# exam1 - k = A/P this may be rewritten as f r =(1 r n(1-kr-1...

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Numerical Analysis/Exam #1/October 6, 2004 NAME: Answer all questions. For full credit, you must show your work. Total points: 50. 1. [10 points] Let f ( x ) = x 2 + 3 x - 1. (a) Show that f has a unique root in the interval [0 , 2]. (b) Use the bisection method to ﬁnd this root with error less than 0 . 1. (c) Find how many iterations of the bisection with initial interval [0 , 2] would be required to ﬁnd this root with error less than 10 - 5 . (You do not have to do this many iterations of the bisection method.)

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2 2. [10 points] Let f ( x ) = x 2 + 3 x - 1. Apply Newton’s method with initial estimate p 0 = 1 to generate a sequence of estimates p 1 , p 2 , . . . for a root of f . Stop when | p n - p n - 1 | < 10 - 5 .
3 3. 10 points The ordinary annuity formula is A = P r (1 - (1 + r ) - n ) . Letting

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Unformatted text preview: k = A/P , this may be rewritten as f ( r ) = (1 + r ) n (1-kr )-1 = 0 . Solve the equation f ( r ) = 0 when k = 100 and n = 360 by using Newton’s Method with starting value 0 . 01. 4 4. [10 points] A function f is given by the following table of values: x f ( x ) 2 1 2 2 4 (a) Find the Lagrange interpolating polynomial for this data. (b) Use this polynomial to estimate f (1 . 8). 5 5. [10 points] Let f ( x ) = sin π 6 x . Then f (0) = 0, f (1) = sin π 6 = 1 2 , f (2) = sin π 3 = √ 3 2 . Use Lagrange interpolation applied to the values f (0), f (1), f (2) to obtain an estimate for f (3 / 2) = sin π 4 = √ 2 2 ....
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exam1 - k = A/P this may be rewritten as f r =(1 r n(1-kr-1...

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