{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 "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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 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 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 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 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 259 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 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 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bulle t Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "Introduction to calculus and equa tions of tangent lines " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone , Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 22.3. 2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "Introduction .. the problem s of differential and integral calculus " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 48 "Calculus can be div ided into two main branches: " }{TEXT 259 21 "differential calculus" } {TEXT -1 5 " and " }{TEXT 259 17 "integral calculus" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Simply s tated the main problem which differential calculus tries to solve is t he " }{TEXT 259 15 "tangent problem" }{TEXT -1 55 " - the problem of d etermining tangent lines to curves. " }}{PARA 0 "" 0 "" {TEXT -1 45 "T he main problem of integral calculus is the " }{TEXT 259 12 "area prob lem" }{TEXT -1 81 " - the problem of calculating the area of a region \+ bounded by a curve or curves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 138 "As an illustration of each of these prob lems with a view to introducing the concepts that will be involved we \+ can apply each of them to a " }{TEXT 259 6 "circle" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 8 "First - \+ " }{TEXT 259 19 "the tangent problem" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 36 "What is a tangent line to a circle? " }}{PARA 0 "" 0 " " {TEXT -1 104 "Roughly speaking, a tangent line to a circle is a line which just touches the circle at a single point. " }}{PARA 0 "" 0 "" {TEXT -1 64 "The description can be made more precise by means of a su itable " }{TEXT 259 16 "limiting process" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "The following ani mation illustrates this. The blue line initially meets the circle at t wo distinct points. Such a line is called a " }{TEXT 259 6 "secant" } {TEXT -1 475 ". The blue line is perpendicular to the middle radius li ne. As the animation plays the blue line moves outwards intil the two \+ points of intersection \"run together\", all the time remaining perpen dicular to the middle radius line. At this final \"limiting\" stage we obtain a tangent line to the circle. Note that this demonstrates that the tangent line to a circle is perpendicular to the radius line whic h joins the centre of the circle to the point of contact of the tangen t. 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j]l7$$\"3WU#=8co?>#FhoF]cm7$$\"3!H!yk,d3!H#FhoFfj]l7$$\"3yjt(>%G5)Q#Fh oF]cm7$$\"3ACpI#)*>h[#FhoFfj]l7$$\"35&[OE7PTe#FhoF]cm7$$\"3_Xg'HEa@o#F hoFfj]l7$$\"3)fg&H.9 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "The equations and slopes of tangent lines " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Example 1 .. introduction to the main ideas " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 271 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 51 "Find the equation o f the tangent line to the graph " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$% \"xG\"\"#" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1,1);" "6#- %!G6$\"\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 92 "Roughly speaking the tangent line is a line which ju st touches the curve at the single point" }{XPPEDIT 18 0 "``(1,1);" "6 #-%!G6$\"\"\"F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 266 292 292 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!\"#\"\"!$\"\"%F *7$$!3smm;HU,\"*=!#<$\"3c-3\"\\\"[$fd$F07$$!3SL$3FH'='z\"F0$\"3%pI8\") >&GEKF07$$!3gmmTgBa*o\"F0$\"3Y@Bk(Q`X&GF07$$!3amm\"H_\">#e\"F0$\"3\\g$ *=:+L.DF07$$!3ML$3_!4Nv9F0$\"3m[!*\\$Hgm<#F07$$!3km;/wfHw8F0$\"3%Q[oOh !>%*=F07$$!3;+]PM.tt7F0$\"3u\\qqk*)QA;F07$$!3em;/,oln6F0$\"39e)=]SAMO \"F07$$!3%)**\\(oWB>1\"F0$\"3n!)G-29oF6F07$$!3eJLLepjJ&*!#=$\"3o)eY0J5 _3*Fhn7$$!3Ulm;z/ot&)Fhn$\"3oaV)ep*z]tFhn7$$!3u)****\\P[_\\(Fhn$\"337S H?[(yh&Fhn7$$!3A*****\\7)Q7kFhn$\"3g+TcY@(=6%Fhn7$$!3e*****\\i^)o`Fhn$ \"3g8l7xnX#)GFhn7$$!3vlmT50A@WFhn$\"3I+HZ&>Fhn7$$!3OKLLeaR%H$Fhn$ \"3oG()eVTI&3\"Fhn7$$!3kJLLLo#)RBFhn$\"3?'o')*4'*yua!#>7$$!3f***\\PfO% H7Fhn$\"3c,;0QV^6:F^q7$$!3MSLLL3`lCF^q$\"3G&R<,HU)yg!#@7$$\"3+L+]i!f#= $)F^q$\"3,$R'3$QV$>p!#?7$$\"3+-+v=xpe=Fhn$\"3Ahko4svaMF^q7$$\"3`\"Fhn7$$\"3) GLL3_?`(\\Fhn$\"3iED]G9QvCFhn7$$\"3!fL$e*)>pxgFhn$\"3)\\aC?*R$Qp$Fhn7$ $\"3w++v$f4t.(Fhn$\"3!*z&G=jsB&\\Fhn7$$\"3OPL$e*Gst!)Fhn$\"3gW.()R,]=l Fhn7$$\"3Y+++]#RW9*Fhn$\"3)p0%p>p2i$)Fhn7$$\"3:++DJE>>5F0$\"3z)H%f>OvQ 5F07$$\"3F+]i!RU07\"F0$\"3s!f(=\\_hb7F07$$\"3+++v=S2L7F0$\"3DvirN:Z?:F 07$$\"3Jmmm\"p)=M8F0$\"3q57(\\Yf+y\"F07$$\"3B++](=]@W\"F0$\"3%fG1L;(zz ?F07$$\"35L$e*[$z*R:F0$\"3UJ\"=_RO:P#F07$$\"3e++]iC$pk\"F0$\"3u:j.OlQ7 FF07$$\"3[m;H2qcZF0$\"3zb'HmX&QCQF07$$\"3w**\\ilAFj?F0$\"3u>t4UC4dUF0 7$$\"3yLLL$)*pp;#F0$\"3wowm3*edp%F07$$\"3)RL$3xe,tAF0$\"39=HZx6gm^F07$ $\"3Cn;HdO=yBF0$\"3?q\"4y]ddl&F07$$\"3a+++D>#[Z#F0$\"3c32YgNuChF07$$\" 3SnmT&G!e&e#F0$\"3iN`L7aA&o'F07$$\"3#RLLL)Qk%o#F0$\"32$3>.y7t?(F07$$\" 37+]iSjE!z#F0$\"3))G[i^ie&y(F07$$\"3a+]P40O\"*GF0$\"3aiK<&fl*f$)F07$$ \"\"$F*$\"\"*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fc[l-F$6$7S7$F($! \"&F*7$F.$!3YLLLe%G?y%F07$F4$!3EnmT&esBf%F07$F9$!3ALL$3s%3zVF07$F>$!3i KL$e/$QkTF07$FC$!3ommT5=q]RF07$FH$!3ILL3_>f_PF07$FM$!3K++vo1YZNF07$FR$ !3;LL3-OJNLF07$FW$!3p***\\P*o%Q7$F07$Ffn$!3Kmmm\"RFj!HF07$F\\o$!33LL$e 4OZr#F07$Fao$!3u*****\\n\\!*\\#F07$Ffo$!3%)*****\\ixCG#F07$F[p$!3#**** **\\KqP2#F07$F`p$!39LL3-TC%)=F07$Fep$!3[mmm\"4z)e;F07$Fjp$!3Mmmmm`'zY \"F07$F`q$!3#****\\(=t)eC\"F07$Feq$!3!ommmh5$\\5F07$F[r$!3S$***\\(=[jL )Fhn7$Far$!3)f***\\iXg#G'Fhn7$Ffr$!3ndmmT&Q(RTFhn7$F[s$!3%\\mmTg=><#Fh n7$F`s$!3vDMLLe*e$\\F_r7$Fes$\"3!=nm\"zRQb@Fhn7$Fjs$\"3_,+](=>Y2%Fhn7$ F_t$\"3summ\"zXu9'Fhn7$Fdt$\"3#4+++]y))G)Fhn7$Fit$\"3H++]i_QQ5F07$F^u$ \"3b++D\"y%3T7F07$Fcu$\"3+++]P![hY\"F07$Fhu$\"3iKLL$Qx$o;F07$F]v$\"3Y+ ++v.I%)=F07$Fbv$\"3?mm\"zpe*z?F07$Fgv$\"3;,++D\\'QH#F07$F\\w$\"3%HL$e9 S8&\\#F07$Faw$\"3s++D1#=bq#F07$Ffw$\"3\"HLL$3s?6HF07$F[x$\"3a***\\7`Wl 7$F07$F`x$\"3enmmm*RRL$F07$Fex$\"3%zmmTvJga$F07$Fjx$\"3]MLe9tOcPF07$F_ y$\"31,++]Qk\\RF07$Fdy$\"3![LL3dg6<%F07$Fiy$\"3%ymmmw(GpVF07$F^z$\"3C+ +D\"oK0e%F07$Fcz$\"35,+v=5s#y%F07$Fhz$\"\"&F*-F][l6&F_[lFc[lFc[lF`[l-F $6&7#7$$\"\"\"F*F`el-%'SYMBOLG6#%'CIRCLEG-F][l6&F_[lF*F*F*-%&STYLEG6#% &POINTG-F$6&F^el-Fcel6#%(DIAMONDGFfelFhel-F$6&F^el-Fcel6#%&CROSSGFfelF hel-%%TEXTG6%7$$\"#:!\"\"F`elQ'P(1,1)6\"Ffel-%+AXESLABELSG6%Q\"xF^glQ \"yF^gl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fhz;F($\"\"'F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "The equation of a straight line with slope " }{TEXT 273 1 "m" }{TEXT -1 5 " and " }{TEXT 274 1 "y" }{TEXT -1 11 " intercept " }{TEXT 275 1 "c" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "y=m*x+c" "6#/%\"y G,&*&%\"mG\"\"\"%\"xGF(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "In order to determine the equation of the tangent line we need to find its " }{TEXT 259 5 "slope" }{TEXT -1 1 " " }{TEXT 276 1 "m" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The constant " } {TEXT 277 1 "c" }{TEXT -1 26 " in the standard equation " }{XPPEDIT 18 0 "y=m*x+c" "6#/%\"yG,&*&%\"mG\"\"\"%\"xGF(F(%\"cGF(" }{TEXT -1 83 " can then be found by using the fact that the tangent line passes thr ough the point" }{XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The idea is to use a limiting pro cess to determine the slope. " }}{PARA 0 "" 0 "" {TEXT -1 91 "We start by consdering secant lines which pass through the point P, and intere ct the curve " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 21 " at another point Q. 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" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following animation illus trates this. 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3)f***\\iXg#G'F\\o7$Fjr$!3ndmmT&Q(RTF\\o7$F_s$!3%\\mmTg=><#F\\o7$Fds$! 3vDMLLe*e$\\Fcr7$Fis$\"3!=nm\"zRQb@F\\o7$F^t$\"3_,+](=>Y2%F\\o7$Fct$\" 3summ\"zXu9'F\\o7$Fht$\"3#4+++]y))G)F\\o7$F]u$\"3H++]i_QQ5F47$Fbu$\"3b ++D\"y%3T7F47$Fgu$\"3+++]P![hY\"F47$F\\v$\"3iKLL$Qx$o;F47$Fav$\"3Y+++v .I%)=F47$Ffv$\"3?mm\"zpe*z?F47$F[w$\"3;,++D\\'QH#F47$F`w$\"3%HL$e9S8& \\#F47$Few$\"3s++D1#=bq#F47$Fjw$\"3\"HLL$3s?6HF47$F_x$\"3a***\\7`Wl7$F 47$Fdx$\"3enmmm*RRL$F47$Fix$\"3%zmmTvJga$F47$F^y$\"3]MLe9tOcPF47$Fcy$ \"31,++]Qk\\RF47$Fhy$\"3![LL3dg6<%F47$F]z$\"3%ymmmw(GpVF47$Fbz$\"3C++D \"oK0e%F47$Fgz$\"35,+v=5s#y%F47$F\\[l$\"\"&F.FgflFcglFbhlF[il-F\\il6$F ^il;$FiglF.F`il" 1 2 0 1 10 0 2 9 1 4 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" }}{TEXT -1 1 " " }}{PARA 258 "" 0 " " {TEXT -1 85 "The tangent line at P is obtained in the limit as the p oint P runs into the point Q. " }}{PARA 0 "" 0 "" {TEXT -1 106 "To fin d the slope of the tangent line at P we need a general expression for \+ the slope of a secant line PQ." }}{PARA 0 "" 0 "" {TEXT -1 30 "Suppose that Q has coordinates" }{XPPEDIT 18 0 "``(x,y);" "6#-%!G6$%\"xG%\"yG " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"# " }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 289 346 346 {PLOTDATA 2 "67-%'CURVESG6$7S7$$\"3U+++++++!)!#=$\"3C,++++++kF *7$$\"3VLLLe,;0$)F*$\"3[OnbD&ov*oF*7$$\"3_nmT!Qy1d)F*$\"3%o;a+z_cM(F*7 $$\"3WLL$3R\"Gp))F*$\"37Z#y*Q_TmyF*7$$\"3eML$etj)p\"*F*$\"3>G^P$4S'3%) F*7$$\"3+nmTlu,p%*F*$\"3a(eOg\"FN7$$\"3aLLeYp$*>6FN$\"3e+CKk(eUD\"FN7$$\"31++]XI8]6FN$\"3706N Ag!GK\"FN7$$\"31++]KJX!=\"FN$\"3yi!H!)fpMR\"FN7$$\"3@++]a@n47FN$\"3[sE P@nIj9FN7$$\"3\\L$3d#e?O7FN$\"3oA_^V[?G:FN7$$\"3)ommr#pvn7FN$\"3bp!zji 2sg\"FN7$$\"3!pmm'[[[%H\"FN$\"3[jvUB5pv;FN7$$\"3;+]PvddD8FN$\"3#G.hi8^ rv\"FN7$$\"3%ommO^'4`8FN$\"3M))oHv,(3$=FN7$$\"3?+]PD6H$Q\"FN$\"3bM7aPV \\8>FN7$$\"36+]7ON/79FN$\"3]%R7z%p'Q*>FN7$$\"3(om;/mV?W\"FN$\"3A**ya=* *[z?FN7$$\"3#omT&RJfp9FN$\"3%H$zy&*Rqf@FN7$$\"3YLLeu*3$*\\\"FN$\"3yEhE ,u#zC#FN7$$\"3iL$3dPv,`\"FN$\"3**RTU!oO9M#FN7$$\"37+]ioY/d:FN$\"3cON4+ \")QCCFN7$$\"3mLL3TU1'e\"FN$\"3)4DVox*f:DFN7$$\"38+++*HWgh\"FN$\"3%=SI j<*f6EFN7$$\"3+++vORPX;FN$\"3y#>O\"z\"FN$\"3yoM FN7$$\"3A+]()[Dxy=FN$\"3!\\E1/H'yHNFN7$$\"3$omm\"4!pv!>FN$\"3+*4LZ_>)Q OFN7$$\"32+]PMirP>FN$\"3/I/'\\?WZv$FN7$$\"3mLLL&f^n'>FN$\"398iuP=6oQFN 7$$\"3WLLeXWW'*>FN$\"3wV0IC/z&)RFN7$$\"3An;/C9*e-#FN$\"3Bub+igB/TFN7$$ \"3++++R,&H0#FN$\"3**=,AtUg9UFN7$$\"3%pm;*zC'R3#FN$\"3@C/q<'**GM%FN7$$ \"3[LLL(G+<6#FN$\"3E$oBN5y#fWFN7$$\"3@+]PvXFT@FN$\"3%yQZr!o0&e%FN7$$\" 37+]iU4ep@FN$\"3m$e,mY\"32ZFN7$$\"3;+++++++AFN$\"3u++++++S[FN-%'COLOUR G6&%$RGBG$\"*++++\"!\")$\"\"!F`[lF_[l-F$6&7$7$$\"\"\"F`[lFe[l7$$\"\"#F `[l$\"\"%F`[l-%'SYMBOLG6#%'CIRCLEG-Fiz6&F[[lF`[lF`[lF`[l-%&STYLEG6#%&P OINTG-F$6&Fc[l-F]\\l6#%(DIAMONDGF`\\lFb\\l-F$6&Fc[l-F]\\l6#%&CROSSGF` \\lFb\\l-F$6$7&Fd[lFg[l7$Fh[lFe[lFd[lF`\\l-F$6$7%7$$\"3'************* \\>FNFe[l7$Fh]l$\"33+++++++6FN7$Fh[lF[^lF`\\l-F$6%7$Fg[l7$FdzFj[lF`\\l -%*LINESTYLEG6#\"\"$-F$6%7$Fc]l7$FdzFe[lF`\\lFb^l-F$6%7$Fc]l7$Fh[l$\"3 w**************fF*F`\\lFb^l-F$6%7$Fd[l7$Fe[lF^_lF`\\lFb^l-F$6&7$7$$\"# @!\"\"$\"#FFj_l7$Fh_l$\"#SFj_l7%7$$\"++++!3#!\"*$\"++++qQFd`lF]`l7$$\" ++++?@Fd`lFe`l-Fc\\l6#%,PATCHNOGRIDGF`\\l-F$6&7$7$Fh_l$\"#BFj_l7$Fh_l$ \"#5Fj_l7%7$Fh`l$\"++++I6Fd`lFcal7$Fb`lFhalFj`lF`\\l-F$6&7$7$$\"#9Fj_l $\"\")Fj_l7$FdalFabl7%7$$\"++++S5Fd`l$\"+++++v!#5Fcbl7$Ffbl$\"+++++&)F jblFj`lF`\\l-F$6&7$7$$\"#;Fj_lFabl7$$\"#?Fj_lFabl7%7$$\"++++g>Fd`lF\\c lFdcl7$FiclFhblFj`lF`\\l-%%TEXTG6%7$$\"#&)!\"#$\"#6Fj_lQ'P(1,1)6\"F`\\ l-F]dl6%7$$\"$&=FbdlFj[lQ'Q(x,y)FfdlF`\\l-F]dl6%7$Fh_l$\"#DFj_lQ&y~-~1 FfdlF`\\l-F]dl6%7$$\"#:Fj_lFablQ&x~-~1FfdlF`\\l-%+AXESLABELSG6%Q\"xFfd lQ!Ffdl-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;Fabl$\"#AF j_lFafl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv e 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Then the slope of the secant line PQ is " } {XPPEDIT 18 0 "(y-1)/(x-1) = (x^2-1)/(x-1);" "6#/*&,&%\"yG\"\"\"F'!\" \"F',&%\"xGF'F'F(F(*&,&*$F*\"\"#F'F'F(F',&F*F'F'F(F(" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We ca n construct a table of values of " }{XPPEDIT 18 0 "(x^2-1)/(x-1)" "6# *&,&*$%\"xG\"\"#\"\"\"F(!\"\"F(,&F&F(F(F)F)" }{TEXT -1 15 " for values of " }{TEXT 278 1 "x" }{TEXT -1 38 " which get progressively closer t o 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "g := x -> (x^2-1)/(x-1);\nxvals := [2,1.5,1.1,1.01,1. 001,1.0001];\nmap(g,xvals);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf *6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"#\"\"\"F2F2!\"\"F2,&F0F 2F2F3F3F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7(\"\"#$\"#: !\"\"$\"#6F)$\"$,\"!\"#$\"%,5!\"$$\"&,+\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$$\"+++++D!\"*$\"+++++@F'$\"++++5?F'$\"++++,?F'$ \"+++5+?F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, `|`, (x^2-1)/(x-1)], [________, ``, ___________________ ____], [2, `|`, (2^2-1)/(2-1) = 3], [1.5, `|`, (1.5^2-1)/(1.5-1) = 2.5 ], [1.1, `|`, (1.1^2-1)/(1.1-1) = 2.1], [1.01, `|`, (1.01^2-1)/(1.01-1 ) = 2.01], [1.001, `|`, (1.001^2-1)/(1.001-1) = 2.001], [1.0001, `|`, \+ (1.0001^2-1)/(1.0001-1) = 2.0001]]);" "6#-%'matrixG6#7*7%%\"xG%\"|grG* &,&*$F(\"\"#\"\"\"F.!\"\"F.,&F(F.F.F/F/7%%)________G%!G%8_____________ __________G7%F-F)/*&,&*$F-F-F.F.F/F.,&F-F.F.F/F/\"\"$7%-%&FloatG6$\"#: F/F)/*&,&*$-F>6$F@F/F-F.F.F/F.,&-F>6$F@F/F.F.F/F/-F>6$\"#DF/7%-F>6$\"# 6F/F)/*&,&*$-F>6$FPF/F-F.F.F/F.,&-F>6$FPF/F.F.F/F/-F>6$\"#@F/7%-F>6$\" $,\"!\"#F)/*&,&*$-F>6$FjnF[oF-F.F.F/F.,&-F>6$FjnF[oF.F.F/F/-F>6$\"$,#F [o7%-F>6$\"%,5!\"$F)/*&,&*$-F>6$F[pF\\pF-F.F.F/F.,&-F>6$F[pF\\pF.F.F/F /-F>6$\"%,?F\\p7%-F>6$\"&,+\"!\"%F)/*&,&*$-F>6$F\\qF]qF-F.F.F/F.,&-F>6 $F\\qF]qF.F.F/F/-F>6$\"&,+#F]q" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "It looks as though the sl ope of the secant line approaches 2 as " }{TEXT 279 1 "x" }{TEXT -1 14 " approaches 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "To see that this is indeed the case note that the express ion " }{XPPEDIT 18 0 "(x^2-1)/(x-1)" "6#*&,&*$%\"xG\"\"#\"\"\"F(!\"\"F (,&F&F(F(F)F)" }{TEXT -1 28 " can be written in the form " }{XPPEDIT 18 0 "(x-1)*(x+1)/(x-1)" "6#*(,&%\"xG\"\"\"F&!\"\"F&,&F%F&F&F&F&,&F%F& F&F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 280 1 "x" }{TEXT -1 30 " is not equal to 1, the factor" }{XPPEDIT 18 0 "``(x-1)" "6#-%!G6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 37 " at the top c an be cancelled with the" }{XPPEDIT 18 0 "``(x-1)" "6#-%!G6#,&%\"xG\" \"\"F(!\"\"" }{TEXT -1 34 " at the bottom. For example, when " } {XPPEDIT 18 0 "x = 1.01" "6#/%\"xG-%&FloatG6$\"$,\"!\"#" }{TEXT -1 2 " , " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(x^2-1)/(x-1)= (x-1)*(x+1)/(x-1)" "6#/*&,&*$%\"xG\"\"#\"\"\"F)!\"\"F),&F'F)F)F*F**(,& F'F)F)F*F),&F'F)F)F)F),&F'F)F)F*F*" }{XPPEDIT 18 0 "`` = (1.01-1)*(1.0 1+1)/(1.01-1);" "6#/%!G*(,&-%&FloatG6$\"$,\"!\"#\"\"\"F,!\"\"F,,&-F(6$ F*F+F,F,F,F,,&-F(6$F*F+F,F,F-F-" }{XPPEDIT 18 0 "`` = ``(`0.01`)*(1.01 +1)/`0.01`;" "6#/%!G*(-F$6#%%0.01G\"\"\",&-%&FloatG6$\"$,\"!\"#F)F)F)F )%%0.01G!\"\"" }{TEXT -1 20 " = 1.01 + 1 = 2.01. " }}{PARA 0 "" 0 "" {TEXT -1 15 "We can evaluate" }{XPPEDIT 18 0 "``(x+1)" "6#-%!G6#,&%\"x G\"\"\"F(F(" }{TEXT -1 22 " to obtain values for " }{XPPEDIT 18 0 "(x^ 2-1)/(x-1)" "6#*&,&*$%\"xG\"\"#\"\"\"F(!\"\"F(,&F&F(F(F)F)" }{TEXT -1 4 " as " }{TEXT 281 1 "x" }{TEXT -1 15 " approaches 1. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Formally, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x^2-1)/(x-1) = x+1;" "6#/*&,&*$%\"xG\"\"#\"\"\"F)!\"\" F),&F'F)F)F*F*,&F'F)F)F)" }{TEXT -1 16 ", provided that " }{XPPEDIT 18 0 "x<>1" "6#0%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 285 14 "Important note" }{TEXT -1 21 ": We cannot evaluate " } {XPPEDIT 18 0 "(x^2-1)/(x-1)" "6#*&,&*$%\"xG\"\"#\"\"\"F(!\"\"F(,&F&F( F(F)F)" }{TEXT -1 6 " when " }{TEXT 282 1 "x" }{TEXT -1 51 " is 1 beca use we obtain the meaningless expression " }{XPPEDIT 18 0 "0/0" "6#*& \"\"!\"\"\"F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 237 "T his is perhaps not too surprising since it would amount to an attempt \+ to calculate the slope of the tangent line at P in the limiting situat ion when the point Q coincides with P in a direct way by dividing a \" rise\" of 0 by a \"run\" of 0." }}{PARA 0 "" 0 "" {TEXT -1 12 "However , if " }{TEXT 283 1 "x" }{TEXT -1 37 " is very close to 1, we can eval uate " }{XPPEDIT 18 0 "(x^2-1)/(x-1)" "6#*&,&*$%\"xG\"\"#\"\"\"F(!\"\" F(,&F&F(F(F)F)" }{TEXT -1 9 " by using" }{XPPEDIT 18 0 " ``(x+1)" "6#- %!G6#,&%\"xG\"\"\"F(F(" }{TEXT -1 20 ". For example, when " }{XPPEDIT 18 0 "x=1.00001" "6#/%\"xG-%&FloatG6$\"',+5!\"&" }{TEXT -1 46 ", the s lope of the secant line PQ is 2.00001. " }}{PARA 0 "" 0 "" {TEXT -1 32 "We can deduce from this that as " }{TEXT 284 1 "x" }{TEXT -1 47 " \+ gets progressively closer to 1, the value of " }{XPPEDIT 18 0 "(x^2-1 )/(x-1) = x+1;" "6#/*&,&*$%\"xG\"\"#\"\"\"F)!\"\"F),&F'F)F)F*F*,&F'F)F )F)" }{TEXT -1 33 " gets progressively closer to 2. " }}{PARA 0 "" 0 " " {TEXT -1 10 "We write: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[(x^2-1)/(x-1)]->2" "6#f*6#7#*&,&*$%\"xG\"\"#\"\"\"F+! \"\"F+,&F)F+F+F,F,7\"6$%)operatorG%&arrowG6\"F*F2F2F2" }{TEXT -1 5 ", \+ as " }{XPPEDIT 18 0 "x->1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\" \"F*F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 55 "where the arrow stands for \"approaches\" or \"ten ds to\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "This says that the slope of the secant line PQ gets progressively closer to 2 as the point Q approaches the point P." }}{PARA 0 "" 0 " " {TEXT -1 127 "Since the secant line becomes the tangent line in the \+ limiting situation when Q has run into the point P, we can conclude th at " }{TEXT 259 29 "the slope of the tangent line" }{TEXT -1 22 " at t he point P is 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The tangent line therefore has an equation of the form " }{XPPEDIT 18 0 "y=2*x+c" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(%\"cGF(" } {TEXT -1 49 ". Since the tangent line passes through the point" } {XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 15 ", we must ha ve " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 15 ". Substituting " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "y=2*x+c" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(%\"cGF(" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "1=2+c" "6#/\"\"\",&\"\"#F$%\"cGF$" }{TEXT -1 29 ", leading to conclusion that " }{XPPEDIT 18 0 "c=-1" "6#/%\"cG,$\" \"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The equatio n of the tangent line is therefore: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=2*x-1" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(F(!\"\"" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 435 274 274 {PLOTDATA 2 "6*-%'CURVESG6$7ao7$ $!35+++++++?!#=$!\"&\"\"!7$$!3qmmmm5\\-8F*$!3sb&zP%efxw!#<7$$!3?++++nT !***!#>$!3$*ftF\\#f4+\"!#;7$$!3WLLLLF#f&pF7$!3G_9f*yBwV\"F:7$$!3wmmm;) >'\\_F7$!31Kvu\\)**[!>F:7$$!30++++pJVNF7$!3?NE%fS9A#GF:7$$!3qmmmTa;!p# F7$!35.9F*\\Vsr$F:7$$!3NLLL$)R,P=F7$!3KkQ>,nhVaF:7$$!3omm;a#Q/T\"F7$!3 +chdz[***3(F:7$$!3-+++]_iQ)*!#?$!3]&))>X;-k,\"!#:7$$!3lmm;/mu0xFfn$!3- @M&[yKxH\"Fin7$$!3GLLLez'Gd&Ffn$!35\"!#97$$\"3SKLLLG#p)HFfn$\"3#o50P8FzM$Fin7$$\"3&=LLLeRR8& Ffn$\"3'Rn>!4>#y%>Fin7$$\"3KJLLLj&4G(Ffn$\"3QC%G;&fWt8Fin7$$\"3-LLL$)* )\\d6F7$\"34*3gMW;$R')F:7$$\"3#HLLLL-pe\"F7$\"3tIXoP]e,jF:7$$\"3qKLLL! 4dW#F7$\"3#R+kx#Qz)3%F:7$$\"3[KLLLd^/LF7$\"3eA2(z,ih-$F:7$$\"30KLLL\"H @-&F7$\"3Me$G*\\t=\"*>F:7$$\"3iJLLLDuRnF7$\"3M8?l.it$[\"F:7$$\"3\")*** ***HU'e,\"F*$\"3\\G5LN^$Q%)*F37$$\"3Ymmm1Uvd8F*$\"3MTuv3L5ltF37$$\"3?L LL`dq\"*>F*$\"3KRd$z'>#3-&F37$$\"3g******fe7[EF*$\"3g$*=4fbDwPF37$$\"3 $HLLLZ'*pK$F*$\"3?5_rOSr0IF37$$\"3y******R**o.SF*$\"3[4d)H\"fp(\\#F37$ $\"3UnmmYBv*p%F*$\"3H'eO9prx7#F37$$\"3(HLLL\\WGJ&F*$\"31w9S')3B#)=F37$ $\"3;,++S5/.gF*$\"3u%3h7OAem\"F37$$\"3y*******frgp'F*$\"3g(H#zVHT$\\\" F37$$\"3O+++g\\$RO(F*$\"3Cf?_D'pzN\"F37$$\"3mLLLt)=/(zF*$\"3'f3c^?RYD \"F37$$\"3Ammm1pe\"p)F*$\"3KYn0$)y`]6F37$$\"3%ymmm#3^-$*F*$\"3%*G'>Ify \\2\"F37$$\"3,+++egJ,5F3$\"3S9BK\"F3$\"39E'[n4& eS$)F*7$$\"3!ommmOGvE\"F3$\"3#>C&)oyp$*)yF*7$$\"3qmmm/')\\I8F3$\"3a71` @$zf^(F*7$$\"3QLLL80U)R\"F3$\"37.`K][#4:(F*7$$\"3ILLL(Gs*o9F3$\"3-&RFJ O![2oF*7$$\"3,+++9yQI:F3$\"377!RYW\"HMlF*7$$\"3QLLLl#=nf\"F3$\"3AJm\\B c%GE'F*7$$\"3))*****>6W_m\"F3$\"3,Pn\"p= F3$\"3qmMg*ov*\\`F*7$$\"3Jmmmi2)Q$>F3$\"3y:+nA'\\4<&F*7$$\"3.+++?h(H+# F3$\"3[t?;b2d#*\\F*7$$\"3-LLL$y'el?F3$\"3^iB5U$R7%[F*7$$\"3')*****fxOS 8#F3$\"3y.A5f[&fo%F*7$$\"39mmm%)GW)>#F3$\"3A5Jc/Sn[XF*7$$\"3)******f#e wlAF3$\"3`h([$p(=NT%F*7$$\"3Wmmm1jeJBF3$\"3atO`9b#*)G%F*7$$\"3]****** \\U\\+CF3$\"3Y3IUq(3e;%F*7$$\"3ULLL*ygoY#F3$\"3G&o!oJ]t`SF*7$$\"32LLLh ,tMDF3$\"3KS^aAJ>XRF*7$$\"3qmmmSv.-EF3$\"3U(pgH4UJ%QF*7$$\"3u*****>.') 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" }}{PARA 0 "" 0 "" {TEXT -1 48 "Let the point Q move along the curve towards P. " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following animation illustrates this. 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(G$Fev7$Fgem$\"3k%yVb:v-6$Fev7$F\\fm$\"3H5T)*f&[]#HFev7$Fafm$\"3)31dPL `mu#Fev7$Fffm$\"35oku[%=Uc#Fev7$F[gm$\"37[Jp;UH$Q#Fev7$F`gm$\"36****\\ PO/fS*\\Fev$\"3)**\\i:?&[ ^()Fev7$$\"3$)***\\(=$f%GcFev$\"3w*\\7.<&)Gf)Fev7$$\"3Q+++Dy,\"G'Fev$ \"3=++vVbuH%)Fev7$$\"33++]7@nFev7$$\"33++vVy!eP\"Fhv$\"3y**\\i!R![glFev7$$\"34+] (=WU[V\"Fhv$\"3?*\\7`*Q*GT'Fev7$$\"3)****\\7B>&)\\\"Fhv$\"33+](=#>q`iF ev7$$\"3)***\\P>:mk:Fhv$\"31+Dc,iM)3'Fev7$$\"3'***\\iv&QAi\"Fhv$\"36+v $4c.W%fFev7$$\"31++vtLU%o\"Fhv$\"3%)**\\il:%*)y&Fev7$$\"3!******\\Nm'[ `Fev7$$\"3&****\\7TW)R>Fhv$\"3c**\\(=(*)Q]^Fe v7$$\"3z*****\\@80+#Fhv$\"3_++]ipr)*\\Fev7$$\"31++]7,Hl?Fhv$\"3()***\\ (=ZxO[Fev7$$\"3()**\\P4w)R7#Fhv$\"3M+Dcw4.!p%Fev7$$\"3;++]x%f\")=#Fhv$ \"3d***\\iI,'HXFev7$$\"3!)**\\P/-a[AFhv$\"3]+D1*[\\'yVFev7$$\"3/+](=Yb ;J#Fhv$\"3!**\\7`Mh3A%Fev7$$\"3')****\\i@OtBFhv$\"3K++v$f%fmSFev7$$\"3 ')**\\PfL'zV#Fhv$\"3N+Dc,;40RFev7$$\"3>+++!*>=+DFhv$\"3a*****\\-X&\\PF ev7$$\"3-++DE&4Qc#Fhv$\"3#***\\P%=w/f$Fev7$$\"3=+]P%>5pi#Fhv$\"3c*\\iS ^CFV$Fev7$$\"39+++bJ*[o#Fhv$\"3k****\\7rw(G$Fev7$$\"33++Dr\"[8v#Fhv$\" 3z**\\(=dH;7$Fev7$$\"3++++Ijy5GFhv$\"3++++vT.tHFev7$$\"31+]P/)fT(GFhv$ \"3#)*\\i!*[+Y\"GFev7$$\"31+]i0j\"[$HFhv$\"3%)*\\PfBfHm#Fev7$Faim$\"3+ +++++++DFevF[imFiimFijm-Fc[n6$Fe[n;FiglF_hl-%*AXESTICKSG6$\"\"%F`hlFb[ n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" " Curve 9" "Curve 10" }}{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 85 " The tangent line at P is obtained in the limit as the point P runs int o the point Q. " }}{PARA 0 "" 0 "" {TEXT -1 106 "To find the slope of \+ the tangent line at P we need a general expression for the slope of a \+ secant line PQ." }}{PARA 0 "" 0 "" {TEXT -1 30 "Suppose that Q has coo rdinates" }{XPPEDIT 18 0 "``(x,y);" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "Then the slope of the secant line PQ is " }{XPPEDIT 18 0 "(y-1/2)/(x-2) = (1/x-1/2)/(x-2); " "6#/*&,&%\"yG\"\"\"*&F'F'\"\"#!\"\"F*F',&%\"xGF'F)F*F**&,&*&F'F'F,F* F'*&F'F'F)F*F*F',&F,F'F)F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We can construct a table \+ of values of " }{XPPEDIT 18 0 "(1/x-1/2)/(x-2);" "6#*&,&*&\"\"\"F&%\" xG!\"\"F&*&F&F&\"\"#F(F(F&,&F'F&F*F(F(" }{TEXT -1 15 " for values of \+ " }{TEXT 264 1 "x" }{TEXT -1 38 " which get progressively closer to 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "g := x -> (1/x-1/2)/(x-2);\nxvals := [3,2.5,2.1,2.01, 2.001,2.0001];\nevalf(evalf(map(g,xvals),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*&\"\"\"F/ 9$!\"\"F/#F/\"\"#F1F/,&F0F/F3F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7(\"\"$$\"#D!\"\"$\"#@F)$\"$,#!\"#$\"%,?!\"$$\"&,+#!\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7($!+nmmm;!#5$!+++++?F&$!+\"Q_4Q#F &$!+*=iv[#F&$!+D1v)\\#F&$!+5]()*\\#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, `|`, (1/x-1/2)/(x-2)], [_ __________, ``, ___________________________], [3, `|`, (1/3-1/2)/(3-2) = -1/6], [5/2 = 2.5, `|`, (2/5-1/2)/(5/2-2) = -1/5], [2.1, `|`, (1/2. 1-1/2)/(2.1-2) = -.2380952381], [2.01, `|`, (1/2.01-1/2)/(2.01-2) = -. 2487562189], [2.001, `|`, (1/2.001-1/2)/(2.001-2) = -.2498750625], [2. 0001, `|`, (2.0001-2)/(2.0001-2) = -.2499875010]]);" "6#-%'matrixG6#7* 7%%\"xG%\"|grG*&,&*&\"\"\"F-F(!\"\"F-*&F-F-\"\"#F.F.F-,&F(F-F0F.F.7%%, ___________G%!G%<___________________________G7%\"\"$F)/*&,&*&F-F-F7F.F -*&F-F-F0F.F.F-,&F7F-F0F.F.,$*&F-F-\"\"'F.F.7%/*&\"\"&F-F0F.-%&FloatG6 $\"#DF.F)/*&,&*&F0F-FDF.F-*&F-F-F0F.F.F-,&*&FDF-F0F.F-F0F.F.,$*&F-F-FD F.F.7%-FF6$\"#@F.F)/*&,&*&F-F--FF6$FUF.F.F-*&F-F-F0F.F.F-,&-FF6$FUF.F- F0F.F.,$-FF6$\"+\"Q_4Q#!#5F.7%-FF6$\"$,#!\"#F)/*&,&*&F-F--FF6$FboFcoF. F-*&F-F-F0F.F.F-,&-FF6$FboFcoF-F0F.F.,$-FF6$\"+*=iv[#F^oF.7%-FF6$\"%,? !\"$F)/*&,&*&F-F--FF6$FepFfpF.F-*&F-F-F0F.F.F-,&-FF6$FepFfpF-F0F.F.,$- FF6$\"+D1v)\\#F^oF.7%-FF6$\"&,+#!\"%F)/*&,&-FF6$FhqFiqF-F0F.F-,&-FF6$F hqFiqF-F0F.F.,$-FF6$\"+5]()*\\#F^oF." }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "It looks as though the \+ slope of the secant line approaches " }{XPPEDIT 18 0 "-1/4=-0.25" "6#/ ,$*&\"\"\"F&\"\"%!\"\"F(,$-%&FloatG6$\"#D!\"#F(" }{TEXT -1 4 " as " } {TEXT 265 1 "x" }{TEXT -1 14 " approaches 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "To see that this is indeed the \+ case note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( 1/x-1/2)/(x-2)=``((2-x)/(2*x))/(x-2)" "6#/*&,&*&\"\"\"F'%\"xG!\"\"F'*& F'F'\"\"#F)F)F',&F(F'F+F)F)*&-%!G6#*&,&F+F'F(F)F'*&F+F'F(F'F)F',&F(F'F +F)F)" }{XPPEDIT 18 0 " ``=(2-x)/(2*x*(x-2))" "6#/%!G*&,&\"\"#\"\"\"% \"xG!\"\"F(*(F'F(F)F(,&F)F(F'F*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 266 1 "x" } {TEXT -1 30 " is not equal to 2, the factor" }{XPPEDIT 18 0 "``(x-2); " "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 79 " at in the denominat or of the last expression can be divided into the numerator" } {XPPEDIT 18 0 "``(2-x);" "6#-%!G6#,&\"\"#\"\"\"%\"xG!\"\"" }{TEXT -1 9 " giving " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 17 "We can evaluate " }{XPPEDIT 18 0 "-1/(2 *x);" "6#,$*&\"\"\"F%*&\"\"#F%%\"xGF%!\"\"F)" }{TEXT -1 22 " to obtain values for " }{XPPEDIT 18 0 "(1/x-1/2)/(x-2);" "6#*&,&*&\"\"\"F&%\"xG !\"\"F&*&F&F&\"\"#F(F(F&,&F'F&F*F(F(" }{TEXT -1 4 " as " }{TEXT 267 1 "x" }{TEXT -1 15 " approaches 2. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Form ally, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1/x-1/2)/( x-2)=-1/(2*x)" "6#/*&,&*&\"\"\"F'%\"xG!\"\"F'*&F'F'\"\"#F)F)F',&F(F'F+ F)F),$*&F'F'*&F+F'F(F'F)F)" }{TEXT -1 16 ", provided that " }{XPPEDIT 18 0 "x<>2" "6#0%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " }{XPPEDIT 18 0 "(1/x-1/2)/(x-2)" "6#*&,&*&\"\" \"F&%\"xG!\"\"F&*&F&F&\"\"#F(F(F&,&F'F&F*F(F(" }{TEXT -1 26 " cannot b e evaluated when " }{TEXT 268 1 "x" }{TEXT -1 51 " is 2 because we obt ain the meaningless expression " }{XPPEDIT 18 0 "0/0" "6#*&\"\"!\"\"\" F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "However, if \+ " }{TEXT 269 1 "x" }{TEXT -1 37 " is very close to 2, we can evaluate \+ " }{XPPEDIT 18 0 "(1/x-1/2)/(x-2);" "6#*&,&*&\"\"\"F&%\"xG!\"\"F&*&F&F &\"\"#F(F(F&,&F'F&F*F(F(" }{TEXT -1 11 " by using " }{XPPEDIT 18 0 "- 1/(2*x);" "6#,$*&\"\"\"F%*&\"\"#F%%\"xGF%!\"\"F)" }{TEXT -1 20 ". For \+ example, when " }{XPPEDIT 18 0 "x = 2.00001;" "6#/%\"xG-%&FloatG6$\"', +?!\"&" }{TEXT -1 38 ", the slope of the secant line PQ is " } {XPPEDIT 18 0 "-1/4.00002;" "6#,$*&\"\"\"F%-%&FloatG6$\"'-+S!\"&!\"\"F +" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "We can deduce from \+ this that as " }{TEXT 270 1 "x" }{TEXT -1 46 " gets progressively clos er to 2, the value of " }{XPPEDIT 18 0 "(1/x-1/2)/(x-2) = -1/(2*x)" "6 #/*&,&*&\"\"\"F'%\"xG!\"\"F'*&F'F'\"\"#F)F)F',&F(F'F+F)F),$*&F'F'*&F+F 'F(F'F)F)" }{TEXT -1 31 " gets progressively closer to " }{XPPEDIT 18 0 "-1/4" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[(1/x-1/2)/(x-2)]->-1/4" "6 #f*6#7#*&,&*&\"\"\"F)%\"xG!\"\"F)*&F)F)\"\"#F+F+F),&F*F)F-F+F+7\"6$%)o peratorG%&arrowG6\",$*&F)F)\"\"%F+F+F3F3F3" }{TEXT -1 5 ", as " } {XPPEDIT 18 0 "x->2" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"#F*F*F *" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "that is, the slope of the secant line PQ approaches " } {XPPEDIT 18 0 "-1/4" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 4 " as " } {TEXT 286 1 "x" }{TEXT -1 15 " approaches 2. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We can conclude that " }{TEXT 259 29 "the slope of the ta ngent line" }{TEXT -1 19 " at the point P is " }{XPPEDIT 18 0 "-1/4" " 6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The tangent line therefore has \+ an equation of the form " }{XPPEDIT 18 0 "y = (-1/4)*x+c;" "6#/%\"yG,& *&,$*&\"\"\"F)\"\"%!\"\"F+F)%\"xGF)F)%\"cGF)" }{TEXT -1 49 ". Since th e tangent line passes through the point" }{XPPEDIT 18 0 "``(2, 1/2);" "6#-%!G6$\"\"#*&\"\"\"F(F&!\"\"" }{TEXT -1 15 ", we must have " } {XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 6 " \+ when " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 15 ". Substit uting " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "y = (-1/4)*x+c;" "6#/%\"yG,&*&,$*& \"\"\"F)\"\"%!\"\"F+F)%\"xGF)F)%\"cGF)" }{TEXT -1 7 " gives " } {XPPEDIT 18 0 "1/2 = -1/2+c;" "6#/*&\"\"\"F%\"\"#!\"\",&*&F%F%F&F'F'% \"cGF%" }{TEXT -1 29 ", leading to conclusion that " }{XPPEDIT 18 0 "c = 1;" "6#/%\"cG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The equation of the tangent line is therefore: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (-1/4)*x+1;" "6#/%\"yG,&*&,$*&\" \"\"F)\"\"%!\"\"F+F)%\"xGF)F)F)F)" }{TEXT -1 2 ". 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y1 := sin(alpha);\n x2 := cos(beta); \+ y2 := sin(beta);\n xm := (x1+x2)/2; ym := (y1+y2)/2;\n x3 := xm-b; y3 := ym+a;\n x4 := xm+b; y4 := ym-a;\n p3 := plot([[x3,y3],[x4,y 4]],color=blue);\n p4 := plot([[x1,y1],[0,0],[x2,y2]],color=black); \n p5 := plot([[[x1,y1],[x2,y2]]$3],style=point,\n sy mbol=[circle,diamond,cross],color=black);\n frms := frms,plots[displ ay]([p1,p2,p3,p4,p5]);\nend do:\nplots[display]([frms],insequence=true ,axes=none);\n\n\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 815 "p1 := plot([cos(t),sin(t),t=0..2*Pi],color =red):\nfrms := NULL:\nfor m from 2 to 5 do\n n := 2^m:\n theta := 2*Pi/n:\n rads := [seq([[0,0],[cos(theta*i),sin(theta*i)]],i=1..n)] :\n p2 := plot(rads,color=black):\n alpha := evalf((Pi+theta)/2): \n hh := sin(theta/2):\n vv := cos(theta/2):\n h := theta/20:\n \+ a := 1.8: b := -.5:\n p11 := plot([seq([a+i*hh,b+((-1)^(i+1)+1)/2* vv],i=-1..n)],color=black):\n c := a+hh: d := b+vv:\n pp := NULL: \n for j from 0 to n/2-1 do\n arc1 := [seq([a+cos(alpha-h*i),b+ sin(alpha-h*i)],i=0..20)]:\n arc2 := [seq([a+hh+cos(h*i-alpha),b+ vv+sin(h*i-alpha)],i=0..20)]:\n pp := pp,plot([arc1,arc2],color=r ed):\n a := a+2*hh:\n end do:\n frms := frms,plots[display]([ p1,p2,pp,p11]);\nend do:\nplots[display]([frms],insequence=true,scalin g=unconstrained,axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 26 "Tangent at (1,1) to y=x^2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "p1 := plot([x^2,2*x- 1],x=-2..3,y=-2..6,color=[red,blue]):\np2 := plot([[[1,1]]$3],style=po int,symbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot] ([1.5,1,P(1,1)],color=black):\nplots[display]([p1,p2,t1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "p1 := plot([x^2,3*x-2],x=-2..3,y=- 2..6,color=[red,blue]):\np2 := plot([[[1,1],[2,4]]$3],style=point,symb ol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[1.5,1 ,P(1,1)],[2.2,4,Q]],color=black):\nplots[display]([p1,p2,t1],view=[-2. .3,-1..6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 808 "p1 := plot(x^2,x=-2..3,y=-2..6,color=red):\np2 := pl ot([[[1,1]]$3],style=point,symbol=[circle,diamond,cross],color=black): \nt1 := plots[textplot]([1.5,1,P(1,1)],color=black):\nfrms := NULL:\nn umfrms := 30:\nh := 1/(numfrms-1):\nfor i from 0 to numfrms-2 do\n \+ a := 1+(numfrms-1-i)*h;\n b := a^2;\n p3 := plot((b-1)/(a-1)*(x- 1)+1,x=-2..3,y=-2..6,color=blue);\n p4 := plot([[[a,b]]$3],style=po int,symbol=[circle,diamond,cross],color=black);\n if i < numfrms-3 \+ then\n t2 := plots[textplot]([a+.2,b,Q],color=black); \n el se\n t2 := plots[textplot]([a-.2,b,Q],color=black);\n end if; \n frms := frms,plots[display]([p1,p2,p3,p4,t1,t2]);\nend do:\np3 : = plot(2*x-1,x=-2..3,y=-2..6,color=blue):\nfrms := frms,plots[display] ([p1,p2,p3,t1]):\nplots[display]([frms],insequence=true,view=[-2..3,-1 ..6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 767 "p1 := plot(x^2,x=.8..2 .2,color=red):\np2 := plot([[[1,1],[2,4]]$3],style=point,symbol=[circl e,diamond,cross],color=black):\np3 := plot([[[1,1],[2,4],[2,1],[1,1]], [[1.95,1],[1.95,1.1],[2,1.1]]],color=black):\np4 := plot([[[2,4],[2.2, 4]],[[2,1],[2.2,1]],[[2,1],[2,.6]],\n [[1,1],[1,.6]]],co lor=black,linestyle=3):\np5 := plottools[arrow]([2.1,2.7],[2.1,4],0,.0 4,.1,arrow,color=black):\np6 := plottools[arrow]([2.1,2.3],[2.1,1],0,. 04,.1,arrow,color=black):\np7 := plottools[arrow]([1.4,.8],[1,.8],0,.1 ,.1,arrow,color=black):\np8 := plottools[arrow]([1.6,.8],[2,.8],0,.1,. 1,arrow,color=black):\nt1 := plots[textplot]([[.85,1.1,P(1,1)],[1.85,4 ,Q(x,y)],\n [2.1,2.5,`y - 1`],[1.5,.8,`x - 1`]],color=black): \nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "p1 \+ := plot([x^2,2*x-1],x=-2..3,y=-2..6,color=[red,blue],\n thi ckness=2):\np2 := plot([[[1,1]]$3],style=point,symbol=[circle,diamond, cross],\n color=black):\nt1 := plots[textplot]([1.5,1,P(1,1)],colo r=black):\nt2 := plots[textplot]([1.8,5.5,y = x],color=red):\nt3 := pl ots[textplot]([2.2,5.6,`2`],color=red,\n font=[HELVETICA,8]): \nt4 := plots[textplot]([2.7,2.9,`y = 2 x - 1`],color=blue):\nplots[di splay]([p1,p2,t1,t2,t3,t4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 28 "Tangent at (2,1/2) to y=1/x " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "p1 := plot([1/x,- 1/4*x+1],x=-.2..3,y=0..2,color=[red,blue]):\np2 := plot([[[2,1/2]]$3], style=point,symbol=[circle,diamond,cross],\n color=black):\nt1 := plo ts[textplot]([2.05,.7,P(2,1/2)],color=black):\nplots[display]([p1,p2,t 1],view=[-.2..3,0..2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "p1 := plot([x^2,3*x-2],x=-2..3,y=0 ..6,color=[red,blue]):\np2 := plot([[[1,1],[2,4]]$3],style=point,symbo l=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[1.5,1, P(1,1)],[2.2,4,Q]],color=black):\nplots[display]([p1,p2,t1],view=[-2.. 3,-1..6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 853 "p1 := plot(1/x,x=0..3.5,y=0..1.8,color=red):\np2 : = plot([[[2,1/2]]$3],style=point,symbol=[circle,diamond,cross],\n \+ color=black):\nt1 := plots[textplot]([2,.65,P(2,1/2)],color=black): \nfrms := NULL:\nnumfrms := 30:\nh := 1/(numfrms-1):\nfor i from 0 to \+ numfrms-2 do\n a := 2+(numfrms-1-i)*h;\n b := 1/a;\n p3 := pl ot((b-1/2)/(a-2)*(x-2)+1/2,x=0..3.5,y=0..1.8,color=blue);\n p4 := p lot([[[a,b]]$3],style=point,symbol=[circle,diamond,cross],color=black) ;\n if i < numfrms-5 then\n t2 := plots[textplot]([a,b-.1,Q], color=black); \n else\n t2 := plots[textplot]([a,b-.1,Q],co lor=black);\n end if;\n frms := frms,plots[display]([p1,p2,p3,p4 ,t1,t2]);\nend do:\np3 := plot(1-1/4*x,x=0..3,y=0..2,color=blue):\nfrm s := frms,plots[display]([p1,p2,p3,t1]):\nplots[display]([frms],insequ ence=true,view=[0..3.5,0..1.8],\n tickmarks=[4,2]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "p1 : = plot([1/x,-x/4+1],x=.2..3.5,y=0..2,color=[red,blue],\n th ickness=2):\np2 := plot([[[2,1/2]]$3],style=point,symbol=[circle,diamo nd,cross],\n color=black):\nt1 := plots[textplot]([2.1,.6,P(2,1/2) ],color=black):\nt2 := plots[textplot]([.9,1.7,y = 1/x],color=red):\nt 3 := plots[textplot]([.8,.6,`y = - x/4 + 1`],color=blue):\nplots[displ ay]([p1,p2,t1,t2,t3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }