{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

211-2006&amp;2007-1-M20-Dec2006

# 211-2006&amp;2007-1-M20-Dec2006 - Kuwait University...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Kuwait University Math. 211 December 16, 2006 Dept. of Math. (E5 Comp. Sc. Second Midterm Exam Duration 90 Minutes Calculators and mobile phones are not allowed Answer the following questions. 1. (4 Points) Find and sketch the domain of the function 2. (a) (2 Points) Find the following limit, if it exists ‘ mgy—yiri:2+1 11m m (Icahn—1‘2) my 4— z + y +1 amends:2 +3;2 (b) (3 Points) Let f(;t._y) = E2+y2 0 if (say) = (0,0) Discuss the continuity of the function f at (O, 0) if (3;, y) 79 (0,0) 6‘2 82 3. (4 Points) Find 5: and a— if z = f (as, y) is deﬁned implicitly by the equation 3 y in (3:2 + y2) + sin (\$2) + cot (yz) + yzz = O 4. (4 Points) Find the linear approximation of the function 2 f(33:y\2)=‘ry2+\$ 1+2: at the point (2, —1, 3) and use it to approximate the number f (2.1, —D.98, 3.04) 5. (4 Points) Find the directional derivative of fin yizl = 2283:“? at the point (1, l, —l) in the direction of the vector 3’ = 2—1" + 7 + 2? 6. (4 Points) Find the local extrema and saddle points, if any, of the function f(m, y) = 3:3 + 3:133;2 — 3:9 00 CO Kuwait University Math. 211 December 16, 2006 Dept; of Math‘ 8; Comp. Sc. Second Midterm Exam Solution 1. D; :{(:c.ly)ER2:4—m2iy2>0‘ Iggy—+«xilglf111dy—f-xil7é0} : (x,y)ER2::L'2+y2§4,051%‘+I;<_‘2andy+x;él} DF 4‘ r'" . _T‘ { if?!) X14411” H\\ 2 J. Jim -—¥; H (ryH—Iz) r+1 (@113 Mir-19 11-1 . . (“1050) tan 1'" _ Law") ;(-""—*"”‘~ 2. h 11:11 31,1 : 1131—" 7 :CUSH 11m - :C053 '2 it"; ( ) (LmeM J) 30+ Hm r ﬂ) 'f'ﬁ-‘ﬂx Thus, ljm f(:::=y) does not exisist —-:> f is noL continuous at (0,0) I, f F? :“ (1 sebum) “-. 3‘ Let f(:r,y,z) = 111(32 + y?) + sin (322) 4— cat. (yz) + ygz : D W) “—~_.z f: (11,32) 2 2222-15-51? + zcos(xz) @ fy (3:,y,zj 1 ‘- 2?! , zcsc2 {93) + 2yz ' (IE? 6; f5: 62 f g __ ,2 2 k : __._ _ : __£ fz(z,y‘2) —ICOS(IZ) yﬁ‘ic (sz+y then ax f; and 6y 1; 4' f(\$:yuz) =3y2+5€2 1+2 f£21_'113) :10 f: (33,3112) : 3;? + 2xx/1+_z f: (2, *1, 3) Z 9 _. r“ :3 7133 fy (I 31:35) : 21y fy (2: _113) Z _4 Q‘N’ﬂ‘ x2 2 1 3 a 1 1 Z 2 :_ a :1 f (3 y 2} 2m f ( J The linearization off at (2, 71,3) is LURE/a3) = H2. 4,3) + f: (27 '1.3)(\$ — 2) + fy£2v_113)(y +1)+ 13(2) *113) (Z — 3) : 93: -— 43,: + z — 15 and the linear approximation of f at (2, --1,3) is ._ \$y2+I21+229x—4y+z—15 If) Now, L(2.1, —0.93,3.04) ; 9(2.1) — 4(—0198) + 3.04 715 : 10.86 Thus, f (2.1, 4198,3041) 2 10‘86 CO 5, f(rc,y,z) :z2eJV’E] f(1,l,—l) : e fr{:c,y,z) :ﬂﬂerv’i fI(1,1,71) 2e . 2 V‘: \‘l Isl-\$33113) : gf/gczﬁ fyUﬁIV'U 2%6 @ Jrz(l'7y,3) :Qzerﬁ fi(]w11_1) :r—Qe \ ‘ xx Wham) =< fIJsz > W(1,L-1J:< ﬁéerge > (/‘i P73" " 5‘. .n’" Cl u: _. 11c; _’ — : 7 ."”:__ ”a“ 3 2,1.2l>@ qu(l,1, 1) VHLL 1) u ge @ 5. I 2 3;:2 +35%! - 3 fl, 2 6:53; fW : 63; f” = 63! {W 2 51- ::l,0j, (Qil) are the C. Pts {a I Faetf: D(I,y):36(\$2—y2) D(U,il)<0 (LL—1,0), (0,1,0) 3z2+3y?—3:0 63y:0 ( 1 M'MH‘\ are 8.13135. CL? 1:} D(:tl,0)>0 fn(ﬁ1,0)<0 f(—1,o)=2 isaL.Max. (s PD fn(1,0)>0 f{1,0):k2 isaL.Min. ® OO 99 ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern