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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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 "Maple Plot" -1 13 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 " -1 256 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" -1 257 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" -1 258 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 35 "Tangent and normal lines to curve s " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Cana da" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 31.1.2006" }}{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 35 "The point-slope equation of a line " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "Find the equation o f the line which passes through the point" }{XPPEDIT 18 0 "``(2,3)" "6 #-%!G6$\"\"#\"\"$" }{TEXT -1 18 " and has gradient " }{XPPEDIT 18 0 "1 /2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 368 206 206 {PLOTDATA 2 "60-%'CURVESG6%7$7$$\"\"!F)$\"\"#F)7 $$\"\")F)$\"\"'F)-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%*THICKNESSG6#\" \"\"-F$6%7%7$F*$\"\"$F)7$F/F@7$F/$\"\"&F)-F26&F4F)F)F)-%*LINESTYLEG6#F +-F$6&7$F?FC-%'SYMBOLG6#%'CIRCLEGFF-%&STYLEG6#%&POINTG-F$6&FM-FO6#%(DI AMONDGFFFR-F$6&FM-FO6#%&CROSSGFFFR-%%TEXTG6&7$$\"#m!\"\"$\"#[F`oQ'P(x, y)6\"FF-%%FONTG6$%*HELVETICAG\"#5-F[o6&7$$\"#BF`o$\"#FF`oQ'A(2,3)FdoFF Feo-F[o6&7$F@$\"#XF`oQ+gradient~=FdoF1Feo-F[o6&7$$\"$&Q!\"#$\"$l%F]qQ \"1FdoF1-Ffo6$FhoF.-F[o6&7$F[q$\"$N%F]qQ\"2FdoF1Faq-F[o6&7$F[q$\"#YF`o Q\"_FdoF1Faq-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FdoFfr-Ffo6#%(DEFA ULTG-%%VIEWG6$FirFir" 1 2 0 1 10 0 2 9 1 1 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" "Curve 11" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$% \"xG%\"yG" }{TEXT -1 48 " is a point on the line which is different fr om " }{XPPEDIT 18 0 "A(2,3)" "6#-%\"AG6$\"\"#\"\"$" }{TEXT -1 29 ". Th en the gradient of AP is " }{XPPEDIT 18 0 "(y-3)/(x-2)" "6#*&,&%\"yG\" \"\"\"\"$!\"\"F&,&%\"xGF&\"\"#F(F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 34 "Since the gradient of the line is " }{XPPEDIT 18 0 "1/2 " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 11 ", we have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-3)/(x-2)=1/2" "6#/*&,&%\"yG\"\" \"\"\"$!\"\"F',&%\"xGF'\"\"#F)F)*&F'F'F,F)" }{TEXT -1 14 " ------- (i) , " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-3=``(1/2)*(x-2)" "6#/,&%\"yG\"\"\"\" \"$!\"\"*&-%!G6#*&F&F&\"\"#F(F&,&%\"xGF&F.F(F&" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Note that the coordinates of the point " }{XPPEDIT 18 0 "A(2,3)" " 6#-%\"AG6$\"\"#\"\"$" }{TEXT -1 97 " do not satisfy the equation (i) b ecause the left hand side of (i) is the meaningless expression " } {XPPEDIT 18 0 "0/0" "6#*&\"\"!\"\"\"F$!\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 107 ". However, they do satisfy equa tion (ii) because the left and right sides of (ii) are both equal to 0 when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y=3" "6#/%\"yG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 168 "Hence (ii) is an equation satisfied by the coordinates o f all points on the line, (and by the coordinates of no other points). It is therefore the equation of the line." }}{PARA 0 "" 0 "" {TEXT -1 47 "Equation (ii) may also be written in the form: " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y-6=x-2" "6#/,&*&\"\"#\"\"\"% \"yGF'F'\"\"'!\"\",&%\"xGF'F&F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2* y=x+4" "6#/*&\"\"#\"\"\"%\"yGF&,&%\"xGF&\"\"%F&" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=x/2+2" "6#/%\"yG,&*&%\"xG\"\"\"\"\"#!\"\"F(F)F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "This argument can be generalised to provide a general fo rmula for the equation of the line which passes through a fixed point " }{XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" } {TEXT -1 18 " and has gradient " }{TEXT 267 1 "m" }{TEXT -1 2 ". " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 384 224 224 {PLOTDATA 2 "6 /-%'CURVESG6%7$7$$\"\"!F)$\"\"#F)7$$\"\")F)$\"\"'F)-%'COLOURG6&%$RGBGF (F($\"*++++\"!\")-%*THICKNESSG6#\"\"\"-F$6%7%7$F*$\"\"$F)7$F/F@7$F/$\" \"&F)-F26&F4F)F)F)-%*LINESTYLEG6#F+-F$6&7$F?FC-%'SYMBOLG6#%'CIRCLEGFF- %&STYLEG6#%&POINTG-F$6&FM-FO6#%(DIAMONDGFFFR-F$6&FM-FO6#%&CROSSGFFFR-% %TEXTG6&7$$\"#m!\"\"$\"#[F`oQ'P(x,y)6\"FF-%%FONTG6$%*HELVETICAG\"#5-F[ o6&7$$\"#AF`o$\"#FF`oQ)A(x~,y~)FdoFFFeo-F[o6&7$$\"$A#!\"#$\"$b#FgpQ\"l FdoFF-Ffo6$Fho\"\"(-F[o6&7$$\"$d#FgpFhpFjpFFF[q-F[o6&7$$\"#MF`o$\"#XF` oQ-gradient~=~mFdoF1Feo-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FdoFbr- Ffo6#%(DEFAULTG-%%VIEWG6$FerFer" 1 2 0 1 10 0 2 9 1 1 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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$%\"xG%\"yG" }{TEXT -1 38 " is a poin t on the line distinct from " }{XPPEDIT 18 0 "A(x[1],y[1])" "6#-%\"AG6 $&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 7 ", then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-y[1])/(x-x[1])=m" "6#/*&,&%\"yG\"\" \"&F&6#F'!\"\"F',&%\"xGF'&F,6#F'F*F*%\"mG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-y[1]=m*(x-x[1])" "6#/,&%\"yG\"\"\"&F%6#F&!\"\"*&%\"mG F&,&%\"xGF&&F-6#F&F)F&" }{TEXT -1 16 " ------- (iii). " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{TEXT 268 24 "___________ " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Equation \+ (iii) is satisfied by the coordinates of all the points on the line (i ncluding " }{XPPEDIT 18 0 "A(x[1],y[1])" "6#-%\"AG6$&%\"xG6#\"\"\"&%\" yG6#F)" }{TEXT -1 24 "), and no other points. " }}{PARA 0 "" 0 "" {TEXT -1 62 "It is therefore the equation of the line. (iii) is called the " }{TEXT 259 20 "point-slope equation" }{TEXT -1 12 " of a line. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Perpendicular lines and the normal line to a curve at a point " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 69 "The following fac t pertains to the gradients of perpendicular lines. " }}{PARA 0 "" 0 " " {TEXT -1 23 "Suppose that the lines " }{XPPEDIT 18 0 "L[1]" "6#&%\"L G6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L[2]" "6#&%\"LG6#\"\"# " }{TEXT -1 16 " have gradients " }{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\" \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m[2]" "6#&%\"mG6#\"\"#" } {TEXT -1 74 " respectively, and suppose that neither line is vertical \+ (parallel to the " }{TEXT 270 1 "y" }{TEXT -1 8 " axis). " }}{PARA 0 " " 0 "" {TEXT -1 4 "Then" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L[1]" "6#&%\"LG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L[2 ]" "6#&%\"LG6#\"\"#" }{TEXT -1 35 " are perpendicular if and only if \+ " }{XPPEDIT 18 0 "m[1]*`.`*m[2]=-1" "6#/*(&%\"mG6#\"\"\"F(%\".GF(&F&6# \"\"#F(,$F(!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 269 28 "____________________________" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 403 259 259 {PLOTDATA 2 "61-%'CU RVESG6$7$7$$!\"$\"\"!$!\"\"F*7$$\"\"$F*$\"\"\"F*-%'COLOURG6&%$RGBG$F*F *F6$\"*++++\"!\")-F$6$7$7$F0F(7$F+F.-F36&F5F7F6F6-F$6$7%7$$!3/+++++++5 !#=$\"3))**************HFG7$$\"35+++++++?FG$\"3A+++++++SFG7$FH$\"3/+++ ++++5FG-F36&F5F*F*F*-%%TEXTG6&7$$!\"#F*$\"#=F,Q-gradient~=~m6\"F?-%%FO NTG6$%*HELVETICAG\"#5-FU6&7$$\"\"&F,$!#GF,Q\"LFgnF?Fhn-FU6&7$$\"#EF,$F /F,FfnF2Fhn-FU6&7$F($!\"%F,FdoF2Fhn-FU6&7$$!#&)FY$\"$l\"FYQ\"lFgnF?-Fi n6$F[o\"\")-FU6&7$$\"\"(F,$!#HF,FgpF?Fhp-FU6&7$$\"$y$FY$\"#:FYQ\"2FgnF 2Fhp-FU6&7$$!$t#FY$!\"&F,FiqF2Fhp-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%, CONSTRAINEDG-%+AXESLABELSG6%Q!FgnF\\s-Fin6#%(DEFAULTG-%%VIEWG6$F_sF_s " 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 271 0 "" }{TEXT -1 105 "In order to prove this fact, it is suffi cient to consider a pair of lines which intersect at the origin. " }} {PARA 0 "" 0 "" {TEXT -1 36 "The following diagram shows a line " } {XPPEDIT 18 0 "y=m*x" "6#/%\"yG*&%\"mG\"\"\"%\"xGF'" }{TEXT -1 27 " w ith (positive) gradient " }{TEXT 272 1 "m" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 89 "Assuming that the gradient of the tan gent line is not zero, the normal line at the point " }{XPPEDIT 18 0 " P(a,phi(a));" "6#-%\"PG6$%\"aG-%$phiG6#F&" }{TEXT -1 15 " has gradient " }{XPPEDIT 18 0 "-1/m = -1/`f '`(a);" "6#/,$*&\"\"\"F&%\"mG!\"\"F(, $*&F&F&-%$f~'G6#%\"aGF(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Since the point" }{XPPEDIT 18 0 "` `(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 160 " lies on both the tangent and normal line at that point, we can use the point-slope for m for the equation of a straight line to obtain the following equation s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 7 "Ta ngent" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-f(a) = `f '`(a)*(x-a);" "6#/,&%\"yG\"\"\"-%\"fG6#%\"aG!\"\"*&- %$f~'G6#F*F&,&%\"xGF&F*F+F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT 288 6 "Normal" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-f(a) = -1/`f '`(a);" "6#/,&%\"yG\"\"\"-%\"fG6#%\"aG! \"\",$*&F&F&-%$f~'G6#F*F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x-a)" "6#-%!G6#,&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{TEXT 289 14 "______________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "In the sp ecial case that the tangent line at the point" }{XPPEDIT 18 0 "``(a,f( a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 78 " has gradient equal to z ero, the tangent line is horizontal, and has equation " }{XPPEDIT 18 0 "y = f(a);" "6#/%\"yG-%\"fG6#%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 28 "The normal line at the point" }{XPPEDIT 18 0 "``(a,f(a) );" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 35 " is vertical, and has the \+ equation " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "Examples of finding the equations of tangent and normal lines to curves" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exam ple 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 292 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the equations of the tangent and normal lines to the curve \+ " }{XPPEDIT 18 0 "y = -x^2+7*x-5;" "6#/%\"yG,(*$%\"xG\"\"#!\"\"*&\"\"( \"\"\"F'F,F,\"\"&F)" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(2 ,5);" "6#-%!G6$\"\"#\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = -x^2+7*x-5;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#!\"\"*&\"\"(\"\"\"F'F.F.\"\"&F+" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "`f '`(x) = -2*x+7;" "6#/-%$f~'G6#%\"xG, &*&\"\"#\"\"\"F'F+!\"\"\"\"(F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "The gradient of the tangent line to the curve " } {XPPEDIT 18 0 "y = -x^2+7*x-5;" "6#/%\"yG,(*$%\"xG\"\"#!\"\"*&\"\"(\" \"\"F'F,F,\"\"&F)" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(2,5 );" "6#-%!G6$\"\"#\"\"&" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(2) = 3;" "6#/-%$f~'G6#\"\"#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The equation of this tangent line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-5 = 3*(x-2);" "6#/,&%\"yG\"\"\"\" \"&!\"\"*&\"\"$F&,&%\"xGF&\"\"#F(F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = 3*x-1;" "6#/%\"yG,&*&\"\"$\"\"\"%\"xGF(F(F(!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 28 "The normal line at the point" } {XPPEDIT 18 0 "``(2,5)" "6#-%!G6$\"\"#\"\"&" }{TEXT -1 14 " has gradie nt " }{XPPEDIT 18 0 "-1/3" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The equation of the normal line is: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-5 = -1/3;" "6#/ ,&%\"yG\"\"\"\"\"&!\"\",$*&F&F&\"\"$F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = -x/3+17/3;" "6#/%\"yG,&*&%\"xG\"\"\"\"\"$!\"\"F**& \"# " 0 "" {MPLTEXT 1 0 100 "plot([7*x-x ^2-5,3*x-1,-x/3+17/3],x=-1..8,y=-1..8,\n color=[red,blue,C OLOR(RGB,0,.7,.2)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 407 385 385 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$!\"\"\"\"!$!#8F*7$$!3k+++DhDQ!)!#=$!37 +6G!\\\"HF6!#;7$$!3g***\\(oKNJjF0$!3Y8@EIw!G$)*!#<7$$!3W****\\([i$! 3[=>\"QG;DR&F97$$\"3#****\\7VsmA\"F0$!3I))3m]lPcTF97$$\"3'))**\\7)R&G2 $F0$!3cSe-HlUVHF97$$\"3A++D\"ex@)\\F0$!3o+hjFmpgVlF07$$\"3F-++vM0V))F0$\"39f!H)[[T\"3%F07$$\"3#****\\P^Pn0 \"F9$\"3ynMGB%o/G\"F97$$\"3G++]#Hb3D\"F9$\"3iSZs%3[8>#F97$$\"3%)****\\ P,xX9F9$\"3)4B8?nR,.$F97$$\"3E++]2ngL;F9$\"3y45rxgdmPF97$$\"3&)**\\73. =/=F9$\"3=z*[Ed&>uVF97$$\"3/++]<)3q+#F9$\"3-^s(*Gt(4-&F97$$\"3/+++q6$) y@F9$\"3bHkjAH^/bF97$$\"3%***\\789qyBF9$\"3ME@2z%*o#*fF97$$\"3u*****\\ W?cb#F9$\"37-!)4cs9ejF97$$\"3U+]7j'G(\\FF9$\"3/,Q1@H4(o'F97$$\"3X+]P*e lX$HF9$\"3$y:\"GFRGIpF97$$\"3s++DJNUFJF9$\"3PzP$\\x'=6rF97$$\"32+]iDt_ /LF9$\"3!fosNV!z6sF97$$\"3[***\\Ppdb\\$F9$\"3ky^\"fE!)*\\sF97$$\"3#*** \\7eX)Rp$F9$\"3w\\!47**pB@(F97$$\"3K+](os:n'QF9$\"3nSDmv&>b6(F97$$\"3u ***\\77qK0%F9$\"3+5?$H<#*Q%pF97$$\"3Y*****\\1**fC%F9$\"3_F\">]R&[$p'F9 7$$\"3t***\\itYXV%F9$\"3#\\ZkxRAmP'F97$$\"3'***\\7.j(ph%F9$\"3#y?d#QRO -gF97$$\"3l***\\PBL&>[F9$\"3:p#H]/K)3bF97$$\"3P*****\\kR:+&F9$\"3)eF$ \\%py`*\\F97$$\"3e++]P.(e>&F9$\"3AOwQ)zBSP%F97$$\"3e**\\7GG'>P&F9$\"3a a#=,1)=2AF97$$\"3n**\\i&Qm\\$fF9$\"3r=j(3qQ4K\"F97$$\"31++](['3?hF9$\" 3k.\"*>!)zY^QF07$$\"3e**\\7y+*QJ'F9$!3+GJq sk@6-&G%oF97$$\"3*3+++**eBV(F9$!3&\\#QBEsW8#)F97$$\"3L**\\78%zCi(F9$!3 JT%R;^O[u*F97$$\"3v**\\(o\"*[W!yF9$!3RO')zZ!Gy7\"F37$$\"\")F*F+-%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*F`[l-F$6$7S7$F($!\"%F*7$F.$!3T++]PoZ6MF9 7$F5$!3k**\\i!)fS**GF97$F;$!31++DY(GNK#F97$F@$!3(****\\PAMQu\"F97$FE$! 3I+]7))[*o;\"F97$FK$!3D++D1F)*>jF07$FP$!3*R.+Dc!Q9yFG7$FU$\"3k++vVF`Y \\F07$FZ$\"3_**\\(oQ8c1\"F97$Fin$\"3o++]Ug\"Hl\"F97$F^o$\"3u***\\7a7-< #F97$Fco$\"3#3++v(ec_FF97$Fho$\"3/****\\7/JPLF97$F]p$\"3w++]A,#3!RF97$ Fbp$\"3c**\\PC4a7WF97$Fgp$\"38++]_k-@]F97$F\\q$\"3n******4N\\ObF97$Faq $\"3Q**\\PRU5OhF97$Ffq$\"3A*****\\Lhom'F97$F[r$\"3O+]P*)f=\\sF97$F`r$ \"3Y+]7onp.yF97$Fer$\"3;-+v$fqAQ)F97$Fjr$\"34,](o(>e8*)F97$F_s$\"3U)** \\73tm[*F97$Fds$\"32+vVn`>35F37$Fis$\"35+D1=Z,g5F37$F^t$\"3$***\\PO5)f 6\"F37$Fct$\"3%)****\\>(*zt6F37$Fht$\"3,+](3-k.B\"F37$F]u$\"3++v$4*G4& G\"F37$Fbu$\"3#)**\\7q*feM\"F37$Fgu$\"3\")****\\$*=Y+9F37$F\\v$\"3=++D ,6we9F37$Fav$\"3))*\\P%[))e6:F37$Ffv$\"3C++vHNMp:F37$F[w$\"3\"**\\PR=' oB;F37$F`w$\"3!**\\(o:**[!o\"F37$Few$\"3-++DYf-OF37$Fix$\"3;+v$\\<>U'>F3 7$F^y$\"3')****\\RQS;?F37$Fcy$\"3#)**\\7aL@w?F37$Fhy$\"3F+++(p2(H@F37$ F]z$\"3!)*\\PR#Qu'=#F37$Fbz$\"3$**\\i]nM8C#F37$Fgz$\"#BF*-Fjz6&F\\[lF` [lF`[lF][l-F$6$7S7$F($\"\"'F*7$F.$\"3&*****\\P&3Y$fF97$F5$\"3O+]ivmP([F97$Ffq$\"3 4+++&=$z9[F97$F[r$\"3;+]iX/4]ZF97$F`r$\"39+](o8y%)o%F97$Fer$\"3N++Dc@> CYF97$Fjr$\"3e+]7ev:lXF97$F_s$\"3Z++vo2[,XF97$Fds$\"3\"4+D1[Q`V%F97$Fi s$\"3[+]PC9wxVF97$F^t$\"3Q++DEmd:VF97$Fct$\"3y+++XOL^UF97$Fht$\"3)4+]7 U%[)=%F97$F]u$\"3-+]ilXnFTF97$Fbu$\"3s++v)eb,1%F97$Fgu$\"3m+++&y'[**RF 97$F\\v$\"3R++]())4Z$RF97$Fav$\"3e+]i!R7g(QF97$Ffv$\"3F++]A0%=\"QF97$F [w$\"3m+]i&zf9v$F97$F`w$\"3S+]7QXM)o$F97$Few$\"3d++]PyjEOF97$Fjw$\"3e+ ]iSm.iNF97$F_x$\"3E+++5!=)*\\$F97$Fdx$\"3U++vt/>OMF97$Fix$\"3E+]i0)*3t LF97$F^y$\"3v+++Xo5:LF97$Fcy$\"3!3+](G=l[KF97$Fhy$\"3++++qO@*=$F97$F]z $\"3#3+Dc>Se7$F97$Fbz$\"3Q+]P%p$=lIF97$Fgz$\"3W+++++++IF9-%&COLORG6&F \\[lF*$\"\"(F)$\"\"#F)-%(SCALINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6$Q\" x6\"Q\"yF^_m-%%VIEWG6$;F(FgzFc_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 294 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the equations of t he tangent and normal lines to the curve " }{XPPEDIT 18 0 "y=3*sqrt(x )-x" "6#/%\"yG,&*&\"\"$\"\"\"-%%sqrtG6#%\"xGF(F(F,!\"\"" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(4,2);" "6#-%!G6$\"\"%\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x) = 3*sqrt(x)-x;" "6#/-%\"fG6#%\"xG,&*&\"\"$\"\"\"-% %sqrtG6#F'F+F+F'!\"\"" }{XPPEDIT 18 0 "``=3*x^(1/2)-x" "6#/%!G,&*&\"\" $\"\"\")%\"xG*&F(F(\"\"#!\"\"F(F(F*F-" }{TEXT -1 8 ". Then " } {XPPEDIT 18 0 "`f '`(x) = 3/2;" "6#/-%$f~'G6#%\"xG*&\"\"$\"\"\"\"\"#! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-1/2)-1 = 3/(2*sqrt(x))-1;" " 6#/,&)%\"xG,$*&\"\"\"F)\"\"#!\"\"F+F)F)F+,&*&\"\"$F)*&F*F)-%%sqrtG6#F& F)F+F)F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "The gradien t of the tangent line to the curve " }{XPPEDIT 18 0 "y = 3*sqrt(x)-x; " "6#/%\"yG,&*&\"\"$\"\"\"-%%sqrtG6#%\"xGF(F(F,!\"\"" }{TEXT -1 13 " a t the point" }{XPPEDIT 18 0 "``(4, 2);" "6#-%!G6$\"\"%\"\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(4) = 3/4-1;" "6#/-%$f~'G6#\"\"%,&*& \"\"$\"\"\"F'!\"\"F+F+F," }{XPPEDIT 18 0 "``=-1/4" "6#/%!G,$*&\"\"\"F' \"\"%!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The equa tion of this tangent line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-2 = -1/4;" "6#/,&%\"yG\"\"\"\"\"#!\"\",$*&F&F&\"\"%F( F(" }{XPPEDIT 18 0 "``(x-4)" "6#-%!G6#,&%\"xG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = -x/4+3;" "6#/%\"yG,&*&%\"xG\"\"\"\" \"%!\"\"F*\"\"$F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "The normal line at the point" }{XPPEDIT 18 0 "``(4, 2);" "6#-%!G6$\"\"%\" \"#" }{TEXT -1 14 " has gradient " }{XPPEDIT 18 0 "4;" "6#\"\"%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The equation of the nor mal line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-2 \+ = 4*(x-4);" "6#/,&%\"yG\"\"\"\"\"#!\"\"*&\"\"%F&,&%\"xGF&F*F(F&" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 4*x-14;" "6#/%\"yG,&*&\"\"%\"\"\" %\"xGF(F(\"#9!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 39 "The following picture shows the curve \+ " }{XPPEDIT 18 0 "y = 3*sqrt(x)-x;" "6#/%\"yG,&*&\"\"$\"\"\"-%%sqrtG6# %\"xGF(F(F,!\"\"" }{TEXT -1 56 " together with the tangent and normal \+ lines at the point" }{XPPEDIT 18 0 "``(4,2);" "6#-%!G6$\"\"%\"\"#" } {TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x/3+3/x;" "6#/-%\"fG6#%\"xG,&*&F'\"\"\"\"\"$!\"\"F**&F+F*F'F,F *" }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x +3*x^(-1)" "6#,&%\"xG\"\"\"*&\"\"$F%)F$,$F%!\" \"F%F%" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "`f '`(x) = 1/3-3*x^(-2) ;" "6#/-%$f~'G6#%\"xG,&*&\"\"\"F*\"\"$!\"\"F**&F+F*)F',$\"\"#F,F*F," } {XPPEDIT 18 0 "``=1/3-3/x^2" "6#/%!G,&*&\"\"\"F'\"\"$!\"\"F'*&F(F'*$% \"xG\"\"#F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 300 1 "y" }{TEXT -1 38 " coordinate of the point on the curve " }{XPPEDIT 18 0 "y=x/3+3/x" "6#/%\"yG,&*&%\"xG\"\"\"\"\"$!\"\"F(*&F)F(F 'F*F(" }{TEXT -1 6 " with " }{TEXT 301 1 "x" }{TEXT -1 17 " coordinate 2 is " }{XPPEDIT 18 0 "y=f(2)" "6#/%\"yG-%\"fG6#\"\"#" }{XPPEDIT 18 0 "``=2/3+3/2" "6#/%!G,&*&\"\"#\"\"\"\"\"$!\"\"F(*&F)F(F'F*F(" } {XPPEDIT 18 0 "``=13/6" "6#/%!G*&\"#8\"\"\"\"\"'!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 "The gradient of the tangent line to t he curve " }{XPPEDIT 18 0 "y = x/3+3/x;" "6#/%\"yG,&*&%\"xG\"\"\"\"\" $!\"\"F(*&F)F(F'F*F(" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``( 2,13/6);" "6#-%!G6$\"\"#*&\"#8\"\"\"\"\"'!\"\"" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "`f '`(2) = 1/3-3/4;" "6#/-%$f~'G6#\"\"#,&*&\"\"\"F*\"\" $!\"\"F**&F+F*\"\"%F,F," }{XPPEDIT 18 0 "`` = -5/12;" "6#/%!G,$*&\"\"& \"\"\"\"#7!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The equation of this tangent line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "y-13/6 = -5/12;" "6#/,&%\"yG\"\"\"*&\"#8F&\"\"'!\"\" F*,$*&\"\"&F&\"#7F*F*" }{XPPEDIT 18 0 "``(x-2);" "6#-%!G6#,&%\"xG\"\" \"\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-13/6 = -5*x/12+5/6; " "6#/,&%\"yG\"\"\"*&\"#8F&\"\"'!\"\"F*,&*(\"\"&F&%\"xGF&\"#7F*F**&F-F &F)F*F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "y=-5*x/12 +3" "6#/%\"yG,&*(\"\"&\"\"\"%\"xG F(\"#7!\"\"F+\"\"$F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 " The normal line at the point" }{XPPEDIT 18 0 "``(2, 13/6);" "6#-%!G6$ \"\"#*&\"#8\"\"\"\"\"'!\"\"" }{TEXT -1 15 " has gradient " }{XPPEDIT 18 0 "-1/``(-5/12) = 12/5;" "6#/,$*&\"\"\"F&-%!G6#,$*&\"\"&F&\"#7!\"\" F.F.F.*&F-F&F,F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The \+ equation of the normal line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-13/6 = 12/5;" "6#/,&%\"yG\"\"\"*&\"#8F&\"\"'!\"\"F**& \"#7F&\"\"&F*" }{XPPEDIT 18 0 "``(x-2)" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\" \"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-13/6 = 12*x/5-24/5;" "6#/,&%\" yG\"\"\"*&\"#8F&\"\"'!\"\"F*,&*(\"#7F&%\"xGF&\"\"&F*F&*&\"#CF&F/F*F*" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 4 "or " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "y=12*x/5-79/30" "6#/%\"yG,&*(\"#7\"\"\"%\"xGF(\"\" &!\"\"F(*&\"#zF(\"#IF+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The following picture shows the cu rve " }{XPPEDIT 18 0 "y = 3*sqrt(x)-x;" "6#/%\"yG,&*&\"\"$\"\"\"-%%sq rtG6#%\"xGF(F(F,!\"\"" }{TEXT -1 56 " together with the tangent and no rmal lines at the point" }{XPPEDIT 18 0 "``(4,2);" "6#-%!G6$\"\"%\"\"# " }{TEXT -1 2 ". 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$F_fm$\"3immE'\\Hv\\\"FH7$Fdfm$\"3]mm'y\"*pb`\"FH7$Fifm$\"3qmmYT*Ghd\" FH7$F^gm$\"3immE-\"\\\\h\"FH7$Fjgl$\"3kmmmmmmc;FH-%&COLORG6&FbhlF\\hl$ \"\"(!\"\"$\"\"#Fban-%(SCALINGG6#%.UNCONSTRAINEDG-%+AXESLABELSG6$Q\"x6 \"Q\"yF]bn-%%VIEWG6$;FfhlFjgl;$FbanF\\hl$\"\"'F\\hl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 290 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 56 "(a) F ind the coordinates of the two points on the curve " }{XPPEDIT 18 0 "y =x^2*(x-2)" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"F'!\"\"F)" }{TEXT -1 40 " w here the tangent line has gradient 4. " }}{PARA 0 "" 0 "" {TEXT -1 68 "(b) Find the equations of the tangent and normal lines to the curve \+ " }{XPPEDIT 18 0 "y=x^2*(x-2)" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"F'!\"\"F )" }{TEXT -1 38 " at the two points found in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x^2* (x-2);" "6#/-%\"fG6#%\"xG*&F'\"\"#,&F'\"\"\"F)!\"\"F+" }{XPPEDIT 18 0 "`` = x^3-2*x^2;" "6#/%!G,&*$%\"xG\"\"$\"\"\"*&\"\"#F)*$F'F+F)!\"\"" } {TEXT -1 8 ". Then " }{XPPEDIT 18 0 "`f '`(x) = 3*x^2-4*x;" "6#/-%$f~ 'G6#%\"xG,&*&\"\"$\"\"\"*$F'\"\"#F+F+*&\"\"%F+F'F+!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We w ish to find the values of " }{TEXT 302 1 "x" }{TEXT -1 26 " for which \+ the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 17 " has the value 4." }}{PARA 0 "" 0 "" {TEXT -1 35 "Thus we need \+ to solve the equation " }{XPPEDIT 18 0 "3*x^2-4*x=4" "6#/,&*&\"\"$\"\" \"*$%\"xG\"\"#F'F'*&\"\"%F'F)F'!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "This equation is equivalent to " }{XPPEDIT 18 0 "3*x ^2-4*x-4=0" "6#/,(*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*&\"\"%F'F)F'!\"\"F,F- \"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "(3*x+2)*(x-2)=0" "6#/*&,&*& \"\"$\"\"\"%\"xGF(F(\"\"#F(F(,&F)F(F*!\"\"F(\"\"!" }{TEXT -1 13 ". Thi s gives " }{XPPEDIT 18 0 "x=-2/3" "6#/%\"xG,$*&\"\"#\"\"\"\"\"$!\"\"F* " }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 296 1 "y" }{TEXT -1 38 " coordinate of the point on the curve " }{XPPEDIT 18 0 "y = x^2*(x -2);" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"F'!\"\"F)" }{TEXT -1 6 " with " } {TEXT 297 1 "x" }{TEXT -1 17 " coordinate 2 is " }{XPPEDIT 18 0 "y=f(2 )" "6#/%\"yG-%\"fG6#\"\"#" }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The equation of this ta ngent line at the point" }{XPPEDIT 18 0 "``(2,0)" "6#-%!G6$\"\"#\"\"! " }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=4*(x-2)" "6#/%\"yG*&\"\"%\"\"\",&%\"xGF'\"\"#!\"\"F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=4*x-8" "6#/%\"yG,&*&\"\"%\"\"\"%\"xGF(F(\" \")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "The normal l ine at the point" }{XPPEDIT 18 0 "``(2, 0);" "6#-%!G6$\"\"#\"\"!" } {TEXT -1 15 " has gradient " }{XPPEDIT 18 0 "-1/4;" "6#,$*&\"\"\"F%\" \"%!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The equati on of the normal line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=-1/4" "6#/%\"yG,$*&\"\"\"F'\"\"%!\"\"F)" }{XPPEDIT 18 0 "``(x-2);" "6#-%!G6#,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=-x/4+1/2" "6#/%\"yG,&*&%\"xG\"\"\"\"\"%!\"\"F**&F(F( \"\"#F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 303 1 "y" }{TEXT -1 38 " coordinate o f the point on the curve " }{XPPEDIT 18 0 "y = x^2*(x-2);" "6#/%\"yG*& %\"xG\"\"#,&F&\"\"\"F'!\"\"F)" }{TEXT -1 6 " with " }{TEXT 304 1 "x" } {TEXT -1 12 " coordinate " }{XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\" \"$!\"\"F(" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = f(-2/3);" "6#/%\"yG-%\"fG6#,$*&\"\"#\"\"\"\"\"$!\" \"F-" }{XPPEDIT 18 0 "`` = 4/9;" "6#/%!G*&\"\"%\"\"\"\"\"*!\"\"" } {XPPEDIT 18 0 " ``(-2/3-2)" "6#-%!G6#,&*&\"\"#\"\"\"\"\"$!\"\"F+F(F+" }{XPPEDIT 18 0 "``=``(4/9)*``(-8/3)" "6#/%!G*&-F$6#*&\"\"%\"\"\"\"\"*! \"\"F*-F$6#,$*&\"\")F*\"\"$F,F,F*" }{XPPEDIT 18 0 "``=-32/27" "6#/%!G, $*&\"#K\"\"\"\"#F!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The equation of this tangent line at the point" }{XPPEDIT 18 0 "`` (-2/3, -32/27);" "6#-%!G6$,$*&\"\"#\"\"\"\"\"$!\"\"F+,$*&\"#KF)\"#FF+F +" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+32/27=4*(x+2/3)" "6#/,&%\"yG\"\"\"*&\"#KF&\"#F!\"\"F&*&\"\"%F&,&% \"xGF&*&\"\"#F&\"\"$F*F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+32/27 \+ = 4*x+8/3;" "6#/,&%\"yG\"\"\"*&\"#KF&\"#F!\"\"F&,&*&\"\"%F&%\"xGF&F&*& \"\")F&\"\"$F*F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=4*x+40/27" "6#/% \"yG,&*&\"\"%\"\"\"%\"xGF(F(*&\"#SF(\"#F!\"\"F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 28 "The normal line at the point" }{XPPEDIT 18 0 "``(-2/3, -32/27);" "6#-%!G6$,$*&\"\"#\"\"\"\"\"$!\"\"F+,$*&\"#KF )\"#FF+F+" }{TEXT -1 15 " has gradient " }{XPPEDIT 18 0 "-1/4;" "6#,$ *&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The equation of the normal line is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+32/27 = -1/4;" "6#/,&%\"yG\"\"\"*&\"#KF&\"# F!\"\"F&,$*&F&F&\"\"%F*F*" }{XPPEDIT 18 0 "``(x+2/3);" "6#-%!G6#,&%\"x G\"\"\"*&\"\"#F(\"\"$!\"\"F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+ 32/27 = -x/4-1/6;" "6#/,&%\"yG\"\"\"*&\"#KF&\"#F!\"\"F&,&*&%\"xGF&\"\" %F*F**&F&F&\"\"'F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "o r " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=-x/4-73/54" " 6#/%\"yG,&*&%\"xG\"\"\"\"\"%!\"\"F**&\"#tF(\"#aF*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "The following picture shows the curve " }{XPPEDIT 18 0 "y = x^2*(x-2);" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"F'!\"\" F)" }{TEXT -1 56 " together with the tangent and normal lines at the p oint" }{XPPEDIT 18 0 "``(2,0);" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 46 " an d the tangent and normal lines at the point" }{XPPEDIT 18 0 "``(-2/3,- 32/27)" "6#-%!G6$,$*&\"\"#\"\"\"\"\"$!\"\"F+,$*&\"#KF)\"#FF+F+" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "plot([x^2*(x-2),4*x-8,4*x+40/27,-x/4+1/2,-x/4 -73/54],x=-1..2.3,y=-2..1.5,\n color=[red,blue$2,COLOR(RGB ,0,.7,.2)$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 380 380 {PLOTDATA 2 " 6)-%'CURVESG6$7W7$$!\"\"\"\"!$!\"$F*7$$!3$****\\i&pMS'*!#=$!3%\\bNP'Qm aF!#<7$$!3')****\\7Rp!G*F0$!3s5(e7j$)>_#F37$$!3W*\\(=A,]P\\Zp3QF0$!3!*3P.Hjs`MF07$$!3e++ ]#R!)34$F0$!3bC[Fko*f?#F07$$!3q++D;\"H'eCF0$!3AE'>P/#fd8F07$$!3'3++v#R 'ou\"F0$!3R,p0c&Ghj'!#>7$$!3W,+]ihGF_p$!33R0e(4$*zc\"Fjp7$$\"3r****\\ (**)pD5F0$!3'>f#*)Qt?'*>F_p7$$\"3#))******G9dl\"F0$!3&fT9>z$))G]F_p7$$ \"3Y)*\\7[=d)Q#F0$!38W$44\\!y/5F07$$\"3Q'****\\'\\FPIF0$!3E'yug.=[c\"F 07$$\"3/+]7)40!\\PF0$!3ylf.3P3%G#F07$$\"3%))*\\P%\\SnU%F0$!3E%pKM#3u^I F07$$\"3o***\\7G')Q8&F0$!3'>*QaG,B=RF07$$\"3=)*\\igoE$y&F0$!3]#[1x_`\\ v%F07$$\"3/(**\\Pa6P['F0$!3_H9$=+Z?o&F07$$\"3j'*\\78nF6sF0$!3-0*z0Md/l 'F07$$\"3U'*\\(=LCY%yF0$!3]`(R5*4>![(F07$$\"3f(**\\76d'G&)F0$!3xMMi'[C SM)F07$$\"3&f****\\!*H`B*F0$!3q8\"p70H8=*F07$$\"3C(**\\iOrm#**F0$!3#fy U.gPh#**F07$$\"3s*\\7y(zbf5F3$!3uB$[Qc*zb5F37$$\"3v**\\P_)GQ8\"F3$!34! 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3!p(\\IY8u;;F37$F\\v$!3p,c%\\14`j\"F37$Fav$!3i^o(f)H*>l\"F37$Ffv$!3#>N *yKm!)p;F37$F[w$!3gw*HWW[fo\"F37$F`w$!3!=Nk\"3df. &oZpu>S=F37$F]z$!3!>gX*)fs%e=F37$Fbz$!3%=&=g#4=[(=F37$Fgz$!3sw\\0)zXA* =F37$Fa[l$!3tE()*eLE*3>F37$F[\\l$!3\"=&=&=&=&o#>F3F\\im-%+AXESLABELSG6 $Q\"x6\"Q\"yF]cn-%%VIEWG6$;F($\"#BF);$!\"#F*$\"#:F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 157 ": The corners of coordinates of the lower right corner o f the rectangle formed by the tangent and normal lines can be found by solving the pair of equations " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([y=4*x-8,``],[y=-x/4-73/54,``])" "6#-%*PIECEW ISEG6$7$/%\"yG,&*&\"\"%\"\"\"%\"xGF,F,\"\")!\"\"%!G7$/F(,&*&F-F,F+F/F/ *&\"#tF,\"#aF/F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Th e coordinates are " }{XPPEDIT 18 0 "x= 718/459" "6#/%\"xG*&\"$=(\"\" \"\"$f%!\"\"" }{TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 9 " 1.5643, \+ " }{XPPEDIT 18 0 "y = -800/459" "6#/%\"yG,$*&\"$+)\"\"\"\"$f%!\"\"F*" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 10 " -1.7429. " }}{PARA 0 " " 0 "" {TEXT -1 115 "The corners of coordinates of the upper left corn er of the rectangle can be found by solving the pair of equations " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([y=4*x+40/2 7,``],[y=-x/4+1/2, ``])" "6#-%*PIECEWISEG6$7$/%\"yG,&*&\"\"%\"\"\"%\"x GF,F,*&\"#SF,\"#F!\"\"F,%!G7$/F(,&*&F-F,F+F1F1*&F,F,\"\"#F1F,F2" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "The coordinates are \+ " }{XPPEDIT 18 0 "x = -106/459" "6#/%\"xG,$*&\"$1\"\"\"\"\"$f%!\"\"F* " }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-0 " "6#,$\"\"!!\"\"" }{TEXT -1 8 ".23094, " }{XPPEDIT 18 0 "y = 256/459 " "6#/%\"yG*&\"$c#\"\"\"\"$f%!\"\"" }{TEXT -1 1 " " }{TEXT 308 1 "~" } {TEXT -1 10 " 0.55773. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Tangents w hich meet a curve at another point" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 312 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 55 "(a) Find the equation of the \+ tangent line to the curve " }{XPPEDIT 18 0 "y=x^2*(x+3)" "6#/%\"yG*&% \"xG\"\"#,&F&\"\"\"\"\"$F)F)" }{TEXT -1 20 " at the point where " } {XPPEDIT 18 0 "x=1/3" "6#/%\"xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 79 "(b) Find the coordinates of the point w here the tangent in (a) meets the curve " }{XPPEDIT 18 0 "y = x^2*(x+3 )" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"\"\"$F)F)" }{TEXT -1 8 " again. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 48 ": It is correct to say that a tangent line to a " }{TEXT 259 6 "circle" }{TEXT -1 34 " is a line which meets the circle " } {TEXT 259 20 "at exactly one point" }{TEXT -1 24 ", but with other cur ves " }{TEXT 259 20 "this may not be true" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x^3 +3*x^2" "6#/-%\"fG6#%\"xG,&*$F'\"\"$\"\"\"*&F*F+*$F'\"\"#F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 314 1 "y" }{TEXT -1 44 " coordinate of the point on the curve where " }{XPPEDIT 18 0 "x =1/3" "6#/%\"xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "f(1/3)=``(1/9)*(1/3+3)" "6#/-%\"fG6#*&\"\"\"F(\"\"$!\"\"*&-%!G6#*&F (F(\"\"*F*F(,&*&F(F(F)F*F(F)F(F(" }{XPPEDIT 18 0 "``=10/27" "6#/%!G*& \"#5\"\"\"\"#F!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Th en " }{XPPEDIT 18 0 "`f '`(x) = 3*x^2+6*x;" "6#/-%$f~'G6#%\"xG,&*&\"\" $\"\"\"*$F'\"\"#F+F+*&\"\"'F+F'F+F+" }{TEXT -1 40 " so the gradient of the tangent line to " }{XPPEDIT 18 0 "y = x^3+3*x^2" "6#/%\"yG,&*$%\" xG\"\"$\"\"\"*&F(F)*$F'\"\"#F)F)" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(1/3,10/27);" "6#-%!G6$*&\"\"\"F'\"\"$!\"\"*&\"#5F'\" #FF)" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(1/3) = 7/3;" "6#/-%$f~' G6#*&\"\"\"F(\"\"$!\"\"*&\"\"(F(F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The equation of the tangent is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-10/27=7/3" "6#/,&%\"yG\"\"\"*&\"#5F& \"#F!\"\"F**&\"\"(F&\"\"$F*" }{XPPEDIT 18 0 "``(x-1/3)" "6#-%!G6#,&%\" xG\"\"\"*&F(F(\"\"$!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-10/27= 7*x/3-7/9" "6#/,&%\"yG\"\"\"*&\"#5F&\"#F!\"\"F*,&*(\"\"(F&%\"xGF&\"\"$ F*F&*&F-F&\"\"*F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=7*x/3-11/27" " 6#/%\"yG,&*(\"\"(\"\"\"%\"xGF(\"\"$!\"\"F(*&\"#6F(\"#FF+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " The following picture shows the curve " }{XPPEDIT 18 0 "y = x^3+3*x^2; " "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*&F(F)*$F'\"\"#F)F)" }{TEXT -1 32 " tog ether with the tangent line " }{XPPEDIT 18 0 "y = 7*x/3-11/27;" "6#/% \"yG,&*(\"\"(\"\"\"%\"xGF(\"\"$!\"\"F(*&\"#6F(\"#FF+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([x^3+3*x^2,7/3*x-11/27],x=-4..2,y=-10..6,color=[ red,blue]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"%\"\"!$!#;F*7 $$!3!******\\2<#pQ!#<$!3_n2uU5H,8F,7$$!3#)***\\7bBav$F0$!3%e*fx*R*Ql5F ,7$$!36++]K3XFOF0$!3Q=T7Z([iD)F07$$!3%)****\\F)H')\\$F0$!3^sx%3'QV.hF0 7$$!3#****\\i3@/P$F0$!3'GX(4+m)y?%F07$$!3;++Dr^b^KF0$!3ilXjy\\ffEF07$$ !3$****\\7Sw%GJF0$!3W)>`1QXuD\"F07$$!3*****\\7;)=,IF0$!3o&\\H!HB>q5!#> 7$$!3!)***\\i83V(GF0$\"3$R1??%=UQ5F07$$!3:+++NkzVFF0$\"3*)G'G]v2)G>F07 $$!3w****\\d;%)GEF0$\"3kQbU8U+lDF07$$!37+++0)H%*\\#F0$\"3mh[Qon8FJF07$ $!3#)*****\\d'[pBF0$\"3AU`'Qe'**RNF07$$!38+++&>iUC#F0$\"3;Gl\"eFMk!QF0 7$$!3!)***\\7YY08#F0$\"3a))\\:U![m%RF07$$!3)******\\XF`*>F0$\"3ELQ)RgM ***RF07$$!3)*******>#z2)=F0$\"3/;wlsP0fRF07$$!3/++D\"RKvu\"F0$\"3w-?3y D([#QF07$$!3<+++qjeH;F0$\"3s!4_fH/#ROF07$$!3()***\\7*3=+:F0$\"3!\\6Ci& pSvLF07$$!3%)***\\PFcpP\"F0$\"3#o=#RI[IxIF07$$!3#)****\\7VQ[7F0$\"3^T$ *GD*G)HFF07$$!3\")***\\i6:.8\"F0$\"3+;&)fLBt)Q#F07$$!31++]P:'H+\"F0$\" 3GI__'e%))3?F07$$!3[++]7'pnq)!#=$\"3%zBK-u$>9;F07$$!3'3++v[G_b(Fds$\"3 ]hf/?2=\"G\"F07$$!3t)****\\_K:J'Fds$\"3T*3[X\"fSO%*Fds7$$!36-+++HnE]Fd s$\"3'o_wFds7$$\"3W.++]&*=jPFds$\"3k/f4ijS\"y%Fds7$$\" 3#f***\\(3/3(\\Fds$\"3w7\"F0$\"3yYc? \"pp$[_F07$$\"3O++v)Q?QD\"F0$\"3au%*RiXG(o'F07$$\"3G+++5jyp8F0$\"3%[8u &oN4*>)F07$$\"3<++]Ujp-:F0$\"3Hq\\4%>_n,\"F,7$$\"3++++gEd@;F0$\"3[2@cc :C:7F,7$$\"39++v3'>$[ -$!3A2u!\\+x3d)F07$FC$!3OuSKvKsr#)F07$FH$!3C3ult%pV*zF07 $FM$!3vS2*pn&=2xF07$FR$!3q2ul$yz,T(F07$FX$!3$4u!*>RET6(F07$Fgn$!3uuS2* 3*f4oF07$F\\o$!3)oS2\\Fr8a'F07$Fao$!3kT2u&G5%RiF07$Ffo$!3NS2u:%4i$fF07 $F[p$!32T2u&>>Sk&F07$F`p$!3)pSdO[#oy`F07$Fep$!3MuS2p9F07$Fex$\"3o#f#4:PqBAF07$Fjx$\"3&o#fn*\\t\"=DF07$F_y$\"3/g#f #\\1w)y#F07$Fdy$\"3sEfUeS)))4$F07$Fiy$\"3+$f#fK@EwLF07$F^z$\"3%)f#4I,0 ?n$F07$Fcz$\"3#)f#4boo]&RF07$Fhz$\"3_f#f#f#f#fUF0-F][l6&F_[lFc[lFc[lF` [l-%+AXESLABELSG6$Q\"x6\"Q\"yF`el-%%VIEWG6$;F(Fhz;$!#5F*$\"\"'F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "By selecting the graph a nd clicking with the mouse you can check that the tangent meets the cu rve at a point with approximate coordinates (-3.66,-8.95)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "We can find the e xact coordinates algebraically by solving the equation: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "7*x/3-11/27 = x^3+3*x^2;" "6#/, &*(\"\"(\"\"\"%\"xGF'\"\"$!\"\"F'*&\"#6F'\"#FF*F*,&*$F(F)F'*&F)F'*$F( \"\"#F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "27*x^3+81*x^2-63*x+11 = 0;" "6#/,**&\"#F\"\"\"*$%\"xG\"\"$F'F'*&\"#\")F'*$F)\"\"#F'F'*&\"#j F'F)F'!\"\"\"#6F'\"\"!" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 21 "We already know that " }{XPPEDIT 18 0 "x=1/3" "6#/%\"x G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 44 " is a solution of this polynomial equation. " }}{PARA 0 "" 0 "" {TEXT -1 18 "This suggests that" } {XPPEDIT 18 0 "``(3*x-1) " "6#-%!G6#,&*&\"\"$\"\"\"%\"xGF)F)F)!\"\"" } {TEXT -1 16 " is a factor of " }{XPPEDIT 18 0 "P(x)=27*x^3+81*x^2-63*x +11" "6#/-%\"PG6#%\"xG,**&\"#F\"\"\"*$F'\"\"$F+F+*&\"#\")F+*$F'\"\"#F+ F+*&\"#jF+F'F+!\"\"\"#6F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 28 " \+ " }{XPPEDIT 18 0 "9*x^2 +30*x-11" "6#,(*&\"\"*\"\"\"*$%\"xG\"\"#F&F&*& \"#IF&F(F&F&\"#6!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 36 " ______________________ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x-1" "6#,&*&\"\"$\"\"\"%\"xGF&F&F&!\"\"" } {TEXT -1 4 " | " }{XPPEDIT 18 0 "27*x^3+81*x^2-63*x+11" "6#,**&\"#F\" \"\"*$%\"xG\"\"$F&F&*&\"#\")F&*$F(\"\"#F&F&*&\"#jF&F(F&!\"\"\"#6F&" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "27*x ^3-9*x^2" "6#,&*&\"#F\"\"\"*$%\"xG\"\"$F&F&*&\"\"*F&*$F(\"\"#F&!\"\"" }{TEXT -1 8 " " }}{PARA 257 "" 0 "" {TEXT -1 18 " _________ \+ " }}{PARA 257 "" 0 "" {TEXT -1 17 " " }{XPPEDIT 18 0 "90*x^2-63*x" "6#,&*&\"#!*\"\"\"*$%\"xG\"\"#F&F&*&\"#jF&F(F&!\"\" " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 17 " " }{XPPEDIT 18 0 "90*x^2-30*x" "6#,&*&\"#!*\"\"\"*$%\"xG\"\"#F&F&*&\"#IF &F(F&!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 27 " \+ __________ " }}{PARA 257 "" 0 "" {TEXT -1 38 " \+ " }{XPPEDIT 18 0 "-33*x+11" "6#,&*&\"#L\"\"\"%\"x GF&!\"\"\"#6F&" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 38 " \+ " }{XPPEDIT 18 0 "-33*x+11" "6#,&*&\" #L\"\"\"%\"xGF&!\"\"\"#6F&" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 48 " ________ " }}{PARA 257 " " 0 "" {TEXT -1 53 " \+ 0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Th is division shows that " }{XPPEDIT 18 0 "P(x)=27*x^3+81*x^2-63*x+11" " 6#/-%\"PG6#%\"xG,**&\"#F\"\"\"*$F'\"\"$F+F+*&\"#\")F+*$F'\"\"#F+F+*&\" #jF+F'F+!\"\"\"#6F+" }{TEXT -1 12 " factors as " }{XPPEDIT 18 0 "(3*x- 1)*(9*x^2+30*x-11)" "6#*&,&*&\"\"$\"\"\"%\"xGF'F'F'!\"\"F',(*&\"\"*F'* $F(\"\"#F'F'*&\"#IF'F(F'F'\"#6F)F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The residual quadratic factors as " }{XPPEDIT 18 0 "9*x^2 +30*x-11=(3*x-1)*(3*x+11)" "6#/,(*&\"\"*\"\"\"*$%\"xG\"\"#F'F'*&\"#IF' F)F'F'\"#6!\"\"*&,&*&\"\"$F'F)F'F'F'F.F',&*&F2F'F)F'F'F-F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "27*x^3 +81*x^2-63*x+11=(3*x-1)^2*(3*x+11)" "6#/,**&\"#F\"\"\"*$%\"xG\"\"$F'F' *&\"#\")F'*$F)\"\"#F'F'*&\"#jF'F)F'!\"\"\"#6F'*&,&*&F*F'F)F'F'F'F1F.,& *&F*F'F)F'F'F2F'F'" }{TEXT -1 36 ", so that the equation (i) becomes: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(3*x-1)^2*(3*x+11 ) = 0;" "6#/*&,&*&\"\"$\"\"\"%\"xGF(F(F(!\"\"\"\"#,&*&F'F(F)F(F(\"#6F( F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows th at " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=1/3" "6#/%\" xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=-11/3" "6 #/%\"xG,$*&\"#6\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "It is perhaps not too surprising that the factor" } {XPPEDIT 18 0 " ``(3*x-1)" "6#-%!G6#,&*&\"\"$\"\"\"%\"xGF)F)F)!\"\"" } {TEXT -1 5 " of " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%\"xG" }{TEXT -1 110 " is repeated, because the geometrical situation suggests that the equation should have exactly two solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor(27*x^ 3+81*x^2-63*x+11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"$\" #6\"\"\"F(),&!\"\"F(*&F&F(F%F(F(\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 310 1 " x" }{TEXT -1 78 " coordinate of the point where the tangent meets the \+ curve again is therefore " }{XPPEDIT 18 0 "x=-11/3" "6#/%\"xG,$*&\"#6 \"\"\"\"\"$!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Th e corresponding " }{TEXT 311 1 "y" }{TEXT -1 15 " coordinate is " } {XPPEDIT 18 0 "f(-11/3)=(-11/3)^2*(-11/3+3)" "6#/-%\"fG6#,$*&\"#6\"\" \"\"\"$!\"\"F,*&,$*&F)F*F+F,F,\"\"#,&*&F)F*F+F,F,F+F*F*" }{XPPEDIT 18 0 "``=``(121/9)*(-2/3)" "6#/%!G*&-F$6#*&\"$@\"\"\"\"\"\"*!\"\"F*,$*&\" \"#F*\"\"$F,F,F*" }{XPPEDIT 18 0 "``=-242/27" "6#/%!G,$*&\"$U#\"\"\"\" #F!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 315 15 "Further remarks" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 97 "If we take a line which is parallel to the tangent, bu t moved up a short distance, there will be " }{TEXT 259 28 "three poin ts of intersection" }{TEXT -1 23 " of the line and curve." }}{PARA 0 " " 0 "" {TEXT -1 22 "For example, the line " }{XPPEDIT 18 0 "y = 7*x/3; " "6#/%\"yG*(\"\"(\"\"\"%\"xGF'\"\"$!\"\"" }{TEXT -1 18 " meets the c urve " }{XPPEDIT 18 0 "y = x^3+3*x^2;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*& F(F)*$F'\"\"#F)F)" }{TEXT -1 20 " at the points with " }{TEXT 263 1 "x " }{TEXT -1 49 " coordinates which are solutions of the equation " } {XPPEDIT 18 0 "7*x/3 = x^3+3*x^2;" "6#/*(\"\"(\"\"\"%\"xGF&\"\"$!\"\", &*$F'F(F&*&F(F&*$F'\"\"#F&F&" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "3*x^ 3+9*x^2-7*x = 0;" "6#/,(*&\"\"$\"\"\"*$%\"xGF&F'F'*&\"\"*F'*$F)\"\"#F' F'*&\"\"(F'F)F'!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "The cubic factors as " }{XPPEDIT 18 0 "x*(3*x^2-9*x-7)" "6#*&% \"xG\"\"\",(*&\"\"$F%*$F$\"\"#F%F%*&\"\"*F%F$F%!\"\"\"\"(F-F%" }{TEXT -1 26 ", so that one solution is " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 70 "There are two more s olutions given by solving the quadratic equation: " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2-9*x-7=0" "6#/,(*&\"\"$\"\"\"*$% \"xG\"\"#F'F'*&\"\"*F'F)F'!\"\"\"\"(F-\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "Completing the square gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*(x^2-3*x+9/4)=7+27/4" "6#/*&\"\" $\"\"\",(*$%\"xG\"\"#F&*&F%F&F)F&!\"\"*&\"\"*F&\"\"%F,F&F&,&\"\"(F&*& \"#FF&F/F,F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*(x-3/2)^2=55/4" "6# /*&\"\"$\"\"\"*$,&%\"xGF&*&F%F&\"\"#!\"\"F,F+F&*&\"#bF&\"\"%F," } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x-3/2 =`` " "6#/,&%\"xG\"\"\"*&\"\"$F &\"\"#!\"\"F*%!G" }{TEXT 316 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqr t(55/12)" "6#-%%sqrtG6#*&\"#b\"\"\"\"#7!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=3/2 " "6#/%\"xG*&\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 317 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1 65)/6;" "6#*&-%%sqrtG6#\"$l\"\"\"\"\"\"'!\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "g := x -> 7/3*x;\nplot([f(x),g(x)],x=-4..2,y=-10..6,color=[red,blu e]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"%\"\"!$!#;F*7$$!3!***** *\\2<#pQ!#<$!3_n2uU5H,8F,7$$!3#)***\\7bBav$F0$!3%e*fx*R*Ql5F,7$$!36++] K3XFOF0$!3Q=T7Z([iD)F07$$!3%)****\\F)H')\\$F0$!3^sx%3'QV.hF07$$!3#**** \\i3@/P$F0$!3'GX(4+m)y?%F07$$!3;++Dr^b^KF0$!3ilXjy\\ffEF07$$!3$****\\7 Sw%GJF0$!3W)>`1QXuD\"F07$$!3*****\\7;)=,IF0$!3o&\\H!HB>q5!#>7$$!3!)*** \\i83V(GF0$\"3$R1??%=UQ5F07$$!3:+++NkzVFF0$\"3*)G'G]v2)G>F07$$!3w**** \\d;%)GEF0$\"3kQbU8U+lDF07$$!37+++0)H%*\\#F0$\"3mh[Qon8FJF07$$!3#)**** *\\d'[pBF0$\"3AU`'Qe'**RNF07$$!38+++&>iUC#F0$\"3;Gl\"eFMk!QF07$$!3!)** *\\7YY08#F0$\"3a))\\:U![m%RF07$$!3)******\\XF`*>F0$\"3ELQ)RgM***RF07$$ !3)*******>#z2)=F0$\"3/;wlsP0fRF07$$!3/++D\"RKvu\"F0$\"3w-?3yD([#QF07$ $!3<+++qjeH;F0$\"3s!4_fH/#ROF07$$!3()***\\7*3=+:F0$\"3!\\6Ci&pSvLF07$$ !3%)***\\PFcpP\"F0$\"3#o=#RI[IxIF07$$!3#)****\\7VQ[7F0$\"3^T$*GD*G)HFF 07$$!3\")***\\i6:.8\"F0$\"3+;&)fLBt)Q#F07$$!31++]P:'H+\"F0$\"3GI__'e%) )3?F07$$!3[++]7'pnq)!#=$\"3%zBK-u$>9;F07$$!3'3++v[G_b(Fds$\"3]hf/?2=\" G\"F07$$!3t)****\\_K:J'Fds$\"3T*3[X\"fSO%*Fds7$$!36-+++HnE]Fds$\"3'o_w Fds7$$\"3W.++]&*=jPFds$\"3k/f4ijS\"y%Fds7$$\"3#f***\\ (3/3(\\Fds$\"3w7\"F0$\"3yYc?\"pp$[_F0 7$$\"3O++v)Q?QD\"F0$\"3au%*RiXG(o'F07$$\"3G+++5jyp8F0$\"3%[8u&oN4*>)F0 7$$\"3<++]Ujp-:F0$\"3Hq\\4%>_n,\"F,7$$\"3++++gEd@;F0$\"3[2@cc:C:7F,7$$ \"39++v3'>$[-0k%)F0 7$F>$!3m++](fpM;)F07$FC$!3#pm;z'eJkyF07$FH$!3!3+]i1ipe(F07$FM$!3ILLep# y(*H(F07$FR$!3E++DwBx-qF07$FX$!3\\LLe%)*=nq'F07$Fgn$!3Inmm\"o\">-kF07$ F\\o$!3X****\\nQ'R8'F07$Fao$!3>MLLyG+KeF07$Ffo$!3\"HLL$3?!)GbF07$F[p$! 3jLLL)y6mB&F07$F`p$!3a***\\i2v7(\\F07$Fep$!3!pmm;1kdl%F07$Fjp$!3gmmmY[ [)Q%F07$F_q$!3sLLezbdxSF07$Fdq$!3Pnmm'>oB!QF07$Fiq$!3=LLez?U+NF07$F^r$ !3GLL3sz*G@$F07$Fcr$!3`mm;Hn*G\"HF07$Fhr$!3Amm\"z$>SPEF07$F]s$!3F++](e V-M#F07$Fbs$!3T++Dw&z:.#F07$Fhs$!3wLL3Zm)Gw\"F07$F]t$!3y****\\A4ps9F07 $Fbt$!3'RLLLM!*G<\"F07$Fgt$!3AQLLe'Rfz)Fds7$F\\u$!3sML$eRY\"efFds7$Fau $!3eNLL33E2GFds7$Ffu$\"3;]BLLL+&R#Fhu7$F]v$\"37pmm;>(o/$Fds7$Fbv$\"3O% ***\\P%)3'y&Fds7$Fgv$\"3/vmm;cx!y)Fds7$F\\w$\"3QKL3(Ga)f6F07$Faw$\"3+n mTb@Ra9F07$Ffw$\"3\"GLL$enNU$F07$Fdy$ \"3=ML$eY\"H1NF07$Fiy$\"3W+++S&pOy$F07$F^z$\"3InmT?CTzSF07$Fcz$\"3Enm \"H4wCO%F07$Fhz$\"3'pmmmmmmm%F0-F][l6&F_[lFc[lFc[lF`[l-%+AXESLABELSG6$ Q\"x6\"Q\"yF`el-%%VIEWG6$;F(Fhz;$!#5F*$\"\"'F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "xcoords := solve(7*x/3=x^3+3*x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xc oordsG6%\"\"!,&#!\"$\"\"#\"\"\"*&#F+\"\"'F+-%%sqrtG6#\"$l\"F+F+,&F(F+* &#F+F.F+*$F/F+F+!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 65 " \nThe line mee ts the curve at the origin and also at points with " }{TEXT 262 1 "x" }{TEXT -1 59 " coordinates which are solutions of the quadratic equati on " }{XPPEDIT 18 0 "3*x^2+9*x-7 = 0;" "6#/,(*&\"\"$\"\"\"*$%\"xG\"\"# F'F'*&\"\"*F'F)F'F'\"\"(!\"\"\"\"!" }{TEXT -1 47 ".\nApproximate coord inates for these points are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(map(x->[x,f(x)],[xcoor ds]),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$$\"\"!F&F%7$$\"%4k!\"% $\"&a\\\"F*7$$!&4k$F*$!%'\\)!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "If we take a line with equation " } {XPPEDIT 18 0 "y = 7*x/3-11/3+delta;" "6#/%\"yG,(*(\"\"(\"\"\"%\"xGF( \"\"$!\"\"F(*&\"#6F(F*F+F+%&deltaGF(" }{TEXT -1 36 " , which is parall el to the tangent " }{XPPEDIT 18 0 "y = 7*x/3-11/3;" "6#/%\"yG,&*(\"\" (\"\"\"%\"xGF(\"\"$!\"\"F(*&\"#6F(F*F+F+" }{TEXT -1 35 ", but moved up a very small amount " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 61 ", there are still three different solutions for the equation " } {XPPEDIT 18 0 "7*x/3-11/27+delta = x^3+3*x^2;" "6#/,(*(\"\"(\"\"\"%\"x GF'\"\"$!\"\"F'*&\"#6F'\"#FF*F*%&deltaGF',&*$F(F)F'*&F)F'*$F(\"\"#F'F' " }{TEXT -1 67 ", and three different points of intersection of the li ne and curve." }}{PARA 0 "" 0 "" {TEXT -1 24 "\nTry the following with " }{XPPEDIT 18 0 "delta = 1/10,1/100,1/1000,1/10000,1/100000;" "6'/%& deltaG*&\"\"\"F&\"#5!\"\"*&F&F&\"$+\"F(*&F&F&\"%+5F(*&F&F&\"&++\"F(*&F &F&\"'++5F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 204 " is taken progressively smaller, the two right-hand points of \+ intersection move closer and closer together. In the limiting situati on, when these two points have run together ,we obtain the tangent lin e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Th is shows up in the algebra by the fact that, in the limiting situation , the cubic polynomial, giving the " }{TEXT 318 1 "x" }{TEXT -1 55 " c oordinates of the points of intersection of the line " }{XPPEDIT 18 0 "y = 7*x/3-11/27" "6#/%\"yG,&*(\"\"(\"\"\"%\"xGF(\"\"$!\"\"F(*&\"#6F( \"#FF+F+" }{TEXT -1 11 " and curve " }{XPPEDIT 18 0 "y = x^3+3*x^2" "6 #/%\"yG,&*$%\"xG\"\"$\"\"\"*&F(F)*$F'\"\"#F)F)" }{TEXT -1 33 " as zero s, has the factorisation " }{XPPEDIT 18 0 "(3*x-1)^2*(3*x+11)" "6#*&,& *&\"\"$\"\"\"%\"xGF'F'F'!\"\"\"\"#,&*&F&F'F(F'F'\"#6F'F'" }{TEXT -1 9 " in which" }{XPPEDIT 18 0 " ``(3*x-1)" "6#-%!G6#,&*&\"\"$\"\"\"%\"xGF )F)F)!\"\"" }{TEXT -1 6 " is a " }{TEXT 259 15 "repeated factor" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "delta := 1/100000;\nxcoords := fsolve(7*x/3-11/ 27+delta=x^3+3*x^2,x);\nevalf(map(x->[x,f(x)],[xcoords]),8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG#\"\"\"\"'++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xcoordsG6%$!+UgmmO!\"*$\"+=)=vJ$!#5$\"+)fT\"\\LF+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$$!)gmmO!\"($!(\\H'*)!\"'7$$\"))=v J$!\")$\")J!pm$F.7$$\");9\\LF.$\")&*oSPF." }}}{PARA 0 "" 0 "" {TEXT -1 391 " \nYou can set up and play the following animation, which will show parallel lines moving downwards towards the tangent. The tangent is obtained in the limiting situation when the two neighbouring point s of contact run together.\nThis limiting process is a little differen t from that involved in the definition of the derivative of a function , but the end result of a tangent line is the same." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "f := x -> x^3+3*x^2:\ncurve := plot(f(x),x=-4..2,y=-10..6,color=red):\nseqn := \+ NULL:\nfor i from 30 by -1 to 0 do\n g := x -> 7*x/3-11/27+i*1/30;\n line := plot(g(x),x=-4..2,y=-10..6,color=blue);\n xcoords := fsol ve(g(x)=f(x),x);\n pts := map(x->[x,f(x)],[xcoords]);\n points := \+ PLOT(POINTS(op(pts),SYMBOL(CIRCLE)));\n frame := plots[display](\{cu rve,line,points\});\n seqn := seqn,frame;\nend do:\nplots[display]([ seqn],insequence=true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 65 "Find the equations of the tangent and normal lines to the curve " }{XPPEDIT 18 0 "y = x^2-5*x+6;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&\" \"&F)F'F)!\"\"\"\"'F)" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "`` (1, 2);" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=-3*x+5" "6#/%\"yG,&*&\"\"$\"\" \"%\"xGF(!\"\"\"\"&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y=x/3+5/3" "6 #/%\"yG,&*&%\"xG\"\"\"\"\"$!\"\"F(*&\"\"&F(F)F*F(" }{TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot([x^2-5*x+6,-3*x+5,x/3+5 /3],x=-.5..5,y=-.5..4,\n color=[red,blue,COLOR(RGB,0,.7,.2)] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "__________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 64 "Find the equations of the tangent and normal lines to the curve " }{XPPEDIT 18 0 "y=2/x+3*sqrt(x)-6" "6#/%\"yG,(*&\"\"#\"\"\"% \"xG!\"\"F(*&\"\"$F(-%%sqrtG6#F)F(F(\"\"'F*" }{TEXT -1 20 " at the poi nt where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 10 " tangent: " }{XPPEDIT 18 0 "y = -x/ 2-1/2" "6#/%\"yG,&*&%\"xG\"\"\"\"\"#!\"\"F**&F(F(F)F*F*" }{TEXT -1 11 ", normal: " }{XPPEDIT 18 0 "y= 2*x-3" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF (F(\"\"$!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "plot([2/x+3*sqrt(x)-6,-x/2- 1/2,2*x-3],x=-.5..5,y=-2..2,\n discont=true,color=[red,blue,C OLOR(RGB,0,.7,.2)]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "______ _____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 64 "Find the equations of the tangent and nor mal lines to the curve " }{XPPEDIT 18 0 "y = (3*x^2-6*x+5)/(2*x);" "6# /%\"yG*&,(*&\"\"$\"\"\"*$%\"xG\"\"#F)F)*&\"\"'F)F+F)!\"\"\"\"&F)F)*&F, F)F+F)F/" }{TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 10 " tangent: " }{XPPEDIT 18 0 "y=7/8*x-1/2" "6#/%\"yG,&*(\"\"(\"\"\" \"\")!\"\"%\"xGF(F(*&F(F(\"\"#F*F*" }{TEXT -1 12 ", normal: " } {XPPEDIT 18 0 "y=-8/7*x+99/28" "6#/%\"yG,&*(\"\")\"\"\"\"\"(!\"\"%\"xG F(F**&\"#**F(\"#GF*F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "plot([(3*x^2-6*x+5)/(2* x),7/8*x-1/2,-8/7*x+99/28],x=0..3,y=0..3.5,\n discont=true,co lor=[red,blue,COLOR(RGB,0,.7,.2)]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 56 "(a) Find the coordi nates of the two points on the curve " }{XPPEDIT 18 0 "y = 1/x;" "6#/% \"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 38 " where the tangent line has gr adient " }{XPPEDIT 18 0 "-1/4;" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "(b) Find the equations of the tangent and normal lines to the curve " }{XPPEDIT 18 0 "y = 1/x;" "6# /%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 38 " at the two points found in \+ part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``( 2,1/2)" "6#-%!G6$\"\"#*&\"\"\"F(F&!\"\"" }{TEXT -1 11 " tangent: " } {XPPEDIT 18 0 "y = -x/4+1;" "6#/%\"yG,&*&%\"xG\"\"\"\"\"%!\"\"F*F(F(" }{TEXT -1 12 ", normal: " }{XPPEDIT 18 0 "y = 4*x-15/2;" "6#/%\"yG,& *&\"\"%\"\"\"%\"xGF(F(*&\"#:F(\"\"#!\"\"F-" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(-2,-1/2)" "6#-%!G6$,$\"\"# !\"\",$*&\"\"\"F+F'F(F(" }{TEXT -1 10 " tangent: " }{XPPEDIT 18 0 "y=- x/4-1" "6#/%\"yG,&*&%\"xG\"\"\"\"\"%!\"\"F*F(F*" }{TEXT -1 13 ", nor mal: " }{XPPEDIT 18 0 "y=4*x+15/2" "6#/%\"yG,&*&\"\"%\"\"\"%\"xGF(F(* &\"#:F(\"\"#!\"\"F(" }{TEXT -1 3 ". 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{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 4 "Q5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 55 "(a) Find the equation of the tangent line to the curve " }{XPPEDIT 18 0 "y = 2-4*x^2+x^3;" "6#/%\"yG,(\"\"#\"\" \"*&\"\"%F'*$%\"xGF&F'!\"\"*$F+\"\"$F'" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(1,-1)" "6#-%!G6$\"\"\",$F&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 79 "(b) Find the coordinates of the point whe re the tangent in (a) meets the curve " }{XPPEDIT 18 0 "y = 2-4*x^2+x^ 3;" "6#/%\"yG,(\"\"#\"\"\"*&\"\"%F'*$%\"xGF&F'!\"\"*$F+\"\"$F'" } {TEXT -1 8 " again. " }}{PARA 0 "" 0 "" {TEXT -1 50 "(c) Find the equa tion of the tangent to the curve " }{XPPEDIT 18 0 "y = 2-4*x^2+x^3;" " 6#/%\"yG,(\"\"#\"\"\"*&\"\"%F'*$%\"xGF&F'!\"\"*$F+\"\"$F'" }{TEXT -1 28 " at the point found in (b). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "y=-5*x+4" "6#/%\"yG,&*&\"\"&\"\"\"%\"xGF(!\" \"\"\"%F(" }{TEXT -1 7 ", (b) " }{XPPEDIT 18 0 "``(2,-6)" "6#-%!G6$\" \"#,$\"\"'!\"\"" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "y=-4*x+2" "6#/% \"yG,&*&\"\"%\"\"\"%\"xGF(!\"\"\"\"#F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plot([ 2-4*x^2+x^3,-5*x+4,-4*x+2],x=-1..4,y=-7.5..2.5,\n color=[red, blue,COLOR(RGB,0,.7,.2)]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 " ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Which of the following curves have the " }{TEXT 309 1 "x " }{TEXT -1 54 " axis as a tangent line, and at which point or points? " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "y = x*(x-2)^2; " "6#/%\"yG*&%\"xG\"\"\"*$,&F&F'\"\"#!\"\"F*F'" }{TEXT -1 6 " (b) " } {XPPEDIT 18 0 "y = x^2*(x-3);" "6#/%\"yG*&%\"xG\"\"#,&F&\"\"\"\"\"$!\" \"F)" }{TEXT -1 9 " (c) " }{XPPEDIT 18 0 "y = x^2*(x-1)^2;" "6#/% \"yG*&%\"xG\"\"#,&F&\"\"\"F)!\"\"F'" }{TEXT -1 8 " (d) " }{XPPEDIT 18 0 "y = x^3*(x-3);" "6#/%\"yG*&%\"xG\"\"$,&F&\"\"\"F'!\"\"F)" } {TEXT -1 8 " ( e) " }{XPPEDIT 18 0 "y = x^3*(x-2)^2;" "6#/%\"yG*&%\" xG\"\"$,&F&\"\"\"\"\"#!\"\"F*" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 147 "Make a very rough sketch by hand of the graph of each of these functions, just to show the general shape, and then check your \+ answers using Maple. " }}{PARA 0 "" 0 "" {TEXT -1 37 "________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 30 "point-slope equation of line " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 541 "p1 := plot([[0,2] ,[8,6]],color=blue):\np2 := plot([[2,3],[6,3],[6,5]],color=black,lines tyle=2):\np3 := plot([[[2,3],[6,5]]$3],style=point,\n symbol =[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[6.6,4.8 ,`P(x,y)`],[2.3,2.7,`A(2,3)`]],\n font=[HELVETICA,10],color=blac k):\nt2 := plots[textplot]([3,4.5,`gradient =`],\n font=[HE LVETICA,10],color=blue):\nt3 := plots[textplot]([[3.85,4.65,`1`],[3.85 ,4.35,`2`],\n [3.85,4.6,`_`]],font=[HELVETICA,8],color=blue):\npl ots[display]([p1,p2,p3,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "p1 := plot( [[0,2],[8,6]],color=blue):\np2 := plot([[2,3],[6,3],[6,5]],color=black ,linestyle=2):\np3 := plot([[[2,3],[6,5]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[6 .6,4.8,`P(x,y)`],[2.2,2.7,`A(x ,y )`]],\n font=[HELVETICA,10],co lor=black):\nt2 := plots[textplot]([[2.22,2.55,`l`],[2.57,2.55,`l`]], \n font=[HELVETICA,7],color=black):\nt3 := plots[textplot]([3.4,4. 5,`gradient = m`],\n font=[HELVETICA,10],color=blue):\nplot s[display]([p1,p2,p3,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 21 "perpendicular lines " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "p1 := plo t([[-3,-1],[3,1]],color=blue):\np2 := plot([[1,-3],[-1,3]],color=red): \np3 := plot([[-.1,.3],[.2,.4],[.3,.1]],color=black):\nt1 := plots[tex tplot]([[-2.,1.8,`gradient = m`],[.5,-2.8,`L`]],\n font=[HE LVETICA,10],color=red):\nt2 := plots[textplot]([[2.6,.3,`gradient = m` ],[-3,-.4,`L`]],\n font=[HELVETICA,10],color=blue):\nt3 := \+ plots[textplot]([[-.85,1.65,`l`],[.7,-2.9,`l`]],\n f ont=[HELVETICA,8],color=red):\nt4 := plots[textplot]([[3.78,.15,`2`],[ -2.73,-.5,`2`]],font=[HELVETICA,8],\n color=blue):\nplots [display]([p1,p2,p3,t1,t2,t3,t4],axes=none,scaling=constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 763 "p1 := plot([[-2,-1],[6,3]],color=blue):\np2 := plot([[-1,2],[4.3, -8.6]],color=red):\np3 := plottools[arrow]([-3,0],[7,0],0,.2,.02,arrow ):\np4 := plottools[arrow]([0,-8.7],[0,4],0,.2,.02,arrow):\np5 := plot ([[4,2.5],[4,-8.5]],color=black,linestyle=2):\nh := evalf(arctan(.5)/8 ):\np6 := plot([seq([1.6*cos(h*i),1.6*sin(h*i)],i=0..8)],color=black): \nt1 := plots[textplot]([[4.35,1.8,`A`],[-.43,-.52,`O`],\n [4.35 ,-.3,`X`],[4.3,-7.8,`B`],[6.7,-.3,`x`],[-.3,4,`y`],\n [4.8,-2.65 ,`x = 1`]],\n font=[HELVETICA,10],color=black):\nt2:= plots[text plot]([2.2,2.3,`gradient = m`],\n font=[HELVETICA,10],color =blue):\nt3:= plots[textplot]([1.15,.32,`q`],font=[SYMBOL,10],color=bl ack):\nplots[display]([p1,p2,p3,p4,p5,p6,t1,t2,t3],axes=none,scaling=c onstrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 26 "tan gent and normal lines " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 661 "p1 := plot(1/x,x=.5..4,color=red,t hickness=2):\np2 := plot([[1.5,-1.5],[2.5,2.5]],color=COLOR(RGB,0,.7,. 2)):\np3 := plot([[0,1],[4,0]],color=blue):\np4 := plot([[1.85,.5375], [1.8125,.3875],[1.9625,.35]],color=black):\np5 := plot([[[2,.5]]$3],st yle=point,\n symbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([2.2,.63,`P`],\n font=[HELVETICA,10],co lor=black):\nt2 := plots[textplot]([3.,-.07,`tangent at P`],\n \+ font=[HELVETICA,10],color=blue):\nt3 := plots[textplot]([1.15,-.6, `normal at P`],\n font=[HELVETICA,10],color=COLOR(RGB,0,.7, .2)):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3],axes=none,scaling=cons trained);" }}}{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 }