{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 " \+ \+ " }{TEXT 256 10 "Calculus I" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 1295 " \+ Implicit Functions and Implicit Diff erentiation\n\n\nUsually when we speak of functions, we are talking ab out explicit functions of the form y = f(x). Sometimes we have instead an equation in x and y, for example, x + y + xy = sin(x + y), which m ay not be solvable for y. The solutions to this equation are a set of \+ points \{(x,y)\} which implicitly define a relation between x and y wh ich we will call an implicit function. Implicit functions are often no t actually functions in the strict definition of the word, because the y often have multiple y values for a single x value. They do have grap hs and derivatives however.\n\nYou are probably familiar with some lin ear implicit functions 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 functions since they are not explicitly solved for y in terms of x. These can be solved for y easily enough and converted into expl icit functions. However, many implicit functions are difficult or impo ssible to convert to explicit functions and must be left in implicit f orm. \nOur first task will be, given an implicit defined function, can we plot some individual points on the graph. Lets start with " } {OLE 1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JBLo>hoYaj]Aj;J:PZ:B:F:YLpfF>:::::::::J?NZ;vyyyyy=J:B:::::: :c:;:=j[vGUMrvC?MoJ::::::::JCNZ;F:f:vYxY:B::::::F:;JmJ:j;vCj^nGGmq>:;: :::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyqj>J?>:Q:S:UJ:n;v;;JBB :]:_J:V:;`:Z@w<WdZfbk;>ZEZ?GHZhV@cYH_WV>Z::j?NHYmmNH=mp> >GD:MMpjsj;^:[MpJJF@[C:s<;:::::::^=N:gVjk=;:kVcgJHUUEB:;j;@j:HJ?dj:DJ= EKKDJ>M]LVjrc:GvDHJA>Z;NZBH@?B\\[KCHBAr=V[KFZ<>klJ::;:::::::jysyAZ:^:;j:J]EZ:F:MZ=Vf;B:AB:s:;jDj\\Fh:fH=MtFGYMq>>Wlj:gml>:;:::::JJ:N:YLpJbNHEms>@[C:>Z::::::::kJ;@j;JZ:>:::::::::J?Z:f:^[<>f>>^GN\\:B:;x yyqNyyyyYjmZ:F[>j?<<:^:b<> c>f:;jCDJZDJaXJHB:qi:;fy>Z:JBA:;B:Cb:@CB:f?=J>JSdJJEJ>iJ;B:]SFKJ:N@NrBF:Fik>;N@Sx:yR;=:m=N@WW:CX;= :kUFK:_;kC;?B:;jRJS>n:kEFK:_;WT:=J:^`j^=VY;sy>Z:JB[:^Zc< ::J]k:J:>@<:UK;^:>x;F:K:_KtAjEaj:Zj>;b<;s:cY;=:<=bZaTXDpql`q?b:B:eT@jPF:C:[Q;JSdJJojDF:faj>;N@^ mAF:V_j>;N@>_=F:V?JSJWF:f_jN;n>N;yayI:>:s:qQBv:>:sg:B:=J;Dlc`qsLqlp@:CZ:>:?J:FZ:J:>@C:US:?J::::WTJWTL>Z:f?=J;[e:QB:[e:>:;:::vYxqyyyyyQ::^Z;J:<::::::::::::: ::::::::::::::::6:" }{TEXT -1 48 ", and find all points where x has th e value 2. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "imp_fun := - 4*x + 10*(x^2) * (y^(-2)) + y^2 = 11;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 339 "Our method will consist of two steps : first, substitute for the value of x, then solve the resulting equation in one variable for y.\n\nWe substitute x= 2 into the implicit 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 t his way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs( x = 2, im p_fun );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s := solve( % ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s := evalf( % );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 869 "GRAPHING IMPLICIT FUNCTIONS\n\nGr aphing implicit functions opens up an exciting new world of graphing p ossibilities. Unfortunately, its quite time-consuming to graph most im plicit functions by hand - often rendering them effectively ungraphabl e without technological aids. Fortunately, Maple graphs these function s effectively.The results are amazing and somewhat a kin to mathematic al tea leaves with many intricacies not usually seen in explicit funct ions.\n\nInstead of using the plot command, we use the implicitplot co mmand. The first thing you may notice about implicit functions is that most of these functions fail the vertical line test.... miserably! Th at failure is one of the reasons they are so interesting. Here is a ti lted ellipse.\n\n T he implicit equation , an x range, and a y rang e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "implicitplot( x^2 + x *y + y^2 = 16, x = -5..5, y = -5..5, grid=[50,50] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Some of the shapes are remarkably simple." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "implicitplot( x^4 + 8*(x^3) + y^4 = 16, x = -12..10, y = -10..10, grid=[50,50],thickness = 2, col or = brown, scaling = constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 289 "On the other hand, some functions can be remarkably complicate d. In fact, this one is so complicated that our first attempt just giv es a hint as to the complexity. By increasing the number of points use d in the production of the graph, a great amount of detail and subtlet y become visible." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "implic itplot( x*y*cos(x^2 + y^2) = 1, x = -10..10, y = -10..10, grid=[30,30] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "implicitplot( x*y*co s(x^2 + y^2) = 1, x = -10..10, y = -10..10, grid=[80,80] );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Many of these graphs are symmetric al in various ways. These are symmetric to the y axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "implicitplot( x^2 + 1.5*y*x^2 + y^2 = 1, x = -10..10, y = -10..10, grid=[50,50] );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 87 "implicitplot( (x^2 + y^2 -2) = (.5 + y*x^2)^2 \+ , x = -5..5, y = -5..5, grid=[100,100] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "This one is symmetrical to the x axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "implicitplot( -4*x + 10*(x^2) + (y^(-2)) \+ + y^2 = 11, x = -5..5, y = -5..5, grid=[100,100] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "And this one is symmetrical to both the x and y axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "implicitplot( x^2 - y^2 = x*y*sin(x*y), x = -4..4, y = -3..3, grid=[100,100] );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "This graph is apparently symmetric al to the line y = -x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 " implicitplot( x - y + sin(2.5*x*y) = sin(x) - sin(y) + sin(x*y), x = - 5..5, y = -5..5, grid=[100,100] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Some of the graphs have symmetries and patterns which are not a s easy to describe." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "impl icitplot( (x*y)*sin(y) = x*cos(x-y), x = -10..10, y = -10..10, grid=[1 00,100]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "This one has more co mplex pattern which hints as a more complex type of symmetry." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "implicitplot( ln( (x + 7*sin (y))^2 ) = exp(y + 2*cos(x)) , x = -9..9, y = -12..3, grid=[100,100]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "This graph creates a Tesslati on-like pattern in the plane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "implicitplot( sin(x + 2*sin(y) ) = cos( y + 3*cos(x)), x = -10.. 10, y = -10..10, grid=[100,100]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "C IMPLICIT DERIVATIVES\n\nJust as we use the implicitplot command rather than the plot command to graph these functions, we use the im plicitdiff command instead of the diff command to find the derivatives of implicit functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " implicitdiff( -4*x + 10*(x^2)*(y^(-2) ) + y^2 = 11, y, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 498 "The format of this command is an equatio n which implicitly defines a function, then the dependent and independ ent variables y, and x. Note that the result of taking an implicit der ivative is a function in both x and y. \n\nSince an implicit function \+ often has multiple y values for a single x value, there are also multi ple tangent lines. In this example, we will go through several steps t o construct all of the tangent lines for the value of x = 2.\n\nLets s tart with the same function we used above." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "imp_fun := -4*x + 10*(x^2)*(y^(-2)) + y^2 =11;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " Step 1 : Choose an x value (c)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c := 2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Step 2 : Find the Corresponding y Values" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "s := evalf( solve( subs( x = c, imp_fun)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "Step 3 : Fiin d the slope for each point - remember the implicit derivative is a fu nction of both x and y. For x = c, there are four different values of \+ y, and thus 4 different slopes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "m1 := evalf( subs ( \{ x = c, y = s[1] \}, implicitdiff( imp_f un, y,x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "m2 := evalf( subs ( \{ x = c, y = s[2] \}, implicitdiff( imp_fun, y,x)));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "m3 := evalf( subs ( \{ x = c , y = s[3] \}, implicitdiff( imp_fun, y,x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "m4 := evalf( subs ( \{ x = c, y = s[4] \}, impli citdiff( imp_fun, y,x)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "We \+ are finding the value of the slope for each of the 4 y values by subst ituting x = c, and y into the implicit derivative for each case.\n\n S tep 4 : Plot the function and its 4 tangent lines" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 148 "implicitplot( \{y - s[1] = m1*(x-c), y - s[ 2] = m2*(x-c), y - s[3] = m3*(x-c), y - s[4] = m4*(x-c), imp_fun \}, x = -5..5, y = -5..5, grid=[100,100]);" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }