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Math241_Exam2V1_2PM_Solns_Spring10 - Math 241 —— Exam 2...

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Unformatted text preview: Math 241 —— Exam 2 — 2PM V1 Name:______-______;; ;____-_-___;__ March 15, 2010 Section Registere In: __________ 50 points possible I. 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. N0 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. l 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 i 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 8 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 be a scalar-function of n—variables. (a) (3pts) Define What it means for f to be continuous at a point [i in the domain. (b) " (4pts) Show that the function g is not continuous at the origin when 2_ 2 9(93 y) '={ :2+:2’ (my) 75 (0,0) “1, ($33!) = (0:0) Wag An 3 w \M 61ml) bk)? 3/ Ma M; (K,\\\A(0,O\ ear/1M --\ , ' a V9 \w ’d—i—X“ :1 “M :25 (flab-VOA Xa +57 - X">’5 Y0 2 W \ 2 l. n X A 0 M59, élxri \ “3V (m‘\\/\Wm§ 0.4— 43%? OHSVM 2. Consider the paths r1(t):— - (t2 + 3 t + 1, 6t ) and r2(t):= (423, 215— 2, t2—~ 7). (a) (5pts) If the functions represent particles 1n motion do the particles collide? MM? first: All: 19 §W m (4am yé (1 m: {Lam-(A mm: (L) H4a\ —— (pa 411W Aft € )(3pts) Consider the image curves defined by these paths. Do the curves intersect? \lg my \A—llerééc'l A‘l‘ die "EVAN {M33 3. Consider the cycloid parametrized by r(t):= (t- sint, 1— cost). (a) (4ptS) Determine if r is a re ular parametri tion ((9 re mhr \J; (a (1A flog-Bis am: 6 {A :40 (:hr M \,,:: [k ”((3% S‘A‘L\ LN: (t QWfl2QVWYM‘1fS‘EF‘”) Vat/3:“— (AA. WWW \5 M5? fzyid- (b) (3pts) Let r(t) represent a particle moving along the cycloid. Find the function that represents the speed of the particle. 5%.: Hr [sat [wigs/i4 1;} MM‘KW‘WM) SPQAW (-i’\:“ \io) "3403+ SW [/t\3:_EL/:.JE.., 5. The length l, Width 112 and height h of a box change With time (a) (2pts) )Find a expression for the surface area SA of the box. (b) (3pts) Compute the differential of SA. ASA (c) (Zpts) At a certain time the dimensions of the box are l = 3, w = 5 and h = 5. At the same time l and w are increasing at a rate of 3 m/s While h is decreasing at a rate of 1 m/s. At this instant, find the rate at which the surface area is changing ASA [9163 sa((¢\1[§\s—(/%\[;s\so>/s\l Na + [(9 (A +Q(<'\]/ «sl\ /€~ to +4? —- [(0 69 Ws/ (a sitar s (grams Mash 6. (6pts) Let T(a:,y) be the temperature at location (cc, y). Assume that VT :2 (y —4,:1:+2y). d Let C(t) 2: (232, t) be a path in the plane. Find the values of t such that it [T(c(t))] : 0. 7. (5pts) Find an equation of the tangent plane to the surface 332 + 2x2y + 3/223 = 11 at the point P(2,1, 1). WM“ 2. (IA-M )aofiw ) MM 5 ...
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