211-2005&2006-3-M20-July2006

211-2005&2006-3-M20-July2006 - Kuwait University...

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Unformatted text preview: Kuwait University Math. 211 July 17, 2006 Dept. of Math. & Comp. Sc. Second Midterm Exam Duration 90 Minutes Calculators and mobile phones are not allowed Answer the following questions. 1. (3 Points) Find and sketch the domain of the function 9 _ x2 _y2 fl”) = W‘ 2. (3 Points) Show that 2 3 lim 25‘ y 4 (mama) x +y does not exist. 3. (3 Points) If x3e?” —ysin(x—z) = 0, then find %, and 3—; 4. (4 Points) Use differentials to approximate 435.8 (3/ 64. l . 5. (4 Points) Letflx,y) = xey +ye‘. Find the directional derivative of fix, y) at the point (1,2) in the direction of the vector 6. (4 Points) Find the equations of the tangent plane and normal line to the surface 2 z 32-“ c0523: at the point (%,0,—1 7. (4 Points) Find the local extrema and saddle points; if any of the function fix,y) : 3x3 +y2 — 9x+4y+ 1. OO 99 Kuwait University Math. 211 July 17, 2006 Dept. of Math. 3: Comp. Sc. Second Midterm Exam Solution 1- D} ={{$,y)ER2:Q—xz—yZEOandac—2gé0} :{(m,y)ER2:$2+y239&$9E2} 2:232 611:2 3 2a3m8 2 3 2. limf(x,aa:2):lim(—)4= im——= a . z—rD 9:40 $3 + ((11-2) 1&0 .138 + 03438 1 + a4 The limit depends on a, so it does not exist. 3. Let f (:12, y, z) = 6:383"+z — gain (at; — z). m = 339834“ — ycos (a: —- z) , fy = .1936!” — sin (:1: u z), fz : $36?” + ycos (a: — z) 82 _ fan ('32; _ _fy 64* f, and 63,— E 4. f (m) = 4% 4%, f4 (4.4) = 64% 4%, 12, (4,4) = 44% 4% f (36,64) = 24 fr (36,64) = 701,- f, (36, 64) : g \/35.8 3/64. 2 24 +§(—0.2) +4011) = 24 — % + 8—10 = 523*; 2 23.946 5. Vf(m,y) :< fmfy >=< 6y +ye$,wey +63 > Vf(1,2) =< €2+2€,€2 + e > €=fimzfi <1,1>, D:f(112)KVf(L2)-W=%(2€+3) 6. f (as, y, z) = .929 cos 21' ~ 2, Vf (my, 2) =< —2€2y sin 233,262y (:05 2:6, —1 > Vf(%,0,—1):< U, —2, —1 > Eq.0ftangent plane —2(y—U)—(z+1)=00r 29+z+1=0 Eqs.0fnormalline ng, y=—-2t, z=—1—t; tER, or:c=%, y:2z+2 7. fm=9x2—9, f =2y+4 fry:0, ffl=18x, fyy=2 94:2 — 9 = 0, 21; + 4 = 0, (—1,—2), (1, —2) are the 0. Pts D (3:,y) = 363:, D(—1,—2) < 0, (—1,—2,2) is a S.Pt. D(1, —2) > 6, fyy (1, 42) > 0, f (1, —2) = —9 is a. L.Min. I OWN)“ ...
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