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# hw6 - Homework 6 Foundations of Computational Math 2 Spring...

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Homework 6 Foundations of Computational Math 2 Spring 2011 Solutions will be posted Monday, 2/28/11 Problem 6.1 Consider a minimax approximation to a function f ( x ) on [ a, b ]. Assume that f ( x ) is contin- uous with continuous Frst and second order derivatives. Also, assume that f ′′ ( x ) < 0 on for a x b , i.e., f is concave on the interval. 6.1.a . Derive the equations you would solve to determine the linear minimax approx- imation, p 1 ( x ) = αx + β , to f ( x ) on [ a, b ] and describe their use to solve the problem. 6.1.b . Apply your approach to determine p 1 ( x ) = αx + β for f ( x ) = - x 2 on [ - 1 , 1]. 6.1.c . How does p 1 ( x ) relate to the quadratic monic Chebyshev polynomial t 2 ( x )? 6.1.d . Apply your approach to determine ˜ p 1 ( x ) = ˜ αx + ˜ β for f ( x ) = - x 2 on [0 , 1]. 6.1.e . How could the quadratic monic Chebyshev polynomial t 2 ( y ) on - 1 y 1 be used to provide and alternative derivation of ˜ p 1 ( x ) on 0 x 1? 6.1.f . Suppose you adapt your approach to derive a constant approximation, p 0 ( x ). What points will you use as the extrema of the error?

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hw6 - Homework 6 Foundations of Computational Math 2 Spring...

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