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2.2 - 1.73193 1.73242 1.73193 1.73218 1.73193 1.73206...

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SECTION 2.2 PAGE 38 1. a b 0.00000 1.00000 0.50000 1.00000 0.50000 0.75000 Estimate=(0 . 75 - 0 . 5) / 2 = 0 . 625. 3(a). a b 0.0000 1.0000 0.5000 1.0000 0.5000 0.7500 0.5000 0.6250 0.5625 0.6250 0.5625 0.5938 0.5781 0.5938 Estimate=0 . 5859. 9. The tricky part here is picking a starting value. Note that f 0 ( x ) = 2 x > 0 for x > 0, so f is increasing for x > 0. Since f (1) = - 2 < 0, f (2) = 1 > 0, the interval [1 , 2] brackets the interval. This is the calculation given below. You can do fewer iterations by taking a smaller bracketing interval. a b 1.00000 2.00000 1.50000 2.00000 1.50000 1.75000 1.62500 1.75000 1.68750 1.75000 1.71875 1.75000 1.71875 1.73438 1.72656 1.73438 1.73047 1.73438 1.73047 1.73242 1.73145 1.73242

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Unformatted text preview: 1.73193 1.73242 1.73193 1.73218 1.73193 1.73206 Estimate=1 . 73199. 11. The number of iterations required depends only on the starting interval, not the function. We want to ﬁnd the smallest n such that b-a 2 n < 10-3 . This gives n > ln 3 × 10 3 ln 2 ’ 11 . 55, so n = 12. The iterations are: a b 1.00000 4.00000 1.00000 2.50000 Date : September 4, 2009. 1 2 SECTION 2.2 PAGE 38 1.00000 1.75000 1.37500 1.75000 1.37500 1.56250 1.37500 1.46875 1.37500 1.42188 1.37500 1.39844 1.37500 1.38672 1.37500 1.38086 1.37793 1.38086 1.37793 1.37939 and the estimate is 1 . 37866....
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