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**Unformatted text preview: **3. (a) (2 pts) What can you say about the acceleration of an object that is moving at a constant speed? (You may wish to give multiple
answers here. Justify each one briefly) If 5/0/260I I5 (II/I,2'/22/,II/ 47.4““ :II/IZI/I (2) / 2% (j ::,~..~~ a”, /\// :{I III ﬂi/ﬂIKZ/‘WIIA I6 III “III /IIIII’I2II aII/'I1€1€1I'222ﬂ/;;”2I pK/Pﬁ/WIHIIIW 20 V (Ill/10114 2/ {MW/l: 2f, IMHO!" ~l/MIUUI Iu2IcIIIA WIII 20/122223?! MIj/IIIWIQ l5 22‘7“?” 2L II? I. ~
(b) (2 pts) If the acceleration vector of an object is always in the same I 5 2221* I“ )
direction as its velocity, what can you say about the path of the
object? (Again justify your answer.) II 27? II 127/ IIIM aI 222022 ION/WM 26 I/\ III 722/227422'2/
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II If) )4:0212IU‘/“72 I TIM)! “II/2 (Ii/gar [5 Mﬂl ///)72W/:/{,Z/ﬁ
k k, (c) (2 pts)m hat can you say about a curve that has constant zero
torsion? Be speciﬁc and explain your answer. 'l m '1 “I w
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III/I III 22.5 I‘I/IIII/I 22/2; IIQ:../I“ZVI§3’[ (I222 III/”2” I III/ 6” ’ "" " ”’WM i2 ‘W . 2 ’2 4 2/
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N V J l/II/ curve I225 III 2. 0/2/62 (d) (2 pts) If a curve has zero torsion everywhere and has constant curvature then what type(s ) of curve could it be? Justify your
answer briefly. I/lJ I/I (Ié’f'lv I’I/ EFF]? {pp/M 15-71722 (II/W: I256 III 22 ”/22 II“ 0:5, //2I2I/W/ pufW/fj’ Mfr/II (Mg/22:42 Curt/WWW 2/6/29 3 2 2 '1 ‘ ~1- 29 . , e2 22/62/“2/2/ 25:22 22222 (22222221222212, ./’I/‘II/II1/\(’ I2“ (I...) “M! 61”“ ( __ (I? If ,_ . I
iMT/Ii 4. (26 pts) An object is moving along a path in 3-dimensiona1 space so that
its position at any time t 2 0 is given by r(t) 2 <3 —s4sin(t2), -—2 — 5 cos(t2), —1 + 38in(t2)) (a) (3 pts) Verify that the speed of the object at time t is 1015. V555 ” <5“? {5% 5555; 555(55 51,) {5555/5525} : 25;< 45/5505: 5/5 5/5/55 5555/5 5:?
((75: [52:5 <55 55/ ‘5, 2/555“ 1555/5 (5/5555 5:55)
55 25 5M 5555 5/57 555512777 5555 55 5 ~— 2552 5/5 5 Note: The remaining parts of this problem may be done in various orders. For example,
you can answer part (d5 even if you don’t get (125 or {05. (b) (4 pts) Find the distance 5 that the object has traveled, as a function
of t fiom the point (3 —7, —1) to its position at time t. 55 (5: ﬁ(/ ”5 Z: {::;O’ (c) (2 pts) Reparametrize the object’s path with respect to arciength
(measured from the point (3, —7, ~1)). " 2 22,222 ,5, EM (d) (4 pts) Find the unit tangent vector T and the unit normal vector
N as functions of t 733 V r <<ycé2122 2332M 23/3302)
“” (Ml (02 /< ”:22 “ 2 2> / _ h- MM 2 ‘
(tuft/33,7533.1%?» Ag 3 3
'2” 5 333/232 332,, 4 at a” 03/234 / (f) (4 pts) Find the normal and tangential components of the object’s
acceleration as functions of t. 3 ,,,,, 3/ ML/M/i/ (g) (3 pts) Find the object’s acceleration. Use this and your answers to
parts (cl) and (f) to verify that a: aTT+aNN at all points on the curve. 1: <:lz¢o3ééz)Nl/(ltgé/Mﬂzﬁlb/ [ﬂy/W [3/03 UM [I {435({3 2H [21: 233/!(3' if} M}< €13,333 {2 .S’fl [M61 9531§L(2)>[’ Zlolf<iﬂﬂtvf0dj4ﬁ§l ) Clv “la! i <XW/l ”(l 3) W236”) t2/ifc336’ﬂ/ ilt/3l’f/[N/2t23’fﬂhi‘2’9 , W/ (h) (2 pts) Find the binomial vector B: T x N as a function of t
Based on this, what can you say about the torsion of this curve? l/
13*” (0/1 1 gm?) 31616166)
515M011 4051,15”) .1 6'11 ) : “'311‘16125’11 K111 15 ”151/1 1111 27'1”) 7 1 ‘1’ 9741/1” ”1 11111; $10126 12> IS 1, {£421,101 1/ f {‘1 ’1 ”A / 1111;”: Bonus: (3 pts) Based on all of your answers above, What kind of curve is this object f011owing? Be speciﬁc and justify your answer/
’ 1 51/11/61 11116 6116 1/1/11 / 1/1 WW1) 17- 111151 11/1 my,» 1 11111” /’WIW// I1 61/111 111 11) 1
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1m 3x111£ :5 11 16, 1316/ “11’ 1/61/1116 111611 1111‘: 11 1
11111,) 6. (10 pts) For each of the following limits, either show that the limit exists and ﬁnd its value, or show that it does not exist. . my —I— :1: — 2y — 2
5 t l ———-——————
(a) ( p S) (aw—133,4) (cc — 2)2 + (y + 1)2 W W i/ Wm (WA) 0 ‘/ p
AM {PM 7 ' A: (A?!) {f} :3 (7:) W i AM Ala/WI, me if 270 WWW 0/)” ((31% l}?
I Z W W 2 J i
W F (ST 7jW/AW {2/2 "/Aze (WI/i (ix/M
2 3 Am‘ 5%: W / / W W W
4Q B lwwmmW W W WW WWW,” HM " ’ (0/ ’A/ MW AM i r“ (“OH/Ari 1A 5/ Am WW ' A) ...

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- Vector Space, Acceleration, II III III