sols_wk1

# sols_wk1 - | p-p n | ≤ b 1-a 1 2 n a condition which is...

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Homework 1 due Thursday October 14 at 12:00 at lecture Prove that the equation x 3 + x + 1 = 0 has one and only one real root, p , and show that it lies between - 1 and 0. Use a bisection method to ﬁnd p correct to three decimal places. How many steps of the bisection method would be needed to guarantee an error no greater than 5 × 10 - 11 in the estimate of p ? Let f ( x ) := x 3 + x + 1. Since f 0 ( x ) = 3 x 2 + 1 1 for all x R it follows that f is monotone increasing. Noting that lim x →±∞ f ( x ) = ±∞ , we conclude from the Intermediate Value Theorem that there is only one point in ( -∞ , ) where f ( x ) = 0. Noting that f ( - 1) = - 1 and f (0) = 1 one concludes that the root p ( - 1 , 0). Using the bisection method n 1 2 3 4 5 6 7 8 a n - 1 - 1 - 0 . 75 - 0 . 75 - 0 . 6875 - 0 . 6875 - 0 . 6875 - 0 . 6875 b n 0 - 0 . 5 - 0 . 5 - 0 . 625 - 0 . 625 - 0 . 65625 - 0 . 671875 - 0 . 6796875 p n - 0 . 5 - 0 . 75 - 0 . 625 - 0 . 6875 - 0 . 65626 - 0 . 671875 - 0 . 6796875 - 0 . 68359375 after several more iterations the answer turns out to be - 0 . 682 correct to three decimal places. From lectures it was proven that
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Unformatted text preview: | p-p n | ≤ b 1-a 1 2 n a condition which is suﬃcient to ensure that the error is smaller than 5 × 10-11 is b 1-a 1 2 n ≤ 5 × 10-11 or n ≥ 35 . • If we wish to compute the cube root of 5, we could proceed by ﬁnding the zeros of f ( x ) = x 3-5. Let the initial interval be deﬁned by a = 0 and b = 3 and calculate an approximation for 5 1 / 3 using 4 steps of the bisection. n 1 2 3 4 5 a n 1 . 5 1 . 5 1 . 5 1 . 6875 b n 3 3 2 . 25 1 . 875 1 . 875 p n 1 . 5 2 . 25 1 . 875 1 . 6875 1 . 78125 f ( a n )-----f ( b n ) + + + + + f ( p n )-+ +-+ The ﬁnal interval is [1 . 6875 , 1 . 78125] and p 6 = 1 . 734375 ≈ p with guaranteed error smaller than (1 . 78125-1 . 6875) / 2 = 0 . 046875. MATH2051 Practical 1 2010-10-10...
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## This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.

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