V o. 'f o. 'NJ. 'Ni c.. l ~ u u. YI \lv\ ~ ~ f fov c.tr '+- Cu w11.t r M o'n ~\'1

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$la~t ~'fr\l. 1.. i~: ~ =-M:X tb Cirili. lx-~~t ly-~)\ fi.$ $t 0 V o. 'f o. 'NJ. 'Ni c.. l ~ u u. YI \lv\ ~ ~ f fov c.$ $'J. ~ lv\ \:it If (.'\ d t ~'4~~ \lcdj.u.. 1.1\ t- i\\j.$

1 la~t ~'fr\l. 1.. i~: ~ =-M:X tb Cirili. lx-~~t ly-~)\ fi. Tu~ll~: ~ Hli'JJ'"~ \WWI 1 we' II 'vj YH'\~ i\i\ M1t. t 0 V o. 'f o. 'NJ. 'Ni c.. l ~ u u. YI \lv\ ~ ~ f fov c.tr '+- Cu w11.t r M o'n ~\'1 ~I \IJ.f>.( M o Y\ l'l V\. 'J. ~ lv\ \:it If (.'\ d t ~'4~~ \lcdj.u.. 1.1\ t- i\\j.f~ ~OllA~ I ti.. l:y1"\) ~-?b\\/.\5 w\\ 1 \o.\u. \A\' -\-LU. ~\.U

$Example 1 Let ~ -?D\'A\ S t~o!-:.j x = 2 + 3t ~~'{'NJ ~ -6\M y = 4. + 5t.... - 'X: "-. 0:.$ $T (UV\ 5 "- ~ - 2. Eliminate t to ge t th e standard equation of the line in th e f orm.$ $If t is in seconds and the axes.. s ~ ~ :. (~~~"''";~ ~\u;~cr.;;.'p-eed of the paclide?$

2 Example 1 Let ~ -?D\'A\ S t~o!-:.j x = 2 + 3t ~~'{'NJ ~ -6\M y = t 'X: "-. 0:.4 X:.5 ~~q 1. What is the slope of the r me.? s\l:)\)e = r i ~e. -:. ll :: ~. T (UV\ 5 "- ~ - 2. Eliminate t to ge t th e standard equation of the line in th e f orm. y = m x + b., x32- : t :: ~~4 - ~ T (x-~) = ':f ~ -I ~ '- t ():?-h4 _, ~:. ~.X.. +} 3. If t is in seconds and the axes.. s ~ ~ :. (~~~"''";~ ~\u;~cr.;;.'p-eed of the paclide? t V\(t;~ '"'Ii ~ 1-0 :: I y \\\ '\'iu~o..e(l1\ Tu r ( WY\'\-: 5 2.) \.. f t~-~) z...-: Jiitif; ~ sy0.. ~-: ~ -=- -rw.. I 2

$Example 2 WO\ \v'll~ 'ou$ $\~oy\ ~1~d the pair of parametric equations for a particle movin.$ g on a straight line with constant speed if at t = 0 ~t t~o x-:.:t.

3 Example 2 WO\ \v'll~ 'oul\t,wu ~ r \UL -equ.o.\~oy\ ~1~d the pair of parametric equations for a particle movin. it is at (2, 5) and at t = 5, it is at (12, 35). g on a straight line with constant speed if at t = 0 ~t t~o x-:.:t. ~ ~ ~+b O ~ ~~ t-=- IS x.::o.tbt NJ.~ '. Q,'v.> J l, cl. ~ ':. 5 5:C..td O 8 1 -: 2. 't 'o-t '6: s +~t ')(-:. \ ~ \"-:'l.+b 5 '&~~ 2.. : \?- '-6: Sc.3 ';5 : 5tcl. 5 ~() : '5d a ~~ uh 3

$Example 3 J.A,\.J....ljj_. \ \J~ \AJ~\...4 ~ U \\n t l \ S, 0 Let lll\v..<..ci 5 x=a+ ~ y=c+&. 1.$ $What is x-~ :: t ::. ~ ~ d sl l~ -o.\:: ~-C... e. ~-:- -t(x-o.\'rc..=}2:.+[~ 2.$ $What is the speed of the particle, in time units per distance? t -: o ( o., c) e, \\u V\ ~ ')'\ hw -= \.. o -: l i-:.$

4 Example 3 J.A,\.J....ljj_. \ \J~ \AJ~\...4 ~ U \\n t l \ S, 0 Let lll\v..<..ci 5 x=a+ ~ y=c+&. 1. If e =I- 0 and d =I- 0 eliminate t to fi d~ l ':: the slope? ' n ie equat10n of this line in standard y = mx + b form. What is x-~ :: t ::. ~ ~ d sl l~ -o.\:: ~-C... e. ~-:- -t(x-o.\'rc..=}2:.+[~ 2. De;mb~t:\lin~~rt '' tb 'lope' s\o~~ ~ -\Y\ \41CJ.~t ~:: C! <!--. Cb n ~ \ o "~ \J o, Y\' c d 3. Describe the line if d = 0. What is the slope? t~o i::c., ( What is the speed of the particle, in time units per distance? t -: o ( o., c) e, \\u V\ ~ ')'\ hw -= \.. o -: l i-:. I ( a.+e. I C:td) c_'y\u.'i\~ \Y\ ~ \\u.yh.l -= J (ei.te -c..) \. _. {c..td -c..) l... s\)q.lc!:: J tl-\dt ~ I ~ f ei +d"j...jet-1di' 4

$: (1,0) t=-i ltos~,si~i\: (0,1) 1.$ What is the -~pe linear s :;;;,;;.

5 Circular Motion Basic Example The U n1 't c ircle Y = sint - i ~o ~- l io~o,smo).: (1,0) t=-i ltos~,si~i\: (0,1) 1. What is the -~pe linear s :;;;,;;..; e d m. cent nneters. per second? t'i\o~ - il\ w\t\~u... t ro i?f- "'" 'ny'm: 5

$I -) ':i,). 4. ~ \.-;; 71.... t' -- "\\\b.l ~:O l { ) 1 0 +:-!!.. '3t:..[ t 0,, ) -r-+--i!$ $\ '{\ ~\CA'/\ U... _ CA 'f W 'M\J W \ f ~\l. ~ - T u 'ti Ul.;- ::. l~l oyv-j.~<a. lwr\~ 2.$

6 Example 2 A particle goes around th e circle. with r a d" ms 7 centered at the o ngm.. with. the position at t1.rn t. x ~ 7 ' '~). e g>ven by - y=7sm~) C.O)t l3t) 't SlY1'-(31:) :- I lt) 1- + Pt )1. ':. I -) ':i,). 4. ~ \.-;; t' -- "\\\b.l ~:O l { ) 1 0 +:-!!.. '3t:..[ t 0,, ) -r-+--i!. *~ ~ /; 0 b - ~ 3t~~rr wi\\ t.aw~\j.\e.. I ~1Jv\u'non t~~ What is the linear speed? \ '{\ ~\CA'/\ U... _ CA 'f W 'M\J W \ f ~\l. ~ - T u 'ti Ul.;- ::. l~l oyv-j.~<a. lwr\~ 2. What is the angular speed? w) SL, w t'icv (eao) \fl \ n-jo\\a\\o~ : ~: 2.ir -:. l rnd/b))\t hvw.. u/:; - unit 6

$G e Y\l '< u.lr~t ~ 9(::. 0. + fw~(w-t+&) ~-:. 'o t rs.$ $.tld1;tjs ~ O-Y\~f" lu,'o) Q..: 1Y1\YHJ uy1o(1.$ $\ l ~ly\ & \'I\ 'NI o~ \? VJ'vl \t. "1'\S & = 0.$

7 G e Y\l '< u.lr~t ~ 9(:: fw~(w-t+&) ~-:. 'o t rs.iyl(wh-&) ica~\j.~: f Cl'fl~u.!Uf s.\)ttk: W i\t..tld1;tjs ~ O-Y\~f" lu,'o) Q..: 1Y1\YHJ uy1o(1. )'I 1 SlYl&- w\jj."' t : o :x.. ~ '(' to~& t (.&.. ~ ~ \:? \ l ~ly\ & \'I\ 'NI o~ \? VJ'vl \t. "1'\S & = 0. W ~ ~W- t\'\oo~\yi~ Hu. CA)lf ~ 1 \JJ t. W..!J. \;J._ Cx'f\ 'wr (u,'ta h.. lo, o).

$f:~ ())\~Y (\,~) ilim,y\a.\e i.$ $. co~ (~t:t f ) t s1y1 l \H~ ) ::.$

8 Example 3 x = 1+2cos (3t + 7r/ 6) Y = 3 + 2sin (3t + 7r/ 6). f:~ ())\~Y (\,~) ilim,y\a.\e i. t.. t.. co~ (~t:t f ) t s1y1 l \H~ ) ::. I t;') ~ -t L~f ~ I l?l - 1\ 'L + l ~ -., \1--; t-:. 0 8

$... v 1.. "Tl"' II In. 16 L Head. St"'t drnwing at th e n. ost pomt.s t = o. : CJ. fl \4 '( lb,5 '((A \I.~':. 4. ~ 1\\'l... - \)-:. Wf, W-:. ~ : JL _. (\ -:.$ $O r ~ 1l ~ ~-;~T4to~(tt-l ~:.5t4~tYl(~t) -I 0! 2 l 4 ' 6 0 ~ t ~ g 2 M:~th Dmw ;. ' ''' l \fo.u.$ $j ': ~ ': JL ": ll r ' S' ~ 1_-: b +1 5C.O) ( i trr) "..lt 1- l 'ry(ati o~u_\ v - " ~ - I - 3. Eye on the left. Start ' drawing at ti ie t op at t = o.... 4.$

9 Example 4 Draw this winking face made u. parameter tis time in seconds. ~o~f c~~c~1~~..:!a~ s~:~~ :rn~.a!line using parametric eqrffl s whhe the - me ies per second.... v 1.. "Tl"' II In. 16 L Head. St"'t drnwing at th e n. ost pomt.s t = o. : CJ. fl \4 '( lb,5 '((A \I.~':. 4. ~ 1\\'l... - \)-:. Wf, W-:. ~ : JL _. (\ -:.O r ~ 1l ~ ~-;~T4to~(tt-l ~:.5t4~tYl(~t) -I 0! 2 l 4 ' 6 0 ~ t ~ g 2 M:~th Dmw ;. ' ''' l \fo.u.. QY\lL ') counterclockw1se starting at the Jtftmost point at t = 0 Ct 'f\\.,d ' 14) 'f CA lii1,1 ~ ':. I.S- 9- ::: 1T l.j.j ': ~ ': JL ": ll r ' S' ~ 1_-: b +1 5C.O) ( i trr) "..lt 1- l 'ry(ati o~u_\ v - " ~ - I - 3. Eye on the left. Start ' drawing at ti ie t op at t = o Eye on the right. Start drawin L \ N t: Xa~ t~ ~~ft point at t =~ ~ ct tit (.(,t {:: 0 :x_:ls L~ :.~] ~o Cl"=-7.) x_-;1.5+bt X-= 1.5 tti t F n1\5n s)>q-ld:. h,t."<di.. 9 :. \\,\=TI 0 ~ l 6 t L'rr"' LI. OV\tl). 1. St 1Tt: 8'.s- -') t ': ~ TT

$&td1)\ I :Xtt$ ~/=5~ t-~o -t

10 Example 5 What if you want to go clockwise? Just put down a negative angular speed! x = 5 cos ( - 2t) y = 5 sin ( - 2t). &td1)\ I :Xtt ~/=5~ t-~o -t -:.1!. { 5 IQ) l 5easf ~) I Ssm/-~)) ": ( Q I -) ) 10

11 Are there any other pairs of parametric equations which trace a circle? Yes, although x = a+ r cos (wt+ 8) y = b + r sin (wt+(}) is the standard form. Example 6 Sketch the circles below. For each of them, mark the point where t = 0 and another t value to see if the motion is clockwise or counterclockwise. x = - isin(t), y = 2cos(t) x = - cos(3t), y = sin(3t) 11

c) LC=Cl+C2+C3; C1:x=dx=O; C2: y=1-x; C3:y=dy=0 1 ydx-xdy (1-x)~x-x(-dx)+ = Jdldx = 1.

c) LC=Cl+C2+C3; C1:x=dx=O; C2: y=1-x; C3:y=dy=0 1 ydx-xdy (1-x)~x-x(-dx)+ = Jdldx = 1. 4. Line Integrals in the Plane 4A. Plane Vector Fields 4A- a) All vectors in the field are identical; continuously differentiable everywhere. b) he vector at P has its tail at P and head at the origin;