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211-2006&amp;2007-3-M20-July2007

# 211-2006&amp;2007-3-M20-July2007 - l KUWAIT UNIVERSITY...

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Unformatted text preview: / l KUWAIT UNIVERSITY Department of Mathematics 84: Computer Science Math 211 (Calculus III) ' _ July 19, 2007 Second Examination Duration: 90 minutes 7% 1. Find and sketch the domain of ﬁx, 3;) = ln(z2 + 23: + yg) + sin—1 (12). [2 points] ‘ :r: 2. Find each of the following limits or show it does not exist. [2 points each] ./ . cos(m + y) _ 1 l —-—-. Q; (a) (mil—{Rom \$2 + 92 ® (b) lim cos{\$2 + y?) — 1. (whom 3:2 + .92 ® 3. (a) Use the deﬁnition to ﬁnd fxﬁ, 0), where [2 points] ms, y) = saw "a— wee: 5°ng + sing 810 6111 @ (‘0) Use the Chain Rule to find — and _.... where - [2 points] - ' 69: By . .w : tr + ue"; t: xyz, r = 5223,12, u : (cyz2 and v = :cyz . . . . . . 62 62 . . . . . 6/) (c) Use imphmt differentiation to ﬁnd a and E where z = z(ac,y) is imphmtly deﬁned by the equation 363W -— 4:322 = 2 — a: cosy. [2 points] 4. Use differentials to approx1mate __ 5V 31.84. [2 paints] 14398) @5. Let f(a:,y) = \$2 + y2 — 4m —- 2y. (3 (3.) Find the directional derivative of f at P(—l, 2) in the direction from P to Qﬂ, 3). [2 points] (99 (b) Find all points at which the direction of fastest cheng of f is i +j. [2 points] 6) 6. Show that'if f(m,y) = Sings + sing and u is a unit vector, then [Duﬂnyﬂ S x/2. [1 point] 7. Let f(3:,y) = I3 + ya + 3:1:2 + 3y2. @ (3) Find and classify the critical points of f. [3 points] 3 (‘0) Find the extreme values of f subject to the constraint 3:2 + y2 = 4 using Lagrange Multipliers method. [3 points] 0W .9 Q _? 1 Q1 (”Pk-21'1”! >0 i7(7(+1)4{lth)-Z‘>l J _..l f— Elf-1"“ _— M? (Q 1. (0k ) {906fo X 421‘!) 9 ': 1/ 1n;- “X":fuwiwi l, 1170 ' '1. who zu Paﬂ’l (9: x In} 435 x . K -7 c, Hm...9.;_1_ In; 0‘: .w‘ t '3” N' P _', -? 'ZX’). Xspd 33:31}; ‘- chﬂméilml 13993Y(¢Zf£9«0)—__I “grillforsfdamgw) Uﬁmgﬁ IIIII i 777777 '1. Deb-End; upm (9 dead- "hd( OXUS+ 2 Y ‘7. (47319460) b? m. 45'}? 1" W I '— g- : lib-1A. 2- S If _ :E 0 ‘t D {Fujﬂ‘3ﬂ“5{f"72+t4?) C9 3.. I“ ‘ ___ {‘ W .___.. aw? F;HM_.__ 5) Y9! f_f,o)__ 1‘13“; £(HRF 0) § ) '§‘(f,°)il-E: Q" - ._. " - ‘9 ' -_ \$10,») - 51L; ﬂiﬁxwme '3‘“? 55:; r..; J‘:‘_...__;éhw— c: a .3 ll . :3” (JP . 4% GP, L‘bdkﬁﬂﬂgjﬂ— D WN—‘ﬂi-.f¢ﬁ§)_:€ : — H w 6’ z —6 MI. W M‘__ﬂ_ \ ._ - __ w, 3W LA" w V W ”ax 31' 95*31 91:?” but 3" ”Eu 3;; 7-‘13+_t2wi+elil+uéyz 1351;: ~ 9W 2 Jr WW4 Z J—LJQV en+xg§> “ltamg W V 3 1 r1012 ’L 31%;]: Q__ ;,¥1Q+_ty 2+9 114:0?»{2— Via—{73(9) 52-4. {2 nzaltﬂjEQ 9th '0") H h______4__7_ .. “.11 -- ""”""' ' "'—-—- -‘ 7;.0. v ............L,,,‘___* C) F51” ;) :_ hull—\‘XQTSH 1. ,. 9:; ; ”J35 : ” C:23_:{:§i%_1§”“” ) )23A17L1E32C31111gﬁxéut9 ) 97L (‘5?— 3 '1") 6:1; 812 7‘1 .52 3;? £19; an; [)1 (1L4 5:13:72: ‘7“ —2x»gx9+5 5: q,% 2.: 3 -¥{-m at 49M 20M (Hum (‘9 32) #9 190132);- “5.2; -0, 1 f7 1 *7‘1971 Jon (9,32) - f * — 9'33“ ) 57(11 2?): :w-an): _q._.%(’xr¢()~%m (—37%) “a? . p i . re _:—- 3‘} S- “y(3“i\$;3!-SH> : F ‘01"31; "‘99)” M; _L(-_ (C) CH- D°Q§+ 1 .43“; i, 72.: . -_——- #—"”;_—'<')'W'.';-) WW". . ”A?”wu————-— um“ ,- @9a)55~~(1) '"dW-"m‘mg’”“3“??? “‘éi‘t? J f131¥““Li_>l-0xa{ Pr—G fa: :33 )/_'_€~_.)4{P;Q ). myf_ ‘ -~ —fl .__. — — G thud @4051. E«@_;_. { Anni; Li, 5) W19: (21-A)l+('m 2)) :{(.})) :5 \$1.1, f :3 vac—2%: . mm M... a . Nagy; ' re: ~ 940”"? éormmcjep ( wet} i x :2 - _y.. 1 :9 L a" S ’ é j_xﬁ'AQ€p{fza{€a/\¢j;_xf JMH “37s., Hr}. ' m; ’3 UVNJ CO '3: @£ (77f— éﬁxl-{nég‘d}, 14:.Uihii-U?3 [33941: [37165:]! :HWfIIMésa 5 “W? I MINT 37 Cr ID“ {15: i®52K+CafEUJ é KI :F ,..,_i...._.—-..-nA—-—w Q (13104595391 “fulﬁéerG } PﬁvQ-ho ' .f9':-‘3'9”'+-e 9 1‘9“? "' 6‘9“» I91 : 538561.70 :3 5'X('>L+2.):o 5-) )z:o/._2 ‘5“) G‘jfo 2) 3‘7(‘9<2).o *5 “9:0,.-1. CY gcaj [Qaln‘IQ war {00) (o *3)( 2 o )( 3"?) {9 0/ M PDIHI 09.] LQfaJﬂ’lrnnmc‘ I—1‘271' 1’! IDOI‘A'I 6*] [ocogj- n1CIrYIYV"\ (195?):sz ( ’2/0)€w4 ﬂu cUQPPomI‘s "g f 7L :‘U u’: U ”x t ‘j 7 '1 9:11 :52 "2xl:‘1 7 1 9x: 5 :2 A3,) )(U_'Q) 2 _D ( 7o) PX ”—1 "’7 I f / J If"? 172)“? "T19 , _ f‘" 7131.) ._."...3‘ , f; if 35. I: 3‘32 . (of?) (2,0,) I10) (mfg) . .7.“ ' _ ...
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211-2006&amp;2007-3-M20-July2007 - l KUWAIT UNIVERSITY...

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