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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,
$1 = 1, and 352 = 2 to ﬁnd approximate value of f (2/ 5).
( NOTE: Algebraic simpliﬁcation is not necessary.) FC d L ' I, Q) {a}: f) :Q') Ill/5.119} “le; @(fris—i )L‘Uocx— .1  (1 f u
0 6% (meme) u (Lumps L 2 (XAQCFXI) g Mk? é ,Lf 1/2332“ W3) 1’ I 1/1 _;L
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pp?) ,igz/ — If ' ' ‘ ' = ln(m) using the nodes at :30 = 1, 2:1 =
2. The quadratic Lagrange approx1matlon of f '
2 and 5122 = 3 is P205) = (:1: # 1)(3 — SE) ln(2) + — 1)(3: — 2) ln(3). Etirnate the error made if we use P2(2.5) as the valéle 03f 1n?) at$=?5.5 )( S a) Z” ll/
_  [an X :4" _, 51654) 2* 4’ 2‘ “7,. Lc?
Egan/42253355.» a; , ( )6I
,‘ 'I i ('g ’5) W 5 .. 6%(éwmu
4 ,l’J—E @WVfg/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$2+B
(—13), (1,7) and (2,1). 4. a. Find the Fourier series representation of the function _ —1 for ~7r<cc<0
ﬂﬂ— 1 for 0<m<7r b. Write a Matlab code that makes uses of what you have found in part (a) to calculate the partial sums 33(53) and graphs it for —7r < mm 5 7r, Where mm = "Tr + mh,
m21,2,...,100 and h = 21r/100. I  IT I 0 ﬂ « _. z...— 5‘ 91 ' vb... '  E ¥6X)hfz!d =9 (info) 0, VJ: Ma” x‘lﬂ‘” n" Mdet—g—‘J @91de 2.. T?" F 0 
z“ E: ‘  {’1 n.— __ ' by.“ I'
be}? If!)me JX: “Eer/mth)o,gﬁlwmﬁl 0.474“) I]
44:sz " ﬁrm): E (l(l}1))&;n 41X 3
’71:! A?“ 5. Cretique the following arguments and make corrections if needed. Taylor’s formula can be used to get the following representations:
1 2 3
f0? + h) = 1%) + hf (33) + hng’TrC) + %3f‘3)(c),
f($ * h) m ffT) — 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)=ﬁq”k 0K .FLX):I. ...
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 Fall '11
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