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winter 2 2009

# winter 2 2009 - Weds Feb 20th 2009 Page 1/6 MATH 32A SECOND...

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Unformatted text preview: Weds Feb. 20th, 2009 Page 1/6 MATH 32A: SECOND MINATION 4W%Lz;é¥_ inter 2009 ‘ LAST NAME: FIRST NAME: ‘m—x ‘ ID NUMBER: Discussion Section: For each problem clearly your ﬁnal answer. Write out your work explicitly, partial credit will only be awarded for clean logical work. Good Luck! Weds Feb. 20th, 2009 Page 2/6 1. (20 points) Let C be a curve deﬁned by the position function r(t) =< t, sin 4t, cos 4t >. (a) Calculate the equation of the Osculating plane at the point (7r, 0, 1) E _\_ “(day-27, p/G‘l- 72' :77— VI”: z~,qrosqe,—wsi~rw> ,\V’\ ‘5‘“- 2 V’ _ Te :1 — E< \,L|cosqf—Vsin'~/t‘> :‘3/: AW; < O) ’/6\$;\‘/f”/‘c°sq£ > .4 a 3/ “4‘ = \g I N = j: —,— (0”S-~¢It,‘c°“/(’>, \rﬁ \T’l e 3* b3: . . OSCA\ P‘M “5 \ A *N 3 —— (-r 'l : a—l Th0 L“\ (a, ‘1, 23>. V\ \V‘.<x.n,v),z-\) (b) Is the Osculating plane perpendicular to the Normal plane? Explain. \323‘. E “Q’M\ +9 05¢H\A +33 f m m\ +9 1qu. ‘?\ad. a; 2 __ —’ 2 T _\__ ‘(5 — ‘T-KN 'f—J) osc- us\\ Nov [ow-t R L \f\r«( . // Weds Feb. 20th, 2009 Page 3/6 2. (20 points) A trajectile is ﬁred from a 500 ft. tall tower at an angle of 60 degrees with respect to the horizontal. Here the gravitational constant g = 32 ft /sec2 . (You may use the back of this sheet if additional space is needed. Do not reduce or simplify numbers. You may use the equations of motion derived in class.) (a) What initial speed 210 must the trajectile have to hit a bunker on the ground located 1000 ft away? (Here, we ignore air resistance.) \m ﬁt 7“*1:' V0 Caset t )iVO‘t \Akt\ "' Vo SKae{-)i3tt : Veg-12 L-J{3tz Va 3‘2 V0 lyltr) = ° 4 9°° +v°f2 1°”-.L (zmx1 , o 1 V° 7' 2 z - V. (b) Find the maximum height the projectile reaches. .—— 1 -; )- C‘Laoo 2 Zaéﬁ “’j V0 7. a V0 : @ \rg/(lz / 5'09 4- ﬁ [09° \/S‘°= '* \r21°°> Us) meimize not): gw+on{ -121” 2:(&\= VQQ _2£ :0 J. {mas V/yMVoflé 7. ’ - M7 “'7” @V 2W“) =V°° “407/2on 0} van/Wt Weds Feb. 20th, 2009 Page 4/6 3. (20 points) Find the curvature of the elliptical helix deﬁned by the function r(t) = (3 cost, 2sint,t) at the points P1 = (3,0,0) and P2 = (0, 2,7r/2). Is the curvature constant? Here, you may us the fact that the curvature H, is given by: IT’(t)l : lr’(t) ><r”(t)| II“(t)| lr’(15)|3 ' \r’ua = <— 3S3v~£ , Inc-St, \ » —-\<v.,o m /. “7"?”— \<~’».°,\3\ 6’9“) haw/cup; —-> Cullen“: N_0_’r. OHM-cm} \— (e :ﬁ/t) —a ISLE/1A Weds Feb. 20th, 2009 Page 5/6 4. (20 points) Let f(a:,y) = —1/(1 + x2 + y2) (a) State the domain and range of f (x, y) .' and; - \tqu = vng . — 2! may 14,03 _ km) —-=. o, 94> \_ loam —5 Q —7 o (b) Draw the level curves of f (:13, y). Speciﬁcally, for z = c (c: constant) draw the contours of f (x, y) = c in the :c —y plane noting the shape of contours for increasing/ decreasing c. Sr-c a ’—’-* = wad-+3" }(‘L’v‘8‘l:—-\.L:V~ c. (c) Graph the function noting intercepts and asymptotic behavior. Note, it may help to look at the projections of f (:L‘, y) onto the :v — z (y=0) and y — 2 (x20) planes. Weds Feb. 20th, 2009 Page 6/6 5. (20 points) Limits and Partial Differentiation. Do as indicated. (a) Recall the Laplace equation in 2—D: uzz + “yy 2 0 where u = u(at, 3/). Functions satisfying the Laplace equation are called Harmonic. Find all values of A and B such that the function u(x,y) = cos(Ax)eBy satisﬁes the Laplace equation and is thus Harmonic. w ~ A W km) (:83 l Hf)! : “A2 CoSlA7‘l 68’; u ’“ﬂ 3 B CoSCAne “3‘3 :2 V52 Cost/{#3989 a. - coS(A7L) (Ba (’14 “'31) cost/xx) =0 7\ A‘ ‘5‘ (b) Calculate the Limit any way you would like (reminder: Don’t forget L’Hospital’s Rule): ‘4’” *’ “‘33 : —-51 o ‘ 0V (x,ylgrlo,0)(\$2 + y2)1n(m2 + 3/2) X:v(asa 7‘Z_\W%1 :- Y2 ‘3: v §;4\e . a. , \{«(K2*1‘)9~(i~1w°") = \\~\ V 2L( V ) id's V‘v~‘0¢A : \w. \r‘ zﬂakﬂ :- L\§w~ 24:1 a v.40» ._. v 40 \ v1 - Vﬂ‘q’ — 3’ 6 Y-Io"' ‘ // ...
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winter 2 2009 - Weds Feb 20th 2009 Page 1/6 MATH 32A SECOND...

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