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1 2 2 2 2 2 2 1 1 1 1 }1 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 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -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 3 3 1 0 1 0 2 2 15 2 }{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 "Exponential and logarithm functio ns" }}{PARA 0 "" 0 "" {TEXT -1 63 "by Peter Stone, Mathematics Dept., \+ Malaspina University-College" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: \+ 9.1.2005 " }}{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 25 "load calculus procedures " }}{PARA 0 "" 0 "" {TEXT -1 35 "RMIT file path to read Maple m-file" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "read \"J:\\\\Class_Notes/Pet er Stone/MapleMath/procdrs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Another file path \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "read \"E:\\\\MapleMath/p rocdrs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Logarithm functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(x) = 2^x;" "6#/-%\"fG6#%\"xG)\"\"#F'" }{TEXT -1 27 " \+ is a one-to-one function. " }}{PARA 0 "" 0 "" {TEXT -1 39 "Its inverse function is the function f" }{XPPEDIT 18 0 "``^(-1);" "6#)%!G,$\"\" \"!\"\"" }{XPPEDIT 18 0 "``(x) = log[2]*x;" "6#/-%!G6#%\"xG*&&%$logG6# \"\"#\"\"\"F'F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "log[2]*x;" "6# *&&%$logG6#\"\"#\"\"\"%\"xGF(" }{TEXT -1 8 " is the " }{TEXT 260 19 "l ogarithm to base 2" }{TEXT -1 4 " of " }{TEXT 269 1 "x" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = log[2]*x;" "6#/%\"yG*&&%$logG6#\"\"#\"\"\"%\"x GF*" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "2^y = x;" "6#/)\"\" #%\"yG%\"xG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 262 16 "________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "For example, " } {XPPEDIT 18 0 "log[2](8) = 3,log[2](1/4) = -2,log[2](sqrt(2)) = 1/2;" "6%/-&%$logG6#\"\"#6#\"\")\"\"$/-&F&6#F(6#*&\"\"\"F2\"\"%!\"\",$F(F4/- &F&6#F(6#-%%sqrtG6#F(*&F2F2F(F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "log[2](1) = 0,log[2](2) = 1; " "6$/-&%$logG6#\"\"#6#\"\"\"\"\"!/-&F&6#F(6#F(F*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 33 "The picture shows the graphs of " } {XPPEDIT 18 0 "y = 2^x;" "6#/%\"yG)\"\"#%\"xG" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "y = log[2]*x;" "6#/%\"yG*&&%$logG6#\"\"#\"\"\"%\"xGF*" }{TEXT -1 25 ", together with the line " }{XPPEDIT 18 0 "y=x" "6#/%\"y G%\"xG" }{TEXT -1 2 ". 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\\UE&)=#F-Fe^m7$$\"3eHL3x[JtUF-Fh^m7$$\"3qlmm\"**HBv'F-F[_m7$$\"3=kmmm 4Q_))F-F^_m7$$\"3y**\\P\\R_H6F1Fa_m7$$\"3wlmm@$edM\"F1Fd_m7$$\"3-+]P*p ,Ie\"F1Fg_m7$$\"3N+]7)\\8*3=F1Fj_m7$$\"3'om;/wGY/#F1F]`m7$$\"3%pmTN&*) 3hAF1F``m7$$\"3yKLe90d%\\#F1Fc`m7$$\"3mK$3xB#4PFF1Ff`m7$$\"3)***\\i5\" 3#[HF1Fi`m7$$\"3ULL3P!>i<$F1F\\am7$$\"3&*)****\\jwlQF1Feam7$$\"3Z***\\7%Gw7TF1Fham7$$\"3*e mm;7:_L%F1F[bm7$$\"3Y****\\7/tsXF1F^bm7$$\"3%GL3xcazy%F1Fabm7$$\"3$4++ vT^K-&F1Fdbm7$$\"3il;/;ukW_F1Fgbm7$$\"3++](o-qgZ&F1Fjbm7$$\"3vlm;HzK-d F1F]cm7$$\"3g)*\\P%)*)>RfF1F`cm7$$\"3;MLLjRLnhF1Fccm7$$\"38LLeH\\j+kF1 Ffcm7$$\"3rm;/YS+KmF1Ficm7$$\"3\"3++]B3Y%oF1F\\dm7$$\"3omm\"ziw#)3(F1F _dm7$$\"3/LLLVl@1tF1Fbdm7$$\"3P**\\P\\feQvF1Fedm7$$\"3o+]i?J*4w(F1Fhdm 7$FhzFhz-%&COLORG6&F][lF*$\"\"'!\"\"$\"\"#F`em-%*LINESTYLEG6#Fbem-%+AX ESLABELSG6%Q\"x6\"Q!Fjem-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FhzFcfm" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "More generally, for any positive constant " }{TEXT 273 1 "b" }{TEXT -1 7 " where " } {XPPEDIT 18 0 "b <> 1;" "6#0%\"bG\"\"\"" }{TEXT -1 26 ", the exponenti al function" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = b^x;" "6#/-%\"fG6#%\"xG)%\"bGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "has the inverse function" }}{PARA 256 "" 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "``^(-1);" "6#)%!G,$\"\"\"!\"\"" }{XPPEDIT 18 0 "` `(x) = log[b]*x;" "6#/-%!G6#%\"xG*&&%$logG6#%\"bG\"\"\"F'F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "log[b] *x;" "6#*&&%$logG6#%\"bG\"\"\"%\"xGF(" }{TEXT -1 8 " is the " }{TEXT 260 18 "logarithm to base " }{TEXT 270 1 "b" }{TEXT 260 4 " of " } {TEXT 271 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = log[b]*x;" "6#/% \"yG*&&%$logG6#%\"bG\"\"\"%\"xGF*" }{TEXT -1 14 " exactly when " } {XPPEDIT 18 0 "x = b^y;" "6#/%\"xG)%\"bG%\"yG" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 16 "________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "log[b]*x;" "6#*&&%$logG6#%\"b G\"\"\"%\"xGF(" }{TEXT -1 22 " is only defined when " }{TEXT 272 1 "x " }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 33 "Logarith ms to base 10 are called " }{TEXT 260 17 "common logarithms" }{TEXT -1 31 " and the base is often omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "The logarithm to base 10 function is \+ denoted by " }{TEXT 0 5 "log10" }{TEXT -1 15 " as well as by " }{TEXT 0 7 "log[10]" }{TEXT -1 10 " in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for k from 2 to 10 do\n \+ print('log[10]'(k)=evalf(log10(k)))\nend do;\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"#$\"+d**H5I!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"$$\"+\\D@rZ!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"%$\"+9**f?g!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"&$\"+T+q*)p!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"'$\"+.D^\"y(!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"($\"++/)4X)!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\")$\"+r)**3.* !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#56#\"\"*$\"+)4DCa *!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"#5F'$\"\"\"\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The standard rules for e xponents, or powers, give rise to the following 3 basic properties of \+ logarithms:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[b] (u*v) = log[b]*u+log[b]*v;" "6#/-&%$logG6#%\"bG6#*&%\"uG\"\"\"%\"vGF,, &*&&F&6#F(F,F+F,F,*&&F&6#F(F,F-F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[b](u/v) = log[b]*u-log[b]*v;" "6 #/-&%$logG6#%\"bG6#*&%\"uG\"\"\"%\"vG!\"\",&*&&F&6#F(F,F+F,F,*&&F&6#F( F,F-F,F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "log[b](u^ p) = p*log[b]*u;" "6#/-&%$logG6#%\"bG6#)%\"uG%\"pG*(F,\"\"\"&F&6#F(F.F +F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "Other useful formu las are:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "log[b]*b = 1;" "6#/*&&%$l ogG6#%\"bG\"\"\"F(F)F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "log[b]*1 = 0;" "6#/*&&%$logG6#%\"bG\"\"\"F)F)\"\"!" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "log[b](b^x) = x;" "6#/-&%$logG 6#%\"bG6#)F(%\"xGF+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "b^(log[b]*x) = x;" "6#/)%\"bG*&&%$logG6#F%\"\"\"%\"xGF*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "The last two formulas say that the functions " }{XPPEDIT 18 0 "f(x) = b^x;" "6#/-%\"fG6#%\"xG)%\"bGF '" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = log[b]*x;" "6#/-%\"gG6#% \"xG*&&%$logG6#%\"bG\"\"\"F'F-" }{TEXT -1 28 " are inverses of each ot her." }}{PARA 0 "" 0 "" {TEXT -1 16 "There is also a " }{TEXT 260 22 " change of base formula" }{TEXT -1 1 " " }{XPPEDIT 18 0 "log[a]*x = log [b]*x/(log[b]*a);" "6#/*&&%$logG6#%\"aG\"\"\"%\"xGF)*(&F&6#%\"bGF)F*F) *&&F&6#F.F)F(F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The last formula can be used to calcula te logs to any base using only logs to base 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For example, we can compu te " }{XPPEDIT 18 0 "log[2]*3;" "6#*&&%$logG6#\"\"#\"\"\"\"\"$F(" } {TEXT -1 20 " using this formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "'log[10](3)/log[10](2)';\nev alf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-&%$logG6#\"#56#\"\"$\"\" \"-F%6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,D'\\e\"!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We c an check the answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "2^1.584962501;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,+++I!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Maple calculates logarithms to base 10 by means of the formula " } {XPPEDIT 18 0 "log[10]*x = log[exp(1)]*x/(log[exp(1)]*10);" "6#/*&&%$l ogG6#\"#5\"\"\"%\"xGF)*(&F&6#-%$expG6#F)F)F*F)*&&F&6#-F/6#F)F)F(F)!\" \"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\" " }{TEXT -1 1 " " }{TEXT 343 1 "~" }{TEXT -1 53 " 2.718281828 is the m athematical constant defined by " }{XPPEDIT 18 0 "exp(1) = Limit((1+t) ^(1/t),t = 0);" "6#/-%$expG6#\"\"\"-%&LimitG6$),&F'F'%\"tGF'*&F'F'F-! \"\"/F-\"\"!" }{TEXT -1 22 " . Logarithms to base " }{XPPEDIT 18 0 " e xp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 12 " are called " }{TEXT 260 18 " natural logarithms" }{TEXT -1 18 " and the notation " }{XPPEDIT 18 0 " log[exp(1)]*x = ln*x;" "6#/*&&%$logG6#-%$expG6#\"\"\"F+%\"xGF+*&%#lnGF +F,F+" }{TEXT -1 19 " is commonly used. " }}{PARA 0 "" 0 "" {TEXT -1 24 "The limit definition of " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\" \"" }{TEXT -1 111 " arises naturally in the discussion of the derivati ve of general logarithm functions given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "' log[10]'(x)=log[10](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6# \"#56#%\"xG*&-%#lnGF)\"\"\"-F-F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The equation " }{XPPEDIT 18 0 "2^x = 3;" "6#/)\"\"#%\"xG \"\"$" }{TEXT -1 18 " has the solution " }{XPPEDIT 18 0 "x = log[2]*3; " "6#/%\"xG*&&%$logG6#\"\"#\"\"\"\"\"$F*" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 5 "solve" }{TEXT -1 32 " gives t he solution in the form " }{XPPEDIT 18 0 "x=ln*3/(ln*2)" "6#/%\"xG*(%# lnG\"\"\"\"\"$F'*&F&F'\"\"#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(2^x=3, x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%#lnG6#\"\"$\"\" \"-F%6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,D'\\e\"!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "'log[2](3)';\n``=eval(%);\n``=evalf(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-&%$logG6#\"\"#6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&-%#lnG6#\"\"$\"\"\"-F'6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+,D'\\e\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "The derivative of the general logarithm function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 45 "We shall find the derivative of the function " } {XPPEDIT 18 0 "f(x) = log[b]*x;" "6#/-%\"fG6#%\"xG*&&%$logG6#%\"bG\"\" \"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 62 "As usual we st art with the limit definition of the derivative " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = Limit((f(x+h)-f(x))/h,h = 0) ;" "6#/-%$f~'G6#%\"xG-%&LimitG6$*&,&-%\"fG6#,&F'\"\"\"%\"hGF1F1-F.6#F' !\"\"F1F2F5/F2\"\"!" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit((log[b](x+h)-log[b]*x)/h,h = 0);" "6#/% !G-%&LimitG6$*&,&-&%$logG6#%\"bG6#,&%\"xG\"\"\"%\"hGF2F2*&&F,6#F.F2F1F 2!\"\"F2F3F7/F3\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Limit(``(1/h )*log[b]((x+h)/x),h = 0);" "6#/%!G-%&LimitG6$*&-F$6#*&\"\"\"F,%\"hG!\" \"F,-&%$logG6#%\"bG6#*&,&%\"xGF,F-F,F,F7F.F,/F-\"\"!" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 19 "(using the formula " }{XPPEDIT 18 0 " log[b]*u-log[b]*v = log[b](u/v);" "6#/,&*&&%$logG6#%\"bG\"\"\"%\"uGF*F **&&F'6#F)F*%\"vGF*!\"\"-&F'6#F)6#*&F+F*F/F0" }{TEXT -1 2 " )" }} {PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(log[b]((1+h/x )^(1/h)),h = 0);" "6#-%&LimitG6$-&%$logG6#%\"bG6#),&\"\"\"F.*&%\"hGF.% \"xG!\"\"F.*&F.F.F0F2/F0\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "(using the formula " }{XPPEDIT 18 0 "p*log[b]*u = log[b]( u^p);" "6#/*(%\"pG\"\"\"&%$logG6#%\"bGF&%\"uGF&-&F(6#F*6#)F+F%" } {TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 83 "In order to procede fu rther in the analysis of this limit we make the substitution " } {XPPEDIT 18 0 "t = h/x;" "6#/%\"tG*&%\"hG\"\"\"%\"xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "h = \+ t*x;" "6#/%\"hG*&%\"tG\"\"\"%\"xGF'" }{TEXT -1 32 ", and, since we are considering " }{TEXT 274 1 "x" }{TEXT -1 19 " to be fixed, as ." }} {PARA 0 "" 0 "" {TEXT -1 16 "We can now write" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = Limit(log[b]((1+t)^(1/(t*x)) ),t = 0);" "6#/-%$f~'G6#%\"xG-%&LimitG6$-&%$logG6#%\"bG6#),&\"\"\"F3% \"tGF3*&F3F3*&F4F3F'F3!\"\"/F4\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 11 "Separating " }{XPPEDIT 18 0 "1/(t*x);" "6#*&\"\"\"F$*&% \"tGF$%\"xGF$!\"\"" }{TEXT -1 25 " into the factored form " } {XPPEDIT 18 0 "1/t;" "6#*&\"\"\"F$%\"tG!\"\"" }{TEXT -1 1 " " }{TEXT 268 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/x;" "6#*&\"\"\"F$%\"xG!\" \"" }{TEXT -1 52 ", and using the power property of logarithms we get \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = Limit( ``(1/x)*log[b]((1+t)^(1/t)),t = 0);" "6#/-%$f~'G6#%\"xG-%&LimitG6$*&-% !G6#*&\"\"\"F0F'!\"\"F0-&%$logG6#%\"bG6#),&F0F0%\"tGF0*&F0F0F:F1F0/F: \"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "If we suppose that the function " }{XPPEDIT 18 0 "f(x) = log[b]*x;" "6#/-%\"fG6#%\"xG*&&%$logG6#%\"bG\"\"\"F'F-" }{TEXT -1 4 " is " }{TEXT 260 10 "continuous" }{TEXT -1 205 ", that is, roughly \+ speaking, small changes in input values lead to corresponding small ch anges in output values, then it is reasonable to interchange the limit and the log in this last expression to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 1/x;" "6#/-%$f~'G6#%\"xG*&\" \"\"F)F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "log[b](Limit((1+t)^(1/t ),t = 0));" "6#-&%$logG6#%\"bG6#-%&LimitG6$),&\"\"\"F.%\"tGF.*&F.F.F/! \"\"/F/\"\"!" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "This is where the number " } {XPPEDIT 18 0 "exp(1);" "6#-%$expG6#\"\"\"" }{TEXT -1 44 " makes its d ebut as the value of the limit: " }{XPPEDIT 18 0 "Limit((1+t)^(1/t),t \+ = 0)" "6#-%&LimitG6$),&\"\"\"F(%\"tGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "We may investigate " } {XPPEDIT 18 0 "Limit((1+t)^(1/t),t = 0)" "6#-%&LimitG6$),&\"\"\"F(%\"t GF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 14 " numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[t, `|`, (1+t)^(1/t)], [____ _____, ``, __________________], [10^(-3), `|`, 2.716923932], [10^(-4), `|`, 2.718145927], [10^(-5), `|`, 2.718268237], [10^(-6), `|`, 2.7182 80469], [10^(-7), `|`, 2.718281693], [10^(-8), `|`, 2.718281815], [10^ (-9), `|`, 2.718281827], [10^(-10), `|`, 2.718281828]]);" "6#-%'matrix G6#7,7%%\"tG%\"|grG),&\"\"\"F,F(F,*&F,F,F(!\"\"7%%*_________G%!G%3____ ______________G7%)\"#5,$\"\"$F.F)-%&FloatG6$\"+KR#pr#!\"*7%)F5,$\"\"%F .F)-F96$\"+Ff9=FF<7%)F5,$\"\"&F.F)-F96$\"+P#o#=FF<7%)F5,$\"\"'F.F)-F96 $\"+p/G=FF<7%)F5,$\"\"(F.F)-F96$\"+$p\"G=FF<7%)F5,$\"\")F.F)-F96$\"+:= G=FF<7%)F5,$\"\"*F.F)-F96$\"+F=G=FF<7%)F5,$F5F.F)-F96$\"+G=G=FF<" } {TEXT -1 13 " " }{XPPEDIT 18 0 "matrix([[t, `|`, (1+t)^(1/ t)], [_________, ``, __________________], [-10^(-4), `|`, 2.718417755] , [-10^(-5), `|`, 2.718295420], [-10^(-6), `|`, 2.718283188], [-10^(-7 ), `|`, 2.718281964], [-10^(-8), `|`, 2.718281842], [-10^(-9), `|`, 2. 718281830], [-10^(-10), `|`, 2.718281829], [-10^(-11), `|`, 2.71828182 8]]);" "6#-%'matrixG6#7,7%%\"tG%\"|grG),&\"\"\"F,F(F,*&F,F,F(!\"\"7%%* _________G%!G%3__________________G7%,$)\"#5,$\"\"%F.F.F)-%&FloatG6$\"+ bxT=F!\"*7%,$)F6,$\"\"&F.F.F)-F:6$\"+?aH=FF=7%,$)F6,$\"\"'F.F.F)-F:6$ \"+)=$G=FF=7%,$)F6,$\"\"(F.F.F)-F:6$\"+k>G=FF=7%,$)F6,$\"\")F.F.F)-F:6 $\"+U=G=FF=7%,$)F6,$\"\"*F.F.F)-F:6$\"+I=G=FF=7%,$)F6,$F6F.F.F)-F:6$\" +H=G=FF=7%,$)F6,$\"#6F.F.F)-F:6$\"+G=G=FF=" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The value of " } {XPPEDIT 18 0 "Limit((1+t)^(1/t),t = 0)" "6#-%&LimitG6$),&\"\"\"F(%\"t GF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 30 " is the mathematical constant \+ " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "~" }{TEXT -1 14 " 2.718281828. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graph of " }{XPPEDIT 18 0 "y=(1+t)^(1/t)" "6#/%\"yG),&\"\"\"F'%\"tGF'*&F'F 'F(!\"\"" }{TEXT -1 6 ". As " }{TEXT 344 1 "t" }{TEXT -1 43 " tends t o 0, the graph approaches the point" }{XPPEDIT 18 0 "``(0,exp(1));" "6 #-%!G6$\"\"!-%$expG6#\"\"\"" }{TEXT -1 8 " on the " }{TEXT 319 1 "y" } {TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "f := x -> (1+t)^(1/t): 'f(t)'=f(t);\np2 := plot(f(t),t=-1..2,y=0..8,color=red):\np1 := plot([[[0,exp(1)]]$2],sty le=point,\n symbolsize=[14,18],symbol=[circle,circle],color =black):\np3 := plot([[-1,0],[-1,8]],color=black,linestyle=3):\nplots[ display]([p1,p2,p3],tickmarks=[4,9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG),&\"\"\"F*F'F**&F*F*F'!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6)-%'CURVESG6&7#7$$\"\"!F)$\"34X!f% G=G=F!#<-%'SYMBOLG6$%'CIRCLEG\"#9-%'COLOURG6&%$RGBGF)F)F)-%&STYLEG6#%& POINTG-F$6&F&-F.6$F0\"#=F2F6-F$6$7`p7$$!3))******p^cz**!#=$\"3Q10#4keg &\\!#:7$$!3W+++N.8f**FE$\"3r+9J0dl-DFH7$$!3\"********\\&pQ**FE$\"3CCI0 FYF$o\"FH7$$!3y******p1E=**FE$\"3K9:T-#QGF\"FH7$$!3m******Re#y*)*FE$\" 3F6\"4'=R9E5FH7$$!3a******45Rx)*FE$\"3Oi0G&R.Sh)!#;7$$!3_+++!=cp&)*FE$ \"30(p(z-IINuF\\o7$$!3(******\\M@l$)*FE$\"3+]&3d8O)\\lF\\o7$$!3a+++5l3 ;)*FE$\"3qciCn(Q+'eF\\o7$$!3U+++!o^cz*FE$\"3F%3f&e[L2`F\\o7$$!3G+++]o@ v(*FE$\"3y)>\\:'[Ua[F\\o7$$!3;+++??ya(*FE$\"3qVs08pUwWF\\o7$$!3/+++!>Z Vt*FE$\"3kVQ9zV5cTF\\o7$$!3]*****\\N7Rr*FE$\"3'>\"H/5x8\")QF\\o7$$!33+ ++?vZ$p*FE$\"3tt`/br[UOF\\o7$$!3%********oUIn*FE$\"3p&pC&f(oLV$F\\o7$$ !3#)******fyg_'*FE$\"3YQ.hi:f[KF\\o7$$!3R+++DIAF\\o7$$!3 R++++Z#yU*FE$\"3m4\")yKb5z?F\\o7$$!39+++S]&pQ*FE$\"3#=avw_mu&>F\\o7$$! 3[*****\\P&3Y$*FE$\"3?s#[(4Cx]=F\\o7$$!3K+++?W'\\F*FE$\"3I>+,fiF$p\"F \\o7$$!3Z+++qM%Q?*FE$\"3gt5MxDRj:F\\o7$$!3^******>DsK\"*FE$\"3C[^PJ*)FE$\"3wr'>kD *p67F\\o7$$!3A+++cx6x()FE$\"37GIL,!*)e4\"F\\o7$$!3m******f=:<')FE$\"3^ `]!oG%pL**F,7$$!3!*******ff=d%)FE$\"3Bg7#)f$e]6*F,7$$!37+++g+A(H)FE$\" 3![U!3+/]X%)F,7$$!3^+++iTDP\")FE$\"31=a^nO9()yF,7$$!3*)******\\;?:yFE$ \"3\\$)ofp'HE+(F,7$$!3Q+++Q\"\\J\\(FE$\"3!=#3.ut:PjF,7$$!3e*****4V0@&o FE$\"3.!y#H`lY-aF,7$$!3k*****f&exdiFE$\"31Okt`T)*4[F,7$$!3h*****f+#QUc FE$\"3'e=S:e@(eVF,7$$!3\"******f!3%f+&FE$\"3FMsUt*>H+%F,7$$!39+++\"oS: P%FE$\"3,T[37(GQs$F,7$$!3<+++v@)*=PFE$\"3GQ3wI`.#\\$F,7$$!3'******zG3U 9$FE$\"3s?x_8Z3ALF,7$$!3-+++D!\\r\\#FE$\"3=k,:9f$)fJF,7$$!3#******\\(G VZ=FE$\"3$Q6awRP5-$F,7$$!3%******\\(4J@7FE$\"3;b'z83<`!HF,7$$!3d****** fIKFl!#>$\"35Jsbh'eE\"GF,7$$\"3#*********>FOB!#?$\"3vfFo!H8^r#F,7$$\"3 #**********Q5'fF\\]l$\"3J_Rn%)=XTEF,7$$\"31+++S!QBE\"FE$\"3!=KHL6MWc#F ,7$$\"3C+++]\"o?&=FE$\"3A#*)H[J!*G]#F,7$$\"31+++Sb4*\\#FE$\"3ft$RE-)[T CF,7$$\"35+++I'=_6$FE$\"3I2'*\\o!p\")Q#F,7$$\"3-+++S%y!ePFE$\"3U+\"['p `=PBF,7$$\"3'*******>WU[VFE$\"35B4A`^.%H#F,7$$\"33+++5B>&)\\FE$\"3yjR& [:i4D#F,7$$\"3W*******=:mk&FE$\"3MP=6Qvf4AF,7$$\"3))******fdQAiFE$\"3% )olYPK3w@F,7$$\"3t******RPBWoFE$\"3Z?+[F6AU@F,7$$\"31+++]Nm'[(FE$\"3$* pDI^,`4@F,7$$\"3U+++!zb^6)FE$\"3W_.ev$z&z?F,7$$\"3b******RVDB()FE$\"3w SGl64L_?F,7$$\"3Y+++5TW)R*FE$\"3_WW[#G%)Q-#F,7$$\"3,+++:K^+5F,$\"3A5Hl $z,)**>F,7$$\"3-+++7,Hl5F,$\"3#z0Ulx'\\v>F,7$$\"31+++4w)R7\"F,$\"3G'Gr ^2EY&>F,7$$\"3)******zZf\")=\"F,$\"3#RX5Q4iH$>F,7$$\"3++++/-a[7F,$\"36 2(y$pMf8>F,7$$\"30+++ial68F,$\"3k&3.Oq>V*=F,7$$\"3/+++i@Ot8F,$\"3)>YgA 3mj(=F,7$$\"30+++fL'zV\"F,$\"3Wgk7`UWe=F,7$$\"3(********)>=+:F,$\"3,!G (4G'o>%=F,7$$\"3++++E&4Qc\"F,$\"3nh%p\\:he#=F,7$$\"3#******R>5pi\"F,$ \"3@`R)=Jx0\"=F,7$$\"3#******\\:$*[o\"F,$\"3[fT0T**4(z\"F,7$$\"31+++r \"[8v\"F,$\"3e)>7@QyAy\"F,7$$\"3++++Ijy5=F,$\"3_87l%o_&pF,$\"3<:`AbL[W " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Referring to (i), th e derivative of " }{XPPEDIT 18 0 "f(x) = log[b]*x;" "6#/-%\"fG6#%\"xG* &&%$logG6#%\"bG\"\"\"F'F-" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(x) = 1/x;" "6#/-%$f~'G6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "log[b]*exp(1);" "6#*&&%$logG6#%\"bG\"\"\"-%$expG6#F(F( " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here are the Maple calculations which give the values in \+ the previous tables. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "t := 't':\nmap(t->evalf[10](evalf[ 50]([t,(1+t)^(1/t)])),[seq(Float(1,-i),i=3..10)]):\n[[t,(1+1/t)^(1/t)] ,op(%)]:\nconvert(%,matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7+7$%\"tG),&\"\"\"F+*&F+F+F(!\"\"F+F,7$$F+!\"$$\"+KR#pr#!\"*7$$F +!\"%$\"+Ff9=FF37$$F+!\"&$\"+P#o#=FF37$$F+!\"'$\"+p/G=FF37$$F+!\"($\"+ $p\"G=FF37$$F+!\")$\"+:=G=FF37$$F+F3$\"+F=G=FF37$$F+!#5$\"+G=G=FF3Q(pp rint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "map(t->evalf[10](evalf[50]([t,(1+t)^(1/t)])),[seq(-F loat(1,-i),i=3..11)]):\n[[t,(1+1/t)^(1/t)],op(%)]:\nconvert(%,matrix); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7,7$%\"tG),&\"\"\"F+*& F+F+F(!\"\"F+F,7$$F-!\"$$\"+;Ak>F!\"*7$$F-!\"%$\"+bxT=FF37$$F-!\"&$\"+ ?aH=FF37$$F-!\"'$\"+)=$G=FF37$$F-!\"($\"+k>G=FF37$$F-!\")$\"+U=G=FF37$ $F-F3$\"+I=G=FF37$$F-!#5$\"+H=G=FF37$$F-!#6$\"+G=G=FF3Q(pprint36\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The valu e of " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 35 " corr ect to 10 digits is given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "exp(1)=evalf(exp(1),10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#\"\"\"$\"+G=G=F!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Maple \"k nows\" the value of " }{XPPEDIT 18 0 "Limit((1+t)^(1/t),t = 0)" "6#-% &LimitG6$),&\"\"\"F(%\"tGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "t := 't':\nLimit((1+t)^(1/t),t=0);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$),&\"\"\"F(%\"tGF(*&F(F(F)!\"\"/F )\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " \nOur final result is that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[log[b]*x] = 1/x;" "6#/7#*&&%$logG6#%\" bG\"\"\"%\"xGF**&F*F*F+!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "log[b]* exp(1);" "6#*&&%$logG6#%\"bG\"\"\"-%$expG6#F(F(" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "In view of the change of base formula, the derivative can also be \+ written as: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/d x" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[log[b] *x] = 1/(x*log[exp(1)]*b);" "6#/7#*&&%$logG6#%\"bG\"\"\"%\"xGF**&F*F** (F+F*&F'6#-%$expG6#F*F*F)F*!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Di ff(log[b](x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Diff G6$*&-%#lnG6#%\"xG\"\"\"-F(6#%\"bG!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&%\"xGF$-%#lnG6#%\"bGF$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "In any case, we shall be mainly concern ed with the special case obtained by taking the base " }{TEXT 275 1 "b " }{TEXT -1 18 " to be the number " }{XPPEDIT 18 0 "exp(1);" "6#-%$exp G6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 19 "Logarithms to base " }{XPPEDIT 18 0 "exp(1);" "6#- %$expG6#\"\"\"" }{TEXT -1 13 " are called " }{TEXT 260 18 "natural lo garithms" }{TEXT -1 15 ", and we write " }{XPPEDIT 18 0 "ln*x = log[ex p(1)]*x;" "6#/*&%#lnG\"\"\"%\"xGF&*&&%$logG6#-%$expG6#F&F&F'F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln*x] = 1/x;" "6#/7#*&%#lnG\"\"\"% \"xGF'*&F'F'F(!\"\"" }{TEXT -1 4 " ]. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 8 "________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Diff(ln(x),x);\nva lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%#lnG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"xG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The following graph \+ shows the graph of " }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%#lnG\"\" \"%\"xGF'" }{TEXT -1 5 " (in " }{TEXT 259 3 "red" }{TEXT -1 37 ") alon g with graph of its derivative " }{XPPEDIT 18 0 "y=1/x" "6#/%\"yG*&\" \"\"F&%\"xG!\"\"" }{TEXT -1 5 " (in " }{TEXT 256 4 "blue" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([ln(x),1/x],x=0..5,y=-3..5,color=[red,blue]);" } }{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&-%'CURVESG6$7^o 7$$\"3(*******R`!eS$!#?$!35S%f/&QF#o&!#<7$$\"3&*******z1h6oF*$!3s%\\** )pm7*)\\F-7$$\"3*******>gT<-\"!#>$!35I8zh:m$e%F-7$$\"3++++O@Ki8F6$!3K \\&R$*[zfH%F-7$$\"3;+++qE!Hq\"F6$!3gR`-Qf$G2%F-7$$\"3*******R?$[V?F6$! 3q%QJ7Q90*QF-7$$\"3;+++QP1%Q#F6$!3AETS,POOPF-7$$\"3)******>FWYs#F6$!3# Rgz(3B$Gg$F-7$$\"3K+++S`!eS$F6$!3?%Rluv)ozLF-7$$\"3)******zSmp3%F6$!3I R9n+sO(>$F-7$$\"3K+++wu7oZF6$!3E\"=W3_;K/$F-7$$\"3/+++X&)G\\aF6$!3V[h. 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\\&Fg[m$\"++dm3 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+ " }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 63 "Find the equation of the tangent and normal lines to the graph " }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%#lnG\"\"\"%\"xG F'" }{TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "x=exp(1)" "6#/ %\"xG-%$expG6#\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%#ln G\"\"\"%\"xGF'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/dx = 1/x; " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&%\"xGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "ln*exp(1) = 1;" "6#/*&%# lnG\"\"\"-%$expG6#F&F&F&" }{TEXT -1 25 ", the point on the curve " } {XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%#lnG\"\"\"%\"xGF'" }{TEXT -1 6 " with " }{TEXT 298 1 "x" }{TEXT -1 12 " coordinate " }{XPPEDIT 18 0 "e xp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 3 " is" }{XPPEDIT 18 0 " ``(exp(1 ),1)" "6#-%!G6$-%$expG6#\"\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The gradient of the tangent line at the point" }{XPPEDIT 18 0 "``(exp(1),1);" "6#-%!G6$-%$expG6#\"\"\"F)" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "eval(dy/dx,x = exp(1) ) = 1/exp(1);" "6#/-%%evalG6$*&%#dyG\"\"\"%#dxG!\"\"/%\"xG-%$expG6#F)* &F)F)-F/6#F)F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The eq uation of the tangent line is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-1 = ``(1/exp(1))*(x-exp(1));" "6#/,&%\"yG\"\"\"F&!\" \"*&-%!G6#*&F&F&-%$expG6#F&F'F&,&%\"xGF&-F.6#F&F'F&" }{TEXT -1 1 "," } }{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/exp(1);" "6#/%\"yG*&\"\"\"F&-%$expG6#F&!\"\" " }{TEXT -1 1 " " }{TEXT 299 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The normal line has gradient " }{XPPEDIT 18 0 "-exp(1)" " 6#,$-%$expG6#\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The equation of the normal line is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1 = -exp(1)*(x-exp(1));" "6#/,&%\"yG\"\"\"F &!\"\",$*&-%$expG6#F&F&,&%\"xGF&-F+6#F&F'F&F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = -exp(1)*x+1+exp(2);" "6#/%\"yG,(*&-%$expG6#\" \"\"F*%\"xGF*!\"\"F*F*-F(6#\"\"#F*" }{TEXT -1 2 ". 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)))*Rx,;L'F-7$Febl$\"3DR2)3rQ\"4gF-7$Fjbl$\"3'>#QW$e^`o&F-7$F_cl$\"3u3 E')*oALP&F-7$Fdcl$\"3qR')eh0(**3&F-7$Ficl$\"3ezimuu.`ZF-7$F^dl$\"3IpsX a4hnWF-7$Fcdl$\"3(o#RFd?fNTF-7$Fhdl$\"3R&eH[2*pTQF-7$F]el$\"3%)zhb$*=D >NF-7$Fbel$\"3\\oy'[#f?7KF-7$Fgel$\"3U@*yxmN=*GF-7$F\\fl$\"3352+jdj(f# F-7$Fafl$\"3m.3n?9I!G#F-7$Fffl$\"3es@=C-o]>F-7$F[gl$\"3Y\\yl.Suj;F-7$F `gl$\"3+]phC]%QN\"F-7$Fegl$\"3^#yIc0*oL5F-7$Fjgl$\"3*[;9S+*y/sF^o7$F_h l$\"3#>!)3Dq>V<%F^o7$Fdhl$\"3gMseSA+&4)F67$Fihl$!38r2sL(RR@#F^o7$F^il$ !3/ey)*Gs5UaF^o7$Fcil$!3Ei\"H(znIn$)F^o7$Fhil$!3%=49NDKl:\"F-7$F]jl$!3 _m(yLhSuX\"F-7$Fbjl$!3qo,C_l(>x\"F-7$Fgjl$!3U;DTCH\\z?F-7$F\\[m$!3`\") G%p1M9S#F-7$Fa[m$!3(z/73l,:r#F-7$Ff[m$!3J-jRF-7$Fj\\m$!3qBLzhNSfUF- 7$F_]m$!3)\\wf(GjAvXF-7$Fd]m$!39%pB[,4v([F-7$Fh]l$!3guXOVIN-_F--%&COLO RG6&F`^lFj]l$\"\"(!\"\"$\"\"$Fhgm-%+AXESLABELSG6$Q\"x6\"Q\"yF_hm-%%VIE WG6$;$!\"&FhgmFh]l;$FhgmFj]l$\"\"#Fj]l" 1 2 0 1 10 0 2 9 1 4 1 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 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 23 "We find the derivative " }{XPPEDIT 18 0 "`f '`(x)" " 6#-%$f~'G6#%\"xG" }{TEXT -1 18 " of the function " }{XPPEDIT 18 0 "f( x) = ln(x^3);" "6#/-%\"fG6#%\"xG-%#lnG6#*$F'\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The easi est way to obtain the derivative of this function is to use properties of logarithms to modify the expression for " }{XPPEDIT 18 0 "f(x)" "6 #-%\"fG6#%\"xG" }{TEXT -1 24 " before differentiation." }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x)=ln(x^3)" "6#/-%\"fG6#%\"xG -%#lnG6#*$F'\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3*ln*x;" "6#/%!G*(\"\"$\"\"\"%#lnGF'%\"xGF'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 3/x;" "6#/-%$f~'G6#%\"xG*& \"\"$\"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "Alt ernatively, the chain rules gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`f '`(x) = 1/(x^3);" "6#/-%$f~'G6#%\"xG*&\"\"\"F)*$F '\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#d xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^3]=1/x^3" "6#/7#*$%\"xG\" \"$*&\"\"\"F)*$F&F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2=3/x" " 6#/*&\"\"$\"\"\"*$%\"xG\"\"#F&*&F%F&F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 8 "We find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" } {TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = ln(cos*x);" "6 #/-%\"fG6#%\"xG-%#lnG6#*&%$cosG\"\"\"F'F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This function c an be differentiated using the chain rule." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = ln(cos*x);" "6#/%\"yG-%#lnG6#*&%$cosG \"\"\"%\"xGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u = cos*x;" "6#/%\" uG*&%$cosG\"\"\"%\"xGF'" }{TEXT -1 11 " so that " }{XPPEDIT 18 0 "y \+ = ln*u;" "6#/%\"yG*&%#lnG\"\"\"%\"uGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "du/dx = -sin*x;" "6#/*&%#duG\" \"\"%#dxG!\"\",$*&%$sinGF&%\"xGF&F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du = 1/u;" "6#/*&%#dyG\"\"\"%#duG!\"\"*&F&F&%\"uGF(" }{TEXT -1 25 " and, by the chain rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 /u" "6#/%!G*&\"\"\"F&%\"uG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-s in*x)=1/(cos*x)" "6#/-%!G6#,$*&%$sinG\"\"\"%\"xGF*!\"\"*&F*F**&%$cosGF *F+F*F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-sin*x)=-tan*x" "6#/-%!G6# ,$*&%$sinG\"\"\"%\"xGF*!\"\",$*&%$tanGF*F+F*F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "More briefly, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 1/(cos*x);" "6#/-%$f~'G6#%\" xG*&\"\"\"F)*&%$cosGF)F'F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[cos*x] \+ = 1/(cos*x);" "6#/7#*&%$cosG\"\"\"%\"xGF'*&F'F'*&F&F'F(F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(-sin*x) = -tan*x;" "6#/-%!G6#,$*&%$s inG\"\"\"%\"xGF*!\"\",$*&%$tanGF*F+F*F," }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff( ln(cos(x)),x)=diff(ln(cos(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$-%#lnG6#-%$cosG6#%\"xGF-,$*&-%$sinGF,\"\"\"F*!\"\"F3" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The follo wing plot shows the graph of " }{XPPEDIT 18 0 "f(x) = ln(cos*x);" "6#/ -%\"fG6#%\"xG-%#lnG6#*&%$cosG\"\"\"F'F-" }{TEXT -1 4 " in " }{TEXT 259 3 "red" }{TEXT -1 33 " and the graph of the derivative " } {XPPEDIT 18 0 "`f '`(x) = -tan*x;" "6#/-%$f~'G6#%\"xG,$*&%$tanG\"\"\"F 'F+!\"\"" }{TEXT -1 5 " in " }{TEXT 256 4 "blue" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "`f '`(x) = 0; " "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" " 6#/%\"xG\"\"!" }{TEXT -1 73 ". This corresponds to fact that there is \+ a maximum point on the graph of " }{XPPEDIT 18 0 "y = ln(cos*x);" "6#/ %\"yG-%#lnG6#*&%$cosG\"\"\"%\"xGF*" }{TEXT -1 15 " at the origin." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 352 "p1 := plot([ln(cos(x)),-tan(x)],x=-Pi/2..Pi/2,\n color=[r ed,blue],discont=true):\np2 := plot([[[-Pi/2,-4],[-Pi/2,4]],[[Pi/2,-4] ,[Pi/2,4]]],\n color=black,linestyle=3):\nplots[display]([p1,p 2],view=[-1.6..1.6,-4..4],font=[SYMBOL,10],ytickmarks=9,\n xtickmark s=[-1.57=`-p/2`,-.7854=`-p/4`,.7854=`p/4`,1.57=`p/2`],\n labelfont=[HE LVETICA,10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 254 434 434 {PLOTDATA 2 "6 *-%'CURVESG6$7ao7$$!3-+++Cjzq:!#<$!3'fP\\E2Z&p>!#;7$$!3o$e(R$Rc'o:F*$! 3WKHSO8)p9'F*7$$!3an^zik^m:F*$!3OK?`)4PQX&F*7$$!3?^F>KlPk:F*$!3_^s>CgP []F*7$$!3%[L!f,mBi:F*$!3+;q7.%*pgZF*7$$!39-bQSn&zb\"F*$!3+1jKo'\\_N%F* 7$$!3Wp1=zon`:F*$!33F[q?!*enSF*7$$!3//5xcr6X:F*$!3#>#fLE]=iOF*7$$!3jQ8 OMubO:F*$!3K-Rkm%)euLF*7$$!3/3?a*)zV>:F*$!3i%ywikn$pHF*7$$!3XxEsW&=B] \"F*$!3eGCF#fF?o#F*7$$!3AEC#*pu_s9F*$!3y3i&*)4e;K#F*7$$!3Av@7&ROFW\"F* $!3Yu!f%G;*z0#F*7$$!3M8RRO+tv8F*$!3\"oA@-ql2k\"F*7$$!3f%eCg^z#38F*$!3g v7%\\]_*[8F*7$$!3/HX?w&\\6C\"F*$!3@/)zJ(3\"z7\"F*7$$!3)o*3wG:\"*y6F*$! 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3-X^([*=lb>Fhp7$Ffw$!3I;!y[!*GXm#Fhp7$F[x$!3Y6p#e))\\IQ$Fhp7$F`x$!3bL' *Hd,O4TFhp7$Fex$!3'\\:v$\\C,i\\Fhp7$Fjx$!3S2^`P2n!y&Fhp7$F_y$!38w9-()= 8CnFhp7$Fdy$!3E*y-mhRkl(Fhp7$Fiy$!3'p9,)*Q*)>y)Fhp7$F^z$!3SiV;FFZp**Fh p7$Fcz$!3([K0Q<\\#Q6F*7$Fhz$!3')>:ufvg)H\"F*7$F][l$!3SI`#y&*y\")\\\"F* 7$Fb[l$!3r7twpL\"Gt\"F*7$Fg[l$!3sw%*ec\"H[.#F*7$F\\\\l$!3zlN\"35RzU#F* 7$Fa\\l$!3%G^z6$\\p>HF*7$Ff\\l$!3)\\V!H#3QKv$F*7$F[]l$!3uF64o1h!)\\F*7 $F`]l$!3o$G*3LNXWvF*7$Fe]l$!3;eA3E/lk**F*7$Fj]l$!3E8U(zI1FY\"F-7$F_^l$ !3!e<))p(Hg^>F-7$Fd^l$!3K!)R<\"Fabl7$F]`l$!3s>w\" e*eii:Fabl7$Fb`l$!3>:!R!3h&RM#Fabl7$Fg`l$!3-wNHw/$zo%Fabl7$F\\al$!3W$G W#**)\\zd$F]bl-F_al6&FaalFealFealFbal-F$6%7$7$$!3c'*[zEjzq:F*$!\"%Ffal 7$Fa_m$\"\"%Ffal-F_al6&FaalFfalFfalFfal-%*LINESTYLEG6#\"\"$-F$6%7$7$$ \"3c'*[zEjzq:F*Fc_m7$Fb`mFf_mFh_mFj_m-%%FONTG6$%'SYMBOLG\"#5-%*AXESTIC KSG6$7&/$!$d\"!\"#%%-p/2G/$!%ayFd_m%%-p/4G/$\"%ayFd_m%$p/4G/$\"$d\"Faa m%$p/2G\"\"*-%+AXESLABELSG6%Q!6\"Fcbm-Ff`m6$%*HELVETICAGFi`m-%%VIEWG6$ ;$F-!\"\"$\"#;F]cm;Fc_mFf_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" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 8 "We find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG " }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = sqrt(x)/(x+ ln*x);" "6#/-%\"fG6#%\"xG*&-%%sqrtG6#F'\"\"\",&F'F,*&%#lnGF,F'F,F,!\" \"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x) = sqrt(x);" "6#/-%\"uG6#%\"xG -%%sqrtG6#F'" }{XPPEDIT 18 0 "``=x^(1/2)" "6#/%!G)%\"xG*&\"\"\"F(\"\"# !\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x) = x+ln*x;" "6#/-%\"vG6 #%\"xG,&F'\"\"\"*&%#lnGF)F'F)F)" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "u*`'`(x) = 1/2;" "6#/*&%\"uG\"\"\"-%\"'G6#%\"xGF&*&F&F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-1/2)=1/(2*sqrt(x))" "6#/)%\"xG,$*& \"\"\"F(\"\"#!\"\"F**&F(F(*&F)F(-%%sqrtG6#F%F(F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v*`'`(x) = 1+1/x;" "6#/*&%\"vG\"\"\"-%\"'G6#%\"xGF&,&F &F&*&F&F&F*!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = (`u \+ '`(x)*v(x)-u(x)*`v '`(x))/(v(x)^2);" "6#/-%$f~'G6#%\"xG*&,&*&-%$u~'G6# F'\"\"\"-%\"vG6#F'F.F.*&-%\"uG6#F'F.-%$v~'G6#F'F.!\"\"F.*$-F06#F'\"\"# F9" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (``(1/(2*sqrt(x)))*`.`*(x+ln*x) -sqrt(x)*`.`*(1+1/x))/((x+ln*x)^2);" "6#/%!G*&,&*(-F$6#*&\"\"\"F+*&\" \"#F+-%%sqrtG6#%\"xGF+!\"\"F+%\".GF+,&F1F+*&%#lnGF+F1F+F+F+F+*(-F/6#F1 F+F3F+,&F+F+*&F+F+F1F2F+F+F2F+*$,&F1F+*&F6F+F1F+F+F-F2" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (x+ln*x-2*x*(1+1/x))/(2*sqrt(x)*(x+ln*x)^2);" "6#/ %!G*&,(%\"xG\"\"\"*&%#lnGF(F'F(F(*(\"\"#F(F'F(,&F(F(*&F(F(F'!\"\"F(F(F /F(*(F,F(-%%sqrtG6#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 64 "by multip lying the top and bottom of the previous expression by " }{XPPEDIT 18 0 "2*sqrt(x)" "6#*&\"\"#\"\"\"-%%sqrtG6#%\"xGF%" }{TEXT -1 3 " ) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (x+ln*x-2*x-2)/(2*sqrt(x)*(x+ln*x)^2);" "6#/%!G*&, *%\"xG\"\"\"*&%#lnGF(F'F(F(*&\"\"#F(F'F(!\"\"F,F-F(*(F,F(-%%sqrtG6#F'F (,&F'F(*&F*F(F'F(F(F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (ln*x-x-2)/ (2*sqrt(x)*(x+ln*x)^2);" "6#/%!G*&,(*&%#lnG\"\"\"%\"xGF)F)F*!\"\"\"\"# F+F)*(F,F)-%%sqrtG6#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 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Diff(sqrt(x)/(x+ln(x)),x)=di ff(sqrt(x)/(x+ln(x)),x);\n``=normal(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"xG#\"\"\"\"\"#,&F(F*-%#lnG6#F(F*!\"\"F( ,&*&F)F**&F*F**&F(#F*F+F,F*F0F*F**(F(F)F,!\"#,&F*F**&F*F*F(F0F*F*F0" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(*(%\"xG#!\"\"F) ,(F+F(-%#lnG6#F+F-F)F(F(,&F+F(F/F(!\"#F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 78 " We find the coordinates of the stationary point on the graph of the fu nction " }{XPPEDIT 18 0 "y = x*ln*x;" "6#/%\"yG*(%\"xG\"\"\"%#lnGF'F& F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 59 "This function can be differentiated using the product r ule." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x) = x;" " 6#/-%\"uG6#%\"xGF'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v(x) = ln*x; " "6#/-%\"vG6#%\"xG*&%#lnG\"\"\"F'F*" }{TEXT -1 11 ", so that " } {XPPEDIT 18 0 "y = u(x)*v(x);" "6#/%\"yG*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F )F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " } {XPPEDIT 18 0 "`u '`(x) = 1;" "6#/-%$u~'G6#%\"xG\"\"\"" }{TEXT -1 6 " \+ and " }{XPPEDIT 18 0 "`v '`(x) = 1/x;" "6#/-%$v~'G6#%\"xG*&\"\"\"F)F' !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Then, by the pro duct rule, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = `u '`(x)*v(x)+u(x)*`v '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&-%$u~' G6#%\"xGF&-%\"vG6#F.F&F&*&-%\"uG6#F.F&-%$v~'G6#F.F&F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1*`.`*ln*x+x *`.`*``(1/x);" "6#/%!G,&**\"\"\"F'%\".GF'%#lnGF'%\"xGF'F'*(F*F'F(F'-F$ 6#*&F'F'F*!\"\"F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = ln*x+1;" "6#/%!G,&*&%#lnG\"\"\"%\"xGF(F(F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff(x*ln(x),x)=diff(x*ln(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&%\"xG\"\"\"-%#lnG6#F(F)F(,&F*F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "The stationary point occurs where " }{XPPEDIT 18 0 "dy/dx =0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 17 ", that is, where \+ " }{XPPEDIT 18 0 "ln(x)=-1" "6#/-%#lnG6#%\"xG,$\"\"\"!\"\"" }{TEXT -1 13 ". This gives " }{XPPEDIT 18 0 "x=exp(-1)" "6#/%\"xG-%$expG6#,$\"\" \"!\"\"" }{TEXT -1 1 " " }{TEXT 301 1 "~" }{TEXT -1 15 " 0.3678794412. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 303 1 "y " }{TEXT -1 15 " coordinate is " }{XPPEDIT 18 0 "y=exp(-1)*ln(exp(-1)) " "6#/%\"yG*&-%$expG6#,$\"\"\"!\"\"F*-%#lnG6#-F'6#,$F*F+F*" }{XPPEDIT 18 0 "``=exp(-1)*(-1)" "6#/%!G*&-%$expG6#,$\"\"\"!\"\"F*,$F*F+F*" } {XPPEDIT 18 0 "``=-exp(-1)" "6#/%!G,$-%$expG6#,$\"\"\"!\"\"F+" }{TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-0" "6#,$\" \"!!\"\"" }{TEXT -1 13 ".3678794412. " }}{PARA 0 "" 0 "" {TEXT -1 23 " The stationary point is" }{XPPEDIT 18 0 "``(exp(-1),-exp(-1))" "6#-%!G 6$-%$expG6#,$\"\"\"!\"\",$-F'6#,$F*F+F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Looking at the gra ph of " }{XPPEDIT 18 0 "y = x*ln(x);" "6#/%\"yG*&%\"xG\"\"\"-%#lnG6#F &F'" }{TEXT -1 47 ". , we see that this point is a minimum point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "p1 := plot(x*ln(x),x=0..3,y):\np2 := plot([[[exp(-1),-exp(-1)]]$3 ],style=point,\n symbol=[circle,diamond,cross],color=bla ck):\nt1 := plots[textplot]([.6,-.8,`minimum point`],color=black):\npl ots[display]([p1,p2,t1],tickmarks=[4,4]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 333 206 206 {PLOTDATA 2 "6*-%'CURVESG6$7fn7$$\"33+++-K[V?!#? $!3+q%*eu%\\bE\"!#>7$$\"3s*****\\Smp3%F*$!3-`8jD?\"yC#F-7$$\"3B+++2'\\ /8'F*$!3VTm&pp\\J7$F-7$$\"3Y******4G$R<)F*$!3g8*GJ?]!HRF-7$$\"3'****** >#**3E7F-$!3[6F#QiQkR&F-7$$\"3*)*****>c'yM;F-$!3#4C!**4F&\\s'F-7$$\"3% )*****H%)z@X#F-$!3A.F%)*pbJ4*F-7$$\"3')*****\\7t&pKF-$!3*=e')H+h$=6!#= 7$$\"3v*****pofV!\\F-$!3qp(*GJooy9FQ7$$\"3s******\\i9RlF-$!3!**\\Ft#HY $y\"FQ7$$\"3=+++XV)RQ*F-$!31=\"3*GiS?AFQ7$$\"31+++WA)GA\"FQ$!3)RC#>&fL (pDFQ7$$\"3/+++Qeui=FQ$!3CS-;lmSIJFQ7$$\"3=+++i3&o]#FQ$!3Q#zfQ(HPoMFQ7 $$\"3')******oX*y9$FQ$!3$Qdpj)y\\QOFQ7$$\"3\")*****R9CAu$FQ$!3wCdK+2Dy OFQ7$$\"3#)*****R*zhdVFQ$!3=2wKcop>OFQ7$$\"33+++%>fS*\\FQ$!3]\\V:D`bnM FQ7$$\"3&)******=$f%GcFQ$!3VRbp+L&\\B$FQ7$$\"3Q+++Dy,\"G'FQ$!3%fP#o1l+ @HFQ7$$\"3/+++7.j[n,!)e#FQ7$$\"3`+++v4&G](FQ$!3s[H)4?%eb@FQ 7$$\"3!)*****\\7nD:)FQ$!3s\"o;5-!=l;FQ7$$\"3[+++D!*oy()FQ$!3+x[BlX\\V6 FQ7$$\"3!******RpnsM*FQ$!3]bV+:3]4jF-7$$\"3(******>FOB+\"!#<$\"3])Q91' *)**QBF*7$$\"3-+++!R5'f5Fds$\"3raeF2-GNhF-7$$\"3&******R!QBE6Fds$\"3g+ B7mr&)Q8FQ7$$\"3!******\\\"o?&=\"Fds$\"3M\"GSGErQ,#FQ7$$\"3/+++a&4*\\7 Fds$\"3K0RtR\")=)y#FQ7$$\"32+++j=_68Fds$\"3aA962CpcNFQ7$$\"35+++Wy!eP \"Fds$\"3QM\"4\\A#R*Q%FQ7$$\"36+++UC%[V\"Fds$\"3k!)Gd[5d!=&FQ7$$\"3'** ****4B>&)\\\"Fds$\"3o8[KTAFds$\"3CUi/yfN&G\"Fds7$$\"3z*****\\@80+#Fds$\"3!)Q>\\vL;(Q\"Fds7 $$\"3-+++7,Hl?Fds$\"3ob5!*=W*y\\\"Fds7$$\"3$)******3w)R7#Fds$\"3!yjg%= )*)**f\"Fds7$$\"3?+++y%f\")=#Fds$\"3j%RqX%=Y8#=Fds7$$\"30+++ial6BFds$\"3xei=oQ3P>Fds7$$\"3#)*****>;iLP#F ds$\"3-vrA:\\J^?Fds7$$\"3$)******eL'zV#Fds$\"3OUJMXFis@Fds7$$\"3>+++!* >=+DFds$\"3;$ff?ev5H#Fds7$$\"3++++E&4Qc#Fds$\"3,:]*G#>\"QT#Fds7$$\"39+ ++%>5pi#Fds$\"38%Q+Ft\"4PDFds7$$\"39+++bJ*[o#Fds$\"3'*HL&HQ5[h%y#Fds7$$\"3++++Ijy5GFds$\"3Wci;is%[!HFds7 $$\"3/+++/)fT(GFds$\"30hj,uSUMIFds7$$\"35+++1j\"[$HFds$\"3#pjEC!\\vfJF ds7$$\"\"$\"\"!$\"37HV+mo$eH$Fds-%'COLOURG6&%$RGBG$\"#5!\"\"$Fe]lFe]lF _^l-F$6&7#7$$\"3MBWr6WzyOFQ$!3MBWr6WzyOFQ-%'SYMBOLG6#%'CIRCLEG-Fi]l6&F [^lFe]lFe]lFe]l-%&STYLEG6#%&POINTG-F$6&Fb^l-Fi^l6#%(DIAMONDGF\\_lF^_l- F$6&Fb^l-Fi^l6#%&CROSSGF\\_lF^_l-%%TEXTG6%7$$\"\"'F^^l$!\")F^^lQ.minim um~point6\"F\\_l-%*AXESTICKSG6$\"\"%Fi`l-%+AXESLABELSG6%Q\"xFe`lQ\"yFe `l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F_^lFc]lFbal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "The exponential function" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"x G-%$expG6#F'" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "exp(1) = Limit((1+ t)^(1/t),t = 0);" "6#/-%$expG6#\"\"\"-%&LimitG6$),&F'F'%\"tGF'*&F'F'F- !\"\"/F-\"\"!" }{TEXT -1 1 " " }{TEXT 304 1 "~" }{TEXT -1 24 " 2.71828 1828, is called " }{TEXT 260 24 "the exponential function" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "Maple represents the exponent ial function by the name " }{TEXT 0 3 "exp" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot (exp(x),x=-2..2,y=0..5,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 335 358 358 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7 $$!3MLLL$Q6G\">!#<$\"3mL4@0mkw9F-7$$!3bmm;M!\\p$=F1$\"3q(zuc7FIf\"F-7$ $!3MLLL))Qj^;Te'f>CF-7$$!3SLL$3WDTL\"F1$\"3>&e(=NU)Qj#F- 7$$!35++]d(Q&\\7F1$\"3OD#)Hb(pj'GF-7$$!3gmmmc4`i6F1$\"3DqR_Tw$p7$F-7$$ !3KLLLQW*e3\"F1$\"3=$*H3()o*fP$F-7$$!3w++++()>'***F-$\"3'*[&ew4$>!o$F- 7$$!3E++++0\"*H\"*F-$\"31/%os_?K,%F-7$$!35++++83&H)F-$\"3e#z1k=QEO%F-7 $$!3\\LLL3k(p`(F-$\"3%*3#z8OJiq%F-7$$!3Anmmmj^NmF-$\"3'Hf#)HA*=]^F-7$$ !3)zmmmYh=(eF-$\"3o![.*pF*)ebF-7$$!3+,++v#\\N)\\F-$\"3KmY(yn#HvgF-7$$! 3commmCC(>%F-$\"372'*[w-GslF-7$$!39*****\\FRXL$F-$\"397`(G1\\W;(F-7$$! 3t*****\\#=/8DF-$\"3smIfnu&yx(F-7$$!3=mmm;a*el\"F-$\"3))R?QF)RRZ)F-7$$ 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2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The in verse function of the exponential function is the " }{TEXT 260 26 "nat ural logarithm function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*x;" "6#*&% #lnG\"\"\"%\"xGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "p1 := plot(exp(x),x=-2..2,c olor=blue):\np2 := plot(ln(x),x=0.1..7,color=red):\np3 := plot(x,x=-2. .7,color=COLOR(RGB,0,.6,.2),linestyle=2):\nplots[display]([p1,p2,p3],l abels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 311 311 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7$$!3MLLL$Q6G\">!#<$\"3 mL4@0mkw9F-7$$!3bmm;M!\\p$=F1$\"3q(zuc7FIf\"F-7$$!3MLLL))Qj^;Te'f>CF-7$$!3SLL$3WDTL\"F1$\"3>&e(=NU)Qj#F-7$$!35++]d(Q&\\7F1$\"3 OD#)Hb(pj'GF-7$$!3gmmmc4`i6F1$\"3DqR_Tw$p7$F-7$$!3KLLLQW*e3\"F1$\"3=$* H3()o*fP$F-7$$!3w++++()>'***F-$\"3'*[&ew4$>!o$F-7$$!3E++++0\"*H\"*F-$ 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$$\"3l***\\PBL&>QF1Febm7$$\"3P*****\\kR:+%F1Fhbm7$$\"3e++]P.(e>%F1F[cm 7$$\"3e**\\7GG'>P%F1F^cm7$$\"3_++]K%yWc%F1Facm7$$\"3S**\\781iXZF1Fdcm7 $$\"3n**\\i&Qm\\$\\F1Fgcm7$$\"31++](['3?^F1Fjcm7$$\"3e**\\7y+*QJ&F1F]d m7$$\"3,,++qfa+bF1F`dm7$$\"31++vy&G9p&F1Fcdm7$$\"3_+]7$eI2)eF1Ffdm7$$ \"3a*****\\YzY0'F1Fidm7$$\"3Q***\\P^WSD'F1F\\em7$$\"3*3+++**eBV'F1F_em 7$$\"3L**\\78%zCi'F1Fbem7$$\"3v**\\(o\"*[W!oF1Feem7$F`\\mF`\\m-%&COLOR G6&F^[lF_[l$\"\"'!\"\"$FhzF]fm-%*LINESTYLEG6#Fhz-%+AXESLABELSG6%%\"xG% \"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F`\\mFjfm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x)=exp (x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(f(x))=x" "6#/-%#lnG6#-% \"fG6#%\"xGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "Differen tiating both sides of this equation with respect to " }{TEXT 300 1 "x " }{TEXT -1 50 " and using the chain rule on the left side gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/f(x)" "6#*&\"\"\"F$ -%\"fG6#%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 1;" "6# /-%$f~'G6#%\"xG\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "s o that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = f(x);" "6#/-%$f~'G6#%\"xG-%\"fG6#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = exp(x);" "6#/-%$f~'G6#%\"xG-%$expG6#F'" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Thus we have: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[exp(x)] = exp(x)" "6#/7 #-%$expG6#%\"xG-F&6#F(" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 8 "________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Remarkable though it may seem a t first sight, the derivative is " }{TEXT 260 33 "the same as the orig inal function" }{TEXT -1 64 ". This means that the gradient of any tan gent line to the curve " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG 6#%\"xG" }{TEXT -1 46 " at any point on the curve is the same as the \+ " }{TEXT 279 1 "y" }{TEXT -1 51 " coordinate of the point of contact o f the tangent." }}{PARA 0 "" 0 "" {TEXT -1 44 "In particular, the tang ent line at the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 41 " has gradient 1, and so its equation is " }{XPPEDIT 18 0 "y=x+1" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The tangent line at the point" }{XPPEDIT 18 0 "``(1,ex p(1))" "6#-%!G6$\"\"\"-%$expG6#F&" }{TEXT -1 14 " has gradient " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 26 ", and so its \+ equation is " }{XPPEDIT 18 0 "y-exp(1)=exp(1)*(x-1)" "6#/,&%\"yG\"\" \"-%$expG6#F&!\"\"*&-F(6#F&F&,&%\"xGF&F&F*F&" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "y=exp(1)*x" "6#/%\"yG*&-%$expG6#\"\"\"F)%\"xGF)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "The tangent line at the \+ point" }{XPPEDIT 18 0 "``(-1,exp(-1))" "6#-%!G6$,$\"\"\"!\"\"-%$expG6# ,$F'F(" }{TEXT -1 14 " has gradient " }{XPPEDIT 18 0 "exp(-1)=1/exp(1) " "6#/-%$expG6#,$\"\"\"!\"\"*&F(F(-F%6#F(F)" }{TEXT -1 27 ", and so it s equation is " }{XPPEDIT 18 0 "y-1/exp(1)=1/exp(1)" "6#/,&%\"yG\"\" \"*&F&F&-%$expG6#F&!\"\"F+*&F&F&-F)6#F&F+" }{XPPEDIT 18 0 " ``(x+1)" " 6#-%!G6#,&%\"xG\"\"\"F(F(" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "y=1/ex p(1)" "6#/%\"yG*&\"\"\"F&-%$expG6#F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x +2/exp(1)" "6#,&%\"xG\"\"\"*&\"\"#F%-%$expG6#F%!\"\"F%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "p1 := plot(exp(x),x=-2..1.65,color=red):\np2 := plot(x+1,x=-1.2..1.2,color=blue):\np3 := plot(exp(1)*x,x=0.1..1.9,col or=magenta):\np4 := plot(exp(-1)*x+2/exp(1),x=-2..0.2,color=COLOR(RGB, 0,.7,.3)):\np5 := plot([[[-1,exp(-1)],[0,1],[1,exp(1)]]$3],style=point ,\n symbol=[circle,diamond,cross],color=black):\nt1 := p lots[textplot]([[1.25,2.76,`(1,e)`],[-1.08,.66,`(1,1/e)`],\n [ -.25,1.23,`(1,0)`]],color=black):\nplots[display]([p1,p2,p3,p4,p5,t1], view=[-2..2,-.2..5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 424 426 426 {PLOTDATA 2 "6.-%'CURVESG6$7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7$$!3mm;H(QS/ #>!#<$\"3Y$>=]7CaY\"F-7$$!3M$3xO*f@^=F1$\"3)e8^6$3Yq:F-7$$!3umT5BfOtF-7$$!3p;/^igpW:F1$\"3qI@^XrwL@F-7$$!3;]P4WJ#)p9 F1$\"3[WziC:m*H#F-7$$!3o;/wk%*Q#R\"F1$\"3Q.\\b%e3[[#F-7$$!3A](=i6/_J\" F1$\"3Md%z')G&>%o#F-7$$!3RL$ez\\4eB\"F1$\"3qm!f[F%*f!HF-7$$!3um\"z\\ny e;\"F1$\"3\\W%=x.([;JF-7$$!3-+]PJJ:(3\"F1$\"3#=d![;-vrLF-7$$!39+]7LV53 5F1$\"3!\\X,Es+\"\\OF-7$$!3&****\\ioh#>$*F-$\"3!R-g%[Y%z$RF-7$$!3>oTgs 4\\F')F-$\"3ix_jhO+?UF-7$$!3wLLe%o3\\!yF-$\"3W[,r%\\5=e%F-7$$!3eMLL)et !3rF-$\"3T<(fOCuC\"\\F-7$$!3O-vV8()[(H'F-$\"3:4tlocDF`F-7$$!3)\\LL3v$) *zbF-$\"3%e3F=bNNs&F-7$$!3a*\\P%)3nFz%F-$\"3U\\ReU6J#>'F-7$$!3X+DJl1:V SF-$\"3I*Qb3-QVn'F-7$$!3OM$3xc/5E$F-$\"30\\Ob3oKg,oxM'[,Yz$)F-7$$!3#))oTgZ[Gj*!#> $\"31VK\"\\[c;3*F-7$$!3AG]il*Rwi#Fds$\"3))o]R4#e1u*F-7$$\"3%HL$eRr83y<6jA\"F17$$\"3C*\\i:Xf*zFF-$\"3)puaG%3[?8F17$$\"3?** \\(oLS9g$F-$\"3R[(\\r(e`L9F17$$\"3bgm;\\udRVF-$\"3Ed9?_ONV:F17$$\"3z)* *\\(ojpF^F-$\"3cb#z)R)4*p;F17$$\"3nJeRZ#\\=%eF-$\"3Q>>)*G&GNz\"F17$$\" 3R)**\\i(pgAmF-$\"3K-()oi7F17$$\"3ci\"HK:RsN(F-$\"3w7*e9D#*p3#F17$$ \"3y)\\7GXT^7)F-$\"3!\\.AVmmND#F17$$\"3cjmT5j!f())F-$\"3-L3'yep#HCF17$ $\"3)p\\i!Rv)=m*F-$\"3_b,QQ(4zi#F17$$\"3?LL$y3))=/\"F1$\"3KuHlqQcMGF17 $$\"3?L3F!f,$>6F1$\"3S]jgEWriIF17$$\"3i;H#)pS2'>\"F1$\"3S!oG%Gz52LF17$ $\"33++D0+im7F1$\"3Cs2t;s$)[NF17$$\"3_mTN3OZZ8F1$\"3CW\"4IY#pZQF17$$\" 3*HLL[.!z>9F1$\"3i'\\b-)=DOTF17$$\"3y\\ilGW*o\\\"F1$\"3C!*R#[X#znWF17$ $\"3!)\\P%=<$pq:F1$\"3Sa#**>=\")*4[F17$$\"3#*************\\;F1$\"3k[)z r#)zp?&F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-F$6$7S7$$!3%******* *******>\"F1$!3a**************>F-7$$!3/+++IooZ6F1$!3Y++++$ooZ\"F-7$$!3 *)****\\?%p@5\"F1$!3'))****\\?%p@5F-7$$!3/+++L.)40\"F1$!3W/+++L.)4&Fds 7$$!3g******4$>X***F-$\"3'))R+++!p![&!#@7$$!3!******\\M%o\"[*F-$\"3)4+ ++bcJ=&Fds7$$!3G*****\\o?i+*F-$\"3=2++]JzP**Fds7$$!3g+++0c!R^)F-$\"3S* ****\\R%4'[\"F-7$$!3]*****\\k_Z+)F-$\"3[+++btC&*>F-7$$!3!******\\aKs\\ (F-$\"34+++buw-DF-7$$!3[******Rd=vpF-$\"3`+++gU\"[-$F-7$$!3;+++ImO:lF- $\"3%)******pLj%[$F-7$$!3c******>#>x*fF-$\"3V+++!y!G-SF-7$$!3g++++j%zZ &F-$\"3S*******p`?_%F-7$$!3i******z([q(\\F-$\"3Q+++?7&H-&F-7$$!3w+++Xe =AXF-$\"3C*****\\:9yZ&F-7$$!3o******>)48)RF-$\"3L+++!=!p=gF-7$$!37+++! 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" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u= a*x" "6#/%\"uG*&%\"aG\"\"\"%\"xGF'" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "y=exp(u)" "6#/%\"yG-%$expG6#%\"uG" }{TEXT -1 7 ". Then \+ " }{XPPEDIT 18 0 "du/dx=a" "6#/*&%#duG\"\"\"%#dxG!\"\"%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du=exp(u)" "6#/*&%#dyG\"\"\"%#duG!\"\" -%$expG6#%\"uG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence, \+ " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF (" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx= exp(a*x)*a" "6#/*&%#duG\"\" \"%#dxG!\"\"*&-%$expG6#*&%\"aGF&%\"xGF&F&F.F&" }{XPPEDIT 18 0 "``=a*ex p(a*x)" "6#/%!G*&%\"aG\"\"\"-%$expG6#*&F&F'%\"xGF'F'" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "In pa rticular, if " }{XPPEDIT 18 0 "y=exp(-x)" "6#/%\"yG-%$expG6#,$%\"xG! \"\"" }{TEXT -1 6 ", the " }{XPPEDIT 18 0 "dy/dx=-exp(-x)" "6#/*&%#dyG \"\"\"%#dxG!\"\",$-%$expG6#,$%\"xGF(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graph of " } {XPPEDIT 18 0 "f(x)=exp(-x)" "6#/-%\"fG6#%\"xG-%$expG6#,$F'!\"\"" } {TEXT -1 5 " (in " }{TEXT 259 3 "red" }{TEXT -1 21 ") and its derivati ve " }{XPPEDIT 18 0 "`f '`(x) = -exp(-x);" "6#/-%$f~'G6#%\"xG,$-%$expG 6#,$F'!\"\"F-" }{TEXT -1 5 " (in " }{TEXT 256 4 "blue" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([exp(-x),-exp(-x)],x=-1.5..2,colour=[red,blue],t hickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 265 491 491 {PLOTDATA 2 "6' -%'CURVESG6$7S7$$!3++++++++:!#<$\"3_k!Q.2*o\"[%F*7$$!3vmmTg*4PU\"F*$\" 3#)\\w#)*\\(\\_TF*7$$!3ALe*[SItN\"F*$\"3'p0gW\"f!e)QF*7$$!3em;H_'zEG\" F*$\"3B-_*QR!H1OF*7$$!3pm;/mS`27F*$\"3Mi`Zo`AXLF*7$$!3DLekLcuK6F*$\"3c LLweZFN$\"3T@]G)oK$4;F*7$$!3Cm;HdV&[4%FN$\"3Y_'o*HE/1 :F*7$$!3VKL$3#o21LFN$\"3_#f_$\"3\"[7N/#plU5F*7$$\"3NF+DJS)3,$Fjq$\"3gj@,d\"*R. 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7$$\"+++++5Fh\\l$\"+7WzyOFi^lFi\\lF]]l-Fc\\l6%F_drFa]lF]]l-Fc\\l6%F_dr Ff]lF]]l-Fc\\l6%7$F`dr$!+7WzyOFi^lFi\\lF`\\l-F(6%7$7$$\"+@y1=!)Fi^l$\" +9l!zS%Fi^l7$$\"+=K>)>\"Fh\\l$\"+5Bo\\HFi^lF\\_lF]]lF`_lFi_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {TEXT -1 8 "We find " }{XPPEDIT 18 0 "`f '`(x)" "6#-% $f~'G6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = exp(sin*x);" "6#/-%\"fG6#%\"xG-%$expG6#*&%$sinG\"\"\"F'F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This function can be differentiated using the chain rule." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = exp(sin*x);" "6#/%\"yG-% $expG6#*&%$sinG\"\"\"%\"xGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u = \+ sin*x;" "6#/%\"uG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 11 " so that " } {XPPEDIT 18 0 "y = exp(u);" "6#/%\"yG-%$expG6#%\"uG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "du/dx = cos*x;" " 6#/*&%#duG\"\"\"%#dxG!\"\"*&%$cosGF&%\"xGF&" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "dy/du = exp(u);" "6#/*&%#dyG\"\"\"%#duG!\"\"-%$expG6#% \"uG" }{TEXT -1 25 " and, by the chain rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&F%F&%#duGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(u)*`.`*cos*x;" "6#/%!G**-%$expG6#%\"uG\"\"\"% \".GF*%$cosGF*%\"xGF*" }{XPPEDIT 18 0 "`` = exp(sin*x)*cos*x;" "6#/%!G *(-%$expG6#*&%$sinG\"\"\"%\"xGF+F+%$cosGF+F,F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "More briefly, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = exp(sin*x);" "6#/-%$f~'G6#% \"xG-%$expG6#*&%$sinG\"\"\"F'F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sin*x] \+ = exp(sin*x)*cos*x;" "6#/7#*&%$sinG\"\"\"%\"xGF'*(-%$expG6#*&F&F'F(F'F '%$cosGF'F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diff(exp(sin(x)),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$cosG6#%\"xG\"\"\"-%$expG6#-%$sinG F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " The following plot shows the graph of " }{XPPEDIT 18 0 "f(x) = exp(si n*x);" "6#/-%\"fG6#%\"xG-%$expG6#*&%$sinG\"\"\"F'F-" }{TEXT -1 4 " in \+ " }{TEXT 259 3 "red" }{TEXT -1 33 " and the graph of the derivative " }{XPPEDIT 18 0 "`f '`(x) = exp(sin*x)*cos*x;" "6#/-%$f~'G6#%\"xG*(-%$e xpG6#*&%$sinG\"\"\"F'F.F.%$cosGF.F'F." }{TEXT -1 5 " in " }{TEXT 256 4 "blue" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot([exp(sin(x)),cos(x)*exp(sin(x ))],x=-Pi..3*Pi,color=[red,blue],\nlegend=[\"function .. f(x)\",\"deri vative .. f'(x)\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 545 277 277 {PLOTDATA 2 "6&-%'CURVESG6%7dq7$$!3*****4tk#fTJ!#<$\"2w1-s$********F*7 $$!3yMay)3PY+$F*$\"3,8***fY?Qs)!#=7$$!3-q3EI:onGF*$\"3\\P3O]!))*HwF27$ $!3'z.'3Js^[FF*$\"3E-`Ogp\"z\"oF27$$!3*e?6>$HNHEF*$\"3MJ$4Y8ja7'F27$$! 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2w1-s$********F*7$F4$!3qpk(pWWbM(F27$F>$!3o%H%\\QMDR`F27$FC$!3()yLv7$Ffn$!3--nmW!R#)>\"Fjq 7$F`o$\"3#>9jYC(pl&*F_^m7$Fjo$\"3%p5&Hr1V\"4#F27$F_p$\"3QYxTnD:&\\$F27 $Fdp$\"3O$yw&3*p,5F*7$Fcr$ \"3S\\KY\"*GFh7F*7$Fhr$\"3'p_[:5;EO\"F*7$F]s$\"3H]kK3dgK9F*7$$\"3]W]PY r'Ql&F2$\"3^A\")*4T3GW\"F*7$$\"3;i(o0Cu:&fF2$\"33$=E:6*e]9F*7$$\"3#)zC wM8G\\iF2$\"3e(>3Ss.eX\"F*7$Fbs$\"3;*p]^g7$e9F*7$$\"39:*\\J_&pWoF2$\"3 _W_0R<)zX\"F*7$$\"3!GjVth-C9(F2$\"3cl#R*[Roa9F*7$$\"3Y]t`6(4,W(F2$\"3! z+\")[7+$[9F*7$Fgs$\"3qY8F*7$Fat$\"3MdkrQ') >u6F*7$Fft$\"3K\"4xK;PEm*F27$F[u$\"3;@SJ87n$3(F27$F`u$\"3U`$*)[s\"G(R& F27$Feu$\"3!Qr,$oK$)3OF27$F_v$\"3S3rZ7`7^o'R<%>CF* $!3w#G(o$y)G`9F*7$$\"3sc!etXxIX#F*$!3m+\"ob_5xX\"F*7$F`z$!3y*Q[wR=$e9F *7$$\"3/h-ryvR?DF*$!3mmBL!R&Hb9F*7$$\"3AjjQRw0aDF*$!3%o%Q9#oN)[9F*7$$ \"3QlC1+xr(e#F*$!3kNpW/C9R9F*7$Fez$!3fd\"))\\YDkU\"F*7$Fjz$!3#)ztSGjEd 8F*7$F_[l$!3'y!*onE:4E\"F*7$Fi[l$!3E*G$4;D?15F*7$Fc\\l$!3s9\\I+F]ttF27 $F]]l$!3;D^`Hf%3M&F27$Fb]l$!37+GLxtpjNF27$Fg]l$!3nM6\"Qm%QH@F27$Fa^l$! 3&yiBB=t8+\"F27$F[_l$!3%HPoBu+;7%Fjq7$Fe_l$\"3fnk?w\"[Q-\"F27$F_`l$\"3 V)yEr[+K6#F27$Fd`l$\"3/%p3:Gqkb$F27$Fi`l$\"3sVAZqrh?_F27$F^al$\"3+$**G 1qA?[(F27$Fhal$\"3,'f)*oB`)Q**F27$Fbbl$\"3'R]\")fdj\"[7F*7$Fgbl$\"33zm QA>*4N\"F*7$F\\cl$\"3aTm+8#[\\U\"F*7$$\"3(*om->'[P$oF*$\"3p8S\">!>-Q9F *7$$\"3m*)\\aEPdnoF*$\"3oa)4)F27$Fjdl$\"3'>'>6Tg!)pmF27$F_el$\"37RnN/-5[]F27$Fdel$\"3%y8 Xe1V/M$F27$F^fl$\"3e*yc5;k^d\"F27$Fhfl$!3uD?Bo)4\\<#F_^m7$Fbgl$!310;*3 )*f@'=F27$F\\hl$!3Mh>\\aE^zMF27$Fahl$!3I&H^p=1i/&F27$Ffhl$!3E=%4!p*>.a 'F27$$\"3/))Q\"*3IXu\")F*$!3J\"pr`l8z8)F27$F[il$!3;?;ano@)e*F27$$\"3nI \"*>pgj8$)F*$!3Wp;.]A/(3\"F*7$F`il$!3m9+Q\"\\Gp>\"F*7$Feil$!3o.h*\\hJ^ M\"F*7$Fjil$!3s'\\.eo\")4V\"F*7$$\"33Y%G&\\QQl')F*$!3n2[HucvV9F*7$$\"3 3x-0Bhc)p)F*$!3w&pu?@%\\_9F*7$$\"363@d'R[ \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 2" }}{PARA 0 "" 0 "" {TEXT -1 8 "We find " }{XPPEDIT 18 0 "`f '`(x )" "6#-%$f~'G6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = exp(-x/3)*cos*2*x;" "6#/-%\"fG6#%\"xG**-%$expG6#,$*&F'\"\"\" \"\"$!\"\"F0F.%$cosGF.\"\"#F.F'F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This function can be di fferentiated using the product rule." }}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{XPPEDIT 18 0 "u(x) = exp(-x/3);" "6#/-%\"uG6#%\"xG-%$expG6#,$*&F '\"\"\"\"\"$!\"\"F/" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v(x) = cos* 2*x;" "6#/-%\"vG6#%\"xG*(%$cosG\"\"\"\"\"#F*F'F*" }{TEXT -1 10 ", so t hat " }{XPPEDIT 18 0 "f(x) = u(x)*v(x);" "6#/-%\"fG6#%\"xG*&-%\"uG6#F' \"\"\"-%\"vG6#F'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We h ave " }{XPPEDIT 18 0 "`u '`(x) = -1/3;" "6#/-%$u~'G6#%\"xG,$*&\"\"\"F* \"\"$!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x/3)" "6#-%$expG6#, $*&%\"xG\"\"\"\"\"$!\"\"F+" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "`v '` (x) = -2*sin*2*x;" "6#/-%$v~'G6#%\"xG,$**\"\"#\"\"\"%$sinGF+F*F+F'F+! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Then, by the prod uct rule, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x )=`u '`(x)*v(x)+u(x)*`v '`(x)" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\"\" -%\"vG6#F'F-F-*&-%\"uG6#F'F--%$v~'G6#F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/3;" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-x/3)*cos*2*x+exp(-x/3)*(-2*sin*2*x);" "6#,&**-%$ex pG6#,$*&%\"xG\"\"\"\"\"$!\"\"F-F+%$cosGF+\"\"#F+F*F+F+*&-F&6#,$*&F*F+F ,F-F-F+,$**F/F+%$sinGF+F/F+F*F+F-F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =-1/3" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x/3)*cos*2*x-2*exp(-x/3)*sin*2*x;" "6#,&**-%$expG6#,$*&%\"xG\" \"\"\"\"$!\"\"F-F+%$cosGF+\"\"#F+F*F+F+*,F/F+-F&6#,$*&F*F+F,F-F-F+%$si nGF+F/F+F*F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Diff(exp(-x/3)*cos(x),x)=dif f(exp(-x/3)*cos(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$* &-%$expG6#,$*&\"\"$!\"\"%\"xG\"\"\"F.F0-%$cosG6#F/F0F/,&*&#F0F-F0F'F0F .*&F(F0-%$sinGF3F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 39 "The following plot shows the graph of \+ " }{XPPEDIT 18 0 "f(x) = exp(-x/3)*cos*2*x;" "6#/-%\"fG6#%\"xG**-%$exp G6#,$*&F'\"\"\"\"\"$!\"\"F0F.%$cosGF.\"\"#F.F'F." }{TEXT -1 4 " in " } {TEXT 259 3 "red" }{TEXT -1 33 " and the graph of the derivative " } {XPPEDIT 18 0 "`f '`(x) = -1/3;" "6#/-%$f~'G6#%\"xG,$*&\"\"\"F*\"\"$! \"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x/3)*cos*2*x-2*exp(-x/3)* sin*2*x;" "6#,&**-%$expG6#,$*&%\"xG\"\"\"\"\"$!\"\"F-F+%$cosGF+\"\"#F+ F*F+F+*,F/F+-F&6#,$*&F*F+F,F-F-F+%$sinGF+F/F+F*F+F-" }{TEXT -1 5 " in " }{TEXT 256 4 "blue" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "plot([exp(-x/3)*cos(2*x ),-1/3*exp(-1/3*x)*cos(2*x)-2*exp(-1/3*x)*sin(2*x)],x=-.1..10,\n col or=[red,blue],legend=[\"function .. f(x)\",\"derivative .. f'(x)\"]); " }}{PARA 13 "" 1 "" {GLPLOT2D 489 342 342 {PLOTDATA 2 "6&-%'CURVESG6% 7bq7$$!3/+++++++5!#=$\"3hI![d/'G85!#<7$$!3kM$ekG4\"[s!#>$\"3PoR^k%4P, \"F-7$$!3Knm\"Hd=i\\%F1$\"3pBvw%y)*4,\"F-7$$!3r+]PfyKW_+ \"F-7$$\"3\"fmmT&Gc25F1$\"35U()yVw \\N,7X)F*7$$\"3]m;H()o.%F*$\"3c[G?!Gg8\"eF*7$$\"37dmr #>*\\F*7$$\"3iLLe>VCr_F*$\"3\")R-+)*Q)H9%F*7$$\"3(QLe*[gO8eF*$\"3KzFfi lrpKF*7$$\"37MLLyx[bjF*$\"3W\\M#*)G&o)Q#F*7$$\"3OM$3x]4w*oF*$\"3g3WSU+ h5:F*7$$\"3iML3P7tRuF*$\"3'Rd/:G_zX'F17$$\"3WF1 7$$\"3G+v$fZ@)=&)F*$!3'HN&>U\"3.)**F17$$\"37$ek`fm$e!*F*$!3cgxZ[%>Qw\" F*7$$\"3#fm\"z9<\"zf*F*$!3)pUQb;R=[#F*7$$\"3#)*\\P*\\m$)f5F-$!3Aj[\")* HIXm$F*7$$\"3;LeRG@))f6F-$!3pNl\"etSbi%F*7$$\"3]m\"zkUtME\"F-$!3o;SI: \\rh`F*7$$\"3%)*\\iXskqO\"F-$!3(pK[&3N/@eF*7$$\"3cm\"H<'ff9\\fF*7$$\"3%)*\\igU:uW\"F-$!3_zXt J:e&)fF*7$$\"3gm\"HK**)>u9F-$!3xrH)H\"4)Q+'F*7$$\"3LLeRgD)4]\"F-$!3lU+ _)e6V+'F*7$$\"3&)*\\iv7mx_\"F-$!3#4\"piE&)=()fF*7$$\"3gm\"HZp\\Xb\"F-$ !3'p(3(zTsG&fF*7$$\"3MLe*=EL8e\"F-$!3\"*))fT7yw,fF*7$$\"3WmTg\\R7)o\"F -$!3EIvB&GC0a&F*7$$\"3y*\\7tj9\\z\"F-$!3cr_sM8Na\\F*7$$\"3X$eke)>w/>F- $!33zh(H0y3;%F*7$$\"3!pm;WL4Y,#F-$!3#**RI.$oADKF*7$$\"3P+D\")QHO6@F-$! 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*evxj&p\"F*7$$\"3dL$ezE-Tu'F-$!3+W(yc43b*=F*7$Fdel$!3S[c-p7O#*>F*7$$\" 3S;H#=$3&>&pF-$!3y'[jf/.L*>F*7$Fiel$!3%Q7,EUF*7$$\"3c;HdH4yerF-$! 3A:gXq`gadQC%)F-$\"3M$=i(*> b.=\"F*7$F\\il$\"35I!f-@!fy6F*7$$\"37](=(H:rP')F-$\"3NIRqT0 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 289 8 "Question" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 63 "Find the coordinates of the station ary points on the graph of " }{XPPEDIT 18 0 "y=x^2*exp(-x)" "6#/%\"yG *&%\"xG\"\"#-%$expG6#,$F&!\"\"\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Give n " }{XPPEDIT 18 0 "y=x^2*exp(-x)" "6#/%\"yG*&%\"xG\"\"#-%$expG6#,$F&! \"\"\"\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=2*x*exp(-x)+x^2*``(-exp(x))" "6#/*&%#dyG\"\"\"%#d xG!\"\",&*(\"\"#F&%\"xGF&-%$expG6#,$F,F(F&F&*&F,F+-%!G6#,$-F.6#F,F(F&F &" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=2*x*exp(-x)-x^2*exp(-x)" "6#/%!G,&*(\"\"#\"\"\"%\"xGF(-%$expG6#,$F) !\"\"F(F(*&F)F'-F+6#,$F)F.F(F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(-x)*(2*x-x^2)" "6#/%!G*&-%$expG6 #,$%\"xG!\"\"\"\"\",&*&\"\"#F,F*F,F,*$F*F/F+F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 34 "The stationary points occur where " } {XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "e xp(-x)*(2*x-x^2) =0" "6#/*&-%$expG6#,$%\"xG!\"\"\"\"\",&*&\"\"#F+F)F+F +*$F)F.F*F+\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 " Sinc e " }{XPPEDIT 18 0 "exp(-x)" "6#-%$expG6#,$%\"xG!\"\"" }{TEXT -1 31 " \+ is positive for all values of " }{TEXT 291 1 "x" }{TEXT -1 13 ", we ob tain: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x*(2-x)=0 " "6#/*&%\"xG\"\"\",&\"\"#F&F%!\"\"F&\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 7 "giving " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 38 "The corresponding points on the curve " }{XPPEDIT 18 0 "y=x^2*exp(-x)" "6#/%\"yG*&%\"xG\"\"#-%$expG6#,$F&! \"\"\"\"\"" }{TEXT -1 20 " given by these two " }{TEXT 292 1 "x" } {TEXT -1 11 " values are" }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(2,4*exp(-2))" "6#-%!G6$\"\"#*&\" \"%\"\"\"-%$expG6#,$F&!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "looking at a graph we se that" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 25 " is a minimum point while" }{XPPEDIT 18 0 " ``(2,4*exp(-2))" "6#-%!G6$\"\"#*&\"\"%\"\"\"-%$expG6#,$F&!\"\"F )" }{TEXT -1 21 " is a maximum point. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "4*exp(-2)" "6#*&\"\"%\"\"\"-%$ expG6#,$\"\"#!\"\"F%" }{TEXT -1 1 " " }{TEXT 293 1 "~" }{TEXT -1 11 " \+ 0.54134. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "f := x -> x^2*exp(-x):\n'f(x)'=f(x);\nDiff(f(x),x)= diff(f(x),x);\nsolve(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" fG6#%\"xG*&)F'\"\"#\"\"\"-%$expG6#,$F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&)%\"xG\"\"#\"\"\"-%$expG6#,$F)!\"\"F+F),&* (F*F+F)F+F,F+F+F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "f := x -> x^2*exp(-x):\n'f( x)'=f(x);\np1 := plot(f(x),x=-1..7,y=0..1):\np2 := plot([[[0,0],[2,4*e xp(-2)]]$3],style=point,\n symbol=[circle,diamond,cross],colo r=black):\nt1 := plots[textplot]([[3.4,.65,`maximum point`],[-1.7,.14, `minimum point`]]):\nplots[display]([p1,p2,t1],view=[-2..7,0..1]); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&)F'\"\"#\"\"\"-%$e xpG6#,$F'!\"\"F+" }}{PARA 13 "" 1 "" {GLPLOT2D 574 240 240 {PLOTDATA 2 "6*-%'CURVESG6$7ao7$$!\"\"\"\"!$\"34X!f%G=G=F!#<7$$!3!RLL$e%G?y*!#=$ \"3KLgL!>*)\\a#F-7$$!3ommm;p0k&*F1$\"3esKK'\\y.Q#F-7$$!3[*****\\P&3Y$* F1$\"3a?w9.E5CAF-7$$!3PLLLLQ6G\"*F1$\"3+l6q]k#e2#F-7$$!3Fnmm\"HU,\"*)F 1$\"3HD.exmAN>F-7$$!3/+++]2<#p)F1$\"3v157[7*>!=F-7$$!3%GLL$3#*>u%)F1$ \"3Kp7g!4>en\"F-7$$!3ummmmwAc#)F1$\"3#[0#HY(>kb\"F-7$$!3#RLL3#f\"p(yF1 $\"3-.j%y\"G'RO\"F-7$$!3)******\\7$$!37MLL$[9cg\"F1$\"3jWT6cv+FIF^q7$$!3SnmmmZR47F1$\"3=5 ho$=n1l\"F^q7$$!3x1+++0vJ\")F^q$\"3cy$>`F>F<(!#?7$$!3cRLLLLbpTF^q$\"3R u\"y.OQD\"=F^r7$$!3QBnmm;ct?F^r$\"3aX)p8%Ge3V!#B7$$\"3i#******\\p_*QF^ q$\"3K=>Lb`Mf9F^r7$$\"3edmmm^*y*zF^q$\"3CfNU/0'\\!fF^r7$$\"3DKLL$3_+@ \"F1$\"3)>&f!GCZtH\"F^q7$$\"3v)*****\\YJ?;F1$\"3@r7WCZpKAF^q7$$\"3Slmm ;H!*oCF1$\"3?uF\"38^>w%F^q7$$\"3/KLL$=\"\\#eY)*yF^q7$$\"3 1)*****\\[A4]F1$\"32d>ldO_?:F17$$\"3qnmmm3Q\\nF1$\"31]!p++i&>BF17$$\"3 aLLLL76#G)F1$\"3%o`'e%=kj*HF17$$\"3&)******f-w+5F-$\"3uA=_)=!f\"o$F17$ $\"3%*********y,u6F-$\"3!pgD_/!pgUF17$$\"3)*******RP)4M\"F-$\"31aH#=&p (Rq%F17$$\"3ILLL=Zg#\\\"F-$\"3y$zqn)e%z+&F17$$\"3cmmmEn*Gn\"F-$\"33(GY rj7ID&F17$$\"3Tmmm1xiD=F-$\"3!QTtNtx)p`F17$$\"3!)*****\\9!H.?F-$\"3&z$ QT&p'R8aF17$$\"3Immm1:bg@F-$\"3-(o]/p)Q!Q&F17$$\"3<+++X@4LBF-$\"3k@ARu nwz_F17$$\"31+++N;R(\\#F-$\"37NA;mQ)H8&F17$$\"3wmmm;4#)oEF-$\"3b!)yZJZF17$$\"3ELLL$G^g*HF-$\"3=#)[22,t '[%F17$$\"3oKLL=2VsJF-$\"3iJd\"H-7r@%F17$$\"3f*****\\`pfK$F-$\"3eXIO-- XvRF17$$\"3!HLLLm&z\"\\$F-$\"3+)*[6#y)=7PF17$$\"3s******z-6jOF-$\"3gT4 m#zi@W$F17$$\"3<******4#32$QF-$\"3)zSTXE%[$=$F17$$\"3O*****\\#y'G*RF-$ \"3$Gg'4W>'4%HF17$$\"3G******H%=H<%F-$\"3?R6m@\"**Go#F17$$\"35mmm1>qMV F-$\"3/Dn$o+HDY#F17$$\"3%)*******HSu]%F-$\"3RKezniGSAF17$$\"3'HLL$ep'R m%F-$\"3%os(ziz/^?F17$$\"3')******R>4N[F-$\"35t!yU1:w&=F17$$\"3#emm;@2 h*\\F-$\"3/p\\MMZU)o\"F17$$\"3]*****\\c9W;&F-$\"32c#Q,YLY_\"F17$$\"3Lm mmmd'*G`F-$\"3aaQ5Vk.x8F17$$\"3j*****\\iN7]&F-$\"3fVjN4mFN7F17$$\"3aLL Lt>:ncF-$\"37WkJ5yVm5(F^q7$$\"3/LLL8-V&\\'F-$\"3gZe$4 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT 285 8 "Question" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = x^x;" "6#/%\"yG)%\"xG F&" }{TEXT -1 6 " find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG! \"\"" }{TEXT -1 67 " and find the coordinates of the stationary point \+ on the graph of " }{XPPEDIT 18 0 "y=x^x" "6#/%\"yG)%\"xGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 286 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 8 "Because " }{XPPEDIT 18 0 "exp(ln*x) = x;" "6#/-%$expG6#*&%#lnG\"\"\"%\"xGF)F*" }{TEXT -1 26 " for all positive n umbers " }{TEXT 282 1 "x" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y=x^x" "6# /%\"yG)%\"xGF&" }{XPPEDIT 18 0 "`` = exp(ln*x)^x;" "6#/%!G)-%$expG6#*& %#lnG\"\"\"%\"xGF+F," }{XPPEDIT 18 0 "`` = exp(x*ln*x);" "6#/%!G-%$exp G6#*(%\"xG\"\"\"%#lnGF*F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "Now we can find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG! \"\"" }{TEXT -1 20 " by the chain rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = exp(x*ln*x);" "6#/*&%#dyG\"\"\"%#dxG! \"\"-%$expG6#*(%\"xGF&%#lnGF&F-F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/d x" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x*ln*x ];" "6#7#*(%\"xG\"\"\"%#lnGF&F%F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(x*ln*x)*(1*`.`*ln*x+x*`.`*`` (1/x));" "6#/%!G*&-%$expG6#*(%\"xG\"\"\"%#lnGF+F*F+F+,&**F+F+%\".GF+F, F+F*F+F+*(F*F+F/F+-F$6#*&F+F+F*!\"\"F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "(using the product rule) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(x*ln*x)*(ln*x+1);" "6#/%!G*&-% $expG6#*(%\"xG\"\"\"%#lnGF+F*F+F+,&*&F,F+F*F+F+F+F+F+" }{TEXT -1 2 ". \+ " }}{PARA 258 "" 0 "" {TEXT -1 7 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = x^x*(ln*x+1);" "6#/*&%#dyG\"\"\"%#dxG !\"\"*&)%\"xGF+F&,&*&%#lnGF&F+F&F&F&F&F&" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "At the minimum poi nt " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 8 ". Since " }{XPPEDIT 18 0 "x^x = exp(x*ln*x);" "6#/)%\"xGF% -%$expG6#*(F%\"\"\"%#lnGF*F%F*" }{TEXT -1 18 ", we can see that " } {XPPEDIT 18 0 "x^x" "6#)%\"xGF$" }{TEXT -1 22 " is never zero. Hence \+ " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 11 " only when " }{XPPEDIT 18 0 "ln*x+1 = 0;" "6#/,&*&%#lnG\"\"\"% \"xGF'F'F'F'\"\"!" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "ln*x = - 1;" "6#/*&%#lnG\"\"\"%\"xGF&,$F&!\"\"" }{TEXT -1 20 ". This happens wh en " }{XPPEDIT 18 0 "x=exp(-1)" "6#/%\"xG-%$expG6#,$\"\"\"!\"\"" } {TEXT -1 21 ". The corresponding " }{TEXT 284 1 "y" }{TEXT -1 38 " co ordinate of the point on the curve " }{XPPEDIT 18 0 "y=x^x" "6#/%\"yG) %\"xGF&" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x=exp(-1)" "6#/%\"xG-%$ expG6#,$\"\"\"!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "exp(-exp(-1)) \+ = exp(-1/exp(1));" "6#/-%$expG6#,$-F%6#,$\"\"\"!\"\"F,-F%6#,$*&F+F+-F% 6#F+F,F," }{TEXT -1 1 " " }{TEXT 283 1 "~" }{TEXT -1 15 " 0.6922006276 . " }}{PARA 0 "" 0 "" {TEXT -1 23 "The stationary point is" }{XPPEDIT 18 0 "``(1/exp(1),exp(-1/exp(1)))" "6#-%!G6$*&\"\"\"F'-%$expG6#F'!\"\" -F)6#,$*&F'F'-F)6#F'F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "Looking at the graph of " }{XPPEDIT 18 0 "y=x^x" "6#/%\"yG)%\"xGF &" }{TEXT -1 45 ", we see that this point is a minimum point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "p1 := plot(x^x,x=0..1.8,y=0..3):\np2 := plot([[[exp(-1),exp(-exp( -1))]]$3],style=point,\n symbol=[circle,diamond,cross],c olor=black):\nt1 := plots[textplot]([.7,.5,`minimum point`],color=blac k):\nplots[display]([p1,p2,t1],tickmarks=[4,4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 364 364 364 {PLOTDATA 2 "6*-%'CURVESG6$7gn7$$\"\"!F)$\"\"\"% *undefinedG7$$\"3#)**\\(=#**3E7!#?$\"3svPpO29=**!#=7$$\"3k***\\P%)z@X# F0$\"3))\\[fjeo`)*F37$$\"3o**\\il(p#yOF0$\"3e]cE?C$fz*F37$$\"3F****\\( ofV!\\F0$\"3&Qrt7hvDu*F37$$\"3M***\\7`RlN(F0$\"3swOw)*e3X'*F37$$\"3`)* ***\\P>(3)*F0$\"3'z%[8[8`c&*F37$$\"3))***\\i!zIr9!#>$\"3ani!HJE\")R*F3 7$$\"3s*****\\(Quh>FP$\"3A%R(yq?wd#*F37$$\"3u****\\7ehUHFP$\"3'4C2(o1[ 9!*F37$$\"3U******\\x[BRFP$\"3)>`>Sv'*o!))F37$$\"3@***\\ig!RIcFP$\"3Br WdpB`/&)F37$$\"3q****\\iMHPtFP$\"3/F>E`c$eD)F37$$\"3.++]-vk<6F3$\"3h<5 $pT(pFyF37$$\"3(*****\\<06/:F3$\"3$[/I.OA1_(F37$$\"3u***\\7uO())=F3$\" 3i\\vMY8T*H(F37$$\"3#****\\i[M`C#F3$\"3ka:b[Qd]rF37$$\"3x***\\izqXh#F3 $\"3!e&zH\"fi;/(F37$$\"3-++D;bV'*HF3$\"3m)Gy3;h*opF37$$\"3+++D\"fvqP$F 3$\"3#y6.0h?3$pF37$$\"3M+++&p5'oPF3$\"3(*>r*H:fF#pF37$$\"3#)****\\F]Z8 TF3$\"3W\\)4R3U\"RpF37$$\"3*******\\e5<]%F3$\"3E3PKZ3p\")pF37$$\"3w*** **\\FS:*[F3$\"31)*)['Q(3%[qF37$$\"3]+++:M@n_F3$\"3$o!f2j%*GMrF37$$\"3$ ****\\ihg$3cF3$\"3H1[G!z)F37$$\"3Q****\\(Q:6**)F3 $\"3?N;XxJ5)3*F37$$\"3'*)**\\i6pzQ*F3$\"3O)RwH6FVU*F37$$\"3&))**\\PXJM t*F3$\"3B]Rr%HW/u*F37$$\"3&****\\U-a1,\"!#<$\"3T1([99o2,\"Faw7$$\"3)** ****H\")*>\\5Faw$\"3-kF))f>o^5Faw7$$\"3)****\\sM4p3\"Faw$\"3/\\Uin-\"[ 4\"Faw7$$\"3&***\\ig_RB6Faw$\"3I&G\"\\55kR6Faw7$$\"3))***\\nk1R;\"Faw$ \"3)*4EvBcA$>\"Faw7$$\"3$)******GzI+7Faw$\"3tL0$*))z,X7Faw7$$\"3.++]nS FHW 4,]\"Faw$\"3]+,Sv#*RP=Faw7$$\"31++v:dGQ:Faw$\"3;p0ft'='R>Faw7$$\"39+]i ;h9w:Faw$\"3oc>v]L_[?Faw7$$\"3&******H*e$4h\"Faw$\"3_,dlItqb@Faw7$$\"3 #****\\F!*33l\"Faw$\"3ytvV;vd(G#Faw7$$\"3!******zzrko\"Faw$\"3sDT6(Q9V T#Faw7$$\"34+]i#)e\\C " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 23 ": Another way to find " }{XPPEDIT 18 0 "d/dx" "6#*&% \"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^x]" "6#7#)%\" xGF%" }{TEXT -1 16 " is as follows. " }}{PARA 256 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "f(x)=x^x" "6#/-%\"fG6#%\"xG)F'F'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 51 "Then take natural logarithms of eac h side to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "l n(f(x))=ln(x^x)" "6#/-%#lnG6#-%\"fG6#%\"xG-F%6#)F*F*" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "ln(f(x)) = x*ln*x;" "6#/-%#lnG6#-%\"fG6#%\"xG*(F*\"\"\" F%F,F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 70 "Then we can \+ differentiate both sides of this equation with respect to " }{TEXT 305 1 "x" }{TEXT -1 12 " to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/f(x)" "6#*&\"\"\"F$-%\"fG6#%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 1*`.`*ln*x+x*`.`*``(1/x);" "6#/-%$f~' G6#%\"xG,&**\"\"\"F*%\".GF*%#lnGF*F'F*F**(F'F*F+F*-%!G6#*&F*F*F'!\"\"F *F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/f(x)" "6#*&\"\"\"F$-%\"fG 6#%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = ln*x+1;" "6#/ -%$f~'G6#%\"xG,&*&%#lnG\"\"\"F'F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = f(x)*(ln*x+1);" "6#/-%$f~'G6#%\"xG*&-%\"fG6# F'\"\"\",&*&%#lnGF,F'F,F,F,F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x^x*(ln*x+1);" "6#/%!G*&)%\"xGF'\" \"\",&*&%#lnGF(F'F(F(F(F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "Diff(x^x,x)=diff(x^x,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$)%\"xGF(F(*&F'\"\"\",&-%#lnG6#F(F*F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }} {PARA 0 "" 0 "" {TEXT 313 8 "Question" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 11 "Show that " }{XPPEDIT 18 0 "y = 1/(1+exp(x));" "6#/% \"yG*&\"\"\"F&,&F&F&-%$expG6#%\"xGF&!\"\"" }{TEXT -1 45 " is a solutio n of the differential equation " }{XPPEDIT 18 0 "dy/dx = y^2-y;" "6#/ *&%#dyG\"\"\"%#dxG!\"\",&*$%\"yG\"\"#F&F+F(" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT 314 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = 1/(1+exp(x));" "6#/%\"yG* &\"\"\"F&,&F&F&-%$expG6#%\"xGF&!\"\"" }{XPPEDIT 18 0 "`` = (1+exp(x))^ (-1);" "6#/%!G),&\"\"\"F'-%$expG6#%\"xGF',$F'!\"\"" }{TEXT -1 2 ", " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (-1)*(1+exp( x))^(-2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,$F&F(F&),&F&F&-%$expG6#%\"xGF &,$\"\"#F(F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%# dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1+exp(x)];" "6#7#,&\"\"\"F% -%$expG6#%\"xGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -1/((1+exp(x))^2);" "6#/%!G,$*&\"\"\"F'*$,&F'F'-%$ expG6#%\"xGF'\"\"#!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp(x)); " "6#-%!G6#-%$expG6#%\"xG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -exp(x)/((1+exp(x))^2);" "6#/%!G,$*&-%$e xpG6#%\"xG\"\"\"*$,&F+F+-F(6#F*F+\"\"#!\"\"F1" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "On the other hand, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2-y=1/(1+exp(x))^2-1/(1+exp(x))" "6#/,&*$%\"yG\"\"#\"\"\"F&!\"\", &*&F(F(*$,&F(F(-%$expG6#%\"xGF(F'F)F(*&F(F(,&F(F(-F/6#F1F(F)F)" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=( 1-(1+exp(x))/(1+exp(x))^2" "6#/%!G,&\"\"\"F&*&,&F&F&-%$expG6#%\"xGF&F& *$,&F&F&-F*6#F,F&\"\"#!\"\"F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-exp(x)/((1+exp(x))^2)" "6#/%!G,$*&- %$expG6#%\"xG\"\"\"*$,&F+F+-F(6#F*F+\"\"#!\"\"F1" }{TEXT -1 15 " ----- -- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Since the expressions (i) and (ii) are the same, this shows that \+ " }{XPPEDIT 18 0 "dy/dx=y^2-y" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*$%\"yG\" \"#F&F+F(" }{TEXT -1 15 ", as required. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 \+ " }}{PARA 0 "" 0 "" {TEXT 287 8 "Question" }{TEXT -1 1 ":" }}{PARA 0 " " 0 "" {TEXT -1 11 "Show that " }{XPPEDIT 18 0 "y=x^3*exp(-x)" "6#/% \"yG*&%\"xG\"\"$-%$expG6#,$F&!\"\"\"\"\"" }{TEXT -1 57 " is a solution of the second order differential equation " }{XPPEDIT 18 0 "d^2*y/(dx ^2)+2" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*$%#dxGF&!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = 6*x*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'%\"yGF'*(\"\"'F'%\"xGF'-%$expG6#,$F-F)F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "d^2*y/(dx^2)=d/dx " "6#/*(%\"dG\"\"#%\"yG\"\"\"*$%#dxGF&!\"\"*&F%F(F*F+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "[dy/dx]" "6#7#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 30 " is the second derivative of " }{TEXT 315 1 "y" }{TEXT -1 17 " with \+ respect to " }{TEXT 316 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "Given " }{XPPEDIT 18 0 "y = x^3*exp (-x);" "6#/%\"yG*&%\"xG\"\"$-%$expG6#,$F&!\"\"\"\"\"" }{TEXT -1 2 ", \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 3*x^2*`.` *exp(-x)+x^3*`.`*(-exp(-x));" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"\"$F&*$ %\"xG\"\"#F&%\".GF&-%$expG6#,$F-F(F&F&*(F-F+F/F&,$-F16#,$F-F(F(F&F&" } {TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 24 "( by the product rule ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=3*x^2*exp(- x)-x^3*exp(-x)" "6#/%!G,&*(\"\"$\"\"\"*$%\"xG\"\"#F(-%$expG6#,$F*!\"\" F(F(*&F*F'-F-6#,$F*F0F(F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(3*x^2-x^3)*exp(-x)" "6#/%!G*&,&*&\"\"$\" \"\"*$%\"xG\"\"#F)F)*$F+F(!\"\"F)-%$expG6#,$F+F.F)" }{TEXT -1 14 " --- ---- (i). " }}{PARA 258 "" 0 "" {TEXT -1 6 "Then " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(dx^2)=d/dx" "6#/*(%\"dG\"\"#% \"yG\"\"\"*$%#dxGF&!\"\"*&F%F(F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[d y/dx] = (6*x-3*x^2)*`.`*exp(-x)+(3*x^2-x^3)*`.`*(-exp(-x));" "6#/7#*&% #dyG\"\"\"%#dxG!\"\",&*(,&*&\"\"'F'%\"xGF'F'*&\"\"$F'*$F/\"\"#F'F)F'% \".GF'-%$expG6#,$F/F)F'F'*(,&*&F1F'*$F/F3F'F'*$F/F1F)F'F4F',$-F66#,$F/ F)F)F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "( by the prod uct rule ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ( 6*x-3*x^2)*exp(-x)+(-3*x^2+x^3)*exp(-x);" "6#/%!G,&*&,&*&\"\"'\"\"\"% \"xGF*F**&\"\"$F**$F+\"\"#F*!\"\"F*-%$expG6#,$F+F0F*F**&,&*&F-F**$F+F/ F*F0*$F+F-F*F*-F26#,$F+F0F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (6*x-6*x^2+x^3)*exp(-x);" "6#/%!G* &,(*&\"\"'\"\"\"%\"xGF)F)*&F(F)*$F*\"\"#F)!\"\"*$F*\"\"$F)F)-%$expG6#, $F*F.F)" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*y/(dx^2)" "6#*(%\"dG\"\"#%\"yG \"\"\"*$%#dxGF%!\"\"" }{TEXT -1 12 " from (ii), " }{XPPEDIT 18 0 "dy/d x" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 14 " from (i) and " }{TEXT 317 1 "y" }{TEXT -1 54 " in the left side of the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(6*x-6*x^ 2+x^3)*exp(-x)+2*(3*x^2-x^3)*exp(-x)+x^3*exp(-x)" "6#,(*&,(*&\"\"'\"\" \"%\"xGF(F(*&F'F(*$F)\"\"#F(!\"\"*$F)\"\"$F(F(-%$expG6#,$F)F-F(F(*(F,F (,&*&F/F(*$F)F,F(F(*$F)F/F-F(-F16#,$F)F-F(F(*&F)F/-F16#,$F)F-F(F(" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=( 6*x-6*x^2+x^3+6*x^2-2*x^3+x^3)*exp(-x)" "6#/%!G*&,.*&\"\"'\"\"\"%\"xGF )F)*&F(F)*$F*\"\"#F)!\"\"*$F*\"\"$F)*&F(F)*$F*F-F)F)*&F-F)*$F*F0F)F.*$ F*F0F)F)-%$expG6#,$F*F.F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=6*x*exp(-x)" "6#/%!G*(\"\"'\"\"\"%\"xGF'-% $expG6#,$F(!\"\"F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 54 "wh ich is the right side of the differential equation. " }}{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 30 "General exponential functions " }}{PARA 0 "" 0 "" {TEXT -1 30 "To find the derivative of the " }{TEXT 260 28 "general ex ponential function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=a^x" "6#/-%\" fG6#%\"xG)%\"aGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a>0" "6#2\" \"!%\"aG" }{TEXT -1 8 ", write " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 14 " in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)=exp(ln(a^x))" "6#/-%\"fG6#%\"xG-%$expG6#-%#lnG6#)% \"aGF'" }{XPPEDIT 18 0 "``=exp(x*ln*a)" "6#/%!G-%$expG6#*(%\"xG\"\"\"% #lnGF*%\"aGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=a^x*ln*a" "6 #/-%$f~'G6#%\"xG*()%\"aGF'\"\"\"%#lnGF+F*F+" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 320 8 "________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 15 "Alternatively, " }{XPPEDIT 18 0 "f(x)=a^x " "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 11 " means that" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(f(x))=ln(a^x)" "6#/-%#lnG6#-% \"fG6#%\"xG-F%6#)%\"aGF*" }{XPPEDIT 18 0 "``=x*ln*a" "6#/%!G*(%\"xG\" \"\"%#lnGF'%\"aGF'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "so \+ that" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1/f(x)" "6#*&\"\"\"F$-%\"fG6# %\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=ln*a" "6#/-%$f~'G 6#%\"xG*&%#lnG\"\"\"%\"aGF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x )=f(x)*ln*a" "6#/-%$f~'G6#%\"xG*(-%\"fG6#F'\"\"\"%#lnGF,%\"aGF," } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=a^x*ln*a" "6#/-%$f~'G6# %\"xG*()%\"aGF'\"\"\"%#lnGF+F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture, the graph of " }{XPPEDIT 18 0 "y=2^x" "6#/%\"yG)\"\"#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 15 ", the graph of " }{XPPEDIT 18 0 "y=3^x" "6#/%\"yG)\"\"$%\"xG" }{TEXT -1 13 " is draw n in " }{TEXT 338 7 "magenta" }{TEXT -1 19 ", and the graph of " } {XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 13 " is dr awn in " }{TEXT 259 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "f := x -> 2^x: 'f (x)'=f(x);\ng := x -> 3^x: 'g(x)'=g(x);\nplot([f(x),exp(x),g(x)],x=-2. .2,y=0..5,color=[blue,red,magenta]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG)\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"g G6#%\"xG)\"\"$F'" }}{PARA 13 "" 1 "" {GLPLOT2D 477 421 421 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"#\"\"!$\"3++++++++D!#=7$$!3MLLL$Q6G\">!#<$\"3 sy?I`^ubEF-7$$!3bmm;M!\\p$=F1$\"3qh**H/68*z#F-7$$!3MLLL))Qj^'***F-$\"3\\0iw.wJ,]F-7$$!3E++++0\"*H\"*F-$\"3QR**)*y)G3J&F-7$ $!35++++83&H)F-$\"3ofo2G1@FcF-7$$!3\\LLL3k(p`(F-$\"3#)opbV`\"3$fF-7$$! 3Anmmmj^NmF-$\"3pM([+*=A8jF-7$$!3)zmmmYh=(eF-$\"3Y[/Jb\"*RcmF-7$$!3+,+ +v#\\N)\\F-$\"3!>&G?*QN\"zqF-7$$!3commmCC(>%F-$\"3%>)Ha*)\\nvuF-7$$!39 *****\\FRXL$F-$\"3v#G:s$=MOzF-7$$!3t*****\\#=/8DF-$\"3.6'RV'fO,%)F-7$$ !3=mmm;a*el\"F-$\"3\"ylH(p6k:*)F-7$$!3komm;Wn(o)!#>$\"3h3RcE!*e:%*F-7$ $!3IqLLL$eV(>!#?$\"3QBb&3:Cj)**F-7$$\"3)Qjmm\"f`@')Fjr$\"3SS=Ol;eh5F17 $$\"3%z****\\nZ)H;F-$\"3G\\dVn5g>6F17$$\"3ckmm;$y*eCF-$\"3Weml=0$e=\"F 17$$\"3f)******R^bJ$F-$\"3ib!=74p$e7F17$$\"3'e*****\\5a`TF-$\"3m%yf7(f iL8F17$$\"3'o****\\7RV'\\F-$\"3qbJ'G>A2T\"F17$$\"3Y'*****\\@fkeF-$\"3# *o$>C(pb,:F17$$\"3_ILLL&4Nn'F-$\"3w>)HH:a\")e\"F17$$\"3A*******\\,s`(F -$\"3/SA%p6Nho\"F17$$\"3%[mm;zM)>$)F-$\"3I6.X\"3J,y\"F17$$\"3M*******p fa<*F-$\"3GJ/gL&**)))=F17$$\"39HLLeg`!)**F-$\"31'H'4WNI(*>F17$$\"3w*** *\\#G2A3\"F1$\"3S$H?>(HF<@F17$$\"3;LLL$)G[k6F1$\"3WU@7j&G:C#F17$$\"3#) ****\\7yh]7F1$\"3O%\\>$oHVzBF17$$\"3xmmm')fdL8F1$\"3P*H]*efE?DF17$$\"3 bmmm,FT=9F1$\"3%QaL)oE\"Hn#F17$$\"3FLL$e#pa-:F1$\"3!>y\"[C[ULGF17$$\"3 !*******Rv&)z:F1$\"3]:G$=H.%*)HF17$$\"3ILLLGUYo;F1$\"3mohu#Hg(yJF17$$ \"3_mmm1^rZF^zG3c$F17$$\"34 ++]2%)38>F1$\"3QMF@QY9mPF17$$\"\"#F*$\"\"%F*-%'COLOURG6&%$RGBG$F*F*F_[ l$\"*++++\"!\")-F$6$7S7$F($\"3-FhOKGN`8F-7$F/$\"3mL4@0mkw9F-7$F5$\"3q( zuc7FIf\"F-7$F:$\"33E,e![-\\t\"F-7$F?$\"3%*R$Rw4#[!*=F-7$FD$\"3A\"pB8% G;Te'f>CF-7$FS$\"3>&e(=NU)Qj#F-7 $FX$\"3OD#)Hb(pj'GF-7$Fgn$\"3DqR_Tw$p7$F-7$F\\o$\"3=$*H3()o*fP$F-7$Fao $\"3'*[&ew4$>!o$F-7$Ffo$\"31/%os_?K,%F-7$F[p$\"3e#z1k=QEO%F-7$F`p$\"3% *3#z8OJiq%F-7$Fep$\"3'Hf#)HA*=]^F-7$Fjp$\"3o![.*pF*)ebF-7$F_q$\"3KmY(y n#HvgF-7$Fdq$\"372'*[w-GslF-7$Fiq$\"397`(G1\\W;(F-7$F^r$\"3smIfnu&yx(F -7$Fcr$\"3))R?QF)RRZ)F-7$Fhr$\"3Kv$eR)3!z;*F-7$F^s$\"3A!4+V*eF!)**F-7$ Fds$\"3I4?O`5/!4\"F17$Fis$\"3'Rk81w=q<\"F17$F^t$\"3e$Ryr\"*o(y7F17$Fct $\"3g:?^hH8$R\"F17$Fht$\"3RF\")*=32\\^\"F17$F]u$\"3@9IJdA&Gk\"F17$Fbu$ \"3O(f&Qv@h(z\"F17$Fgu$\"31MZg0t1\\>F17$F\\v$\"3Rf[`T-*[7#F17$Fav$\"3i $[nV+syH#F17$Ffv$\"3B(Qnc1SJ]#F17$F[w$\"3)QXji7'*Hr#F17$F`w$\"3K'Go$pk =^HF17$Few$\"3MP8ivaE/KF17$Fjw$\"3%o$fA,+]#\\$F17$F_x$\"3+9#RDb)e%z$F1 7$Fdx$\"3))**e)[!)e08%F17$Fix$\"3YrqL[\"=J\\%F17$F^y$\"3WGb\\BUEa[F17$ Fcy$\"3uo!4=y:SI&F17$Fhy$\"3\\_g\"H.p9u&F17$F]z$\"3qQjuyzpZiF17$Fbz$\" 3;#Gn\"Gt(Rx'F17$Fgz$\"3S]1$*)4c!*Q(F1-F\\[l6&F^[lF`[lF_[lF_[l-F$6$7S7 $F($\"3066666666F-7$F/$\"3f]!QD^/GA\"F-7$F5$\"3%p1V\\V%3H8F-7$F:$\"3AS vSi>of9F-7$F?$\"3>+0)*y%3Tg\"F-7$FD$\"3S'*3R'yM?w\"F-7$FI$\"3.i))))*45 B#>F-7$FN$\"3Gv'4#o\\k.@F-7$FS$\"3_K2xoQ>4BF-7$FX$\"34B=6u$pS`#F-7$Fgn $\"37\\i)zQ]#)y#F-7$F\\o$\"3%H\\E&>&pJ.$F-7$Fao$\"3.LfnvcsMLF-7$Ffo$\" 3?3W!H_*onOF-7$F[p$\"3DEr7o8)*>SF-7$F`p$\"3ifDY\"oG\"pVF-7$Fep$\"3>UE@ A!zR#[F-7$Fjp$\"3-31F%o`hC&F-7$F_q$\"3'yH\\*fl%Ry&F-7$Fdq$\"3dT[k8l!eI 'F-7$Fiq$\"3#Gg=h>%pKpF-7$F^r$\"3'*ysO&zxue(F-7$Fcr$\"3#)fk*\\5\"pO$)F -7$Fhr$\"3;[D24Rp*3*F-7$F^s$\"3?#)>rjHLy**F-7$Fds$\"3Q$)Q*Q(zM*4\"F17$ Fis$\"3)[0b6+*3'>\"F17$F^t$\"3l(Ruo@c,J\"F17$Fct$\"3A!ym5#[VR9F17$Fht$ \"3a$*\\Y`WCy:F17$F]u$\"3xqJ#[My_s\"F17$Fbu$\"3,VEcNnj/>F17$Fgu$\"3?Fs tY\"[;3#F17$F\\v$\"39Uf%>ZU))G#F17$Fav$\"31e)*)\\vZV\\#F17$Ffv$\"3O-(y l/!>SFF17$F[w$\"3!3vih&=f$*HF17$F`w$\"3weu\"y\\`NG$F17$Few$\"3U![\\)4w <%f$F17$Fjw$\"3;aS`]F!4&RF17$F_x$\"31\"*or(H-zK%F17$Fdx$\"3uc*ew#)p1v% F17$Fix$\"3-^iz`?r5_F17$F^y$\"3/VlBa*=En&F17$Fcy$\"3!>wg'Rqe_iF17$Fhy$ \"3-Oi)4cw8#oF17$F]z$\"3_]rR\")4$\\[(F17$Fbz$\"3e5$*pE8T!=)F17$Fgz$\" \"*F*-F\\[l6&F^[lF`[lF_[lF`[l-%+AXESLABELSG6$Q\"x6\"Q\"yFg^m-%%VIEWG6$ ;F(Fgz;F_[l$\"\"&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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The gradients of the tangent lines to " }{XPPEDIT 18 0 "y=2^x" "6#/%\"yG)\"\"#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=exp (x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3^ x" "6#/%\"yG)\"\"$%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 19 " are respectively: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln*2" "6#*&%#lnG\"\" \"\"\"#F%" }{TEXT -1 1 " " }{TEXT 321 1 "~" }{TEXT -1 15 " 0.693147180 6, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln*exp(1) = 1 " "6#/*&%#lnG\"\"\"-%$expG6#F&F&F&" }{TEXT -1 4 ", \n " }{XPPEDIT 18 0 "ln*3" "6#*&%#lnG\"\"\"\"\"$F%" }{TEXT -1 1 " " }{TEXT 322 1 "~" } {TEXT -1 14 " 1.098612289. " }}{PARA 0 "" 0 "" {TEXT -1 53 "Using the \+ definition of the derivative of a function " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 20 ", the derivative of " }{XPPEDIT 18 0 "f(x )=a^x" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 14 " is given by: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=Limit((f(x+h )-f(x))/h,h=0)" "6#/-%$f~'G6#%\"xG-%&LimitG6$*&,&-%\"fG6#,&F'\"\"\"%\" hGF1F1-F.6#F'!\"\"F1F2F5/F2\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Limit((a^(x+h)-a^x)/h,h=0)" "6#/%!G- %&LimitG6$*&,&)%\"aG,&%\"xG\"\"\"%\"hGF.F.)F+F-!\"\"F.F/F1/F/\"\"!" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=L imit((a^x*(a^h-1)/h,h=0)" "6#/%!G-%&LimitG6$*()%\"aG%\"xG\"\"\",&)F*% \"hGF,F,!\"\"F,F/F0/F/\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=a^x" "6#/%!G)%\"aG%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((a^h-1)/h,h=0)" "6#-%&LimitG6$*&,&)%\"aG% \"hG\"\"\"F+!\"\"F+F*F,/F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Since we know that " }{XPPEDIT 18 0 "`f '`(x)=a^x*ln*a" " 6#/-%$f~'G6#%\"xG*()%\"aGF'\"\"\"%#lnGF+F*F+" }{TEXT -1 18 ", it follo ws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((a^ h-1)/h,h = 0)=ln*a" "6#/-%&LimitG6$*&,&)%\"aG%\"hG\"\"\"F,!\"\"F,F+F-/ F+\"\"!*&%#lnGF,F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 " Note that this limit is " }{XPPEDIT 18 0 "`f '`(0)" "6#-%$f~'G6#\"\"! " }{TEXT -1 60 ", so it gives the gradient of the tangent line to the \+ graph " }{XPPEDIT 18 0 "y=a^x" "6#/%\"yG)%\"aG%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Some texts make an investigation of the limit " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Lim it((a^h-1)/h,h = 0)" "6#-%&LimitG6$*&,&)%\"aG%\"hG\"\"\"F+!\"\"F+F*F,/ F*\"\"!" }{TEXT -1 13 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 183 " the starting point when considering the derivatives of logarithm and e xponential functions, interpreting this limit geometrically as the gra dient of the tangent line to the graph of " }{XPPEDIT 18 0 "y=a^x" "6 #/%\"yG)%\"aG%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ``(0 ,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "Without saying that the value of the limit (i) is " }{XPPEDIT 18 0 "ln*a" "6#*&%#lnG\"\"\"%\"aGF%" }{TEXT -1 56 ", it can be estimat ed numerically for various values of " }{TEXT 323 1 "a" }{TEXT -1 16 " , starting with " }{XPPEDIT 18 0 "a=2" "6#/%\"aG\"\"#" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "a=3" "6#/%\"aG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[h, `|`, (2^h-1)/h], [_________, ``, __________________], [ 10^(-3), `|`, .6933874626], [10^(-4), `|`, .6931712038], [10^(-5), `|` , .6931495828], [10^(-6), `|`, .6931474208], [10^(-7), `|`, .693147204 6], [10^(-8), `|`, .6931471830], [10^(-9), `|`, .6931471808], [10^(-10 ), `|`, .6931471806]]);" "6#-%'matrixG6#7,7%%\"hG%\"|grG*&,&)\"\"#F(\" \"\"F.!\"\"F.F(F/7%%*_________G%!G%3__________________G7%)\"#5,$\"\"$F /F)-%&FloatG6$\"+EY(Q$p!#57%)F6,$\"\"%F/F)-F:6$\"+Q?rJpF=7%)F6,$\"\"&F /F)-F:6$\"+Ge\\JpF=7%)F6,$\"\"'F/F)-F:6$\"+3UZJpF=7%)F6,$\"\"(F/F)-F:6 $\"+Y?ZJpF=7%)F6,$\"\")F/F)-F:6$\"+I=ZJpF=7%)F6,$\"\"*F/F)-F:6$\"+3=ZJ pF=7%)F6,$F6F/F)-F:6$\"+1=ZJpF=" }{TEXT -1 14 " " } {XPPEDIT 18 0 "matrix([[h, `|`, (3^h-1)/h], [_________, ``, __________ ________], [10^(-3), `|`, 1.099215984], [10^(-4), `|`, 1.098672638], [ 10^(-5), `|`, 1.098618323], [10^(-6), `|`, 1.098612892], [10^(-7), `|` , 1.098612349], [10^(-8), `|`, 1.098612295], [10^(-9), `|`, 1.09861228 9], [10^(-10), `|`, 1.098612289]]);" "6#-%'matrixG6#7,7%%\"hG%\"|grG*& ,&)\"\"$F(\"\"\"F.!\"\"F.F(F/7%%*_________G%!G%3__________________G7%) \"#5,$F-F/F)-%&FloatG6$\"+%)f@*4\"!\"*7%)F6,$\"\"%F/F)-F96$\"+QEn)4\"F <7%)F6,$\"\"&F/F)-F96$\"+B$=')4\"F<7%)F6,$\"\"'F/F)-F96$\"+#*Gh)4\"F<7 %)F6,$\"\"(F/F)-F96$\"+\\Bh)4\"F<7%)F6,$\"\")F/F)-F96$\"+&H7')4\"F<7%) F6,$\"\"*F/F)-F96$\"+*G7')4\"F<7%)F6,$F6F/F)-F96$\"+*G7')4\"F<" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a := 2:\nmap(h->evalf[10](evalf[40]([h,(a^h-1)/h ])),[seq(Float(1,-i),i=3..10)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7* 7$$\"\"\"!\"$$\"+EY(Q$p!#57$$F&!\"%$\"+Q?rJpF*7$$F&!\"&$\"+Ge\\JpF*7$$ F&!\"'$\"+3UZJpF*7$$F&!\"($\"+Y?ZJpF*7$$F&!\")$\"+I=ZJpF*7$$F&!\"*$\"+ 3=ZJpF*7$$F&F*$\"+1=ZJpF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a := 3:\nmap(h->evalf[10](evalf[40] ([h,(a^h-1)/h])),[seq(Float(1,-i),i=3..10)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*7$$\"\"\"!\"$$\"+%)f@*4\"!\"*7$$F&!\"%$\"+QEn)4\"F*7$ $F&!\"&$\"+B$=')4\"F*7$$F&!\"'$\"+#*Gh)4\"F*7$$F&!\"($\"+\\Bh)4\"F*7$$ F&!\")$\"+&H7')4\"F*7$$F&F*$\"+*G7')4\"F*7$$F&!#5FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 61 "One can now suggest that there is some choice for the number " }{TEXT 324 1 " a" }{TEXT -1 37 " somewhere between 2 and 3 such that " }{XPPEDIT 18 0 "Limit((a^h-1)/h,h = 0)=1" "6#/-%&LimitG6$*&,&)%\"aG%\"hG\"\"\"F,!\" \"F,F+F-/F+\"\"!F," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "T his leads to the correct choice being the number " }{XPPEDIT 18 0 "exp (1)" "6#-%$expG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 325 1 "~" }{TEXT -1 14 " 2.718281828. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We could attempt to calculate " }{XPPEDIT 18 0 "exp(1)" " 6#-%$expG6#\"\"\"" }{TEXT -1 17 " by this method. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(a)=Limit((a^h-1)/h,h = 0)" "6#/- %$phiG6#%\"aG-%&LimitG6$*&,&)F'%\"hG\"\"\"F/!\"\"F/F.F0/F.\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Then, the calculations \+ above give " }{XPPEDIT 18 0 "phi(2)" "6#-%$phiG6#\"\"#" }{TEXT -1 1 " \+ " }{TEXT 326 1 "~" }{TEXT -1 18 " 0.6931471806 and " }{XPPEDIT 18 0 "p hi(3)" "6#-%$phiG6#\"\"$" }{TEXT -1 1 " " }{TEXT 327 1 "~" }{TEXT -1 14 " 1.098612289. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 328 1 "E" }{TEXT -1 25 " be the number such t hat " }{XPPEDIT 18 0 "phi(E)=1" "6#/-%$phiG6#%\"EG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Suppose that the graph of " } {XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 80 " is a straight line, or a t least approximately a straight line, in the interval " }{XPPEDIT 18 0 "2<=a" "6#1\"\"#%\"aG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 487 397 397 {PLOTDATA 2 "6.-%'CURVESG6$7S7$$\"3/+++++++=!#<$\"3(Q7$Q*eT07'!#=7$$\" 32++DJdpK=F*$\"39\"pU9P6JD'F-7$$\"33+v=7T9h=F*$\"3%z%f'*H'f%ojF-7$$\"3 ++](=HPJ*=F*$\"3!)4Jd53=)\\'F-7$$\"3(***\\7VDMD>F*$\"3RF/Pn=wGmF-7$$\" 3'**\\P%GZRd>F*$\"3k%y)>&HA(enF-7$$\"37+v=276()>F*$\"3C8%))yC7#zoF-7$$ \"3&**\\(o**3)y,#F*$\"3U(3[[hsR+(F-7$$\"3/+vofHq\\?F*$\"3&=LH['***H8(F -7$$\"35+v$f'HU\"3#F*$\"3+7GYANhhsF-7$$\"3z***\\7*309@F*$\"3w*e/^P3RR( F-7$$\"35+]i&e*yU@F*$\"3nf5K`PV5vF-7$$\"3C++v[D9v@F*$\"3QO?BIPhTwF-7$$ \"3()***\\iNGw?#F*$\"3I>0)HBF*$\"3SXD0'z;(o#)F-7$$\"3/+v=-p6jBF*$\"3,=IhXTy.%)F-7$$ \"3w****\\2Mg#R#F*$\"3mkcDO;MB&)F-7$$\"3u*\\(=xZ&\\U#F*$\"3SeNL_^^a')F -7$$\"3u*\\i:$4wbCF*$\"3+?TZfLUz()F-7$$\"35+](=#R!z[#F*$\"3^'*Q*)p>v4* )F-7$$\"3)**\\P4A@u^#F*$\"3rr#[%=VVH!*F-7$$\"3/+]i:'f#\\DF*$\"3K[y))** y_e\"*F-7$$\"3!)*\\(of2L#e#F*$\"3MpM\"*\\)>EH*F-7$$\"3-+D\"yG>6h#F*$\" 3\"3D)f'GZ$4%*F-7$$\"33+](oo6Ak#F*$\"3rQ/2;hTN&*F-7$$\"3))****\\xJLuEF *$\"3E[!fFbdcm*F-7$$\"3y**\\P*yddq#F*$\"3!zeq#*QtIz*F-7$$\"3#)*\\(='GF*$\"3VIhw\"fbE/\"F*7$$\"3!****\\(Q(zS*GF*$\"3[L&*[=`mb5F*7$ $\"3s*\\(=-,FCHF*$\"3/WXYdk!z1\"F*7$$\"31+v$4tFe&HF*$\"3M+t\"[(>q!3\"F *7$$\"3a***\\73\"o')HF*$\"31ds$=$>@$4\"F*7$$\"3v*\\(oz;)*=IF*$\"3`wYZ( p3j5\"F*7$$\"3q*****\\*44]IF*$\"3?I5PRC#*=6F*7$$\"3%)**\\7jZ!>3$F*$\"3 PhJI7=#=8\"F*7$$\"3\"**\\(=(4bM6$F*$\"3whY\">P9Y9\"F*7$$\"3*)****\\xlW UJF*$\"3?&y8cSpj:\"F*7$$\"3')**\\i&3uc<$F*$\"3Q`yy+?%)p6F*7$$\"3#)**** *\\;$R0KF*$\"3u7&>00#*==\"F*7$$\"3')*\\(=-*zqB$F*$\"3]P)Qo\"*RZ>\"F*7$ $\"33+D\"G:3uE$F*$\"3sMsDkp.27F*7$$\"3#)*************H$F*$\"32f05@=D?7 F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-F$6%7%7$$\"\"#F`[lF_[l7$F e[l$\"3'GX*f0=ZJpF-7$F_[lFh[l-Fiz6&F[[lF`[lF`[lF`[l-%*LINESTYLEG6#Ff[l -F$6%7%7$$\"\"$F`[lF_[l7$Fd\\l$\"3y4\"o')G7')4\"F*7$F_[lFg\\lF[\\lF]\\ l-F$6%7%7$$\"3')*****4<#zcFF*F_[l7$F^]l$\"\"\"F`[l7$F_[lFa]l-Fiz6&F[[l F_[lF_[l$\"*++++\"!\")F]\\l-F$6&7(Fd[lFg[lFj[lFc\\lFf\\lFi\\l-%'SYMBOL G6#%'CIRCLEGF[\\l-%&STYLEG6#%&POINTG-F$6&F[^l-F]^l6#%(DIAMONDGF[\\lF`^ l-F$6&F[^l-F]^l6#%&CROSSGF[\\lF`^l-F$6&F\\]lF\\^l-%&COLORG6&F[[lF_[lF_ [l$\"#&)!\"#F`^l-F$6&F\\]lFf^lF`_lF`^l-F$6&F\\]lF[_lF`_lF`^l-%+AXESLAB ELSG6%Q\"a6\"Q!F^`l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F_[l$\"#NF^[l;F_[l$ \"#9F^[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur ve 8" "Curve 9" "Curve 10" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Then, for " }{XPPEDIT 18 0 "2<=a" "6#1\"\"#%\"aG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(a)" "6#-%$phiG6#%\"aG" }{TEXT -1 1 " " } {TEXT 329 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(2)+``((phi(3)-phi( 2))/(3-2))*(a-2);" "6#,&-%$phiG6#\"\"#\"\"\"*&-%!G6#*&,&-F%6#\"\"$F(-F %6#F'!\"\"F(,&F1F(F'F4F4F(,&%\"aGF(F'F4F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "phi(E)=1" "6#/-%$phiG6#%\"E G\"\"\"" }{TEXT -1 6 " then " }}{PARA 256 "" 0 "" {TEXT -1 3 " E " } {TEXT 330 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2+(1-phi(2))/(phi(3)-p hi(2))" "6#,&\"\"#\"\"\"*&,&F%F%-%$phiG6#F$!\"\"F%,&-F)6#\"\"$F%-F)6#F $F+F+F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 331 1 "~" }{TEXT -1 14 " 2.756792171. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "To obtai n a better approximation repeat this process using " }{XPPEDIT 18 0 "a [1]=2.756792171" "6#/&%\"aG6#\"\"\"-%&FloatG6$\"+r@zcF!\"*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[2]=3" "6#/&%\"aG6#\"\"#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "a[1]<=a" "6#1 &%\"aG6#\"\"\"F%" }{XPPEDIT 18 0 "``<=a[2]" "6#1%!G&%\"aG6#\"\"#" } {TEXT -1 13 " assume that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "phi(a)=phi(a[1])+``((phi(a[2])-phi(a[1]))/(a[2]-a[1]))* (a-a[1])" "6#/-%$phiG6#%\"aG,&-F%6#&F'6#\"\"\"F-*&-%!G6#*&,&-F%6#&F'6# \"\"#F--F%6#&F'6#F-!\"\"F-,&&F'6#F8F-&F'6#F-F=F=F-,&F'F-&F'6#F-F=F-F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "We need to estimate \+ " }{XPPEDIT 18 0 "phi(a[1])" "6#-%$phiG6#&%\"aG6#\"\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 42 "This can be done by calculating val ues of " }{XPPEDIT 18 0 "(a^h-1)/h;" "6#*&,&)%\"aG%\"hG\"\"\"F(!\"\"F( F'F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a=a[1]" "6#/%\"aG&F$6#\" \"\"" }{TEXT -1 51 ", for a sequence of successively smaller values of " }{TEXT 342 1 "h" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "a := 2.756792171:\nmap(h->ev alf[10](evalf[40]([h,(a^h-1)/h])),[seq(Float(1,-i),i=3..10)]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7*7$$\"\"\"!\"$$\"+(3#e95!\"*7$$F&!\"% $\"+l\">T,\"F*7$$F&!\"&$\"+*)G295F*7$$F&!\"'$\"+h#oS,\"F*7$$F&!\"($\"+ )znS,\"F*7$$F&!\")$\"+_x195F*7$$F&F*$\"+Zx195F*7$$F&!#5FF" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Alternatively, we can just let Maple evaluate the limit. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "a := 'a': phi := a - > Limit((a^h-1)/h,h=0):\na1 := 2.756792171:\n'phi'(a1)=phi(a1);\n``=ev alf(evalf[16](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#$ \"+r@zcF!\"*-%&LimitG6$*&,&)F'%\"hG\"\"\"F1!\"\"F1F0F2/F0\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+Zx195!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 340 1 "E" }{TEXT -1 1 " " }{TEXT 339 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a[1]+(1-phi(a[1]))*(a[2]-a[1])/(phi(a[2])-phi(a[1]))" "6#,&&%\"a G6#\"\"\"F'*(,&F'F'-%$phiG6#&F%6#F'!\"\"F',&&F%6#\"\"#F'&F%6#F'F/F',&- F+6#&F%6#F3F'-F+6#&F%6#F'F/F/F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 341 1 "~" } {TEXT -1 14 " 2.716323725. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "a := 'a': phi := a -> Limit((a^h-1 )/h,h=0):\na1 := 2.756792171: a2 := 3:\na1+(1-phi(a1))*(a2-a1)/(phi(a2 )-phi(a1)):\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+DPK;F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Repeating this process with " }{XPPEDIT 18 0 "a[1]=2.7163 23725" "6#/&%\"aG6#\"\"\"-%&FloatG6$\"+DPK;F!\"*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[2]=2.756792171" "6#/&%\"aG6#\"\"#-%&FloatG6$\"+r@zcF !\"*" }{TEXT -1 7 " gives " }{TEXT 333 1 "E" }{TEXT -1 1 " " }{TEXT 332 1 "~" }{TEXT -1 15 " 2.718295668. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "a := 'a': phi := a -> \+ Limit((a^h-1)/h,h=0):\na1 := 2.716323725: a2 := 2.756792171:\na1+(1-ph i(a1))*(a2-a1)/(phi(a2)-phi(a1)):\nevalf(evalf[14](%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+ocH=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Repeating this process with " } {XPPEDIT 18 0 "a[1]=2.716323725" "6#/&%\"aG6#\"\"\"-%&FloatG6$\"+DPK;F !\"*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[2]=2.718295668" "6#/&%\"aG 6#\"\"#-%&FloatG6$\"+ocH=F!\"*" }{TEXT -1 7 " gives " }{TEXT 335 1 "E " }{TEXT -1 1 " " }{TEXT 334 1 "~" }{TEXT -1 15 " 2.718281833. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "a := 'a': phi := a -> Limit((a^h-1)/h,h=0):\na1 := 2.716323725: a 2 := 2.718295668:\na1+(1-phi(a1))*(a2-a1)/(phi(a2)-phi(a1)):\nevalf(ev alf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+L=G=F!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Repeating this process with " }{XPPEDIT 18 0 "a[1]=2.718281833" "6#/&%\"aG6#\" \"\"-%&FloatG6$\"+L=G=F!\"*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[2]= 2.718295668" "6#/&%\"aG6#\"\"#-%&FloatG6$\"+ocH=F!\"*" }{TEXT -1 7 " g ives " }{TEXT 337 1 "E" }{TEXT -1 1 " " }{TEXT 336 1 "~" }{TEXT -1 15 " 2.718281828. " }}{PARA 0 "" 0 "" {TEXT -1 37 "This agrees with the \+ known value for " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 15 " to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "a := 'a': phi := a -> Limit((a^h-1 )/h,h=0):\na1 := 2.718281833: a2 := 2.718295668:\na1+(1-phi(a1))*(a2-a 1)/(phi(a2)-phi(a1)):\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Summary of standard derivatives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "mat rix([[f(x), `|`, `f '`(x)], [________, ``, _________], [ln*x, `|`, 1/x ], [ln(u(x)), `|`, `u '`(x)/u(x)], [log[a]*x, `|`, 1/(x*ln*a)]]);" "6# -%'matrixG6#7'7%-%\"fG6#%\"xG%\"|grG-%$f~'G6#F+7%%)________G%!G%*_____ ____G7%*&%#lnG\"\"\"F+F7F,*&F7F7F+!\"\"7%-F66#-%\"uG6#F+F,*&-%$u~'G6#F +F7-F>6#F+F97%*&&%$logG6#%\"aGF7F+F7F,*&F7F7*(F+F7F6F7FKF7F9" }{TEXT -1 25 " " }{XPPEDIT 18 0 "matrix([[f(x), `|`, \+ `f '`(x)], [________, ``, _________], [exp(x), `|`, exp(x)], [exp(a*x) , `|`, a*exp(a*x)], [exp(u(x)), `|`, exp(u(x))*`u '`(x)], [a^x, `|`, a ^x*ln*a]]);" "6#-%'matrixG6#7(7%-%\"fG6#%\"xG%\"|grG-%$f~'G6#F+7%%)___ _____G%!G%*_________G7%-%$expG6#F+F,-F66#F+7%-F66#*&%\"aG\"\"\"F+F?F,* &F>F?-F66#*&F>F?F+F?F?7%-F66#-%\"uG6#F+F,*&-F66#-FH6#F+F?-%$u~'G6#F+F? 7%)F>F+F,*()F>F+F?%#lnGF?F>F?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 53 ": The derivative of the general exponential function " }{XPPEDIT 18 0 "f(x) =a^x" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 26 ", can be found by writing " } {XPPEDIT 18 0 "f(x)=exp(ln(a^x))" "6#/-%\"fG6#%\"xG-%$expG6#-%#lnG6#)% \"aGF'" }{XPPEDIT 18 0 "``=exp(x*ln*a)" "6#/%!G-%$expG6#*(%\"xG\"\"\"% #lnGF*%\"aGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "`f '`(x) = exp(x*ln*a)*ln*a;" "6#/-%$f~'G6#%\"xG*(-%$ex pG6#*(F'\"\"\"%#lnGF-%\"aGF-F-F.F-F/F-" }{XPPEDIT 18 0 "``=a^x*ln*a" " 6#/%!G*()%\"aG%\"xG\"\"\"%#lnGF)F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Some graphs i nvolving the exponential function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+ " }}{PARA 0 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "y = 2-exp (-x);" "6#/%\"yG,&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F-" }{TEXT -1 14 " h as the line " }{XPPEDIT 18 0 "y = 2" "6#/%\"yG\"\"#" }{TEXT -1 34 " as a horizontal asymptote, since " }{XPPEDIT 18 0 "Limit(2-exp(-x),x = i nfinity) = 2;" "6#/-%&LimitG6$,&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F//F.% )infinityGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Limit(2-exp(-x),x = infinit y);\nvalue(%);\nplot([2-exp(-x),2],x=-2..4,y=-2..3,color=[red,black],l inestyle=[1,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,&\"\"# \"\"\"-%$expG6#,$%\"xG!\"\"F./F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 328 244 244 {PLOTDATA 2 "6&-%'CURVESG6%7U7$$!\"#\"\"!$!3S]1$*)4c!*Q&!#<7$$!3&***** \\P&3Y$>F-$!31%GVA3M8#\\F-7$$!3!******\\2<#p=F-$!3!=+!)y`=K[%F-7$$!3') **\\78.K7=F-$!3\\j@@X?kCTF-7$$!31++D^NUb)e_bZ#F-7$$!3#**** \\i3@/P\"F-$!36)*=[m\"3q$>F-7$$!3;++Dr^b^7F-$!3)3do9Evd\\\"F-7$$!3$*** *\\7Sw%G6F-$!3%*zD)HbV44\"F-7$$!3*****\\7;)=,5F-$!3&G&[!40N^@(!#=7$$!3 M++]i83V()Fjn$!3_!*R>J<;sRFjn7$$!3B******\\V'zV(Fjn$!3MbrX3s2R5Fjn7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 9 "The point" }{XPPEDIT 18 0 " ``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 20 " is a maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(exp(-x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"x G\"\"\"-%$expG6#,$*$)F%\"\"#F&!\"\"F&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "exp(-x^2 );" "6#-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 82 " is always positive , the derivative has the opposite sign to that of the variable " } {TEXT 276 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 78 "The is \+ consistent with the graph shown, since the function is increasing when " }{TEXT 277 1 "x" }{TEXT -1 34 " is negative, and decreasing when " }{TEXT 278 1 "x" }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(exp(-x^2),x=infinity)=0" "6#/-%&LimitG6 $-%$expG6#,$*$%\"xG\"\"#!\"\"/F,%)infinityG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(exp(-x^2),x=-infinity)=0" "6#/-%&LimitG6$-%$expG 6#,$*$%\"xG\"\"#!\"\"/F,,$%)infinityGF.\"\"!" }{TEXT -1 9 ", so the " }{TEXT 308 1 "x" }{TEXT -1 33 " axis is a horizontal asymptote. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Note that the graph is symmetrical about the " }{TEXT 309 1 "y" }{TEXT -1 78 " \+ axis. This arises because the value of the function at any negative va lue of " }{TEXT 310 1 "x" }{TEXT -1 66 " is the same as the value at t he corresponding positive value for " }{TEXT 312 1 "x" }{TEXT -1 15 ". For example, " }{XPPEDIT 18 0 "f(-1)=f(1)" "6#/-%\"fG6#,$\"\"\"!\"\"- F%6#F(" }{XPPEDIT 18 0 "``=exp(-1)" "6#/%!G-%$expG6#,$\"\"\"!\"\"" } {TEXT -1 13 ". Formally, " }{XPPEDIT 18 0 "f(-x)=f(x)" "6#/-%\"fG6#,$ %\"xG!\"\"-F%6#F(" }{TEXT -1 21 " for any real number " }{TEXT 311 1 " x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{PARA 0 "" 0 "" {TEXT -1 43 "The tangent line to this graph at the point" } {XPPEDIT 18 0 "``(1,1/exp(1));" "6#-%!G6$\"\"\"*&F&F&-%$expG6#F&!\"\" " }{TEXT -1 14 " has gradient " }{XPPEDIT 18 0 "-2/exp(1);" "6#,$*&\" \"#\"\"\"-%$expG6#F&!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The equation of this tangent is " }{XPPEDIT 18 0 "y-1/exp(1) = \+ -2/exp(1)(x-1);" "6#/,&%\"yG\"\"\"*&F&F&-%$expG6#F&!\"\"F+,$*&\"\"#F&- -F)6#F&6#,&%\"xGF&F&F+F+F+" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "y = (- 2/exp(1))*x+3/exp(1);" "6#/%\"yG,&*&,$*&\"\"#\"\"\"-%$expG6#F*!\"\"F.F *%\"xGF*F**&\"\"$F*-F,6#F*F.F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot([exp(- x^2),(-2/exp(1))*x+3/exp(1)],x=-3..3,y=-0.5..1.5,\n color=[red,blu e],tickmarks=[6,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 429 162 162 {PLOTDATA 2 "6'-%'CURVESG6$7co7$$!\"$\"\"!$\"1'zm3/)4M7!#>7$$!1+++vq@p 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"> " 0 "" {MPLTEXT 1 0 28 "diff(x*exp(x),x);\nfactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &-%$expG6#%\"xG\"\"\"*&F'F(F$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &-%$expG6#%\"xG\"\"\",&F(F(F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "The derivative is negative when " } {XPPEDIT 18 0 "x < -1;" "6#2%\"xG,$\"\"\"!\"\"" }{TEXT -1 19 " and pos itive when " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 49 " Find the de rivatives of the following functions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (a) " }{XPPEDIT 18 0 "f(x) = exp(-x )*sin*2*x;" "6#/-%\"fG6#%\"xG**-%$expG6#,$F'!\"\"\"\"\"%$sinGF.\"\"#F. F'F." }{TEXT -1 9 " (b) " }{XPPEDIT 18 0 "f(x) = exp(cos*x);" "6#/ -%\"fG6#%\"xG-%$expG6#*&%$cosG\"\"\"F'F-" }{TEXT -1 7 " (c) " } {XPPEDIT 18 0 "f(x)=ln(sqrt(x))" "6#/-%\"fG6#%\"xG-%#lnG6#-%%sqrtG6#F' " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (d) " }{XPPEDIT 18 0 "f(x) = 1/(ln*x);" "6#/-%\"fG6#%\"xG*&\"\"\"F)*&%#lnGF)F'F)!\"\" " }{TEXT -1 10 " (e) " }{XPPEDIT 18 0 "f(x) = ln*x/(1+x^2);" "6#/ -%\"fG6#%\"xG*(%#lnG\"\"\"F'F*,&F*F**$F'\"\"#F*!\"\"" }{TEXT -1 8 " ( f) " }{XPPEDIT 18 0 "f(x) = exp(2*x)*ln*x;" "6#/-%\"fG6#%\"xG*(-%$ex pG6#*&\"\"#\"\"\"F'F.F.%#lnGF.F'F." }{TEXT -1 9 " (g) " }{XPPEDIT 18 0 "f(x)=(1+exp(x))/(1-exp(x))" "6#/-%\"fG6#%\"xG*&,&\"\"\"F*-%$expG 6#F'F*F*,&F*F*-F,6#F'!\"\"F1" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (h) " }{XPPEDIT 18 0 "f(x)= (exp(3*x)-exp(x))/exp(2*x) " "6#/-%\"fG6#%\"xG*&,&-%$expG6#*&\"\"$\"\"\"F'F/F/-F+6#F'!\"\"F/-F+6# *&\"\"#F/F'F/F2" }{TEXT -1 8 " (i) " }{XPPEDIT 18 0 "f(x) = ln(x^3* sin*x);" "6#/-%\"fG6#%\"xG-%#lnG6#*(F'\"\"$%$sinG\"\"\"F'F." }{TEXT -1 9 " (j) " }{XPPEDIT 18 0 "f(x) = sqrt(ln*x);" "6#/-%\"fG6#%\"xG -%%sqrtG6#*&%#lnG\"\"\"F'F-" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (k) " }{XPPEDIT 18 0 "f(x) = ln(exp(3*x)-1);" "6#/-%\"f G6#%\"xG-%#lnG6#,&-%$expG6#*&\"\"$\"\"\"F'F1F1F1!\"\"" }{TEXT -1 7 " \+ (l) " }{XPPEDIT 18 0 "f(x) = ln(x+ln*x);" "6#/-%\"fG6#%\"xG-%#lnG6#,& F'\"\"\"*&F)F,F'F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "`f '`(x) = 2*exp(-x)*cos*2*x-exp(-x)*sin*2 *x;" "6#/-%$f~'G6#%\"xG,&*,\"\"#\"\"\"-%$expG6#,$F'!\"\"F+%$cosGF+F*F+ F'F+F+**-F-6#,$F'F0F+%$sinGF+F*F+F'F+F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "`f '`(x) = -exp(cos*x)*sin*x;" "6#/-%$f~'G6#%\"xG,$*(-%$expG6#*&%$cosG\"\"\"F'F/F/%$sinGF/F'F/!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "`f '`(x) = 1/(2*x);" "6#/-%$f~'G6#%\"xG*&\"\"\"F)*&\"\"#F)F'F)!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "`f \+ '`(x) = -1/(x*(ln*x)^2);" "6#/-%$f~'G6#%\"xG,$*&\"\"\"F**&F'F**$*&%#ln GF*F'F*\"\"#F*!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }{XPPEDIT 18 0 "`f '`(x) = (1+x^ 2-2*x^2*ln*x)/x/((1+x^2)^2);" "6#/-%$f~'G6#%\"xG*(,(\"\"\"F**$F'\"\"#F ***F,F**$F'F,F*%#lnGF*F'F*!\"\"F*F'F0*$,&F*F**$F'F,F*F,F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " ( f) " }{XPPEDIT 18 0 "`f '`(x) = 2*exp(2*x)*ln*x+exp(2*x)/x;" "6#/-%$f ~'G6#%\"xG,&**\"\"#\"\"\"-%$expG6#*&F*F+F'F+F+%#lnGF+F'F+F+*&-F-6#*&F* F+F'F+F+F'!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " (g) \+ " }{XPPEDIT 18 0 "`f '`(x) = 2*exp(x)/((1-exp(x))^2);" "6#/-%$f~'G6#% \"xG*(\"\"#\"\"\"-%$expG6#F'F**$,&F*F*-F,6#F'!\"\"F)F2" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " (h) \+ " }{XPPEDIT 18 0 "`f '`(x) = exp(x)+exp(-x);" "6#/-%$f~'G6#%\"xG,&-%$e xpG6#F'\"\"\"-F*6#,$F'!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " (i) " }{XPPEDIT 18 0 "`f '`(x) = (3*sin*x+x*cos*x)/(x*sin*x);" "6#/-%$f~'G6#%\"xG*&,&*(\"\"$\" \"\"%$sinGF,F'F,F,*(F'F,%$cosGF,F'F,F,F,*(F'F,F-F,F'F,!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " (j) " }{XPPEDIT 18 0 "`f '`( x) = 1/(2*x*sqrt(ln*x));" "6#/-%$f~'G6#%\"xG*&\"\"\"F)*(\"\"#F)F'F)-%% sqrtG6#*&%#lnGF)F'F)F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " (k) " }{XPPEDIT 18 0 "`f '`(x ) = 3*exp(3*x)/(exp(3*x)-1);" "6#/-%$f~'G6#%\"xG*(\"\"$\"\"\"-%$expG6# *&F)F*F'F*F*,&-F,6#*&F)F*F'F*F*F*!\"\"F3" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 7 " (l) " }{XPPEDIT 18 0 "`f '`(x) = (x+1)/x/(x+ln*x );" "6#/-%$f~'G6#%\"xG*(,&F'\"\"\"F*F*F*F'!\"\",&F'F**&%#lnGF*F'F*F*F+ " }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 43 "____________________ _______________________" }}{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 43 "_______________________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 68 "(a) \+ Find the equations of the tangent and normal lines to the graph " } {XPPEDIT 18 0 "y = x*ln*x;" "6#/%\"yG*(%\"xG\"\"\"%#lnGF'F&F'" }{TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "x=exp(1)" "6#/%\"xG-%$exp G6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Plot the graphs of " }{XPPEDIT 18 0 "y = x*ln*x;" "6#/%\"yG*(%\"xG\"\"\"%#lnGF 'F&F'" }{TEXT -1 55 " and the tangent line found in (a) in the same pi cture." }}{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=2*x-exp(1)" "6#/%\"yG,&*&\"\"#\"\"\"%\"xGF(F(-%$expG6#F(!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y=-x/2+3*exp(1)/2" "6#/%\"yG,&*&%\"xG \"\"\"\"\"#!\"\"F**(\"\"$F(-%$expG6#F(F(F)F*F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot([x*ln(x),2*x-exp(1),-x/2+3*exp(1)/2],x=0..4,y,\n \+ color=[red,blue,COLOR(RGB,0,.7,.3)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 466 466 466 {PLOTDATA 2 "6'-%'CURVESG6$7fn7$$\"3#)******pUkCF!#?$!3-'3 q(oh,4;!#>7$$\"3k******R&)G\\aF*$!3&*4VyQkJSGF-7$$\"3Y******4G$R<)F*$! 3g8*GJ?]!HRF-7$$\"3%******zqd)*3\"F-$!3s&*o0!3,_#\\F-7$$\"3*)*****>c'y M;F-$!3#4C!**4F&\\s'F-7$$\"3')*****fT:(z@F-$!3ZW.4l&Q&R$)F-7$$\"3y**** *R7t&pKF-$!3zIXu-5O=6!#=7$$\"3y*****H$3VfVF-$!3AimA%*\\tl8FL7$$\"3k*** 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________________________________" }}{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 43 "____________________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 23 "By using the fact that " }{XPPEDIT 18 0 "exp(ln*x) = x;" "6#/-%$expG6#*&%#lnG\"\"\"%\"xGF)F*" }{TEXT -1 30 " for any positive r eal number " }{TEXT 306 1 "x" }{TEXT -1 11 ", to write " }{XPPEDIT 18 0 "x^r" "6#)%\"xG%\"rG" }{TEXT -1 13 " in the form " }{XPPEDIT 18 0 "e xp(ln*x)^r;" "6#)-%$expG6#*&%#lnG\"\"\"%\"xGF)%\"rG" }{TEXT -1 70 ", \+ deduce the general power rule for differentiatiion, namely that if " } {XPPEDIT 18 0 "y=x^r" "6#/%\"yG)%\"xG%\"rG" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "dy/dx=r*x^(r-1)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"rGF&)% \"xG,&F*F&F&F(F&" }{TEXT -1 21 ", for the case where " }{TEXT 307 1 "x " }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 43 "________ ___________________________________" }}{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 43 "____________________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" } {TEXT -1 10 " if (a) " }{XPPEDIT 18 0 "f(x)=x^(sqrt(x))" "6#/-%\"fG6 #%\"xG)F'-%%sqrtG6#F'" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "f(x) = x ^(sin*x);" "6#/-%\"fG6#%\"xG)F'*&%$sinG\"\"\"F'F+" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "f(x) = x^(ln*x);" "6#/-%\"fG6#%\"xG)F'*&%#lnG\"\" \"F'F+" }{TEXT -1 2 ". " }}{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 "`f '`(x) = x^sqrt(x)*``((ln*x+2)/(2*sqrt(x)));" "6#/-% $f~'G6#%\"xG*&)F'-%%sqrtG6#F'\"\"\"-%!G6#*&,&*&%#lnGF-F'F-F-\"\"#F-F-* &F5F--F+6#F'F-!\"\"F-" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "`f '`(x) \+ = x^(sin*x)*(cos*x*ln*x+sin*x/x);" "6#/-%$f~'G6#%\"xG*&)F'*&%$sinG\"\" \"F'F,F,,&**%$cosGF,F'F,%#lnGF,F'F,F,*(F+F,F'F,F'!\"\"F,F," }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "`f '`(x) = 2*x^(ln*x)/x*ln*x;" "6#/-%$f~'G 6#%\"xG*,\"\"#\"\"\")F'*&%#lnGF*F'F*F*F'!\"\"F-F*F'F*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "____ _______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "____________ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 15 "(a) Show that " }{XPPEDIT 18 0 "y=1/(x+1+exp(x))" "6# /%\"yG*&\"\"\"F&,(%\"xGF&F&F&-%$expG6#F(F&!\"\"" }{TEXT -1 45 " is a s olution of the differential equation " }{XPPEDIT 18 0 "dy/dx=x*y^2-y " "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%\"xGF&*$%\"yG\"\"#F&F&F-F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "(b) Show that " }{XPPEDIT 18 0 "y=x*exp(x^2)" "6#/%\"yG*&%\"xG\"\"\"-%$expG6#*$F&\"\"#F'" }{TEXT -1 45 " is a solution of the differential equation " }{TEXT 294 1 "x " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y*(1+2*x^2)" "6#/*&%#dyG\"\"\" %#dxG!\"\"*&%\"yGF&,&F&F&*&\"\"#F&*$%\"xGF-F&F&F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 14 "(c) Show that " }{XPPEDIT 18 0 "y=6*x*exp (-x)" "6#/%\"yG*(\"\"'\"\"\"%\"xGF'-%$expG6#,$F(!\"\"F'" }{TEXT -1 45 " is a solution of the differential equation " }{XPPEDIT 18 0 "d^2*y/ (dx^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*$%#dxGF&!\"\"F(\"\"$F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 6*exp(-x);" "6#/,&*&%#dyG\" \"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'*&\"\"'F'-%$expG6#,$%\"xGF)F'" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "______________________ _____________________" }}{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 43 "_______________________________ ____________" }}{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 18 "Code for picture " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 13 "Esti mating e " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "e := 2.756792171: a := 'a':\nf := x -> ln(2)+(ln(3)- ln(2))*(x-2):\np1 := plot(f(a),a=1.8..3.3):\np2 := plot([[[2,0],[2,ln( 2)],[0,ln(2)]],[[3,0],[3,ln(3)],[0,ln(3)]],\n [[e,0],[e,1],[0,1] ]],\n color=[black$2,blue],linestyle=2):\np3 := plot([[[2,0],[2 ,ln(2)],[0,ln(2)],[3,0],[3,ln(3)],[0,ln(3)]]$3],\n color=black, style=point,symbol=[circle,diamond,cross]):\np4 := plot([[[e,0],[e,1], [0,1]]$3],color=COLOR(RGB,0,0,.85),\n style=point,symbol=[circ le,diamond,cross]):\nplots[display]([p1,p2,p3,p4],view=[0..3.5,0..1.4] );" }}}{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 }