Key_Test_1B

# Key_Test_1B - Closed book, no notes, calculators OK. Show...

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Unformatted text preview: Closed book, no notes, calculators OK. Show the whole work. There are 10 problems each 10 pts for total 100 pts. The bonus problem is extra 5 pts (30 you can score 105 % ill). 1. Find the point in which the line with parametric equations a; = 2 — t, y m 1 + 3t, 2 2 6t intersects the plane 23; w y + z — 5 m0. Is the line perpendicular to the plane ? «be ’- eer) +6€~95>0 :7 eat- #3.? 7‘61"; ’"0 at W WWW/M rx‘rgaﬂsrwz waltz No7" L 74 “Ha Mae l 2. Find the domain of the function f (32,31) 2 M332 —— y. Sketch the domain and the level curves corresponding to the values 1/2,1,2 of f(2:,1). For the sketches introduce all necessary notations. 9mm \$69,).- 25770} I fﬁﬁmmwf i. ‘-» ' N 3. Find an equation of the plane that passes through the line of interseéi?”dm_gf .' 3‘ + y "i" Z =3 2 and a: - 31+ 2 m 2 and passes through the point P(—1,2,2)_ "an? >< M ’% 2-:2. 935.1qu on, ‘l‘f‘e rub—(Mm 6% ﬁe Wakes “’3 “5'”; i=2 {:0 ‘120 i=4 Kn} 3:0 4. A curve in R3 is given by the vector function r(t) m (4 cos t,33in t,2), O S t < 2%. Find the curvature at the points P1(4,0,2) and P2(2\/§,3/2,2). WWW "m [\$41913 ' 5. A particle starts at the origin with initial velocity (1, 2, 3). Its acceleration is a(t) 2: (t2, 1, t). Here t is the time. Find its position as a. vector function r(t). 1%): 23%) “inﬂow? jag/[10 .1: 5%) Wm VW 970).: <0, w, 0; WW v.3”? . .5 (5 + 4 W+)= Wm— jﬁrocl5= (7,273>+46%2 Mojave») 0 - _ a @- x<¢,2,3; + (gig—52%?) W9 .. I . 2- W) —-—- (Mgisﬂzré) 5+é+ ) *7 75 l 3 ‘ 6. Find the limit, if it exists, or show that the limit does not exist (fully justify your answer) (3 + \$02 1i . (am—ﬁes) as? + 2 (2‘10) hfﬁﬂ)=é‘m ﬁxgmﬂ 5-1 KW x—w V I £4102 @w W) Am [£9,935 2"” x70 2. (2x) 7:- 53 2. XXL 7. Find the partial derivatives fm, fy, fm, fyy of the functionﬂm, y) m in #212 ~+ 212 in R2 except the point (0, 0). 7%}: {i { ﬁH/i J 5 @2772) f: i 2”; .~.-.. 5 a 7” : xz+yzéxfzx52 «ﬁt-9% 6% xz-yz XH xx 62%).,” 62W); 8. Suppose that over certain region in [£3 the electrical potential V is given by the expression V(:u,y, z) L: 51‘? — Bray + :cyz. (8.) Find the rate of change of the potential at the point P(3,4, 1) in the direction of the vector v := i + j w» 2k. (b) In which direction does V change most rapidly at P ? What is the maximum rate of change at P i? W :Dgi/{Qéﬂx <22f6272> . < 4, 0—95,: m7}; (22——e-24):~§— 9. Given the surface deﬁned by the equation 9:2 + 312 — z2 —» 2233} + 432 — 4 m 0. 'E ‘f’fx’y/ .3.) (8.) Find an equation of the tangent plane at the point P(1, 1, 2) (‘0) Find a parametric equation of the normal line at the point P(1, 1, 2). [Va-rm {194% WW (MW; VFKEvai gxxszyMzr ﬁC1,1,L)=8 as 2w amw 5:: *22«+4;< Fgf M2 1):" O 10. Find the local maximum and minimum values of the function f(m,y) m 332 + y2 + 12223; + 8, _ deﬁned for all (22,91) 6 122. 9%: Zﬂjmz 2y+xi~19 9:9 Myzo jz—If x252 W W [0; 0) KNEE” We (ﬁr/I) CY) AM WW w M)?» "M MKS/e 725% la W m 12* law 2 5M, 3470 ﬂy: ﬁx/OIDJ-“HQ o i :DC—l/ifl) _._— 0,02” gala)":- wg z: D {gel/#1) WW6 8 ...
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## Key_Test_1B - Closed book, no notes, calculators OK. Show...

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