Math241_Exam3V1_2PM_Solns_Spring10 - Math 241 — Exam 3...

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Unformatted text preview: Math 241 — Exam 3 —- 2PM V1 Name: _____________ . _______________ April 16, 2010 Section Registered I 50 points possible l. No hats or dark sunglasses. All hats are to be removed. 2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now. 3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time—piece. 4. No music systems — IPODs, MP3 players, etc. — or calculators; same rules as with cell phones. 5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam. 6. Every exam is worth a total of 50 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 7 pages. 7. Be sure to print your proper name clearly and then write down the title of the dis- cussion section (i.e. AD7) for which you are registered. 8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID. 9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready. 10. Good luck. You have fifty minutes to complete the exam. 1. Let f(a:, y Z) '= 6_z sin(:c+y) )(3pts) Determine the gradient field F associated with f. VLW (C [S/>{+\(\) 6 2&S/%w\) ”‘6 3w [KM/‘5 (b) (3pts) Determine if F IS incompre8511b.le ms \akl \8 AN (V£\‘=— O $21”qu 2 \ <7° V“ M: 1—— g; (ézag[¢+1\\1§{6{as(¢+7k\ Sz/ 6 96qu : ”'6 29A [>‘H?\+ +L 6 gfl/>l+7\\ + 3431/16! ’73 Man/1V : “fivgém (>l+\(\ %O 1'(€WTULL€\ mm) 3: \S «ft/Jr {/1 [imprffl LL” 2. (6pts)Consider a fence Whose base takes the shape of the curve y = x3 for 0 3 cc 3 2 in the rug-plane. Let the height of the fence be dictated by h(:c, y) := ‘/1 + 9mg. Calculate the area of one—side of the fence. Pam he. wai “A": “meta, 64* 4%) ' flit»: ). 31% :gi MHL’W Cur ; g; [HSfiUKL b a“ 3. (a) (7pts) The force field F(a:, y, z) :: (m2, —z, —yz‘1) acts on a particle. Calculate the work performed moving the particle from P(O, 0,0) to Q(1,1,l) along the path r(t) := (t2,t3,t) for O S t S 1. (Assume that F is given in newtons and the unit of distance is meter.) g www. « m mwmm WW ‘nfiwmmxmamqmmmrmwm\r (b) (3pts) Determine the work performed moving the particle from Q(1, 1, 1) to P(0, O, 0) along the path p(s) 2: (32, ~33, —s) for ——1 < t < 0. Justify your answer. (Assume that F is given in newtons an t e umt o Istance is meter. T Doll/AMé : ’- (/5/L\\ :34, w t W W M Wt J m.-mmmmmmmwmmwwmwwamwwm 4. (a) (5pts) Find a potential function for F(m, y, z) := (2153/ + 5, x2 ~~42, ~4y). Q : 9x714? --=; «067.3: >537+C8<+€ (71$ Q 1 8,42, ——>s fiwgg: K97 47’2— +(/><(2\ £2:qu ,3, wa— P412 +-,(§(><y\ (b) (3pts) Evaluate f (2:1:y+5) dsc+ (932— ~—42) dy+ (—4y) dz when r(t) := (t2, sin(7rt), 69—22:) 3 mm" wwwzmwwwmwmwammwmmxmwmmfixmmmmwzwmmwgmmmxmawmwwmmwmwaawsvmmmmvmwmnmwxmmwmsmwmwmm»om;wwmmummsss,wmmmssmawrmwmwmwmwmwwahmNevmmwpmm1mmaa(SR-WWW»mmmm:(;.<m,~s.w;wa: «W «HumWHVWVMAQW“.WW/‘wmummy“ MW 1 e 5. Consider / / f (23y) dydx. 0 e?” (a) (4pts) Sketch the region of integration. \ #3119; (FM YC‘fx “h; Vice mm >490 AW 56"- I (b) (4pts) Write the iterated double integral that reverses the role of integration gf g: Yfirmle 37/ (c) (2pts) What theorem makes this “reversa ” possible? fiwni fiemwwmvwamhmwmwmsmmwxw\mszmwmks!)vamwmwsmwntxmxwwwmms:«nxmszrrwAW1WZx <kwmwam’xmwmmswmumwww‘aflmwé'nmamu\w:\wq2r 6. (a) (4pts) Sketch the region D bounded by the curves y = a: and y = quadrant. j \3/5 in the first We ng)::§::J/J¥:d:l 6;” we MA Jr» 4» its mt >< 70ml Bhw\mmmmfiwwm‘wwwm>flw«immWfimzw‘wunwmem‘xx‘xumw‘wmmmtamwamawwuyflwwmfiw:(Mwwurzmxflww<wumrtavygm.7,:p3-(axm-3rvmaxxx ,, L: m‘ t A wtmwmxa ...
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