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Exam1-solns - Math 2110Q Vectors and Vector Functions V1.0...

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Unformatted text preview: Math 2110Q Vectors and Vector Functions V1.0 — Falé - 2010 1. Intersection of iine and plane: (6 points) Find all points (if any) at which the line marl—1275, ymélt, 2w2~3t intersects the piano 9: -i- 2y — z + 5 = 0. 11121:) .1131 [1): )1) —— (11'519115 0 11, 11,1 1,1 P‘W‘W 1111 t g” 1/43 '1’ %% .11 '2" if g-«b .1, 5 :p- 0 UN § 1‘1113 123*, Jr "‘ :21- _ "Li L) O /"\7 4:1 115 O 5 7:. ~— 52 .1 1.“ 37C. l W :3 1'21 1 1 11 w- —— :21 1'5 ,3 2. Tangent line: ( 6 points) Find a vector equation of the tangent line to the curve r(t) = (St sin 5t, F2, 6* cos 5t) MachlIOQ Vectors and Vector Functions V1.0 - Fail — 2010 3. Curvature: (6 points) Find the curvature of r(t) 2 (2t, t+ 1J2 «i— 1) at t = 0. K (H W ”(a 34 "max _s./ \ jl:[é}: {123/52't7 ”é‘p/CJC): <G/\0)1> V (0”) :> (1, U W :33“; \f @331?“ 4} y! ‘f. ‘\ A A Y" (OM/(61%;? figfiflfla r Li’i‘xgg ’rlfifla : $3431; {($765) riflfiofl W ' My («M2 w 5 6’ ’2; M 5 4. Vector algebra: (4 points) If a = 4i —— 2k and b = —2j _ k, caiculate compab. ”g k—V—uw/ WW"‘?“?R\0 H; . " “$501338 - 61 [9 --— QE'MQWPQEQ 1 w 3"" — W W ”“3" {Ii-(0MP 1‘ Mm 3"? Math 2110Q Vectors and Vector Functions V1.0 — Fall - 2010 6. Reparametrization: (10 points) Reparametrize the following curve with respect to arc length. r(t) = 3‘: — etj + 4k {in \ g (”351 Z: 36 ”I"; ' fl an x “m {L ‘\ .6 TE}? m m M» a EWb 3 W, My Jo w—u—r—A' "‘~. r W (“I i J 22w 0;) {W 1 m } v 2 UK \‘ (AW JL 3:1» 0 (9m. 6 Vii“ -: “\l 3% Q/ C;\JU\ : f\ :2»— );3/ j ‘5 Ji (6 f E) a W M 5 ~ :3 , h - 3”“ ,_, -~.- :L. M ';:':1=--r.\__i,r v, ) ”h \ é"; mm 0.4 mfg-fin m, .12 (M i A“ E :M 1 E Math 2110Q Vectors and Vector Functions V1.0 — Fall ~ 2010 6. Equation of a Plane: {10 points) Given that the two curves r105) = (t + 1, L4 ~+« 2t ~1— t2) and 1-203) 2 (3 — s, 3 — 2, s2) intersect at the point (1,0, 4), find an equation of the plane containing both of their tangent lines. ’fi ’ M. i J... , 2 \t—‘r \ “’ t M? 1:j;~~ O W ..:"“\ .- 5 fl “8 '3 i 1:111); S - 1/ TM J 2 “- OJW‘ vac?» E? t R ewe “r“ f" «E :3 , if: (“luv 23;: K x 2} \ 2%)? W,” “3) ff.“ '1 RN) .4 x3 ) _- 1 m .\ i 1’ CM“ , ”(fl r<W1 1 35> EEK) .. A” ’) ’9 w \M3 w E Ll) ; \ x F “1,. —...\ . w ’ ,i 1C3 (3-9“ W lat-)1 u\\ ‘1 JR”; ,. 1 I ‘ .r t .. a"), if} Q Fanfnpfikfi‘ ‘ E“; "~ ‘f\r" , if} e "gin :1 <E j '5; } 2/7 7Q <_M1 \ I :1; “1 _, / (j; :r FM is a? 0;: t j j I M ”“17"" ‘ a u «x \ R \ i Q N ( INS” n .-"‘ I I: Math 211%) Vectors and Vector Functions V1.0 — Fali — 2010 7. Features of graphs, mostly: { 8 points) For each part, circle “True” or “False” as appropriate and provide a one sentence justification, which may be accompanied by a short computation if necessary. Only parts with correct ream soning wili receive any credit. You do not need to make any sketches as part of your answers on this page (though you are welcome to if it helps you). (a) The surface 2 = 3x2 + 4y4 stretches infinitely far both upwards and downwards. TRUE @324?) I - a 0 Vi h; m eon W‘Yfi Q QiGKdVVQ ff: \f'fik,lfij»._figj {fig/we Era-ya £31k“)??? I I (”at g “ \ M ; . I: h ,, A ,.-‘|'\ '- ‘ fan-‘5 -. r‘-' .4 a in“ {ca t: a a: s r { 0 .HK- a»... 5,? :9 .-. , , M-‘r ‘p‘ 1‘ W .' . ' . ..'~ . . r. r. , a. 5%. ix} 5er v‘Lé‘v‘£’-’t.+..g,-; of?” f . " , : .r‘- ' .- ‘-....c / r " e (b) The traces of m = 23:2 — 222 in all planes parafiei to the coordinate planes are hyperbolas. TRUE @ (c) The curve r(t) = (sin t, cos 12, int) can be contained in some sphere of finite radius. TRUE €1.39 0"“. S k; “mt? m ‘Z/LQN w r i ‘ ’ _. I" l? w .: :_ €3.35 r." :1,- :3 c - r ,7 -«_ J f c. Us} r~ arm 1‘ Q) \‘r {1% mm : 2‘ Wk 5 .4.“ a; A . V ‘ (“can i r r a cm i V\ Lari/W1 one” «3.63%er (6) (More diflicult, save for last!) if r(t) is adifferentiable vector function; then £11123] = |r’{t)|. (Hint: see if it’s true for a parametrization of the unit circie.) as =2 am, my ® ”0%" Qfilkafl’ ...
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