Homework 12-solutions

# Homework 12-solutions - wtm369 Homework 12 Helleloid...

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Unformatted text preview: wtm369 Homework 12 Helleloid (58250) 1 This print-out should have 16 questions. Multiple-choice 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 ) &lt; 0 for x &lt; x , (ii) f ( x ) &gt; 0 for x &gt; 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 x-intercept of this tangent line pro- vides an approximate value for the x-intercept of the graph of f . Since the graph of f crosses the x-axis 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.

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Homework 12-solutions - wtm369 Homework 12 Helleloid...

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