Finalkey - Final Exam Math 2401 All/A12 May 2, 2008 Name:...

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Unformatted text preview: Final Exam Math 2401 All/A12 May 2, 2008 Name: K6 y by writing my name I agree to uphold the honor code Read all of the following information before starting the exam: 0 This test has 10 problems, each worth 10 points. o You have 170 minutes to complete the test. 0 Show all of your work. You must show all work to receive full credit. a Circle or otherwise indicate your final answers. 0 Please do not write on the back of the test pages. If you need more space to write your answer, you may use paper provided by the Instructor. Please write your name on any additional sheets and indicate that your work continues on another page. a You may use a scientific calculator during this exam. You may NOT use any other kinds of calculators. a You may use ONE 8”X11” notesheet during the exam. You may not share your notesheet with other students. You may not use any books. You may not talk to other students. You may not email, phone or otherwise communicate with anyone besides the Instructor or TA. If you have questions about the exam or about the suitability of behavior or devices, please consult the Instructor. 0 Good luck! 1 . (10 points) Use Green’s Theorem to evaluate / xzy dx + 2y2 dy, C where C is the square with vertices (0, 0), (1, O), (1, 1), (0, 1) oriented in the counterclockwise direction. \ r... m. -nummm- l QX’PY: oflx E l 33 ~28 ax47 : -x’zéx'iy IL ° ‘3. ': "hi ‘ Q] 3 L Y 2 «1 5 2. (10 points) Find an equation of the plane tangent to the surface z2 z 932 + y? at the point (1, 1, 3,, ‘2. ,2“ M45 Zzzxziyzég [L'ICVcifiuff-fiflafl n: “azxftngj +422sz ML.“ tséfvf'cecafi 3. (10 points) a. (5 pts) Find the unit tangent for 7‘05) : 2ti — 3j + #2 k, t E R. Tm: fife)ch WW: ‘2" + “Q Hr‘Ur') 3‘ an , __3:_., ' MK Tm:- ‘ ‘3’ WW”. b. (5 pts) Sketch the curve with parametrization r05) = (t —- 1)i + (t + 1)j, t 6 [0,1]. xi 4. (10 points) Use spherical coordinates to find the voiume of the solid bounded above by the sphere $2 + y2 + Z2 = 4 and below by the cone 2 = Mm? + yz. 5. (10 points) Sketch the surface with equation y = 2 — 2:2 —— Z2. 6. (10 points) Find the surface area, (E the part of the paraboloid z = 1 — 3:2 - 392 that lies above the my—piane. 2 Z a «9 x 24, Y 23 f m): mar xzh/LQJ if M 5—» r” 5 S rflrlJ/l dCQ 6‘ b _, Lira“ ml 3:; m3] *grdr r:0 39 Mil g 7T 2.. 3511:?“ j ,5; Liv. : 9 «3m ; 7. (10 points) Minimize$+2y+4z on the ellipsoid $2+y2+4z22 1. jbgylzgl xqfiwyg‘w“ LIZZLI Vj ; 3X15 , ¥ ’:92>X V¥z ti) gxrél‘ky Xafiqbaf 4f: 322. I XL+/2“+ 41/2 2": g Xf :A z y:- “s E } I W 3"; :lg 25;); :1 }l 2’! all)“, F"! r" xi+\[1+L‘Z;“f A _, 2 5: :muié NJ. ,-i hii'x“5 335 2*} '2 ~~r “we: mi >1 i'xv \/-« 3 Zn» 3 8. (10 points) Find the direction in which the function f (23,34) = m + sin(a: + 2y) decreases the most rapidly at the point (0, 0). Find the rate of change of f in this direction. V79: (ngixtlfl)f + (emsawmj ai' (0,33 deéfaaic> magi” (afflouy i/k \ ‘ “Whom : -— [1" 4—23] : $21-92) <Jv~eofia4cflifi bu H/{A {‘"zfiéél - 03} :2: ' J 41+] :3: 9. (10 points) Use the Divergence Theorem to find the flux of v($, y, z) : :1: i—y j—z2 k into the cylinder 1:2 + if = 4, 8 g 2 g 4, including top and base. 4:)“ {\(hflw : “flux @g‘i' (31L fofmd/(f T: "' 012‘ Ci; 5mm: ()me X’Mfi'i Oezeq L, I I S\-’i ‘AZ :1: 4x671 &55}L’0L O {yaw}? L \ ‘ 2 ‘* a 4k : 321g; éx dy ‘ 2:10 k / 4,1)lt D éhL O-Efké. anfié> % y: 10 10. (10 points) Find and classify the critical points of f (3:, y) = y + 111:1: — my? VJL ( ix-YZM' +(l vim/U VJC Ufldfl-gwe/é WM Xj'a (bLH X30 V137LIA 4N1 Clo/“tuft 0% V£;O H: «kw/1:0 “3 ‘lxyv‘u 3E;74. :7 Xzixf’” :37 iixy: : ~g—l :O 7 IE “7; :9 7f“ :7 Y:Q\ :13 X34, (fill) 0/117 Cn'fluJ Ifmnjr a. :_2 Ma ~27, M x vi 71%;er jpxx: 'Mo JEXY‘” dH 7PVV: ‘1‘ D: ‘yxx‘FYy" 4‘2ka : 3/ *Ho lw5 <0 11 ...
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Finalkey - Final Exam Math 2401 All/A12 May 2, 2008 Name:...

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