Exam1solutionsSummer2016.pdf

Exam1solutionsSummer2016.pdf - Short answer questions(7...

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Unformatted text preview: Short answer questions (7 points each): Let the diagonal vector in the picture be denoted n and the horizontal vector be denoted v, with the magnitudes and angle between the vectors as indicated. /_#..__#_s 1. Find [v x u] N Km :. mini 3.“ 4‘: 2. Regarding the page as a plane in 3-space, does v x 11 point up out of the page toward you, or down toward the desk or table? Mr 3. Suppose v is taken to be a displacement vector with magnitude in feet and u is taken to be a. force with magnitude in pounds. Find the work done by the force 11 dragging a box across levei ground, so that its displacement is given by v. w {$.21 a -\; a WWW; 4r" :2. \‘3‘ aft ’3‘? W12. {Hi-5, More short answer questions. 4. Find an equation for the plane which is perpendicular to the line given parametrically by r(t) = t(2, «é, m2) + (2, 5, %) and passes through the origin. \ JURQJHWV‘ 'vqycfil‘wfi 0% IM ,3 a Adm“) mm... \. a.) ix. (w “ix <1, haw-”L? himrooox. 06‘6“ Tc“ 1.x ”fire ”2%“ "’3‘“? 5. Find (3,—%,1)><(%,0,w1) “9.33 all; “’x 3 \‘1 fl. is ...- .._ we .9 of “g“ $ 9* 3%\ w, BMW-45m «rel: 1/1” 0 ~-\ F... l '3» ._ (e. ”*2. e5 6. Give the coordinates in cylindrical and spherical coordinates for the point given in rect— angular coordinates by (33,31, 2) = (41,4,4‘5) If the angles in your answer are integer multiples of ar / 4 (45°) or 1r / 6 (30°), you should not leave your answer in terms of inverse trig functions. If they are not, lacking a. calulator, you answers will have to involve inverse Howw...“ trig functions. Y" 1 (“hair 9"” 1. J3: r2, (3 (1. (r‘éfl‘el'i- (afifmm 6“— MW.) W Jif/anwfimr/Ei [Pie‘k’fl gmfi .' on Law L.\(. M... f) 1:, [2:49 gZHIZWéGZE: :3 {U7 k l5 e32 :7. B 3 . . q: ._ {it w .i, (i) ”i can” (fi;-.\ (“as ‘ ( ’12,) ,.. M? b m. ,. ‘75 T?" “T’ , Ku— ( 8 ( Al“ ‘4? ) Yet more short anwer questions: 7. Find the angle between the planes 43; + By —— 122: = 1 and 4m — 3y 2 0. As you are not allowed calculators, the correct answer will likely be in the form of some inverse trig function applied to a number. VAL ‘DENW’A ("”3734 '; fl {30%ng Wef!‘M\ wad-00 93‘) Z3 3 Wm 3: .1, <3 Wm» 1m»: 1???th fit? :13 mt»— from“? ; film My («3‘3 0 .3, Wm \6 Wei «\ o w; ”it 3:; W‘C L03 {51A, 8. What is the magnitude of the torque applied to the bolt when the force is applied to the wrench as illustrated? The last short answer questions: Consider the equation m2+y23422_1 9. What type of quadric surface does it represent? 1;, 4. 1.. X +3; m «m. em ”X331“ M122: i mmm e we we 10. Express the same surface by an equation in cyiindrical coordinates. r “ ,2” as"; r .2“! 9 r1”: 4}“?er \ of F were: WJ 0/47;th r13, 1 11. If the term involving 3;, and only the term involving y, were negated, what type of quadric surface would the new equatiOn represent? 3M, w x‘“ +53» 417‘“:- J W 1, L. +~ on Ix‘za‘o‘hw «w WW Long questions, point values follow the question number in parentheses. 12. (2Q) (21) Find an equation for the plane passing through the points (1,0 1), (1,2, 3) and ( 1, 2 ,.3) p if, 'f {1m ““1 VEW) M {9W2 ‘ FM MIME e «A ”3" O '1 2 > w) .._ hi 3 It" % ” V ”’ < l l V\ “" O L ”2“,“ .2, 1 “5"”? w— {VL12' > ”2»: Z, L, [042 “S— 35 21 ““2- % 7‘20”“2. (b) Find the distance from the point 121,0, 1) to the plane 111 part (a). Am» 1” (4,0,2, 20,0. l) 1 0,0,”‘0 13. (211:) Find 3 / 3i JV sin(t)j _ tea—tk dt 0 and give a physical interpretation of the value if the integrand represents velocity in Ill/Sec and the variable of integration represents time elapsed in seconds. T \3 («at pp 9% ”3 w" “ W “7E ‘1 k" ' we : g£\ mm +Q<1 M, 3 0 9, flag 0m, 1‘ SC” A Vme’fi‘ “m CT] W (593(3)? —+- 46:3 I“: . :2, u, {mat at 'W W} (ue' H Fwfi' Mfld Pu,“ q"? + (Museum + may) k “3: <61 New» I 4’@’3«)> H wrmfina“) M Fogjkowx «3% M {with “*9 v . 4w :w mm 63%; AN“? »% W R) (/{V‘AGI (”ka /\ POM h I fix MG ”RC; ‘5? +9.“ '"%WQ (:3 4-2an 7 14. (25) Consider the vector—valued finiction r(t) : cos(1rt)i + 35in(7rt) j + tk as a trajectory (i.e. giving the position of a particle at time t, say in seconds after some chosen time t = 0 as a displacement from the origin in some units, say cm). (a) Find the velocity function v(t). “(II it); ”(ti/{t \ a: m. ’Ti‘ gmflt’t) T +31? (03 (TM) 59 'i' Ic; (b) Find the acceleration function a(t). WM: FWHH: m:— "it"LCeglntlT wBTJ‘ SMW‘t‘fi Vail“ (c) Find the v(1) and a(1), the velocity and acceleration at time t = 1. “.23 WT: "TV- 0 T + Bet-«ct :— n T: «an: H: $40533 J) fitllm 'w/t’tm‘i “' “3:: ct 2:: xJJalf (T130 0 ((1) Find the Speed of the particle as a function of time v(t). m new», ._ , ~ , - m W, w». WEI/till Wi‘tSmfl'm—T tfiirl’mgtm-H J J (e) Give a. definite integral expressing the length of the trajectory the particle travels between time t = 0 and time t = 1. DO NOT ATTEMPT TO EVALUATE THE INTEGRAL. .1, Jr“ JJ w , My»; ...
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