EXAM 1 ECE 2331 FALL 2010

EXAM 1 ECE 2331 FALL 2010 - Page I of 7 DO NOT OPEN THIS...

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Unformatted text preview: Page I of 7 DO NOT OPEN THIS BOOKLET UNTIL YOUAREINSTRUCTED TO DO SO ECE 2331 ~ Exam 1 — Fall 2010 I agree to abide by the provisions of the University of Houston Academic Honesty Policy while taking this exam. Signature: flat M222&Z INSTRUCTIONS: (1) Read and sign the Academic Honesty statement above. Unsigned exams will not be graded. (2) Including this cover page, this exam has 7 pages. Verify this now and raise your hand if you have a problem. Do NOT look at the questions! (3) This exam is closed book and closed notes except for the reference sheet we will give you and One page (two sides) of notes which must be handed in with your exam. (4)Communication devices of any kind (eg. cell phone, pager, etc.) may not be used during the exam. Turn them off or give them to the room monitor. No such devices should be visible on your desk. (5)The use of a calculator is required, but all steps must be shown. Unless otherwise indicated, round your results to 4 significant digits. (6)You will have 90 minutes for this exam. (7)When you finish the exam, unless the hire-minute warning has been given, put your crib sheet inside the exam, hand in both the reference sheet and your exam, and leave the room. (8)0nce the two-minute warning is given, stay seated until time is called, at which point you must stop writing immediately and stand up. Continuing to work after time is called will result in substantial penalty. Problem 1. l O /15 pts Problem 4 \Li /20pts Problem 2. 17" /25pts Problem 5. \fl /15pts Problem 3. “i f5pts Problem 6. \ ‘3 /20pts TOTAL SCORE 95'7“ /100pts possible I \ £09; 33“ 6 PageZof? Problem 1.( lSpt total ) Consider the function f (x) = x Error) + 2. For n > 2 its n- -th derivative; 15 given by fini(x):(—1)"gr_wl«2)t We): 1-04 0|" ,— ’Zlfi IrQr) Fit”: x ”3):: '3 “33333 F z)=-— 3—: A. ( 7pts) Find the Taylor series expansion centered at x= 3, including terms so that 13% truncation error is Firm 0014)- "p H 14‘ 01511:: O f 13-” 4"" 46111”: 1" ”C If” Féjt-fit ” 3"? m ,_ "2.2ng F” cw— Hangs)" TU)” (:01) +W+ 333 2 _ 1:” .qu0 To) «- 5. 3% + F3. on; ‘th— a): Lot—3) + 761—3) 4— M (:53) 11.3: 9.101“ ).ou[§-BJ+ .lGLGbr-BJA- .0t‘i'a'0’333‘l 514053613 ,3 B. ( 8pts) Based on the remainder expression, find an upper bound on the error in using this expansion to approximateflx) 111 the interval 2 75 < x < 3. 25. in. on! 5 MM //,>. 5’ 1R“)! 5 .0 3H . hue-Bl I10 (0mg), 5 . 0337s . .00047“ (90 Page 3 of 7 Profile!!! 2.( 25pt total )Consider the equation x — e" = l: A.(8 pts)Use the bisection method to solve the equation on the interval ] s x s 3. Do four iterations and circle your final approximation. M : 7:01.34- L“ [r and m x. ear.) W ‘9'" ".3B‘T‘l® 1,450“) l mat-m 1-3. A?“ w: —.oscs' LET ~37” [a -.oa; 137"”?‘3 B (5 pts)How many iterations of bisection would be required for the solution to be accurate to four decimals? Use the error bound, NOT the “true“ solution. ._ t} . S‘ X t 0 ,lLl—a 4 crraf at am "f 09210 A 4 m '3'"! M 3 ”£26 ,sxm'” .23/34 "mlfil [ /\ Page 4 of 7 .(lZpts) Solve the same equation (x—e"‘ 1) w1th simple iteration (also called successive substitution or Picard iteration), beginning at x = 1. Do three iterations and Wm Verify whether your (MW’V’MF. ' iteration scheme meets the converlgence criterion for this method. &’ __ -x__ ....[7 '8' +12% [lift Ykléi Yrs W U .367qéf'Ye5 g 1: to) en Problem 3. ( 5pts total ) In the list of root- -finding methods shown below, circle those that are W on using, a straight-line approximation to the function whose root is sought. false position secant "l l . fa 4\ PageSof’l Problem 4. (20 pts total). Consider the function fix): ln(x) and the three sample points x— = l .0 1. 4 and 1.8, (X 00 “)‘5’ .33L§+ é GEN)" 916;.) aka/“'09) + 0'! ’)Q-‘(‘ Ix) +08 0(343) PM = 43.103 Lx—DQ-l-sj Jc 1.3370 ~0é~~0 / B.(10pts)Using the correct upper bound on polynomial interpolation error, determine the number of equispaced points in the interval [1 .0,1 .8] that are needed to guarantee that the interpolation polynomial 13.00 based on the points satisfies the condition jp..(x)-f(x)l< 0.01 at every point x in the interval 15 x s 1.8. L“ .7 41 I) 7, ,. 2 "—fi .fi 1"“ MK)?’ #64 I k "1 i '2 Y ,, . 2 0 ‘ 35.40 ‘30” M3 , (net) "I“ ("I “”3 fivL‘ c. 'l . 8 “WWW“ Z 37,77“ a ry m; ' o ltll and (I: . If g g. 555": 7“ EV) ': 3):; m“; /—————-————-—"’ ad'- Lf an (14/ [m J» 3 Tu"! r320 ( 5—3 ”501 1a 9‘) ' :“fi q .._._‘ l i i” L l\ Page 60f? ‘ I ‘ L .. q L h 3 . — q S (“'— 0' ) + _._) Problem 5.(15 pts total) Consider the following definite integral Ixexdx : L f- I )- 0 a A.(10 pts)Use the Manes-Legendre! formula to compute an estimate of this integral. Keep at least four decimal places in your calculations. 77“? 16 L1 ”—E- “S‘GLO‘QB-l (7. sets ((2? 3 Fl S'S‘SC PUP/40] 15 “aw”! ”W"? 4’ Jaw“? ; In: 3.662 B.(5pts)If we used three rows of the Romberg table instead, at what values of x would we need to evaluate the integrand? (do NOT actually do it, just tell me the values of x to use!) 0 0 l3 Cfldt’o‘fn‘bl‘ (' D .-: (k2!) [FLO +980] Maggy—5,9, n , 4;: ta D+Q°—q)l3é+‘~fl alts-z I36") 1—1111] + '° Jétlekl-l PG 13%)] oust: '1. at 60 Md X MM 5* 3.1931 / Page 7 of 7 Q Problem 6(20 pts total) The remaining questions refer to the tabulated function below: x O 0.] 0.2 0.4 0.6 0.3 1.0 f(x) 0 0.2 aw 9x231 (3X. '1 A 3: I: . 3 1 A. ( llpts )Use Simpson's 1/3 rule to obtain the best possible approximation to I f (x)dx Linear!“ I) + {3):} + .. 5130) + “C‘Q‘D‘m‘cflyfigg sat-rm] .-.3-[o+.r.4+.4] +1331: “Hat?“ ”gr-"fl +:[n (Fifi. 8f? / B.(6pts )Find the best 0(h2) approximation to the first derivative of f(x) at the following points. Use only the given data. 4' ‘l :0) 1)atx=0: -' .1 /’2_ 91.4,] fl:2_3l’%[email protected] 14146:] C (3ptd)Find the [email protected] to [email protected] K?!) at X: 0.8. Use only the given data. Hts) mafia ‘04— EU?) / W" 30.3)- 30.0 + dawnq 3'04 2)at x = 0.4 ...
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