{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
258 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 
0 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
263 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
273 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 
0 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
303 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 1 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 307 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
308 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
313 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
318 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 322 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
323 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
328 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 1 12 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
333 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
338 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times
" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 
257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 
0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "T
imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 1" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 
2 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 0 "" }{TEXT 258 23 "Space Curve Student Lab" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Notes to Student
:" }}{PARA 0 "" 0 "" {TEXT -1 307 "Three-dimensional curves and surfac
es are an integral (no pun intended) part of advanced calculus. Visual
ization of such objects can be tricky. You'll want to rerun the animat
ions frequently and examine the outputs carefully in order to better u
nderstand them. The objects can be quite interesting so enjoy." }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Introduction:" }}{PARA 0 "" 0 "" 
{TEXT -1 421 "The use of vectors to model three dimensional curves (pa
ths) is critical to the sciences. Whether we model the flight path of \+
an object or the structure of deoxyribonucleic acid (DNA), vectors and
 vector-valued functions play a key role in the process. The ability t
o view how an object changes when we let a parameter vary is paramount
 in ascertaining accurate graphs and valuable information about the mo
del in general." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Objectives De
fined:" }}{PARA 258 "" 0 "" {TEXT -1 111 "The purpose of this lab is t
o do the following things with regards to space curves and vector-valu
ed functions." }}{PARA 258 "" 0 "" {TEXT -1 64 "1. View several exampl
es of space curves over specified domains." }}{PARA 258 "" 0 "" {TEXT 
-1 93 "2. Determine the appropriate domains of several vector function
s and use MAPLE to graph them." }}{PARA 258 "" 0 "" {TEXT -1 182 "3. E
xamine the intersection of two surfaces and determine the vector funct
ion which represents the curve of  the intersection while commenting o
n any interesting domain observations." }}{PARA 258 "" 0 "" {TEXT -1 
73 "4. Determine if the trajectories of two seperate particles will co
llide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 323 55 "5. Provide add
itional practice pen-and-pencil problems." }}{PARA 0 "" 0 "" {TEXT 
267 28 "Time allocation: 50 minutes." }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 29 "Viewing Space Curves:(Obj.1) " }{TEXT 324 8 "Animated" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 285 "Below \+
you will find three examples of animated space curves and their respec
tive domains. Examine them carefully paying particular attention to th
eir shapes. You run the animations by clicking on the picture and then
 using the controls located at the top. Place the cursor in the red \"
" }{TEXT 338 7 "restart" }{TEXT -1 40 "\" word and press \"Enter\"to g
et started. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools):" }{TEXT 309 
53 "*Place the cursor in the red restart and press Enter." }{MPLTEXT 
1 0 703 "\nn:=50:\nfor i from 1 to n do a[i]:=spacecurve([cos(4*t),t,s
in(4*t)],t=0..i*2*Pi/n,axes=boxed,thickness=4, title=\"The Domain is [
0,2pi]\",titlefont=[TIMES,BOLD,14],orientation=[20,60]):\nb[i]:=spacec
urve([t^3,ln(3-t),sqrt(t)],t=0..i*3/n,axes=boxed,thickness=4,title=\"T
he Domain is [0,3]\",titlefont=[TIMES,BOLD,14],orientation=[20,60]):\n
c[i]:=spacecurve([(2+cos(1.5*t))*cos(t),(2+cos(1.5*t))*sin(t),sin(1.5*
t)],t=0..i*4*Pi/n,axes=boxed,thickness=4,title=\"The Domain is [0,4pi]
\",titlefont=[TIMES,BOLD,14],orientation=[20,60]):\nend do:\ndisplay(s
eq(a[i],i=1..n),insequence=true,labels=[x,y,z]);\ndisplay(seq(b[i],i=1
..n),insequence=true,labels=[x,y,z]);\ndisplay(seq(c[i],i=1..n),insequ
ence=true,labels=[x,y,z]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 338 "
You will notice that the last curve, the Trefoil Knot,  is difficult t
o discern if it crosses over itself. MAPLE has a tubeplot option to he
lp you see if the graph does, indeed, cross itself. Position the curso
r in the red \"restart\" code line and press \"Enter\" to see the tube
plot. You should click on the image to animate and rotate it.." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 319 "restart:with(plots):with(plottools):\nn:=50:\nfor i from 1 to n
 do\nd[i]:=tubeplot([(2+cos(1.5*t))*cos(t),(2+cos(1.5*t))*sin(t),sin(1
.5*t)],t=0..i*4*Pi/n,radius=0.2,axes=boxed,labels=[x,y,z],title=\"Tube
plot\",titlefont=[TIMES,BOLD,14],orientation=[20,60]): \nend do:\ndisp
lay(seq(d[i],i=1..n),insequence=true,labels=[x,y,z]);" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 293 27 "Procede to t
he next section" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Determining Appropri
ate Domians:(Obj.2)" }}{PARA 0 "" 0 "" {TEXT 292 236 "Space curves and
 vector-valued functions require their domains to be specified for the
 parametric/component functions. Find the appropriate domains for the \+
two vector functions shown below and write your solutions in the space
 provided." }}{PARA 0 "" 0 "" {TEXT -1 58 "                           \+
                               " }{TEXT 271 1 " " }{TEXT 260 0 "" }
{TEXT 261 9 "Problem 1" }{TEXT 262 30 "                              \+
" }{TEXT 263 9 "Problem 2" }{TEXT 264 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
48 "                                                " }{OLE 1 6216 1 "
[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:
I:K:M:O:wAyA::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ;B:Abm?:;JZC:bK
B::>fuQoIa_X^r;V:>bFZ:j:vCSmlJ::::::::::OJ;@jyyyyyI:;Z:::::::^<>:F:Alq
fG[maNFO=;::::::::_J;@j:j<>:yayA:<::::::=J:FG>:VZ:vCj^nGGmq<J:>:::::::
:N<<jyyiy=J:B::::::V:;J<>:;:wyyN::wyyyq<J=>:IZ:>;F;N;;j?J@j@>:wAY:[Z:F
<N<;jCJDjDJE>:i:k:m:o:q:s:u:w:y:;C:<jJ<:?;A;C;E;G;I;K;M;O;Q;S;U;W;Y;[[
:nYvY::::::::::::::::::::::::::::::::::::::::::::::::::::F:D:<::::::::
::=Jyyy;d:yayYZHQ:>:;`:Z@o<fc[_hb_ds?h_?^nn_j>^?GhoGfnGgioGSZ::RSPN]v;
;j\\@AJQLncpNdL]nIalHH<J::::::j;J::::::::::JHB:?JfhWdD:;dGU<;jvUm]^wdA
IER:LZ>WdG[;V\\QGghGbf_g<GfmGg]Wbi_f_?d[wf_WH]R;cTTUUSaEBWTSiEB_tUUURW
mBHjc`qpDpql`KfZ=wgoOhcgfl?^HgF;R>CRMcDBEDXceV=MC>[[v_?wk>w_;?dnA\\Ivb
YGbJOxYn];?dni\\IFb:^x?_x_^xIPbYGB<[=we;kAQNteli\\NtyrZ<ZJBJcy]=tYKDw[
r==dDyDKyTKcyQE\\;w[FF:=j:DJ;DJ;<J;F:<j;F:CJ:dJBL:;B:Ky:?DUeUSxj:@Z=rA
?j:N:AfVN:gvo;k\\<jrfH=MtFGYMq>>Wlj:gmlJ::::::>^:N:yay=J:B::::::^:>h<B
:;B:N:YLpJbNHEms>@[C:>Z::::::::kJ;@j;>:C:yayA:<::::::W:SG<B:::::::::::
:jysy:>:<:::::::::::::::::::vYxI:;Z::::::::J;LJ=q]wNZ>njcV^[Q:AFWHZVj?
@j:@Z<B\\_;Ob:DRPV:OZ;Z=L;LZLN^gN:KVF;j?B:Ob:::\\rTNZ>:VJ<j?:Cb[OsB]i:
@RJNZ:FJ::::V:y:rZ_M:DZP\\<^:u[=r<<j:j;Ma<JCZBP@@::h\\eN:K^:>Z=:Zc?:<J
E[>:DJ;Ia<>Z<HG?Zhc:>??B:AFW>Z<H?A:Z=PGDJ:TC^::SM;Njh;K;KVHDJ<N<rZcm;^
=j:N:D=dZ:NjcZY:VjeK;<JEK:<:::vYxy;<Z:^:<j:>Z:vYxY;>:;:_eCfb>Js\\:=j>r
:mC;>;eS:<Z:^:<J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<FZ:j<j?D:<j<J@DJjIJjElCJMT
jA^=yyyxY:<Jv>\\:B:uwyyQVyyyyyHE<kv:V[:B:G;Sj`@@RS;B:;H:MZ:Ny<B:;B:jyE
JK^:f_;j<B:cb:=b:IH;EB:;jCDjoAJqMj<:Cf:QT;E:vW]W;E:JM\\j=JxEJ:B:yQ:^:f
?=j>Jwg:>:yyy<JK^:f??JGB:Cs:QJ\\MJxAjvMjiAJ`PJhDJOQJmo^BnVGkv?q?FYeT;C
f:?T;Cf:CI;iY:OH;Ib:C:UK::c:QH<EB:V<QQs]j<:Cna`j<:YqTaj<:>fF^:;jPN:JLJ
^]JxAjx]jiAJb`JhDJQaJm?fFnV;mv_qCFYmT<Cf:GT<Cf:KI<iY:WH<Ib:C:UK:F;NY;j
C<Z:>:yi:?;f?Aj<JD>Z<^[Lf:V\\<>Z<ntNf:F[<jyg:^<cW:Wv:e:qi:?sy>Z:JBAZ:>
Z:b:;B>cTTUUSZZ:WM:^:f?CJ?JSd:<J;>Z:N`lF^W^q;f:^<WU:IB?e:qQ:uI<:[><Z<B
>:::C:Uk<^Z:JrAJ<<j>>:_c<B:=B:N`:>Z=fZ:JDDJjn_>f<VYZ:JBA:<Z<:VH::C:UK<
^:>XEjMJSd:<jD>:SSFSSG;CB;CBrVMeUKSDSeUKSHMK_HJW?k;nc=>F;Mj:nmAnLSMZOZ
;>q<nF;H:wmaHZ=fZ:JD^k=N[Nf<VYZ:JB[:b:;:VH:J:^:f?EJ<Jr_Z:F;;B:_;<j:JS<
:<:cb:WmGDkDjw;<:[V:=J:>Z<:VH::C:UK<^:>XEj>>:_c<B:=:cM:HZ:JD>FukD>:qAB
:>l;:;b::a=:J<B:Uk<^:>XCB:gZ:N@B:OB:Nhn_hnOhc_hd_he_r<fp<N]@>x=^c?N]>n
d;nn?fq;r:EB:^\\<^Kkb>eB:VYZ:JB;;b::a=:J<jP^:C:[q<F;;JSZ:F:^HHZ:JDnGIi
>[Z:VYZ:JBAJ:b:DJ:<kc`a\\wgfS:CB:;B:Uk<^:>XCj>>:_c<B:=:UC::cb:;m_@jDJS
Z:F[:jX@xPplrqmOhlPN`;fZ>Na>>p;Nf>ny>nh?^s=r:JD^KCi>MJ:N@B:=:Q[=:cJ?@J
ZF<VYZ:JBA:@Z<j`LOJP>@`;^Z:>:US:CB:C:[q<v[:>:_;<j<B:_eV_eV<j;yjUPj^PJQ
YZ=f`;JMJjDj`@PtZ_gdO_d<fZ;B;LncZ;:l@=]];:]JdRSa=:F<bRY]KfZ=;EMJt>;::N
<K:JbLYBvB<OsKbLyDB:P>rQB>H^;kAQ^dZID:<[:_X=tYKLBxjZ:xo\\RJl:<j:DJ;DJ;
<J;F:<j;<j:^:;b<=B:[B;>Z:>Wl`_s==r:xK;F:?j;EA?JEK;LJ=q=?ZcV>?j;MAl[:V[
;F:DZB\\:N[<bZS;OZ;Z=L;LZLRE>kkj?B:O::ZBh@?:j;C:Q:^:O;;RZNZ:FJ::::VZ:v
=Z=PH;b:TC^:u[=:=:AFWD:_:\\RSR::rDB:K^:>Z=:Zc?:<JE[>:DZmK:DrnNZ:bUJNB:
AFW>Z<H?A:Z=PGDJ:TC^::S]:Njh;K;KVHDJ<N<rZcm;^=j:N:D=dZ:NjcZY:VjeK;<JEK
:<j?<Z:n>^;yayI:SL:<J:F;;JwC:<:JK^:<jPf:IJ:^Y;Z::C:UK=>\\:>Z:VY[j=J:^q
<Z:FZ<bBaTXaEWEUU^:f?IJ<JrAJ<V:;B:;:::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::5:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 10 "Solutions:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 258 "" 0 "" {TEXT -1 46 "Using your answers found above, ent
er in your " }{TEXT 265 10 "Problem 1 " }{TEXT -1 44 "endpoints for \"
a\" and \"b\" in the code lines." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "restart:with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:= \+
1:" }{TEXT 326 32 "* Enter left-hand endpoint here." }{TEXT -1 0 "" }
{MPLTEXT 1 0 7 "\nb:= 5:" }{TEXT 270 33 "* Enter right-hand endpoint h
ere." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "spacecurve([t^2,sq
rt(t-1),sqrt(5-t)],t=a..b,axes=boxed,thickness=4, title=\"Space-Curve \+
Problem 1\",titlefont=[TIMES,BOLD,14]);\n" }}}{EXCHG {PARA 258 "" 0 "
" {TEXT -1 46 "Using your answers found above, enter in your " }{TEXT 
266 9 "Problem 2" }{TEXT -1 45 " endpoints for \"a\" and \"b\" in the \+
code lines." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:= -3;" }
{TEXT 268 32 "* Enter left-hand endpoint here." }{MPLTEXT 1 0 1 "\n" }
{TEXT -1 0 "" }{MPLTEXT 1 0 6 "b:= 3;" }{TEXT 269 33 "* Enter right-ha
nd endpoint here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "space
curve([(t-2)/(t+2),sin(t),ln(9-t^2)],t=a..b,axes=boxed,thickness=4, ti
tle=\"Space-Curve Problem 2\",titlefont=[TIMES,BOLD,14]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 337 28 "Procede to the next section." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 39 "Examining Surface Intersections:(Obj.3)" }}{EXCHG 
{PARA 258 "" 0 "" {TEXT -1 179 "The intersection of two surfaces can b
e traced by a space curve. The following example below shows the inter
section of a cylinder and a plane. The surfaces are defined as follows
:" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 37 "
                                     " }{TEXT 278 0 "" }{TEXT 279 9 "C
ylinder:" }{TEXT 280 20 "   x^2+y^2=1        " }{TEXT 281 6 "Plane:" }
{TEXT 282 8 "   y+z=2" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT 283 1 " " }{TEXT 284 0 "" }{TEXT 285 0 "" }{TEXT -1 307 "
We can parameterize their intersection as: x(t) = cos(t),  y(t) = sin(
t), and z(t) = 2 - sin(t). Following this example you will be asked to
 determine the parameterization of the intersection of two surfaces an
d an appropriate domain.Once agian, place the cursor in the red \"rest
art\" word and press \"Enter\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 373 "re
start:with(plots):with(plottools):\np1:=[implicitplot3d(\{x^2+y^2=1,y+
z=2\},x=-2..2,y=-2..2,z=-2..5,numpoints=1000,orientation=[30,80],axes=
boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light3)]:\np2:=[spac
ecurve([cos(t),sin(t),2-sin(t)],t=0..2*Pi,color=black,thickness=4)]:\n
display3d(p1,p2,title=\"Intersection of Surfaces\\nCylinder and Plane
\",titlefont=[TIMES,BOLD,14]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 
50 "Examine the figure by clicking it and rotating it." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 115 "Pro
blem 3: The code below will produce the intersection of a cone and a p
lane. The surfaces are defined as follows:" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "                                            " }{TEXT 288 2 "  \+
" }{TEXT 286 5 "Cone:" }{TEXT 289 24 "  sqrt(x^2+y^2)=z       " }
{TEXT 287 6 "Plane:" }{TEXT 290 7 "  y+1=z" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "restart:with(plots):with
(plottools):\np1:=[implicitplot3d(\{sqrt(x^2+y^2)=z,y+1=z\},x=-4..4,y=
-4..4,z=-1..4,numpoints=2000,orientation=[27,82],axes=boxed,labels=[x,
y,z],style=patchnogrid,lightmodel=light1)]:\ndisplay3d(p1,title=\"Inte
rsection of Cone and Plane\",titlefont=[TIMES,BOLD,14]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
291 229 "Determine algebraically the vector component functions for Pr
oblem 3's intersection and write them in the space below along with an
 appropriate domain. Then place your three functions and domain endpoi
nts in the code as directed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 0 "" }{TEXT 273 0 "" }{TEXT 274 22 
"Solution to Problem 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "restart:with(plots):with(plottools):\nxt:= t:" }{TEXT 275 32 "
* Enter your x(t) function here." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" 
}{MPLTEXT 1 0 14 "yt:=(t^2-1)/2:" }{TEXT 276 32 "* Enter your y(t) fun
ction here." }{MPLTEXT 1 0 41 "                        \nzt:=(t^2-1)/2
+1:" }{TEXT 277 32 "* Enter your z(t) function here." }{MPLTEXT 1 0 8 
"\na:=-4 :" }{TEXT -1 0 "" }{TEXT 306 45 "*Enter your left-endpoint of
 the domain here." }{TEXT -1 0 "" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" 
}{MPLTEXT 1 0 7 "b:=4 : " }{TEXT -1 0 "" }{TEXT 307 46 "*Enter your ri
ght-endpoint of the domain here." }{MPLTEXT 1 0 334 "p1:=[implicitplot
3d(\{sqrt(x^2+y^2)=z,y+1=z\},x=-4..4,y=-4..4,z=-1..4,numpoints=2000,or
ientation=[27,82],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmod
el=light1)]:\np2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=
4)]:\ndisplay3d(p1,p2,title=\"Intersection of Cone and Plane\\nWith Pa
rametric Curve\",titlefont=[TIMES,BOLD,14]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 328 48 "Question: What do you observe about your domain?" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 134 "Problem 4: The code
 below will produce the intersection of an ellipsoid and a parabolic c
ylinder. The surfaces are defined as follows:" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "                                   " }{TEXT 296 1 " " }
{TEXT 294 10 "Ellipsoid:" }{TEXT 297 25 " x^2+4*y^2+4*z^2=16      " }
{TEXT 295 19 "Parabolic Cylinder:" }{TEXT 298 7 "  y=x^2" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "restart:wit
h(plots):with(plottools):\np1:=[implicitplot3d(\{x^2+4*y^2+4*z^2=16,y=
x^2\},x=-5..5,y=-5..5,z=-5..5,numpoints=2000,orientation=[44,26],axes=
boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:\ndisplay3d
(p1,title=\"Intersection of Ellipsoid and Parabolic Cylinder\",titlefo
nt=[TIMES,BOLD,14]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 258 "" 0 "" {TEXT -1 246 "Determine the parametric curve
 for the top-half of the intersection. Write your equations for x(t), \+
y(t), and z(t) along with an appropriate domain interval in the space \+
below. Once done, substitute your answers in the red code lines as dir
ected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 304 0 "" }{TEXT 305 22 "Solution to Problem 4:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart:with(plots):with(plotto
ols):\nxt:= t:" }{TEXT -1 0 "" }{TEXT 299 31 "*Enter your x(t) functio
n here." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 8 "yt:=t^2:
" }{TEXT -1 0 "" }{TEXT 300 31 "*Enter your y(t) function here." }
{MPLTEXT 1 0 52 "                        \nzt:=sqrt((16-t^2-4*t^4)/4):
" }{TEXT -1 0 "" }{TEXT 301 31 "*Enter your z(t) function here." }
{TEXT -1 0 "" }{MPLTEXT 1 0 8 "\na:= -5:" }{TEXT -1 0 "" }{TEXT 302 
45 "*Enter your left-endpoint of the domain here." }{MPLTEXT 1 0 7 "\n
b:= 5:" }{TEXT -1 0 "" }{TEXT 303 46 "*Enter your right-endpoint of th
e domain here." }{MPLTEXT 1 0 333 " p1:=[implicitplot3d(\{x^2+4*y^2+4*
z^2=16,y=x^2\},x=-5..5,y=-5..5,z=-5..5,numpoints=2000,orientation=[44,
26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:\n
p2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=5)]:\ndisplay3
d(p1,p2,title=\"Intersection of Ellipsoid and Parabolic Cylinder\",tit
lefont=[TIMES,BOLD,14]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 48 "Ques
tion: What do you observe about your domain?" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 258 "" 0 "" {TEXT -1 116 "Problem 5: The code below will produce
 the intersection of a sphere and a cone. The surfaces are defined as \+
follows:" }}{PARA 0 "" 0 "" {TEXT -1 46 "                             \+
                 " }{TEXT 312 1 " " }{TEXT 310 7 "Sphere:" }{TEXT 313 
20 "  x^2+y^2+z^2=1     " }{TEXT 311 5 "Cone:" }{TEXT 314 12 " z^2=x^2
+y^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 289 "restart:with(plots):with(plottools):\np1:=[implicitplot3d(\{x^2
+y^2+z^2=1,z^2=x^2+y^2\},x=-1..1,y=-1..1,z=-1..1,numpoints=2000,orient
ation=[44,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=l
ight1)]:\ndisplay3d(p1,title=\"Intersection of Sphere and Cone\",title
font=[TIMES,BOLD,14]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 246 "Determine the parametric curve for the top-half of the interse
ction. Write your equations for x(t), y(t), and z(t) along with an app
ropriate domain interval in the space below. Once done, substitute you
r answers in the red code lines as directed." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 315 0 "" }{TEXT 316 22 
"Solution to Problem 5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart:with(plots):with(plotto
ols):\nxt:= t:" }{TEXT -1 0 "" }{TEXT 317 31 "*Enter your x(t) functio
n here." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 18 "yt:=sqr
t(1/2-t^2):" }{TEXT -1 0 "" }{TEXT 318 30 "Enter your y(t) function he
re." }{MPLTEXT 1 0 39 "                        \nzt:=sqrt(1/2):" }
{TEXT -1 0 "" }{TEXT 319 31 "*Enter your z(t) function here." }
{MPLTEXT 1 0 16 "\na:=-sqrt(0.5) :" }{TEXT -1 0 "" }{TEXT 320 45 "*Ent
er your left-endpoint of the domain here." }{MPLTEXT 1 0 15 "\nb:=sqrt
(0.5) :" }{TEXT -1 0 "" }{TEXT 321 46 "*Enter your right-endpoint of t
he domain here." }{MPLTEXT 1 0 317 " p1:=[implicitplot3d(\{x^2+y^2+z^2
=1,z^2=x^2+y^2\},x=-1..1,y=-1..1,z=-1..1,numpoints=2000,orientation=[4
4,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
\np2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=3)]:\ndispla
y3d(p1,p2,title=\"Intersection of Sphere and Cone\",titlefont=[TIMES,B
OLD,14]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 322 48 "Question: What do y
ou observe about your domain?" }{TEXT -1 1 " " }{TEXT 330 58 "Does you
r parameterization trace the entire intersection? " }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 28 "Proceed to the next
 section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 50 "Determining If Two Particles will collide
:(Obj.4) " }{TEXT 325 8 "Animated" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 329 142 "The following problem seeks to determine if two particl
es following different space curves will collide. The two paths are de
fined as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 70 "                                                         \+
             " }{OLE 1 5152 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:
nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::fyyyyya:nYf::G:I:K:wAyA:::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::
::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JjhYOyVPtK\\Aj;JZ]:<:=ja^GE=;:::::::::N
;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N<?R:=:EJ:vYxY:B::::::F
:;jnJ:j;<ja:MLqnFYE:;J::::::::JCB:yyyxI:;Z::::::j;>:C:;:wyyN::wyyyq<J=
>:IZ:>;F;N;;j?Jyk@JAjA>:[Z:F<N<;jCJDjDJE>:i:k:m:o:q:s:u:w:y:;C:<jJ<:?;
wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:D:<
::::::::::=Jyyy;d:yayYZHQ:>:;H:<:TNEj`@Pt\\Pd`QrPPJLMQ<MJPnrPqjLqnxPq^
;<:Z_SKGlAj\\@AJQLncpNdL]nIalHH<J::::::j;J::::::::::JHB:?jy\\W`AJ:>bpf
C>:miGILpeXvfZ;B;Lncp>@jCXontpZlPpDnjHqnHp[xPlPPb@pmPpsF\\?^dcgg_WhZnc
_whZNdigg[oG]r:aTXUeRYEU@k<HnqPQs`plDQJtnl>ZK^ZG_dZfbr_hlGF_J>@lQPnAMn
Q@NbLYBxj]xoZ<OsyKI@NbLyDxjZ<JtQNtQLtyR[xoZB>Hni@>YVKciDyDLcy]=D:<[:_x
IHby?cnA\\IFbdve>we?_xYgB@nAlj:F:=b:?b:?B:?j:B:Aj:^:;b<Cj:@JBL:;B:Ky:?
DUeUSxj:@Z<rZ_M;Njbb:HRtfB]mtFFcmnvGWMJnC==nHE=;:::::JJ<J;vYxI:;Z:::::
:J<JnAZ:>Z:J;vCS=[LsfFaMR>`:J:<:::::::>=?R:AJ:^:vYxY:B::::::^;>p?J:<::
::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::N:?VDDJ<TrXfa<^Z@H
_]Fv<rZV:sZEFZ;F:>klJ:dJ;Njfb:v@?j;c:CJCDlPDZBH`:N:KfFDZ<<A^=::\\B;:Cb
[OsB]qAF:;r==:HRsN:?FD>Z=`H?B:?VDjX:M;\\RTN:A>CJC:aC:?B:?FErZV::J>E=B:
?>f@?j;c::n@?B;d<<J:<::::jysyAB:^:F:vYxY;J:JCy:iw;nh?B:=j>r:qg:N:]A;J<
J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<FZ:j<j?D:;B:E:Sb:XJJtjCJMTjA^=yyyxY:<Jv><
<JZyIewyyyy;Y=<jTU:QB:;JMJ@fc[;PP:>l;f:<j>DJ:vy<f:^\\<^X]R;e:qi:?sy>Z:
JB?Z:B:DJ:<Kd`ppP@BBJq>:C:US:N[:B:_c<B:?J:<JSDmUTJXAj<JDVn?FwBf<VYZ:JB
K:<Z<B>:::C:Uk:^:>XF[:>Z:N`DZ:FZ:B:_C:sW:E:cJJAJng<VY;fyB:>l;Z:b::::^:
f?;J<JrGZ:N<;JSd:<j=B:KSDSsG]cFSK;@j]IJ?Oj<>RoO:Hj<<:cjaTJng<VYtI;B:>L
>Z:b::a=:J<jPF:C:[Q:V<N@>Z:>;>?csFSCEecG?R:Gg:of:WoRAJ<Om<r:EB:^<;W:ok
Djw;<:[n:>Z:>Z<:VH::C:UK:^:>X=jA>:_;<j<JsLQtLQt<JP@JIHjuDJZIZ=fZ:JDvCo
kDjw;<:[>;B:;b::a=:>:C:Uk:^:>x;>:YB:;JSZ:f:NH^h:JlIJCHjlDJ:HZ:JD>NWh:[
Z:VYZ:JB?:;b:DZJVdscRYEU@J<<J:f?;J<JrAj:f;<JSd:<J<jX@xP@YS@j]G^>r:E:cb
:YLaEjAJSZ:f:fAplrO`;n\\<vh?^t;r:]Q:<JMjeAj`@Pt:eTOQX:ER:BKa\\nR:a:l@=
=::c\\_WHZiKqF<bRY]KfZ=;E=>:j\\bV=MC>;:LYBvB<OsKbLyDB:P>:<[=SJvk?cDr]<
ZJBJcqZxQ^>\\IFBZYgB@>F:=b:?J;<:=Z:VZ:F:CJ:dJ<FZ;F:[B;>Z:>w=:xj:DZ=PH?
J;]\\<rZcM;<J;a<fAjNHljE:VZ:^=;r<=j:N:KfFb<?J;m\\<>:i;A^<^:_:F@>klb:DB
VJH::B\\>:^:O;YZ:>ZIFZ;Z=PH?J;]<HRtNZ:Njc:u;F?B\\cK;VJ^:_:V`:B:?FErZV:
:J>E=B:?>f@?j;c::n@bDB:;B:QB:n>^;yayI:SL:<J:><<jw?\\:v:>:sg:<Z:FZ<bBaT
XaEWEUU^:f?=J<Jrk;<J:<J:::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 331 107 "
Problem 6: Determine algebraically if the two paths cross for some tim
e(s), t, in the solution space below;" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 332 0 "" }{TEXT 333 22 "Solut
ion to Problem 6:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 334 106 "Exa
mine the animation of the two space curves. Rotate the figure to see i
f it appears that they intersect." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 407 "restart:with(plots):with(pl
ottools):\nn:=20:\nfor i from 1 to n do \np1[i]:=spacecurve([t^2,7*t-1
2,t^2],t=0..5*i/n,color=blue,thickness=3):\np2[i]:=spacecurve([4*t-3,t
^2,5*t-6],t=0..5*i/n,color=red,thickness=3):\np3[i]:=display3d(([p1[i]
,p2[i]])):\nend do:\ndisplay3d(seq(p3[i],i=1..n),insequence=true,label
s=[x,y,z],title=\"Particle Intersection Problem\",titlefont=[TIMES,BOL
D,14],axes=framed,orientation=[25,66]);" }}}{EXCHG {PARA 258 "" 0 "" 
{TEXT -1 51 "Next examine the tubeplot of the same trajectories." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "re
start:with(plots):with(plottools):\np1:=[tubeplot([t^2,7*t-12,t^2],t=0
.5..5,radius=1.0,color=blue,thickness=2)]:\np2:=[tubeplot([4*t-3,t^2,5
*t-6],t=0.5..5,radius =1.0,color=red,thickness=2)]:\ndisplay3d(p1,p2,t
itle=\"Particle Intersection Problem\",titlefont=[TIMES,BOLD,14],style
=patchnogrid,orientation=[25,66],axes= framed,labels=[x,y,z]);" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 28 "Procede to the next section." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 259 "" 0 "" 
{TEXT 336 32 "Extra Practice Problems: (Obj.5)" }}{PARA 258 "" 0 "" 
{TEXT -1 0 "" }{TEXT 335 78 "Below you will find 5 extra problems whic
h can be done with pencil and paper. " }}{PARA 258 "" 0 "" {TEXT -1 0 
"" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 112 
"#1. Determine an appropriate domain for the given vector function.\n
\n                                            " }{OLE 1 5152 1 "[xm]Br
=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:K:wA
yA::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JJHLTy
VPtK\\Aj;JZ]:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkN
G>::::::::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<ja:MLqnFYM:>::::::::N<<jyyiy
=J:B::::::V:;J<J:JyyI?:Jyyyyg:n:v:<J>j>Jyk?J@>:UJ:n;v;;JBB:]:_J:V<^<f<
n<;jEJFjFJGjGJHjHJIjIJJ<Z:F^:JKJyky;::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::=Z<Z::::::::::j:nyyYZDjysyAbm?:;JZC:bKg:UTR
cETcTX[US;CGWCF;SK]UW=EWMuUWM@B::P`?oBY:ETV:WCKaTMcDGLyVDurB:;::::::A:
;::::::::::sZ:N:yEp;U:<JZTm`J:FypvBSutkm<@Z>BKaTMR:arOMeU=DUSeJ=uVMuRA
tUCUS[TRIUSamBPJd`ppPps<LaPpu<Lcxpp@PqF\\=Vdsgg\\wgfS>ErJYUW_UTEeV;cME
M:@K<pNd<l\\\\QtDqjN<KRBYTKYFKYSJ[Dw[r=ItQ=DN_yAwRJ[Dwer==D:cYKcYCcy]?
tQ=\\JrJyTJvk?_xdveB_xIHb:B><NtyrZxQ^LYBxjZdliLniPNtyo\\RJYB==j:FZ<NZ<
NZ:N:=Z:V:=J<>ZCBb=J;Vjlc:DRNN:AFWDZ<H??B;GVIpM;LJ=alK@x\\FHemj^HMmqnG
;KaFFJufF>::::::;C:?jysy:>:<::::::CZ:^i;B:;B:N:YLpJb<JsfFaMR>`:J:<::::
:::>=?R:AJ:^:vYxY:B::::::N;;B:[b;B::::::::::::jysy:>:<::::::::::::::::
:::vYxI:;Z::::::::J;VJHZVj>@JJAj:@Z<B\\c;MB:DJ:DRPV:OZ;b:HB?B;h\\eN:KV
F;j>@:<j>Z<<QZ;ZBX`:R:rZV:s:j:N:K>;B:`J:B;dlK@I;B:jysyAB:^:F:;jysy?Z:>
:NbJ>:gI;nt?B:=j>>Z=Vw<B:KJZ?Z:J<B:?R:=j<B:=b:yyyyI:C:WS:mj<jRDj:<:E:Q
b:>Z:f:^[<>j>>rHV<n^@v;sjyyiyAZ:>I[B:;B:;xyvoyyyyAjHjQL:QB:<JMJ@fc[;PP
:>VMZ:Ny<B:<:b:?C:CZ:f_;j<B:cb:]d:Ku;EJ:V\\<v`<nlAf:JoEj;\\j<:_R:mE<E:
jAij=JxEJ:B:DJ:<J<jPF:M:oq<J:vY<JK^:f??JGB:Cs:QjlXj_DJKYjNDj;\\JYEjr\\
j=@jAiZ;vKuR:=F<uR:WB<Oi:ky;Oi:Qv;Gd:Su;kd:CB:f?;j>Jw?:g:;Z<N>jPV:E:cJ
bDj\\lj<>:aJbDj[qj<j>D:;b:E:cjuAJxQjDjwEjhui:>Z:JBAZ:>Z:b:;B>cTTUUSnC<
<nG;J<jP^:MJ:N`DZ:FZ:JT<J]Dj<JD^d=nxFf<VYZ:JBA:<Z<:::J<jPf:C:[Y:CB:>:_
;<j:jSnb<:cb:KW:cF=e:qQJtI<:[V:B:D::::C:UK<^:>XEj>>:_;<j:jS>r=fZ:JD>r<
>\\;f<VYZ:JBA:<Z<:VH::C:Uk<^:>XCB:YJ:<JSZ:f:>_c^_k^?uS:YE;sy:CQZC:^\\<
jVf<;jw;<:[n:<J:>Z<:VhZ::^:f?CJ<JrAj<f;N@B:C:_EWcEW_U:Cw:Ys:;x:EB:^<uB
;_D=e:qAB:>LJ:D:js::^:f?EJ<Jr_Z:F;;JSZ:F:^H;`:JDDJZ_c=>\\:B:qAB:>l;>:D
Z<B>aTXDpql`;^Z:>:UK<^:>XEj@JSd:<J<jX@xO@Yj@jbMjcLJZIJZ=Z:n>fh;fc[_Hjd
Pon@j<@:LncBG@:BU:^F:^DPps:JqF<bRY]KfZ=wGfF;cM>:j\\^H=MC>;:LYBvB<OsKbL
yDB:P>:<[=SJvk?cDr]<ZJBJcqZxQ^>\\IFBvExo\\RJj:FZ<N:?B:?j:B:AZ:^:;b<=B:
MJJ?Z=PH?J;]<;rZcE:?B:?VDb[u;F_]Fv<rZN[=BA<j>@:@Z<B\\cK:<j><:DRPN[=b<F
;@Z<rZNZ>rDJ>A=MR:B:MZ<bZZEj><Z;ZBX`:R:rZN;l;@:@Z<B\\B:;j><Z<^:O;;B:QZ
:n>^;yayI:SL:<J:F;;JwC:<:JK^:f?Ej=Jx?:<:^:f?GJB<Z:VY[j=J:^q<>:<j:DZ\\V
dsWhngfgK<jPv:C:[Y:Cj;>Z:>::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::5:" }{TEXT -1 99 " \n  \n#2. Sketch the give
n vector function.\n          \n                                      \+
       " }{OLE 1 4128 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy
]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyy
iy=?Z:VZHQ:>:;`:Z@[::JJHLTyVPtK\\Aj;J:T:<:=ja^GE=;:::::::::N;?R:yyyyyy
:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<j
a:MLqnFYM:>::::::::N<<jyyiy=J:B::::::V:;J<J:JyyI?:Jyyyyg:n:v:<J>j>Jyk?
J@j@>:W:YJ:><<jBJC>:a:c:e:gJ:v<nYvY:::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::=Z<B:<::::::::::=Jyyy;d:yayYZHQ:>:;`
:Z@o<fc[_hb_ds?h_?^nn_j>^?GhoGfnGgioGSZ::RSPN]v;fbk;n_>WdG_dnBwmcf]]:>
::::::V:>::::::::::^=<J;viRa_;>:;dGU<;jvUm]^geAIER:LZ>WdG[;V\\QGghGbf_
g<GfmGg]Wbi_f_?d[wf_WH]R;cTTUUSaEBWTSiEB_tUUURWmBHjc`qpDpql`KfZ=wgoOhc
gfl?^HgF;R>CRMcDBEDXceV=MC>[[vc?wk>w_;?dnA\\IvbYGbJOxYn];?dni\\IFb:^x?
_x_^xIPbYGB<[=we;kAQNteli\\NtyrZ<ZJBJcy]=tYKDw[r==dDyDKyTKcyQE\\;w[FF:
=j:DJ;DJ;<J;F:<j;F:CJ:dj:<J;Vjlc:DRNN:AFWDZ<H??B;GVIpM;LJ=alK@X;fb:FHe
mj^HMmqnG;KaFFJufF>::::::;C:?jysy:>:<::::::CZ:>c;B:;B:N:YLpJbNHEms>@[C
:>Z::::::::kJ;@j;>:C:yayA:<::::::OB:;B:ou:B::::::::::::jysy:>:<:::::::
::::::::::::vYxI:;Z::::::::J>E]<NZ<XF?B:?vUD:aM;VJbZSK;>kl:a=?VTDJ:nG?
j;[<`?>km>Z<^:OsB]A;:::::::::jysyAB:^:F:;jysy?Z:>:>jKjwDZ:>b=j:F;HJJAJ
:<J;FN;B:;J<>:?R:=j<B:=b:yyyyI:C:WS:mj<jRDj:<:E:Qb:>Z:f:^[<>b<>vIV<<JM
TjA^=yyyxY:<Jv><;B:;xyvtyyyyAnIjEDZ:V[:JMJ@fc[;PP:<Jjf:<j>D:;b:E:cb:;V
:CR:e:qi:;dyB:>l;Z:b:;B>cTTUUSZZ:WM:^:f_;jGJSd:<j@>:KSD_sRYuVScFaUTWEE
akP@jqEj<@J\\?v:vf;^MwqQoIGW:WN:Ij<<:cb:FAe:qA<J:<:[n:>Z:>Z<:VH:J:^:f?
=J<JrA:U:_;<J<JsLQtLQi@J=XjpLJ:Ij<<:c:Is:[B:<jw;<:[N:>Z<b:;B>aTXDpql`;
^Z:jP>:C:[q:^;;B:_c<B:AB:fa[Ix:Fd;niD>j=FN<JMjU@j`@PtZ_gdOW_;fZ;>:LncZ
;:l@==:I=^DPps:JqF<bRY]KfZ=;E=>:j\\jjN<Kja:LYBvB<OsKbLyDB:P>:<[=SJvk?c
Dr]<ZJBJcqZxQ^>\\IFBZYgB@>j:DJ;NZ:j:B:Aj:^:;b<=B:?j;EADRNB:AFWbZMK;LZw
:`lK@X;>klb:?jkB:x@VHVJbZS;K>VHNjcK:nGVJbZS;KvF;J<N?>:Q:G;Sjysy=J`>Z:>
:[B:<jw?\\:v:>:sg:B:=Z\\VdsWhngfgK<B:Uk:^:>X;j;>Z:>::::::::::3:" }
{TEXT -1 134 " \n                \n#3. Find a parameterization for the
 intersection of the following two surfaces\n\n                       \+
            " }{OLE 1 4640 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:n
yyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::fyyyyya:nYf::G:I:wAyA::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::
::gj<vyyuyJ;B:Abm?:;JZC:bKB::>^]^pIa_X^r;V:>ZAZ:j:vCSmlJ::::::::::OJ;@
jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<::::::=J:F
G>:VZ:vCj^nGGmq>:;::::::::_Z:vyyuy:>:<::::::AJ:^:>:nyyM;:nyyyYE:G:IZ:>
;F;N;;JyK@j@>:W:YJ:><<jBJC>:a:c:e:gJ:v<>=F=N=V=nYvY:::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::F:DZ:B::::::::::F:wyyAb
<vYxy;tV;J:>R<:TNEj`@Pt\\Pd`QrPPJLMQ<MJPnrPqjLqnxPq^;<:Z_SKGlAj\\@AJQL
ncpNdL]nIalHH<J::::::j;J::::::::::JHB:?jy\\W:AZ:>bpfC<J:FypvBSutkm<@Z>
BKaTMR:arOMeU=DUSeJ=uVMuRAtUCUS[TRIUSamBPJd`ppPps<LaPpu<Lcxpp@PqF\\=Vd
sgg\\wgfS>ErJYUW_UTEeV;cMEM:@K<pNd<l\\\\QtDqjN<KRBYTKYFKYSJ[Dw[r=ItQ=D
N_yAwRJ[Dwer==D:cYKcYCcy]?tQ=\\JrJyTJvk?_xdveB_xIHb:B><NtyrZxQ^LYBxjZd
liLniPNtyo\\RJYB==j:FZ<NZ<NZ:N:=Z:V:=J<>ZDFZ:N:AFX;r:l;QR:=R:DZBP@N[<^
Z@X?Qc:Hrv<j\\FHemj^HMmqnG;KaFFJufF>::::::;C:?jysy:>:<::::::CZ:^m;B:;B
:N:YLpJbNHEms>@[C:>Z::::::::kJ;@j;>:C:yayA:<::::::QB:^a>B:<:::::::::::
:vYxI:;Z::::::::::::::::::::yay=J:B::::::::V:s:=:>K<J:d::AfXH:::;:fAuc
:\\rR<J;FjZb::::jjN:t`<JlN:<<::::::AVXDJ<TBrZoM:<J:::::::yayYZ:B:C:=J:
vYxY;J:jU`;Ci:NZ>>:=j>r:=b:<Z:VS>:CZ:NZ;F:EZ:FZ<vyyyy:^:n_;F=E:]k:<:E:
Qb:>Z:f:^[<>n<>vPV<;JMTjA^=yyyxY:<Jv>\\:>Z:>rymQyyyyYJU;Yd:V[:JMJ@fc[K
dRS;B:;m<B:Mb:>Z<f:^\\<v\\;N_;f<Vy<fyx]:JBAZ:B:DJ:<Kd`ppP@BBJq>:C:US:^
[:>Z:N`DZ:VZ:>:_cF_K;IJfHJ=Dj<JD>v;FbDf<<jw?bx]:JBK:<Z<:::J<jPF:C:[Y:F
^:>Z:N`DZ:F<;jR<LJ<LJFfh_fZ>^Z>>B:QPr>H:CPj=Jj?H[=:>j=fZ:JDR:WD:eZ:VYJ
:<:[n:J:D:c<a=:J<jP>:C:[q:v;;B:_;<j<Jv`Qw`QwVi=fg=N[FNn<>j=fZ:JDR:;d:[
Z:VYZ:JB?:;b:DZJVdscRYEU@J<<:Uk:^:>x;>:U:_c<<Z:^:V_efawNj=>Eqt<;v:QH:n
>^k;fc[_HRSeTO;V:ER:BKa<@:BUjn:@@F<c\\_WH:nG]:UeRY]KrJ:t>;:fBFF_J>vCZn
A<I\\JOX>dni<<Z?;ZJrJ@>YVKd<xb:B><NtGby?C[r==<rQE\\;;=j:DJ;NZ:N:=Z:V:=
J<>ZDFZ:j;]Q:>ZVB:QR:b:\\RSB:Ob:V?rZmE:A::Z:::AfXHJ:::>:jXZBH@FJD::::F
FbUD:CM;BB::::::Vjsc:TBrZoM:V[:JMJ@vYxy:^C<J:>\\:>Z:Vy<><IZ:>:sg:B:=b:
Dlc`qsLqlp@CZ:f?;J<JrG:AJ:<J::::::::::::::::::::::::::::::::::::::::::
:::::::::::1:" }{TEXT -1 3 ".\n " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 140 "#4. \+
Determine if two particles will collide if their respective paths are \+
given by the space curves\n\n .\n                                    \+
" }{OLE 1 5152 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyy
yy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::fyyyyya:nYf::G:I:K:wAyA::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=
?Z:VZHQ:>:;`:Z@[::JJHLTyVPtK\\Aj;JZ]:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:
<::::::JDJ:j:VBYmp>HYLkNG>::::::::N<?R:=:EJ:vYxY:B::::::F:;jnJ:j;<ja:M
LqnFYM:>::::::::N<<jyyiy=J:B::::::V:;J<J:JyyI?:Jyyyyg:n:v:<J>j>J?>:Q:S
J:nYn;v;;JBB:]:_J:V<^<f<n<;jEJFjFJGjGJHjHJIjIJJ<Z:F^:JKJyky;::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::=Z<Z::::::::::j:nyy
YZDjysyAbm?:;JZC:bKg:UTRcETcTX[US;CGWCF;SK]UW=EWMuUWM@B::P`?oBY:ETV:WC
KaTMcDGLyVDurB:;::::::A:;::::::::::sZ:N:yEpsb:>bpfC>:miGILphXvfZ;B;Lnc
p>@jCXontpZlPpDnjHqnHp[xPlPPb@pmPpsF\\?^dcgg_WhZnc_whZNdigg[oG]r:aTXUe
RYEU@k<HnqPQs`plDQJtnl>ZK^ZG_dZfbr_hlGF_J>@laPnAMnQ@NbLYBxj]xoZ<OsyKI@
NbLyDxjZ<JtQNtQLtyR[xoZB>Hni@>YVKciDyDLcy]=D:<[:_xIHby?cnA\\IFbdve>we?
_xYgB@nAlj:F:=b:?b:?B:?j:B:Aj:^:;b<=B:?j;Ea=r@V[;b:\\rR<J:@j:N[<bZKK;V
jnc:ED:]mtFFcmnvGWMJnC==nHEM:>::::::;C:?jysy:>:<::::::CZ:>a<J:<:?ja^G>
D_mlVH[KR<:;B:::::::JFNZ;V:;J<jysy;Z::::::j@>:kU;>Z:::::::::::::yay=J:
B:::::::::::::::::::jysy:>:<::::::::DrNNZ>njwRI?B;GVDASr?j;K:DRPN:A^=B
AV;=R:DZBP@N[<:VJ::\\rS<:^Z@P_]Fx<RZ;K;FJ::::Z[c:nG?Z^c:>>::::::HRsV:q
:<J;>klj:b<<J;]<;rZcM;Njc:u;F?=aBH@LZerD?J>E]<rZJ;m<\\rR:J>U]<rZJ;[BH@
::;JTN:A>CJOB:<J::jysyAB:<J<j:<J:vYxY;B:;:cfD<JlE:wS;F:MZ=vm<J;>];B:<J
<J;@j:f:<j:Djyyyy=J<JQ@jFf:F`<FZ:j<j?DZ:>Z:f:^[<>n<>n`V<;JMTjA^=yyyxY:
<Jv<JBB:;xyvoyyyyAf`B:Yd:V[:B:G;Sj`@@RS<Jjf:<j>D:;b:E:cb:iR:or;e:qi:ui
yB:>l;B:<Z<>ZJ^dcgg_;<<nG;J<jP@:OB:<JSd:<J;>:_sFuX:Gb:E:cJbDjJg\\:jw;<
:[>;B:DZJ:::^:f?=J<JrA:O:_c<B:?J:F`lvcMnZ<f:^<;Y:IT:eZ:VY;dyB:>L>Z:b:<
;::J<jP>:C:[q:FA<JSZ:F^:>Z:>_c^_f>^Z>^ZZ[og^?^Z>^Z:KSD]cFSSFgCE]SFcC:U
S:cu:EY:Qf:[Mr>H:^R;I:;Qr>H:JQAjP@JYEJvDJp@joFh<^g;j]DJj?j=fZ:JDR:mkDB
:qA>Z:JBG:;J:D:c<a=:J<jPF:C:[Q:n<<JSZ:N;Nhn_hn_hl_hnCW=W:qu:=S:EY:]r<]
X:Mg;KD;wI;;v:EB:^<@JoHJB<Z:VYZ:JB?:;b:DZJVdscRYEU@J<<:UK:^:>X=jDB:_c<
B:MB:;B:uSr]YI=yDQsD]qb@jmMjOpjb@JBAjaLJ^LJwHJ:IJF@:G;Ec:UTRc=fdOG]<fZ
;Z>WDR:Zf;::^DPps::]r:bRY]KfZ=;E]HK::E<=MC>;Y<>dnA\\IvB<OsKbLyDB:P>:<[
=SJvk?cD^xID:<[:_X=tYKLBxjZ:xo\\RJj:FZ<N:?B:?j:B:Aj:^:;b<=B:?j;Ea=r@V[
;ZBH@;R:=J?bZK;AFWDJ:DrNN:p=RDASr?j;;S[:VJHZVB:Qj:b:\\RSJ?D:j;;:B\\aC:
J<N?DZ;@>?j:;::::@@nG?Z^c:>>::::::HRsV:q:<J;>klj:b<<J;]<HRtN:?VDjX:Mkj
C\\][>rDJ>E]<rZJ;m<\\rR:J>U]<rZJ;[BH@::;JTj;K<N?<J:<j?<Z:n>^;yayI:SL:<
J:>\\:B:qQBv:<J:^q<J:<j:DZ\\VdsWhngfgK<B:Uk:^:>X;j;<J:<J::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::3:" }{TEXT -1 3 "\n \+
\n" }}{PARA 258 "" 0 "" {TEXT -1 214 "#5. Show that the curve with par
ametric equations:\n\n                                    x(t) = sin t
\n                                    y(t) = cos t\n                  \+
                  z(t) = t \n\nlies on the cone " }{OLE 1 4128 1 "[xm]
Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=?Z:VZHQ:>:;`:Z@[::JJHLTyVP
tK\\Aj;J:QZ:B:F:YLpfF>:::::::::J?NZ;vyyyyy=J:B:::::::c:;:=j[vGUMrvC?Mo
J::::::::JCNZ;F:f:vYxY:B::::::F:;jnJ:j;<ja:MLqnFYM:>::::::::N<<jyyiy=J
:B::::::V:;J<J:JyyI?:Jyyyyg:n:v:<J>>:wAO:Q:S:UJ:n;v;;JBB:]:_J:V<^<f<nY
vY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::F:DZ:B::::::::::F:wyyAb<vYxy;tV;J:>R<:TNEj`@Pt\\Pd`QrPPJLMQ<MJPnrPqj
LqnxPq^;<:Z_SKGlAj\\@AJQLncpNdL]nIalHH<J::::::j;J::::::::::JHB:?jy\\WD
@:;dGU<;jvUm]^wdAIER:LZ>WdG[;V\\QGghGbf_g<GfmGg]Wbi_f_?d[wf_WH]R;cTTUU
SaEBWTSiEB_tUUURWmBHjc`qpDpql`KfZ=wgoOhcgfl?^HgF;R>CRMcDBEDXceV=MC>[[v
c?wk>w_;?dnA\\IvbYGbJOxYn];?dni\\IFb:^x?_x_^xIPbYGB<[=we;kAQNteli\\Nty
rZ<ZJBJcy]=tYKDw[r==dDyDKyTKcyQE\\;w[FF:=j:DJ;DJ;<J;F:<j;F:CJ:dj:<J;Vj
ts:l;QR:=R:DZBP@N[<^Z@hAuc:HRv<j\\FHemj^HMmqnG;KaFFJufF>::::::;C:?jysy
:>:<::::::CZ:>`;B:;B:N:YLpJbNHEms>@[C:>Z::::::::kJ;@j;>:C:yayA:<::::::
M:ws:B::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z::::::::j;^=:>K>ZD
ZerD?j;aQ:H:::;J::::yayYZ:J<j:jysy?:;:EW<^v<J<HZ:F:MZ=Nj;J;fL<J:^Z:J;@
j:f:FZ<vyyyy:^:n_;F=E:]c:=B:<j<j?DJ:>Z:f:^[<>n<>rBV<<JMTjA^=yyyxY:<Jv>
\\:B:;xyvoyyyyAjBjaD:QB:n>^;UTR^DPP:<Jjf:<j>D:;b:E:cb:iR:sR:e:qi:uiyB:
>l;B:<Z<>ZJ^dcgg_;<<nG;J<jP@:SB:;B:_c<>Z:VZ:>:_cF_kiHJ;IJ=Dj<JD>v;nc:f
<<jw?bx]:JBG:;B:;b::::^:f?=J<JrA:S:_c<B:A:oEXmmbHjwHJ:Ij<<:cZ;N`<><<jw
;<:[N:>Z<b:<kc`a\\wgfS:CB:f?;J<JrG:OJ:N`DZ:N:;jXhliHJ:IjD?Z:n>>^;fc[_H
RSeTOcR:ER:BKa<@:BU:Z[;^DPpsnC:WmBjpDpqR>H>:>:j\\jjN<vCZnA<I\\JOX>dni<
<Z?;ZJrJ@>YVKd<xb:B><NtGby?C[r==<rQE\\;;FZ<N:?B:FZ:B:Aj:^:;b<=B:Vjts:l
[:V[;Z<B\\J?D:u;>i:V::::h<Vjs?:::>:;j?<:G;Sjysy=J`B:;JB<J:<jw?<I:;JXE:
<j:DZ\\VdsWhngfgK<B:Uk:^:>X;j;>Z:>:::::::::::::::::::::::::::::::1:" }
{TEXT -1 28 " . Then sketch the graph.\n\n\n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }