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0 {PARA 3 "" 0 "" {TEXT -1 37 "Exponential and logarithm functio ns " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca nada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 51 "Building up the domain of the exponential function " }{XPPEDIT 18 0 "f(x)=2^x" "6#/-%\"fG6#%\"xG)\"\"#F'" }{TEXT -1 24 " .. rules for exponents " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 23 "It is easy to evaluate " }{XPPEDIT 18 0 "f(x)=2^x" "6#/-%\"fG6#%\"xG)\"\"#F'" }{TEXT -1 6 " when " } {TEXT 303 1 "x" }{TEXT -1 6 " is a " }{TEXT 261 16 "positive integer" }{TEXT -1 49 ", since we simply need to multiply 2 by itself \"" } {TEXT 273 1 "x" }{TEXT -1 8 " times\"." }}{PARA 0 "" 0 "" {TEXT -1 33 "It is also easy to interpret the " }{TEXT 261 8 "sum rule" }{TEXT -1 108 " for powers or exponents when multiplying two powers of 2, where \+ only positive integers powers are involved." }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{XPPEDIT 18 0 "2^3*`.`*2^4 = 2^7;" "6#/*(\"\"# \"\"$%\".G\"\"\"F%\"\"%*$F%\"\"(" }{TEXT -1 77 ", since on both the le ft and right sides we have 7 two's multiplied together." }}{PARA 0 "" 0 "" {TEXT -1 31 "Extension of the definition of " }{XPPEDIT 18 0 "2^x " "6#)\"\"#%\"xG" }{TEXT -1 125 " to successively larger sets of numbe rs than just the positive integers is made in such a way as to ensure \+ that the sum rule " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^u*`.`*2^v = 2^(u+v);" "6#/*()\"\"#%\"uG\"\"\"%\".GF()F&%\"vGF()F&,& F'F(F+F(" }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 36 " applies in the wider context, where " }{TEXT 304 1 "u" }{TEXT -1 5 " a nd " }{TEXT 305 1 "v" }{TEXT -1 39 " belong to the enlarged set of num bers." }}{PARA 0 "" 0 "" {TEXT -1 56 "To start with we know that the s um rule (i) apples when " }{TEXT 308 1 "u" }{TEXT -1 5 " and " }{TEXT 306 1 "v" }{TEXT -1 19 " belong to the set " }{XPPEDIT 18 0 "\{1,2,3,` . . . `\}" "6#<&\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Now we extend the definition of " }{XPPEDIT 18 0 "2^x" "6#)\"\"#%\"xG" }{TEXT -1 5 " for " }{TEXT 307 1 "x" }{TEXT -1 12 " in the set " }{XPPEDIT 18 0 "\{0,1, 2, 3, ` . . . `\}" "6#<'\" \"!\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 76 " so that the rule (i) sti ll applies. Thus we need a suitable definition for " }{XPPEDIT 18 0 "2 ^0" "6#*$\"\"#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "If the rule (i) is to apply then, for example, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^0*`.`*2^4=2^(0+4)" "6#/*(\"\"#\"\"!% \".G\"\"\"F%\"\"%)F%,&F&F(F)F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2^4 " "6#*$\"\"#\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "The \+ only way this can be made to be true is by setting" }}{PARA 257 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "2^0=1" "6#/*$\"\"#\"\"!\"\"\"" } {TEXT -1 15 " ------- (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {TEXT 262 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "With this definition the sum rule (i) now holds for any pair of numbers " }{TEXT 274 1 "u" }{TEXT -1 5 " an d " }{TEXT 275 1 "v" }{TEXT -1 12 " in the set " }{XPPEDIT 18 0 "\{0, \+ 1, 2, 3, ` . . . `\}" "6#<'\"\"!\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 26 " of non-negative integers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Now we try to extend the definition of " }{XPPEDIT 18 0 "2^x" "6#)\"\"#%\"xG" }{TEXT -1 7 " where " }{TEXT 276 1 "x" }{TEXT -1 33 " is any member of the set of all " }{TEXT 261 8 "i ntegers" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Z = \{ ` . . . `,-3,-2,-1,0, 1, 2, 3, ` . . . `\}" "6#/%\"ZG<+%( ~.~.~.~G,$\"\"$!\"\",$\"\"#F),$\"\"\"F)\"\"!F-F+F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "If the rule (i) is to apply then, for e xample, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(-4)*`. `*2^4 = 2^(-4+4);" "6#/*()\"\"#,$\"\"%!\"\"\"\"\"%\".GF*F&F()F&,&F(F)F (F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2^0=1" "6#/*$\"\"#\"\"!\"\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The only way this can be made to be true is by setting " }{XPPEDIT 18 0 "2^(-4) = 1/(2^4); " "6#/)\"\"#,$\"\"%!\"\"*&\"\"\"F**$F%F'F(" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "1/16" "6#*&\"\"\"F$\"#;!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 "More generally, when " }{TEXT 309 1 "n" }{TEXT -1 32 " is any positive integer, we set" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2^(-n)=1/2^n" "6#/)\"\"#,$%\"nG!\"\"*&\"\"\"F*) F%F'F(" }{TEXT -1 16 " ------- (iii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "With this definition the \+ sum rule (i) now holds for any pair of numbers " }{TEXT 310 1 "u" } {TEXT -1 5 " and " }{TEXT 311 1 "v" }{TEXT -1 28 " in the set of all i ntegers " }{XPPEDIT 18 0 "Z = \{` . . . `, -3, -2, -1, 0, 1, 2, 3, ` . . . `\}" "6#/%\"ZG<+%(~.~.~.~G,$\"\"$!\"\",$\"\"#F),$\"\"\"F)\"\"!F-F +F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Now we try to extend the definition of " }{XPPEDIT 18 0 "2^x" "6#)\"\"#%\"xG" }{TEXT -1 7 " where " }{TEXT 277 1 "x" } {TEXT -1 33 " is any member of the set of all " }{TEXT 261 16 "rationa l numbers" }{TEXT -1 16 " or \"fractions\" " }}{PARA 257 "" 0 "" {TEXT -1 7 " Q = \{ " }{XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"qG!\"\" " }{TEXT -1 2 " " }{TEXT 312 1 ":" }{TEXT -1 2 " " }{XPPEDIT 18 0 "p , q*epsilon*Z" "6$%\"pG*(%\"qG\"\"\"%(epsilonGF&%\"ZGF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q<>0" "6#0%\"qG\"\"!" }{TEXT -1 4 " \}. " }} {PARA 0 "" 0 "" {TEXT -1 53 "We start by considering rational numbers \+ of the form " }{XPPEDIT 18 0 "x=1/n" "6#/%\"xG*&\"\"\"F&%\"nG!\"\"" } {TEXT -1 8 ", where " }{TEXT 313 1 "n" }{TEXT -1 23 " is a positive in teger " }}{PARA 0 "" 0 "" {TEXT -1 47 "If the rule (i) is to apply the n, for example, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 ^(1/2)*`.`*2^(1/2) = 2^(1/2+1/2);" "6#/*()\"\"#*&\"\"\"F(F&!\"\"F(%\". GF()F&*&F(F(F&F)F()F&,&*&F(F(F&F)F(*&F(F(F&F)F(" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "2^``(1) = 1;" "6#/)\"\"#-%!G6#\"\"\"F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "2^(1/2)" "6#)\" \"#*&\"\"\"F&F$!\"\"" }{TEXT -1 120 " must have the property that, whe n it is multiplied with itself, or squared, the resulting number is 2. This means that " }{XPPEDIT 18 0 "2^(1/2)=sqrt(2)" "6#/)\"\"#*&\"\"\" F'F%!\"\"-%%sqrtG6#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 " Similarly, " }{XPPEDIT 18 0 "2^(1/3)" "6#)\"\"#*&\"\"\"F&\"\"$!\"\"" } {TEXT -1 8 " is the " }{TEXT 261 9 "cube root" }{TEXT -1 26 " of 2, si nce we must have " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2^(1/3)*`.`*2^(1/3)*`.`*2^(1/3) = 2^(1/3+1/3+1/3);" "6#/*,)\"\"#*&\" \"\"F(\"\"$!\"\"F(%\".GF()F&*&F(F(F)F*F(F+F()F&*&F(F(F)F*F()F&,(*&F(F( F)F*F(*&F(F(F)F*F(*&F(F(F)F*F(" }{TEXT -1 5 " = 2." }}{PARA 0 "" 0 "" {TEXT -1 23 "More generally, we set " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(1/n) = ``;" "6#/)\"\"#*&\"\"\"F'%\"nG!\"\"%!G" } {TEXT -1 2 " " }{TEXT 321 1 "n" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt( 2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 16 " ------- (iv), " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{TEXT 264 14 "______________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "the " } {TEXT 261 9 "n th root" }{TEXT -1 56 " of 2. This is the number which, when multiplied itself " }{TEXT 323 1 "n" }{TEXT -1 17 " times, gives 2. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(1/n)*`.`*2^(1/n)*`. . . . `*2^(1/n) = 2^(1 /n+1/n+` . . . `+1/n);" "6#/*,)\"\"#*&\"\"\"F(%\"nG!\"\"F(%\".GF()F&*& F(F(F)F*F(%).~.~.~.~GF()F&*&F(F(F)F*F()F&,**&F(F(F)F*F(*&F(F(F)F*F(%(~ .~.~.~GF(*&F(F(F)F*F(" }{TEXT -1 5 " = 2," }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2^(1/n))^n = 2;" "6#/))\"\"#*&\"\"\"F(%\"nG!\"\"F)F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Now we define " }{XPPEDIT 18 0 "2^(m/n)" "6#)\" \"#*&%\"mG\"\"\"%\"nG!\"\"" }{TEXT -1 8 ", where " }{TEXT 278 1 "m" } {TEXT -1 5 " and " }{TEXT 279 1 "n" }{TEXT -1 27 " are positive intege rs, by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(m/n) = \+ (2^(1/n))^m" "6#/)\"\"#*&%\"mG\"\"\"%\"nG!\"\"))F%*&F(F(F)F*F'" } {TEXT -1 5 " = ( " }{TEXT 322 1 "n" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sq rt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 3 " )" }{XPPEDIT 18 0 "``^m" "6# )%!G%\"mG" }{TEXT -1 14 " ------- (v)," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 265 23 "_______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "2^(-m/n);" "6#)\"\"#,$*&% \"mG\"\"\"%\"nG!\"\"F*" }{TEXT -1 8 ", where " }{TEXT 280 1 "m" } {TEXT -1 5 " and " }{TEXT 281 1 "n" }{TEXT -1 28 " are positive intege rs, by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(-m/n) \+ = 1/(2^(m/n));" "6#/)\"\"#,$*&%\"mG\"\"\"%\"nG!\"\"F+*&F)F))F%*&F(F)F* F+F+" }{TEXT -1 16 " = ------- (vi)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 266 17 "_________________" }{TEXT -1 2 " " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 59 "The formulas (iv), (v) and (vi) complete the definition of " }{XPPEDIT 18 0 "2^x" "6#)\" \"#%\"xG" }{TEXT -1 7 " where " }{TEXT 282 1 "x" }{TEXT -1 81 " is a r ational number, and it turns out that the sum rule (i) still applies w hen " }{TEXT 314 1 "u" }{TEXT -1 5 " and " }{TEXT 315 1 "v" }{TEXT -1 26 " are any rational numbers." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 261 12 "product rule" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2^u)^v = 2^(u*v)" "6#/))\"\"#%\"uG%\"vG)F&*& F'\"\"\"F(F+" }{TEXT -1 15 " ------- (vii) " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 267 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "also applies when " }{TEXT 283 1 "u" }{TEXT -1 5 " and " }{TEXT 284 1 "v" }{TEXT -1 26 " are any rational numbers." } }{PARA 0 "" 0 "" {TEXT -1 45 "It is easy to see why this rule applies \+ when " }{TEXT 285 1 "u" }{TEXT -1 5 " and " }{TEXT 286 1 "v" }{TEXT -1 23 " are positive integers." }}{PARA 0 "" 0 "" {TEXT -1 12 "For exa mple " }{XPPEDIT 18 0 "(2^3)^2=2^6" "6#/*$*$\"\"#\"\"$F&*$F&\"\"'" } {TEXT -1 146 ", because the left and right sides we have 6 two's multi plied together. Note also that the definition (v) is also an example o f the formula (vii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 22 "Now we need to define " }{XPPEDIT 18 0 "2^x" "6#)\"\"#% \"xG" }{TEXT -1 5 " for " }{TEXT 316 1 "x" }{TEXT -1 1 " " }{TEXT 261 15 "any real number" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Th is is the hardest part of the definition." }}{PARA 0 "" 0 "" {TEXT -1 77 "In order to do this we need to recognise that any real number is r ealised as " }{TEXT 261 43 "the limit of a sequence of rational number s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 206 "If we think of a re al number as an infinite decimal, then, if we \"chop off\", or truncat e the decimal expansion after any fixed number of decimal places, we h ave a rational approximation for our real number." }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 60 " is known to be irrational. The first few decimal digits of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 82 " are 3.14159265359, so we c an start to construct the sequence of rational numbers " }}{PARA 257 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "3, 31/10, 314/100, 3141/1000, \+ 31415/100000, 314159/1000000, ` . . . .`" "6)\"\"$*&\"#J\"\"\"\"#5!\" \"*&\"$9$F&\"$+\"F(*&\"%TJF&\"%+5F(*&\"&:9$F&\"'++5F(*&\"'fTJF&\"(+++ \"F(%)~.~.~.~.G" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "whic h give successively closer rational approximations for the number " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 134 "We can get Maple to construct this sequence as far as we want, except that Maple reduces the rational numbers to their \"lowest term s\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sq := [seq(trunc(evalf(Pi*10^n,n+2))/10^n,n=0..11)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sqG7.\"\"$#\"#J\"#5#\"$d\"\"#]# \"%TJ\"%+5#\"%$G'\"%+?#\"'fTJ\"'++5#\"'*p#R\"'+]7#\")jzq:\"(+++&#\")`= $G'\")+++?#\"+`EfTJ\"+++++5#\"+2`=$G'\"+++++?#\"-f`EfTJ\"-+++++5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Denote the " }{TEXT 287 1 "i" } {TEXT -1 29 " th term in this sequence by " }{XPPEDIT 18 0 "x[i]" "6#& %\"xG6#%\"iG" }{TEXT -1 23 ", so that for example, " }{XPPEDIT 18 0 "x [2]=31/10" "6#/&%\"xG6#\"\"#*&\"#J\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "2^x[i];" "6#)\"\"#&% \"xG6#%\"iG" }{TEXT -1 60 " is defined by the formulas above for every rational number " }{XPPEDIT 18 0 "x=x[i]" "6#/%\"xG&F$6#%\"iG" } {TEXT -1 32 " in this sequence. For example, " }{XPPEDIT 18 0 "2^x[2] \+ = 2^(31/10);" "6#/)\"\"#&%\"xG6#F%)F%*&\"#J\"\"\"\"#5!\"\"" }{TEXT -1 46 " is the 10th root of 2 raised to the power 31." }}{PARA 0 "" 0 "" {TEXT -1 79 "Now look at a corresponding sequence of decimal approxima tions for the numbers " }{XPPEDIT 18 0 "2^x[i];" "6#)\"\"#&%\"xG6#%\"i G" }{TEXT -1 71 " (without worrying about how the computations are act ually performed). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 35 "evalf(evalf(map(x->2^x,sq),15),12);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7.$\"\")\"\"!$\"-H+x=u&)!#6$\"-,F4C:)) F)$\"-b/LN@))F)$\"-[#36W#))F)$\"-1&fh\\#))F)$\"-1HQ(\\#))F)$\"-F*\\x\\ #))F)$\"-70y(\\#))F)$\"-ZBy(\\#))F)$\"-`Ey(\\#))F)$\"-3Fy(\\#))F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "These computations suggest that as " }{TEXT 288 1 "x" }{TEXT -1 49 " follows the sequence of rational approximations " }{XPPEDIT 18 0 "x[1],x[2],x[3], ` . . . `" "6&&%\"xG 6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 6 " for " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 38 ", the corresponding sequenc e of values" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2^x[1 ],2^x[2],2^x[3], ` . . . `" "6&)\"\"#&%\"xG6#\"\"\")F$&F&6#F$)F$&F&6# \"\"$%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "is ap proaching some fixed real number. " }}{PARA 0 "" 0 "" {TEXT -1 8 "We t ake " }{XPPEDIT 18 0 "2^Pi" "6#)\"\"#%#PiG" }{TEXT -1 34 " to be the l imit of this sequence." }}{PARA 0 "" 0 "" {TEXT -1 56 "The last decima l value agrees with a 12 digit value for " }{XPPEDIT 18 0 "2^Pi" "6#) \"\"#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(2^Pi,12);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"-3Fy(\\#))!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "When the definition of " }{XPPEDIT 18 0 "2^x" "6#)\"\"#% \"xG" }{TEXT -1 127 " is extended to the set of real numbers in the ma nner suggested, the sum and product rules for exponents still apply, t hat is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE ([2^u*`.`*2^v = 2^(u+v), ``],[(2^u)^v = 2^(u*v), ``])" "6#-%*PIECEWISE G6$7$/*()\"\"#%\"uG\"\"\"%\".GF,)F*%\"vGF,)F*,&F+F,F/F,%!G7$/))F*F+F/) F**&F+F,F/F,F2" }{TEXT -1 2 ", " }}{PARA 259 "" 0 "" {TEXT -1 29 "for \+ any pair of real numbers " }{TEXT 289 1 "u" }{TEXT -1 5 " and " } {TEXT 290 1 "v" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "We also have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^(-u)=1/2^ u" "6#/)\"\"#,$%\"uG!\"\"*&\"\"\"F*)F%F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 47 "which, in conjunction with the sum rule gives: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2^u/2^v=2^(u-v)" "6#/*&)\"\"#%\"uG\"\"\")F&%\"vG!\"\")F&,&F'F(F*F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 62 "More ge nerally, these rules still apply when 2 is replaced by " }{TEXT 261 24 "any positive real number" }{TEXT -1 1 " " }{TEXT 317 1 "a" }{TEXT -1 3 " : " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS E([a^u*`.`*a^v = a^(u+v), ``],[a^u/(a^v) = a^(u-v), ``],[(a^u)^v = a^( u*v), ``]);" "6#-%*PIECEWISEG6%7$/*()%\"aG%\"uG\"\"\"%\".GF,)F*%\"vGF, )F*,&F+F,F/F,%!G7$/*&)F*F+F,)F*F/!\"\")F*,&F+F,F/F8F27$/))F*F+F/)F**&F +F,F/F,F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "f(x)=2^x" "6#/-%\"fG6#%\"xG)\"\"#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "Let f be the function defined by " }{XPPEDIT 18 0 " f(x)=2^x" "6#/-%\"fG6#%\"xG)\"\"#F'" }{TEXT -1 65 ". We compute some values of the f in order to sketch its graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(1) \+ = 2^``(1);" "6#/-%\"fG6#\"\"\")\"\"#-%!G6#F'" }{XPPEDIT 18 0 "``=2" "6 #/%!G\"\"#" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "f(2)=2^2" "6#/-%\"f G6#\"\"#*$F'F'" }{XPPEDIT 18 0 "``=4" "6#/%!G\"\"%" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "f(3)=2^3" "6#/-%\"fG6#\"\"$*$\"\"#F'" }{XPPEDIT 18 0 "``=8" "6#/%!G\"\")" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "f(0)= 2^0" "6#/-%\"fG6#\"\"!*$\"\"#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "f(-1)=2^(-1)" "6#/-%\"fG6#,$\" \"\"!\"\")\"\"#,$F(F)" }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#! \"\"" }{XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" }{TEXT -1 10 ".5, " } {XPPEDIT 18 0 "f(-2)=2^(-2)" "6#/-%\"fG6#,$\"\"#!\"\")F(,$F(F)" } {XPPEDIT 18 0 "``=1/2^2" "6#/%!G*&\"\"\"F&*$\"\"#F(!\"\"" }{XPPEDIT 18 0 "``=1/4" "6#/%!G*&\"\"\"F&\"\"%!\"\"" }{XPPEDIT 18 0 "``=0" "6#/% !G\"\"!" }{TEXT -1 4 ".25," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(-3)=2^(-3)" "6#/-%\"fG6#, $\"\"$!\"\")\"\"#,$F(F)" }{XPPEDIT 18 0 "``=1/2^3" "6#/%!G*&\"\"\"F&*$ \"\"#\"\"$!\"\"" }{XPPEDIT 18 0 "``=1/8" "6#/%!G*&\"\"\"F&\"\")!\"\"" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 22 ".125, \+ " }{XPPEDIT 18 0 "f(1/2)=2^(1/2)" "6#/-%\"fG6#*&\"\"\"F(\"\"#!\"\")F )*&F(F(F)F*" }{XPPEDIT 18 0 "``=sqrt(2)" "6#/%!G-%%sqrtG6#\"\"#" } {TEXT -1 1 " " }{TEXT 324 1 "~" }{TEXT -1 26 " 1.41421, \+ " }{XPPEDIT 18 0 "f(3/2) = 2*`.`*2^(1/2);" "6#/-%\"fG6#*&\"\"$\"\"\" \"\"#!\"\"*(F*F)%\".GF))F**&F)F)F*F+F)" }{XPPEDIT 18 0 "``=2*sqrt(2)" "6#/%!G*&\"\"#\"\"\"-%%sqrtG6#F&F'" }{TEXT -1 1 " " }{TEXT 325 1 "~" } {TEXT -1 13 " 2.82843, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(5/2)=2^(5/2)" "6#/-%\"fG6 #*&\"\"&\"\"\"\"\"#!\"\")F**&F(F)F*F+" }{XPPEDIT 18 0 "``=2^2*`.`*2^(1 /2)" "6#/%!G*(\"\"#F&%\".G\"\"\")F&*&F(F(F&!\"\"F(" }{XPPEDIT 18 0 "`` =4*sqrt(2)" "6#/%!G*&\"\"%\"\"\"-%%sqrtG6#\"\"#F'" }{TEXT -1 1 " " } {TEXT 327 1 "~" }{TEXT -1 44 " 5.65685, \+ " }{XPPEDIT 18 0 "f(-1/2)=2^(-1/2)" "6#/-%\"fG6#,$*&\"\"\"F)\"\"#! \"\"F+)F*,$*&F)F)F*F+F+" }{XPPEDIT 18 0 "``=1/2^(1/2)" "6#/%!G*&\"\"\" F&)\"\"#*&F&F&F(!\"\"F*" }{XPPEDIT 18 0 "``=1/sqrt(2)" "6#/%!G*&\"\"\" F&-%%sqrtG6#\"\"#!\"\"" }{XPPEDIT 18 0 "``=sqrt(2)/2" "6#/%!G*&-%%sqrt G6#\"\"#\"\"\"F)!\"\"" }{TEXT -1 1 " " }{TEXT 328 1 "~" }{TEXT -1 10 " 0.707107," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(-3/2)=2^(-3/2)" "6#/-%\"fG6#,$*&\"\"$\"\"\" \"\"#!\"\"F,)F+,$*&F)F*F+F,F," }{XPPEDIT 18 0 "``=1/2^(3/2)" "6#/%!G*& \"\"\"F&)\"\"#*&\"\"$F&F(!\"\"F+" }{XPPEDIT 18 0 "``=1/(2*sqrt(2))" "6 #/%!G*&\"\"\"F&*&\"\"#F&-%%sqrtG6#F(F&!\"\"" }{XPPEDIT 18 0 "``=sqrt(2 )/4" "6#/%!G*&-%%sqrtG6#\"\"#\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 326 1 "~" }{TEXT -1 37 " 0.353553, " } {XPPEDIT 18 0 "f(-5/2)=2^(-5/2)" "6#/-%\"fG6#,$*&\"\"&\"\"\"\"\"#!\"\" F,)F+,$*&F)F*F+F,F," }{XPPEDIT 18 0 "``=1/2^(5/2)" "6#/%!G*&\"\"\"F&) \"\"#*&\"\"&F&F(!\"\"F+" }{XPPEDIT 18 0 "``=1/(4*sqrt(2)" "6#/%!G*&\" \"\"F&*&\"\"%F&-%%sqrtG6#\"\"#F&!\"\"" }{XPPEDIT 18 0 "``=sqrt(2)/8" " 6#/%!G*&-%%sqrtG6#\"\"#\"\"\"\"\")!\"\"" }{TEXT -1 1 " " }{TEXT 329 1 "~" }{TEXT -1 10 " 0.176777 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "These values are collected together in the foll owing table." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x,`|`,-3, -5/2,-2,-3/2, -1, -1 /2, 0, 1/2, 1,3/2,2,5/2,3],[ f(x),`|`,1/8, sqrt(2)/8, 1/4, sqrt(2)/4, \+ 1/2, sqrt(2)/2, 1, sqrt(2), 2, 2*sqrt(2), 4, 4*sqrt(2), 8],[``,`|`,0*. 125000, 0*.176777, 0*.250000, 0*.353553, 0*.500000, 0*.707107, 1, 1.41 421, 2, 2.82843, 4, 5.65685, 8]])" "6#-%'matrixG6#7%71%\"xG%\"|grG,$\" \"$!\"\",$*&\"\"&\"\"\"\"\"#F,F,,$F1F,,$*&F+F0F1F,F,,$F0F,,$*&F0F0F1F, F,\"\"!*&F0F0F1F,F0*&F+F0F1F,F1*&F/F0F1F,F+71-%\"fG6#F(F)*&F0F0\"\")F, *&-%%sqrtG6#F1F0FAF,*&F0F0\"\"%F,*&-FD6#F1F0FGF,*&F0F0F1F,*&-FD6#F1F0F 1F,F0-FD6#F1F1*&F1F0-FD6#F1F0FG*&FGF0-FD6#F1F0FA71%!GF)*&F8F0-%&FloatG 6$\"'+]7!\"'F0*&F8F0-Fen6$\"'xn7$$!3Wmmm^$zF*$\"3)plW3 `.9a#F37$$!3qmm;X0-Y=F*$\"3e=o5*o&e\"y#F37$$!3s******yPN*p\"F*$\"33[0( p**R#zIF37$$!3#******\\y%3_:F*$\"3uy#fgG9TF37$$!3>LLL#yP!G6F*$\" 3=)>Re,w`d%F37$$!3AMLL$\\k@)**F3$\"3[x6-Q^=1]F37$$!39++]nP.s%)F3$\"3e/ DoM@hebF37$$!36MLL$>7`8(F3$\"3%o,.%zmD)4'F37$$!3U****\\nnrocF3$\"3w!)y xg.z]nF37$$!3w****\\-6Y&\\pV(F37$$!3%4LL$3A-:GF3$\"3f,o' 4\"*QtA)F37$$!3cIL$3l/pZ\"F3$\"3;,)[q#Q%p-*F37$$!392pmm\"4kN$!#?$\"3U2 N>'=in(**F37$$\"3IJL$e5hcY\"F3$\"3*)y()eD;$p5\"F*7$$\"3]++]Z5uqFF3$\"3 Mu*)>+\\t67F*7$$\"3ILLLQJE!=%F3$\"3()>wuZ%)4O8F*7$$\"3s'*****ztVOcF3$ \"3m:*olS(*zZ\"F*7$$\"31+++&y>51(F3$\"3f9iB*H!RJ;F*7$$\"3V'***\\7lPR%) F3$\"3ymy(y_U\\z\"F*7$$\"3'>++]l1)p**F3$\"355qa!p=e*>F*7$$\"3Ymmm?m\\M 6F*$\"3+dR,x%>a>#F*7$$\"3#******\\DC8G\"F*$\"34`!y)>)>1V#F*7$$\"36LLe9 >P99F*$\"3GWbm)oOam#F*7$$\"3y+++\\\"G)f:F*$\"3wl@RJs=[HF*7$$\"3!em;*H6 p'p\"F*$\"3vd\\U5kcTKF*7$$\"3w***\\-Q_(R=F*$\"3]'*\\,fe[zNF*7$$\"3%fmm ;!4iz>F*$\"3\"Gdu4M%*Q%RF*7$$\"3&****\\7G]g7#F*$\"3%e1CJk2_O%F*7$$\"33 MLLx\"zqE#F*$\"3=P_wniZ8[F*7$$\"3,LL$Gf,8T#F*$\"3BRR&pNS&>`F*7$$\"3Enm \"RxHVb#F*$\"39NZ)z1XR(eF*7$$\"3M+++=yv&o#F*$\"3^UNG%y'=MkF*7$$\"3=nm; )=*QOGF*$\"3QcsBm)F*7$$\"3)******zD1N=$F*$\"3cmlqIV7&3*F*7$$\"3))*** \\FH]AD$F*$\"3^*Q9'\\x]G&*F*7$$\"3!***\\PY^7ELF*$\"3E,(f,T8H+\"!#;7$$ \"3!**************R$F*$\"3O:$='Gjgb5F_[l-%&COLORG6&%$RGBG$\"#$*!\"#$\" \"!F]\\lF\\\\l-%*THICKNESSG6#\"\"#-F$6&7/7$$!\"$F]\\l$\"3+++++++]7F37$ $!3++++++++DF*$\"3++++_pwn " 0 "" {MPLTEXT 1 0 19 "seq(1.1^n,n=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6,$\"#6!\"\"$\"$@\"!\"#$\"%J8!\"$$\"&TY\"!\"%$\"'^5;!\"&$\"(h:x\"!\"'$ \")rr[>!\"($\"*\"))eV@!\")$\"+\"pZzN#!\"*$\"+gCu$f#F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Now suppose that loan shark A offers a \+ loan of " }{TEXT 294 1 "L" }{TEXT -1 79 " dollars for a period of one \+ year at an annual rate of 100% interest, so that " }{XPPEDIT 18 0 "2* L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 55 " dollars must be repaid at t he end of the loan period. " }}{PARA 0 "" 0 "" {TEXT -1 150 "A greedie r loan shark B offers the same annual interest rate but compounds the \+ loan on a monthly basis with an effective annual interest rate of 100% ." }}{PARA 0 "" 0 "" {TEXT -1 77 "This means that, at the end of each \+ month, the amount owing is multiplied by " }{XPPEDIT 18 0 "1+1/12" "6# ,&\"\"\"F$*&F$F$\"#7!\"\"F$" }{TEXT -1 75 ". The amount which must be \+ repaid to B at the end of the year is therefore " }{XPPEDIT 18 0 "(1+1 /12)^12" "6#*$,&\"\"\"F%*&F%F%\"#7!\"\"F%F'" }{TEXT -1 30 " times the \+ amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(1+1/12.)^12.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"GNIh#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "An even greedier loan shark C offers the same annual interest rat e as A and B, but compounds the interest weekly." }}{PARA 0 "" 0 "" {TEXT -1 35 "The amount to repaid to C would be " }{XPPEDIT 18 0 "(1+1 /52)^52" "6#*$,&\"\"\"F%*&F%F%\"#_!\"\"F%F'" }{TEXT -1 30 " times the \+ amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(1+1/52.)^52.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Bpf#p#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "This is not a huge amount more than would have to be repaid to loa n shark B." }}{PARA 0 "" 0 "" {TEXT -1 154 "Is there a limit to what c an be obtained by shortening the period over which the interest is com pounded, while the effective annual rate is kept at 100% ?" }}{PARA 0 "" 0 "" {TEXT -1 113 "Let's calculate the amount to repaid if the inte rest is compounded daily when the effective annual rate is 100%. " }} {PARA 0 "" 0 "" {TEXT -1 30 "The amount to repaid would be " } {XPPEDIT 18 0 "(1+1/365)^365;" "6#*$,&\"\"\"F%*&F%F%\"$l$!\"\"F%F'" } {TEXT -1 30 " times the amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(1+1/365.)^365.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+buc9F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "When compounding every ho ur, the amount to repaid would be " }{XPPEDIT 18 0 "(1+1/(365*`.`*24)) ^(365*`.`*24);" "6#),&\"\"\"F%*&F%F%*(\"$l$F%%\".GF%\"#CF%!\"\"F%*(F(F %F)F%F*F%" }{TEXT -1 30 " times the amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "(1+1/(3 65*24.))^(365*24.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+827=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "When c ompounding every minute, the amount to repaid would be " }{XPPEDIT 18 0 "(1+1/(365*`.`*24*`.`*60))^(365*`.`*24*`.`*60);" "6#),&\"\"\"F%*&F%F %*,\"$l$F%%\".GF%\"#CF%F)F%\"#gF%!\"\"F%*,F(F%F)F%F*F%F)F%F+F%" } {TEXT -1 30 " times the amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Digits := 16:\namn t := (1+1/(365*24*60.))^(365*24*60.):\nDigits := 10:\nevalf(amnt);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V#z#=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "When compounding every se cond, the amount to repaid would be " }{XPPEDIT 18 0 "(1+1/(365*`.`*24 *`.`*60^2))^(365*`.`*24*`.`*60^2);" "6#),&\"\"\"F%*&F%F%*,\"$l$F%%\".G F%\"#CF%F)F%\"#g\"\"#!\"\"F%*,F(F%F)F%F*F%F)F%F+F," }{TEXT -1 30 " tim es the amount of the loan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Digits := 20:\namnt := (1+1/(365*24 *60*60.))^(365*24*60*60.):\nDigits := 10:\nevalf(amnt);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+&y\"G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "These numbers are approaching the mat hematical constant " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" } {TEXT -1 48 ", which has an approximate value of 2.718281828." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Two descriptions of " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 54 "F ollowing the ideas in the last section we can define " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 44 " to be the limit of the seq uence of numbers " }{XPPEDIT 18 0 "(1+1/n)^n" "6#),&\"\"\"F%*&F%F%%\"n G!\"\"F%F'" }{TEXT -1 5 " as ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "We can plot some points " }{XPPEDIT 18 0 "``(n,(1+1/n)^n);" "6#-%!G6$%\"nG),&\"\"\"F)*&F)F)F&!\"\"F)F&" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\" #\"\"$%)~.~.~.~~G" }{TEXT -1 48 " to illustrate the convergence of thi s sequence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "terms := [seq([n, (1+1/n)^n],n=1..20)]:\nplot([(1+1/x)^x,exp(1),terms],x=0.001..20,\n s tyle=[line$2,point],color=[grey,navy,COLOR(RGB,.4,0,.9)],\n symbol= circle,linestyle=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7gn7$$\"3-+++++++5!#?$\"3iX3GvE$p+\"!#<7$$ \"3i\"z>'Q!3X#G!#>$\"35+;ijc'o5\"F-7$$\"3;$eRs2;!\\bF1$\"33_Ntjccx6F-7 $$\"3/v$fe6CNF)F1$\"3i=0#H8!4P7F-7$$\"3i;zW:K!)*4\"!#=$\"39'4(>>o\\*G \"F-7$$\"3+v=e\"F-7$$\"3Ym;zhG@pVFA$\"3s?0cY< H#o\"F-7$$\"3gu=-jnnliFA$\"3kwL;V?)z\"=F-7$$\"3G$3_UmS@;)FA$\"3h6Y1K'y 4#>F-7$$\"3V7`hm7\\H5F-$\"3!y#o\\n.@6?F-7$$\"3k;a!oYoFC\"F-$\"3)>&oG6s v#3#F-7$$\"3im;s1%fuX\"F-$\"3h&3B^.E89#F-7$$\"3i;zjY.:s;F-$\"3uAG_>q!* *=#F-7$$\"3#zqZ$\\9\\*4#F-$\"3-lB&)e`glAF-7$$\"3cT&G]N\"p&\\#F-$\"3%4B vMW&f=BF-7$$\"3luoc3M$f!HF-$\"3WQZSJ'fFj?ILF-$\"33g\">= 4`vR#F-7$$\"3eu$*=(f=Jv$F-$\"3\"Q[1#e]rECF-7$$\"3gLeS\\e8)=%F-$\"3)*3. 11Fk^CF-7$$\"3!o\"H%>G*HrXF-$\"3[v\"p8e5.Z#F-7$$\"3W+v'\\b]E+&F-$\"3#> `=V?B%)[#F-7$$\"3A+Dw(*HxNaF-$\"3(*HIuZq7/DF-7$$\"3u+D.Fn;`eF-$\"3!)\\ /A7RN!zF-$\"3)R=9J#z3kDF-7$$\"39u=)fs8LL)F-$\"3#>O'*f _%GrDF-7$$\"3@Cca8>/W()F-$\"3))Hi0+$Rvd#F-7$$\"3#Q3_!*o$fs\"*F-$\"3%eD -J(>_$e#F-7$$\"3S3FN)*z8m&*F-$\"3)Q'GU6Ye)e#F-7$$\"3U:H#>qG1***F-$\"3W \"y/?RNOf#F-7$$\"37a$*>Cb:V5!#;$\"3CEQTy7[)f#F-7$$\"3W7384$Q:3\"Fgv$\" 3O[c!f$GS-EF-7$$\"3oTvQoF*H7\"Fgv$\"3?aoS?'fjg#F-7$$\"3!**\\@6G>e;\"Fg v$\"3]!4#**)=t,h#F-7$$\"3!\\PZ'omr27Fgv$\"3X#=C@fgOh#F-7$$\"3W(=-aaa#[ 7Fgv$\"3o#**o$3r#oh#F-7$$\"3)[i>g%\\E$H\"Fgv$\"3%)R/]vc7?EF-7$$\"3?LG* Gz3PL\"Fgv$\"3zp5^\"z6Hi#F-7$$\"3%*\\i*>B\"*oP\"Fgv$\"3>2+!z2C6bc'31j#F-7$$ \"3k?ld^=0*\\\"Fgv$\"3ya+c)zFGj#F-7$$\"3^Pz1'eE6a\"Fgv$\"3FX**fL/.NEF- 7$$\"3;$e%fHBE#e\"Fgv$\"35nf%*>o2PEF-7$$\"3f(o/3kF`i\"Fgv$\"3b-uv%49\" REF-7$$\"3#omLRl/om\"Fgv$\"3I9Q2k=)4k#F-7$$\"3y\"z:0*3A4#** y\"H!Gk#F-7$$\"3#4_V#**eG^Fgv$\"3m;kz`?e]EF-7$$\"3[7)RlPYl&>Fgv$ \"3S;^O3)=>l#F-7$$\"#?\"\"!$\"3?AW90xH`EF--%'COLOURG6&%$RGBG$\")=THv! \")Fb^lFb^l-%&STYLEG6#%%LINEG-F$6%7S7$F($\"34X!f%G=G=FF-7$FTF]_l7$FhnF ]_l7$FboF]_l7$F\\pF]_l7$FapF]_l7$FfpF]_l7$F[qF]_l7$F`qF]_l7$FeqF]_l7$F jqF]_l7$F_rF]_l7$FdrF]_l7$FirF]_l7$F^sF]_l7$FcsF]_l7$FhsF]_l7$F]tF]_l7 $FbtF]_l7$FgtF]_l7$F\\uF]_l7$FauF]_l7$FfuF]_l7$F[vF]_l7$F`vF]_l7$FevF] _l7$F[wF]_l7$F`wF]_l7$FewF]_l7$FjwF]_l7$F_xF]_l7$FdxF]_l7$FixF]_l7$F^y F]_l7$FcyF]_l7$FhyF]_l7$F]zF]_l7$FbzF]_l7$FgzF]_l7$F\\[lF]_l7$Fa[lF]_l 7$Ff[lF]_l7$F[\\lF]_l7$F`\\lF]_l7$Fe\\lF]_l7$Fj\\lF]_l7$F_]lF]_l7$Fd]l F]_l7$Fi]lF]_l-F_^l6&Fa^l$\")!\\DP\"Fd^lFabl$\")viobFd^lFe^l-F$6%767$$ \"\"\"F[^l$\"\"#F[^l7$F[cl$\"3+++++++]AF-7$$\"\"$F[^l$\"3Cq.Pq.PqBF-7$ $\"\"%F[^l$\"3++++]iSTCF-7$$\"\"&F[^l$\"3')*********>$)[#F-7$$\"\"'F[^ l$\"3d7@urji@DF-7$$\"\"(F[^l$\"3+82/(p*\\YDF-7$$\"\")F[^l$\"3!zM]R^%yl DF-7$$\"\"*F[^l$\"33(>8/EF-7$$\"#7F[^l$\"31yYA!HNIh#F-7$$\"#8F[^l$\"34Kd)y)3g?EF- 7$$\"#9F[^l$\"3Rp3Ic::FEF-7$$\"#:F[^l$\"3:>zs<(yGj#F-7$$\"#;F[^l$\"3'* *fmt\\Gzj#F-7$$\"#F[^l$\"3qW/kEV.]EF-7$Fi]l$\"3U?W90xH`EF--%&COLORG6&Fa^l$Fgcl!\"\"F[^ l$F`elF\\il-Ff^l6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\" Q!6\"-%*LINESTYLEG6#Fbcl-%%VIEWG6$;$Fjbl!\"$Fi]l%(DEFAULTG" 1 2 4 3 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Formally," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(1 )=Limit((1+1/n)^n,n=infinity)" "6#/-%$expG6#\"\"\"-%&LimitG6$),&F'F'*& F'F'%\"nG!\"\"F'F./F.%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "Maple \"knows\" this formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Limit((1+1/n)^n,n \+ = infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$ ),&\"\"\"F(*&F(F(%\"nG!\"\"F(F*/F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "There is an alternative (and better) way to compute " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "For any real number " } {TEXT 291 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(1+x)^n" "6#),&\"\"\" F%%\"xGF%%\"nG" }{TEXT -1 46 " can be expanded, by the binomial theore m, as " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x)^n = 1 +n*x+``(n*(n-1)/2!)*x^2+``(n*(n-1)*(n-2)/3!)*x^3+` . . . `+n*x^(n-1)+x ^n;" "6#/),&\"\"\"F&%\"xGF&%\"nG,0F&F&*&F(F&F'F&F&*&-%!G6#*(F(F&,&F(F& F&!\"\"F&-%*factorialG6#\"\"#F1F&*$F'F5F&F&*&-F-6#**F(F&,&F(F&F&F1F&,& F(F&F5F1F&-F36#\"\"$F1F&*$F'F?F&F&%(~.~.~.~GF&*&F(F&)F',&F(F&F&F1F&F&) F'F(F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " } {XPPEDIT 18 0 "2! = 2*`.`*1;" "6#/-%*factorialG6#\"\"#*(F'\"\"\"%\".GF )F)F)" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "3! = 3*`.`*2*`.`*1" "6#/-%*f actorialG6#\"\"$*,F'\"\"\"%\".GF)\"\"#F)F*F)F)F)" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "4! = 4*`.`*3*`.`*2*`.`*1" "6#/-%*factorialG6#\"\"%*0F' \"\"\"%\".GF)\"\"$F)F*F)\"\"#F)F*F)F)F)" }{TEXT -1 6 ", etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " } {XPPEDIT 18 0 "x=1/n" "6#/%\"xG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 18 ", i t follows that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1 +1/n)^n = 1+n*``(1/n)+``(n*(n-1)/2!)*(1/n)^2+``(n*(n-1)*(n-2)/3!)*(1/n )^3+` . . . `+n*(1/n)^(n-1)+(1/n)^n;" "6#/),&\"\"\"F&*&F&F&%\"nG!\"\"F &F(,0F&F&*&F(F&-%!G6#*&F&F&F(F)F&F&*&-F-6#*(F(F&,&F(F&F&F)F&-%*factori alG6#\"\"#F)F&*$*&F&F&F(F)F8F&F&*&-F-6#**F(F&,&F(F&F&F)F&,&F(F&F8F)F&- F66#\"\"$F)F&*$*&F&F&F(F)FCF&F&%(~.~.~.~GF&*&F(F&)*&F&F&F(F),&F(F&F&F) F&F&)*&F&F&F(F)F(F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 5 " = " }{XPPEDIT 18 0 "1+1+(1-1/n)/2!+ (1-1/n)*(1-2/n)/3!+(1-1/n)*(1-2/n)*(1-3/n)/4!+` . . . `+1/(n^(n-2))+1/ (n^n);" "6#,2\"\"\"F$F$F$*&,&F$F$*&F$F$%\"nG!\"\"F)F$-%*factorialG6#\" \"#F)F$*(,&F$F$*&F$F$F(F)F)F$,&F$F$*&F-F$F(F)F)F$-F+6#\"\"$F)F$**,&F$F $*&F$F$F(F)F)F$,&F$F$*&F-F$F(F)F)F$,&F$F$*&F5F$F(F)F)F$-F+6#\"\"%F)F$% (~.~.~.~GF$*&F$F$)F(,&F(F$F-F)F)F$*&F$F$)F(F(F)F$" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "As , " } {XPPEDIT 18 0 "1/n, 2/n, ` . . . `" "6%*&\"\"\"F$%\"nG!\"\"*&\"\"#F$F% F&%(~.~.~.~G" }{TEXT -1 58 " all tend to 0, and the number of terms te nds to infinity." }}{PARA 0 "" 0 "" {TEXT -1 34 "This leads to the inf inite series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp( 1) = 1+1+1/2!+1/3!+1/4!+ ` . . . `" "6#/-%$expG6#\"\"\",.F'F'F'F'*&F'F '-%*factorialG6#\"\"#!\"\"F'*&F'F'-F+6#\"\"$F.F'*&F'F'-F+6#\"\"%F.F'%( ~.~.~.~GF'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 38 "The sum of the 5 terms shown is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Sum(1/i!,i=0..4);\nvalue(%); \nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'-%*f actorialG6#%\"iG!\"\"/F+;\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ##\"#l\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLL3F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Adding 11 terms gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sum(1/i!,i=0..10);\nvalue(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'-%*factorialG6#%\"i G!\"\"/F+;\"\"!\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(,T')*\"(+)G O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,=G=F!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " . . . and adding 13 te rms gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "Sum(1/i!,i=0..12);\nvalue(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'-%*factorialG6#%\"i G!\"\"/F+;\"\"!\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"*pATg#\")?.! e*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To say that " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 35 " is the sum o f the infinite series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+1+1/2!+1/3!+1/4!+` . . . `;" "6#,.\"\"\"F$F$F$*&F$F$-%*factori alG6#\"\"#!\"\"F$*&F$F$-F'6#\"\"$F*F$*&F$F$-F'6#\"\"%F*F$%(~.~.~.~GF$ " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 11 "means that " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 30 " is the limit of the sequence " }{XPPEDIT 18 0 "a[1],a[2],a[3], ` . . . `" "6&&%\"a G6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 16 " of finite sums " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a[1]=1" "6#/&%\" aG6#\"\"\"F'" }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "a[2]=1+1" "6#/&%\"aG6#\"\"#,&\"\"\"F)F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a[3]=1+1+1/2! " "6#/&%\"aG6#\"\"$,(\"\"\"F)F)F)*&F)F)-%*factorialG6#\"\"#!\"\"F)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a[4 ]=1+1+1/2!+1/3!" "6#/&%\"aG6#\"\"%,*\"\"\"F)F)F)*&F)F)-%*factorialG6# \"\"#!\"\"F)*&F)F)-F,6#\"\"$F/F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a[5]=1+1+1/2!+1/3!+1/4!" "6#/&%\"aG6# \"\"&,,\"\"\"F)F)F)*&F)F)-%*factorialG6#\"\"#!\"\"F)*&F)F)-F,6#\"\"$F/ F)*&F)F)-F,6#\"\"%F/F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 15 " is irrational " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 16 "This proof \+ that " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 98 " is i rrational comes from: \"What is Mathematics\", by Courant and Robbins, Oxford University Press." }}{PARA 0 "" 0 "" {TEXT -1 53 "We need the \+ result that an infinite geometric series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+a*r+a*r^2+a*r^3+` . . . `" "6#,,%\"aG\"\"\" *&F$F%%\"rGF%F%*&F$F%*$F'\"\"#F%F%*&F$F%*$F'\"\"$F%F%%(~.~.~.~GF%" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a bs(r)<1" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT -1 14 ", has the sum " } {XPPEDIT 18 0 "a/(1-r)" "6#*&%\"aG\"\"\",&F%F%%\"rG!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "W e have " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(1)" " 6#-%$expG6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1+1+1/2!+1/3!+1/4 !+1/5!+` . . . ` < 1+1+1/2+1/(2^2)+1/(2^3)+1/(2^4)+` . . . `;" "6#2,0 \"\"\"F%F%F%*&F%F%-%*factorialG6#\"\"#!\"\"F%*&F%F%-F(6#\"\"$F+F%*&F%F %-F(6#\"\"%F+F%*&F%F%-F(6#\"\"&F+F%%(~.~.~.~GF%,0F%F%F%F%*&F%F%F*F+F%* &F%F%*$F*F*F+F%*&F%F%*$F*F/F+F%*&F%F%*$F*F3F+F%F8F%" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 81 "Using the formula for the sum of an infi nite geometric series, the right side is " }{XPPEDIT 18 0 "1 + 1/(1-1/ 2)=3" "6#/,&\"\"\"F%*&F%F%,&F%F%*&F%F%\"\"#!\"\"F*F*F%\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Thus we see that " }{XPPEDIT 18 0 "2=2" "6#1\"\" #%\"qG" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(q+1)+1/((q+1)*(q+2))+` . . . `< 1/3+1/(3^2)+1/(3^3)+1/(3^4)+` . . . `" "6#2,(*&\"\"\"F&,&%\"qGF&F&F&!\"\"F&*&F&F&*&,&F(F&F&F&F&,&F( F&\"\"#F&F&F)F&%(~.~.~.~GF&,,*&F&F&\"\"$F)F&*&F&F&*$F2F.F)F&*&F&F&*$F2 F2F)F&*&F&F&*$F2\"\"%F)F&F/F&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/3 / (1-1/3) = 1/2" "6#/*(\"\"\"F%\"\"$!\"\",&F%F%*&F%F%F&F'F'F'*&F%F%\"\"# F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "We now have a cont radiction to the earlier statement that " }{XPPEDIT 18 0 "q!*exp(1);" "6#*&-%*factorialG6#%\"qG\"\"\"-%$expG6#F(F(" }{TEXT -1 47 " is an int eger, so the original assertion that " }{XPPEDIT 18 0 "exp(1)" "6#-%$e xpG6#\"\"\"" }{TEXT -1 27 " is rational must be false." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "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 12 " is called " }{TEXT 258 3 "the" }{TEXT -1 1 " " }{TEXT 261 20 "exponential function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maple function which computes " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 0 3 "exp" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 82 "plot([exp(x),x+1],x=-2..1.5,y=0..4,color=[red, COLOR(RGB,.5,0,1)],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 314 345 345 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7 $$!3vmmTg*4P#>!#<$\"3uiS#zjS1Y\"F-7$$!3ALe*[SIt&=F1$\"3M>'GZo()3c\"F-7 $$!3em;H_'zEy\"F1$\"3)pSlego=o\"F-7$$!3pm;/mS`2F-7$$!3sm\"HK=2Mc\"F1$\"3%3$4VuJ@%4#F-7$$!3 %)*\\iSB6;\\\"F1$\"3mU?D)>)4]AF-7$$!3em\"H2wftT\"F1$\"3WQRcH/`BCF-7$$! 36+D\"GTYLM\"F1$\"3P%)f#Q*)3(4EF-7$$!37LL3(e9sE\"F1$\"3YX)oBW\\h\"GF-7 $$!3gm;aLw:+7F1$\"3C%RPMPn9,$F-7$$!3&)***\\iQnY7\"F1$\"37mT$>p/wC$F-7$ $!3'****\\(or')[5F1$\"3.SV_TRM.NF-7$$!3E****\\Ph>e(*F-$\"3wp!==GL)oPF- 7$$!3Cm;HdV&[4*F-$\"3o_1;>SJFSF-7$$!3VKL$3#o21$)F-$\"3q+3yuQ%yN%F-7$$! 3qJLL$yyyj(F-$\"3Ev!eH9$)*eYF-7$$!3[**\\i:cggoF-$\"3?U9^U*eb.&F-7$$!3g LLLereshF-$\"3K.GT#>6UR&F-7$$!38)*\\il=sn^cuxF-7$$!3?m;H2chX$\"3CoIcv]gd'*F-7$$\"3!))*****\\Z26SF^t$\"3N:mwZg#4/\"F17$$\"3# >+](=%[V8\"F-$\"3dTrm,*=,7\"F17$$\"3!G+vVt'zV=F-$\"3x&>BKGsC?\"F17$$\" 36***\\78=:j#F-$\"3=$*>^;U-,8F17$$\"3@kmmT3KRLF-$\"3#)pQS*H[kR\"F17$$ \"3K/+]780&4%F-$\"3scZb#Hsg]\"F17$$\"3uJ$3FWb)zZF-$\"3!Hk%po@#Gh\"F17$ $\"3'\\++vBF&GbF-$\"3LnBEpX?QE%>lo1l=F17$$ \"3?,](=s8$ppF-$\"3PF=W@Fe2?F17$$\"3Snm;H_A*o(F-$\"3cz')Q3/Wd@F17$$\"3 Q)*\\Pfe!HW)F-$\"3!HiJ+!pKEBF17$$\"3ARLL$))*yo\"*F-$\"3cb)*yt5Z,DF17$$ \"37NLeR666**F-$\"3#[X;CSEUp#F17$$\"3;nT5g&GZ1\"F1$\"3pcY[(o^+!HF17$$ \"3Y++]Z`PK6F1$\"3a)*eC+&=I5$F17$$\"3\"pm\"z*>1*47F1$\"3!RHU(4,<`LF17$ $\"3[LLL=2Dz7F1$\"3]`tsQe%Rf$F17$$\"33+vVQk=`8F1$\"3E#3nMeO(pQF17$$\"3 I+DccB&RU\"F1$\"3/%*)=7n;HK=2McF-7$FN$!3Z)*\\iSB6;\\ F-7$FS$!3%em\"H2wftTF-7$FX$!37,]7GTYLMF-7$Fgn$!3AJL$3(e9sEF-7$F\\o$!3- mmTNjd,?F-7$Fao$!3Y)***\\iQnY7F-7$Ffo$!3u'****\\(or')[F^t7$F[p$\"3W2++ D'Q!=CF^t7$F`p$\"3aPL3FkX^!*F^t7$Fep$\"3cnm;zJ#Rp\"F-7$Fjp$\"3Iomm;77i BF-7$F_q$\"3^+]P%Q%RRJF-7$Fdq$\"3SmmmTGTFQF-7$Fiq$\"3'=+vV8yAe%F-7$F^r $\"3t-]7.%)3,`F-7$Fcr$\"3[omT5:4^gF-7$Fhr$\"3;o;a)[G)RnF-7$F]s$\"3-LLe kVs#[(F-7$Fbs$\"3!QL3FR%Qa#)F-7$Fgs$\"3)4+Dcr;h#*)F-7$F\\t$\"3#[L$3Fgg ^'*F-7$Fbt$\"3******\\Z26S5F17$Fgt$\"3>+](=%[V86F17$F\\u$\"3G+vVt'zV= \"F17$Fau$\"3#***\\78=:j7F17$Ffu$\"3Umm;%3KRL\"F17$F[v$\"3V++DJ^]49F17 $F`v$\"3=L3FWb)zZ\"F17$Fev$\"3]++vBF&Gb\"F17$Fjv$\"3emT50pHB;F17$F_w$ \"35+v=s8$pp\"F17$Fdw$\"3umm\"H_A*oF17$Fcx$\"3_L$eR666*>F17$Fhx$\"3;nT5g&GZ1#F17$F]y$\"3Y++] Z`PK@F17$Fby$\"3\"pm\"z*>1*4AF17$Fgy$\"3[LLL=2DzAF17$F\\z$\"33+vVQk=`B F17$Faz$\"3I+DccB&RU#F17$Ffz$\"3++++++++DF1-%&COLORG6&F][l$\"\"&F[\\lF a[l$Fe[lF*-Fc[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yFiel-%%VIEWG6$;F($\"# :F[\\l;Fa[l$\"\"%F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG 6#%\"xG" }{TEXT -1 5 " has " }{TEXT 318 1 "y" }{TEXT -1 11 " intercept " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$ex pG6#%\"xG" }{TEXT -1 30 " also passes through the point" }{XPPEDIT 18 0 "``(1,exp(1));" "6#-%!G6$\"\"\"-%$expG6#F&" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 32 " 2.718281828, and also the point" }{XPPEDIT 18 0 " ``(-1,exp(-1));" "6#-%!G6$,$\"\"\"!\"\"-%$expG6#,$F'F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "exp(-1)" "6#-%$expG6#,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 269 1 "~" }{TEXT -1 14 " 0.3678794412." }}{PARA 0 "" 0 "" {TEXT -1 103 "One interesting feature of the graph is that the ta ngent line at the point where the graph crosses the " }{TEXT 297 1 "x " }{TEXT -1 77 " axis has gradient 1. This can be appreciated by zoomi ng in towards the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\" " }{TEXT -1 38 ", where it can be seen that the graph " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 35 " is indistiguishab le from the line " }{XPPEDIT 18 0 "y=x+1" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 33 " in a small region near the point" }{XPPEDIT 18 0 "``(1, 0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([exp(x),x+1],x=- 0.1..0.1,color=[red,COLOR(RGB,.5,0,1)],thickness=[1,2]);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!3/++++++ +5!#=$\"3=&ff.=u$[!*F*7$$!3_nmm;p0k&*!#>$\"3%fdlt/1z3*F*7$$!3ILL$3uCS9;*F*7$$!3Enmm\"4m(G$ )F0$\"34qySiV'3?*F*7$$!3DLL$3i.9!zF0$\"3`MA>X&p-C*F*7$$!3#zmmT!R=0vF0$ \"3uP%=C%R&pF*F*7$$!3U,+]P8#\\4(F0$\"3w2oM\"*>4:$*F*7$$!37om;/siqmF0$ \"3CBOgY$*pa$*F*7$$!3k++](y$pZiF0$\"3!*yxCzsM%R*F*7$$!3OLLL$yaE\"eF0$ \"34(yh8Q0`V*F*7$$!3.nmm\">s%HaF0$\"37/()Q'=H:Z*F*7$$!37+++]$*4)*\\F0$ \"3fie9U]Z7&*F*7$$!3)4+++Db\\c%F0$\"3#35;\"Hrw`&*F*7$$!3i+++]1aZTF0$\" 3W+[*QHHPf*F*7$$!3Inm;/#)[oPF0$\"3&H*)4$oN;I'*F*7$$!3WMLL$=exJ$F0$\"37 &4FddnOn*F*7$$!3VLLLL2$f$HF0$\"3D$za5!\\n5(*F*7$$!3y++]PYx\"\\#F0$\"3a H\\.\"Q,Rv*F*7$$!37NLLL7i)4#F0$\"3%z6s!eYK#z*F*7$$!3S++]P'psm\"F0$\"3s ]Z$zBbY$)*F*7$$!3U++]74_c7F0$\"3e+I4NS8v)*F*7$$!3mLLL$3x%z#)!#?$\"3_g& )*Q.Zv\"**F*7$$!3')RLL3s$QM%Fdr$\"31vu1Oelc**F*7$$!3/Hrmm;zr)*!#A$\"3; mybpG,****F*7$$\"3\\ALLezw5VFdr$\"3B&*Qos+K/5!#<7$$\"3]#****\\PQ#\\\") Fdr$\"3V72!Q`#=35Fgs7$$\"3%GLL$e\"*[H7F0$\"3jZbZ%yqB,\"Fgs7$$\"3[)**** **pvxl\"F0$\"3ZLJ\\Ifr;5Fgs7$$\"3/*****\\_qn2#F0$\"36@sma[)4-\"Fgs7$$ \"3s)***\\i&p@[#F0$\"3#pN]'=B8D5Fgs7$$\"3y)****\\2'HKHF0$\"3?!Q1>6d(H5 Fgs7$$\"3gjmmmZvOLF0$\"3#ex]\")[IR.\"Fgs7$$\"3s+++]2goPF0$\"3XD9EI^SQ5 Fgs7$$\"3)=LLeR<*fTF0$\"3;\"G]KawC/\"Fgs7$$\"3o******\\)Hxe%F0$\"3-wzB Tf%p/\"Fgs7$$\"3+km;H!o-*\\F0$\"3;Wx&>zo60\"Fgs7$$\"3X++]7k.6aF0$\"3K7 z@'4,c0\"Fgs7$$\"3#emmmT9C#eF0$\"3ucsE^D&*f5Fgs7$$\"3U(***\\i!*3`iF0$ \"3?'*[KUt_k5Fgs7$$\"3SMLLL*zym'F0$\"3Kyx!\\2_*o5Fgs7$$\"3?KLL3N1#4(F0 $\"3Et[cCg\\t5Fgs7$$\"3Ynm;HYt7vF0$\"3iHX#3\"Fgs7$$\"30nmmT6KU$)F0$\"33UTwU<+(3\"Fgs7$$\"3[JLLLb dQ()F0$\"3Agw5!e<84\"Fgs7$$\"3/,+]i`1h\"*F0$\"3)*y,]R!Qf4\"Fgs7$$\"3M* ***\\P?Wl&*F0$\"3'e36)H(y.5\"Fgs7$$\"3/+++++++5F*$\"3rZc2=4<06Fgs-%'CO LOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fc[lFb[l-%*THICKNESSG6#\"\"\"-F$6%7S7 $F($\"3A+++++++!*F*7$F.$\"3nLLL3VfV!*F*7$F4$\"3%pm;H[D:3*F*7$F9$\"3nLL $e0$=C\"*F*7$F>$\"3%QLL3RBr;*F*7$FC$\"3Smm\"zjf)4#*F*7$FH$\"3?LLe4;[\\ #*F*7$FM$\"3e***\\i'y]!H*F*7$FR$\"3>LLezs$HL*F*7$FW$\"3G++D@1Bv$*F*7$F fn$\"3Dmmm@Xt=%*F*7$F[o$\"3yLL$3y_qX*F*7$F`o$\"3T+++l+>+&*F*7$Feo$\"3x *****\\Z/Na*F*7$Fjo$\"3m*****\\$fC&e*F*7$F_p$\"3bLLez6:B'*F*7$Fdp$\"3) pmm;=C#o'*F*7$Fip$\"3QmmmEpS1(*F*7$F^q$\"31++DOD#3v*F*7$Fcq$\"3!pmmm(y 8!z*F*7$Fhq$\"3#)***\\i.tK$)*F*7$F]r$\"3o***\\(3zMu)*F*7$Fbr$\"3!omm\" H_?<**F*7$Fhr$\"3)om;zihl&**F*7$F]s$\"3'GLL3#G,****F*7$Fcs$\"3TL$ezw5V +\"Fgs7$Fis$\"3#***\\PQ#\\\"35Fgs7$F^t$\"3TLLe\"*[H75Fgs7$Fct$\"33+++d xd;5Fgs7$Fht$\"3$****\\_qn2-\"Fgs7$F]u$\"3.+]i&p@[-\"Fgs7$Fbu$\"32++vg HKH5Fgs7$Fgu$\"3hmmmZvOL5Fgs7$F\\v$\"31++]2goP5Fgs7$Fav$\"3BL$eR<*fT5F gs7$Ffv$\"3+++])Hxe/\"Fgs7$F[w$\"3mm;H!o-*\\5Fgs7$F`w$\"3/+]7k.6a5Fgs7 $Few$\"3mmm;WTAe5Fgs7$Fjw$\"3%***\\i!*3`i5Fgs7$F_x$\"3KLLL*zym1\"Fgs7$ Fdx$\"3MLL3N1#42\"Fgs7$Fix$\"3im;HYt7v5Fgs7$F^y$\"3!******p(G**y5Fgs7$ Fcy$\"3kmmT6KU$3\"Fgs7$Fhy$\"3ELLLbdQ(3\"Fgs7$F]z$\"33+]i`1h\"4\"Fgs7$ Fbz$\"3/+]P?Wl&4\"Fgs7$Fgz$\"33+++++++6Fgs-%&COLORG6&F^[l$\"\"&!\"\"Fb [l$Fg[lFc[l-Fe[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F\\fl-%%VIEWG6$;$FcelF cel$Fg[lFcel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Logarithm functions (scrap) " }}{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 38 "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 261 19 "log arithm to base 2" }{TEXT -1 4 " of " }{TEXT 298 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "y = log[2]*x;" "6#/%\"yG*&&%$logG6#\"\"#\"\"\"%\"xGF *" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "2^y = x;" "6#/)\"\"#% \"yG%\"xG" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 270 17 "_________________" }{TEXT -1 1 " " }}{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#\"\"#\"\"\"\"\")F)\"\"$/-&F&6#F(6#* &F)F)\"\"%!\"\",$F(F3/-&F&6#F(6#-%%sqrtG6#F(*&F)F)F(F3" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "log[2]*1 = 0,log[2]*2 = 1;" "6$/*&&%$logG6#\"\"#\"\"\"F)F)\"\"!/*&&F&6#F(F)F(F )F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The picture shows \+ the graphs of " }{XPPEDIT 18 0 "y = 2^x;" "6#/%\"yG)\"\"#%\"xG" } {TEXT -1 5 " 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#/%\"yG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 43 "Note that the brackets around the argument " }{TEXT 320 1 "x" }{TEXT -1 51 " are usually omitted in the mathematical notat ion: " }{XPPEDIT 18 0 "log[2]*x" "6#*&&%$logG6#\"\"#\"\"\"%\"xGF(" } {TEXT -1 41 ", but must be present in the Maple form: " }{XPPEDIT 18 0 "log[2](x)" "6#-&%$logG6#\"\"#6#%\"xG" }{TEXT -1 4 " .. " }{TEXT 319 9 "log[2](x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "p1 := plot(2^x,x=-3..3,colo r=blue):\np2 := plot(log[2](x),x=1/8..8,color=red):\np3 := plot(x,x=-3 ..8,color=black,linestyle=2):\nplots[display]([p1,p2,p3],view=[-3..8,- 3..8]);" }}{PARA 13 "" 1 "" {GLPLOT2D 403 387 387 {PLOTDATA 2 "6'-%'CU RVESG6$7S7$$!\"$\"\"!$\"3+++++++]7!#=7$$!3!******\\2<#pG!#<$\"3Q!=__k4 'o8F-7$$!3#)***\\7bBav#F1$\"3oJ=7B6$4[\"F-7$$!36++]K3XFEF1$\"3<0DJizH= ;F-7$$!3%)****\\F)H')\\#F1$\"3Sm6\"RlY%pF-7$$!3;++Dr^b^AF1$\"3+UY&>7w**4#F-7$$!3$****\\7Sw%G@F1$\"35P2l $f#*pG#F-7$$!3*****\\7;)=,?F1$\"3e&y^5#>%z\\#F-7$$!3/++DO\"3V(=F1$\"3# )p#>U1xvs#F-7$$!3#******\\V'zVxMB$F-7$$!3!******\\!)H%*\\\"F1$\"3ND_$y_Jp`$F-7$$!3/+++vl[ p8F1$\"3C*QURf*GqQF-7$$!3\"******\\>iUC\"F1$\"36!H94?P7A%F-7$$!3-++Dhk aI6F1$\"3Q()QH#)oUnXF-7$$!3s******\\XF`**F-$\"3U5jBf+A;]F-7$$!3u****** *>#z2))F-$\"3P!4\"p+.uIaF-7$$!3S++]7RKvuF-$\"3m')[FsW@cfF-7$$!3s,+++P' 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F-F^_m7$$\"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$$\"3%pmTN&*)3hAF1 F``m7$$\"3yKLe90d%\\#F1Fc`m7$$\"3mK$3xB#4PFF1Ff`m7$$\"3)***\\i5\"3#[HF 1Fi`m7$$\"3ULL3P!>i<$F1F\\am7$$\"3&*)****\\jwlQF1Feam7$$\"3Z***\\7%Gw7TF1Fham7$$\"3*emm;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-dF1F]cm 7$$\"3g)*\\P%)*)>RfF1F`cm7$$\"3;MLLjRLnhF1Fccm7$$\"38LLeH\\j+kF1Ffcm7$ $\"3rm;/YS+KmF1Ficm7$$\"3\"3++]B3Y%oF1F\\dm7$$\"3omm\"ziw#)3(F1F_dm7$$ \"3/LLLVl@1tF1Fbdm7$$\"3P**\\P\\feQvF1Fedm7$$\"3o+]i?J*4w(F1Fhdm7$FhzF hz-F[[l6&F][lF*F*F*-%*LINESTYLEG6#\"\"#-%+AXESLABELSG6%Q\"x6\"Q!Feem-% %FONTG6#%(DEFAULTG-%%VIEWG6$;F(FhzF^fm" 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 28 "The domain of the function f" } {XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 8 " where f" } {XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{XPPEDIT 18 0 "``(x) = \+ log[2]*x;" "6#/-%!G6#%\"xG*&&%$logG6#\"\"#\"\"\"F'F-" }{TEXT -1 11 " i s the set" }{XPPEDIT 18 0 "``(0,infinity);" "6#-%!G6$\"\"!%)infinityG " }{TEXT -1 45 " of positive real numbers, and the range of f" } {XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" }{TEXT -1 32 " is the se t of all real numbers." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "log[2]" "6#&%$logG6#\"\"#" }{TEXT -1 6 " as ." }} {PARA 0 "" 0 "" {TEXT -1 42 "More generally, for any positive constant " }{TEXT 299 1 "a" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "a <> 1;" "6# 0%\"aG\"\"\"" }{TEXT -1 27 ", the exponential function " }{XPPEDIT 18 0 "f(x) = a^x;" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 27 " has the inve rse function f" }{XPPEDIT 18 0 "``^(-1)" "6#)%!G,$\"\"\"!\"\"" } {XPPEDIT 18 0 "``(x) = log[a]*x;" "6#/-%!G6#%\"xG*&&%$logG6#%\"aG\"\" \"F'F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "log[a]*x;" "6#*&&%$logG 6#%\"aG\"\"\"%\"xGF(" }{TEXT -1 8 " is the " }{TEXT 261 18 "logarithm \+ to base " }{TEXT 300 1 "a" }{TEXT 261 4 " of " }{TEXT 301 1 "x" } {TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = \+ log[a]*x;" "6#/%\"yG*&&%$logG6#%\"aG\"\"\"%\"xGF*" }{TEXT -1 14 " exac tly when " }{XPPEDIT 18 0 "x = a^y;" "6#/%\"xG)%\"aG%\"yG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 271 16 "______________ __" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "log[a]*x;" "6#*&&%$logG6#%\"a G\"\"\"%\"xGF(" }{TEXT -1 22 " is only defined when " }{TEXT 302 1 "x " }{TEXT -1 13 " is positive." }}{PARA 0 "" 0 "" {TEXT -1 33 "Logarith ms to base 10 are called " }{TEXT 261 17 "common logarithms" }{TEXT -1 31 " and the base is often omitted." }}{PARA 0 "" 0 "" {TEXT -1 48 "The logarithm to base 10 function is denoted by " }{TEXT 0 5 "log10" }{TEXT -1 15 " as well as by " }{TEXT 0 7 "log[10]" }{TEXT -1 10 " in \+ Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for k from 1 to 10 do\n print(log[10](k)=evalf(log1 0(k)))\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"#$\"+d**H5I!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"$$\"+ZD@rZ!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"%$\"+8**f?g!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"&$\"+V+q*)p!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"'$\"+/D^\"y(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"($\"++/)4X)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\")$\"+q)**3.*!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"\"*$\"+%4DCa*!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&log10G6#\"#5$\"+++++5!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The standard rules for exponent s or powers lead to the following 3 basic properties of logarithms: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([log[a](u*v) = log[a]*u+log[a]*v, `-------`*` `(i)],[log[a](u/v) = log[a]*u-log[a]*v, `-------`*``(ii)],[log[a](u^p) = p*log[a]*u, `-------`*``(iii)]);" "6#-%*PIECEWISEG6%7$/-&%$logG6#% \"aG6#*&%\"uG\"\"\"%\"vGF0,&*&&F*6#F,F0F/F0F0*&&F*6#F,F0F1F0F0*&%(---- ---GF0-%!G6#%\"iGF07$/-&F*6#F,6#*&F/F0F1!\"\",&*&&F*6#F,F0F/F0F0*&&F*6 #F,F0F1F0FF*&F:F0-F<6#%#iiGF07$/-&F*6#F,6#)F/%\"pG*(FYF0&F*6#F,F0F/F0* &F:F0-F<6#%$iiiGF0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 27 "Other useful formulas are: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([log[a]*a = 1, `` ],[log[a]*1 = 0, ``]);" "6#-%*PIECEWISEG6$7$/*&&%$logG6#%\"aG\"\"\"F,F -F-%!G7$/*&&F*6#F,F-F-F-\"\"!F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC EWISE([log[a](a^x) = x, ``],[a^(log[a]*x) = x, ``]);" "6#-%*PIECEWISEG 6$7$/-&%$logG6#%\"aG6#)F,%\"xGF/%!G7$/)F,*&&F*6#F,\"\"\"F/F7F/F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The last two formulas say that the functions " }{XPPEDIT 18 0 "f(x) = a^x;" "6#/-%\"fG6#%\"xG)%\"aGF'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "g(x) = log[a]*x;" "6#/-%\"gG6#%\"xG*&&%$logG6#%\"aG\"\" \"F'F-" }{TEXT -1 28 " are inverses of each other." }}{PARA 0 "" 0 "" {TEXT -1 16 "There is also a " }{TEXT 261 22 "change of base formula" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log [b]*x = log[a]*x/(log[a]*b);" "6#/*&&%$logG6#%\"bG\"\"\"%\"xGF)*(&F&6# %\"aGF)F*F)*&&F&6#F.F)F(F)!\"\"" }{TEXT -1 15 " ------- (iv). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Justifications for some \+ of the formulas " }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "log[a](u*v) = log[a] *u+log[a]*v" "6#/-&%$logG6#%\"aG6#*&%\"uG\"\"\"%\"vGF,,&*&&F&6#F(F,F+F ,F,*&&F&6#F(F,F-F,F," }{TEXT -1 14 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 41 "follows from the sum rule for exponents: " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^x*`.`*a^y = a^(x+y);" "6#/*()% \"aG%\"xG\"\"\"%\".GF()F&%\"yGF()F&,&F'F(F+F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x=log[a]*u" "6#/%\"x G*&&%$logG6#%\"aG\"\"\"%\"uGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y= log[a]*v" "6#/%\"yG*&&%$logG6#%\"aG\"\"\"%\"vGF*" }{TEXT -1 10 ", so t hat " }{XPPEDIT 18 0 "u=a^x" "6#/%\"uG)%\"aG%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "v = a^y;" "6#/%\"vG)%\"aG%\"yG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u*`.`*v=a^x*`.`*a^y" "6#/*(%\"uG\"\"\"%\".GF&%\"vGF&*( )%\"aG%\"xGF&F'F&)F+%\"yGF&" }{XPPEDIT 18 0 "`` = a^(x+y)" "6#/%!G)%\" aG,&%\"xG\"\"\"%\"yGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+y=log[ a](u*`.`*v)" "6#/,&%\"xG\"\"\"%\"yGF&-&%$logG6#%\"aG6#*(%\"uGF&%\".GF& %\"vGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u*v) = log[a]* u+log[a]*v" "6#/-&%$logG6#%\"aG6#*&%\"uG\"\"\"%\"vGF,,&*&&F&6#F(F,F+F, F,*&&F&6#F(F,F-F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "T he formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a] (u/v) = log[a]*u-log[a]*v" "6#/-&%$logG6#%\"aG6#*&%\"uG\"\"\"%\"vG!\" \",&*&&F&6#F(F,F+F,F,*&&F&6#F(F,F-F,F." }{TEXT -1 15 " ------- (ii) \+ " }}{PARA 0 "" 0 "" {TEXT -1 57 "is proved in a similar way using the \+ rules for exponents " }{XPPEDIT 18 0 "a^x/a^y = a^(x-y)" "6#/*&)%\"aG% \"xG\"\"\")F&%\"yG!\"\")F&,&F'F(F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "To prove the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a](u^p) = p*log[a]*u" "6#/-&%$logG6#%\"aG6#)% \"uG%\"pG*(F,\"\"\"&F&6#F(F.F+F." }{TEXT -1 16 " ------- (iii) " }} {PARA 0 "" 0 "" {TEXT -1 4 "let " }{XPPEDIT 18 0 "x=log[a]*u" "6#/%\"x G*&&%$logG6#%\"aG\"\"\"%\"uGF*" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "u=a^x" "6#/%\"uG)%\"aG%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " u^p=(a^x)^p" "6#/)%\"uG%\"pG))%\"aG%\"xGF&" }{XPPEDIT 18 0 "``=a^(x*`. `*p)" "6#/%!G)%\"aG*(%\"xG\"\"\"%\".GF)%\"pGF)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*`.`*p=log[a](u^p)" "6#/*(%\"xG\"\"\"%\".GF&%\"pGF &-&%$logG6#%\"aG6#)%\"uGF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lo g[a](u^p)=p*log[a]*u" "6#/-&%$logG6#%\"aG6#)%\"uG%\"pG*(F,\"\"\"&F&6#F (F.F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 21 "To prove the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[b]*x = log[a]*x/(log[a]*b);" "6#/*&&%$log G6#%\"bG\"\"\"%\"xGF)*(&F&6#%\"aGF)F*F)*&&F&6#F.F)F(F)!\"\"" }{TEXT -1 15 " ------- (iv), " }}{PARA 0 "" 0 "" {TEXT -1 4 "let " }{XPPEDIT 18 0 "y=log[b]*x" "6#/%\"yG*&&%$logG6#%\"bG\"\"\"%\"xGF*" }{TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=b ^y" "6#/%\"xG)%\"bG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a]*x= log[a](b^y)" "6#/*&&%$logG6#%\"aG\"\"\"%\"xGF)-&F&6#F(6#)%\"bG%\"yG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "log[a]*x = y*log[a]*b;" "6#/*&&%$l ogG6#%\"aG\"\"\"%\"xGF)*(%\"yGF)&F&6#F(F)%\"bGF)" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=log[a]*x/(log[a]*b)" "6#/%\"yG*(&%$logG6#% \"aG\"\"\"%\"xGF**&&F'6#F)F*%\"bGF*!\"\"" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "log[b]*x = log[a]*x/(log[a]*b)" "6#/*&&%$logG6#%\"bG\" \"\"%\"xGF)*(&F&6#%\"aGF)F*F)*&&F&6#F.F)F(F)!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The last formula can be used to calculate logs to any base using only logs to base 10." }}{PARA 0 "" 0 "" {TEXT -1 28 "For example, we can \+ compute " }{XPPEDIT 18 0 "log[2]*3;" "6#*&&%$logG6#\"\"#\"\"\"\"\"$F( " }{TEXT -1 21 " using this formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'log[2](3)'=evalf(log[10 ](3)/log10(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$logG6#\"\"#6# \"\"$$\"+,D'\\e\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We can 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#$\"+,+++I!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Maple can solve the equat ion " }{XPPEDIT 18 0 "2^x = 3;" "6#/)\"\"#%\"xG\"\"$" }{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# $\"+,D'\\e\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The symbo lic answer is given in terms of " }{TEXT 261 18 "natural logarithms" } {TEXT -1 49 ", which are logarithms to the base of the number " } {XPPEDIT 18 0 "exp(1);" "6#-%$expG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The notation " }{XPPEDIT 18 0 "ln*x=ln(x)" "6#/* &%#lnG\"\"\"%\"xGF&-F%6#F'" }{XPPEDIT 18 0 "`` = log[exp(1)]*x;" "6#/% !G*&&%$logG6#-%$expG6#\"\"\"F,%\"xGF," }{TEXT -1 9 " is used." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG* &%#lnG\"\"\"%\"xGF'" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "x = exp(y);" "6#/%\"xG-%$expG6#%\"yG" }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{TEXT 272 15 "_______________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The pictu re shows the graphs of " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$exp G6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = ln*x;" "6#/%\"yG*&%# lnG\"\"\"%\"xGF'" }{TEXT -1 25 ", together with the line " }{XPPEDIT 18 0 "y = x" "6#/%\"yG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "p1 := plot(exp(x) ,x=-3..3,color=blue):\np2 := plot(ln(x),x=0.01..8,color=red):\np3 := p lot(x,x=-3..8,color=black,linestyle=2):\nplots[display]([p1,p2,p3],vie w=[-3..8,-3..8]);" }}{PARA 13 "" 1 "" {GLPLOT2D 371 314 314 {PLOTDATA 2 "6'-%'CURVESG6$7V7$$!\"$\"\"!$\"3W%R'yOoqy\\!#>7$$!3!******\\2<#pG!# <$\"3M-!*)o\\LVn&F-7$$!3#)***\\7bBav#F1$\"3aWlPp$3#ejF-7$$!36++]K3XFEF 1$\"3S4#R.nViA(F-7$$!3%)****\\F)H')\\#F1$\"3SKY*=ja(>#)F-7$$!3#****\\i 3@/P#F1$\"3(yI7;6PTM*F-7$$!3;++Dr^b^AF1$\"3#Hb!4\"QaB0\"!#=7$$!3$**** \\7Sw%G@F1$\"3it^1A\\=!>\"FM7$$!3*****\\7;)=,?F1$\"3[tt(eyXFM7$$!3!******\\!)H%*\\\"F1$\"3]/HHTU dKAFM7$$!3/+++vl[p8F1$\"3pJ?;zXPUDFM7$$!3\"******\\>iUC\"F1$\"3'Q$)HwI M:)GFM7$$!3-++DhkaI6F1$\"3G9X)[zn&GKFM7$$!3s******\\XF`**FM$\"3K!*[tlR -'p$FM7$$!3u*******>#z2))FM$\"3rMf7&HVk\"F17$$\"3A)****\\d6.B'FM$\"3crX*RHrX'=F17$$\"3s****\\(o3lW(F M$\"32%Q9)Qhq0@F17$$\"35*****\\A))oz)FM$\"3G]rQ0'\\,T#F17$$\"3e******H 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YL$eR%)4;b#F1Fg_m7$$!3smm\"H>$*pJ#F1Fj_m7$$!3gmmT]8#33#F1F]`m7$$!3[L$e 9*>xX=F1F``m7$$!3!om\"HZ6&yi\"F1Fc`m7$$!34+]iNn?-9F1Ff`m7$$!3ym;Hi\\%) o6F1Fi`m7$$!3+,+DJeJi$*FMF\\am7$$!3-LLL38gppFMF_am7$$!3Glm;aq4i[FMFbam 7$$!3o(****\\UY&*[#FMFeam7$$!3_S++](QD2\"F-Fham7$$\"3=*****\\UE&)=#FMF [bm7$$\"3eHL3x[JtUFMF^bm7$$\"3qlmm\"**HBv'FMFabm7$$\"3=kmmm4Q_))FMFdbm 7$$\"3y**\\P\\R_H6F1Fgbm7$$\"3wlmm@$edM\"F1Fjbm7$$\"3-+]P*p,Ie\"F1F]cm 7$$\"3N+]7)\\8*3=F1F`cm7$$\"3'om;/wGY/#F1Fccm7$$\"3%pmTN&*)3hAF1Ffcm7$ $\"3yKLe90d%\\#F1Ficm7$$\"3mK$3xB#4PFF1F\\dm7$$\"3)***\\i5\"3#[HF1F_dm 7$$\"3ULL3P!>i<$F1Fbdm7$$\"3&*)****\\jwlQF1F[em7$$\"3Z***\\7%Gw7TF1F^em7$$\"3*emm;7:_L%F1Fa em7$$\"3Y****\\7/tsXF1Fdem7$$\"3%GL3xcazy%F1Fgem7$$\"3$4++vT^K-&F1Fjem 7$$\"3il;/;ukW_F1F]fm7$$\"3++](o-qgZ&F1F`fm7$$\"3vlm;HzK-dF1Fcfm7$$\"3 g)*\\P%)*)>RfF1Fffm7$$\"3;MLLjRLnhF1Fifm7$$\"38LLeH\\j+kF1F\\gm7$$\"3r m;/YS+KmF1F_gm7$$\"3\"3++]B3Y%oF1Fbgm7$$\"3omm\"ziw#)3(F1Fegm7$$\"3/LL LVl@1tF1Fhgm7$$\"3P**\\P\\feQvF1F[hm7$$\"3o+]i?J*4w(F1F^hm7$Fi^mFi^m-F \\\\l6&F^\\lF*F*F*-%*LINESTYLEG6#\"\"#-%+AXESLABELSG6%Q\"x6\"Q!F[im-%% FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fi^mFdim" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The domai n of the function " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 7 " where \+ " }{XPPEDIT 18 0 "g(x) = ln*x;" "6#/-%\"gG6#%\"xG*&%#lnG\"\"\"F'F*" } {TEXT -1 11 " is the set" }{XPPEDIT 18 0 "``(0,infinity);" "6#-%!G6$\" \"!%)infinityG" }{TEXT -1 44 " of positive real numbers, and the range of " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 32 " is the set of all r eal numbers." }}{PARA 0 "" 0 "" {TEXT -1 19 "Note that ln as ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 pi ctures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "Code for pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> 2^ x:\nmap(f,[10,20,50,100]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"%C 5\"(w&[5\"1CE%o!***e7\"\"@w`?.n\\,%H#G-g]wE\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "f := x -> 2 ^x:\nmap(f,[-3, -5/2,-2,-3/2, -1, -1/2, 0, 1/2, 1,3/2,2,5/2,3]);\neval f(%):\nevalf[6](%);\nmap(evalf@(x->[x,f(x)]),[seq(j/2,j=-6..6)]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(2^Pi);\n2^3.141593;\n2^3.14159;\n2^3141593:\nevalf(%);\nlog[ 10](%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 673 "f := x -> 2^x:\np1 := plot(f(x),x=-3.4..3.4,thicknes s=2,color=COLOR(RGB,.93,0,0)):\np2 := plot([map(evalf@(x->[x,f(x)]),[s eq(j/2,j=-6..6)])$4],style=point,\n symbol=[circle$2,diamond,cross], \n symbolsize=[15,10$3],color=[black,COLOR(RGB,0,1,0)$3]):\np3 := p lot([seq([[i/2,-.3],[i/2,8.4]],i=-6..6),seq([[-3.4,j/2],[3.4,j/2]],j=0 ..16)],\n color=COLOR(RGB,.6,.6,.6)):\nt1 := plots[textplot]([[3. 39,-.19,`x`],[-.2,8.39,`y`]]):\nplots[display]([p||(1..3),t1],view=[-3 .39..3.39,-.19..8.39],\n scaling=unconstrained,font=[HELVETICA,8],\n \+ xtickmarks=[-3=`-3`,-2=`-2`,-1=`-1`,0=0,1=`1`,2=`2`,3=`3`],\n ytic kmarks=[1=`1`,2=`2`,3=`3`,4=`4`,5=`5`,6=`6`,7=`7`,8=`8`],labels=[``,`` ]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }