{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Lab 14 Maple Activities fo r November 7: (Lesson 11 - Implicit Differentiation)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "In this lab, we will lear n the Maple command for finding the derivative of an implicit function ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 8 "Examples" }{TEXT -1 52 " Differentiate the following functions using Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{XPPEDIT 18 0 "3*x^2+4*y^2 = 5;" "6#/,&*&\"\"$\" \"\"*$%\"xG\"\"#F'F'*&\"\"%F'*$%\"yG\"\"#F'F'\"\"&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dy/dx = implicitdiff(3*x^2 + 4*y^2 = 5, y, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 5 " NOTE:" }{TEXT -1 24 " The first entry in the " }{TEXT 266 12 "implicit diff" }{TEXT -1 210 " command is the implicit function which is given \+ as an equation. The second entry in the command is the dependent vari able in the equation of the implicit function. In our equation, the d ependent variable is " }{TEXT 267 1 "y" }{TEXT -1 150 ". The third en try in the command is the independent variable in the equation of the \+ implicit function. In our equation, the independent variable is " } {TEXT 270 1 "x" }{TEXT -1 89 ". The derivative answer, which Maple gi ves, is the derivative of the dependent variable " }{TEXT 268 1 "y" } {TEXT -1 57 " differentiated with respect to the independent variable \+ " }{TEXT 269 1 "x" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{XPPEDIT 18 0 "5*x^2-x*y-3*y ^4 = x;" "6#/,(*&\"\"&\"\"\"*$%\"xG\"\"#F'F'*&F)F'%\"yGF'!\"\"*&\"\"$F '*$F,\"\"%F'F-F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dy/dx = implicitdiff(5*x^2 - x*y - 3*y^4 = x, y, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "3. " }{XPPEDIT 18 0 "(y^2-9)^4 = (4*x^2+3 *x-1)^3;" "6#/*$,&*$%\"yG\"\"#\"\"\"\"\"*!\"\"\"\"%*$,(*&\"\"%F)*$%\"x G\"\"#F)F)*&\"\"$F)F2F)F)\"\"\"F+\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "dy/dx \+ = implicitdiff((y^2 - 9)^4 = (4*x^2 + 3*x - 1)^3, y, x); factor(%);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 271 5 "NOTE:" }{TEXT -1 111 " The factored answer above allow s us to compare this Maple answer with the answer which we obtained in lecture." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. " }{XPPEDIT 18 0 "6*x^2/(y^4+5)+y = 3*x+2;" "6#/,&*(\" \"'\"\"\"*$%\"xG\"\"#F',&*$%\"yG\"\"%F'\"\"&F'!\"\"F'F-F',&*&\"\"$F'F) F'F'\"\"#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dy/dx = implicitdiff((6*x^2) /(y^4 + 5) + y = 3*x + 2, y, x); factor(%);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 5 "NOTE:" } {TEXT -1 121 " Again, the factored answer above allows us to compare t his Maple answer with the answer which is given in lecture notes." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5. \+ " }{XPPEDIT 18 0 "x*tan*y-y^3 = x^2;" "6#/,&*(%\"xG\"\"\"%$tanGF'%\"y GF'F'*$F)\"\"$!\"\"*$F&\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dy/dx = implicitd iff(x*tan(y) - y^3 = x^2, y, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 5 "NOTE:" }{TEXT -1 33 " Remember, the Maple expression " }{XPPEDIT 18 0 "tan(y)^2;" "6#* $-%$tanG6#%\"yG\"\"#" }{TEXT -1 34 " is the mathematical expression \+ " }{XPPEDIT 18 0 "tan^2*y;" "6#*&%$tanG\"\"#%\"yG\"\"\"" }{TEXT -1 156 " . This answer does not look like the answer which we obtained w hen we worked this problem in lecture. This is because Maple has repl aced the expression " }{XPPEDIT 18 0 "sec^2*y;" "6#*&%$secG\"\"#%\"yG \"\"\"" }{TEXT -1 39 " in the answer by the expression 1 + " } {XPPEDIT 18 0 "tan^2*y;" "6#*&%$tanG\"\"#%\"yG\"\"\"" }{TEXT -1 49 " \+ which follows from the trigonometric identity " }{XPPEDIT 18 0 "sec^2 *y;" "6#*&%$secG\"\"#%\"yG\"\"\"" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "ta n^2*y;" "6#*&%$tanG\"\"#%\"yG\"\"\"" }{TEXT -1 6 " = 1. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "6. " } {XPPEDIT 18 0 "sec(x^2*y) = cos^3*y+4*x;" "6#/-%$secG6#*&%\"xG\"\"#%\" yG\"\"\",&*&%$cosG\"\"$F*F+F+*&\"\"%F+F(F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "d y/dx = implicitdiff(sec(x^2*y) = cos(y)^3 + 4*x, y, x); simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 274 5 "NOTE:" }{TEXT -1 254 " This answer does not look like \+ the answer which is given for this problem in the lecture notes. Let' s ask Maple to simplify the answer from the lecture notes and see if t he two answers match. The answer given in the lecture notes for this \+ problem is " }{XPPEDIT 18 0 "(4-2*x*y*sec(x^2*y)*tan(x^2*y))/(x^2*sec (x^2*y)*tan(x^2*y)+3*cos^2*y*siny);" "6#*&,&\"\"%\"\"\"*,\"\"#F&%\"xGF &%\"yGF&-%$secG6#*&F)\"\"#F*F&F&-%$tanG6#*&F)\"\"#F*F&F&!\"\"F&,&*(F) \"\"#-F,6#*&F)\"\"#F*F&F&-F16#*&F)\"\"#F*F&F&F&**\"\"$F&*$%$cosG\"\"#F &F*F&%%sinyGF&F&F5" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "dy/dx = simplify((4-2*x*y* sec(x^2*y)*tan(x^2*y))/(x^2*sec(x^2*y)*tan(x^2*y) + 3*cos(y)^2*sin(y)) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 5 "NOTE:" }{TEXT -1 276 " The Maple answer and the answer given in the lecture notes are the same. In order to compare \+ your answer with the answer given by Maple, ask Maple to simplify its \+ answer and ask Maple to simplify your answer. These two answers shoul d be the same if your answer is correct." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT 262 7 "Example" }{TEXT 263 0 "" }{TEXT -1 70 " Find the equation of t he tangent line to the graph of the equation " }{XPPEDIT 18 0 "2*x^3- x^2*y+y^3-1 = 0;" "6#/,**&\"\"#\"\"\"*$%\"xG\"\"$F'F'*&F)\"\"#%\"yGF'! \"\"*$F-\"\"$F'\"\"\"F.\"\"!" }{TEXT -1 24 " at the point (2, - 3)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "We would do the following in order to find the slope of the tangent line:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "dy/dx = implicitdiff(2*x^3 - x^2*y + y^3 - 1 = 0, y, \+ x); subs(x=2, y=-3, %);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "Now, to find the equation of the tangent \+ line:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y + 3 = -36/23*(x \+ - 2); y = solve(%, y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 7 "Example" } {TEXT 260 0 "" }{TEXT -1 66 " Find the slope of the normal line to th e graph of the equation " }{XPPEDIT 18 0 "y^4+3*y-4*x^3 = 5*x+1;" "6# /,(*$%\"yG\"\"%\"\"\"*&\"\"$F(F&F(F(*&\"\"%F(*$%\"xG\"\"$F(!\"\",&*&\" \"&F(F.F(F(\"\"\"F(" }{TEXT -1 24 " at the point (1, - 2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "We would do the following in order to fin d the slope of the tangent line:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "dy/dx = implicitdiff(y^4 + 3*y - 4*x^3 = 5*x + 1, y, x); implicitdiff(y^4 + 3*y - 4*x^3 = 5*x + 1, y, x); subs(x=1, y=-2, \+ %); mtan := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Now, the slope of the normal line would be given by: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "-1/mtan;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 18 "Practice Problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "In Problems 1 - 5, find the derivative of the following implici t functions using the " }{TEXT 264 12 "implicitdiff" }{TEXT -1 9 " com mand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{XPPEDIT 18 0 "x^3+ 2*x^2*y^3+y^5 = x^2+y;" "6#/,(*$%\"xG\"\"$\"\"\"*(\"\"#F(*$F&\"\"#F(% \"yG\"\"$F(*$F-\"\"&F(,&*$F&\"\"#F(F-F(" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{XPPEDIT 18 0 "x^2*y+x*y^2+y^3 = x+3*y;" "6 #/,(*&%\"xG\"\"#%\"yG\"\"\"F)*&F&F)*$F(\"\"#F)F)*$F(\"\"$F),&F&F)*&\" \"$F)F(F)F)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. \+ " }{XPPEDIT 18 0 "sqrt(9-x^2*y^6) = 2*x*y;" "6#/-%%sqrtG6#,&\"\"*\"\" \"*&%\"xG\"\"#%\"yG\"\"'!\"\"*(\"\"#F)F+F)F-F)" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "4. " }{XPPEDIT 18 0 "sin(y^5) = x^4*y^6; " "6#/-%$sinG6#*$%\"yG\"\"&*&%\"xG\"\"%F(\"\"'" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "5. " }{XPPEDIT 18 0 "x^3+y^2 = cot(2*x+3* y);" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yG\"\"#F(-%$cotG6#,&*&\"\"#F(F&F(F(* &\"\"$F(F*F(F(" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "6. Find the equation of the tangent line to the graph of the equation \+ " }{XPPEDIT 18 0 "x^4-2*x^2*y^3+4*y = 220;" "6#/,(*$%\"xG\"\"%\"\"\"* (\"\"#F(*$F&\"\"#F(%\"yG\"\"$!\"\"*&\"\"%F(F-F(F(\"$?#" }{TEXT -1 24 " at the point (2, - 3)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 } {VIEWOPTS 1 1 0 1 1 1803 }