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157IQsol6.pdf

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Department
Mathematics
Course
MATH 157
Professor
Stephen Choi
Semester
Fall

Description
MATH 157 - D100A SSIGNMENT #6 Due date: Wednesday Feb 29, 2012 Instructor Questions accompanying paper Assignment # 6 Solution Directions: Print double sided, write your solutions to both questions directly on this page, place immediately after cover page and staple together! Marks: An answer without a detailed solution process will be awarded zero marks! Q1: Let f(x) = x ▯ 5x + 8. Then a) Show that f(x) = 0 has a root between x = 1 and x = 2. 4 3 4 3 Solution: Since f(1) = (1) ▯ 5 ▯ (1) + 8 = 4 > 0 and f(2) = (2) ▯ 5 ▯ (2) + 8 = ▯16 < 0, so there is a root of f(x) = 0 between x = 1 and x = 2 by the Intermediate Value Theorem because f is a polynomial and hence continuous. b) Use the Newton-Raphson method to find the zero of f. (Find the root in the precision of 4 decimal places) 4 3 0 3 2 Solution: Since f(x) = x ▯ 5x + 8, so f (x) = 4x ▯ 15x . The iterative formula is f(xn) xn+1 = x ▯n 0 f (xn) 4 3 = x ▯ x n▯ 5x n + 8 n 4x ▯ 15x 2 n n 4x ▯ 15x ▯ x + 5x ▯ 8 3 = n n n n x (4x n 15) n 3xn ▯ 10x n ▯ 8 = : xn(4x n 15) Now take x =01, we have x = 1:363636364▯▯▯ 2 x3= 1:294864839▯▯▯ x4= 1:292186672
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