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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Module  8 :  Differential Calculu
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 38 "805 \+
: Implicit Functions & Derivatives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 253 "In \+
this project we will use the following command packages. Type and exec
ute this line before beginning the project below. If you re-enter the \+
worksheet for this project, be sure to re-execute this statement befor
e jumping to any point in the worksheet." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):
" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has be
en redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 63 "_________________________________________________________
______" }}{PARA 4 "" 0 "" {TEXT -1 17 "A. FINDING POINTS" }}{PARA 0 "
" 0 "" {TEXT -1 73 "__________________________________________________
_______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 568 "Usually when we speak of functions, we are talking \+
about explicit functions of the form y = f(x). Sometimes we have inste
ad an equation in x and y, for example, x + y + xy = sin(x + y), which
 may not be solvable for y. The solutions to this equation are a set o
f points \{(x,y)\} which implicitly define a relation between x and y \+
which we will call an implicit function. Implicit functions are often \+
not actually functions in the strict definition of the word, because t
hey often have multiple y values for a single x value. They do have gr
aphs and derivatives however." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 452 "You are \+
probably familiar with some linear implicit functions from algebra. Li
nes in point-slope and standard form (such as y -3 = 2(x + 5) and 7x +
 9y = 63) are both examples of implicit functions since they are not e
xplicitly solvable for y. These can be solved for y easily enough and \+
converted into explicit functions. However, many implicit functions ar
e difficult or impossible to convert to explicit functions and must be
 left in implicit form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 216 "Our first task wi
ll be, given an implicit defined function, can we plot some individual
 points on the graph. Lets start with this one -4*x + 10*(x^2)*(y^(-2)
) + y^2 = 11, and find all points where x has the value 2. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "im
p_fun := -4*x + 10*(x^2) * (y^(-2)) + y^2 = 11;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(imp_funG/,(%\"xG!\"%*&*&\"#5\"\"\")F'\"\"#F,F,*$)%\"
yGF.F,!\"\"F,*$F0F,F,\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 173 "For later reference, we are naming the equation \+
which defines the implicit function a name. Note that we use := to ass
ign the equation name but we use = within the equation." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "Our method will con
sist of two steps : first, substitute for the value of x, then solve t
he resulting equation in one variable for y." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "We substitute x= 2 into the im
plicit equation.\nThis gives us an equation with only one unknown.\nWe
 solve this equation and get exact radical solutions.\nWe can convert \+
these to decimal answers in this way." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs( x = 2, imp_fun );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(!\")\"\"\"*&\"#SF&*$)%\"yG\"\"#
F&!\"\"F&*$F*F&F&\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s \+
:= solve( % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6&,$*$-%%sqrtG
6#,&\"#Q\"\"\"*&\"\"#F--F)6#\"$,#F-F-F-#!\"\"F/,$F'#F-F/,$*$-F)6#,&F,F
-*&F/F-F0F-F4F-F3,$F8F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
s := evalf( % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6&$!+hc#H2%!
\"*$\"+hc#H2%F($!+o&GGb\"F($\"+o&GGb\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 5 "" 0 "" {TEXT -1 29 "B GRAPHING IMPLICIT FUNCTIONS" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 427 "Graphing
 implicit functions opens up an exciting new world of graphing possibi
lities. Unfortunately, its quite time-consuming to graph most implicit
 functions by hand - often rendering them effectively ungraphable with
out technological aids. Fortunately, Maple graphs these functions effe
ctively.The results are amazing and somewhat a kin to mathematical tea
 leaves with many intricacies not usually seen in explicit functions.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 284 "Inst
ead of using the plot command, we use the implicitplot command. The fi
rst thing you may notice about implicit functions is that most of thes
e functions fail the vertical line test.... miserably! That failure is
 one of the reasons they are so interesting. Here is a tilted ellipse.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "    \+
                                                The implicit equation \+
 ,             an  x range,         and  a y range." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 59 "implicitplot( x^2 + x*y + y^2 = 16, x = -5.
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