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Math 128a, fall 2011, Chorin, theory homework 1, due in the week of Sept. 5.
1. Use the formula for the error in interpolation to show that if two polyno
mials
P
n
,Q
n
of degree
≤
n
coincide at
n
+1 distinct points, then they are
identical and coincide at all points.
2. Suppose you represent the function
f
(
x
) =
e
x
in the interval [0
,
3] by the
ﬁrst 7 terms of its Taylor series. Find a bound for the diﬀerence between
f
and the sum of these terms in that interval.
3. Find the maximum norm of the following functions on the interval [0
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Unformatted text preview: , 1]: (i) f ( x ) = x 3 ; (ii) f (1 / 2) = 1 ,f ( x ) = 0 when x 6 = 1 / 2. (iii) f ( x ) =x 100 . 4. Find the least squares norm of each of the functions in the previous problem. 5. Consider the function f ( x ) = x n , where n is a positive integer; ﬁnd its maximum norm and its least squares norm. Find what happens to these norms as n → ∞ . 1...
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This note was uploaded on 01/21/2012 for the course MATH 128A taught by Professor Rieffel during the Fall '08 term at University of California, Berkeley.
 Fall '08
 Rieffel
 Polynomials, Numerical Analysis

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