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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 64 "MAT 155                 Project IV                     Fa
ll 2002" }}{PARA 0 "" 0 "" {TEXT 274 62 "                           In
verse Functions                  " }{TEXT -1 9 "         " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 264 31 "If you cannot recall a command " }
{TEXT -1 28 "from a previous lab you may " }{TEXT 265 26 "consult the \+
command index " }{TEXT -1 106 "which can be opened up as a second wind
ow.  The name of the file with the command index is 155.00.00.html." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 256 25 "Do not work on this file." }{TEXT -1 38 "  This is ju
st the list of problems.  " }{TEXT 257 18 "Open your own file" }{TEXT 
-1 82 " by selecting \"File\" on the Maple V bar and choosing \"New\".
  Save the new file as " }{TEXT 258 14 "yourname4.mws " }{TEXT -1 7 "a
nd as " }{TEXT 259 14 "yourname4b.mws" }{TEXT -1 41 " regularly using \+
the \"save as\" command.  " }{TEXT 260 81 "Be careful not to save this
 file with that name or you will overwrite your work! " }{TEXT -1 69 "
 This window should always have 155.00.04.mws printed on the top.    \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 266 30 "Do not hit return on this file" }{TEXT -1 122 " or the c
omputer will remember it when working on your file and may overwrite v
ariables and functions that you've defined." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 89 "Sign your name as a comment at
 the top of your file by backspacing in front of the prompt" }{TEXT 
-1 103 " and typing it in.  Also write Project IV and the names of any
 other students who are working with you." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 262 28 "At the start of each problem" }
{TEXT -1 160 " in your file, backspace in front of the prompt and type
 the Problem Number 1 or Problem Number 2 etc.  Then hit return and on
 the first line with a prompt type" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "This \+
will clear all previous work." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT 267 27 "If you fix or change a line" }{TEXT -1 
162 " you must go back up to the restart to clear your previous work a
nd then hit enter on all the lines after to get the computer to redo t
he computations correctly. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 30 "Do all the problems
 in order. " }{TEXT -1 56 " Ask a student or the teacher if you are ha
ving trouble." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 257 "" 
0 "" {TEXT -1 18 "Problem 1: Review:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "Define f(x)=sin(x)/x using the command" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:= x -> sin(x)/x;" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Notice that this
 function is defined everywhere except where x=0.  At x=0, we get f(0)
=0/0 which is not defined.  " }}{PARA 0 "" 0 "" {TEXT -1 33 "Now you c
an plot f(x)  by typing." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"plot(f(x), x=-10..10);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 127 "Notice that the graph crosses the x axis somewhere betwe
en 2 and 4.  To get a decimal approximation of the crossing point type
:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fso
lve(f(x)=0, x=2..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Note tha
t this is Pi.  Recall that sin(Pi)=0 because in calculus we use radian
s not degrees.  Now find out the next number where the graph crosses t
he x axis." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 
"Notice that near x=0, the function's graph appears  to be perfectly n
ice.  You might even think that f(x) is defined at zero.  Look at " }
{MPLTEXT 1 0 0 "" }{TEXT -1 199 " x values near 0 and see where it app
ears to cross.  Focus the graph very close to x=0.  Try plotting it fr
om x=-0.1 to 0.1.  Write a comment estimating the number where the gra
ph appears to cross.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Now plot the graph f
rom x=-0.01 to 0.01.  " }{TEXT 269 92 "Remember you can cut and paste \+
your plot command onto a new line and just edit the x values." }{TEXT 
-1 126 "   Then write a comment telling where it appears to cross.  Wh
at is the largest y value on the graph and what is the smallest?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "It appea
rs that as x gets closer and closer to zero, the graph gets closer and
 closer to 1.  We say that the " }{TEXT 270 37 "limit of f(x) as x app
roaches 0 is 1 " }{TEXT -1 9 "and write" }{XPPEDIT 18 0 "limit(f(x),x \+
= 0);" "6#-%&limitG6$-%\"fG6#%\"xG/F)\"\"!" }{TEXT 273 1 "=" }{TEXT 
-1 1 "1" }{TEXT 275 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 50 "Problem 2:  Inverses and the horizontal
 line test:" }}{PARA 0 "" 0 "" {TEXT -1 115 "Sometimes you don't just \+
want to solve f(x)=0.  Suppose you want to solve f(x)=2 and f(x)=4 and
 f(x) =15 and so on." }}{PARA 0 "" 0 "" {TEXT -1 87 "Rather than solvi
ng the problem repeatedly, it is easier to find an inverse function,  \+
" }{XPPEDIT 18 0 "f^(-1);" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 206 " = i
nvf, which will just cancel everything f does, so that you need only e
valuate invf(2), invf(4) and invf(15) and so on.  For example, if f(x)
=x+2, then invf(x)=x-2 and if f(x)=x^3 then invf(x)=x^(1/3).  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "To get Ma
ple to find an inverse for you, you just solve f(y)=x for y " }{TEXT 
272 53 "(you may copy and paste the commands into your file):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:= x -> x^3;" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(f(y)=x,y);" }
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 120 " You will notice that Maple gives 3 answers but 2 are \+
imaginary.  So the only real solution is y=x^(1/3).  So we define:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "invf:=x->x^(1/3);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Suppose you want to find the inve
rse of x^2?  Let g(x)=x^2 and solve  g(y)=x for y.  You will get two a
nswers, so there is no inverse for g(x)!  " }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "If you graph a function f(x) you \+
can tell whether it has an inverse or not.  It has an inverse if for e
very y value, you can draw a horizontal line through y and find exactl
y one x value such that f(x)=y.  That is the horizontal line can only \+
cross the graph once." }}{PARA 0 "" 0 "" {TEXT -1 88 "Use the plot com
mand to check if f(x)=x^3 and g(x)=x^2 satisfy the horizontal line tes
t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The
n let h(x)=2^x and graph it to see if it has an inverse." }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 517 "Use the solve command to try to find the
 inverse of h(x).  You will get a strange fraction with the function l
n(x) in it.  ln(x) is called the natural log function.  It is in fact \+
the inverse of e^x.  The natural log function is used to find inverses
 of exponential functions.  Remember that to solve x=(5y)^3 for y you \+
take the cube root of both sides of the equation.  So if x=2^y, you ta
ke the natural log of both sides  ln(x)=ln(2^y).  Use Maple to find ln
(x)=ln(2^y)=yln(2), so y=ln(x)/ln(2) (see P.6 Example 6)." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 11 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 268 42 "Problem 3: graphing functions and inverses" }
{TEXT -1 2 ", " }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 80 "Let f(x)=x^3 again and invf(x)=x^(1/3).  Graph them together us
ing the command (" }{TEXT 271 35 "You may wish to copy and paste it).
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot
(\{x^3, x^(1/3)\}, x=0..1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "You may notice that the graphs hav
e an interesting symmetry around the y=x line:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "plot(\{x^3,x, x^(1/3)\}, x=0..1);" }}{PARA 13 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Why does this \+
happen?  Compare with other functions like h(x) and invh(x) from probl
em 2 using the command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "
plot(\{2^x, ln(x)/ln(2)\}, x=-5..5, y=-5..5);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Compare exp(x), \+
which is Maple's special way of taking e^x, and ln(x) which is its inv
erse.  Use the x= and y= parts of the command to make a nice square pl
ot.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 133 "Now define g(x) to be the function which describes a lin
e of slope 3 through (1,2) and solve for its inverse and graph them to
gether." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "xer
cises 21 through 25 on page 44 of the text. " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 260 "" 0 "" {TEXT -1 12 "Exploration:" }{MPLTEXT 1 0 
0 "" }}{PARA 0 "" 0 "" {TEXT 278 2 "A:" }{TEXT -1 397 "  If you graph \+
f(x)=cos(x) on [-10,10], you will see that it fails the horizontal lin
e test.  However, it passes the horizontal line test if you graph f(x)
=cos(x) on [0, Pi].  So we could define an inverse for this restricted
 definition of cosine.  It is called arccos(x).  Now graph sin(x) on [
-10,10].  Choose a restricted domain for sin(x) that includes x=0 and \+
passes the horizontal line test." }{TEXT 279 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "36 3 1" 397 }
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