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Unformatted text preview: wtm369 Homework 12 Helleloid (58250) 1 This printout should have 16 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Use Newtons method to estimate the solu tion to x 3 + 2 x 13 = 0 starting with the initial guess x = 2 and applying one iteration. 1. estimate = 9 5 2. estimate = 15 8 3. estimate = 17 8 4. estimate = 29 14 correct 5. estimate = 27 14 6. estimate = 11 5 Explanation: If x n is one estimate of a solution to the equation f ( x ) = 0, then Newtons method says that x n +1 = x n f ( x n ) f ( x n ) will usually be a better estimate. But when f ( x ) = x 3 + 2 x 13 . then f ( x ) = 3 x 2 + 2 , so Newtons method gives the iteration for mula x n +1 = x n x 3 n + 2 x n 13 3 x 2 n + 2 . Consequently, with an initial guess of x = 2, x 1 = 2 2 3 + 2 2 13 3 2 2 + 2 = 29 14 . 002 10.0 points A differentiable function f has the proper ties (i) f ( x ) < 0 for x < x , (ii) f ( x ) > 0 for x > x , (iii) f (5) = 5 , f (5) = 7. Use the tangent line at (5 , 5) to the graph of f to compute an approximate value for the value of x . 1. x 31 7 2. x 32 7 3. x 4 4. x 29 7 5. x 30 7 correct Explanation: The tangent line at (5 , 5) to the graph of f is given by y 5 = f (5)( x 5) , i . e ., y = 7 x 30 . The basic idea underlying Newtons Method is that the xintercept of this tangent line pro vides an approximate value for the xintercept of the graph of f . Since the graph of f crosses the xaxis at x = x , x 30 7 . 003 10.0 points Use Newtons method to compute the value of 24 . 04 correct to three decimal places. wtm369 Homework 12 Helleloid (58250) 2 Correct answer: 4 . 90306. Explanation: Given a function f , the idea underlying Newtons method is that in general the se quence { x n } of values defined recursively by x n +1 = x n f ( x n ) f ( x n ) satisfies f ( x ) = f parenleftBig lim n x n parenrightBig = 0; in other words, x = lim n x n is root of f . Thus x n is an approximation to the root x . Now, when f ( x ) = x 2 24 . 04, x n +1 = x n x 2 n 24 . 04 2 x n . Beginning with x 1 = 4 . 8, we obtain recur sively x 2 = 47 . 08 9 . 6 = 4 . 8 + 1 9 . 6 , x 3 = 4 . 8 + 31 301 , and so on. Since the approximation has to be correct to the first three decimal places, we continue computing x n until the digit in the third decimal place does not change from x n to x n +1 . Consequntly, 24 . 04 4 . 90 . 004 10.0 points What will be the result of applying New tons method successively often to the func tion f whose graph is 1 2 3 4 5 6 7 x y when the initial guess is x = 7....
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This note was uploaded on 12/04/2009 for the course M 408k taught by Professor Schultz during the Fall '08 term at University of Texas at Austin.
 Fall '08
 schultz
 Calculus

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