2311_MG8_Lab_Newtons_Method_solutions_F11

2311_MG8_Lab_Newtons_Method_solutions_F11 - MAC2311 MGS...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAC2311 MGS (4.8) Name: 50 id?!“ 0 W5 0 Fall 2011 . Section: (circle yours) MWF 9:00 MWF 10:00 Directions: Use all available resources, including the math lab, Teague’s office hours, and fellow students. To justify answers and earn partial credit, show work in the spaces previded on this document. MGB in Teague’s unit, 13—230, by 3:00PM on Monday 11/7/11; The solutions will he posted shortly after the deadline, so no latepapers can be accepted. “Newton’s Method” 1. Consider f (x) z x3 — 4x2 “1. The graph is shown below, in part b). a) Starting with initial guess x0 = 3 , use CONVERGE to find the first two Newton iterations. Record the results below without round—off. x1: [0. 322318 b) When CONVERGE finds a Newton iteration, the graph window shows the tangent line that produced it. Sketch the tangent line at x0 2 3 , extending it until it intersects the X-axis. The x— x, =0 5.00 m value at this intersection point is the result of the first Newton iteration, so label it 36,. Repeat this process, sketching the tangent line for the second Newton iteration. Label it as 362 . 0) Continue asking CONVERGE for “One More” estimate until CONVERGE reaches the “Last Estimate”. Record the last estimate below. Last estimate = . 0(9 0(0 5” d) Repeat part a) using the calculator as follows: Enter the formula for f in Y1 , the formula for f ’ in Y2 , and return to the home screen. To get the first Newton iteration starting with x0 :3 , use . - He) the functlon notation feature of the calculator to author 3 — _ Y2 (3) second Newton iteration, use the 2““t ANS feature to author Ans — Y1 (Ans) / Y2 (Ans) , and press enter. Record answers without round-off. x1: ta.%33333333 x2 : 5.00azsa057 and press enter. To get the If you have just completed #1 with CONVERGE and want to proceed to #2 in the same CONVERGE session, click the radial button for “Apply Newton’s Method to a different function”, and click Type equation(s) yourself. This restarts CONVERGE’S Newton iteration process. 2. Consider f (x) z: x3 — 12x + 2. Launch the ‘Newton’s Method” feature of CONVERGE as shown in class, enter the function f(x) = x3 —— 12x + 2 , and use window dimensions x-min: ~8, x-rnax: 8, y—min: 7 20, y-max: 20. a) To estimate the root in the interval [3,4] , start at x0 =3 and give CONVERGE’S “Last estimate” . Record all digits given by CONVERGE. .. do NOT round off. ‘ Last estimate: 3. 37 25? b) If initial value x0 is too far from the desired root, Newton’s Method might not converge to the desired root. Use CONVERGE starting at .150 = 1.8 to get a “Last estimate”. Did it produce the largest of the three roots, the middle one, or the smallest one? .5 Malleai Gist-1 ! -3.- 5 417%! c) Newton’s Method has weaknesses. If any Newton iteration produces a horizontal tangent line, this tangent line won’t intercept the x-axis, so we simply don’t get the next Newton estimate. To H fixe) f’h‘e)’ see an example of this, start with x0 :2 2 and calculate the fist Newton iteration, x1 = 3:0 by hand in the space below. a. ' w, H2!) “in” 2» I ' ta) i “la . rim) 113% well X t Z ,3, 2112(2) +2 . \ to): X:le +2, yaw 3(2)?"-w l2” win-a. . melt-t X! f: m some over“ gr Nexuion‘s Melhoti. 2. Moria, "tarmac,ij ail X3 :52, , 3. Perhaps the most intricate weakness of Newton’s Method is the “cycling” problem. If a particular Newton iteration turns out to be equal to a previous iteration, the process becomes an infinite loop that never converges to the desired root. Consider f (x) z x3 —' 6x2 + 7x + 2 . a) Using are = l, compute Newton Iterations x, , x2 , and x3 by hand in the spare below. 7‘ . “flexiéxzwwz Kelvfll: wit-as . I 5: (K) :1" Bxl’wtla +7 Answers: x1= 3 x2: i x3: 3 b) The graph of f is shown. Use a straight—edge to sketch what happened in part a). That is, sketch the tangent lines generated by initial value are :1 and the first three Newton iterations, .rl , x2 , and x3. Extend each tangent line until it intercepts the x—axis. This illustrates the “cycling” problem. c) CONVERGE was designed to overcome the horizontal tangent and cycling problems. From the image above it appears that f has a root at x = 2. Starting with initial guess x0 = 1, report CONVERGE’S last estimate, recording all digits shown by CONVERGE. Last estimate : Q- . 00000 4. Newton’s Method can be uSed to approximate square roots. To approximatex/E we take advantage of the fact J; is a root (or “zero”) of the function f(x) = x2 — It. For instance, to approximate V15 we enter the function f (x) = x2 ""15 into CONVERGE, start with a reasonably close initial guess for «[15 , and generate Newton iterations to the last estimate. a) Use CONVERGE starting at x0 = 4 to apprOXimate «HS . Last estimate = :5. g 7 22g 8 b) The ideas of part a) can be generalized to estimate 41/; because {I}; is a root for the function f (x) = x" — k . Use CONVERGE to estimate 3/? starting with x0 2 2 . In the spaces below write the function you are using in CONVERGE and the last estimate. f(x) = Last estimate : \*O‘ I 3 ...
View Full Document

This note was uploaded on 12/12/2011 for the course MAC 2311 taught by Professor Teague during the Spring '11 term at Santa Fe College.

Page1 / 4

2311_MG8_Lab_Newtons_Method_solutions_F11 - MAC2311 MGS...

This preview shows document pages 1 - 4. Sign up to view the full document.

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