thq_07 - Use T.C(First Name(Last Name 511/0326 W m M ble...

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Unformatted text preview: Use T.C. (First Name) (Last Name) 511/0326 W m M ble Calculus 2016—03~03 (Thu) McCary 1varia O. A07/005/01 Q 165 MATH 2451 thquY Mult a.” O 3 meeeeeee @ a . ®[email protected]©@[email protected]é 5 @[email protected]@weeeeeee“weave we me @mfiaenfieea a a a u a a o o I. [email protected] C ® me @ © my fl. @ @ me © ® © Q ® ® ® ® my ® ® ® @ may we 5 @ ma @ @ aw. 0 me @ ® @ Q ® © @ ® ® a...» ® 3 @ me Q Q mm aw ® ® @ “mm/e ® mu m ‘ .\ [email protected]©®@@®@®®@®®@@®®©®@©®®®®®®®®®®®®®®®n .r>@@@@@@@@@@@@@®ee©@@MEQnw®®®®®®®®®®@fl [email protected]@e‘eeeeeeeengage? @e @eeeeeami Net ID Exam Course Due Date Instructor mew @ e 3 n...“ O 2016-02-24 09:38 O. AO7/005/02 Q 1. (10 points) In lecture, we discussed Newton’s method, which is a way to solve a nonlinear equation f = iteration. For a. function with signature R —> R, the underlying method is: fern) In+1 = 5511 _ f,($n) The primary idea is to start with an initial guess $0 and then iterate until (hopefully) the method converges. Your calculator (and computer) uses the following approximation to compute square roots. To compute x/E, your machine uses Newton’s method with $60 = 1 to solve the equation f = $2 — b = 0. A little bit of algebra yields the following, which is known as the divide and average formula. / ‘x (79/ 1 Lil-a =58 _m%—b__ 322+?) m l b mn+1 — $11 _ n 2&1 H 2m“ 5 ($11 + A little bit of work can show that no “hope” is necessary in this case: the divide and average formula converges1 and the rate at which it converges is incredible. Use the divide and average formula to fill out the following tables. N ewton’s for V10 Newton’s for x/fi l Grader Only if 5r“ O. AOT/OOS/OL Q 7 .0 points) Computers are machines which very quickly perform arithmetic: +, —, and *. Division is more complicated. One strategy computers take is when a, division 0/1; is requested, the computer first computes 1/b and then multiplies by a. To evaluate 1/1), many computers solve f I: 1/9: — b = 0 with Newton’s method. (a) Use Newton’s method to determine an update formula (analogous to the divide and average formula) which uses only +: —, and *. The name for this formula. is the Newton—Raphson division, and you can easily look it up to check your answer. However, you should try to figure it out before looking it up! k, * (“-2 7' ‘- 7' 79A "‘ 1P i _- I ‘ 7' 74,“ ’ _ ZECJ' ’ i _‘:' ..l 1" .-. _' l/ __ I. l '-' +7.? 4 L .— ' r_ I a “a my. - An A x. (b) Use the Newton-Raphson formula to fill out the following tables. You’ll find that the value of $0 is quite important (thus the “hope”)! 3 ! Newton 5 for 1/ 13.8 Newton 5 for 1/133 (Mug? "J R J E'Hm’w’sf ’ 3/, l, Grader Only J, 0 Ge 0 O) O. AOT/OOS/O4 Q 3. (10 points) Let flx) = 233 + 3.1: +1. (5.) Show f(:c) is invertible by showing f is monotone. ig :a-\,;:f_,i , 77‘ ‘ 3, _ '- , ' l ‘ r J.‘ V N __ - /.- I -.__ ’— __ " us“ ix _‘I ,‘ ’d 27- _;, -- i h; / (j) V ,*< _; ’fl; g 7" 7' . .‘..7| ‘7 .‘ .; «j _ '-.. r . w ‘ ; ‘ . (13) Even though f is invertible, that doesn’t mean it is easy to compute f ‘10)) for a specific 5: no easy analytic formula exists! In general this must be solved numerically, and Newton’s method is one of the most common techniques. Observe: f‘1(b)=w => b=f(sc) => f(z)—b=0 So setting 9(3) = fins] — b, we can use New’mn‘s method on g to compute f‘1(b). In the same manner as the previous two exercises, find an update formula. (c) Use your update formula to fill out the following tables. You can check to make sure you’re on the right track by plugging your 2:9 into f. Newton’s for f'1(7] Newton‘s for f‘1(13.8) J. Grader Only l . O® @@®@@®@®®®® . O. A07/005/05 . -0 points) Newton’s method also applies to functions with signature R” —> R”. In this case, to solve f = l3, you start with an initial guess £0 and then iterate ifn+1 = in _ [D And in the same manner as the previous exercise, Newton’s method can be used to compute the inverse fimction when {V5 2 Nu the function is invertible. Let f 2: (I + 3'1”)? 3; my — 45: 5 (a) Use Newton‘s method to compute f ‘1 (6 ). Your answer must be accurate to 4: places past the decimal. 5 (b) Use Newton’s method to compute f’1 (1 ). Your answer must be accurate to 4 places past the decimal. This can be done by hand, but by all means, use a symbolic calculator. To encourage you to use a. symbolic calculator: you can use the free, web—based version of SAGE at the following URL: . Here is a script which solves the first part of this exercise: . t Grader Only J, 0 . \.../‘ \_, my a, Q. AO7/005/06 0 . GL0 f’r A TY?;'%'N;£F;1E; MA; mfg Wk; [Irv irwxjwfifg [1 Emir/{Np 1- 5. (10 points) For each function and point: determine if the inverse flmction theorem guarantees a local inverse. E. your 72 ‘fl’ 1? ’Mé \?&rft‘¢/h1v€ \fi' INV€’3’Z['[EJLL Jr; is » $29 THé mm?ng i; Loififl lwvéri'méLE 1: 1 / (a)F()= —2:n at(). y 2 1 y r m 1': 4:] pr lzxx/ ’>< ; ‘2. 0 f _ 9 :2 M, P = Wig [F f x ' / ‘ k 1 ‘3.’ fig —Tvz O " “\JVC’f’Jé: T'rr/ [Hf/"n (711.4511 5A1wTC3t'g A 91;}; grin/873:5 same, ‘ ° Wé WWW/Hm; le’ [wvész‘rr'tgz/é” $ (0) PC - f L; A~ J1 :7 |Nvérd§é I’M “IF-1m .QLL/anfiwfléé‘; A Lac/51,. ifiiVEZrzifi/é . A4ij [2L r: ' ivéwE/ip mvazég THé/ PémV-m-Wg £9 (:7. 1‘3“ JaVbfigi-g"it~:§.Lé . J, Grader0n1y¢ . O® @@®@@®©®@@® . .. AOT/OOS/OT . .0 points) Let E“ = R2 — {5} denote the punctured plane. Does f : E“ —> R2 have a local inverse at every point? 7P? —‘> iP—j' we) P f/ '1. _- 1 I; M7 - we :1" ./’ 7 {* "/1 Vekpvw 7mg ._..J%?(72 —z+fl;7i V/ ,7 , aw 1" y M f f 0 fit: _-\\/ ngxr 3" f f i=3.) FIT'QKAA. 3—- *5! 7 rain k 7 x f. a, . .I' It" [f5- ET‘<;a;._.,.;,a| j‘ .- H: 16 éw‘.;éz;.:;fl;al 7 OF... NOT INK/Giggmafgi THC agrbfinig THé’PflfTEid-Tiflvi’; U065: W57!“ Hfi‘VE: 1!: L031! if"; “I; 1i .3 INT; ‘ I / i Grader Only J. O o This page won’t be graded- O. AOT/OOS/OS . ...
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