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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Maple Project 1" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "In this \+
session we will have fun with functions and their inverses.  Suppose w
e wanted to determine whether y = 6 - 2" }{XPPEDIT 18 0 "x^2;" "6#)%\"
xG\"\"#" }{TEXT -1 159 "  is 1-1 or not.  One way to do it is just use
 the horizontal line test.  We can use Maple to plot the graph over so
me interval and then we can apply the test." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "plot(6-2*x^2,x=-5..5);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 24 "So we see that y = 6 - 2" }{XPPEDIT 18 0 "x^2;" "6#*$%\"x
G\"\"#" }{TEXT -1 148 " is not one-to-one since it clearly fails the h
orizontal line test on the interval [-5,5].  Huzzah!  Let's plot it on
 a smaller interval, say [1,3]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "plot(6-2*x^2,x=1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 
"Our friend clearly passes the horizontal line test on the interval [1
,3].  Let's find its inverse on this interval.  First let's define it \+
as a function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=x->(4*
x-5)/(x+3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Remember, to find
 the inverse of y = f(x), we need to solve for x as a function of y.  \+
In the following command, we solve for x as a function of y and put th
e answer in the variable called \"solution\":" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "solution:=solve(y=f(x),x);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 21 "So we see either x = " }{XPPEDIT 18 0 "1/2;" "6#*&
\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(-2*y+12);
" "6#-%%sqrtG6#,&*&\"\"#\"\"\"%\"yGF)!\"\"\"#7F)" }{TEXT -1 8 "or x = \+
-" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "
sqrt(-2*y+12);" "6#-%%sqrtG6#,&*&\"\"#\"\"\"%\"yGF)!\"\"\"#7F)" }
{TEXT -1 139 ".  In our case, we want x to be positive since we are in
 the interval [1,3], so we would like the first solution.  We can refe
r to it with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solution[1
];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 "(When finding inverses, yo
u don't always get two solutions.  In your project, you will only get \+
one solution, so you won't have to worry about the command we just did
.)  So, continuing like we did in class, we can rewrite it by substitu
ting x for y.  We do this by using the \"subs\" command." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(y=x,solution[1]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 196 "Let's take this last output and make it \+
into a function we can actually plug values into.   We do this by usin
g the \"unapply\" command and our friend \"%\", which always refers us
 to our last output:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=
unapply(%,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 404 "So, this functi
on \"g\" is indeed the inverse of f.  Now we can plot y = f(x), its in
verse y = g(x), and the line y = x all on the same graph by using the \+
plot command.  The braces just mean that all 3 curves will be on the s
ame plot.  I also added in the \"y=0..6\" to restrict the y values bet
ween 0 and 6, so we could see the plots a little better. We are also p
lotting these graphs on the interval [0,6]." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "plot(\{f(x),g(x),x\},x=-2..3);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 259 "Doesn't look bad, eh?   The green curve is y=f(x
), the red one is y=x, and the yellow one is y=g(x).  Remember that th
e graphs of f and f inverse should be symmetrical about the line y=x, \+
i.e. one should be the reflection of the other across the line y = x. \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "Now
 let's find g'(2) using the inverse differentiation theorem.  The form
ula is:  g'(2) = 1 / f'(g(2)), where g is f inverse.  We really only n
eed to find the derivative of f and then plug things in.  This command
 finds the derivative of f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "diff(f(x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Here, our d
erivative is just an expression.  Let's make it into a function we can
 plop values into by using that \"unapply\" thingy again." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fprime:=unapply(%,x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "Now lets plug things into our formula; so
 we see that g'(2) is equal to:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "1/fprime(g(5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "Suppo
sedly this is g'(2), the derivative of f inverse at 2.  Let's doublech
eck by actually taking the derivative of g (which is f inverse) and ev
aluating it at 2, instead of using the formula." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "diff(g(x),x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "gprime:=unapply(%,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "gprime(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "simplify(g(f(x)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Well, t
here you have it.  The formula works, darn it all!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 482 "So here's the project pr
oblems.  Do them using Maple and the techniques we have valiantly pion
eered in the above section.  (Start a new worksheet by choosing \"New
\" from the \"File\" menu.  When you are finished, print out your Mapl
e sheet and PUT YOUR NAME ON IT, and give it to me.)  You can put text
 into your worksheet by clicking on the \"T\" button in the menu up to
p.  When you are done, click on the button to the right of it to get t
he Maple prompt back for your Maple commands." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "1) Graph  y = " }
{XPPEDIT 18 0 "x^3;" "6#)%\"xG\"\"$" }{TEXT -1 104 " + x.  Is it one-t
o-one?  Why or why not?  If not, find an interval on which the functio
n is one-to-one." }}{PARA 0 "" 0 "" {TEXT -1 46 "2) a) Find the invers
e of f(x) = (4x-5)/(x+3)." }}{PARA 0 "" 0 "" {TEXT -1 92 "    b) Plot \+
f, its inverse, and the line y=x all on the same graph over the interv
al [-2,3]." }}{PARA 0 "" 0 "" {TEXT -1 75 "    c) Verify the inverse d
ifferentiation theorem for this function at x=5." }}}}{MARK "27 7 0" 
75 }{VIEWOPTS 1 1 0 1 1 1803 }