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Unformatted text preview: CSC 302 Introduction to Numerical Methods Class Test #1. Time allowed: 75 minutes . Read the directions carefully . Total points = 125. This test is closed book, closed notes . Show all work for partial credit on a problem. You will receive zero credit for an unjustified answer. Read every question carefully before starting. If you notice an error, or if a question is confusing, ask about it at the start. CALCULATORS ARE NOT ALLOWED ANSWER ALL 5 QUESTIONS. Question 1. (a) State Taylors theorem for f ( x + h ). Be sure to include any conditions that the function must satisfy and give the error term. [9 points] (b) Derive the mean value theorem for derivatives from Taylors theorem. [8 points] (c) Develop the first two terms and the error in the Taylor series in terms of h for ln(3- 2 h ). [8 points] 1 Answer: (a) If the function f ( x ) possesses continuous derivatives of order 0 , 1 , 2 ,... ( n +1) in a closed interval I = [ a,b ], then for any x I , f ( x + h ) = n X k =0 f ( k ) ( x ) k ! h k + E n +1 where h is any value such that x + h is in I and where E n +1 = f ( n +1 ( ) ( n + 1)! h n +1 for some between x and x + h . (b) The mean value theorem is obtained by setting n = 0. f ( x + h ) = n X k =0 h k k ! f ( k ) ( x ) + h n +1 ( n + 1)! f ( n +1) ( ) Let n = 0: f ( x + h ) = X k =0 h k k ! f ( k ) ( x ) + hf ( ) = f ( x ) + hf ( ) f ( x + h )- f ( x ) h = f ( ) This is sufficient to get full points. To put into standard form, then change variables: a = x and b = x + h : h = b- x = b- a ....
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This note was uploaded on 02/16/2011 for the course CS 302 taught by Professor Stewart during the Spring '11 term at N.C. State.
- Spring '11