practicefinalexam_solutions

practicefinalexam_solutions - MATH 210: CALCULUS 3 E. Kim:...

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Unformatted text preview: MATH 210: CALCULUS 3 E. Kim: Summer 2006 Session 1 PRACTICE FINAL NAME ........ ..................................... .. ID. NUMBER ................................................. .. With my signature, I certify that the work on this examination is wholly my own, and that: (1) I have not received any assistance from others, (2) I have not assisted anyone else, and (3) I have not used any reference material [books, notes, etc] or tool (calculator, slide rule, pager, cell phone, laptop, etc.] not specifically authorized for this examination. SIGNATURE ................................................. .. o All cell phones and pagers must be off during the test. 0 Read directions carefully. 0 You must show your work to earn credit: Even if you do not fully solve a problem, relevant work leading to a successful solution earns partial credit. A correct answer with no work shown can earn zero creditll SHOW ALL OF YOUR WORK. 0 Cross out irrelevant scratch work. Ensure that your answers are both thorough and clear: unclear answers are also liable for reduced credit! 0 Do not begin until told to do so. In fairness to other students, you must return this exam at 9:40. Page numbers are printed at the bottom. INITIALS ............. .. 1. [10 pts.] What is the area of the triangle With vertices A = (—3,8)7 B = (4, 2), .—. _' _. v I andC' ( O: 57:52: ‘5‘.§+§/{+6’>: <+l}+/L/) A a: ?=<9~-£1+6>=<+7/+X> .3, ., u ( J A E7 any}: <+2fi/¢/0>x(+q/4—K)0> 2/11 174 0 i") 44’ 0 : f/‘W 0 9(+Z 0/+E)+1 w/ RNA 0% Vamllglol’fi‘“: llfixall‘ oifg’um’y— 1H0” . l ,, -V 7. Sn (M00 0? f/mnglq, 1 1-H0 —- 55' 2a. [5 pts.] What is the derivative of the vector function r(t) = (cos(3t), t, sin(4t)) ?’C+)= <—33}n 3t/ 1} 4 C05 4t> C 2b. [5 pts.] State the Angle Formula for the angle 6 between two vectors u and v. .9 J- v W a "N M 2c. [5 pts.] Compute the cross product u x v if u = 1+ 3j + 4k and v : 2i — 3j+1k. fi=<tm7 17:<?—,'§A> Cos c. I? l2 ,5 r, l M ->I3 ' 3H :l\'3\\'3\gl\+L1»Z 1’34, .—3 aw : (gufi? 'jh’g) + IZ('3’6> : lg: +73 -61: INITIALS ............. .. 3. [15 pts. each] Find the gradient V f for each function below: f(a:,y, z) : Zacy + yzzg + x10 — z 7px: 2g + ’qu 7Q= (ix + .2723 7&2: 37121 "( + )OXQJ 2X+£yzzj 3y121" f(:v, y, z) : cos(sin y) + sin(cosx) + 3323;222 2— " +02X 7-27— 74 2 0 + C03{c11§><)- gm: x) + afllez : (C03(wx))( Sm X> 7 7g: 'SMKS/hj)’%(5/}If/) +O+ Qxijal: »Cosabz‘5M(5/"J) +02X2yzz W = <-(smx)(mzm xD‘Jrixlej r (cos Wm (w m +ng a: 15%? f(a:, y, z) = siny 00830 + xyesy INITIALS ............. .. 4. [5 pts.] What is the length of the vector w = (W, \/—, —80>? mm 5. [10 pts.] Let u : (1,1./1> and V = (17 —27 3). There are two vectors of length one that are perpendicular to both u and v. What are the two vectors? 2 3’ 1’ A, i \ »\=rl.‘12.\~j'.é +1?“ Ll=52~ozj—3k=<s—2,-3> l '7. 3 . <§rglg> ,‘S {wwwbw +0 [,2 owl \7 (0W03mal leMTW'PM>) ém; not, 07f no/m 0/11,. 30 5 5: :2. —s’ 2— 3 > <fi/W/‘EE WA) W2€EM§2§ 6a. [15 ptsl In which direction should you travel for the greatest increase in the output of -5 ‘9: "v (A w(:1:, y, 2) = mg + 33/22 if you start at (—1,—1,9)? Vow/2): <3M+323 €72? VW(’l,'l)5l)= (A) ~\ + aw ) 59%)) W fl mldl 17ml la d/NCHM 6b. [15 pts.] If y is a function of a: determined by the equation cos(:L‘ — y) = my, what is 93? dz {50%} (03/x*g) » xcg ¢ 0 INITIALS ............. .. 7a. [10 pts.] If f is the function 1 Hm What is the domain of f ? Is it open or closed? Bounded or unbounded? “(widths #34120 _ 00mm is Mascot and Wham/Jed. f(93,y7 2) = 7b. [10 pts.] If h is the function 1 W What is the domain of h? Is it open or closed? Bounded or unbounded? We, 2/: Z) = 1&3?de l; 1419 +g7 flaw/A I} OfP/I and M/Ié’OM/Ydfl‘il- 2 7c. [5 pts.] What is the range of 9(56, y) : (y — sinx) Squaw? dorm a 1H ‘fb alums la; >/0 QAgd’Oml £20 I wom- ij g: 1%,,- "Why $X10. I So I’d/"pa In; a” NWIS7’O 2 7d. [15 pts.] Draw ve level sets1 of the function 9(33, 3/) = (y — sin 3:) 1You are permitted to draw each level set separately. INITIALS ............. .. 8. [10 pts.] Does the series 00 , 1 E TLSlD — TL n21 converge or diverge? Give reason(s) for your answer. (Hint: A successful solution uses one of: the ratio test, the direct comparison test, or the Nth term test.) . A I ‘ y‘- ' ..L-§V"" ' ,‘ S/Aln ~ ’07 ' l ' v n w r J. Altar-Jr. . cesamwoswr A n ‘ (i 9. [25 pts.] Let 2 = 95, a: = set. and y = 1+ 36“. (Write out the Chain Rule formula you use and a dependency diagram.) 2- .Z \3 :éUseatheaChaian R: to find %. .r - 2' 1: ,L J— r ’ I\ a' 552*2‘95 (0(5) + (wtleme/‘Jfl ‘ b. Use the Chain Rule to find %. are +< zero.» SM 10. [10 pts.] Use the Chain Rule to find % if z = :0 ln(:z:+2y), a: : sint, and y 2 cost. Write out the Chain Rule formula you use and a dependency diagram. 2 dz dad; aid /\ X‘Xaflaj'i Xx Z3 :Qv‘fl + mimosa + (m; 4345;. t) t INITIALS ............. .. 11. [15 pts.] Let f(:r, y, 2) = ~29:+cos(yz) +6WZ. What is the directional derivative of f at P0 : (1,4, —1) when traveling from P0 towards (3, 37 ~1)? 5 . X 2 . v a V4; <QQ+ gzgxy j —‘ZS‘/fl/y2)+r)(2€ y j {fly/fl v > WWH>=< 4—46“, WM * 68 4am «0+ 4e"’> 0°=<3“/3“//"-('D>= (1,-1,0> IIBH=Wsfi a: <42?) :7L;) 0) (amt—m 62-4/6ng?) + (Sin6'v'ei‘9("%) + O 12. [8 pts.] Describe (in wOrds) the level surfaces of the function w(:1:7 y, z) = m + 3g + 52 712% CW, Maw OJ/ flO/ml Vector <l413/5’> 13. [10 pts.] Does the series 00 Z lnn 71:2 n converge or diverge‘.7 Give reason(s) for your answer. (Hint: One successful solution uses one of: the Nth term test, the integral test, or the root test.) few I 4% mm WK =: _,_ ‘01:: : (@1le for flu 9~+C r? +C INITIALS .......... 14a. [10 pts.] What are the parametric equations of the line segment joining (2, 0, 8) and (—1,3,0)? 4’ V: <’-/‘;/ 2‘0) O’J)>: <“3/3/—lf> P= (2,022) ><= :2. —3«t 7‘ 0 +3t§ Oété' E ‘ do ~J’t 14b. [5 pts.: What is the length of the line segment joining (I, 1, 3) and (4, O, —5)? 74-402? ((—0)l + (3+5)? 15. [20 pm] Find the local minimum and maximum values and saddle points of Ma y) = (23? - x2)(2y — 1/2) / Illa): (240(le 7PM): (amaze—a) ' 0:(2-2a>(2a»r) 01(2X’XQ)(2’23) ()fifllvgflvMfle—j} 0: ><(o2' KLZ/l-y) (r1930) Xfl (0%“) M 7:1 Xzo V Fl or f" (\, 15 - ' ) - ,, m 2, a L £02200) 4(- 2{2y/y) 4y: ‘x/er) 74%: (1’30U'Zy) I (0);) 17(14): VKZ‘M‘WK‘XI) ” [34/072,ng (2,2) BMW ‘/ >0 MW *2 <0 /oc‘z/mmum flea): ~le <0 Sada/4e 9/2»); -Ié <0 gag/(e, Wm) 46 < 0 Saddle, mp): -/é <0 Sela/Xe. INITIALS ............. .. 16. [20 pts.] Find the maximum and minimum values of f (as, y, z) = xyz, subject to the constraint 2:2 + 21/2 + 322 = 6 W= <32, v2, Xg> (960w); A, ’ V9: <1” 49/6z7 \ ‘VF=)\VSL IVIUA3 y%:lx/\ xzz4}/\ ngézx XgE‘QXQ'A {WE‘LLle X'j%=éia)\ // mom Gm, @ (1674) 3%) ('13-, m4 [.fir/bé) and mm M @ ('fiF'I’TVE) V CE; l; ' ’ (:52 ’/%Z') I l}- 17. [20 pts.] Find the local maximum and minimum values and saddle points of the function f (cc, y) = 9 — 2x + 43/ — x2 — 43/2. What is the global maximum and global minimum? _2 _2X 7%: ¥_g 3%) ’2’2K30 4d?!ij x=’l y= é God/040W (vb £4819; 49"? 2 61%!) ~ 0‘ : 9>0 fix<o g0 (4)11) is «local mafimum 3% $5 3514qu mm! Vow mdfiifi. 1mm A49 no Wary 9 (.411; {54-h glow maximum UMX no MI’I’L ...
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practicefinalexam_solutions - MATH 210: CALCULUS 3 E. Kim:...

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