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Math 122 Test 2 Fall 2007 Answers

# Math 122 Test 2 Fall 2007 Answers - i Math 122 Test 2 Si...

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Unformatted text preview: i . Math 122 Test 2 Si: ' —"i October 16, 2007 ii ""1 W: “a, Name 54‘.” 2:“ «f F—-‘ DON as OI 6 i i‘ i 7 i Style Points ‘ Total ‘ 1. No books7 notes or rooting for the Red Sox. You may 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. Directions: to . 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. 3. 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. 4. On this test, explanations count. If I can’t follow What you are doing, you will not get much credit. 0. 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. (1.5 points) Find the general solution for the. following differential equa— 1310115. . dv _ wgy ~ 411 r W x e,” \(L) at; —— W w- 25,, 72 23» j If f x} v?“ *‘ ~34: Mi *‘ U 0% f “if: 3V. “W QQJ‘R i e “um mail/YR “‘6 :1?” a? gag. “{g‘ :9 WW‘ a“ \ :3!» 2.. 9 , {My 0’ (WM. 2. (15 points) Dean Anion has recruited a certain Chemistry professor (we will just call him Prof. K) to make her a love potion. A tank initially contains 50 gallons of pure water, Water containing 1 lb of magic chei'nicials per gallon enters the tank at 2 gal/min, and the mixed solution leaves the tank at 1 gal/min. \ (a) Write a differential equation (with initial condition) for how much of the magic Chemicals are in the tank. k :‘ 3; \g M r: 4‘“ , rm 1.ij w “w' . ”:17: M wm ﬁ' 1 WW v u (in: ' & t R o M \g" in? @ £2; sit; “g :1 “rt :57‘ "‘5 m . «it, Wt (V w “it i» m (3;: A "2‘ W ,«q r» :11? w 7» ‘ 6m; mm a“ ﬁt ”We? MM .\ M if in \$3,; a“; f} E? {:3 A? M 9;} *WEFE-w— W E E2,” ‘3. (10 points) (a) Find a second order differential equation (with initial conditions) that has 9 Z BEE—21' + 2851‘ as the solution. V” » _ _ it“ 1 W W at, 3 “.2 K ” w Mi 4’ x w ”1 it - mt 13 Egg M 3} “’2” M w a rm g": m ”£34 in,» ”J“ iii} “5:" {My l’ i g we M. :52 a“ W“ is” “w m: (T f} E i ” ' '2 W ; f m} E x: a. Z?” é é “a; to 3 w mtg I (b) Find the general solution of: yl/N __ 139.” + 36y = O 4. (15 points) (a) Sketch a graph of the area of the region outside 7‘ = 2 \$1129 and inside 7" = 2. (b): Find the area of the region outside 7' = 28in 29 and inside 7" 2 2. ﬂ 2‘ w Ex W a “2 E “é? s :1 i ‘ I! LL W63 W § § ? r W a i “if 3 3 “:5 32/ 2:2: #4 i ’5’; 1 5. (15 points} A curve is. given by the parametric equations: a: = acos t y = bsint Where a and b are positive constants. Find the area that lies lmtwecn the curve and the x~axis for 0 g t g 7r W ma :2 *2; m”: {:ﬁp 1‘ m “M W} «E WW MW 3 ~33; N 692‘“, ”’1‘ ":3 33‘ y» W\ «:13 w W! 1 g M :3 Mn: an x. 1 W t i: 4"‘« L“. mg 6. (15 points} Consider the parametric curve given by szt—i—l y=t3—2t. I ‘. dig Ea) Fmd dx' , ”W“ ‘i ’ h FE “EM '5‘. f» “m f N V . x535 it“ {c} Find the points on the curve Where the tangent line is perpendic— ular to the line 317 -l~ 5y : 8. 5”“ 1‘4”» 1% «f; ,5._.m,.. psi“ “+5 Mym My 7 A «M, “ "f r“ 6:” 6:, m“ w M» f}; k“ a L‘. “.2 f” 5’ , ‘42:: Ex“ ” { 5§E1WLEE J \Q’N. J” 7. (10 points) lnolieate Whether the following statements are true or false by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. ~ , ‘ d2 H t a) If gr : ftt) and y :96), then ﬂy; 2 ﬁll/(75>) The area of the region enclosed by the circle 7' = sinﬁ is / \ '27T ‘ 5i s bI A: g / s1n26d9 T F e _ The curve :1: :2 t3, 3/ = t2 has a horizontal tangent at the {I 321‘ e) , . dy ‘ T F 's origin because 3? z 0 when t 2 O. 3 If the graph of a, polar curve passes through the four d) points (1, O), (—170), (0., 1) and (O, —1), it must be a cir— T é: F i ole. / r 1 x «. , \ . ‘ . ‘ : 2 = ’2 .L MW,” e) 1hr, g1 aph of the parametric equamon x t , y r is T {r F the line y : .17, § / FORMULA PAGE 1 b f<C> = f / f<cc>dx . . ) "‘ CL 81112 9 + cos2 6 = 1 ‘1 ta1129+1zseC29 /secmdcp:1n1secm+tanm «1» C 1 ; 2 : F’ 9 .1 1 + con 0 CSC /sec‘5 3: dsc : 3 [sec 3: tan 36 + 1n 1 sec :6 + tanle—C sin(a + 8) : sin oz cos 6 + cos a sin [3’ sin(o¢ —, 1: sinoxcosﬂ — 0050581115 /CSCLL’dJl=1nICSC\$ _ 00133311" 0 cos(a —— B) : cosoacosﬂ — sings/1x15 1 + 1 = 2 coda w [3) = cos oz cos ,8 + SinOz sin [/3 tan a + tan [3’ mum + 5) = W 51112 l“ = £20331 2 2 1+ cos 23: cos 1' = WKM” 5111251; 2 2 sin 2: cos :1: , / (8111 :r) = COS a: , \/ - (cos :13) r: n sun: (tan :L')’ = 5662 x (sec 3:)’ = sec {I} tan a: (08(1ch : — cscrr, cot (L (cot :C)’ = w 0802 :r; w"! _ :r (e ) - e @1151: I = _, 1 > :5 (drosin :r)’ ~ ———}———— ( 4L / — ‘ V1 - 1'2 / 1 (arctan 931 = 2 1 + it (arcsoc m)’ = 1 mm”: W .1 ﬁx”) vcn—H ”* 33m “‘ f/(an) f (b) - f (a) b—~ a f’QC) = ...
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