{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" 0 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 0 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 264 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 265 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 266 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 267 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 268 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 269 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 266 270 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 266 282 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 283 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 284 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 266 285 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 286 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 287 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 266 288 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 21 "Hyperbolic Functions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{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 40 "The hyperbolic sine and cosine functions" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Let f be the exponential function: " }{XPPEDIT 18 0 "f(x) = exp(x)" "6#/-%\"fG6#% \"xG-%$expG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Then t he even part of f is the " }{TEXT 265 17 "hyperbolic cosine" }{TEXT -1 10 " function:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " cosh(x) = (exp(x)+exp(-x))/2" "6#/-%%coshG6#%\"xG*&,&-%$expG6#F'\"\"\" -F+6#,$F'!\"\"F-F-\"\"#F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "Then the odd part of f is the " }{TEXT 265 15 "hyperbolic sine" } {TEXT -1 10 " function:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh(x) = (exp(x)-exp(-x))/2;" "6#/-%%sinhG6#%\"xG*&,&-%$expG6#F '\"\"\"-F+6#,$F'!\"\"F1F-\"\"#F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Since the Maclaurin serie s expansion of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)=1+x+ x^2/2!+x^3/3+x^4/4!+x^5/5!+x^6/6!+x^7/7!+x^8/8!+x^9/9!" "6#/-%$expG6#% \"xG,6\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$F1F/F)*&F' \"\"%-F-6#F3F/F)*&F'\"\"&-F-6#F7F/F)*&F'\"\"'-F-6#F;F/F)*&F'\"\"(-F-6# F?F/F)*&F'\"\")-F-6#FCF/F)*&F'\"\"*-F-6#FGF/F)" }{TEXT -1 11 " + . . . , " }}{PARA 0 "" 0 "" {TEXT -1 73 "the hyperbolic sine and cosine fun ctions have Maclaurin series expansions" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(x)=1+x^2/2!+x^4/4!+x^6/6!+x^8/8!" "6#/-%%co shG6#%\"xG,,\"\"\"F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"%-F-6#F1F /F)*&F'\"\"'-F-6#F5F/F)*&F'\"\")-F-6#F9F/F)" }{TEXT -1 9 " + . . . " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh(x) = x + x^3/3! +x^5/5!+x^7/7!+x^9/9!" "6#/-%%sinhG6#%\"xG,,F'\"\"\"*&F'\"\"$-%*factor ialG6#F+!\"\"F)*&F'\"\"&-F-6#F1F/F)*&F'\"\"(-F-6#F5F/F)*&F'\"\"*-F-6#F 9F/F)" }{TEXT -1 9 " + . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "cosh(x)=taylor(cosh(x),x,10) ;\nsinh(x)=taylor(sinh(x),x,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%coshG6#%\"xG+/F'\"\"\"\"\"!#F)\"\"#F,#F)\"#C\"\"%#F)\"$?(\"\"'#F)\"& ?.%\"\")-%\"OG6#F)\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#% \"xG+/F'\"\"\"F)#F)\"\"'\"\"$#F)\"$?\"\"\"&#F)\"%S]\"\"(#F)\"'!)GO\"\" *-%\"OG6#F)\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=cosh(x)" "6#/%\"yG-%%cos hG6#%\"xG" }{TEXT -1 58 " can be constructed by adding ordinates along the graphs " }{XPPEDIT 18 0 "y=exp(x)/2" "6#/%\"yG*&-%$expG6#%\"xG\" \"\"\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=exp(-x)/2" "6#/ %\"yG*&-%$expG6#,$%\"xG!\"\"\"\"\"\"\"#F+" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot( [exp(x)/2,exp(-x)/2,cosh(x)],x=-2..2,\n color=[brown,blue,red],lines tyle=[2,2,1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 378 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\\o7$Fas$!1$y:&RsO?lFcs7$Fgs$!1E69\"Hp2[\"Fis7$F]t$\"1^FfI)e1Z'Fcs7$Fb t$\"1T%)HMAVD7F\\o7$Fgt$\"17E&4(eqa=F\\o7$F\\u$\"1J7E@+P7DF\\o7$Fau$\" 1xh*pG%ylJF\\o7$Ffu$\"1)[#\\Ag()4QF\\o7$F[v$\"1V<9*4X;a%F\\o7$F`v$\"1! QWq)3u;_F\\o7$Fev$\"1b*y81:)efF\\o7$Fjv$\"1v.K;'fFl'F\\o7$F_w$\"1sb(Gu YxV(F\\o7$Fdw$\"1x\")\\WjF/#)F\\o7$Fiw$\"1EZ`B3bP!*F\\o7$F^x$\"1V*eA0@ q))*F\\o7$Fcx$\"1c)ROPu;3\"F*7$Fhx$\"1D;NQ?[v6F*7$F]y$\"1a^x)eChF\"F*7 $Fby$\"1*=@voO5Q\"F*7$Fgy$\"1&4`uEzA[\"F*7$F\\z$\"1\")ykbkY/;F*7$Faz$ \"1[5&3eS(>F*7$Fj[l$\"1< [4b%z#H@F*-F]\\l6&F_\\lFc\\lFc\\lF`\\l-%%TEXTG6%7$$Fb\\l!\"\"$\"#=Fafl %*y=cosh(x)GF\\\\l-F]fl6%7$$\"#\"*!\"#$\"$i\"Fjfl%\"AG-F]\\l6&F_\\lFc \\lFc\\lFc\\l-F]fl6%7$$\"#&*Fjfl$\"#%*Fjfl%\"BGF^gl-F]fl6%7$F`fl$!#9Fa fl%*y=sinh(x)GFjel-F$6%7$7$$\"1+++pet8))F\\oFc\\l7$Fbhl$\"1+++iN@99F*F ^gl-%*LINESTYLEG6#\"\"#-F$6%7$7$Fc\\l$\"\"\"Fc\\l7$FbhlF_ilF^glFghl-F$ 6%7$7$Fc\\lFehlFdhlF^glFghl-%+AXESLABELSG6$Q\"x6\"%!G-%*AXESTICKSG6$\" \"$\"\"&-%%VIEWG6$;$F*Fafl$\"#:Fafl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 177 "In the picture, the point B on the gra ph y = sinh(x) has y coordinate 1 and the point A on the graph y = cos h(x) is directly above B, that is, it has the same x coordinate as B. " }}{PARA 0 "" 0 "" {TEXT -1 79 "We shall compute the common x coordin ate of A and B and the y coordinate of A. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Let u be the common x coordinat e of A and B." }}{PARA 0 "" 0 "" {TEXT -1 27 "Then sinh(u) = 1, that i s, " }{XPPEDIT 18 0 "(exp(u)-exp(-u))/2=1" "6#/*&,&-%$expG6#%\"uG\"\" \"-F'6#,$F)!\"\"F.F*\"\"#F.F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 " Thus " }{XPPEDIT 18 0 "(exp(u)-exp(-u))=2" "6#/,&-%$expG6 #%\"uG\"\"\"-F&6#,$F(!\"\"F-\"\"#" }{TEXT -1 6 ", or " }{XPPEDIT 18 0 "v - 1/v = 2" "6#/,&%\"vG\"\"\"*&F&F&F%!\"\"F(\"\"#" }{TEXT -1 8 ", \+ where " }{XPPEDIT 18 0 "v = exp(u)" "6#/%\"vG-%$expG6#%\"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 " Then " }{XPPEDIT 18 0 "v^2 \+ - 1 = 2*v" "6#/,&*$%\"vG\"\"#\"\"\"F(!\"\"*&F'F(F&F(" }{TEXT -1 8 " o r " }{XPPEDIT 18 0 "v^2 - 2*v = 1" "6#/,&*$%\"vG\"\"#\"\"\"*&F'F(F& F(!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 " This gives \+ " }{XPPEDIT 18 0 "v^2-2*v+1=2" "6#/,(*$%\"vG\"\"#\"\"\"*&F'F(F&F(!\"\" F(F(F'" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "(v-1)^2 = 2" "6#/*$ ,&%\"vG\"\"\"F'!\"\"\"\"#F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "v - 1 \+ = sqrt(2)" "6#/,&%\"vG\"\"\"F&!\"\"-%%sqrtG6#\"\"#" }{TEXT -1 4 " or \+ " }{XPPEDIT 18 0 "v - 1 = -sqrt(2)" "6#/,&%\"vG\"\"\"F&!\"\",$-%%sqrtG 6#\"\"#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "v = exp(u)" "6#/%\"vG-%$expG6#%\"uG" }{TEXT -1 30 " can not be negative, we have " }{TEXT 259 1 "v" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "exp(u) = 1+sqrt(2)" "6#/-%$expG6#%\"uG,&\"\"\"F)-%%sqrt G6#\"\"#F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 " and " } {XPPEDIT 18 0 "u = ln(1+sqrt(2))" "6#/%\"uG-%#lnG6#,&\"\"\"F)-%%sqrtG6 #\"\"#F)" }{TEXT -1 1 " " }{TEXT 260 1 "~" }{TEXT -1 13 " 0.881373587. " }}{PARA 0 "" 0 "" {TEXT -1 16 "B is the point (" }{XPPEDIT 18 0 "ln( 1+sqrt(2))" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\"#F'" }{TEXT -1 4 ",1). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 26 "__________________ ________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(u) = cosh(ln(1+sqrt(2)))" "6#/-%%coshG6#%\"uG-F%6#-%#lnG6#, &\"\"\"F.-%%sqrtG6#\"\"#F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[exp(ln(1+sqrt(2)))+exp(-ln(1+sqrt(2)))];" "6#7#,&- %$expG6#-%#lnG6#,&\"\"\"F,-%%sqrtG6#\"\"#F,F,-F&6#,$-F)6#,&F,F,-F.6#F0 F,!\"\"F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[1+sqrt(2)+1/(1+sqrt(2))];" "6#7#,(\"\"\"F%-%%sqrtG6#\" \"#F%*&F%F%,&F%F%-F'6#F)F%!\"\"F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 4 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[1+sqrt(2) + 1-sqrt(2)]" "6#7#,*\"\"\"F %-%%sqrtG6#\"\"#F%F%F%-F'6#F)!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 261 1 "~" } {TEXT -1 13 " 1.414213562 " }}{PARA 0 "" 0 "" {TEXT -1 16 "A is the po int (" }{XPPEDIT 18 0 "ln(1+sqrt(2)),sqrt(2)" "6$-%#lnG6#,&\"\"\"F'-%% sqrtG6#\"\"#F'-F)6#F+" }{TEXT -1 2 ")." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 26 "__________________________" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(sinh(u)=1);\ncosh(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+qet8))!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "The hyperbolic function s: tanh, sech, cosech, coth and their graphs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 108 "By analogy with the trigonometric functions we define the functions tanh, sech, cosec h [or csch] and coth by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tanh(x) = sinh(x)/cosh(x)" "6#/-%%tanhG6#%\"xG*&-%%sinhG6#F'\"\" \"-%%coshG6#F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``= (exp(x)-exp(-x))/(exp(x)+exp(-x))" "6#/%!G*&,&-% $expG6#%\"xG\"\"\"-F(6#,$F*!\"\"F/F+,&-F(6#F*F+-F(6#,$F*F/F+F/" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+ (exp(2*x)-1)/(exp(2*x)+1)" "6#/%!G*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,F ,!\"\"F,,&-F(6#*&F+F,F-F,F,F,F,F." }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{TEXT 267 9 "_________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sech(x) = 1/cosh(x)" "6#/-%%sechG6#%\"xG*&\"\"\"F)-%%coshG6#F'!\"\" " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `= 2/(exp(x)+exp(-x))" "6#/%!G*&\"\"#\"\"\",&-%$expG6#%\"xGF'-F*6#,$F, !\"\"F'F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=2*exp(x)/(exp(2*x)+1)" "6#/%!G*(\"\"#\"\"\"-%$expG6# %\"xGF',&-F)6#*&F&F'F+F'F'F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 268 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosech(x) = 1/sinh(x);" "6#/-%'cosechG6#%\"xG*&\"\"\"F)-%%sinhG6#F' !\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/(exp(x)-exp(-x));" "6#/%!G*&\"\"#\"\"\",&-%$expG6#%\"xGF' -F*6#,$F,!\"\"F0F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 2*exp(x)/(exp(2*x)-1);" "6#/%!G*(\"\"#\"\"\"-%$ expG6#%\"xGF',&-F)6#*&F&F'F+F'F'F'!\"\"F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 269 10 "_________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "co th(x) = cosh(x)/sinh(x);" "6#/-%%cothG6#%\"xG*&-%%coshG6#F'\"\"\"-%%si nhG6#F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (exp(x)+exp(-x))/(exp(x)-exp(-x));" "6#/%!G*&,&-%$ expG6#%\"xG\"\"\"-F(6#,$F*!\"\"F+F+,&-F(6#F*F+-F(6#,$F*F/F/F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp( 2*x)+1)/(exp(2*x)-1);" "6#/%!G*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,F,F,F ,,&-F(6#*&F+F,F-F,F,F,!\"\"F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 270 9 "_________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Obt aining the formulas using " }{TEXT 0 15 "convert(..,exp)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 15 " convert(..,exp)" }{TEXT -1 32 " provides the conversions above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "convert(tanh(x),exp);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)-%$expG6#%\"xG\"\"#\"\"\"F,F,!\"\"F,,&F%F,F,F,F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#,$%\"xG\"\"#\"\"\"F+!\"\" F+,&F%F+F+F+F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "convert(sech(x),exp);\nsimplify(normal(%));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&-%$expG6#%\"xGF%*&F%F%F '!\"\"F%F,\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#%\"xG \"\"\",&-F&6#,$F(\"\"#F)F)F)!\"\"F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "convert(csch(x),exp);\nsim plify(normal(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&-%$ expG6#%\"xGF%*&F%F%F'!\"\"F,F,\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&-%$expG6#%\"xG\"\"\",&-F&6#,$F(\"\"#F)F)!\"\"F/F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "convert (coth(x),exp);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$ )-%$expG6#%\"xG\"\"#\"\"\"F,F,F,F,,&F%F,F,!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#,$%\"xG\"\"#\"\"\"F+F+F+,&F%F+F+!\"\"F-" } }}{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 4 "" 0 "" {TEXT -1 13 "The graph o f " }{XPPEDIT 18 0 "y=tanh(x)" "6#/%\"yG-%%tanhG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "plot([tanh(x),1,-1],x=-3..3,y=-1.2..1.2,linestyle=[1 ,3$2],\n color=[red,black$2],ytickmarks=3,title=`y = tanh(x)`);" }} {PARA 13 "" 1 "" {GLPLOT2D 460 175 175 {PLOTDATA 2 "6)-%'CURVESG6%7S7$ $!\"$\"\"!$!3k/t'o`Z0&**!#=7$$!3!******\\2<#pG!#<$!3wAAHc0\"e$**F-7$$! 3#)***\\7bBav#F1$!3zAN-A>Z>**F-7$$!36++]K3XFEF1$!3?N$R#G`5'*)*F-7$$!3% )****\\F)H')\\#F1$!3e[k9H\"yd')*F-7$$!3#****\\i3@/P#F1$!3>W@G9d)o#)*F- 7$$!3;++Dr^b^AF1$!379S?ug$4y*F-7$$!3$****\\7Sw%G@F1$!3Op33)3\\1s*F-7$$ !3*****\\7;)=,?F1$!3#=upcG96k*F-7$$!3/++DO\"3V(=F1$!3R%o6U\"4$)R&*F-7$ $!3#******\\V'zViUC\"F1$!3seflB#omY)F-7$$!3-++DhkaI6F1$!31i%oO/i?6 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6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%*LINESTYLEG6#\"\"\"-F$6%7S7$F($Fe[lF* 7$F/Fj[l7$F5Fj[l7$F:Fj[l7$F?Fj[l7$FDFj[l7$FIFj[l7$FNFj[l7$FSFj[l7$FXFj [l7$FgnFj[l7$F\\oFj[l7$FaoFj[l7$FfoFj[l7$F[pFj[l7$F`pFj[l7$FepFj[l7$Fj pFj[l7$F_qFj[l7$FdqFj[l7$FiqFj[l7$F^rFj[l7$FcrFj[l7$FhrFj[l7$F]sFj[l7$ FcsFj[l7$FhsFj[l7$F]tFj[l7$FbtFj[l7$FgtFj[l7$F\\uFj[l7$FauFj[l7$FfuFj[ l7$F[vFj[l7$F`vFj[l7$FevFj[l7$FjvFj[l7$F_wFj[l7$FdwFj[l7$FiwFj[l7$F^xF j[l7$FcxFj[l7$FhxFj[l7$F]yFj[l7$FbyFj[l7$FgyFj[l7$F\\zFj[l7$FazFj[l7$F fzFj[l-F[[l6&F][lF*F*F*-Fc[l6#Fgz-F$6%7S7$F($!\"\"F*7$F/Fc_l7$F5Fc_l7$ F:Fc_l7$F?Fc_l7$FDFc_l7$FIFc_l7$FNFc_l7$FSFc_l7$FXFc_l7$FgnFc_l7$F\\oF c_l7$FaoFc_l7$FfoFc_l7$F[pFc_l7$F`pFc_l7$FepFc_l7$FjpFc_l7$F_qFc_l7$Fd qFc_l7$FiqFc_l7$F^rFc_l7$FcrFc_l7$FhrFc_l7$F]sFc_l7$FcsFc_l7$FhsFc_l7$ F]tFc_l7$FbtFc_l7$FgtFc_l7$F\\uFc_l7$FauFc_l7$FfuFc_l7$F[vFc_l7$F`vFc_ l7$FevFc_l7$FjvFc_l7$F_wFc_l7$FdwFc_l7$FiwFc_l7$F^xFc_l7$FcxFc_l7$FhxF c_l7$F]yFc_l7$FbyFc_l7$FgyFc_l7$F\\zFc_l7$FazFc_l7$FfzFc_lF[_lF]_l-%*A XESTICKSG6$%(DEFAULTGFgz-%&TITLEG6#%,y~=~tanh(x)G-%+AXESLABELSG6$Q\"x6 \"Q\"yFacl-%%VIEWG6$;F(Ffz;$!#7Fd_l$\"#7Fd_l" 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 37 "For a sam ple point on the graph let " }{XPPEDIT 18 0 "u = ln(1+sqrt(2))" "6#/% \"uG-%#lnG6#,&\"\"\"F)-%%sqrtG6#\"\"#F)" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 14 " 0.8813735869." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }{XPPEDIT 18 0 "sinh(u)=1" "6#/-%%sinhG6#%\"uG\"\"\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "cosh(u)=sqrt(2)" "6#/-%%coshG6#%\"uG-%%sqrtG6#\" \"#" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "tanh(u)=1/sqrt(2)" "6#/-%%tan hG6#%\"uG*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " } {TEXT 274 1 "~" }{TEXT -1 14 " 0.7071067810." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(tanh(x),x=infinity) = Limit((exp(x )-exp(-x))/(exp(x)+exp(-x)),x=infinity)" "6#/-%&LimitG6$-%%tanhG6#%\"x G/F*%)infinityG-F%6$*&,&-%$expG6#F*\"\"\"-F26#,$F*!\"\"F8F4,&-F26#F*F4 -F26#,$F*F8F4F8/F*F," }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit((1-exp( -2*x))/(1+exp(-2*x)),x=infinity) = 1" "6#/-%&LimitG6$*&,&\"\"\"F)-%$ex pG6#,$*&\"\"#F)%\"xGF)!\"\"F1F),&F)F)-F+6#,$*&F/F)F0F)F1F)F1/F0%)infin ityGF)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(tanh(x),x=-infinity) = Limit((exp(x)-exp(-x))/(exp(x)+exp( -x)),x=-infinity)" "6#/-%&LimitG6$-%%tanhG6#%\"xG/F*,$%)infinityG!\"\" -F%6$*&,&-%$expG6#F*\"\"\"-F46#,$F*F.F.F6,&-F46#F*F6-F46#,$F*F.F6F./F* ,$F-F." }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit((exp(2*x)-1)/(exp(2*x )+1),x = -infinity) = -1;" "6#/-%&LimitG6$*&,&-%$expG6#*&\"\"#\"\"\"% \"xGF.F.F.!\"\"F.,&-F*6#*&F-F.F/F.F.F.F.F0/F/,$%)infinityGF0,$F.F0" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "the graph " }{XPPEDIT 18 0 "y=tanh(x)" "6#/%\"yG-%%tanhG6 #%\"xG" }{TEXT -1 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ecmpF17$F^y$\"3WOSslMf?zF17$Fcy$\"3%HV6^!pCP*)F17$Fhy$\"3.Pg3)opR,\"Fc ^l7$F]z$\"3SYfbhm:Z6Fc^l7$Fbz$\"3I;3KxHP08Fc^l7$Fgz$\"3glqT6#\\$y9Fc^l 7$F\\[l$\"3c=zfd5(*y;Fc^l7$Fa[l$\"30D_Z+/\"[!>Fc^l7$Ff[l$\"3kS_^?8-R@F c^l7$F[\\l$\"3$pOWBHxIW#Fc^l7$F`\\l$\"3+z'31Wo9v#Fc^l7$Fehl$\"3e/)pcO# [JHFc^l7$Fe\\l$\"3yp&[IktK7$Fc^l7$F]il$\"3)y/ z&\\z5ENFc^l7$Feil$\"3G;Ep4#4Ow$Fc^l7$F_]lFcjlF[dmF\\jl-%+AXESLABELSG6 $Q\"x6\"Q\"yF\\^n-%&TITLEG6#%,y~=~sech(x)G-%%VIEWG6$;F(F_]l;Fh]l$\"\"% F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "For a sample point on the graph let " } {XPPEDIT 18 0 "u = ln(1+sqrt(2))" "6#/%\"uG-%#lnG6#,&\"\"\"F)-%%sqrtG6 #\"\"#F)" }{TEXT -1 1 " " }{TEXT 278 1 "~" }{TEXT -1 14 " 0.8813735869 ." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "cosh(u)=sqrt(2 )" "6#/-%%coshG6#%\"uG-%%sqrtG6#\"\"#" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "sech(u) = 1/sqrt(2);" "6#/-%%sechG6#%\"uG*&\"\"\"F)-%%sqrtG6#\" \"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*&-%%sqrtG6# \"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{TEXT 279 1 "~" }{TEXT -1 18 " 0. 7071067810.\nAs " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)op eratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " sech(x) = 2/(exp(x)+exp(-x))" "6#/-%%sechG6#%\"xG*&\"\"#\"\"\",&-%$exp G6#F'F*-F-6#,$F'!\"\"F*F2" }{TEXT -1 1 " " }{TEXT 280 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2/exp(x) = 2*exp(-x)" "6#/*&\"\"#\"\"\"-%$expG6# %\"xG!\"\"*&F%F&-F(6#,$F*F+F&" }{TEXT -1 19 ", so the graph of " } {XPPEDIT 18 0 "y=sech(x)" "6#/%\"yG-%%sechG6#%\"xG" }{TEXT -1 26 " app roaches the graph of " }{XPPEDIT 18 0 "y=2*exp(-x)" "6#/%\"yG*&\"\"# \"\"\"-%$expG6#,$%\"xG!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)op eratorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sech(x)=2/(exp(x)+exp(-x))" "6#/-%%sechG6#%\"xG*&\"\"# \"\"\",&-%$expG6#F'F*-F-6#,$F'!\"\"F*F2" }{TEXT -1 1 " " }{TEXT 281 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2/exp(-x) = 2*exp(x)" "6#/*&\"\"# \"\"\"-%$expG6#,$%\"xG!\"\"F,*&F%F&-F(6#F+F&" }{TEXT -1 19 ", so the g raph of " }{XPPEDIT 18 0 "y=sech(x)" "6#/%\"yG-%%sechG6#%\"xG" } {TEXT -1 26 " approaches the graph of " }{XPPEDIT 18 0 "y = 2*exp(x); " "6#/%\"yG*&\"\"#\"\"\"-%$expG6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(sech(x),x = infinity) = Limit(2/(exp(x)+exp(-x)),x = infin ity);" "6#/-%&LimitG6$-%%sechG6#%\"xG/F*%)infinityG-F%6$*&\"\"#\"\"\", &-%$expG6#F*F1-F46#,$F*!\"\"F1F9/F*F," }{TEXT -1 10 " = 0 and " } {XPPEDIT 18 0 "Limit(sech(x),x = -infinity) = Limit(2/(exp(x)+exp(-x)) ,x = -infinity);" "6#/-%&LimitG6$-%%sechG6#%\"xG/F*,$%)infinityG!\"\"- F%6$*&\"\"#\"\"\",&-%$expG6#F*F3-F66#,$F*F.F3F./F*,$F-F." }{TEXT -1 21 " = 0, the graph of " }{XPPEDIT 18 0 "y = sech(x);" "6#/%\"yG-%%s echG6#%\"xG" }{TEXT -1 21 " has the x axis as a " }{TEXT 265 20 "horiz ontal asymptote" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=cosech(x)" "6#/%\"yG-% 'cosechG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "plot([csch(x),2*exp(-x),-2* exp(x)],x=-3..3,y=-3..3,\n linestyle=[1,2$3],color=[red,black$2],disco nt=true,\n title=`y = cosech(x)`);" }}{PARA 13 "" 1 " " {GLPLOT2D 426 306 306 {PLOTDATA 2 "6(-%'CURVESG6&7gn7$$!\"$\"\"!$!3A tA)o'p:#)**!#>7$$!3*GyIw`3Y$H!#<$!3!>)))*)*4Ug1\"!#=7$$!3Ew&pex6x(GF1$ !30d%)R&fS)G6F47$$!3;\\Di;as8GF1$!3D#o3m#R&R?\"F47$$!3?q8D9\\J\\FF1$!3 Wz\"QA)3p%G\"F47$$!3yyXvV0@&o#F1$!3G-=,*\\70P\"F47$$!3sWMP'exdi#F1$!35 $=H#HQHb9F47$$!3[Bl\\,#QUc#F1$!3#\\>-(eJt[:F47$$!3p6Qi\"3%f+DF1$!3!=#y ;k<%=l\"F47$$!3*=p]#pS:PCF1$!3Ea/O.[khF47$$!3Sq0+/\\r\\AF1$!3(o+h*HmGK@F47$ 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')G$F17$Fb[n$!36V\"*)zeU\"HPF17$Fg[n$!38o(GwF79@%F17$F\\\\n$!3d+Vx5#*H ?[F17$Fa\\n$!3Q6;!QxY@W&F17$Ff\\n$!3GgNA(R1\\>'F17$F[]n$!3!fb8G&ecmpF1 7$F`]n$!3WOSslMf?zF17$Fe]n$!3%HV6^!pCP*)F17$Fj]n$!3.Pg3)opR,\"Fez7$F_^ n$!3SYfbhm:Z6Fez7$Fd^n$!3I;3KxHP08Fez7$Fi^n$!3glqT6#\\$y9Fez7$F^_n$!3c =zfd5(*y;Fez7$Fc_n$!30D_Z+/\"[!>Fez7$Fh_n$!3kS_^?8-R@Fez7$F]`n$!3$pOWB HxIW#Fez7$Fb`n$!3+z'31Wo9v#Fez7$$\"31+]PMh%\\o#F1$!3e/)pcO#[JHFez7$Fg` n$!3yp&[IktK7$Fez7$$\"39+++5h(*3GF1$!3)y/z&\\z5 ENFez7$$\"31+]i0j\"[$HF1$!3G;Ep4#4Ow$Fez7$Fe`m$!3fLvj%Q2r,%FezFi`mFcan -%+AXESLABELSG6%Q\"x6\"Q\"yF^\\o-%%FONTG6#%(DEFAULTG-%&TITLEG6#%.y~=~c osech(x)G-%%VIEWG6$;F(Fe`mF[]o" 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 37 "For a sample point o n the graph let " }{XPPEDIT 18 0 "u = ln(1+sqrt(2))" "6#/%\"uG-%#lnG6 #,&\"\"\"F)-%%sqrtG6#\"\"#F)" }{TEXT -1 1 " " }{TEXT 275 1 "~" }{TEXT -1 14 " 0.8813735869." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sinh(u) = 1;" "6#/-%%sinhG6#%\"uG\"\"\"" }{TEXT -1 5 ", so " } {XPPEDIT 18 0 "cosech(u) = 1;" "6#/-%'cosechG6#%\"uG\"\"\"" }{TEXT -1 6 " also." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x -> 0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6 \"\"\"!F*F*F*" }{TEXT -1 3 "+, " }{XPPEDIT 18 0 "sinh(x)->0" "6#f*6#-% %sinhG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 6 "+, s o " }{XPPEDIT 18 0 "cosech(x)->infinity" "6#f*6#-%'cosechG6#%\"xG7\"6$ %)operatorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x ->0" "6#f*6#%\"xG7\"6$%)oper atorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 3 "-, " }{XPPEDIT 18 0 "sinh(x) ->0" "6#f*6#-%%sinhG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" } {TEXT -1 6 "-, so " }{XPPEDIT 18 0 "cosech(x)->-infinity" "6#f*6#-%'co sechG6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F-F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "It follows that the grap h of " }{XPPEDIT 18 0 "y= cosech(x)" "6#/%\"yG-%'cosechG6#%\"xG" } {TEXT -1 21 " has the y axis as a " }{TEXT 265 18 "vertical asymptote " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " \nAs " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF* F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cosech(x) = 2/(exp(x)-exp(-x)); " "6#/-%'cosechG6#%\"xG*&\"\"#\"\"\",&-%$expG6#F'F*-F-6#,$F'!\"\"F2F2 " }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2/e xp(x) = 2*exp(-x)" "6#/*&\"\"#\"\"\"-%$expG6#%\"xG!\"\"*&F%F&-F(6#,$F* F+F&" }{TEXT -1 19 ", so the graph of " }{XPPEDIT 18 0 "y = cosech(x) ;" "6#/%\"yG-%'cosechG6#%\"xG" }{TEXT -1 26 " approaches the graph of \+ " }{XPPEDIT 18 0 "y=2*exp(-x)" "6#/%\"yG*&\"\"#\"\"\"-%$expG6#,$%\"xG !\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infin ityG!\"\"F*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cosech(x) = 2/(exp(x )-exp(-x));" "6#/-%'cosechG6#%\"xG*&\"\"#\"\"\",&-%$expG6#F'F*-F-6#,$F '!\"\"F2F2" }{TEXT -1 1 " " }{TEXT 277 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-2/exp(-x) = -2*exp(x);" "6#/,$*&\"\"#\"\"\"-%$expG6#,$ %\"xG!\"\"F-F-,$*&F&F'-F)6#F,F'F-" }{TEXT -1 19 ", so the graph of " }{XPPEDIT 18 0 "y=sech(x)" "6#/%\"yG-%%sechG6#%\"xG" }{TEXT -1 26 " ap proaches the graph of " }{XPPEDIT 18 0 "y = -2*exp(x);" "6#/%\"yG,$*& \"\"#\"\"\"-%$expG6#%\"xGF(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Li mit(cosech(x),x = infinity) = Limit(2/(exp(x)-exp(-x)),x = infinity); " "6#/-%&LimitG6$-%'cosechG6#%\"xG/F*%)infinityG-F%6$*&\"\"#\"\"\",&-% $expG6#F*F1-F46#,$F*!\"\"F9F9/F*F," }{TEXT -1 10 " = 0 and " } {XPPEDIT 18 0 "Limit(cosech(x),x = -infinity) = Limit(2/(exp(x)-exp(-x )),x = -infinity);" "6#/-%&LimitG6$-%'cosechG6#%\"xG/F*,$%)infinityG! \"\"-F%6$*&\"\"#\"\"\",&-%$expG6#F*F3-F66#,$F*F.F.F./F*,$F-F." }{TEXT -1 21 " = 0, the graph of " }{XPPEDIT 18 0 "y = cosech(x);" "6#/%\"y G-%'cosechG6#%\"xG" }{TEXT -1 21 " has the x axis as a " }{TEXT 265 20 "horizontal asymptote" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=coth(x)" "6#/% \"yG-%%cothG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plot([coth(x),1,-1],x=-2.. 2,y=-5..5,linestyle=[1,3$2],\n color=[red,black$2],discont=true,title =`y = coth(x)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 306 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*\\nZ)H;FcsFiam7$$\"3ckmm;$y*eCFcsFiam7$$\"3f)******R^bJ$FcsFiam7$$\"3 'e*****\\5a`TFcsFiam7$$\"3'o****\\7RV'\\FcsFiam7$$\"3Y'*****\\@fkeFcsF iam7$$\"3_ILLL&4Nn'FcsFiam7$$\"3A*******\\,s`(FcsFiam7$$\"3%[mm;zM)>$) FcsFiam7$$\"3M*******pfa<*FcsFiam7$$\"39HLLeg`!)**FcsFiam7$$\"3w****\\ #G2A3\"F-Fiam7$$\"3;LLL$)G[k6F-Fiam7$$\"3#)****\\7yh]7F-Fiam7$$\"3xmmm ')fdL8F-Fiam7$$\"3bmmm,FT=9F-Fiam7$$\"3FLL$e#pa-:F-Fiam7$$\"3!*******R v&)z:F-Fiam7$$\"3ILLLGUYo;F-Fiam7$$\"3_mmm1^rZF-Fiam7$Fe`mFiam-Fj`m6&F\\amF*F*F*-Fbam6#\"\"$-F$6 %7S7$F($!\"\"F*7$F[bmFa[n7$F^bmFa[n7$FabmFa[n7$FdbmFa[n7$FgbmFa[n7$Fjb mFa[n7$F]cmFa[n7$F`cmFa[n7$FccmFa[n7$FfcmFa[n7$FicmFa[n7$F\\dmFa[n7$F_ dmFa[n7$FbdmFa[n7$FedmFa[n7$FhdmFa[n7$F[emFa[n7$F^emFa[n7$FaemFa[n7$Fd emFa[n7$FgemFa[n7$FjemFa[n7$F]fmFa[n7$F`fmFa[n7$FcfmFa[n7$FffmFa[n7$Fi fmFa[n7$F\\gmFa[n7$F_gmFa[n7$FbgmFa[n7$FegmFa[n7$FhgmFa[n7$F[hmFa[n7$F ^hmFa[n7$FahmFa[n7$FdhmFa[n7$FghmFa[n7$FjhmFa[n7$F]imFa[n7$F`imFa[n7$F cimFa[n7$FfimFa[n7$FiimFa[n7$F\\jmFa[n7$F_jmFa[n7$FbjmFa[n7$FejmFa[n7$ Fe`mFa[nFhjmFjjm-%&TITLEG6#%,y~=~coth(x)G-%+AXESLABELSG6%Q\"x6\"Q\"yF[ _n-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fe`m;$!\"&F*$\"\"&F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Fo r a sample point on the graph let " }{XPPEDIT 18 0 "u = ln(1+sqrt(2)) " "6#/%\"uG-%#lnG6#,&\"\"\"F)-%%sqrtG6#\"\"#F)" }{TEXT -1 1 " " } {TEXT 271 1 "~" }{TEXT -1 14 " 0.8813735869." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sinh(u)=1" "6#/-%%sinhG6#%\"uG\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh(u)=sqrt(2)" "6#/-%%coshG6#%\"u G-%%sqrtG6#\"\"#" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "coth(u) = sqrt(2 );" "6#/-%%cothG6#%\"uG-%%sqrtG6#\"\"#" }{TEXT -1 2 " " }{TEXT 272 1 "~" }{TEXT -1 13 " 1.414213562." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x -> 0" "6#f*6#%\"xG7 \"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 3 "+, " }{XPPEDIT 18 0 "cosh(x)->1" "6#f*6#-%%coshG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\" F-F-F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sinh(x)->0" "6#f*6#-%%sinh G6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 7 "+ , so " }{XPPEDIT 18 0 "coth(x)->infinity" "6#f*6#-%%cothG6#%\"xG7\"6$%)operat orG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x ->0" "6#f*6#%\"xG7\"6$%)operatorG%& arrowG6\"\"\"!F*F*F*" }{TEXT -1 3 "-, " }{XPPEDIT 18 0 "cosh(x)->1" "6 #f*6#-%%coshG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F-F-F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sinh(x)->0" "6#f*6#-%%sinhG6#%\"xG7\"6$%) operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 6 "-, so " }{XPPEDIT 18 0 " coth(x)->-infinity" "6#f*6#-%%cothG6#%\"xG7\"6$%)operatorG%&arrowG6\", $%)infinityG!\"\"F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "It follows that the graph of " }{XPPEDIT 18 0 "y = coth(x);" "6#/% \"yG-%%cothG6#%\"xG" }{TEXT -1 21 " has the y axis as a " }{TEXT 265 18 "vertical asymptote" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(coth(x), x = infinity) = Limit((exp(x)+exp(-x))/(exp(x)-exp(-x)),x = infinity); " "6#/-%&LimitG6$-%%cothG6#%\"xG/F*%)infinityG-F%6$*&,&-%$expG6#F*\"\" \"-F26#,$F*!\"\"F4F4,&-F26#F*F4-F26#,$F*F8F8F8/F*F," }{TEXT -1 4 " = \+ " }{XPPEDIT 18 0 "Limit((1+exp(-2*x))/(1-exp(-2*x)),x = infinity) = 1; " "6#/-%&LimitG6$*&,&\"\"\"F)-%$expG6#,$*&\"\"#F)%\"xGF)!\"\"F)F),&F)F )-F+6#,$*&F/F)F0F)F1F1F1/F0%)infinityGF)" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(coth(x),x = -infinity) \+ = Limit((exp(x)+exp(-x))/(exp(x)-exp(-x)),x = -infinity);" "6#/-%&Limi tG6$-%%cothG6#%\"xG/F*,$%)infinityG!\"\"-F%6$*&,&-%$expG6#F*\"\"\"-F46 #,$F*F.F6F6,&-F46#F*F6-F46#,$F*F.F.F./F*,$F-F." }{TEXT -1 4 " = " } {XPPEDIT 18 0 "Limit((exp(2*x)+1)/(exp(2*x)-1),x = -infinity) = -1;" " 6#/-%&LimitG6$*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF.F.F.F.F.,&-F*6#*&F-F.F/ F.F.F.!\"\"F4/F/,$%)infinityGF4,$F.F4" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "the graph " }{XPPEDIT 18 0 "y=coth(x)" "6#/%\"yG-%%cothG6#%\"xG" }{TEXT -1 15 " has the line s " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y=-1" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 4 " as " } {TEXT 265 21 "horizontal asymptotes" }{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 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "Some indentities involving hyperbolic functions " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 115 "There are analagous identities involving hyperbolic functions \+ to many identities involving trigonometric functions." }}{PARA 0 "" 0 "" {TEXT -1 27 "The most basic identity is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(x)^2-sinh(x)^2=1" "6#/,&*$-%%coshG 6#%\"xG\"\"#\"\"\"*$-%%sinhG6#F)F*!\"\"F+" }{TEXT -1 14 " ------- (i), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 14 "______________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "which may be compared and/or contrasted with the trigonom etric identity " }{XPPEDIT 18 0 "cos(x)^2+sin(x)^2=1" "6#/,&*$-%$cosG6 #%\"xG\"\"#\"\"\"*$-%$sinG6#F)F*F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 47 "The identity (i) can be proved by substituting " } {XPPEDIT 18 0 "cosh(x)=(exp(x)+exp(x))/2" "6#/-%%coshG6#%\"xG*&,&-%$ex pG6#F'\"\"\"-F+6#F'F-F-\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sinh(x)=(exp(x)-exp(-x))/2" "6#/-%%sinhG6#%\"xG*&,&-%$expG6#F'\"\"\"- F+6#,$F'!\"\"F1F-\"\"#F1" }{TEXT -1 30 " in the left hand side to get \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(x)^2-sinh(x) ^2 = ((exp(x)+exp(x))/2)^2 -((exp(x)+exp(x))/2)^2" "6#/,&*$-%%coshG6#% \"xG\"\"#\"\"\"*$-%%sinhG6#F)F*!\"\",&*$*&,&-%$expG6#F)F+-F66#F)F+F+F* F0F*F+*$*&,&-F66#F)F+-F66#F)F+F+F*F0F*F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (exp(2*x)+2+exp(-2*x))/4 - (exp(2*x)-2+exp(-2*x))/4" "6#/%!G,&*&,(-%$expG6#*&\"\"#\"\"\"%\"xGF-F -F,F--F)6#,$*&F,F-F.F-!\"\"F-F-\"\"%F3F-*&,(-F)6#*&F,F-F.F-F-F,F3-F)6# ,$*&F,F-F.F-F3F-F-F4F3F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/2+1/2;" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'*&F'F'F(F )F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 "= 1. " }}{PARA 0 "" 0 "" {TEXT -1 33 "On dividing both sides of (i) by " }{XPPEDIT 18 0 "cosh(x)^2" "6#*$-%%coshG6#%\"xG\"\"#" }{TEXT -1 26 " we obtain the \+ identity: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-tanh (x)^2 = sech(x)^2;" "6#/,&\"\"\"F%*$-%%tanhG6#%\"xG\"\"#!\"\"*$-%%sech G6#F*F+" }{TEXT -1 15 " ------- (ii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 283 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 37 "and on dividing both sides of (i) by " }{XPPEDIT 18 0 " sinh(x)^2" "6#*$-%%sinhG6#%\"xG\"\"#" }{TEXT -1 26 " we obtain the ide ntity: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "coth(x)^2 -1 = cosech(x)^2;" "6#/,&*$-%%cothG6#%\"xG\"\"#\"\"\"F+!\"\"*$-%'cosec hG6#F)F*" }{TEXT -1 15 " ------- (iii)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Corresponding to the do uble-angle trigonometric formulas " }{XPPEDIT 18 0 "sin(2*x)=2*sin(x)* cos(x)" "6#/-%$sinG6#*&\"\"#\"\"\"%\"xGF)*(F(F)-F%6#F*F)-%$cosG6#F*F) " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(2*x) = cos(x)^2-sin(x)^2" "6 #/-%$cosG6#*&\"\"#\"\"\"%\"xGF),&*$-F%6#F*F(F)*$-%$sinG6#F*F(!\"\"" } {TEXT -1 23 ", we have the formulas:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "PIEC EWISE([sinh(2*x) = 2*sinh(x)*cosh(x) , ``],[cosh(2*x) = cosh(x)^2+sinh (x)^2 ,`` ])" "6#-%*PIECEWISEG6$7$/-%%sinhG6#*&\"\"#\"\"\"%\"xGF-*(F,F --F)6#F.F--%%coshG6#F.F-%!G7$/-F36#*&F,F-F.F-,&*$-F36#F.F,F-*$-F)6#F.F ,F-F5" }{TEXT -1 14 " ------- (iv) " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 285 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "These can be proved in the same manner as (i) by substitu ting " }{XPPEDIT 18 0 "cosh(x)=(exp(x)+exp(x))/2" "6#/-%%coshG6#%\"xG *&,&-%$expG6#F'\"\"\"-F+6#F'F-F-\"\"#!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sinh(x)=(exp(x)-exp(-x))/2" "6#/-%%sinhG6#%\"xG*&,&-%$e xpG6#F'\"\"\"-F+6#,$F'!\"\"F1F-\"\"#F1" }{TEXT -1 2 " ." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sinh(x)*cosh(x) = 2*[(exp(x)-e xp(-x))/2]*[(exp(x)+exp(-x))/2]" "6#/*(\"\"#\"\"\"-%%sinhG6#%\"xGF&-%% coshG6#F*F&*(F%F&7#*&,&-%$expG6#F*F&-F36#,$F*!\"\"F8F&F%F8F&7#*&,&-F36 #F*F&-F36#,$F*F8F&F&F%F8F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (exp(2*x)-exp(-2*x))/2" "6#/%!G*&,&-%$exp G6#*&\"\"#\"\"\"%\"xGF,F,-F(6#,$*&F+F,F-F,!\"\"F2F,F+F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= sinh(2*x)" " 6#/%!G-%%sinhG6#*&\"\"#\"\"\"%\"xGF*" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(x)^2+sinh(x)^2 = ((exp(x)+exp(x))/2)^2-((exp(x)+exp(x))/2)^2" "6 #/,&*$-%%coshG6#%\"xG\"\"#\"\"\"*$-%%sinhG6#F)F*F+,&*$*&,&-%$expG6#F)F +-F56#F)F+F+F*!\"\"F*F+*$*&,&-F56#F)F+-F56#F)F+F+F*F9F*F9" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (exp(2*x) +2+exp(-2*x))/4+(exp(2*x)-2+exp(-2*x))/4" "6#/%!G,&*&,(-%$expG6#*&\"\" #\"\"\"%\"xGF-F-F,F--F)6#,$*&F,F-F.F-!\"\"F-F-\"\"%F3F-*&,(-F)6#*&F,F- F.F-F-F,F3-F)6#,$*&F,F-F.F-F3F-F-F4F3F-" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (exp(2*x)+exp(-2*x))/2" "6#/ %!G*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,-F(6#,$*&F+F,F-F,!\"\"F,F,F+F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= cosh(2*x)" "6#/%!G-%%coshG6#*&\"\"#\"\"\"%\"xGF*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 77 "Alternatively, the formulas (vi) are spec ial cases of two of the identities: " }}{PARA 256 "" 0 "" {TEXT -1 22 " " }{XPPEDIT 18 0 "PIECEWISE([sinh(x+y) = sinh(x )*cosh(y)+cosh(x)*sinh(y),``], [sinh(x-y) = sinh(x)*cosh(y)-cosh(x)*si nh(y),``], [cosh(x+y) = cosh(x)*cosh(y)+sinh(x)*sinh(y),``], [cosh(x-y ) = cosh(x)*cosh(y)-sinh(x)*sinh(y) ,``])" "6#-%*PIECEWISEG6&7$/-%%sin hG6#,&%\"xG\"\"\"%\"yGF-,&*&-F)6#F,F--%%coshG6#F.F-F-*&-F46#F,F--F)6#F .F-F-%!G7$/-F)6#,&F,F-F.!\"\",&*&-F)6#F,F--F46#F.F-F-*&-F46#F,F--F)6#F .F-FAF;7$/-F46#,&F,F-F.F-,&*&-F46#F,F--F46#F.F-F-*&-F)6#F,F--F)6#F.F-F -F;7$/-F46#,&F,F-F.FA,&*&-F46#F,F--F46#F.F-F-*&-F)6#F,F--F)6#F.F-FAF; " }{TEXT -1 13 " ------- (v)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 286 20 "____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The following proof of t he first of these formulas provides the method of proof for the other \+ three." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh(x)*cosh(y)+cosh(x)*sinh(y) = [(exp(x)-exp(- x))/2]*[(exp(y)+exp(-y))/2]+[(exp(x)+exp(-x))/2]*[(exp(y)-exp(-y))/2] " "6#/,&*&-%%sinhG6#%\"xG\"\"\"-%%coshG6#%\"yGF*F**&-F,6#F)F*-F'6#F.F* F*,&*&7#*&,&-%$expG6#F)F*-F:6#,$F)!\"\"F?F*\"\"#F?F*7#*&,&-F:6#F.F*-F: 6#,$F.F?F*F*F@F?F*F**&7#*&,&-F:6#F)F*-F:6#,$F)F?F*F*F@F?F*7#*&,&-F:6#F .F*-F:6#,$F.F?F?F*F@F?F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp(x+y)+exp(x-y)-exp(-x+y)-exp(-x-y))/ 4+(exp(x+y)-exp(x-y)+exp(-x+y)-exp(-x-y))/4;" "6#/%!G,&*&,*-%$expG6#,& %\"xG\"\"\"%\"yGF-F--F)6#,&F,F-F.!\"\"F--F)6#,&F,F2F.F-F2-F)6#,&F,F2F. F2F2F-\"\"%F2F-*&,*-F)6#,&F,F-F.F-F--F)6#,&F,F-F.F2F2-F)6#,&F,F2F.F-F- -F)6#,&F,F2F.F2F2F-F9F2F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= ((exp(x+y)-exp(-x-y))/2" "6#/%!G*&,&-%$ex pG6#,&%\"xG\"\"\"%\"yGF,F,-F(6#,&F+!\"\"F-F1F1F,\"\"#F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sinh(x+y)" "6#/%!G-%%sinhG6#,&%\"xG\"\"\"%\"yGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure \+ " }{TEXT 0 6 "expand" }{TEXT -1 24 " 'knows' these formulas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "' sinh(x+y)'=expand(sinh(x+y));\n'sinh(x-y)'=expand(sinh(x-y));\n'cosh(x +y)'=expand(cosh(x+y));\n'cosh(x-y)'=expand(cosh(x-y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%sinhG6#,&%\"xG\"\"\"%\"yGF),&*&-F%6#F(F)-%%c oshG6#F*F)F)*&-F0F.F)-F%F1F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%% sinhG6#,&%\"xG\"\"\"%\"yG!\"\",&*&-F%6#F(F)-%%coshG6#F*F)F)*&-F1F/F)-F %F2F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%coshG6#,&%\"xG\"\"\"%\" yGF),&*&-F%6#F(F)-F%6#F*F)F)*&-%%sinhGF.F)-F3F0F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%coshG6#,&%\"xG\"\"\"%\"yG!\"\",&*&-F%6#F(F)-F%6#F *F)F)*&-%%sinhGF/F)-F4F1F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 43 "Geometrical interpretation of the identity " }{XPPEDIT 18 0 "cosh(x)^2-sinh(x)^2=1" "6#/,&*$-%%coshG6#%\"xG\"\"#\"\"\"*$-%%si nhG6#F)F*!\"\"F+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The identity " }{XPPEDIT 18 0 "cosh(x)^2-sinh(x)^2 = 1" "6#/,&*$-%%coshG6#%\"xG\"\"#\"\"\"*$-%%sin hG6#F)F*!\"\"F+" }{TEXT -1 36 " means that, for any t, the point ( " } {XPPEDIT 18 0 "cosh(t),sinh(t)" "6$-%%coshG6#%\"tG-%%sinhG6#F&" } {TEXT -1 15 " ) lies on the " }{TEXT 265 21 "rectangular hyperbola" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$% \"yGF'!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "This exp lains why the functions are called hyperbolic." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6$7X7$$ \"3W;.TqJJWT!#<$!3CNt:Un&=-%F*7$$\"3;q^xwZ\\kRF*$!3*zw>fv-j$QF*7$$\"3= py!f[%)Hz$F*$!3w'ou(H#)yeOF*7$$\"3Yp=L@MB]OF*$!3PX#45d%e5NF*7$$\"3\"po #*))HtK^$F*$!3[/,M2.&zO$F*7$$\"3I\"e#Gp:FDKF*$!3QMK<:&Hj1$F*7$$\"3yp^R GL^hHF*$!3WrdzMEd(y#F*7$$\"3M6BG*4QHs#F*$!3H3EMy\\mKDF*7$$\"36UP%G#GR@ DF*$!3'oe\"3+0h9BF*7$$\"3%QDpB,%4JBF*$!3eW$RZO1d5#F*7$$\"3ye/fTI[_@F*$ !3:q5D!4\"41>F*7$$\"3AwGPYUY\"*>F*$!3DOGD\"F*$!3_fp%*=objlFgp7$$\"3'\\6Rdsf+9\"F*$!3yQ) *H$p;[Z&Fgp7$$\"39@,q$)[p)4\"F*$!3/)=yehe6b%Fgp7$$\"3)Gi;k0B>1\"F*$!3y ')Q&Q\">CtNFgp7$$\"3r\"3rU*f,N5F*$!3Ghx(GW@%pEFgp7$$\"3(3yLXL`^,\"F*$! 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In this last case the parameter t can be visualised as the angle which the radius \+ line joining the origin to the point ( " }{XPPEDIT 18 0 "(cos(t),sin(t ))" "6$-%$cosG6#%\"tG-%$sinG6#F&" }{TEXT -1 74 " ) makes with the posi tive direction of the x axis (measured in radians). 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "The derivatives of the hyperbol ic functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ sinh(x)] = cosh(x)" "6#/7#-%% sinhG6#%\"xG-%%coshG6#F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ cosh(x)] = sinh(x)" "6#/7#-%%coshG6#%\"xG-%%si nhG6#F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 287 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "These are easy to check from the defi nitions " }{XPPEDIT 18 0 "sinh(x) = (exp(x)-exp(x))/2;" "6#/-%%sinhG6# %\"xG*&,&-%$expG6#F'\"\"\"-F+6#F'!\"\"F-\"\"#F0" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cosh(x) = (exp(x)+exp(-x))/2;" "6#/-%%coshG6#%\"xG*&,&- %$expG6#F'\"\"\"-F+6#,$F'!\"\"F-F-\"\"#F1" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 102 "Then we can use the differentiation rules to obta in the derivatives of the other hyperbolic functions." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[tanh(x)] = sech(x)^2;" "6#/7#-%%tanhG 6#%\"xG*$-%%sechG6#F(\"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sech(x)] = -sech(x)*tanh(x);" "6#/7#-%%sechG6#% \"xG,$*&-F&6#F(\"\"\"-%%tanhG6#F(F-!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[cosech(x)] = -cosech(x)*coth(x); " "6#/7#-%'cosechG6#%\"xG,$*&-F&6#F(\"\"\"-%%cothG6#F(F-!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#* &%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[coth(x)] = -c osech(x)^2;" "6#/7#-%%cothG6#%\"xG,$*$-%'cosechG6#F(\"\"#!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 288 14 "_____ _________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "alias(cosech=csch):\nDiff(sinh(x), x)=diff(sinh(x),x);\nDiff(cosh(x),x)=diff(cosh(x),x);\nDiff(tanh(x),x) =diff(tanh(x),x);\nDiff(sech(x),x)=diff(sech(x),x);\nDiff(cosech(x),x) =diff(cosech(x),x);\nDiff(coth(x),x)=diff(coth(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%%sinhG6#%\"xGF*-%%coshGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%%coshG6#%\"xGF*-%%sinhGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%%tanhG6#%\"xGF*,&\"\"\"F, *$)F'\"\"#F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%%sec hG6#%\"xGF*,$*&F'\"\"\"-%%tanhGF)F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'cosechG6#%\"xGF*,$*&F'\"\"\"-%%cothGF)F-!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%%cothG6#%\"xGF*,&\"\"\"F, *$)F'\"\"#F,!\"\"" }}}{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 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 23 "Code for first pic ture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "with(plots):\naa := evalf(ln((1+sqrt(2)))): bb := ev alf(sqrt(2)):\np1 := plot([cosh(x),sinh(x)],x=-1.5..1.5,color=[red,blu e]):\np2 := plot([[[aa,0],[aa,bb]],[[0,1],[aa,1]],\n [[0,bb],[aa, bb]]],color=black,linestyle=2):\nt1 := textplot([-.8,1.8,`y=cosh(x)`], color=red):\nt2 := textplot([-.8,-1.4,`y=sinh(x)`],color=blue):\nt3 := textplot([[.91,1.62,`A`],[.95,.94,`B`]],color=black):\ndisplay(\{p1,p 2,t1,t2,t3\},tickmarks=[3,5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "Code for 2nd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 402 "f := x->sqrt(x^2-1):\n p1 := plot([cosh(t),sinh(t),t=-2.1..2.1],color=red):\np2 := plot([-cos h(t),sinh(t),t=-2.1..2.1],color=red):\np3 := plot([x,-x],x=-3..3,color =black,linestyle=4):\npt := [1.5,f(1.5)]:\nall := [circle,diamond,cros s]:\np4 := plot([[pt]$3],style=point,symbol=all,color=red):\nt1 := plo ts[textplot]([2.5,1.2,`(cosh(t),sinh(t))`],color=red):\nplots[display] ([p1,p2,p3,p4,t1],view=[-3..3,-3..3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "Code for 3rd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 418 "f := x->sqr t(1-x^2):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=red):\npt := [.5 5,f(.55)]:\np2 := plot([[0,0],pt],color=red):\nall := [circle,diamond, cross]:\np3 := plot([[pt]$3],style=point,symbol=all,color=red):\np4 := plot([.25*cos(t),.25*sin(t),t=0..arccos(.55)],color=red):\nt1 := plot s[textplot]([[.15,.1,`t`],[.9,.9,`(cos(t),sin(t))`]],color=red):\nplot s[display]([p1,p2,p3,p4,t1],tickmarks=[3,3],scaling=constrained);" }}} {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 }