Homework 5 – Math 104A, Fall 2010
Due on Tuesday, November 9th, 2010
1. Given
x
i
, i
= 0
,
1
, . . . , n
, consider the Lagrange polynomials
L
n,j
for
j
= 0
,
1
, . . . , n
. Prove that
n
X
j
=0
L
n,j
(
x
) = 1 for all
x
∈
R
.
2. The following data is taken from a polynomial. What is its degree?
x
2
1
0
1
2
3
p
(
x
)
5
1
1
1
7
25
3. Consider the function
e
x
on [0
, b
] and its approximation by an interpo
lating polynomial. For
n
≥
1, let
h
=
b/n
,
x
j
=
jh
,
j
= 0
,
1
, . . . , n
; and
let
p
n
(
x
) be the
n
th degree polynomial interpolating
e
x
on the nodes
x
0
, . . . , x
n
. Prove that
max
0
≤
x
≤
b

e
x

p
n
(
x
)
 →
0
as
n
→ ∞
.
4. Let
x
0
, x
1
, . . . , x
n
be distinct real points, and consider the following
interpolation problem. Choose a function
P
n
(
x
) =
n
X
j
=0
c
j
e
jx
such that
P
n
(
x
i
) =
y
i
, i
= 0
,
1
, . . . , n,
with the
{
y
j
}
given data. Show there is a unique choice of
c
0
, c
1
, . . . , c
n
.
5. Show there is a unique cubic polynomial