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Numerical Methods  Homework 6
September 28, 2007
1.
[1pt]
Section 4.1 #1
Solution:
p
(
x
) = 7
(
x

2)(
x

3)(
x

4)
(0

2)(0

3)(0

4)
+11
(
x

0)(
x

3)(
x

4)
(2

0)(2

3)(2

4)
+28
(
x

0)(
x

2)(
x

4)
(3

0)(3

2)(3

4)
+63
(
x

0)(
x

2)(
x

3)
(4

0)(4

2)(4

3)
2.
[2pt]
Section 4.1 #3
Solution:
Let
{
x
0
, x
1
, x
2
, x
3
}
=
{
1
,
1
,
3
,
4
}
. We ﬁnd the four corresponding Lagrange polynomials to be
`
0
=
(
x

1)(
x

3)(
x

4)
(

1

1)(

1

3)(

1

4)
`
1
=
(
x
+ 1)(
x

3)(
x

4)
(1 + 1)(1

3)(1

4)
`
2
=
(
x
+ 1)(
x

1)(
x

4)
(3 + 1)(3

1)(3

4)
`
3
=
(
x
+ 1)(
x

1)(
x

3)
(4 + 1)(4

1)(4

3)
2
1
0
1
2
3
4
5
3
2
1
0
1
2
3
4
Lagrange Polynomials
The key property your graph should show is that
`
i
(
x
j
) =
±
1
i
=
j
0
i
6
=
j
3.
[1pt]
Section 4.1 #12
Solution:
q
(
x
) =
p
(
x
) +
c
(
x
+ 2)(
x
+ 1)(
x
)(
x

1)(
x

2)
.
When
x
= 3, 30 = 61 +
c
(5)(4)(3)(2)(1) and
c
= 31/120. Thus
q
(
x
) =
x
4

x
3
+
x
2

x
+ 1

31
120
(
x
+ 2)(
x
+ 1)(
x
)(
x

1)(
x

2)
.
4.
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This homework help was uploaded on 04/09/2008 for the course CS 257 taught by Professor Thomaskerkhoven during the Fall '05 term at University of Illinois at Urbana–Champaign.
 Fall '05
 ThomasKerkhoven

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