MAC2313_Test2_Spring2010_Solutions

MAC2313_Test2_Spring2010_Solutions - Page 1 of 4 MAC 2313...

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Unformatted text preview: Page 1 of 4 MAC 2313 Name Test 2 Date: February 11, 2010 Clearly show all work for full credit. 1. (8 points) Find r(t) if r"(t)=2i+6tj+12t2k and r’(0)=i and ’(0):j’k' “PM-<0 0 '> . ‘ ) Q. .2: < a, (at, ‘1’?” K0 F”(ol = 0,0107 i We): weal-3) 49> + C ~‘) *7 F5103)»: <o.oio> + C. g<h0i0> 1‘) C»: <"°IO> F"({:) : < at“, 3259, Ht3> F49): < taint, t3) t“l >+ (L 73(0) : (0,0) o> HZ):- <011,«:1> => Q: «Din-w F" (t) : < taut, t3+() t“~i> 2. (8 points) Find the length ofthe curve given by r(t)=4sin(t)i—3tj+4cos(t)k, Where 0933- 2 <45m<cij —3t) “legs (é)> Flog) : < were). —3, «we» 73ml 2: may“) +6? 1— lefn’hE) 2 W I 5— \3 ‘ s = fl F’Wsllﬁ = 3.) 5v“ = ’5 3. (8 points) A plastic bucket (right circular cylinder) has diameter 20 inches and height 20 inches. The plastic is 0.5 inches thick. Use differentials to estimate the amount of plastic (in cubic inches) needed to make the bucket. .. a raw" V“ ”FA [‘th AV: WW + arer Amos” 11 {EA AA = 203,5"): 1(1)?) .1 01V: nciooim + anomboxom : 300“— ML that“ I! no £0?) am (IA: 0'5- ” ital/k alV = “WORDS + angioyaoi (0.9 :— QSOTF Calm; Mala Page 2 of4 4. (8 points) On the same 132- plane, draw the level curves associated with K - -l ,0, l, and 2, for the function f(x, y): 11y x Label one point on each curve. K2‘l:(40 Qua/Q Curve) K: 9 K20: («51:1Jvex2123 1/278z K=lf 1:371:12» 74% Ken: 921F131 => ”:93” —,—xy——— (x y) i (0 0) 5. (8 points) Determine whether the ﬁJnction f (x y)- _ x + xy + y %m(x’y) :(O’O) is continuous. Explain your reasoning ’KV aﬂyz X‘QKLS Z, 52 :1 O E ‘ «9 3 Ma ’Y (M X +’X"f f'Y «(:0 . (19111-310101 9‘ “7 ° 1 :2 Q X 1 i“ 0? XY 1 a ' ’YI’X [(2—1 3% 3: “:3 ‘4” 7i +7<y +7 7<->o (IX/Y) ‘7 (0)0) v x _ ‘ ‘Qck. x3+7211 MIR DAN; 1”:> 10 13 nut 034514ch («name/o) 4,6 (0, 01 6. (8 points) Suppose r(t) :< If ,t2 >. Determine an equation ofthe normal plane at the point (1,1,1). ’V 15:1 91(11- <1,ac,;zt> =5 "r3111: <1 51 a> if“) 15 0» Kora-14,6 Veeﬁyr ﬁx a“ “WAVE fﬂﬁa‘é ‘L6 (I, I) I) 1 -> S ”Rte, F), (I\ (is {n (11¢ Sic/11¢ per/"6,065“ 115 T (l) ) F7 (Cl) [3 (£50 6L. (11%me veL/éb/ gr 0113 Pﬂazte, , (L éﬁuaﬁioq gr Ila, [\W‘I‘mﬂ F/zQ/ie 4,1": (’1 (I 1‘) kg I I 1 1 1 «+215, +32 :5— Page 3 of4 7. (8 points) A sequence of periodic payments of 1 forms an annuity. The present value of this annuity, z, is a function of the interest rate used for discounting the payments, I, and the number of payments, N. Symbolically, z = f ( I , N ). The table below shows present values for the given values of l and N. Use the table to determine a linear approximation of the present value when the interest rate is 5.75% and there are 16 payments. I as a 5nercent ii-I-i 10 8353 7572 702 64— “94 10-38 8814 12 46 10 59 1741 1409 11.65 CUrqulif, pl: (6)15) : ZzPCBJS): “‘6“! >41: 4‘ 1 -0173 Ave—Fag" 24‘5"” _._ l0:8: =~01¢Ll 194676? ’0‘“ 2: £07.53 = ﬁlo » , z {3(5193277; 4. 2 o 53; mm“ guts, lSl . 2 use» 2,01383M> p (5 )5) : 0.97% z: {Raga-411% A} = 0 ‘~H N I LC: N3 : «on: (1:53 + 0%“,va + 10.38 I L(5’75 IQ :: ~o.7i(o.7§) + OH’HUl + “938 r; [0.3; ‘ ) 8. (8 points) Determine the curvature of r(t) :< t2 ,0,t > at the point (0,0,0). ”v C 1 0 PW): 424,0)» 2—4» '(m: <0,0,i> ? 177%): «320,07 2) ‘r’-"(o\= 920,0) F’YOB x F”(o) :: <0,9,o> I s 22 : _- - a I ~)[(O)13 (3 K __ (PM x F7”(o)\ 4‘ 9. (8 points) Given f(x,t)=e"sin(2xt),detennine f,(1,0), 1'? ‘Ft (1,150: a“? C03 (3909 ' 9X " a 5m (2115) Q (1,03: «mic»; - Memo): Page 4 of4 10. (8 points) Given 1110912) 2 z — x, ﬁnd 232;. Simplify answer completely. 3 , l ( “DE _, 32 6“- «W. 1V;+‘X%va f‘O V xv E V D V a 2~ . i 2?: :> m 5:, f 7&3 ‘1 ”We 61/ _ /U (-6; «’5 ”Z O M> E __ lg...— : ﬁg...— : (CuiicéZ/Q scam M By “ xyg~><y 70/(24) “((3—0 «#0 ll. (6 points each) You are given u(x,t) : sin(x — at). (a) Determine u". Mb : COS (Xedél » (‘al : «@605 (7(»6sz> aw : asc‘n we» (—0 : wags/A (V~aé> (b) Determine urn. (“(1 T- C05 ('X ‘ ké) M X X 1‘ —‘ SM (”XL“ aﬁ) 12. (8 points) Suppose f is a differentiable function of x and y, and g(u,v) = f(2u — v,v2 — 4u). Use the table below to calculate gv(1,2). <00 { Willi-:27 ax (1:2) 6 3 2 5 x: wa/ ET: *1 ...
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