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MAD 4401 Review for Test 1 James Keesling February 9, 2009 1 Bisection Method Problem 1.1. State the Intermediate Value Theorem . Problem 1.2. Using the Intermediate ValueTheorem, explain how the Bisection Method works. Problem 1.3. Determine a solution to cos ( x ) = x using the Bisection Method. Problem 1.4. Determine the real roots of the polynomial p ( x ) = x 10 - 25 · x 5 + 10 · x - 15 using the Bisection Method. Problem 1.5. Solve the equation x - cos( x ) = 0 using the Bisection Method. Problem 1.6. Show that if p ( x ) is an odd-degree polynomial, then there is a real number x 0 such that p ( x 0 ) = 0 . Problem 1.7. Find a real root of p ( x ) = x 11 + 2 · x 2 - 20 · x + 30 using the Bisection Method. 2 Newton’s Method Problem 2.1. State the Mean Value Theorem . Problem 2.2. Suppose that z is such that f ( z ) = z . Let > 0 and suppose that x 0 [ z - , z + ] with x 0 6 = z . Suppose also that | f ( x ) | < 1 for all x [ z - , z + ] . Show that | f ( x 0 - z | < | x 0 - z | . [ Use the Mean Value Theorem. ] Problem 2.3. Approximate 6 5 using Newton’s Method.

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