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Unformatted text preview: Exercises UF Calculus Set 29 1. The graph of a function f ( x ) is shown below. (a) We can see that there is a root between x = 2 and x = 4 . Draw the linearization at x 1 = 4 and label x 2 as would correspond to the first iteration of Newton’s method. Then repeat the process once more, drawing the linearization at your x 2 and labeling x 3 . (b) What seems to happen if you apply Newton’s method with x 1 = 3 or x 1 = 4 to find the negative root? 2. Suppose you are attempting to find a single root to a differentiable function with Newton’s method. At some step in the process you have x n = 1 . 125 and the next approximation turns out to be x n +1 = 1 . 13358 (a) What is the midpoint x of these two approximations? (b) Suppose f ( x n ) and f ( x n +1 ) have opposite sign. What is the guaranteed accuracy of approximating the root by x ? 3. For each function below, demonstrate that the function has a root on [ 1 , 2 ] . Choosing x 1 = 1 , find the exact value of the third approximation...
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 Spring '08
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 Calculus, Geometry, ObjectOriented Programming, Derivative, Decimal, Continuous function, 17 inches

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