{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 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 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 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 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE " " -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 " " 0 "" {TEXT -1 32 "Lesson 12: Linear Approximation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Assume that f(x ) is differentiable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Recall that : " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 12 "y = f (x + " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 11 "x) - f (x) " }} {PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 15 "x = dx =( x + " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 9 "x ) - x" } }{PARA 0 "" 0 "" {TEXT -1 65 " \+ dy = f ' (x) dx " }}{PARA 0 "" 0 "" {TEXT -1 23 "For sufficie ntly small " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 3 "x, " }} {PARA 0 "" 0 "" {TEXT -1 57 " \+ f ( x + " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 51 "x ) - f (x) is approximately equal to f ' (x) dx " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "and hence for sufficien tly small " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 3 "x " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 " \+ dy approximates " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 1 "y" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 7 "Example" }{TEXT -1 1 " " }{TEXT 258 1 "1" }{TEXT -1 5 "\n Let " }{XPPEDIT 18 0 "f (x) = x^5" "6#/-%\"fG6#%\"xG*$F'\"\"&" }{TEXT -1 15 ", x = 2, and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 10 "x = 0.4. " }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate, " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 71 "y, dy, plot f(x) a nd the tangent line when x = 2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:= x -> x^5;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"&\"\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&ar rowGF&,$*$)9$\"\"%\"\"\"\"\"&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f( 2 + 0.4) - f(2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"(CEw%!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(2.4);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(CE' z!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "df:= x -> 5*x^4;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGR6#%\"xG6\"6$%) operatorG%&arrowGF(,$*$)9$\"\"%\"\"\"\"\"&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "df(2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "80 * .4;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$?$!\"\"" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Hence, " }{XPPEDIT 18 0 "Delta; " "6#%&DeltaG" }{TEXT -1 5 "y = " }{XPPEDIT 18 0 "47.62624;" "6#$\"(C Ew%!\"&" }{TEXT -1 37 " and dy = f '(2) (0.4) = 32; " }}{PARA 0 "" 0 "" {TEXT -1 12 "note that " }{XPPEDIT 18 0 "Delta;" "6#%&Delt aG" }{TEXT -1 27 "x = 0.4 is large. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "tl:= x -> 32 + 80*(x - 2);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tlGR6#%\"xG6\"6$%)operatorG%&arrowG F(,&!$G\"\"\"\"*&\"#!)F.9$F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "tl(2.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$S'!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots): " }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "A:= plot(\{f( x),tl(x)\}, x= 1.5..2.7, color=[blue,brown]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "B:= plot([t,32,t = 2..2.4], color = magenta):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "C:= plot([2.4,t,t= 32 ..79 .62624], color = magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "F:= plot([t, 64, t= 2.4..2.6], color = black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "G:= plot([2.6,t,t=64..79.62624], color = bl ack):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "H:= plot([t,79.626 24, t= 2.4..2.6], color = black): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K:= textplot([2.65,74,'dy'],color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "L:= textplot([2.5,50,'deltay'] , co lor = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "M:= textplo t([2.2,20,'deltax'],color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(\{A,B,C,F,G,H,K,L,M\}, axes = boxed);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/ -%%TEXTG6%7$$\"#A!\"\"$\"#?\"\"!Q'deltax6\"-%'COLOURG6&%$RGBG$\"*++++ \"!\")$F,F,F6-%'CURVESG6$7S7$$\"3++++++++:!#<$F5F,7$$\"32+++&ech_\"F>$ !3J)******>tu!fF>7$$\"3/++v*G:*[:F>$!3?(*****>ox'3%F>7$$\"31++]L)4Xd\" F>$!3c()****>L@R?F>7$$\"3%*****\\MSF+;F>$\"3#y`*****fF#>#!#>7$$\"3$*** *\\Fy:fi\"F>$\"3'*z****>EEt?F>7$$\"3-++vl*)o\\;F>$\"3W!*****fs6vRF>7$$ \"31++v>ZIu;F>$\"3E3++!exV%fF>7$$\"3+++vnBw*p\"F>$\"3W9++?%*)4)zF>7$$ \"3/++vs$Q^s\"F>$\"3)))****>)p5,5!#;7$$\"31+++82C^$\"3x*****RqD*47F ho7$$\"39++]o;Bu$\"3!=++![L&QR\"Fho7$$\"36+++RS6+=F>$\"3#)*****>J74 g\"Fho7$$\"3&******\\o-h#=F>$\"3H+++![@)3=Fho7$$\"35+++hv9^=F>$\"3A,++ )[!=4?Fho7$$\"3+++v22*Q(=F>$\"3!*)****>mD6>#Fho7$$\"30+++4X$4!>F>$\"30 +++sgZ2CFho7$$\"39+++cT%Q#>F>$\"3Y,++[Kv!f#Fho7$$\"37++v@N\\]>F>$\"3O, ++u\"[R!GFho7$$\"3-+++EF3u>F>$\"3s*****z!=m#*HFho7$$\"3?++v@Q'***>F>$ \"3M-++u0r*>$Fho7$$\"37++DX(3Y-#F>$\"3a*****>'*poR$Fho7$$\"3S++]PJK]?F >$\"3c/+++^e-OFho7$$\"3A++vwp$R2#F>$\"3-,++9e\\\"z$Fho7$$\"3C++]#p2%*4 #F>$\"3M,++S:E&*RFho7$$\"33++v2Y'e7#F>$\"3#******>'o\"p?%Fho7$$\"33++D Ia*)[@F>$\"3G,++UM;\"R%Fho7$$\"35++]\\$pP<#F>$\"3i,++'za,f%Fho7$$\"37+ ++UlY*>#F>$\"3Q+++OBt&z%Fho7$$\"39++]JigCAF>$\"3c-++_)\\o*\\Fho7$$\"3E ++vt,$*[AF>$\"3!G+++RT9>&Fho7$$\"33++]kx$fF#F>$\"3d+++;@]2aFho7$$\"3=+ ++'G0-I#F>$\"3<-++)GU;g&Fho7$$\"3-+++Xg6EBF>$\"3v)*****f$G*3eFho7$$\"3 %****\\P/&f\\BF>$\"3c******\\.w'*fFho7$$\"31+++\"zj_P#F>$\"3b+++G.6-iF ho7$$\"33++v\"3;%*R#F>$\"3?-++a'G`R'Fho7$$\"3G++v%=iYU#F>$\"3a,++yuH(f 'Fho7$$\"3/+++l[M\\CF>$\"3-,++?*eZz'Fho7$$\"3A++vV`=vCF>$\"3S-++]F[,qF ho7$$\"3V+++'zs+]#F>$\"3'[++!oBe+sFho7$$\"3>++]5Q_DDF>$\"3$H++S[!>/uFh o7$$\"3C++vxSw]DF>$\"3c+++AE61wFho7$$\"3U+++is&Rd#F>$\"3I.++'4e;z(Fho7 $$\"3R++]o#R0g#F>$\"37.++[TJ/!)Fho7$$\"3`+++KXJCEF>$\"3%G++gD;X>)Fho7$ $\"3?++v@Rm\\EF>$\"33.++u8J(R)Fho7$$\"3Q++DAl#Rn#F>$\"3k,++y@T\"f)Fho7 $$\"3;+++++++FF>$\"#))F,-F06&F2F6F6F3-F86$7S7$F<$\"3++++++v$f(F>7$FA$ \"3qHh'HON$z#)F>7$FF$\"3@\"p7,q[`\"*)F>7$FK$\"3E7Hh+8pw'*F>7$FP$\"3KD5 *QEu%\\5Fho7$FV$\"3c'eZ^u#HO6Fho7$Fen$\"3mIT%=xG=A\"Fho7$Fjn$\"3powQgt u:8Fho7$F_o$\"3TIiB8\\')=9Fho7$Fdo$\"3gM0_iq)z_\"Fho7$Fjo$\"39_16:^8Z; 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$\"#:F)$\"#FF)Fj_o" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 79 "The tangent line is the linear approximation to a funct ion. We see here, that " }}{PARA 0 "" 0 "" {TEXT -1 13 "for larger \+ " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 50 "x , the linear a pproxiamtion is NOT a good one. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 7 "Example" }{TEXT -1 1 " " }{TEXT 260 1 "2" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x) = sqrt( x + 3)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&F'\" \"\"\"\"$F," }{TEXT -1 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 54 "a) Fi nd the linear approximation to f(x) when x = 1. " }}{PARA 0 "" 0 "" {TEXT -1 54 "b) Plot f(x) and the approximation on the same axes. " } }{PARA 0 "" 0 "" {TEXT -1 60 "c) Use the linear approximation to est imate, sqrt(3.98). " }}{PARA 0 "" 0 "" {TEXT -1 43 "Compare this esti mate to the actual value. " }}{PARA 0 "" 0 "" {TEXT -1 14 "d) Calculat e " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 31 "y and dy, \+ for x = 1 and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 10 " x = 2.98. " }}{PARA 0 "" 0 "" {TEXT -1 52 "Plot f(x), the linear appro ximation and show dy and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" } {TEXT -1 4 " y ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= x - > sqrt(x + 3);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f GR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&9$\"\"\"\"\"$F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,$*& \"\"\"F,-%%sqrtG6#,&9$F,\"\"$F,!\"\"#F,\"\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "df:= x -> (0.5) * ( 1 / (sqrt(x + 3)));" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGR6#%\"xG6\"6$%)o peratorG%&arrowGF(,$*&\"\"\"F.-%%sqrtG6#,&9$F.\"\"$F.!\"\"$\"\"&F5F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "df(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++D!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Thus the linea r approximation (i.e., the tangent line) passes through the point (1, 2) " }}{PARA 0 "" 0 "" {TEXT -1 21 "and has slope .25 . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "L:= x -> 2 + 0.25 * ( x - 1);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGR6#%\"xG6\"6$%)o peratorG%&arrowGF(,&$\"$v\"!\"#\"\"\"*&$\"#DF/F09$F0F0F(F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "a) Thus, linear approximation is: L(x) = 1.75 + 0.25 x . 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L(3.98);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"& ]u#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f(3.98);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j*o>k#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "c) The linear approximation function giv es 2.7450 as an estimate for sqrt(3.98), " }}{PARA 0 "" 0 "" {TEXT -1 35 "the actual value is approximately, " }{XPPEDIT 18 0 "2.641968963; " "6#$\"+j*o>k#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "d) We now do part (d). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(3.98) - f(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*j*o>k!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 " Thus, " }{XPPEDIT 18 0 "D elta;" "6#%&DeltaG" }{TEXT -1 6 " y = " }{XPPEDIT 18 0 ".641968963;" "6#$\"*j*o>k!\"*" }{TEXT -1 20 " (for x = 1 and " }{XPPEDIT 18 0 " Delta;" "6#%&DeltaG" }{TEXT -1 20 "x = 2.98). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "df(1) * 2.98;" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]u!#5" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 14 "Thus, dy = " }{XPPEDIT 18 0 ".7450000000;" "6#$\"+++ +]u!#5" }{TEXT -1 19 " (for x = 1 and " }{XPPEDIT 18 0 "Delta;" "6# %&DeltaG" }{TEXT -1 13 " x= 2.98). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L(3.98);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&]u#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$+#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f(3.98);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ j*o>k#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots): \+ " }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has be en redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:= plot( \{f(x),L(x)\}, x = 0..6, color=[blue,brown]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "B:= plot([t,2, t = 1..3.98], color = magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "C:= plot([3.98,t,t = 2..2 .7450], color = magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "F:= plot([t,2.641968963, t = 3.98..4.2],color = black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "G:= plot([4.2,t,t = 2.641968 963..2.7450], color = black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "H:= plot([t,2.7450, t = 3.98..4.2],color = black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "K:= textplot([4.5,2.7,'dy'], align= RIGHT, color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L:= textplot([4.2,2.3,'deltay'], align=RIGHT, color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M:= textplot([2.3,1.9,'deltax'], co lor = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display( \{A ,B,C,F,G,H,K,L,M\}, axes = boxed);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$\"3)************ *zR!#<$\"31+++j*o>k#F*7$$\"3gmm\"RP&z%)RF*F+7$$\"3aL37.y'*))RF*F+7$$\" 3gm;9O,m$*RF*F+7$$\"3em;*Hd$Q)*RF*F+7$$\"3eL3,%F*F+7$$\"3%o;a25Bm,%F*F+7$$\"3@+vLo`F@SF*F+7$$ \"3yLLQ(zgg-%F*F+7$$\"3zm;*e!eFISF*F+7$$\"3%)***\\r!4-NSF*F+7$$\"30++D #\\&yRSF*F+7$$\"3#)***\\G0xV/%F*F+7$$\"3-nTvHma[SF*F+7$$\"35LL)*fY]`SF *F+7$$\"3_LL$>w/x0%F*F+7$$\"3[+v)*y/fiSF*F+7$$\"3iLLVm^\"p1%F*F+7$$\"3 U+v)R.g;2%F*F+7$$\"3)**\\i*p#yh2%F*F+7$$\"3#QL3_d#*33%F*F+7$$\"3SL32z< A&3%F*F+7$$\"3-n;H59*)*3%F*F+7$$\"3-nTvW=u%4%F*F+7$$\"3#**\\7A;k*)4%F* F+7$$\"3\"omT2QCN5%F*F+7$$\"3>++qKbB3TF*F+7$$\"3z**\\xvW%G6%F*F+7$$\"3 -+v=lQI%F*F+7$$\"3K+DTi)>_>% F*F+7$$\"3;+++++++UF*F+-%'COLOURG6&%$RGBG\"\"!F[uF[u-F$6$7S7$$F[uF[u$ \"3+++++++]F*7$$ \"3e****\\(G[W[(Ffu$\"32+v=276P>F*7$$\"3i****\\()fB:()Ffu$\"3#**\\(o** 3)y'>F*7$$\"39++](Q=\"))**Ffu$\"3++vofHq**>F*7$$\"3(****\\P'=pD6F*$\"3 $)*\\Pf'HUJ?F*7$$\"33+++lN?c7F*$\"3'****\\7*30k?F*7$$\"3-++]U$e6P\"F*$ \"3%)**\\i&e*y#4#F*7$$\"36+++&>q0]\"F*$\"3(****\\([D9D@F*7$$\"3'****** \\U80j\"F*$\"3/++Dc$Gw:#F*7$$\"35+++0ytb@F*$\"3y*****\\>0)zAF*7$$\"3'****\\(3wY_AF*$\"3x*\\(=-p68BF*7 $$\"3#)******HOTqBF*$\"3&*****\\2MgUBF*7$$\"37++v3\">)*\\#F*$\"3#**\\( =xZ&\\P#F*7$$\"3:++DEP/BEF*$\"3$**\\i:$4w0CF*7$$\"3=++](o:;v#F*$\"3#)* *\\(=#R!zV#F*7$$\"3=++v$)[opGF*$\"3;+v$4A@uY#F*7$$\"3%*****\\i%Qq*HF*$ \"3?+]i:'f#*\\#F*7$$\"3&****\\(QIKHJF*$\"3***\\(of2LKDF*7$$\"3#****\\7 :xWC$F*$\"3w*\\7yG>6c#F*7$$\"37++]Zn%)oLF*$\"3C+](oo6Af#F*7$$\"3y***** *4FL(\\$F*$\"31++]xJLCEF*7$$\"3#)****\\d6.BOF*$\"3'***\\P*yddl#F*7$$\" 3(****\\(o3lWPF*$\"3)**\\(= FF*7$$\"3e******Hk-,SF*$\"3!*****\\2mD]FF*7$$\"36+++D-eITF*$\"3!)***\\ i0XEy#F*7$$\"3u***\\(=_(zC%F*$\"3$**\\(o/Q*>\"GF*7$$\"3M+++b*=jP%F*$\" 33++vQ(zS%GF*7$$\"3g***\\(3/3(\\%F*$\"3!**\\(=-,FuGF*7$$\"33++vB4JBYF* $\"3C+v$4tFe!HF*7$$\"3u*****\\KCnu%F*$\"3r***\\73\"oOHF*7$$\"3s***\\(= n#f([F*$\"3%**\\(oz;)*oHF*7$$\"3P+++!)RO+]F*$\"3()*****\\*44+IF*7$$\"3 0++]_!>w7&F*$\"3,+]7jZ!>.$F*7$$\"3O++v)Q?QD&F*$\"33+v=(4bM1$F*7$$\"3G+ ++5jyp`F*$\"31++]xlW#4$F*7$$\"3<++]Ujp-bF*$\"3/+]i&3uc7$F*7$$\"3++++gE d@cF*$\"3++++lJRbJF*7$$\"39++v3'>$[dF*$\"3-+v=-*zq=$F*7$$\"37++D6EjpeF *$\"3D+D\"G:3u@$F*7$$\"\"'F[u$\"3+++++++]KF*-Fht6&FjtF`uF`u$\"*++++\"! 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