BENG449 Problem 6 1

BENG449 Problem 6 1 - BENG449 Problem Set 6, Question 1...

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BENG449 – Problem Set 6, Question 1 Alex Lemon March 10, 2008 1 Remarks The derivative of a function can be used to construct the best linear approxi- mation of a function in a given neighborhood of the domain. In particular, the tangent line to a curve, f ( x ), at a point x 0 Dom ( f ) is given by y - f ( x 0 ) = f ( x 0 )( x - x 0 ) . To approximate a zero of f ( x ), Fnd the zero of the tangent line by setting y = 0 and solving for x : x = x 0 - f ( x 0 ) f ( x 0 ) . The Newton-Raphson method Fnds the zeros of f ( x ) by iteratively using the tangent-line approximation outlined above. In general, if the initial guess is suf- Fciently close to a zero of the function, then the sequence of Newton-Raphson sequence will converge geometrically, halving the distance between the approxi- mation and the zero at each step. Additionally, in the case of multiple zeros, the Newton-Raphson method will follow the slope of f ( x ) to the nearest zero. This phenomenon was observed in all but one of the experiments. The problematic
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BENG449 Problem 6 1 - BENG449 Problem Set 6, Question 1...

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