{VERSION 5 0 "IBM INTEL LINUX" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
274 "" 0 1 0 0 0 0 1 0 0 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 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 18 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 18 0 0 0 0 0 1 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
284 "" 1 14 0 0 0 0 0 1 0 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 "" -1 287 "" 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 "" 1 14 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 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 295 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
296 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 24 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 
0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 
258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "
" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 296 0 "" }{TEXT 
297 25 "Calculus II - worksheet 3" }}{PARA 261 "" 0 "" {TEXT 256 0 "" 
}}{PARA 260 "" 0 "" {TEXT 257 30 "Drawing well-labelled diagrams" }
{TEXT 295 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 280 0 "" }
{TEXT 281 45 "Simplifying inverse trigonometric expressions" }}{PARA 
259 "" 0 "" {TEXT -1 0 "" }{TEXT 282 17 "Maple assumptions" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 42 "Author: Carl De
vore <devore@math.udel.edu>" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 16 "13 February 2002" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 9 "Audience:" }}{PARA 0 
"" 0 "" {TEXT -1 36 "1.  Second-semester calclus students" }}{PARA 0 "
" 0 "" {TEXT -1 73 "2.  Anyone who wants to learn to draw simple label
led diagrams with Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 284 11 "Objectives:" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "1. Draw simple labelled diagrams with Maple." }}{PARA 0 "
" 0 "" {TEXT -1 125 "2. Learn how a reference triangle can be used to \+
simplify a composition of trigonometric and inverse trigonometric func
tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 291 34 "New Maple commands/concepts  used:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 292 19 "textplot, convert, " }{TEXT -1 24 "plot
ting line segments, " }{TEXT 293 6 "assume" }{TEXT -1 2 ", " }{TEXT 
294 8 "assuming" }{TEXT -1 30 ", verifying inequalities with " }{TEXT 
298 2 "is" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 360 "The primary purpo
se of this worksheet is to show you how you can use Maple to draw well
-labelled diagrams.  To illustrate this, I will simplify an arctrig ex
pression by drawing a reference triangle.  Of course, if your purpose \+
was merely to do this problem, it would be far easier to just draw the
 triangle by hand.  This problem is just to illustrate how you " }
{TEXT 269 5 "could" }{TEXT -1 269 " use Maple to draw diagrams.   If y
our purpose is to display the solution on the world-wide web, then it \+
may be easier to do it in Maple and use the \"Export as HTML\" feature
 than to draw it by hand, scan it in, and then cut-and-paste the scann
ed images into a document." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "First, I will jus
t draw a right triangle.  The plot command can be used to draw line se
gments by merely specifying the coordinates of the endpoints of the se
gments.  Each point (" }{XPPEDIT 18 0 "x,y;" "6$%\"xG%\"yG" }{TEXT -1 
258 ") is represented in Maple as a list of two numbers so it goes in \+
square brackets.  Then the two points themselves are considered a list
 so they go into another pair of square brackets.  So the following co
mmand will draw the line segment from (0,0) to (1,1)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot( [ [0,0], [1,1] ] );" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "That's going to be the hypothenus
e of the triangle.  The blank spaces that I put in the command are not
 necessary; I just put them there to emphasize the extra square bracke
ts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "
To finish the triangle, I need line segments from (1,1) to (1,0) and f
rom (1,0) back to (0,0).  I can make this into a single list and draw \+
all the line segments at once with the following command:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot( [ [0,0], [1,1], [1,0], [0,0] \+
] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "The x-axis is obscuring \+
the bottom of the triangle.   The axes are not a useful part of this p
icture anymore.  I just want to see the triangle.  So I add the \"axes
= none\" option to the plot command:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "plot([[0,0], [1,1], [1,0], [0,0]], axes= none );" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "Now that I am satisfied with the \+
triangle, I'd like to save it in a variable for future use.  The symbo
l \"%\" can always be used to refer the results of the last command th
at was actually executed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "Triangle:= %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "All that in
fo is Maple's internal representation of the drawing.  Usually, there \+
is no need to see this.  So I could have ended the above command with \+
a colon instead of a semicolon.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 414 "Note that the % always refers to the las
t command that was actually executed.  It does not matter where your c
ursor is at the time that you use the percent sign.  Therefore, it is \+
possible for the percent sign to refer to a command that is not the pr
evious command on the screen.  This can cause great confusion, especia
lly after you save and reload the worksheet, so be careful where you p
lace your percent signs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 29 "Now on to the trig problem.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 
272 8 "Problem:" }{TEXT 273 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 270 9 "Evaluate " }{XPPEDIT 18 0 "sin(arctan(sqrt(x^2-2*x)));" "
6#-%$sinG6#-%'arctanG6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*&F/F0F.F0!\"\"" 
}{TEXT -1 5 ",    " }{XPPEDIT 18 0 "2 <= x;" "6#1\"\"#%\"xG" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 4 "Let " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 29 " be the
 angle represented by " }{XPPEDIT 18 0 "arctan(sqrt(x^2-2*x));" "6#-%'
arctanG6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*&F,F-F+F-!\"\"" }{TEXT -1 8 ".
  Then " }{XPPEDIT 18 0 "tan(theta) = sqrt(x^2-2*x);" "6#/-%$tanG6#%&t
hetaG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*&F.F/F-F/!\"\"" }{TEXT -1 35 ".  I
n the diagram, I am going make " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" 
}{TEXT -1 46 " the lower left angle of the triangle.  Since " }
{XPPEDIT 18 0 "tan(theta) = opp/adj;" "6#/-%$tanG6#%&thetaG*&%$oppG\"
\"\"%$adjG!\"\"" }{TEXT -1 44 ",  I define these two sides of the tria
ngle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "opp:= sqrt(x^2-2*x
); adj:= 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Note above that mo
re than one command can be placed on a line." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Now have Maple apply the Pythag
orean theorem to compute the hypothenuse." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "hyp:= sqrt(opp^2+adj^2);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 49 "Sometimes Maple might print the above answer as  " }
{XPPEDIT 18 0 "-1+x;" "6#,&\"\"\"!\"\"%\"xGF$" }{TEXT -1 40 "  and oth
er times it might print it as  " }{XPPEDIT 18 0 "x-1;" "6#,&%\"xG\"\"
\"F%!\"\"" }{TEXT -1 187 ".  It is very difficult to predict which ord
er Maple will use.  Sometimes Maple chooses to represent things in an \+
order that appears strange to us.  I could force Maple to print that a
s  " }{XPPEDIT 18 0 "x-1;" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 148 ", b
ut there's no need right now to delve into the complexities of that.  \+
But remember not to rely on Maple printing an answer in any specific o
rder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 201 "Now we label the sides of the triangle.  Maple has over \+
2500 commands, but all of these are not immediately available when you
 start Maple.   Many of the commands are grouped together into sets ca
lled " }{TEXT 258 8 "packages" }{TEXT -1 139 ".  There are many comman
ds for plotting other than the \"plot\" command that we've used up unt
il now.   Some of these are in a package named " }{TEXT 259 5 "plots" 
}{TEXT -1 30 ".  To load a package, use the " }{TEXT 260 4 "with" }
{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi
th(plots);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The list that Mapl
e has printed above is all the new commands that are now available.  I
n this worksheet, we are going to use two of them: " }{TEXT 261 8 "tex
tplot" }{TEXT -1 5 " and " }{TEXT 262 7 "display" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "M
ove you mouse pointer back up to the picture of the triangle.  Click i
n several places within and near the triangle.  Notice the numbers tha
t appear on the left side of the second toolbar.  These numbers are th
e  " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"y;" "6#%\"yG" }{TEXT -1 225 " coordinates of the point that you just \+
clicked on.  We can use this technique to figure out good coordinate p
ositions to place the labels.  By doing this, I've decided that (0.2, \+
0.1) would be a good place to put the symbol " }{XPPEDIT 18 0 "theta;
" "6#%&thetaG" }{TEXT -1 36 " in the lower right of the triangle." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Greek
 letter " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 41 " is repre
sented by the letter `q` in the " }{TEXT 263 6 "SYMBOL" }{TEXT -1 47 "
 font.  So here's the command for printing the " }{XPPEDIT 18 0 "theta
;" "6#%&thetaG" }{TEXT -1 24 " at position (0.2, 0.1):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "textplot([.2,.1,`q`], font= [SYMBOL
,24]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The 24 that appears in \+
the above command is to specify the size of the " }{XPPEDIT 18 0 "thet
a;" "6#%&thetaG" }{TEXT -1 23 " that I'd like printed." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Save that plot in a \+
variable:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Theta:= %:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "
The display command is used to combine two or more plots into a single
 plot.  I'd like to put the plots Theta and Triangle together." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display( [Theta,Triangle] );
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Notice that I used square bra
ckets to specify the " }{TEXT 264 4 "list" }{TEXT -1 24 " of plots tha
t I wanted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 42 "Now I am going to label the adjacent side:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 52 "textplot([.5, -.1, `1`], font= [HELVETICA,BO
LD,16]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "However, to be gener
al, I don't won't to have to refer to the specific value `1`.  I alrea
dy set the value of the variable adj to be 1.  Now we can convert the \+
value of any variable into a symbol for the purposes of printing by us
ing the " }{TEXT 265 7 "convert" }{TEXT -1 9 " command." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert(adj, symbol);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 193 "Now I will refer to it this way in the t
extplot command.  There's no need for us to see the \"raw\" plot of th
e symbol `1`.  So this time, I am going to save the plot into a variab
le immediately." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Adj:= te
xtplot([.5, -.1, convert(adj,symbol)], font= [HELVETICA,BOLD,16]):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Notice that I ended that command \+
with a colon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display([A
dj,Triangle,Theta]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 276 "Now we'l
l label the opposite side.  To do this, I choose a point near the cent
er of the opposite side.  But I want the text to go to the right of th
is point, so I use the option \"align= RIGHT\".  Unfortunately, there \+
is no way to get the text to print vertically or on a slant." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Opp:= textplot( [1.1, 0.5, \+
convert(opp,symbol)]\n               ,font= [HELVETICA,BOLD,16]\n     \+
          ,align= RIGHT\n              ): " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "display([Opp,Adj,Triangle,Theta]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "Finally, we label the hypothenuse.  This \+
time, I want the label to go to the " }{TEXT 268 4 "left" }{TEXT -1 
47 " of a point near the center of the hypothenuse." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 132 "Hyp:= textplot( [.4, .5, convert(hyp,sym
bol)]\n               ,font= [HELVETICA,BOLD,16]\n               ,alig
n= LEFT\n              ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "display([Hyp,Opp,Adj,Triangle,Theta], view= [0..2, -1..1]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Finally, to answer the problem, we
 note that " }{XPPEDIT 18 0 "sin(theta) = opp/hyp;" "6#/-%$sinG6#%&the
taG*&%$oppG\"\"\"%$hypG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "opp/hyp;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 
"It looks the denominator can be simplified.  However, unless you tell
 it otherwise, Maple assumes that all variables represent complex numb
ers (this is called a " }{TEXT 274 7 "default" }{TEXT -1 14 " assumpti
on). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "If you ever see \"csgn\" (which s
tands for \"complex sign\") appear in an answer, you know that Maple i
s using the complex assumption.  Let's change the assumption." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(%) assuming real;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "As you should all know from alg
ebra, for real numbers, " }{XPPEDIT 18 0 "sqrt(x^2) = abs(x);" "6#/-%%
sqrtG6#*$%\"xG\"\"#-%$absG6#F(" }{TEXT -1 91 ".  So that answer makes \+
sense.  But the original problem contained the stronger assumption " }
{XPPEDIT 18 0 "2 <= x;" "6#1\"\"#%\"xG" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(%) assuming 2<=x;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The command modifier " }{TEXT 275 
8 "assuming" }{TEXT -1 80 " is a feature new to release 7.   In older \+
releases you need to use the command " }{TEXT 276 6 "assume" }{TEXT 
-1 44 " (as a separate command) before simplifying." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "assume(x>=2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "Maple has no printed response to a correctly entered assu
me command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(opp
/hyp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "The ~ next to the vari
able names indicate that an assumption has been made about that variab
le.  Assumptions made with " }{TEXT 277 7 "assume " }{TEXT -1 120 "sta
y in effect until they are explicitly removed (for example, by assigni
ng a value to the variable or by using another " }{TEXT 285 7 "assume \+
" }{TEXT -1 46 "on the same variable) whereas those made with " }
{TEXT 278 8 "assuming" }{TEXT -1 61 " are only in effect for the singl
e command that contains the " }{TEXT 279 8 "assuming" }{TEXT -1 1 "." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 416 "Finally, note that this refere
nce triangle technique ignores the sign of the answer -- it always giv
es a positive answer.  To determine the sign of the answer, it is best
 to consider quadrants.  An arctrig function applied to a positive num
ber (in the domain of the function) always gives an angle in the first
 quadrant.   Since the square root symbol always denotes a nonnegative
 number,  and since we are assuming " }{XPPEDIT 18 0 "2 <= x;" "6#1\"
\"#%\"xG" }{TEXT -1 40 ", the above answer is nonnegative.  The " }
{TEXT 287 2 "is" }{TEXT -1 84 " can sometimes verify simple inequaliti
es involving variables that have assumptions." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "is(%>=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "
But " }{TEXT 289 3 "is " }{TEXT -1 18 "will often return " }{TEXT 290 
4 "FAIL" }{TEXT -1 177 " for more complicated inequalities -- which me
ans that it could not determine whether the inequality would be true f
or all values of the variables using the current assumptions." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 2" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }