m251test104

m251test104 - MATH 251 Test I October 1 2004 There are a...

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Unformatted text preview: MATH 251 Test I October 1, 2004 There are a total of 4 problems. No calculators are allowed. 1 _______ Name 2 Section 502 (MWF 10:20—11:10) 3. 507 (TR 11:10—12:25) 4. Total __ 1. Given two lines " L1: $=—1-3t, y=2—t, z=5+t, L22 $=3+$, 3/228, Z:-—3+48,} 8,t€R7 (a) show that L1 and L2 are skew lines; (10%) (b) compute the distance between them. (15%) (a) First, not, W» Of A! ‘42,: web) W410.) (—3) -'-l, (1,2,4), Wt, m 0mm «'6 z. WA 1 W . wt —[~;t:3+§ (x Z) ; ~2—6t: é+z$ 1-1:: ZS -) 2-t’:7,} '+-;+=4 => -¢,—t=lo =2) f=-—7_ =’-> s: ~l-3t~3= “64:2, W557 s: 2,)t=~z 4m 3:“ t=~3+4s, we 06%; 3+(-L)="3++(L) ‘3? 5=§, “- Xv 41ml 1-; 4M Ala-W hm. (lam-5,194 a mm W E? ‘7‘ T: A l‘3 "l I :(‘4-'7-)i +(12,+I)T+ (—é+1)’]::_é€+nf_§i 'é (“+l)+!301'l)r§(}-g): o I. P l 3:: +3” $5 1“? Hé “15:7 A '— lVl 0K (310)—3, M A; W £52114 f‘vfi'IoL/nm): E ~(o(><~3)+:3 («1—0)°-$(3'+3)= 0, \z-) - ~éx H33 v53=—18+I§:~3. The. 014mm W A. M494; he and WWW 75» Hm». d’ (,3m’7l It) .. __________,___ __,.___ flee)» +13”+(—§)7’ ,J a} o a a- 2. Let n be a. positive integer greater than 2, and 2—11, f($1,$2,~-a$n) : ($%+$g+“'+$i)T- (a) Show that f satisfies the Laplace equation 62f 62f (92f —0, for (xlyx27-Hamn) 82f(x17$27 ‘ "7xn) Evaluate W. (a) i=l,1,~~',""- 34;_I Z-m 7, 7. 7. 29;.“ 4 ’3 L 1. xi- Q?) (math + +79.) 3;; (m. +---+—xn) = CZ") (m’vr ~ - - +76)? 2x{ ’1 ——‘ ‘ - -—- «mule-wax ..._._| 3X; __."—..‘ ._£_ . z :(z—n) [Uh-"Md? 1—- nxf' (x."+~-- Hm.) ’ ‘ 'L 1,’ n 7— 9 9 2‘il:L+-~'+ i : 4 '2 l 3%: :22; 97‘: M \ -.:L—' 42-“): L 1:. . n '1 " 1 -%-I = _ f- :"i. n X; (‘x."+.--+x..1 (2.. (x + > +x ) ) ) ~ I - “'2': ' L ‘— “12.4.” ; (HM n-(x.‘+--~+x§ —n- (“I +""’"~) ii; = (ii-n) («Hf-MD"? +9113)?“ Lx,"+-..J,x})1 . “J 9") Y! I— .——~ ’5 2. L 1 ii; : (I‘M) 7‘. (Och-u +792) 7' - 7334‘“ Ha +-'~+Xw) "" 2767,, :-(2—n) n 9cm, (991+ v +79%) 2. 3. (a) Let flay) = muggy If (my) 75 (0,0)- Show that lim f (2:, 3;) does not exist. (12%) ($,y)~>(0,0) (b) Let f( >— ‘W 'f (a: >7é (0 0) $3 y — + I 7y 7 ' Prove that lim 3:, =0. (27y)—'(0:0)f( y) (You must check the definition for a limit rigorously by an (5,6)—argument.) (13%) ( a) Aim7 d4 \jwc) we hm L 1. e~ ’96: ‘ {x09 ;L; ~5x‘x x—>o +0 ) if; XL+3x+ x90 x"([4—’§x”) = —$‘X _ ~$=0 ’ x90 1+3x" _ 113.0 ‘0. z _- 1‘7— _ + 5— b (1', >2: ‘ M :_ ~ ‘5 :4 jaw + ‘1 ‘1 '1-90 C1311“??? Tao U (Hg) 7+0 4" 6" A“ &%"L€I;) {‘meawbflmO allmfft+ww)wcm M97 4/?0’ iULCX'V‘Dk 6/ 51L $<J (x—o)‘+(j_oy—<$ .. X 1' - 7.37, _l \ "flix mm: [J42 _ W“ M <1 l (“VI/j"- : _}|x[£ <»}$:9/} "ELI, “moi/«946%, 4. (a) Let a > 0 be a given constant. Find the tangent plane to the surface 2 = 0,5102 + y2 at the point (1,1,a + 1). What is the name of the surface given in part (a)? (b) Use differentials to compute an approximation value for with 4 decimal place accuracy. (a) 313— : lax '—‘ 24' 3" MW :3 727%:- x:{,‘1'—f; 2L) x:! :2” ‘1“ +087): 3 %:é(éx1+4mxjt 3. X°:[o) ‘36: g ______.__z +([o)l>:i)éao+q.n;\l 1000 ——I0_ 5 24 [9 _ __fl ’1‘” 3m are (8%) (2%) (15%) ...
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m251test104 - MATH 251 Test I October 1 2004 There are a...

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