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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 29 "
Lab 7: Implicit Relationships" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Goals" }}{PARA 0 "" 0 "" {TEXT -1 
64 "1.  Students will differentiate implicitly with Maple using the " 
}{TEXT 261 12 "implicitdiff" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "
" {TEXT -1 48 "2.  Students will graph non-functions using the " }
{TEXT 256 12 "implicitplot" }{TEXT -1 9 " command." }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 10 "Discussion" }}{PARA 0 "" 0 "" {TEXT -1 38 "A re
lationship that is not solved for " }{TEXT 284 1 "y" }{TEXT -1 18 ", b
ut instead has " }{TEXT 286 1 "x" }{TEXT -1 217 "'s and y's intermingl
ed throughout is called an implicit relationship.  Occasionally these \+
are functions, but more often they are not.  In this lab we will look \+
at implicit relationships, their graphs and derivatives." }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 26 "Implicit Differentiation  " }}{PARA 0 "" 
0 "" {TEXT -1 14 "First use the " }{TEXT 271 11 "with(plots)" }{TEXT 
-1 59 " command to load more graphing options into Maple's memory." }}
{EXCHG {PARA 0 "> " 0 "with(plots)" {MPLTEXT 1 0 12 "with(plots);" }}}
{PARA 0 "" 0 "" {TEXT -1 48 "Here's an example of how to plot the equa
tion   " }{XPPEDIT 18 0 "y^3+y^2-5*y-x^2 = -4;" "6#/,**$%\"yG\"\"$\"\"
\"*$F&\"\"#F(*&\"\"&F(F&F(!\"\"*$%\"xGF*F-,$\"\"%F-" }{TEXT -1 22 " im
plicitly using the " }{TEXT 264 12 "implicitplot" }{TEXT -1 9 " comman
d." }}{EXCHG {PARA 0 "> " 0 "implicitplot" {MPLTEXT 1 0 49 "implicitpl
ot(y^3+y^2-5*y-x^2=-4,x=-5..5,y=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Note that  the above equation is \+
not a function because its graph does not pass the vertical line test.
" }}{PARA 0 "" 0 "" {TEXT -1 183 "Recall that the derivative of a func
tion is undefined wherever vertical tangents occur.  Let's take the de
rivative and find out exactly where the vertical tangents occur.  We u
se the " }{TEXT 258 12 "implicitdiff" }{TEXT -1 20 " command as follow
s." }}{EXCHG {PARA 0 "> " 0 "implicitdiff" {MPLTEXT 1 0 37 "implicitdi
ff(y^3+y^2-5*y-x^2=-4,y,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 73 "The derivative is undefined when the denominato
r is zero.  Thus, to find " }{TEXT 265 1 "y" }{TEXT -1 131 "-values wh
ere the vertical tangents occur, set the above denominator equal to 0 \+
and have have Maple solve this equation as follows." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "solve(3*y^2+2*y-5=0,y);" }}}{PARA 0 "" 0 "
" {TEXT -1 69 "(Note that we could have solved this equation by hand r
ather easily.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 12 "To find the " }{TEXT 262 1 "x" }{TEXT -1 32 " c
oordinates that go with these " }{TEXT 263 1 "y" }{TEXT -1 28 " coordi
nates, first set the " }{TEXT 260 8 "original" }{TEXT -1 12 " equation
,  " }{XPPEDIT 18 0 "y^3+y^2-5*y-x^2 = -4;" "6#/,**$%\"yG\"\"$\"\"\"*$
F&\"\"#F(*&\"\"&F(F&F(!\"\"*$%\"xGF*F-,$\"\"%F-" }{TEXT -1 23 " equal \+
to zero to get  " }{XPPEDIT 18 0 "y^3+y^2-5*y-x^2+4 = 0;" "6#/,,*$%\"y
G\"\"$\"\"\"*$F&\"\"#F(*&\"\"&F(F&F(!\"\"*$%\"xGF*F-\"\"%F(\"\"!" }
{TEXT -1 16 ".  Then use the " }{TEXT 257 4 "subs" }{TEXT 267 0 "" }
{TEXT 268 0 "" }{TEXT -1 5 " (or " }{TEXT 266 4 "eval" }{TEXT -1 44 ")
 command to evaluate the this equation for " }{XPPEDIT 18 0 "y=-5/3" "
6#/%\"yG,$*&\"\"&\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 "." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 33 "subs(y=-5/3,y^3+y^2-5*y-x^2+4=0);" }}}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Solve th
e resulting equation for " }{TEXT 269 1 "x" }{TEXT -1 23 " to obtain t
he desired " }{TEXT 270 1 "x" }{TEXT -1 8 "-values." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "solve(283/27-x^2=0,x);" }}}{PARA 0 "" 0 "" 
{TEXT -1 39 "To get decimal approximations of these " }{TEXT 287 1 "x
" }{TEXT -1 17 "-values, use the " }{TEXT 259 5 "evalf" }{TEXT -1 9 " \+
command." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Next, we \+
repeat the process for " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }
{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(y=1,y^
3+y^2-5*y-x^2 +4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve
(1-x^2=0,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "So the points whe
re the vertical tangent lines occur are (" }{XPPEDIT 18 0 "1/9*sqrt(84
9);" "6#*(\"\"\"F$\"\"*!\"\"-%%sqrtG6#\"$\\)F$" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "-5/3" "6#,$*&\"\"&\"\"\"\"\"$!\"\"F(" }{TEXT -1 4 "), (
" }{XPPEDIT 18 0 "-1/9*sqrt(849);" "6#,$*(\"\"\"F%\"\"*!\"\"-%%sqrtG6#
\"$\\)F%F'" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "-5/3" "6#,$*&\"\"&\"\"\"
\"\"$!\"\"F(" }{TEXT -1 21 "), (1, 1) and (-1, 1)" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "The equations of the associated vertical tangent lines ar
e " }{TEXT 272 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/9*sqrt(849);
" "6#*(\"\"\"F$\"\"*!\"\"-%%sqrtG6#\"$\\)F$" }{TEXT -1 3 " , " }{TEXT 
273 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-1/9*sqrt(849);" "6#,$*(\"
\"\"F%\"\"*!\"\"-%%sqrtG6#\"$\\)F%F'" }{TEXT -1 3 " , " }{TEXT 274 1 "
x" }{TEXT -1 9 " =1, and " }{TEXT 275 1 "x" }{TEXT -1 6 " = -1." }}
{PARA 0 "" 0 "" {TEXT -1 100 "In the final example, we graph the origi
nal equation, along with all of it's vertical tangent lines." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "implicitplot(\{y^3+y^2-5*y-x
^2=-4,x=1,x=-1,x=3.238,x=-3.238\},x=-5..5,y=-5..5);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 19 "Now it's
 your turn." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Lab Assignment 7 \+
" }}{PARA 0 "" 0 "" {TEXT -1 295 "Start the lab by putting your name o
n the first line.  Do the assignment in a new file (go to the file men
u in Maple and click on \"New\").  Copy the below questions to your ne
w file.  When you complete the lab, hand in a printout of it.  ANSWER \+
ALL QUESTIONS COMPLETELY IF YOU WANT FULL CREDIT!!!" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "1.  a)  Find the
 point (both the x and y coordinates) at which " }{XPPEDIT 18 0 "y^3+y
^2-5*y-x^2 = -4;" "6#/,**$%\"yG\"\"$\"\"\"*$F&\"\"#F(*&\"\"&F(F&F(!\"
\"*$%\"xGF*F-,$\"\"%F-" }{TEXT -1 51 " has a horizontal tangent line  \+
         using the " }{TEXT 276 12 "implicitdiff" }{TEXT -1 116 " comm
and (keep in mind that only real numbers work for the coordinates of a
 point in the           Cartesian plane)." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "     b)  Plot both the above equat
ion and its associated horizontal tangent line using " }{TEXT 278 12 "
implicitplot" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 31 "2.  a)  Take the derivative of " }{XPPEDIT 18 0 "x^3-x*
y+y^2=4" "6#/,(*$%\"xG\"\"$\"\"\"*&F&F(%\"yGF(!\"\"*$F*\"\"#F(\"\"%" }
{TEXT -1 7 " using " }{TEXT 277 12 "implicitdiff" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 71 "     b)  Find the equation(s) of the vert
ical tangent(s) to the curve. " }}{PARA 0 "" 0 "" {TEXT -1 87 "     c)
  Plot the above equation and its associated vertical tangent line(s) \+
using the " }{TEXT 288 12 "implicitplot" }{TEXT -1 9 " command." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "3.
  a)  Graph the equation " }{XPPEDIT 18 0 "y(y^2-1)(y-2)=x(x-1)(x-2)" 
"6#/--%\"yG6#,&*$F&\"\"#\"\"\"F+!\"\"6#,&F&F+F*F,--%\"xG6#,&F1F+F+F,6#
,&F1F+F*F," }{TEXT -1 36 ".  Use your pointer to estimate the " }
{TEXT 279 1 "x" }{TEXT -1 61 "-coordinates of the points where horizon
tal tangents occur.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 21 "b)  Locate the exact " }{TEXT 289 1 "x" }{TEXT -1 89 "
-values of the points where horizontal tangents occur  (you estimated \+
these in part (a))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 81 "c)  Find the tangent line equations to the curve at the
 points (0, 1) and (0, 2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 245 "d)  \"Play\" with the equation in part (a) to de
sign at least two other interesting looking graphs  (i.e. Be creative \+
in changing the above curve by throwing in additional exponents or fac
tors to the given equation and graph these new equations). " }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "4.  (#34
 page 231 of your text).  " }}{PARA 0 "" 0 "" {TEXT -1 28 "a)  The cur
ve with equation " }{XPPEDIT 18 0 "2y^3+y^2-y^5=x^4-2x^3+x^2" "6#/,(*&
\"\"#\"\"\"*$%\"yG\"\"$F'F'*$F)F&F'*$F)\"\"&!\"\",(*$%\"xG\"\"%F'*&F&F
'*$F1F*F'F.*$F1F&F'" }{TEXT -1 136 " has been likened to a bouncing wa
gon.  Use a computer algebra system to graph this curve and discover w
hy.  [Graph the curve using the " }{TEXT 283 12 "implicitplot" }{TEXT 
-1 112 " command.  Make sure you plot your graph using an appropriate \+
domain and range so that you can see the \"wagon.\"]" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "b)  At how many points \+
does this curve have horizontal tangent lines?  Find the " }{TEXT 281 
1 "x" }{TEXT -1 29 "-coordinates of these points." }}{PARA 0 "" 0 "" 
{TEXT -1 83 "[There are actually 9 distinct points with horizontal tan
gents but only 3 distinct " }{TEXT 285 2 "x-" }{TEXT -1 71 "coordinate
s.  For instance, 3 points with horizontal tangents occur at " }{TEXT 
282 1 "x" }{TEXT -1 168 " = 0.  Can you find the others?  In part (b) \+
you will need to plot the graph again, adjusting the domain and range \+
in order to find these tangents. Be sure to find all " }{TEXT 280 1 "x
" }{TEXT -1 86 "-coordinates (there are only 3 distinct ones) of the p
oints with horizontal tangents.]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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