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Exam2 sol - MATH 4334 EXAM II — 1 For f 7— cos(7r:c/2...

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Unformatted text preview: MATH 4334 EXAM II — Nov 10, 2011 _,\_. 1. - For f 7— cos(7r:c/2) use quadratic Lagrange interpolation based on the nodes :30 = 0, \$1 = 1, and 352 = 2 to ﬁnd approximate value of f (2/ 5). ( NOTE: Algebraic simpliﬁcation is not necessary.) FC d L ' I, Q) {a}: f) :Q') Ill/5.119} “le; @(fris—i )L‘Uocx— .1 - (-1 f u- 0 6% (meme) u (Lumps L 2 (XAQCFXI) g Mk? é ,Lf 1/2332“ W3) 1’ I 1/1 _;L ' . L, “04-1) Mos—I) : 06 pp?) ,i-gz/ — If ' ' ‘ ' = ln(m) using the nodes at :30 = 1, 2:1 = 2. The quadratic Lagrange approx1matlon of f ' 2 and 5122 = 3 is P205) = (:1: # 1)(3 -— SE) ln(2) + — 1)(3: — 2) ln(3). Etirnate the error made if we use P2(2.5) as the valéle 03f 1n?) at\$=?5.5 )( S a) Z” ll/ _ - [an -X :4" _, 5165-4) 2* 4’ 2‘ “7,. Lc? Egan/42253355.» a; , ( )6-I ,‘ 'I i (--'g ’5) W 5 .. 6%(éwmu 4 ,l’J—E @WVf-g/é?» a; W . Lin/F. 3. By reducing the problem to the least—squares best method, ﬁnd the constants A and B such that the function y m 7 is a least—squares best fit to the points (33,31) : A\$2+B (—13), (1,7) and (2,1). 4. a. Find the Fourier series representation of the function _ —1 for ~7r<cc<0 ﬂﬂ— 1 for 0<m<7r b. Write a Matlab code that makes uses of what you have found in part (a) to calculate the partial sums 33(53) and graphs it for —7r < mm 5 7r, Where mm = "Tr + mh, m21,2,...,100 and h = 21r/100. I - IT I 0 ﬂ « _. z...— 5‘ 91 ' vb... ' - E ¥6X)hfz!d =9 (info) 0, VJ: Ma” x‘lﬂ‘” n" Mde-t—g—‘J @91de 2.. T?" F 0 - z“ E:- ‘ --- {’1 n.— __ ' by.“ I' be}? If!)me JX: “Eer/mth)o,gﬁlwmﬁl 0.474“) I] 44:sz "- ﬁrm): E (l-(-l}1))&;n 41X 3 ’71:! A?“ 5. Cretique the following arguments and make corrections if needed. Taylor’s formula can be used to get the following representations: 1 2 3 f0? + h) = 1%) + hf (33) + hng’TrC) + %3f‘3)(c), f(\$ * h) m ff-T) — hf’lxl + %f”(\$) * %f(3)(0)- Adding above quantities results in: ﬁx + h) + f(:1: — h) = 2f(:c) + hgf”(:r), which can be solved to obtain an exact formula PM}: f (5“er (213)” (13—h). 1722C 05PM Mojweenmfw’f" szi ¥C\$ih):£(*l j memo 11.3.4? (Ci) MM. ‘f t .; cx+kl+~mw=zﬂll+ I . _ L 7 I ’7. WJ+OCh2),WOCK)=ﬁ-q”k 0K .FLX):I. ...
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