# exam_1_a - 1. (20 pts) Suppose you know the following...

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Unformatted text preview: 1. (20 pts) Suppose you know the following information about f f0) = —2 f(4) = 8 f(5) =12 (a) Is there a root for this function within the interval [1,5]? @or No? (b) Can there be more than one root within the interval [1,5]? r No? (C) Is it possible to have a root in the interval [4,5]? 1' No? ((1) Consider the interval [1,4]. Estimate the root using bisection. IM-cw-l CIIH] G.‘|'lD m== T. =Z._>" (9) Use f(4) and f (5) to estimate a. root using the seam: method. 2. (20 pts) Use Newton—Raphson to derive the following iterative expression for esti— mating the cube mat of a. 1 535+; ‘: pri + Use this expression to estimate the cube root of 11 with an initial guess 3:0 =1. Find x1, x2 and 5133. Estimate relative error from \$2 and \$3. What is the absolute error in ﬁx; _' #‘pcxcl pCXJ: X3, Q :30 H' .Hxa Sol-HM X=VZ XL)” ‘1 POW-3)!" CH: x. — MT 31‘; UL __ 1. C q . =iL x~'xc+7:1]' 3 [ZXf/X‘] X02! xi: é-[z-r—f-L] =42333 ._ I ” riqul xi- 31 M's-12H mg .--—L 713.0 9 + ” 2:2.W-UJ— X3 3 i 8 0 (3.1334013 Exact magnetr- 3 H 2 2.27.70 AbsaL‘t fire/s 2.2240-2-9wr) y/Do)o 2.22Ho i 9.3190 ‘QaLH‘t em" \ 14415—307“ ) x/oo’h 8.4‘HJ' '3: 20-5, 70 .. (20 pts) Estimate a solution for the root, f(:r) = 0, of the equation below by the following approaches: 2 ( f(m)=exp(—x)*w F15):-C¥y(-X}'2X (3) Use Newton-Raphson. Use an initial guess of 30:0.25 and ﬁnd the ﬁrst two iterations: 1:1 and 3:2. 32 _ - ._ -?~ exﬂ—xU-ixa XI : 0_7_J‘ f. 0M 20.8“), XL: 0. 7077 (3)) Approximate the function exp(—m) by using a Taylor series expansion up to powers of 2:2, 323., exp(—x) = a0 + 0,132 + 323:2. Now solve the problem analytically. ear/(“‘0 3’ z—v+—§_x¢ . m L:- ¥cn = crpd—zd—Y‘: l-x—tﬂ O xZ+Lx —L=° (uni—=3 X:Gi\ )(.:\B—|—_=c>.732 4. (15 pts] Expand the following functions around a: = 0 up to and including \$2: (a) a: exp(—:c) (b) 1 i I (c) l-n(1 + :22) F(")3 £60) + \$29)}: 4-i- C'Ya) xat ~-——- G13 3? =x awn—x) limp-=0 F's- C¥ﬂ(—-XJ* xawt—N duel-x).- Q—XF(—‘p) + x exyl—x) Var—w ll 13" 5. (25 pts) The volume, V, of liquid in a sphemlca! tank is given by [312 u h] 3 V: we? where h is the depth of the liquid and R is the tank radius. If R=3 In and V230 1113, ﬁnd h. Use Newton—Raphael} with an initial guesa for ho corresponding to the tank being half full. Perform two iterations, i.e., ﬁnd hl and hg. Neal h SJIVc. ﬁl‘ZEDQ-lﬁ] _v-___,° .3 3 2—9! -L3 +3Ia I,“ — 3.3.":30 I”): _ L +32L. 1T 71 1 R=3 ~r=3o {TWP-3'“ “Eh 00 L3: 911: +13. (.523 H e w L 3+ (3)L-q(3)’~1—Z8.f-J‘3 ____ W_____________ I any—+180) lqa 7- 2.06l1m hi: 2,0‘3‘2.+ ...
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## This note was uploaded on 04/18/2008 for the course CHE 348 taught by Professor Chelikowsky during the Spring '08 term at University of Texas at Austin.

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exam_1_a - 1. (20 pts) Suppose you know the following...

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