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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 93 "Lab # 3:  Plotting Implic
it Functions, Their Tangent Lines, The \"solve\" and \"fsolve\" Comman
ds" }}{PARA 256 "" 0 "" {TEXT -1 186 "In this lab we investigate how M
aple plots implicitly defined functions and how to combine the \"solve
\" and \"fsolve\" command to find tangent lines to curves generated by
 these functions. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 59 "Plotting i
mplicitly defined functions using \"implicitplot\"." }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 218 "Here we try to plot functions that are difficult
 or impossible to describe y explicitly in terms of x. The command imp
licitplot is located in a library called plots so we must first intsru
ct Maple to load this library:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "with(plots);" }{TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Wa
rning, the name changecoords has been redefined\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7W%(animateG%*animate3dG%-animatecurveG%&arrowG%-change
coordsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG%,conto
urplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,dens
ityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+g
radplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%-listcontplotG%/l
istcontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%
(logplotG%+matrixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3dG%*p
olarplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhed
raplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG
%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+t
extplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "This di
splays all of the commands available in this library.  impicitplot sho
uld be one of them.  Recall that the equation for a circle is " }
{XPPEDIT 18 0 "r^2 = x^2+y^2;" "6#/*$%\"rG\"\"#,&*$%\"xGF&\"\"\"*$%\"y
GF&F*" }{TEXT -1 96 ".   We can express the right hand side of this eq
uation as a function of x and y called circ by " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "circ := (x,y) -> x^2 + y^2;" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%circGf*6$%\"xG%\"yG6\"6$%)operatorG
%&arrowGF),&*$)9%\"\"#\"\"\"F2*$)9$F1F2F2F)F)F)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 52 "Now, suppose we want to plot a circle of radius 2.  \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "implicitplot(circ(x,y) \+
= 4, x = -3..3, y = -3..3);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
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{PARA 0 "" 0 "" {TEXT -1 287 "Notice that the resulting plot does not \+
look like a circle.  If you want to constrain the x and y scaling to b
e equal, move the mouse arrow anywhere on the plot and click the right
 button. Go to \"projection\" and click on \"constrained\". Or you can
 include the \"scaling\" option as follows:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 73 "implicitplot(circ(x,y) = 4, x = -3..3, y = -3..3, \+
scaling = constrained);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 
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&%$RGBG\"\"\"\"\"!Fhhl-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG
%\"yG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve \+
1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Now we try a more difficul
t implicit function where it is very difficult to express y explicitly
 in terms of x. Consider the " }{TEXT 264 11 "trisectrix " }{TEXT -1 
49 "(number 34 in Anton section 3.6) described by:   " }{XPPEDIT 18 0 
"y^3+y*x^2+x^2-3*y^2;" "6#,**$%\"yG\"\"$\"\"\"*&F%F'*$%\"xG\"\"#F'F'*$
F*F+F'*&F&F'*$F%F+F'!\"\"" }{TEXT -1 100 " = 0, and express the left h
and side of the equals sign by a function of x and y called trisectix.
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "trisectix := (x,y) ->
 y^3 + y * x^2 + x^2 - 3*y^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*trisectixGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),*
*$)9%\"\"$\"\"\"F2*&F0F2)9$\"\"#F2F2*$F4F2F2*&F1F2)F0F6F2!\"\"F)F)F)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "implicitplot(trisectix(x,
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 an expression for dy/dx from the equation for a circle. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "slope := (x,y) -> -x/y;" }{TEXT -1 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&slopeGf*6$%\"xG%\"yG6\"6$%)o
peratorG%&arrowGF),$*&9$\"\"\"9%!\"\"F2F)F)F)" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 733 "Suppose we wanted to find out the slope of the tangen
t line to the circle at  x = 0.8.  You should notice that this does no
t imply a unique a value.  This is because a circle is not the graph o
f a function.  Suppose we choose the upper choice which should lead us
 to a negative slope.  I go into a brief digression here on the abilit
y of maple to solve equations for variables. All of the below work can
 be done easily enough without maple. The goal  is once we have x and \+
y we can find the slope using the above definition of slope.  We will \+
then have a point and a slope and from there can draw the tangent line
.  We can use maple's \"solve\" function to determine the appropriate \+
value of y in the first quadrant for the  x = 0.8.  " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "solve(circ(x,y) = 4,y);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"!\"\"\"\"%F,F,,$F#F
-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "This yields two options for
 our function described explictly in terms of x.  This is not always p
ossible but if you have a number for x you can always solve for the co
rresponding y value with the function \"fsolve\":" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "fsolve(circ(.8,yvalue) = 4, yvalue);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+y-.L=!\"*$\"+y-.L=F%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 264 "I use the term \"yvalue\" as opposed to \+
\"y\" here so that my \"y\" remains a variable and is not assigned a s
pecific number.  Now I want to use the second of these two numbers. I \+
can do this by calling it \"ynot\" and accessing the previous expressi
on using the % symbol:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y
not := %[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ynotG$\"+y-.L=!\"*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "This could easily have been \+
done by letting \"ynot = +sqrt(4-(0.8)^2)\"  but I just want you to se
e the solve options and the % symbol used as a reference to the previo
us output.  In any case, you can easily evalute the slope:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m := slope(0.8,ynot);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"mG$!+0yNkV!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "And now define and plot the tangent line." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "line := x -> m*(x-0.8) + ynot;" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lineGf*6#%\"xG6\"6$
%)operatorG%&arrowGF(,&*&%\"mG\"\"\",&9$F/$\"\")!\"\"F4F/F/%%ynotGF/F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(line(x), x = 0
..4,labels = ['x','y'], title = \"Tangent Line\");" }{TEXT -1 0 "" }}
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" {TEXT -1 242 "So this is the plot of the tangent line to the circle \+
at the point (0.8, ynot).  To plot this line and the circle, we may no
 longer use the implicitplot command.  Instead we must plot both the l
ower and upper functions defining the circle.   " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 125 "plot([line(x),sqrt(4-x^2),-sqrt(4-x^2)], x =
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Assignment   " }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 256 11 "Problem # 1" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "Plot the circle " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%
\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"%" }{TEXT -1 64 " and the tangent line t
o this circle in the upper hemisphere at " }{XPPEDIT 18 0 "x = -.75;" 
"6#/%\"xG,$-%&FloatG6$\"#v!\"#!\"\"" }{TEXT -1 136 ".  I want a final \+
graph of the circle (that looks circular) and the tangent line on one \+
graph.  What is the slope of the tangent line?  " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 257 11 "Problem # 2" }{TEXT -1 73 " This i
s number 34 from section 3.6 in Calculus (by Anton) 7th edition.  " }}
{PARA 0 "" 0 "" {TEXT -1 35 "Curves with equations of the form  " }
{XPPEDIT 18 0 "y^2 = x(x-a)(x-b);" "6#/*$%\"yG\"\"#--%\"xG6#,&F)\"\"\"
%\"aG!\"\"6#,&F)F,%\"bGF." }{TEXT -1 26 " , where a < b are called " }
{TEXT 258 18 "bipartite cubics. " }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }
{TEXT 259 1 "a" }{TEXT -1 76 ") Use the implicit plotting capability o
f Maple to grah the bipartite cubic " }{XPPEDIT 18 0 "y^2 = x(x-1)(x-2
);" "6#/*$%\"yG\"\"#--%\"xG6#,&F)\"\"\"F,!\"\"6#,&F)F,F&F-" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 260 1 "b" }{TEXT -1 85 "
) At what points, P=(x,y), does the curve in part (a) have a horizonta
l tangent line." }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 261 1 "c" }
{TEXT -1 139 ") Solve the equation in part (a) for y in terms of x, an
d use the result to explain why the graph consists of two seperate par
ts (i.e., is " }{TEXT 262 9 "bipartite" }{TEXT -1 3 "). " }}{PARA 0 "
" 0 "" {TEXT -1 1 "(" }{TEXT 263 1 "d" }{TEXT -1 198 ") Graph the equa
tion in part (a) without using the implicit plotting capability of Map
le.  It should result in the same graph as in part (a),  so show me th
e command you use to generate this graph. " }}}}}{MARK "3" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }