MAC2313_Test2_Spring2008_Solutions

MAC2313_Test2_Spring2008_Solutions - Page 1 of 4 MAC 2313...

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Unformatted text preview: Page 1 of 4 MAC 2313 Name Test 2 (Happy Valentines Day) Date: February 14, 2008 Clearly show all work for full credit. 1. (8 points) Find r(t) if r’(t) = cos(t)i—sin(t)j+3t2k and r(0) = j+k . .—.. r65): SF’GE) do: <~Sih(fi)‘¢o§(tl)fi3> f C, do): < o} 50> + 53 = <Oilif‘7 2. (8 points) Find the length of the curve given by r(t) = 23in(t)i — 2 cos(t) j + Stk , where 0 S t S 3 . rim :Qw w), 98‘51’23 i g > =3 ir’ltll r JHQQM Moira +— 95“ :- J31? 3 L L: 3 Jamie : 3J5? O 3. (5 points each) Determine each of the following limits: (a)(xy1)i_rf(163)XyCOS(x—2y) 9(3).CQS (o) .: IS) . W4 (b) (x.y1)1—13(10.0) x2 + y8 0 : O we 4:, 1/ flaw? (X a" «(.30 5" Y .. i a" “if “1‘! ‘ “8’ W6 Z” isms 51 7.90 ‘ DNE‘ Page 2 of4 2 4. (5 points each) Suppose r(t) =<t ,%t3,t >. (a) Find the unit tangent vector T at the point (1,%,l). N f = i r:(t):4;¢, at?) l) :2» r’m: 451» 3, ‘7 (l { lr‘mi: JHWH =3 ’1 . Tm: " l = +3 <3,9u>=<%)%,é> J (b) Given that the unit normal vector at the point (l,%,l) is N =< —§,%,—% > ’ determine the binormal vector B. . . ‘ ' V L (1 k k 3 B 2 Tx N 9/3 33 3/3 8’3 “Y 7 "3’ “9’3 3'3 .. -a I ‘2 3 3 3 " fifi< Egg, 3) 5. (8 points) Find all first partial derivatives of the function f (x, y, z) = x2e”. Z)? ‘- 3181/2 H z. ’D'X “a? 2 “IE ’~ : e. 2.x! ’X Z 79?, 2 7? ai‘xr‘ie‘ 6. (8 points) A function u = u(xy) is a solution of Laplace’s equation if um + uyy = 0. Determine whether u = ln(x2 + y2 ) is a solution of Laplace’s equation. _ ‘ I _ 22% M" q 98+? (“A 73+»21 2 D. a .: QM“ : 91,:355 xx Hf”) (xq’rwl) 9% “v i: 78*? . Q a - (fix/lira» MM : 22%sz W - (KW/“3; Eras/l) um + am 1 (DvaflvflxL (9761-9?) .: O a 2 O CKRHPY CE”? .6. (A: Qt ( (78+ 4133 {5 cc $42,841er of éfiuqfiba. Page 3 of 4 7. (8 points) Given sin(xyz) =x+2 y+3z, use implicit differentiation to find 1 , , y TALL 232i "Heal-125, 7: CLS Cc {Maths/1 c‘fl‘y‘ 61w( 'x a; a; ézavzsi‘ént ' 29} v 3 ~ 0 r 22 ‘1’ 3 2‘2 605(X41a)‘(0(y as + X2 _ by m, 2331:; as (my = 22—- xz cosCWl rag Qw- Kz cosbvvfl 3‘1 ‘ xvmbcva '“ 3 8. (8 points) The dimensions of a rectangular cardboard box are measured to be 12 inches by 12 inches by 10 inches. The cardboard is 0.25 inches wide. Use differentials to estimate the amount of cardboard (in cubic inches) needed to make the box. V: Gm, 2 iii/7:7) (jg: it 1‘5“ , Y1; ~fllv: 4,12 Jr 1- XZDIY '1' 9‘7 0‘? zzro AZ 26’ 51v: Gallows) + Qataamw + gagging) = (go + (90 + 752 a: ma 9. (8 points) Use the attached table to find a linear approximation of the wind chill index when the actual temperature is -l4°C and the wind speed is 42 km/h. Lam : was i Q[43El(x-4\ % I; (QWVD use («,Lvtmo f. PMD '3 "'37 . «3H3? : Ave,an -L _ I- {'(‘Lflkflo‘l'POEM) 595- ‘ 5» L2 “a gt (a; ‘ get: A: “3:231 : L {;(alg):l‘3 Ml u 5 H -r- Pafiin-WPc-rmv) A * : -oz2' mm \t 4)L\ "‘~ h LL55 we Ark” «alarm I; ~04 mm: ’0'“; 1T5" £(~ILI)‘~Q\§: LGHfiQB: ~97 + new is) + (~o.t-§)(‘~l3"’-Ib) 1.». ——:2(, Page 4 of 4 10. (8 points) If u = x3y +yzz3 , where x : rse’, y = rsze", and z = rzssin(t), find the valueofa—uwhenr=2,s=l,andt=0. O as 1X: z 34 v: 22 z: 0 29: i 95’ i: 9“ 9‘5 if"? 53 '92 25 7g ‘ v =(3«3~)~(re¢)+ an mf>~Camé¢> + <3v235‘CF we} ’3& : (an (:23 1+ (9)00 + (0') Lo) : 30 11. (8 points) Find the directional derivative of f (x, y, z) = x2 + y2 + 22 at the point P(0,4,3) 1n the dlrectlon from P to the origin. 0 : (OI 0/ 0.) (<5 i .3 v‘>'~ ‘ ~_ __~ “3“ . Ni “.3 T: ‘ uh ~\) M ‘ t: 5.) K gfihq‘z) “I < 91, 3'1) '9?7 ":5 $€(OJH,3) ': <0) 8)é:§ ._ A . a 5» . ~> ' ' Du (019,133 r (05%33 ° LA 7:: (2 ~ %(S\-- $10 = —]0 12. (8 points) Find the maximum rate of change of f (x, y) = xe” + ye” at the point (0,0) and the direction in which it occurs. _.) ~ ~. 74 VHWA: < 83*- ve“) “Key/1L e x > “:5 $470M: < I) s > ‘3 Aakl‘flufl“ (“ml-e of fi/jUi/lie, (LS i VP(O)O)$ 2 [a (la Amie“ 0f {fflom v: <m7 (La 2%) ...
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MAC2313_Test2_Spring2008_Solutions - Page 1 of 4 MAC 2313...

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