::::l<r/ L 11>(=ft\iir(~1J~:::: Fo. l. AG =(0,.2,L}> M  &c == < ) I) ~...::.1 ( \ I 0. /:rf!:,t f1c = <I _,, 2...


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1 Math 3298 Exam 1 NAME: SCORE: l. Given three points A(I, l, 1), B(l,;2, 3), C(2,  l, 2). (a) Find vectors AD, AC, nc. (b) Find AB+ DC, AB  AC, and 2AD. >,,. /:rf!:,t f1c = <I _,, 2...J I) ~..::. M  &c == < ) I)..) l. AG =(0,.2,L}> (c) Find length of AB and area of the parallelogram spanned by AB, AC. ~...::.1 ( \ I 0 (}:rev.. = I A ~ >' kc. = ( _ v 1 J , ::::l<r/ L 11>(=ft\iir(~1J~:::: Fo 2. Given two unit vectors u, v. The angle between them is 7r/4. What is u v? 3. Find the equation of a line passing point P(l, 1, 5) and parallel to another line whose equation is x = l + t, y = 3  St, z = 9 + 2t. Jv;( v = <. l / a/ r. v >
2 4. find equation of the plane which pa:;s through point P(2, 1, L) n.nd parallel to another plane x + 2y  z = o. ~ ~\ JJ< n ~(1,, 2) \> LI ( x  v) + z..{ 'd I ) I. ( i' ~1) ~ 5. Fin<l the point in which the line x = l  t, y = l + 3t, z = t intersects the plane 2x y + z = 2. (n tk. 1 \J.e~\qn_; 2(1t).._ (I 15 f:) t f ~ 1 ~ t== i. L+ l ~ 0 >< " ~  i poi..\. C:.L 1:;= Make a rough sketch of tlie quadratic surfaces, and give their names (i) z = 4x 2 + 9y 2. (ii) x2 + y2  z2 = 0. :/::, l / 7. ind a vector function r(t) = (x(t), y(t), z(t)) for the curve which is the intersection of surfaces 4 = x 2 + y 2 and z 2 = x 2 + y 2.
3 8. Suppose a =f. 0. If ax (b  c) = 0, can we claim that b = c? If Yes, prove it. If 110, give a.11 example to support your conclusion. U. A particle moves with position function r(t) = (cosl,sint,t). Find its velocity, speed, an<l acceleration. vb < = Sv\,,\,. ('.,oyt,, 1/ / / \ n x ~ 11 L /;) lta T ~ / ~ I? c.."' ~ k' v,_ "' t ~ I s~itj ~ l (sf (I ~ (: ~ :_~ ; ~ r; k ::: ( ~ f ;!;v l.;f ' 0 ) 11. particle starts at the origin with initial \I locity v(o) = (0, 0, 0). Its acceleration is a(t) = (2, 1, 0). Find the distance it travels from t = 0 to t = 1. Vu,= ) ' ~(h~(; ~ ) \>,I, o).iv~ ~ <!. = (lk, t ' o) + c::...:... ~ ~ S"vL...> \} Ci. ; :::. Ct> ; o ) ~ "=3> G :: ( o; li 1 o> ~ \J (:h ~z)/t ' {; / D> \/ lh ~ J~t)"'+t 1t l)~ ~ (? ~ I S:::: ( fr\;~\ :; Jitvl Jo v o I
4 L2. Fi11d the local max am! min values a11d :;addle poi11t::; of the f1111ctio11 J(.c, y) =.c 1 +xy+ y 1 1.i:. 2 { ' ~ *' u = ( 0, L3. (i) Find V' J for J(x, y, z) = :;in x ~ os ~ ez. (ii) Find the directional derivative of J along 72). (3) At point (rr, Jl', 0), find the direction which will yield the largest directional derivative of J c c :t; \7 f = (l<s\x Of\'(} e /  s. h.;< s,k ~ ~ ; Svtv:>< (,../) 'O.e, ) j) ~ J_?.L ~. 7(; IA f ::.. \7 f. la ::. r1))~ K (ra d  ~ f rv S ..ivx (fa. D e = it 6<5 j ('_ t(c.ri,x + <;,), x) "Tf, <f '" )J < ~ lig( ( t1) (I) ~, 0 ' 0 > ~ (I / D, 0 > j)j, r rul M"'XI ~" la = i1}~(if,of, o) = < \ 1 D, 0,) 14. Find absolute max and min off (x, y) = x + y  xy on the closed region D bounded by the triangle with vertices (0, 0),(0, 2),(4,0) '. ('. f \\, \f( D I ), =/ I, l ) ::c I + I  I ~
5 Math 3298 Exam 2 NAME: SCORE: 1. Evaluate the iterated integral r212y Jo Y x 2 ydxdy. 2. Sketch the region D of the double integral associated with the iterated integral in last problem. Then change of order of the iterated integral to dydx. Evaluate it. ~ ) ~ f: \. JA )74 (clyd# j) 0 f"i ~  "'f 1\ I\  4'. l r 1. ;:;  1
6 4. A plate lies on the region D bounded between the circles x 2 + y 2 = 11 and x 2 + y 2 = 4. The density is p(x, y) = h Find the mass of the plate. Hint: switch to polar coordinates. x +y L)\ 2)T ( z._ 2, Ry,/wo\,(9. 5. Find the volume of the solid under the surface z = 9  x 2  y 2, and over the rectangle D = [O, 2] x [O, 2]. ~ ~ \'J1<'1") JA ~ \, \. (1~~~')h ""7< j) l ~ \ 0 ( 9 '6  )<"'" f J' ) \ J.,.  )}c~:)  t)e  ~ 'l Ji } 6. Find the surface area of the paraboloid z = x 2 + y 2 between z = 1 and z = 4. _j) ll J_ ( I + ~ ~ 1) 2. \ i.
7 7. Given the following double integrals f 1 Jx xydydx + r.,/2 rx xydydx + J 2 11;../2 ~ 11 lo.,/2 lo Sketch all three regions of integration on one XY plane. { J 4 x 2 xydydx Use an easy way to calculate the integral in last problem. 9. Rewrite the integral x2 lo rly J(x, y, z)dzdydx as an iterated integral in the order dxdydz. Sketch the region first. /
8 10. Calculate the triple integral j j f 2 1 dv by converting it to spherical coordinates. }EX +y 2 +z 2 Here E is the region between sphere x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 4. \L: ~~ \ L t1. ( S 'k c\( Jc~ ctr C ~~ s, ~4  f \ ~ l~ ct~  f ~ ' If (;v) ~ \ cj1t ~ ~ ~ Yi\ ( l + I)  ~If
9 Math 32D8 Exam [[I NAME: SCORE: l. Compute the line in tegral 1 yz cos xcl.s, c C : :i; = t, y = 3 cost, z = :3 sin t, 0 ~ t ~ 1r. 2. Use Green's Theorem to evaluate k F cir = k J l + x:lclx + 2xycly where C is the triangle with vertices (0, 0), (1, 0), an<l (1, 3) with positive orientation.
10 :.l. Find a parametric equation for the curve!j = sin x starting from (0, 0) and ending at (7r, 0). Specify the parameter region. '' Find a parametric equation for the surface which is the pnrt of the sphere x 2 + y 2 + z 2 = 4 and above the cone z = j x 2 + y 2. Specify the parameter region. x ~ 2 (' 0 \} $ " f d ~ 0~4) S Vf c 9 / q ) Ej) ~ [ G) vrr) X ( d, l j t: ~ ~~1 5. Given a parametric surface ) z_ 6L= tt ~ :C :::fil r ~ ~ 1 ~\ == ~ r (u,v) = (ucosv,usinv,sinu), 'U E [Jr, Jr], v E [0, 27rj. Find its Cartesian equation 'l '}, , \ v '){ +J = t.t ~ \J.+\j\ "'\!\ \) 6. Given a vector field F = (p(x, y, z), Q(x, y, z), R(x, y, z)). Define curlf and divf. > 9rz w JL.d.R ~ _ ll> c(a,'< \ f :::: ~  oc ) 0 c  ())I). / 'O )\ 60 t
11 7. Suppose that yon know that J(:c, y) = e.i: +.1: 2 y'2 a nd 'V f = F = (c'" + '2.c.1/, 2.c 2 y). Eval11ate the li11c integral ( F dr where C: is r(l) = (l +sin rct, '2t +cos rct), O :::; t :::; 1. le L~Jy " f(yl1l]f(ylo\) =:,f ( f) I)  f ( o 1 1) 8. Evaluate the surface integral j h F ds, where F(x, y, z) = (xz,  2y, 3x), S: r(u,v) = (2s in ucosv,2s in usi ~ v,2cosu) au<l 0 :::; u ~ 71', and 0 :::; v :::; 27!'. ) J ~~r.js s F. (j If i '(.i:) J.A  ~) <4»lu. C,.,,U."" I/ I  (j. S.\.I<_". v' G ~ 'kvl "''">. v
12 F = (l11y 1'2x!i,3. i:~.1/ 1:!.:) y is conservati\'p. If so, fi11cl the potential fu11ctio 11 f (:i:, y) such that F = 'V f. P 1 ~ df'.l f L :::: {V~ t ))('0 ~= V\ t b '/.. J ..) u & == 3 x~ir ~ ~ ~ ::: l t t J ir\ > r ~~ivt_ fu<, ~l "" ) (.e"' ~+v\<'.'j') J..1< = lz.q"' ~ + i<. ~ ~" + ~ { ~ ),... ol v i. i. I 1!,.,_, ;L ~ qr ft = 1 + j x ~ + d { ~ ) = Cl ;;= 3 i 0 t 'a ( u l 'J ) ~ 0 ~ cri~)~lj. 10. Find the surface area of the upper half sphere :i: 2 + y 2 + z 2 = 4 and z ~ 0 by surface integral. s ; 0( (! 1f \ c=q [0~,\Cf ' 2. s.\,, ~),\cf " AC'>)= ))1Js = c):~y6 n~\clt\ 2.(,U\{\J >., tj. E:[ 0, ), TI] ' p E[o,~ l "' \": \ ~~ 23,'Ml s, ve. ~ It ' '1, 0 > ){ < µ..;~ un1 u... ~ ~ :''..~A"" ;r =: [ \~ } 51~ q jfj cl Cf ' $TI (  Gr> <I')\: :=: g Tl
13 t l l. Use th0 Lagnu1µ;p M11lt.ipli<'r nwt.hod to find t.h1 ~ whose s111n is 100. maxirn11n1 procl11rt. of thrc'c' posit.ivc' mtmlwrs ~~' 1 h :: )( ~ i ~t< X~c ~,t ~ ~ t' ~ ={,(JO,X. 1  =t?v 12 3 L~~ ~I 4t_,_,, """ ~ ~?; =\ l (L_ Xe ~ ~., Q2 Xb ~ ;\I Q Xt~ t l =(ITT> G x. t: ~ 1 l :) re )( J) = o? x ~ ~ 't:=o w,~l\ ~\J.l(_ X~t=V vw~ ~146 (  'fil = )( ~ ;> )( ( l  ~ ) ~ ~ ~ ~ l ')( =1> W "vi,(!r V "l 'l( ~ ~ ~ 0 I VU. f 'i;vu'')<. ')( =: ~ = r e (QI) X=~=:i~3 ( (J1j 0 tl lrb ~7
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