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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 257 "
" 0 "" {TEXT -1 59 "Lesson 11:  Implicit Functions and Implicit Differ
entiation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 351 "Usually when we speak of funct
ions, we are talking about explicit functions of the form y = f(x). So
metimes we have instead an equation in x and y, for example, x + y + x
y = sin(x + y), which may not be solvable for y. The solutions to this
 equation are a set of points \{(x,y)\} which implicitly define a rela
tion between x and y which we will call an " }{TEXT 256 17 "implicit f
unction" }{TEXT -1 200 ". Implicit functions are often not actually fu
nctions in the strict definition of the word, because they often have \+
multiple y values for a single x value. They do have graphs and deriva
tives however." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 464 "You are probably familiar with some linear implicit func
tions from algebra. Lines in point-slope and standard form (such as y \+
-3 = 2(x + 5) and 7x + 9y = 63) are both examples of implicit function
s since they are not explicitly solved for y in terms of x. These can \+
be solved for y easily enough and converted into explicit functions. H
owever, many implicit functions are difficult or impossible to convert
 to explicit functions and must be left in implicit form. " }}{PARA 0 
"" 0 "" {TEXT -1 126 "Our first task will be, given an implicit define
d function, can we plot some individual points on the graph. Lets star
t with  " }{XPPEDIT 18 0 "-4*x + 10*(x^2)/y^(2) + y^2 = 11" "6#/,(*&\"
\"%\"\"\"%\"xGF'!\"\"*(\"#5F'*$F(\"\"#F'*$%\"yGF-F)F'*$F/F-F'\"#6" }
{TEXT -1 47 ", and find all points where x has the value 2. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "re
start; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name \+
changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "imp_fun := -4*x + 10*(x^2) * (y^(-2)) + y^2 = 11;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(imp_funG/,(%\"xG!\"%*&*&\"#5\"\"\")
F'\"\"#F,F,*$)%\"yGF.F,!\"\"F,*$F0F,F,\"#6" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "Our method will consist of two
 steps : first, substitute for the value of x, then solve the resultin
g equation in one variable for y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 204 "We substitute x= 2 into the implicit equ
ation.\nThis gives us an equation with only one unknown.\nWe solve thi
s equation and get exact radical solutions.\nWe can convert these to d
ecimal answers in this way." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "subs( x = 2, imp_fun );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(!\")
\"\"\"*&\"#SF&*$)%\"yG\"\"#F&!\"\"F&*$F*F&F&\"#6" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "s := solve( % );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sG6&,$*$-%%sqrtG6#,&\"#Q\"\"\"*&\"\"#F--F)6#\"$,#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&GG
b\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 27 
"GRAPHING IMPLICIT FUNCTIONS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 427 "Graphing implicit functions opens up an exciti
ng new world of graphing possibilities. Unfortunately, its quite time-
consuming to graph most implicit functions by hand - often rendering t
hem effectively ungraphable without technological aids. Fortunately, M
aple graphs these functions effectively.The results are amazing and so
mewhat a kin to mathematical tea leaves with many intricacies not usua
lly seen in explicit functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "Instead of using the " }{TEXT 258 4 "plot
" }{TEXT -1 21 " command, we use the " }{TEXT 257 12 "implicitplot" }
{TEXT -1 226 " command. The first thing you may notice about implicit \+
functions is that most of these functions fail the vertical line test.
... miserably! That failure is one of the reasons they are so interest
ing. 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 73 "implicitp
lot( x^2 + x*y + y^2 = 16, x = -5..5, y = -5..5, grid=[50,50] );" }}
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