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{SECT 0 {PARA 0 "" 0 "" {TEXT 256 23 " Worksheet #6  Math 151" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 35 "1. Plot \+
the graph of the function  " }{XPPEDIT 18 0 "f(x) = x/(4+x^2);" "6#/-%
\"fG6#%\"xG*&F'\"\"\",&\"\"%F)*$)F'\"\"#F)F)!\"\"" }{TEXT -1 25 " on t
he interval [-3,3]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }{TEXT 258 185 "2. Use Maple to find the largest inte
rval  I containing the origin in which this function is one-to-one. Hi
nt: find the endpoints of  I by differentiating  f  and  solving the e
quation " }{XPPEDIT 18 0 "df/dx = 0;" "6#/*&%#dfG\"\"\"%#dxG!\"\"\"\"!
" }{TEXT 260 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 262 61 "3. Use Maple to find  J, the domain of the inverse fun
ction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 
23 "4. In order to obtain  " }{XPPEDIT 18 0 "f^`-1`;" "6#)%\"fG%#-1G" 
}{TEXT 265 37 "  , use Maple to solve the equation  " }{XPPEDIT 18 0 "
y = x/(4+x^2);" "6#/%\"yG*&%\"xG\"\"\",&\"\"%F'*$)F&\"\"#F'F'!\"\"" }
{TEXT 305 7 "  for  " }{TEXT 306 1 "x" }{TEXT 307 20 "  as a function \+
of  " }{TEXT 308 1 "y" }{TEXT 309 57 ". You will obtain two solutions,
 which are functions of  " }{TEXT 310 1 "y" }{TEXT 311 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 134 "5. In problem
 4, determine which of  the two solutions is the correct inverse.  (Hi
nt: it should be the one which looks like  0/0  as " }{TEXT 312 1 "y" 
}{TEXT 313 13 " goes to 0.) " }}{PARA 0 "" 0 "" {TEXT 269 6 "      " }
}{PARA 0 "" 0 "" {TEXT 268 11 "6. Replace " }{TEXT 314 1 "y" }{TEXT 
315 6 "'s by " }{TEXT 316 1 "x" }{TEXT 317 253 "'s in the function whi
ch you chose in problem 5 and call the resulting expression finv. By t
aking the reciprocal of the rationalization of the reciprocal, put fin
v in a form which shows that finv(0)=0. Hint: use the command: finv:=1
/rationalize(1/finv);" }}{PARA 0 "" 0 "" {TEXT 270 1 " " }}{PARA 0 "" 
0 "" {TEXT 280 37 "7. Plot f and finv on the same graph." }}{PARA 0 "
" 0 "" {TEXT 287 9 "         " }}{PARA 0 "" 0 "" {TEXT 284 29 "8. Defi
ne a new expression h(" }{TEXT 318 1 "x" }{TEXT 319 9 ")=finv(f(" }
{TEXT 320 1 "x" }{TEXT 321 13 ")) using the " }{TEXT 283 4 "subs" }
{TEXT 285 27 " command. Theoretically, h(" }{TEXT 322 1 "x" }{TEXT 
323 2 ")=" }{TEXT 324 1 "x" }{TEXT 325 36 " identically on I. I can't \+
reduce h(" }{TEXT 326 1 "x" }{TEXT 327 5 ") to " }{TEXT 328 2 "x " }
{TEXT 329 81 "using Maple commands (Maybe you or one of your friends c
an) . However, using the " }{TEXT 282 6 "normal" }{TEXT 286 84 " comma
nd you can reduce h(x) to a form which can be seen by inspection to re
duce to " }{TEXT 347 1 "x" }{TEXT 348 10 ".  Plot h(" }{TEXT 349 1 "x
" }{TEXT 350 79 ") on the interval [-2,2] using the 1:1 option to see \+
that its graph is that of " }{TEXT 330 1 "y" }{TEXT 331 1 "=" }{TEXT 
332 1 "x" }{TEXT 333 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 2 "9." }{TEXT 288 24 "Define the expression g(" }
{TEXT 334 1 "x" }{TEXT 335 9 ")=f(finv(" }{TEXT 336 1 "x" }{TEXT 337 
18 ")). Try to plot g(" }{TEXT 338 1 "x" }{TEXT 339 6 ") for " }{TEXT 
340 1 "x" }{TEXT 341 125 " in [-2,2] using the 1:1 option. Explain wha
t happens. What x interval should it be plotted over? Do so using the \+
1:1 option." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
291 87 "10. The inverse trigonometric functions appear on Maple in the
 arc- form, e.g., arcsin(" }{TEXT 342 1 "x" }{TEXT 343 27 ") and are d
escribed in the " }{TEXT 289 7 "invtrig" }{TEXT 292 8 " box in " }
{TEXT 290 4 "Help" }{TEXT 293 26 ". Enter  the expression   " }
{XPPEDIT 18 0 "g(x) = sin(arctan(x));" "6#/-%\"gG6#%\"xG-%$sinG6#-%'ar
ctanG6#F'" }{TEXT 294 56 " . Differentiate it and simplify. Enter the \+
expression  " }{XPPEDIT 18 0 "h(x) = cos(arctan(x))/(1+x^2);" "6#/-%\"
hG6#%\"xG*&-%$cosG6#-%'arctanG6#F'\"\"\",&F/F/*$)F'\"\"#F/F/!\"\"" }
{TEXT 295 28 "  . What is its relation to " }{XPPEDIT 18 0 "g(x);" "6#
-%\"gG6#%\"xG" }{TEXT 344 11 " ? Explain." }}{PARA 0 "" 0 "" {TEXT 
299 1 " " }}{PARA 0 "" 0 "" {TEXT 297 25 "11. Plot the expression  " }
{XPPEDIT 18 0 "t(x) = arctan(tan(x));" "6#/-%\"tG6#%\"xG-%'arctanG6#-%
$tanG6#F'" }{TEXT 345 26 "  on the interval   [ 0 , " }{XPPEDIT 18 0 "
3*Pi;" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT 346 70 " ].  (Suggestion: use t
he option discont=true). What happens and why ?" }}{PARA 0 "" 0 "" 
{TEXT 302 2 "  " }}{PARA 0 "" 0 "" {TEXT 300 21 "12. Do the same for  \+
" }{XPPEDIT 18 0 "t2(x) = arcsec(sec(x));" "6#/-%#t2G6#%\"xG-%'arcsecG
6#-%$secG6#F'" }{TEXT 301 25 " . What is happening now?" }}{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 
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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 2 "  " }}
{PARA 0 "" 0 "" {TEXT 271 4 "    " }}{PARA 0 "" 0 "" {TEXT 272 4 "    \+
" }}{PARA 0 "" 0 "" {TEXT 273 4 "    " }}{PARA 0 "" 0 "" {TEXT 274 5 "
     " }}{PARA 0 "" 0 "" {TEXT 275 4 "    " }}{PARA 0 "" 0 "" {TEXT 
276 5 "     " }}{PARA 0 "" 0 "" {TEXT 277 5 "     " }}{PARA 0 "" 0 "" 
{TEXT 278 4 "    " }}{PARA 0 "" 0 "" {TEXT 279 3 "   " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 4 "    " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 20 "                    " 
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