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(MATH231)[2007](f)midterm~ma_yxf^_10486.pdf
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Remark: Use 5-digit rounding arithmetic in all calculations.
[points]
1. Verify that the bisection method may be applied to .nd a root of x3 + 14x . 6 = 0 in [0, 2]; and carry out 3 iterations of the bisection method. 15 points
2. Let f(x) = .x3 . cos x.
(a) Use Newtons method to .nd p2 with p0 = .1; (b) Use the Secant method to .nd p3 with p0 = .1 and p1 = 0. 10 points 10 points
3. Given the following data:
i 0 1 2 3 4
xi 0 2 1 -1 4
yi 0.11 -0.21 -2.1 1.2
(a) Find the Lagrange basis functions L4,i(x), i = 0, 1, 2, 3, 4 such that .i = 0, 1, 2, 3, 4, and j = 0, 1, 2, 3, 4, L4,i(xi) = 1, and L4,i(xj ) = 0, if i .= j; 15 points
(b) Construct the Lagrange interpolating polynomial P4(x) of degree at most 4 which interpolates the given data (xi, yi), i = 0, 1, 2, 3, 4; 10 points
(c) Use Newtons divided-di.erence formula to construct the interpolating polynomials of degree one, two, and three for the given data (xi, yi), i = 0, 1, 2, 3; 20 points
(d) Show that 2L4,1(x) + L4,2(x) . L4,3(x) + 4L4,4(x) x; 10 points
4. Assume that g C1[p . , p + ], where > 0; g(p) = p; and .1 g.(x) 1, .x (p . , p + ). Prove that g(x) [p . , p + ], .x [p . , p + ].
(Hint: Apply the technique used several times in class for proving the properties of the .xed point iteration and the convergence of Newtons method.) 10 points
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