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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Maple Project 3" }}
{PARA 0 "" 0 "" {TEXT -1 240 "In this project we will explore the inve
rse trigonometric functions.  In Maple the inverse sine is just \"arcs
in(x)\" and the inverse cosine is \"arccos(x)\".  Let's just make sure
 we understand what these things are.  Here is how sin(x) looks:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(sin(x),x=-10*Pi..10*Pi)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "It is clearly not 1-1 by the
 horizontal line test.  But it is 1-1 on the interval  [-" }{XPPEDIT 
18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 
18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(sin(x),x=-Pi/2..Pi/2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 349 "So if we restrict sine to this in
terval, it passes the horizontal line test, and it will have an invers
e on this interval.  It is also 1-1 on infinitely many other intervals
, but we just pick this one for simplicity's sake.  Thus its inverse a
rcsin(x) is only defined on this interval.  Its domain is [-1,1] (the \+
range of sin(x)) and its range is [-" }{XPPEDIT 18 0 "Pi/2;" "6#*&%#Pi
G\"\"\"\"\"#!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT -1 144 " ] (the domain of our restricted sine
 function).  The story for cosine is much the same except the interval
 we take to define its inverse is [0," }{XPPEDIT 18 0 "Pi;" "6#%#PiG" 
}{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 620 "In the following series of commands we will plot the cur
ves y=sin(x) and  y=arcsin(x) on their domains as specified above, alo
ng with the line y=x to show their inverse relationship.  The followin
g commands are used when we wish to plot 2 different functions on TWO \+
DIFFERENT INTERVALS, but on the same graph.  (Our plot command in earl
ier labs graphed 2 different functions on the SAME interval.)  The fir
st command loads up a special plots package.  The COLON at the end of \+
the first 4 commands just means that we don't want to see the output o
f the command.  Don't forget the COLON, or you will get lots of gibber
ish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "A:=plot(sin(x),x=-Pi/2..Pi/2,color=red,thickness=2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "B:=plot(arcsin(x),x=-1..1,color=gre
en,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "C:=plot
(x,x=-Pi/2..Pi/2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "display([A,B,C],title=`Yo.  Here's the plot.`);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "It sure \+
is peachy keen.  Now: problems.  By the way, please answer any asked q
uestions, and comment your answer into your Maple sheet using the \"T
\" button." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 20 "1)  Find arcsin(sin(" }{XPPEDIT 18 0 "5*Pi/2;" "6#*(\"\"&\"\"\"
%#PiGF%\"\"#!\"\"" }{TEXT -1 37 ")) on Maple.  Why is your answer not \+
" }{XPPEDIT 18 0 "5*Pi/2" "6#*(\"\"&\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 
-1 66 " ?  If you think about it, since the range of the arcsine is  [
-  " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " \+
, " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 17 " \+
], as x goes to " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 
-1 6 " (or -" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 77 
"), the values of arcsin(sin(x)) will forever oscillate in the interva
l [-    " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 
-1 3 " , " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 
-1 163 " ], since the values of sin(x) will forever oscillate between \+
-1 and 1.  To see this, plot arcsin(sin(x)) on the interval [-20,20]. \+
 Now plot it on the interval (-" }{XPPEDIT 18 0 "infinity;" "6#%)infin
ityG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }
{TEXT -1 332 ").  (In your plot command, just use \"x=-infinity..infin
ity\".)  Of course Maple can't really plot it on this interval, it wou
ldn't fit on the screen, but it does give you a \"bird's eye view\".  \+
What do you think the limit of arcsin(sin(x)) is as x goes to infinity
?  Evaluate this limit on Maple.  What do you think this output means?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 452 "2) W
e know cosine and arccosine are inverses of each other, so if we plot \+
them both on the same graph they should be symmetric about the line y=
x.  Unfortunately, no matter how we use our plot command like we've do
ne in other labs (y'know, that brace stuff), we won't be able to plot \+
both cosine and arccosine in such a way that nicely shows this symmetr
y.  For example, plot the cosine and inverse cosine together on the sa
me plot on the interval [-1," }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 
-1 395 "] using our old plot command with the brace notation.  Note ho
w we have a little bit more of the cosine than we really want.  No mat
ter what interval we take, we will not get the nice plot we want in or
der to show off this symmetry.   But we can do it using the display co
mmand as we did above for sine and arcsine.  SO, using the 5 commands \+
as above, plot y=arccos(x) on [-1,1], y=cos(x) on [0," }{XPPEDIT 18 0 
"Pi;" "6#%#PiG" }{TEXT -1 57 "], and y=x on [-1,1] on the same graph. \+
 Voila, symmetry!" }}{PARA 0 "" 0 "" {TEXT -1 22 "3) Evaluate the limi
t " }{XPPEDIT 18 0 "limit(arctan(1/(x-2)),x = 2);" "6#-%&limitG6$-%'ar
ctanG6#*&\"\"\"\"\"\",&%\"xGF+\"\"#!\"\"F//F-\"\"#" }{TEXT -1 190 "   \+
on Maple (an example from the text).  The reason why you get this answ
er is because for a limit to be defined, its left and right-sided limi
ts must be equal.  So let's evaluate the limit " }{XPPEDIT 18 0 "limit
(arctan(1/(x-2)),x = 2);" "6#-%&limitG6$-%'arctanG6#*&\"\"\"\"\"\",&%
\"xGF+\"\"#!\"\"F//F-\"\"#" }{TEXT -1 710 "   FROM THE RIGHT.  Of cour
se, we don't know how to take right-sided limits on Maple, so we will \+
use the \"Help\" menu.  Go and click on the \"Help\" menu at the top o
f the window.  Click on \"Topic Search\".  Type in the word \"limit\".
  We want a directional limit, so in the list of topics click on \"lim
it, dir\" and click the \"ok\" button.  Now peruse the window for the \+
command we need to take a right-hand limit (just look at the examples)
 and use this syntax to evaluate our limit.  Then evaluate the left-ha
nd limit.  Note the right and left-sided limits are not equal, explain
ing our answer from our original limit. Now that you can use the help \+
menu, you don't need to go to Maple lab anymore.  (Just kidding.)" }}}
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