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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 
258 11 "Homework 4b" }}{PARA 257 "" 0 "" {TEXT -1 14 "Graphing Roots" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "This is " }{TEXT 259 8 "rootplot" }{TEXT 
-1 28 ".  It takes in two numbers, " }{TEXT 260 1 "m" }{TEXT -1 5 " an
d " }{TEXT 261 1 "n" }{TEXT -1 16 ", and it graphs " }{TEXT 262 7 "z^(
m/n)" }{TEXT -1 106 ".  All branches.  The color is defined by the ima
ginary portion, curves are the level curves of the graph." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "roo
tplot := proc(m, n)\n        plot3d([r*cos(t), r*sin(t), r^(m/n)*cos(m
*t/n)], r=0..1, t=0..n*2*Pi, color=r^(m/n)*sin(m*t/n), style=patchcont
our, grid=[25, 150], scaling=constrained);\n        end proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rootplot(1, 2);   # the grap
h of z^(1/2)" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 12 "Exerci
se 1) " }{TEXT -1 42 "Graph some of these for various values of " }
{TEXT 264 1 "m" }{TEXT -1 5 " and " }{TEXT 265 1 "n" }{TEXT -1 60 ".  \+
Determine the relation between the graph and the numbers " }{TEXT 266 
1 "m" }{TEXT -1 5 " and " }{TEXT 267 1 "n" }{TEXT -1 82 ".  In other w
ords, you should be able to look at a graph and be able to determine \+
" }{TEXT 268 1 "m" }{TEXT -1 5 " and " }{TEXT 269 1 "n" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 270 12 "Exercise 2) " }{TEXT -1 233 "The drawings for these grap
hs obviously have intersections.  But do they have intersections as a \+
complex function?  For a colored graph to have a point of intersection
, there must be a point where the graph intersects itself and the " }
{TEXT 271 6 "colors" }{TEXT -1 147 " at both points are the same.  Loo
k at a few examples.  Is there any point of intersection on these grap
hs?  Justify your answer by thinking about " }{TEXT 272 7 "z^(m/n)" }
{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 -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 16 "Final function: " }{TEXT 273 9 "rootp
lot2" }{TEXT -1 29 ".  This takes on two values: " }{TEXT 281 1 "n" }
{TEXT -1 5 " and " }{TEXT 282 1 "k" }{TEXT -1 36 ".  What this functio
n does is graph " }{TEXT 283 3 "z^n" }{TEXT -1 27 " as theta ranges fr
om 0 to " }{TEXT 284 6 "2*Pi*k" }{TEXT -1 1 "." }{TEXT 280 2 "  " }
{TEXT -1 11 "The number " }{TEXT 285 1 "n" }{TEXT -1 24 " can be any r
eal number." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 196 "rootplot2 := proc(n, k)\n        plot3d([r*cos(
t), r*sin(t), r^(n)*cos(n*t)], r=0..1, t=0..k*2*Pi, color=r^(n)*sin(n*
t), style=patchcontour, grid=[25, k*17], scaling=constrained);\n      \+
  end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rootplot2(1/
2, 2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 12 "Exercise \+
3) " }{TEXT -1 4 "Use " }{TEXT 287 9 "rootplot2" }{TEXT -1 10 " to gra
ph " }{TEXT 288 7 "z^(1/2)" }{TEXT -1 5 " and " }{TEXT 289 7 "z^(2/3)
" }{TEXT -1 23 " for various values of " }{TEXT 290 1 "k" }{TEXT -1 
14 ".  Try making " }{TEXT 292 1 "k" }{TEXT -1 62 " big: start with 3,
 then 5, 7, 10, 15, etc.  Keep in mind how " }{TEXT 293 1 "k" }{TEXT 
-1 60 " affects your range of theta.  What happens to the plots of " }
{TEXT 294 7 "z^(1/2)" }{TEXT -1 5 " and " }{TEXT 295 8 "z^(2/3) " }
{TEXT -1 87 "(you might want to do Exercise 4 before answering this)? \+
 Explain why this is the case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 12 "Exercise 4) " }{TEXT -1 
8 "Now use " }{TEXT 297 9 "rootplot2" }{TEXT -1 10 " to graph " }
{TEXT 298 11 "z^(sqrt(2))" }{TEXT -1 23 " for various values of " }
{TEXT 299 1 "k" }{TEXT -1 21 ".  Again, try making " }{TEXT 300 1 "k" 
}{TEXT -1 74 " big: start with 3, then 5, 7, 10, 15, etc.  What happen
s to the graph as " }{TEXT 301 1 "k" }{TEXT -1 32 " gets big?  Explain
 your result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 303 8 "WARNING:" }{TEXT -1 215 " If you submit th
is, please delete the pictures first (highlight a picture, then use CT
RL-X, the universal command for \"cut\").  Your pictures will be very \+
large and will take up a lot of disk space if you save them." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 3 2" 203 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }