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Test 2 Fall 2006 - Math 122 Test 2 Name 1 2 3 4 5 6 7 8...

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Unformatted text preview: Math 122 Test 2 October 17, 2006 Name _ 1 2 3 4 5 6 7 8 Style Points Total 1 Directions: 1. No books, notes or opinions (this is not a SAGES class). You may 0! use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. . You should Show your work, and explain how you arrived at your an— swers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. . You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. . On this test, explanations count. If I can’t follow what you are doing, you will not get much credit. . You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. 1. (10 points) (a) For the differential equation (5% = F(SC,y), where F(:c,y) is given below: a: 0 1 2 3 4 0 1 0 —1 —1 O 1 0 1 0 1 0 y 2 -1 0 —1 1 1 3 -1 1 0 1 0 4 1 1 «1 —1 1 i. Sketch the slope field at the given points. ii. Use Euler’s method with h r: 1 and y(0) :: 0 to approximate 9(4)- 2. (15 points) Find the solution for the following differential equations. (a) 642% = 33y2 + 96 (b) (1+m)% +3; 2 0083:, 31(0): 1 3. (15 points) A tank initially contains 60 gallons of pure water. Brine containing 1 lb of salt per gallon enters the the tank at 2 gal per minute and the mixed solution leaves the tank a 3 gal per minute. (a) W'rite a differential equation for the amount of salt in the tank (with initial condition). (b) Solve the differential equation from part (a). (c) How much salt will the tank contain after 20 minutes? 4. (10 points) Match each of the following differential equations with the correct solution. y" + 23/ + y = 0 a) y = 6162”“ + 6265”” y” + 391’ + 23,! : 0 b) y = 8‘333 [01 cos 29: + C2 y” + 6y’ + 13y = O c) y 2 (316” + 621116—97 y” + 4y 2 0 d) y = 016“?“ + 626—233 y”—7y’+10y=0 c) y=C1COSZJI+CQSin2$ f) none of these 5. (10 points) (a) Which of the following is not a polar representation for the point (—12). a» We b> [Ml—2e] c> we] 01) [2V2 347:] e) None of these (13) Find an equation in rectangular coordinates for the curve whose equation in polar coordinates is 'r' : ~4c089+3sin0 6. (10 Points) (a) Find the points of intersection (:13, y) of 7:2 and T=1+2cosfi (b) Set up an integral to find the area of the region that lies inside the iimagon 7‘ z 1 + ‘2 0086 and outside 7" :2 2. Do not solve the integral. 7. (15 points) For the curve given parametrically by‘ x=1+2t2 y=4+t3 a) Find iii attzl d2: 2 (b) Find %% at t = 1 (c) The equation of the tangent line throught the point (3,3). 8. (10 points) Find the area under :pzt‘Sint y=1~cost for0£t§27r FORMULA PAGE fl S=/ rug/)2 d0 $1112 6 + (3052 6 = 1 ”can2 6 +1 : sec?” 6 gynfl : m” — £2337») $71 1 + cot2 6 = CSC2 6 f/(C) _ 116(1))” f(a) 5111(01 + 6) __ 531110: cos [3’ + cos a sin 6 — b ._ a 8111((1—13‘) — sin a cos 6 — cos a sin [1' 1 b cos(oa+[7’) =cosa0056—sinasin6 (C) : b—ah/f 003(04—13): cos oz cos 8 ~~ 8111a sin 6 fscc 3: d3: 2 111 1 seen: + tanm1+ C’ tan ()5 —— tan [5 1an(a'+/n) = -—j—- 1 1— tanatanfi /sec3 xda: = —2- [secmtanx +111|seca: + taan—l-C , 2 1—008233 S111 113: —— 2 /cscxdm=1n|cscw—cotxi+0 1 '2, l 008236: W 1+1==2 sin 23: = 2 sin .7: cos 2: cos 2:1: = COSZ .1: — sin2 a: (sin m)’ : cos a: / . (C0833) = — S1113: (tan 3:) = $602 :0 (800 x)’ 2 sec :1: tan :1: cscx = —CSC:ECO 3E ( )l t (cot m)’ = — 0ch :1: 1 ' ’ = * (m) a: 1 (arcsinajy = ——-—— \/ 1 -— x2 1 (arctan :L')’ = 1 + m2 1 ‘LI‘CSCC a: I : ——-——-—— (( > [fix/932 — 1 b 1 (1T 2 (1y 2 $111141) 1 ...
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