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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 Xaxis. 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 roundoff. 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 xmin: ~8, xrnax: 8, y—min: 7
20, ymax: 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 Gist1 ! 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 xaxis, 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.
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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‘ .
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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 ...
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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.
 Spring '11
 Teague
 Calculus

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