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midterm2solutions - Maths 32a — Midterm 2 Instructor R NI...

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Unformatted text preview: Maths 32a. — Midterm 2 Instructor : R. NI Fernandez 15th May 2009 Name (please print legibly): ___._—_.__—...—._.___ Student ID Number: Section Number: Signature: Rami Mohieddine (1a, Tues.) Rami Mohieddine (1b, Thurs.) Patrick (Siwei) Zhu (10, Tues.) Patrick (Siwei) Zhu (1d, Thurs.) Haokun Xu (1e, Tues.) Haokun Xu (1f, Thurs.) 0 There are five questions on this examination, and nine numbered pages. 0 Do not turn this page until told that y\ou may do so by a proctor. Calcula— ' tors, notes and books may not be used in this examination. 0 You may not receive full credit for a correct answer if insufficient work is shown. 0 Where applicable, put final answers in the spaces provided. Please indicate clearly if you are using the backs of pages. — —-10-l l_--| Part A 1. (10 points) (a) Calculate the length of the path with parametric equation $(t) = sin 2t y(t) = cos 2t, z(t)=3t—-1,for1<t<3. W 3:: :1: H J7? ANSWER: 2 (b) Reparameterize the curve 6‘ cos it r(t) = et sint at with respect to arc—length measured from the point where t = 0 in the direction of increasing t. 511 :- Wfitet" abll’lf {(2 7514!— erb'fCéWJ & 27-" arr 0h; :— 6b OH’ I L I": L Z(' L e (“x ‘44)) +/"‘) = e (Mae/If) +e Mme; a?) ”F 2? . + CLL' H: e {(4)164 1h 16 "WI/flat + Wi£+Jl11€ 4' Lav)l’J,/z(‘ +1. ’34}? f 9w ”(W mitt???" 3 6 J? at 1‘: => 5: jet/353$ 7 5’ Nb") 0 $62“ 5 =5 13/4. /Jé1+l) 2. (10 points) A projectile is fired from the origin with initial speed 110 and angle of inclination a. If a: and y are the coordinates of the particle, you may assume that a: = ('00 cos (1)15 and y = ('00 sin a)t — 9132/2, where g = 10 m/sz. (21) Suppose that a = 1r/4 and 110 x 5 m/s. Find the time 151- when the particle hits the ground after launch (you may not quote any formulas except those given above). :Voal/10(_ tr {'1' fl 91 , ‘ _ 7r 5‘)? a ()1 V0510“ ’ if} % lil 7.21/15!“ (X ‘ 5m/4 T __ L a; .3) 0: VaSm o( ’0)“ Pl? (—9 J34. (b) In this part of the question, a and '00 are arbitrary. Show that the projectile will hit the top of a tower of height 11 located d metres away if 5d2sec2a * 2 __ ”0—dtana—h' L \ #04,me =5 at: vomit, W t, a, m m. M m X’bOflf/(J/Mt [5 X W! Fer hohlh M :l' Yd; fit’gflblx {/0 i l ,k yfl‘): k. NIHI 19H”; Vogl/l IX f ’ flL mat L ANSWER: . L =2) b(£.l= Vué'wh'fllL" mm 19 ’ {-4— l M Dde merx t': ifwm 00"“ AM % We)" “Md ‘ £341 136 0 M)Lo(.(§1l'5ulo( 3 MM- h 3. (10 points) Determine, with justification, whether the following limits exist. If a. limit exists, calculate it. (a) _ :1: + 3} 11m 2 2. (mm—40.0) :17 + y wring“; 6% 1+3 =—’— k—hfiwfirfim M ANSWER: M (b) , 23:23; 11m . (mm—40,0) :64 + 112 You may find it helpful to divide top and bottom by 11:2y. N s: L? p ll“ \I\ i5 ‘9 5:: IW \— \L w ('5. f L 1:1 ~ , 2 W1" 23“}. Mt w 14 MW (7’1 - 4L. 2x3 = A»: =9 IIWM._/ “"46“” T» ‘2 )t-rfl 4. (10 points) (3) Suppose that f(-'r, y) = HUM-'132 + ’9)- Calculate 3 f / 83:. {5.1L 7 [ab/2+3) + 3:.Zx. = b/llfi) +_22_Zf' 3—32 .117“; 11?}, ' L ANSWER: I” 11+ 7' 1W? (b) Suppose that sin(a:yz) = a: + 2y + 32. Calculate (92/ 8:1: (here z is a function of :1: and y). w; (lit) 2 (x5e) , l—f— 3?}; I); fifth/474$"), 21 >1 H-flL’ x; g m {ML/)3 («2+ 191;) 914— 39;— 97C 27¢ 5. (10 points) (a) For 0 S t S 277, the path of a particle is given by r(t) : 2cos(t) ' 2 sm(t) _ Sketch this path. You must indicate the coordinates of some of the points on the curve and the direction (with an arrow). (b) Calculate a(t) at t = 0, where a is the acceleration vector of the particle in part (a) and sketch the vector. «LSME‘J ("[(fl: ”Local" r'(él: L): (’ W 01/0): (2;) ML, (c) In this part of the question, r(t) is the position vector of a particle but is not the function in parts (a) and (b). Let v(t) denote the particle’s velocity vector at time 15. Suppose that v05) and r(t) are orthogonal for all t and let D(t) = |r(t)|2. If |r(0)| = 2, Show that D(t) = 4 for all t. M): {(6)166} b‘m ; We) [M + 5(qu [We] 2- zMWr’M ...
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