Spring 2005 Final - Prob.l .Prob.2 Prob.3 Prob.4 Prob.5...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

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

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

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

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Prob.l .Prob.2 Prob.3 Prob.4 Prob.5 Prob.6 I.D.‘ Total MATH 102-A Spring 2005 Final Exam June 10, 2005 Name: T l 3 Student No: Follow the directions. No work = No credit!! Problem 1 (8 points each) Answer the following questions. Show your work! 2. Determine if the following function has any discontinuity or not. fix y) = :32 if(x,y) t (0,0) ’ ° if(x,y)=(o.0) “3% (X'vltmn); (my): 25;: .m “Mi jwdw is MN x *37— eve/DWJMLLQQJMM “a :7 GEL, can. «dun (x-yHEBj.) Ta [Xdllt (0,0)} £(0/0):0 .3 ,Vxfi TB/ “aka-Fla, £VWW15' MON 0' “4 (X'Y)—) (91°) MW) 2M9 "M \u @4le 40 find—.0. elk—“£3 b“ {(7‘17). '33 you WqucL. (0,0) @lmé, PM Q1414 YIX (Xtyl '5 (0/0) L» m =4 in W “0 X‘W‘ wax: 5 Ml‘fiMDGW-t b. Consider the triangle formed by the points (0,0,0),(1,1,0),(1,0,—1) in IK’. Find the angle at the origin. - M 414 La ’ll/m Vdc'lv Aryan/t The Drift-H4” (Molgflfl fl 0 ’I 4 c, r. ,, (Ala/p1) éfiz7—k mild. at m owe/WW..L'L::(W1lHWl.Co/29, when 61" “W: M c we aw. WWW I”. :> A: :7 0076=lz 7') a=0rccoifilar>tpz // 'u f’ 2 Problem 2 (7+5+5 points) Answer the following questions. Show Xour work! 3. Determine the interval of convergence for the following series: (—1)"n M ’+ g 2" I" a 7 .' “‘1 ‘ _———___... _ n l Mm )M (W Z r fxl 1%» 4‘;— = m was 2"“ mm new "Ln, 2. Wu l/ 2 ——'> 'Tlu sen/a Canveféaa M (-1; 3)? Efl& OWL \‘ N (x) Z(—‘Y\vx~(-2)n, in mm. aw; 1""7‘” - X='7. =7 mg 2" mo ‘A—*A"\"" x: :7 m (if—fl 2y: Zc—U’M :div. 37W “:3 V (to b. Evaluate the infinite sum °° m gUM : Z ‘ MO @2ch m SeomfiL Mm. Azo 4-x . r ' ‘ + - “M‘swww 3 r) M X- ’ M c a : T = _ We 3 x J gm HE) L} c. Find the Taylor series of g(x) = xez" aboutx = 0. Make sure you write the general term of the series. ' We Wow ex: Z |xl<i. Problem 3 (4 pts. each) Answer the following questions. Show your work! 3. Find the equation of the tangent plane to the surface 2x2 + y2 + 3zZ = 6 at the point (1,1,1). w “E r “X? f 2315+ 6%}: gm“) - -1 :9 W («4,435 Lt? +aj’tett ‘7 VVFUI't/O (OH)? 1- (th)§+(2:?:i:l:0 'i” 2:114. k 4(X'l)+2( 'l)fé(%—l>:o Wem I). Find the directional derivative of flx, y, z) = 2x2 — y3 + z in the direction of the vector ?+ 2?+i€at the poigt (1,0,4). [Mr-«w... .Mw'fi—a—Z vgztxt-zgmc fl WHO, 2): cm +8. (J .4 1 1% “mad Now fin Weir)! 0 mt om; c. For a function g(x,y), suppose that Vg(0, 0) =72 Describe the vectors at the origin along which the function g increases. DNQt'k’ow diam/ABM “AU/Ti he $0/q i751 mum:ch ‘wt (Ii/Mu,th « ii 2" g. 5 So; lg (35 \‘v‘ (MM?) '14 340,0) : Vole»). «a: W. UM [[1 M? 'u U U A a) W W W (W t” W “1‘ “W W ' W 1" I 44 WWI/7i W =) 005970 ‘ (Were/Make Problem 4 (8 points each) Answer the following questions. Show your work! a.Evaluate I: If xcos(yz)dxdy “‘ 1 x43 '1 g ‘ ‘r (I) (A :30 [Cm(:)1)% : jco‘da‘) (740% t (jam: iaf x23 0 3 7 "flaw—m... 2 2 3:0 ‘f of 4* 3$IW0:’\ a fi‘3=-('<'3) amen-Bl Given the region D. Write the double integral flx, y)dA as iterated integrals D I Iflx,y)dxdy and I Iflx,y)dydx (i.e. in both orders of the variables). Show your work in getting the limits of the integrals. 1:. Sketch the region R in the first quadrant (in R2) which is bounded from above by y = J4 —x2 and from below by y = [2— . Then find the following integral: II di I? l? A??? >JX=SX‘u—x1 AX :fi LN all \ o I 1 11 3 «to = —.\ % u’a c l V 8 " 3 s s 3— A. A” ,3 -5)’ _ «fwwmf’jwflc ‘” —— M r u, g S 59091? rdrAG TVV \E/XMG' x V 33 you work ‘Haza while 5mm sow M'Mf (0/? «pale—all) Problem 5 (S+2+8 points) Given the function flx,y) = x2 + Zyz. 3. Determine the critical point(s) off and state the type (i.e. local maxima, local minima, or saddle points) Qua/£3, at?) —.3 (0,07 :9 m “LB came Wat fixd \ \ \> ’9 'Fxx 413‘ £x3: Y 1?) V) a Mm. A1040: ixx=3~>0 b. In the next step, you will find the global min and max of the function f on the region x2 + y2 S 1. Without any computation (i .e. before solving part 6), how would you be sure that both global min and max exist on this region? (Hint: This is a one line answer! !) xhylgt dextvibe; ‘ ‘ l ‘ ~ ‘ GOAL 1:) U (- Maw Lg'h‘ N’Wh /NOX. ’HM Umi Ohm; é?) , #M s on N?ij c. What are the global min and max values of f on the region at2 + y2 S l, and where do they occur? Emmi; m um AxbL (Le, xii-71(1)] w: Maw Tuiw (mi. Pall! (0.0)) Cum KlicyL: ( (Le onii circle)/ murky LajrnUlt; ixam 6‘? 6097): X17" —2 We WM'i' V-(=—?\V% (Mot x261: \- mmtwag’ fl mammmmaé v r " "’ szklx 1 135%11‘203 U9: XZvJ I x1+kal=| ID. 1:) X:0 or )x’:\ ,.«~~~«\_\ \ r .— :i’fl '3' (0(4) (or X-0w)[bflm)/%) )0! 0 (b m) :t‘ ade/(rlph )\:l two-.1? 9A., :4: '9 a . cw‘km VAN” "lb 4“?) 0+ my S Yo‘ws“ f, 312% mm;2 MW“) Whom“) /' 4(4‘0):£(’|,o):4 /' -?(0,-I)~_]€(0,1)-;Z 9H»! min :0 M” (0,0) ______———~ Problem 6. (8+4+4 points) Let Wbe the region bounded from below by the cone z = ‘/x2 + y2 , from the sides by the cylinder x2 + y2 = lY and from above by the plane 1 = .5 (i.e. close the top of the given pictures by horizontal plane z = J5 ). i. Set up triple integrals calculating the volume of the region Win Cartesian coordinates and cylindrical coordinates. ii. Choose one of the integrals you have set up above and integrate to find the volume. iii. Set up an integral in spherical coordinates. Do not integrate. NOTE: Do not use any short-cut formula for volume of any shape. All volumes should be formulated/found using integrals. (for instance: even if you lmow that the volume of the sphere of radius ris éflr’, you are expected to get this result using an integral of specified sort in the problem!) Z—lemQI m) 14 when m (AM i: leyL on?! “We (UQW XLe [Si New} gay 3 :7 €:\Yl——:/l_ -4 3 I :7 (Mira.er W: S g 1. Arab Ax % t H1. ' ; r LIMMiPA L, %: Lvumotwt; qufiw‘ find cm a: 2‘ u 7 ——-_ f 26? lg to ‘ f) Val: A 2 Ar 0k \ ,-w:~_._,._-,~'__j ém flag} 4 A9 a (’11) Somme THE C‘ILINO‘K‘LAL: m 71? r {,5 1Q l)d ‘3‘“ we § § [my lardfizg 3 3W ' (9348 o O 0 0 E:( Noxe m ‘66 WIT}. Wm.A osésg , {.3 towmea 2=H film W ‘ 'vm ' r _ .— 53- 1 Caerau €w5¢_fi —7 P_ a = ( J? ) OS f—w 3:34 :4 =9 _ .L ‘ ""‘”“‘“ v) 61 .4 vb s g é/O?////;n”“ 03344 W T‘ VJ: g (23,44 gleam 15g? ...
View Full Document

This note was uploaded on 03/16/2012 for the course FENS 101 taught by Professor Selçukerdem during the Fall '12 term at Sabancı University.

Page1 / 8

Spring 2005 Final - Prob.l .Prob.2 Prob.3 Prob.4 Prob.5...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online