211-2008&amp;2009-1-M20-December2008

# 211-2008&amp;amp;2009-1-M20-December2008 - Kuwait...

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Unformatted text preview: Kuwait University Math 211 December 22, 2008 Math. 85 Comp. Sci. Dept. Second Exam Time: 90 minutes Calculators and mobile phones are NOT allowed. 1. (3 pts.) Describe and sketch the domain of 3+3; E’y f(\$:y) = 2. (3 pts.) If possible, ﬁnd a constant a which makes —q——g—Ec““\$‘:;°5 if (2;, y) # (0,0) 0e if (\$,y)= G'(5t:,y) = { continuous at (0,0). 3. (3 pts.) If 2 2 f(%, is dilferentiable, Show that Bz 62: __ m:0 \$B\$+y3y 4. (3-pts.) Find equations for the tangent plane and normal line to the surface \$2~2\$+2y2+322=5 at the point P“), 1,1). cos’1 my 5. (2+2 pts.) Let f(z,y) : W (3) Find man, y). (b) Use the deﬁnition of partial derivative to ﬁnd fy(0, 1.). 6. (1+2+1 pts.) Let f(.4c,_y,z) = em“ + (a) Show that is differentiable at the point (2,1,0). (b) Find the linear approximation to f at (2,1, 0). (c) Use your linear approximation to approximate the value f(1.99, 1.01,‘—0.01). 7. (3+2 pts.) ILct K(:1:,y) = 3:3 + 332042 — 1). Find the critical points of K and classify them as local maximum, local minimum or saddle _ points. . (b) Find the absolute maximum and minimum of K over the region R={(\$ayl=03\$£2,wigyg1} OO 99 ‘ IL Hoﬁcew an Sand Enema 93/ '1! 200% 13\$ : {(31,311 xﬁ‘f'ho and X15811 L. m\m\auzx,un GL¢,1)=UM\ 9° :0. X —’>¢> X ---"0 1X1 (Mama 511.7" ,Umrx 6C7,'Lz') : Kim M _ x-bo K—bo 570' _-_-_ Um 13v“ 7.?- -%~=nx : F}— - L-K. K—Jpo m x ’0 ' 85 a -PO-‘re vulc) Kim GOV-#31 b-N-E . LViLF-E.‘%i£ ,‘ “(n—I; (9,0) we“: is no c‘naice cg a" meat:va _u.>1\\ mat/(e G. 5. ’2: qu,u\ wwe LL: {£5 3 \l: '3‘; r2 3- 2A _ 9}; 'BH ax v - d "‘ 'DX '- r-DW FOX + V 3:8); 'ELALJIKJS\*?V(;‘L) \3 E}. — 1: UK 1:, EV .._. -54 \— FDB __ % % K5 + %V TEX -- ?V in mm, x%— ﬁnsg; : QWKQJVKAFRK: +g(§_1=0. ‘JL ‘3 .. TL=GF'\ :<zx—1,u.‘j,e=t7\ :4—129J6> Cay”) Cabiai] —- x 4. “L‘s-«~11: :2 6 Homaﬂ Q‘me: x:o-':, \5: \+-2_\:,’1:.= “at, tel/K. -\ 5. Q\ ¥xtxfﬁ : “‘3 , x*\\$ .— 675 \ . kS\.—7€Iﬂ1’ 1A7; *3 7—4.“:5 -H—L b\ \$3K0,\\ : \‘mr'x. gm : &\ "*V“ "' \ R—qu w \m kﬁo R : UM -“LL\+\'\\L_ _\__ LVZ h—‘bo \ " d 1. 5'}; 30:31: 703?: 34-: byte? +\_ﬂ \ E3 :x’tc ﬂail , ?%= x3: . cu since. £315) 9:, 91 346%}:— omd‘ NLM. {rm aokrsw Lou-9. Geeks: QLJL,0\, 5? 1.4 dx'fgzrwiodok cit—2.150), b) Lc«)\a,%] : 3 + Can—L] -62. L‘d—\\ .u—lL'i- CH c) Qﬁl.qa}1.o\’_o.ol] E LLL.01°[)1.ol)—o.ol) :- 3+(__o.o\} -9. Laban +74 0.00: SLQEJ. OO 99 141 1.. chpﬁ : x3+7>3£(‘d1-\\- K1 :- 33LZ+3Kﬂ1—\\ L103 : 6X53. pct =0 mug“ =7 (o,—L—\\ amok {the} “‘05P: “K 1-? 1‘3 :0} our. ﬂag %:uf"(‘_¢1hCﬁ—O ?o:w—3Ts 08 K. Xxx : 6% Mars: 67C Wynn: 53. ©0439 _ 390-— ‘5le Cs-ncmHm Coy, cs) "5 a swat: 130:4. Lorne] is ax Walk gov»? Kk\,o\ 1.1; is :3 Quad? m'nnhvu/m LCL-\’o\-.L{- as Q Qami MK'IMVM OnL..: l/(L‘X,—\\ : 78’ Joex 3.7. wean; J V<(o,_\\: 0 I K(1,.—ﬂ:.8. OnL-L: Kai, “1'3 -: 8+ 60-312Q : 6ﬁ1+1f4§jzﬂl erA-c KC1,t\\:<B , K'Q‘LJO]:'2.. Uan: K0130 :56“), 64;er {toga-:0 ‘ may: , LC C '2”? = 8. OnL—u: Kﬂo)~a_n :o, -\é"j 1:1. We. QaCaJl M'\¢\'\rr\.us\ a): CHOW baman YD Vac. ;A\E.r.'o(" «:00 Q. Watg—yc‘ m dean}: Mxn'.mdl'¥‘\ is %k\,o):;.?. and Mia qbgakuJ—e mxxmum is will a: Q = 8 . OO 99 ...
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## This note was uploaded on 02/23/2010 for the course CAL C 0410211 taught by Professor Deparpment during the Spring '10 term at Kuwait University.

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211-2008&amp;amp;2009-1-M20-December2008 - Kuwait...

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