.
 Problem
4






































/


27



.
 Problem
5






































/


17



.
 
 TOTAL











































/
100
 
 1
 1
.

(a)
‐
Loss
of
significant
figures
may
result
in
the
computation
of
the
functions

 
 for
certain
values
of
x.
Explain
and
specify
these
values
then
propose
alternative
functions
 that
would
remedy
the
loss
of
significant
figures.
 Note
that
:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (b)
 ‐
 Let
 f(x)
 =
 g(x)
 –
 h(x).
 Using
 the
 results
 in
 (a),
 find
 a
 way
 to
 calculate
 accurately
 f(x)
 for
 values
of
x
close
to
zero.
Find
then
 .







































































(3
points)
 
 
 
 
 
 
 (10
points) 
 2
 (c)
 –Solving
 the
 quadratic
 equation
 
 ,
 using
 a
 machine
 that
 carries
 only
 8
 decimal
 digits
 causes
 a
 problem.
 Investigate
 this
 example,
 observe
 the
 difficulty,
 and
 propose
a
remedy
to
approximate
the
2
solutions
 .











































(6
points)
 

 
 
 
 
 
 
 
 
 
 
 
 2
.

The
Reciprocal
of
the
cubic
root
of
a
positive
number
 

(i.e.
 an
iterative
formula
that
does
not
use
division
by
the
iterate

 
 i

–
Establish
this
formula
by
applying
Newton’s
method
to
some
appropriate
function
f(x).
 

Determine
 also
 any
 necessary
 restrictions
 on
 the
 choice
 of
 the
 initial
 value
 
 of
 the
 iterative
procedure.



















































































































(9

points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 3
 
),
can
be
computed
by
 ii
–
Let
a
=
2.
Plot
the
graph
of
the
function
f,
and
give
a
graphic
justification
of
the
necessity
 of
this
restriction
on
 .
















































































































(4
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iii
 –
 For
 a
 suitable
 choice
 of
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4
 ,
 approximate
 
 by
 computing
 the
 first
 3
 iterates
 
by
Newton’s
formula.




























































































(5
points)
 iv
–
Prove
that
Newton’s
method
is
quadratic.




















































































































 
(4
points)
 
 
 
 
 
 
 
 
 
 
 
 
 3.
 Based
 on
 a
 set
 of
 data 
 i. Give
 the
 formula
 for
 the
 composite
 Trapezoid
 Rule
 T(h)
 to
 approximate
 the
 integral

I,
and
derive
the
expression
of

I
–
T(h)
in
terms
of
the
powers
of
h.
 (8
points)
 
 
 
 
 
 
 
 ,
 where
 ,
 and
 nh=2,
 let
 
 5
 ii. Obtain
 an
 upper
 bound
 on
 the
 absolute
 error
 when
 we
 compute
 
 by

 means

of
the
composite
Trapezoid
Rule
using

51

equally
spaced
points.
 (7
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 4.
Based
on
the
following
table
of
values
for
a
function
f(x)

 
 i 0 1 2 3 4 5 6 7 8 x ( i) 0 .0 0 0 0 .1 2 5 0 .2 5 0 0 .3 7 5 0 .5 0 0 0 .6 2 5 0 .7 5 0 0 .8 7 5 1 .0 0 0 f ( x ( i) ) 1 .0 0 0 0 0 0 0 0 .9 8 4 4 9 6 4 0 .9 3 9 4 1 3 1 0 .8 6 8 8 1 5 1 0 .7 7 8 8 0 0 8 0 .6 7 6 6 3 3 8 0 .5 6 9 7 8 2 8 0 .4 6 5 0 4 3 2 0 .3 6 7 8 7 9 4 
 
 6
 
 Consider
the
approximation
the
integral
 composite
trapezoid
rule
T(h).
 i. Derive
 Romberg
 subsequent
 integration
 formulae,
 and
obtain
expressions
of
the
errors
 ,
 
and
 ,
 .
 (12
points)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7
 ,
using
Romberg
integration
based
on
the
 ii. Prove
that
the
Simpson’s
composite
rule
S(h)
is
equivalent
to

R1
(h).
 (5
points)
 
 
 
 
 
 
 
 
 
 
 
 iii. Using
 the
 table
 and
 the
 results
 above,
 fill
 the
 empty
 slots
 in
 the
 following
 table.



 (10
points)

 
 
 h
 1
 0.5
 0.25
 0.125
 M(h)
 

 

 

 

 T(h)
 

 

 

 

 
 
 
 
 
 8
 R1(h)
 

 

 

 

 R2(h)
 

 

 

 

 R3(h)
 

 

 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9
 5.
(a)
–
Fill
in
the
missing
statements
in
the
following
Matlab
program
 (8
points)
 function
p
=
Neville(X,
Y,
x)
 %
Input
arguments:
a
vector
X
representing
the
distinct
xi
data
 %

































:
a
vector
Y
representing
the
corresponding
yi
=
f(xi
)data
 %

































:
x
is
a
vector
of
numbers
at
which
we
want
to
interpolate
 %
Output
arguments:
p
=
p(x)
the
value
of
the
interpolation
polynomial
at
the
X
data
 %
Assume
length
(X)
=
length
(Y)
>
0
 %
Place
the
base
statement
of
the
recursion
 n
=
length(X);
 if
n
=
1
 




……………………………………………………………………………………………………………………….
 else
 %
Place
the
statements
of
the
recursion
 





X1
=
X(1
:
n­1)
;
Y1
=
Y(1
:
n­1);
 





X2
=
X(2
:
n)
;
Y2
=
Y(2
:
n);
 





p2
=
…………………………………………………………………………………………………………………
;
 





p
=



………………………………………………………………………………………………………………….;
 end
 
 
 
 
 
 
 
 
 
 
 
 
 





p1
=
…………………………………………………………………………………………………………………
;



































 
 10
 (b)
 –
 Show
 the
 various
 steps
 of
 the
 execution
 process
 of
 this
 program,
 when
 n
 =
 3
 ,
 in
 the
 form
of
a
diagram:


























































































































(9
points)
 
 
 
 p1
=
……….


























































































p2
=
……….
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11
 
 
 
 
 
 12
 ...
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