Homework 12-solutions

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ) < 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 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....
View Full Document

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.

Page1 / 9

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online