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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 hireminute 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 twominute 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): 104 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
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,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’
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(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“
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[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}
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.(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 straightline 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—DQlsj 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
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Problem 5.(15 pts total) Consider the following deﬁnite integral Ixexdx : L f I )
0 a A.(10 pts)Use the ManesLegendre! formula to compute an estimate of this integral. Keep at least four decimal places in your calculations. 77“? 16
L1 ”—E “S‘GLO‘QBl (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
altsz I36") 1—1111] + '° Jétlekll 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 satrm]
..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?) /
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30.3) 30.0 + dawnq 3'04 2)at x = 0.4 ...
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