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-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 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 17 "Given a func tion " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 6 ", the \+ " }{TEXT 263 9 "even part" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 17 " is the function " }{XPPEDIT 18 0 "ph i[even](x);" "6#-&%$phiG6#%%evenG6#%\"xG" }{TEXT -1 11 " defined by" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[even](x) = (phi (x)+phi(-x))/2;" "6#/-&%$phiG6#%%evenG6#%\"xG*&,&-F&6#F*\"\"\"-F&6#,$F *!\"\"F/F/\"\"#F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Sim ilarly, the " }{TEXT 263 8 "odd part" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 17 " is the function " } {XPPEDIT 18 0 "phi[odd](x)" "6#-&%$phiG6#%$oddG6#%\"xG" }{TEXT -1 11 " defined by" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[o dd](x) = (phi(x)-phi(-x))/2;" "6#/-&%$phiG6#%$oddG6#%\"xG*&,&-F&6#F*\" \"\"-F&6#,$F*!\"\"F3F/\"\"#F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the exponential \+ function: " }{XPPEDIT 18 0 "f(x) = exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F '" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "Then the even part o f " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 8 " is the " }{TEXT 263 17 "hyperbolic cosine" }{TEXT -1 10 " function:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*x \+ = (exp(x)+exp(-x))/2;" "6#/*&%%coshG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6 #,$F'!\"\"F&F&\"\"#F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 " Then the odd part of " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\" xG-%$expG6#F'" }{TEXT -1 8 " is the " }{TEXT 263 15 "hyperbolic sine" }{TEXT -1 10 " function:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sinh*x = (exp(x)-exp(-x))/2;" "6#/*&%%sinhG\"\"\"%\"xGF &*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F0F&\"\"#F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 263 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "sinh*x" "6#*&%%sinhG\"\"\"%\"xGF%" }{TEXT -1 34 " is usually pronounced as \"s hine\" " }{TEXT 285 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Since the Maclaurin series expansi on of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 3 " is" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = 1+x+x^2/2!+ x^3/3+x^4/4!+x^5/5!+x^6/6!+x^7/7!+x^8/8!+x^9/9!+` . . . `;" "6#/-%$exp G6#%\"xG,8\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$F1F/F) *&F'\"\"%-F-6#F3F/F)*&F'\"\"&-F-6#F7F/F)*&F'\"\"'-F-6#F;F/F)*&F'\"\"(- F-6#F?F/F)*&F'\"\")-F-6#FCF/F)*&F'\"\"*-F-6#FGF/F)%(~.~.~.~GF)" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 73 "the hyperbolic sine and cosine functions have Maclaurin series expansions" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*x = 1+x^2/2!+x^4/4!+x^6/6!+x^8/8 !+` . . . `;" "6#/*&%%coshG\"\"\"%\"xGF&,.F&F&*&F'\"\"#-%*factorialG6# F*!\"\"F&*&F'\"\"%-F,6#F0F.F&*&F'\"\"'-F,6#F4F.F&*&F'\"\")-F,6#F8F.F&% (~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sinh*x = x+x^3/3!+x^5/5!+x^7/7!+x^9/9!+` . . . `;" "6#/ *&%%sinhG\"\"\"%\"xGF&,.F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F&*&F'\"\"& -F,6#F0F.F&*&F'\"\"(-F,6#F4F.F&*&F'\"\"*-F,6#F8F.F&%(~.~.~.~GF&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "cosh(x)=taylor(cosh(x),x,10);\nsinh(x)=taylor(si nh(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)\"$?\"\"\"&#F)\"%S]\"\"(#F)\"'!)GO\"\"*-%\"OG6#F)\"#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The grap h of " }{XPPEDIT 18 0 "y = cosh*x;" "6#/%\"yG*&%%coshG\"\"\"%\"xGF'" } {TEXT -1 58 " can be constructed by adding ordinates along the graphs \+ " }{XPPEDIT 18 0 "y=exp(x)/2" "6#/%\"yG*&-%$expG6#%\"xG\"\"\"\"\"#!\" \"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=exp(-x)/2" "6#/%\"yG*&-%$ex pG6#,$%\"xG!\"\"\"\"\"\"\"#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 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(3wiu#F17$Fhy$\"3'odDEC?y&HF17$F]z$\"3I&=-hYyQ?$F17$$\"33+++S2ls=F1$\" 3'>iA5$ohHLF17$Fbz$\"3Q(>PQa+3Y$F17$$\"3/++v.Uac>F1$\"3[!=M2\"343OF17$ FgzFcel-F\\[l6&F^[lF]elF\\elF\\el-Fe[l6#Fhz-%+AXESLABELSG6$Q\"x6\"Q\"y F_`m-%*AXESTICKSG6$\"\"&\"\"%-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Th e graph of " }{XPPEDIT 18 0 "y = sinh*x;" "6#/%\"yG*&%%sinhG\"\"\"%\"x GF'" }{TEXT -1 58 " can be constructed by adding ordinates along the g raphs " }{XPPEDIT 18 0 "y=exp(x)/2" "6#/%\"yG*&-%$expG6#%\"xG\"\"\"\" \"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = -exp(-x)/2;" "6#/% \"yG,$*&-%$expG6#,$%\"xG!\"\"\"\"\"\"\"#F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "p lot([exp(x)/2,-exp(-x)/2,sinh(x)],x=-2..2,y,\n color=[brown,blue,red ],thickness=[1,1,2],tickmarks=[5,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 288 381 381 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$!\"#\"\"!$\"37N1$=;knw'!#> 7$$!3MLLL$Q6G\">!#<$\"3IoY0EIB$Q(F-7$$!3bmm;M!\\p$=F1$\"3a))RPGc8lzF-7 $$!3MLLL))Qj^)[5CX*F-7$$!3 wmm;C2G!e\"F1$\"3hX=m?keH5!#=7$$!3OLL$3yO5]\"F1$\"3p'o#y9Z\\96FH7$$!3& *****\\nU)*=9F1$\"3]*4e?H)z47FH7$$!3SLL$3WDTL\"F1$\"3f#z$f<@%pJ\"FH7$$ !35++]d(Q&\\7F1$\"3o7\"\\w([=L9FH7$$!3gmmmc4`i6F1$\"37&)>w?)oMc\"FH7$$ !3KLLLQW*e3\"F1$\"3e'\\TNW)*zo\"FH7$$!3w++++()>'***FH$\"3[u#H)[l4S=FH7 $$!3E++++0\"*H\"*FH$\"3.-Ujj-h1?FH7$$!35++++83&H)FH$\"3I'R.K4>8=#FH7$$ !3\\LLL3k(p`(FH$\"3[/'*o!o:JN#FH7$$!3Anmmmj^NmFH$\"3['H\"\\6Y4vDFH7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 309 1 "u" }{TEXT -1 15 " be the common " }{TEXT 288 1 "x" }{TEXT -1 23 " coordinate of A and B." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sinh*u) = 1" "6#/*&%%sinhG\"\"\"%\"uGF&F&" } {TEXT -1 11 ", that is, " }{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 " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(exp(u)-exp(-u))=2" "6#/,&-%$expG6#%\"uG\"\"\"-F&6#, $F(!\"\"F-\"\"#" }{TEXT -1 7 ", 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 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "v^2 - 1 = 2*v" "6#/,&*$%\"vG\"\"#\"\"\"F(!\"\"*&F'F(F&F (" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "v^2 - 2*v = 1" "6#/,&*$%\"vG \"\"#\"\"\"*&F'F(F&F(!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v^2-2*v+1=2" "6#/,(*$%\"vG\"\"#\"\"\"*&F'F(F&F(!\"\"F(F(F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{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&!\"\",$-%%sqrtG6#\"\"#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "v = exp(u)" "6#/%\"vG-%$expG6#%\"uG" }{TEXT -1 30 " cannot be negative, we have " }{TEXT 259 1 "v" }{TEXT -1 3 " \+ = " }{XPPEDIT 18 0 "exp(u) = 1+sqrt(2)" "6#/-%$expG6#%\"uG,&\"\"\"F)-% %sqrtG6#\"\"#F)" }{TEXT -1 7 ", 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 14 "B is the point" }{XPPEDIT 18 0 "``(ln(1+sqrt(2),1)" "6#-%!G6#-% #lnG6$,&\"\"\"F*-%%sqrtG6#\"\"#F*F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 291 20 "____________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We h ave" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*u = cosh( ln(1+sqrt(2)));" "6#/*&%%coshG\"\"\"%\"uGF&-F%6#-%#lnG6#,&F&F&-%%sqrtG 6#\"\"#F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"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#F0F,!\"\"F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\" \"" }{XPPEDIT 18 0 "``(1+sqrt(2)+1/(1+sqrt(2)));" "6#-%!G6#,(\"\"\"F'- %%sqrtG6#\"\"#F'*&F'F',&F'F'-F)6#F+F'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(1+sqr t(2)+1-sqrt(2));" "6#-%!G6#,*\"\"\"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 point (" }{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 292 20 "________________ ____" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fso lve(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 functions: tanh, s ech, cosech, coth and their graphs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 98 "By analogy with the trigonometric functions we define the functions tanh, sech, csch and \+ coth by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tanh*x = sinh*x/(cosh*x);" "6#/*&%%tanhG\"\" \"%\"xGF&*(%%sinhGF&F'F&*&%%coshGF&F'F&!\"\"" }{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/" }{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 265 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#/*&%%sechG\"\"\"%\"xGF&*&F&F&*&%%coshGF&F'F&!\"\" " }{XPPEDIT 18 0 "``= 2/(exp(x)+exp(-x))" "6#/%!G*&\"\"#\"\"\",&-%$exp G6#%\"xGF'-F*6#,$F,!\"\"F'F0" }{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 266 9 "_____ ____" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "csch*x = 1/(sinh*x);" "6#/*&%%csch G\"\"\"%\"xGF&*&F&F&*&%%sinhGF&F'F&!\"\"" }{XPPEDIT 18 0 "`` = 2/(exp( x)-exp(-x));" "6#/%!G*&\"\"#\"\"\",&-%$expG6#%\"xGF'-F*6#,$F,!\"\"F0F0 " }{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 267 9 "_________" }{TEXT 293 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "coth*x = cosh*x/(sinh*x);" "6#/*&%%cothG\"\"\"%\"xGF&*( %%coshGF&F'F&*&%%sinhGF&F'F&!\"\"" }{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/" }{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 268 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 "Obtaining the formul as 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 38 " provides the conversions given above." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tanh(x);\n ``=convert(%,exp);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%tanhG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&*$)-% $expG6#%\"xG\"\"#\"\"\"F.F.!\"\"F.,&F'F.F.F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&-%$expG6#,$*&\"\"#\"\"\"%\"xGF-F-F-F-!\"\"F-,&F 'F-F-F-F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "sech(x);\n``=convert(%,exp);\n``=simplify(normal(rh s(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sechG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\",&-%$expG6#%\"xGF(*&F(F(F *!\"\"F(F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"-%$ expG6#%\"xGF(,&-F*6#,$*&F'F(F,F(F(F(F(F(!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "csch(x);\n`` =convert(%,exp);\n``=simplify(normal(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%cschG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G ,$*&\"\"#\"\"\",&-%$expG6#%\"xGF(*&F(F(F*!\"\"F/F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"-%$expG6#%\"xGF(,&-F*6#,$*&F'F(F, F(F(F(F(!\"\"F2F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "coth(x);\n``=convert(%,exp);\n``=simplify(rh s(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%cothG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&*$)-%$expG6#%\"xG\"\"#\"\"\"F.F.F.F.,&F 'F.F.!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&,&-%$expG6#,$*& \"\"#\"\"\"%\"xGF-F-F-F-F-F-,&F'F-F-!\"\"F0" }}}{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 of " }{XPPEDIT 18 0 "y \+ = tanh*x;" "6#/%\"yG*&%%tanhG\"\"\"%\"xGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "plo t([tanh(x),1,-1],x=-3..3,y=-1.2..1.2,linestyle=[1,3$2],thickness=[2,1$ 2],\n color=[red,black$2],ytickmarks=3,title=`y = tanh x`);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 234 234 {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$$!3-++DhkaI6F1$!31i%oO/i?6)F-7$ $!3s******\\XF`**F-$!3?-;3K\"[if(F-7$$!3u*******>#z2))F-$!3]xreFZ4oqF- 7$$!3S++]7RKvuF-$!36a'y6/XnL'F-7$$!3s,+++P'eH'F-$!3)QpITLtwd&F-7$$!3q) ***\\7*3=+&F-$!3H&o3)oSfAYF-7$$!3[)***\\PFcpPF-$!3KZ'fs]71g$F-7$$!3;)* ***\\7VQ[#F-$!3+=s^AH*RV#F-7$$!32)***\\i6:.8F-$!3?tJ!>@CeH\"F-7$$!3Wb+ ++v`hH!#?$!3y0QxT)G:'HF_s7$$\"3]****\\(QIKH\"F-$\"3'G?4jF-7$$\"35*****\\A))oz)F-$\"3E\" o;*\\RjiqF-7$$\"3e******Hk-,5F1$\"3P:f]L*[-i(F-7$$\"36+++D-eI6F1$\"3a/ /z$\\x@6)F-7$$\"3u***\\(=_(zC\"F1$\"3?K\\%3&)[rZ)F-7$$\"3M+++b*=jP\"F1 $\"3+8#\\h7V7!))F-7$$\"3g***\\(3/3(\\\"F1$\"3[z^%pn#>Y!*F-7$$\"33++vB4 JB;F1$\"31KE#>nI5D*F-7$$\"3u*****\\KCnu\"F1$\"3()*Q1Bv:+T*F-7$$\"3s*** \\(=n#f(=F1$\"3N*[V!QSGT&*F-7$$\"3P+++!)RO+?F1$\"3$=`=a'G`S'*F-7$$\"30 ++]_!>w7#F1$\"3bc2U#Qw,s*F-7$$\"3O++v)Q?QD#F1$\"3!p%3(o[:>y*F-7$$\"3G+ ++5jypBF1$\"3Aks(Hqnm#)*F-7$$\"3<++]Ujp-DF1$\"3)*z0\"\\4eo')*F-7$$\"3+ +++gEd@EF1$\"3;C>$[FF1$\"3cCngFVK=**F-7$$\"37++ D6EjpGF1$\"3\\kE@4P'e$**F-7$$\"\"$F*$\"3k/t'o`Z0&**F--%'COLOURG6&%$RGB G$\"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#\"\"\"-F$6& 7S7$F($Fi[lF*7$F/F^\\l7$F5F^\\l7$F:F^\\l7$F?F^\\l7$FDF^\\l7$FIF^\\l7$F NF^\\l7$FSF^\\l7$FXF^\\l7$FgnF^\\l7$F\\oF^\\l7$FaoF^\\l7$FfoF^\\l7$F[p F^\\l7$F`pF^\\l7$FepF^\\l7$FjpF^\\l7$F_qF^\\l7$FdqF^\\l7$FiqF^\\l7$F^r F^\\l7$FcrF^\\l7$FhrF^\\l7$F]sF^\\l7$FcsF^\\l7$FhsF^\\l7$F]tF^\\l7$Fbt F^\\l7$FgtF^\\l7$F\\uF^\\l7$FauF^\\l7$FfuF^\\l7$F[vF^\\l7$F`vF^\\l7$Fe vF^\\l7$FjvF^\\l7$F_wF^\\l7$FdwF^\\l7$FiwF^\\l7$F^xF^\\l7$FcxF^\\l7$Fh xF^\\l7$F]yF^\\l7$FbyF^\\l7$FgyF^\\l7$F\\zF^\\l7$FazF^\\l7$FfzF^\\l-F[ [l6&F][lF*F*F*-Fc[lFh[l-Fg[l6#Fgz-F$6&7S7$F($!\"\"F*7$F/Fh_l7$F5Fh_l7$ F:Fh_l7$F?Fh_l7$FDFh_l7$FIFh_l7$FNFh_l7$FSFh_l7$FXFh_l7$FgnFh_l7$F\\oF h_l7$FaoFh_l7$FfoFh_l7$F[pFh_l7$F`pFh_l7$FepFh_l7$FjpFh_l7$F_qFh_l7$Fd qFh_l7$FiqFh_l7$F^rFh_l7$FcrFh_l7$FhrFh_l7$F]sFh_l7$FcsFh_l7$FhsFh_l7$ F]tFh_l7$FbtFh_l7$FgtFh_l7$F\\uFh_l7$FauFh_l7$FfuFh_l7$F[vFh_l7$F`vFh_ l7$FevFh_l7$FjvFh_l7$F_wFh_l7$FdwFh_l7$FiwFh_l7$F^xFh_l7$FcxFh_l7$FhxF h_l7$F]yFh_l7$FbyFh_l7$FgyFh_l7$F\\zFh_l7$FazFh_l7$FfzFh_lF__lFa_lFb_l -%&TITLEG6#%+y~=~tanh~xG-%+AXESLABELSG6$Q\"x6\"Q\"yFbcl-%*AXESTICKSG6$ %(DEFAULTGFgz-%%VIEWG6$;F(Ffz;$!#7Fi_l$\"#7Fi_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" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{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 271 1 "~" }{TEXT -1 14 " 0.8 813735869." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "sinh* u = 1;" "6#/*&%%sinhG\"\"\"%\"uGF&F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*u = sqrt(2);" "6#/*&%%coshG\"\"\"%\"uGF&-%%sqrtG6#\"\"#" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "tanh*u = 1/sqrt(2);" "6#/*&%%tanhG \"\"\"%\"uGF&*&F&F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " } {TEXT 272 1 "~" }{TEXT -1 14 " 0.7071067810." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(tanh*x,x = infinity) = Limit((exp(x)-ex p(-x))/(exp(x)+exp(-x)),x = infinity);" "6#/-%&LimitG6$*&%%tanhG\"\"\" %\"xGF)/F*%)infinityG-F%6$*&,&-%$expG6#F*F)-F26#,$F*!\"\"F7F),&-F26#F* F)-F26#,$F*F7F)F7/F*F," }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit((1-ex p(-2*x))/(1+exp(-2*x)),x=infinity) = 1" "6#/-%&LimitG6$*&,&\"\"\"F)-%$ expG6#,$*&\"\"#F)%\"xGF)!\"\"F1F),&F)F)-F+6#,$*&F/F)F0F)F1F)F1/F0%)inf inityGF)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " } {XPPEDIT 18 0 "Limit(tanh*x,x = -infinity) = Limit((exp(x)-exp(-x))/(e xp(x)+exp(-x)),x = -infinity);" "6#/-%&LimitG6$*&%%tanhG\"\"\"%\"xGF)/ F*,$%)infinityG!\"\"-F%6$*&,&-%$expG6#F*F)-F46#,$F*F.F.F),&-F46#F*F)-F 46#,$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*6#*&F-F.F/F.F.F.F.F0/F/,$%)infini tyGF0,$F.F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "It follows that the graph " }{XPPEDIT 18 0 "y = tanh*x;" "6#/%\"yG*&%%tanhG\"\"\"%\"xGF'" }{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 263 21 "horizontal asymptotes" }{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 \+ = sech*x;" "6#/%\"yG*&%%sechG\"\"\"%\"xGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "plo t([sech(x),cosh(x),2*exp(-x),2*exp(x)],x=-3..3,y=0..4,\n linestyle=[1, 2$3],thickness=[2,1$3],color=[red,blue,black$2],title=`y = sech x`);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 306 306 {PLOTDATA 2 "6)-%'CURVESG6&7e n7$$!\"$\"\"!$\"3o?L%>u#zK**!#>7$$!3!******\\2<#pG!#<$\"3QB?FwVAJ6!#=7 $$!3#)***\\7bBav#F1$\"3W5<)3`@lE\"F47$$!36++]K3XFEF1$\"310TxV\"F47 $$!3%)****\\F)H')\\#F1$\"39EC$z@=Hj\"F47$$!3#****\\i3@/P#F1$\"3'o>0hQ^ E&=F47$$!3;++Dr^b^AF1$\"3kvg(RZb;3#F47$$!3$****\\7Sw%G@F1$\"3_;sz>>7ZB F47$$!3*****\\7;)=,?F1$\"3[&**elMz\\l#F47$$!3/++DO\"3V(=F1$\"3qk(>*HSg )*HF47$$!3#******\\V'zViUC\"F1$\"3$HB3M!QB@`F47$$!3-++DhkaI6F1$\" 3C?QJ@?gZeF47$$!3s******\\XF`**F4$\"3RS\">.[:O]'F47$$!3u*******>#z2))F 4$\"3!R78mkRS2(F47$$!3S++]7RKvuF4$\"3@>!z2#y*ft(F47$$!3s,+++P'eH'F4$\" 3U#3SY]t**H)F47$$!3q)***\\7*3=+&F4$\"3cGc_QuWn))F47$$!3[)***\\PFcpPF4$ \"3_OGOfmGH$*F47$$!3;)****\\7VQ[#F4$\"33H7%yuh#*p*F47$$!35)**\\P9(\\$* =F4$\"3t%)))*H)HPB)*F47$$!32)***\\i6:.8F4$\"3DdwjUlo:**F47$$!3%p)*\\i: sw%)*F-$\"3K@%eL$oq^**F47$$!39$***\\(oKQm'F-$\"3?EOBvw$y(**F47$$!3O*** \\(=K**zMF-$\"313[_xy%R***F47$$!3Wb+++v`hH!#?$\"37S%Qm9c*****F47$$\"3l 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l7$Ff[l$\"3kS_^?8-R@Fg^l7$F[\\l$\"3$pOWBHxIW#Fg^l7$F`\\l$\"3+z'31Wo9v# Fg^l7$Fihl$\"3e/)pcO#[JHFg^l7$Fe\\l$\"3yp&[IktK7$Fg^l7$Fail$\"3)y/z&\\z5ENFg^l7$Fiil$\"3G;Ep4#4Ow$Fg^l7$F_]lFfjlF^d mF`jlFajl-%&TITLEG6#%+y~=~sech~xG-%+AXESLABELSG6$Q\"x6\"Q\"yFc^n-%%VIE WG6$;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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{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)-%%sqrtG 6#\"\"#F)" }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 14 " 0.881373586 9." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "cosh*u = sqrt (2);" "6#/*&%%coshG\"\"\"%\"uGF&-%%sqrtG6#\"\"#" }{TEXT -1 5 ", so " } {XPPEDIT 18 0 "sech*u = 1/sqrt(2);" "6#/*&%%sechG\"\"\"%\"uGF&*&F&F&-% %sqrtG6#\"\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*& -%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{TEXT 277 1 "~" }{TEXT -1 18 " 0.7071067810.\nAs " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"x G7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sech*x = 2/(exp(x)+exp(-x));" "6#/*&%%sechG\"\"\"%\"xGF &*&\"\"#F&,&-%$expG6#F'F&-F,6#,$F'!\"\"F&F1" }{TEXT -1 1 " " }{TEXT 278 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*&%%sechG\"\"\"% \"xGF'" }{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\",$%)infinityG!\"\"F*F*F*" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "sech*x = 2/(exp(x)+exp(-x));" "6#/*&%% sechG\"\"\"%\"xGF&*&\"\"#F&,&-%$expG6#F'F&-F,6#,$F'!\"\"F&F1" }{TEXT -1 1 " " }{TEXT 279 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 graph of " }{XPPEDIT 18 0 "y = sech*x;" "6#/%\" yG*&%%sechG\"\"\"%\"xGF'" }{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 = infinity);" "6#/-%&LimitG6$*&%%sechG\"\"\"%\"x GF)/F*%)infinityG-F%6$*&\"\"#F),&-%$expG6#F*F)-F36#,$F*!\"\"F)F8/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$*&%%sechG \"\"\"%\"xGF)/F*,$%)infinityG!\"\"-F%6$*&\"\"#F),&-%$expG6#F*F)-F56#,$ F*F.F)F./F*,$F-F." }{TEXT -1 21 " = 0, the graph of " }{XPPEDIT 18 0 "y = sech*x;" "6#/%\"yG*&%%sechG\"\"\"%\"xGF'" }{TEXT -1 9 " has the " }{TEXT 310 1 "x" }{TEXT -1 11 " axis as a " }{TEXT 263 20 "horizont al 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 = csch*x;" "6#/%\"yG*& %%cschG\"\"\"%\"xGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "plot([csch(x),2*exp(-x),- 2*exp(x)],x=-3..3,y=-3..3,linestyle=[1,2$3],thickness=[2,1$2],\n co lor=[red,black$2],discont=true,title=`y = cosech x`);" }}{PARA 13 "" 1 "" {GLPLOT2D 441 386 386 {PLOTDATA 2 "6(-%'CURVESG6'7gn7$$!\"$\"\"!$ !3AtA)o'p:#)**!#>7$$!3*GyIw`3Y$H!#<$!3!>)))*)*4Ug1\"!#=7$$!3Ew&pex6x(G F1$!30d%)R&fS)G6F47$$!3;\\Di;as8GF1$!3D#o3m#R&R?\"F47$$!3?q8D9\\J\\FF1 $!3Wz\"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! 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3)y/z&\\z5ENFez7$$\"31+]i0j\"[$HF1$!3G;Ep4#4Ow$F ez7$Fe`m$!3fLvj%Q2r,%FezFi`mFganFian-%+AXESLABELSG6%Q\"x6\"Q\"yFc\\o-% %FONTG6#%(DEFAULTG-%&TITLEG6#%-y~=~cosech~xG-%%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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {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#/*&%%sinhG\"\"\"%\"uGF&F&" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "csch*u = 1;" "6#/*&%%cschG\"\"\"%\"uGF&F& " }{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$%)operato rG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 8 "+, sinh " }{XPPEDIT 18 0 "x->0 " "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 11 "+, so csch " }{XPPEDIT 18 0 "proc (x) options operator, arrow; infinity \+ end proc;" "6#f*6#%\"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$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 8 "- , sinh " }{XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\" \"\"!F*F*F*" }{TEXT -1 11 "-, so csch " }{XPPEDIT 18 0 "proc (x) optio ns operator, arrow; -infinity end proc;" "6#f*6#%\"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 = csch*x ;" "6#/%\"yG*&%%cschG\"\"\"%\"xGF'" }{TEXT -1 9 " has the " }{TEXT 311 1 "y" }{TEXT -1 11 " axis as a " }{TEXT 263 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 3 ", " }{XPPEDIT 18 0 "csch*x = 2/(exp(x)-exp(-x));" "6#/*&%%cschG\"\"\"%\"xGF&*&\"\"#F&,&-%$expG6#F'F&-F,6#,$F'!\"\"F1F1" }{TEXT -1 1 " " }{TEXT 274 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 = csch*x;" "6 #/%\"yG*&%%cschG\"\"\"%\"xGF'" }{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 "csch*x = 2/(exp(x)-e xp(-x));" "6#/*&%%cschG\"\"\"%\"xGF&*&\"\"#F&,&-%$expG6#F'F&-F,6#,$F'! \"\"F1F1" }{TEXT -1 1 " " }{TEXT 275 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 = csch*x;" "6#/%\"yG*&%%cschG\"\"\"%\"xGF'" }{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(csch*x,x = infinity) = Limit(2/(exp(x)-exp(-x)),x = infini ty);" "6#/-%&LimitG6$*&%%cschG\"\"\"%\"xGF)/F*%)infinityG-F%6$*&\"\"#F ),&-%$expG6#F*F)-F36#,$F*!\"\"F8F8/F*F," }{TEXT -1 11 " = 0 and " } {XPPEDIT 18 0 "Limit(csch*x,x = -infinity) = Limit(2/(exp(x)-exp(-x)), x = -infinity);" "6#/-%&LimitG6$*&%%cschG\"\"\"%\"xGF)/F*,$%)infinityG !\"\"-F%6$*&\"\"#F),&-%$expG6#F*F)-F56#,$F*F.F.F./F*,$F-F." }{TEXT -1 21 " = 0, the graph of " }{XPPEDIT 18 0 "y = csch*x;" "6#/%\"yG*&%%c schG\"\"\"%\"xGF'" }{TEXT -1 9 " has the " }{TEXT 312 1 "x" }{TEXT -1 11 " axis as a " }{TEXT 263 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*&%%cothG\"\"\"%\"xGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot([coth(x),1,-1],x=-2..2,y=-5..5,linestyle=[1,3$2 ],thickness=[2,1$2],\n color=[red,black$2],discont=true,title=`y = co th x`);" }}{PARA 13 "" 1 "" {GLPLOT2D 427 360 360 {PLOTDATA 2 "6(-%'CU RVESG6'7gn7$$!\"#\"\"!$!3;[vs?ZJP5!#<7$$!3EbQv\"p0k&>F-$!369Kf/NyS5F-7 $$!3S%QYs^u%=>F-$!3D\\DcV'oS/\"F-7$$!3wK]TWp\"e(=F-$!3!R][Mx(3[5F-7$$! 3Y84]4m(G$=F-$!3#>GDSL8D0\"F-7$$!3__I]i.9!z\"F-$!3Kf@&4^Lt0\"F-7$$!3)) Hc\"4R=0v\"F-$!3m%)G\")y(3A1\"F-7$$!3m:5LM@\\4&3x1\"F-7$ $!3GTD3@F1n;F-$!3!fJ$)[mER2\"F-7$$!3Th/]z$pZi\"F-$!3;BGV3`r!3\"F-7$$!3 G-3%=Z'[6\"F-7$$ !3#=\\qh1aZT\"F-$!30H,jDJ\\D6F-7$$!31pHm@)[oP\"F-$!3tk#[a0\\g8\"F-7$$! 3v\"yp'>exJ8F-$!31x#e8nW)\\6F-7$$!3EZhuuIf$H\"F-$!3YB1wkKqi6F-7$$!3?X; 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1 end proc;" "6#f*6#%\"xG7\"6$%)oper atorG%&arrowG6\"\"\"\"F*F*F*" }{TEXT -1 10 " and cosh " }{XPPEDIT 18 0 "proc (x) options operator, arrow; 0 end proc;" "6#f*6#%\"xG7\"6$%)o peratorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 12 "+ , so coth " }{XPPEDIT 18 0 "proc (x) options operator, arrow; infinity end proc;" "6#f*6#%\" xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x ->0" "6#f*6#%\"xG7 \"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 8 "-, cosh " } {XPPEDIT 18 0 "proc (x) options operator, arrow; 1 end proc;" "6#f*6#% \"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F*F*" }{TEXT -1 10 " and sinh \+ " }{XPPEDIT 18 0 "proc (x) options operator, arrow; 0 end proc;" "6#f* 6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 11 "-, so cot h " }{XPPEDIT 18 0 "proc (x) options operator, arrow; -infinity end pr oc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "It follows that the gra ph of " }{XPPEDIT 18 0 "y = coth*x;" "6#/%\"yG*&%%cothG\"\"\"%\"xGF' " }{TEXT -1 9 " has the " }{TEXT 313 1 "y" }{TEXT -1 11 " axis as a " }{TEXT 263 18 "vertical asymptote" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Li mit(coth*x,x = infinity) = Limit((exp(x)+exp(-x))/(exp(x)-exp(-x)),x = infinity);" "6#/-%&LimitG6$*&%%cothG\"\"\"%\"xGF)/F*%)infinityG-F%6$* &,&-%$expG6#F*F)-F26#,$F*!\"\"F)F),&-F26#F*F)-F26#,$F*F7F7F7/F*F," } {TEXT -1 4 " = " }{XPPEDIT 18 0 "Limit((1+exp(-2*x))/(1-exp(-2*x)),x \+ = infinity) = 1;" "6#/-%&LimitG6$*&,&\"\"\"F)-%$expG6#,$*&\"\"#F)%\"xG F)!\"\"F)F),&F)F)-F+6#,$*&F/F)F0F)F1F1F1/F0%)infinityGF)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(coth*x,x = -infinity) = Limit((exp(x)+exp(-x))/(exp(x)-exp(-x)),x = -infinity) ;" "6#/-%&LimitG6$*&%%cothG\"\"\"%\"xGF)/F*,$%)infinityG!\"\"-F%6$*&,& -%$expG6#F*F)-F46#,$F*F.F)F),&-F46#F*F)-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 14 "the graph of " }{XPPEDIT 18 0 "y = coth*x;" "6#/%\"yG*&%%cothG\"\"\"%\"x GF'" }{TEXT -1 15 " has the lines " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\" \"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=-1" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 4 " as " }{TEXT 263 21 "horizontal asymptotes" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Some identities involving hyperb olic functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 115 "There are analagous identities involving hyperbolic functions to many identities involving trigonometric funct ions." }}{PARA 0 "" 0 "" {TEXT -1 28 "The most basic identity is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh^2*x-sinh^2*x = 1 ;" "6#/,&*&%%coshG\"\"#%\"xG\"\"\"F)*&%%sinhGF'F(F)!\"\"F)" }{TEXT -1 14 " ------- (i), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 23 " ___________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "which may be compared and/or contrasted with the tri gonometric identity " }{XPPEDIT 18 0 "cos^2*x+sin^2*x = 1;" "6#/,&*&%$ cosG\"\"#%\"xG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 47 "The identity (i) can be proved by substituting " } {XPPEDIT 18 0 "cosh*x = (exp(x)+exp(x))/2;" "6#/*&%%coshG\"\"\"%\"xGF& *&,&-%$expG6#F'F&-F+6#F'F&F&\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sinh*x = (exp(x)-exp(-x))/2;" "6#/*&%%sinhG\"\"\"%\"xGF&*&,&-%$e xpG6#F'F&-F+6#,$F'!\"\"F0F&\"\"#F0" }{TEXT -1 30 " in the left hand si de to get " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh^2* x-sinh^2*x = ((exp(x)+exp(-x))/2)^2-((exp(x)-exp(-x))/2)^2;" "6#/,&*&% %coshG\"\"#%\"xG\"\"\"F)*&%%sinhGF'F(F)!\"\",&*$*&,&-%$expG6#F(F)-F26# ,$F(F,F)F)F'F,F'F)*$*&,&-F26#F(F)-F26#,$F(F,F,F)F'F,F'F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``= (exp(2*x)+2+exp(-2*x))/4 - (exp(2*x)-2+exp(-2*x ))/4" "6#/%!G,&*&,(-%$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-F4F 3F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/2+1/2; " "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'*&F'F'F(F)F'" }{XPPEDIT 18 0 "``= 1" "6#/%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Note th at (i) implies that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh^2*x = 1+sinh^2*x" "6#/*&%%coshG\"\"#%\"xG\"\"\",&F(F(*&%%sinhG F&F'F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh^2*x = cosh^2*x-1" "6#/ *&%%sinhG\"\"#%\"xG\"\"\",&*&%%coshGF&F'F(F(F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "On dividing both sides of (i) by " } {XPPEDIT 18 0 "cosh^2*x;" "6#*&%%coshG\"\"#%\"xG\"\"\"" }{TEXT -1 26 " we obtain the identity: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1-tanh^2*x = sech^2*x;" "6#/,&\"\"\"F%*&%%tanhG\"\"#%\" xGF%!\"\"*&%%sechGF(F)F%" }{TEXT -1 15 " ------- (ii), " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 294 23 "___________ " }}{PARA 0 "" 0 "" {TEXT -1 37 "and on dividing both sides of (i) by " } {XPPEDIT 18 0 "sinh^2*x;" "6#*&%%sinhG\"\"#%\"xG\"\"\"" }{TEXT -1 26 " we obtain the identity: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "coth^2*x-1 = csch^2*x;" "6#/,&*&%%cothG\"\"#%\"xG\"\"\" F)F)!\"\"*&%%cschGF'F(F)" }{TEXT -1 15 " ------- (iii)." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 295 23 "___________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Corresponding t o the double-angle trigonometric formulas " }{XPPEDIT 18 0 "sin*2*x = \+ 2*sin*x*cos*x;" "6#/*(%$sinG\"\"\"\"\"#F&%\"xGF&*,F'F&F%F&F(F&%$cosGF& F(F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*2*x = cos^2*x-sin^2*x;" "6#/*(%$cosG\"\"\"\"\"#F&%\"xGF&,&*&F%F'F(F&F&*&%$sinGF'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^2*x+s inh^2*x, ``]);" "6#-%*PIECEWISEG6%7$/*(%%sinhG\"\"\"\"\"#F*%\"xGF**,F+ F*F)F*F,F*%%coshGF*F,F*%!G7$F/F/7$/*(F.F*F+F*F,F*,&*&F.F+F,F*F**&F)F+F ,F*F*F/" }{TEXT -1 14 " ------- (iv) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 296 24 "_______________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "These can be proved in the same manner as (i) by substituting " }{XPPEDIT 18 0 "cosh*x = (exp(x)+exp (x))/2;" "6#/*&%%coshG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#F'F&F&\"\"#! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sinh*x = (exp(x)-exp(-x))/2; " "6#/*&%%sinhG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F0F&\"\"#F0 " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "2*sinh*x*cosh*x = 2*``((exp(x)-exp(-x ))/2)*``((exp(x)+exp(-x))/2);" "6#/*,\"\"#\"\"\"%%sinhGF&%\"xGF&%%cosh GF&F(F&*(F%F&-%!G6#*&,&-%$expG6#F(F&-F16#,$F(!\"\"F6F&F%F6F&-F,6#*&,&- F16#F(F&-F16#,$F(F6F&F&F%F6F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= ( exp(2*x)-exp(-2*x))/2" "6#/%!G*&,&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,-F(6# ,$*&F+F,F-F,!\"\"F2F,F+F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sinh*2*x; " "6#/%!G*(%%sinhG\"\"\"\"\"#F'%\"xGF'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh^2*x+sinh^2*x = ((exp(x)+exp(-x))/2)^2+((exp(x)+exp(-x))/2)^ 2;" "6#/,&*&%%coshG\"\"#%\"xG\"\"\"F)*&%%sinhGF'F(F)F),&*$*&,&-%$expG6 #F(F)-F16#,$F(!\"\"F)F)F'F6F'F)*$*&,&-F16#F(F)-F16#,$F(F6F)F)F'F6F'F) " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = cosh*2*x;" "6#/%!G*(%%coshG\"\"\"\"\"#F'%\"xGF'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Alternatively, the formulas (vi) are special 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*s inh*y, ``],[sinh(x-y) = sinh*x*cosh*y-cosh*x*sinh*y, ``],[cosh(x+y) = \+ cosh*x*cosh*y+sinh*x*sinh*y, ``],[cosh(x-y) = cosh*x*cosh*y-sinh*x*sin h*y, ``]);" "6#-%*PIECEWISEG6&7$/-%%sinhG6#,&%\"xG\"\"\"%\"yGF-,&**F)F -F,F-%%coshGF-F.F-F-**F1F-F,F-F)F-F.F-F-%!G7$/-F)6#,&F,F-F.!\"\",&**F) F-F,F-F1F-F.F-F-**F1F-F,F-F)F-F.F-F9F37$/-F16#,&F,F-F.F-,&**F1F-F,F-F1 F-F.F-F-**F)F-F,F-F)F-F.F-F-F37$/-F16#,&F,F-F.F9,&**F1F-F,F-F1F-F.F-F- **F)F-F,F-F)F-F.F-F9F3" }{TEXT -1 13 " ------- (v)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 281 23 "_______________________" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The following proof of the first of these formulas provides the metho d 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#/,&**%%sinhG\"\"\"%\"xGF'%%coshGF' %\"yGF'F'**F)F'F(F'F&F'F*F'F',&*&-%!G6#*&,&-%$expG6#F(F'-F46#,$F(!\"\" F9F'\"\"#F9F'-F/6#*&,&-F46#F*F'-F46#,$F*F9F'F'F:F9F'F'*&-F/6#*&,&-F46# F(F'-F46#,$F(F9F'F'F:F9F'-F/6#*&,&-F46#F*F'-F46#,$F*F9F9F'F:F9F'F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (exp(x+y)+exp(x-y)-exp(-x+y)-exp(- x-y))/4+(exp(x+y)-exp(x-y)+exp(-x+y)-exp(-x-y))/4;" "6#/%!G,&*&,*-%$ex pG6#,&%\"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,F2 F.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*&, &-%$expG6#,&%\"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 proce dure " }{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)-%%coshG6#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#,&%\"x G\"\"\"%\"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+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cosh^2*x = 1+sinh^2*x;" "6#/*&% %coshG\"\"#%\"xG\"\"\",&F(F(*&%%sinhGF&F'F(F(" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sinh^2*x = cosh^2*x-1;" "6#/*&%%sinhG\"\"#%\"xG\"\"\",& *&%%coshGF&F'F(F(F(!\"\"" }{TEXT -1 15 ", the identity " }{XPPEDIT 18 0 "cosh*2*x = cosh^2*x+sinh^2*x;" "6#/*(%%coshG\"\"\"\"\"#F&%\"xGF&,&* &F%F'F(F&F&*&%%sinhGF'F(F&F&" }{TEXT -1 37 " can be re-written in the \+ two forms: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([cosh*2*x = 2*cosh^2*x-1, ``] ,[cosh*2*x = 2*sinh^2*x+1, ``]);" "6#-%*PIECEWISEG6$7$/*(%%coshG\"\"\" \"\"#F*%\"xGF*,&*(F+F**$F)F+F*F,F*F*F*!\"\"%!G7$/*(F)F*F+F*F,F*,&*(F+F **$%%sinhGF+F*F,F*F*F*F*F1" }{TEXT -1 13 "------- (vi)." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 284 13 "_____________" }{TEXT -1 25 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The formulas (vi) give rise to: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([cosh^2*x = (cosh*2*x+1)/2, ` `],[sinh^2*x = (cosh*2*x-1)/2, ``]);" "6#-%*PIECEWISEG6$7$/*&%%coshG\" \"#%\"xG\"\"\"*&,&*(F)F,F*F,F+F,F,F,F,F,F*!\"\"%!G7$/*&%%sinhGF*F+F,*& ,&*(F)F,F*F,F+F,F,F,F0F,F*F0F1" }{TEXT -1 15 " ------- (vii)." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 297 13 "_____________" }{TEXT -1 25 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "2*cosh(x)^2-1=combine(2*cosh (x)^2-1,trig);\n1+2*sinh(x)^2=combine(1+2*sinh(x)^2,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)-%%coshG6#%\"xG\"\"#\"\"\"F+F,!\"\"-F( 6#,$F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"\"F%*&\"\"#F%)-%%si nhG6#%\"xGF'F%F%-%%coshG6#,$F,F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cosh(x)^2=combine(cosh(x)^2, trig);\nsinh(x)^2=combine(sinh(x)^2,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%%coshG6#%\"xG\"\"#\"\"\",&-F'6#,$F)F*#F+F*F0F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%%sinhG6#%\"xG\"\"#\"\"\",&#!\" \"F*F+*&#F+F*F+-%%coshG6#,$F)F*F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Geometrical interpretation of the identity " } {XPPEDIT 18 0 "cosh^2*x-sinh^2*x = 1;" "6#/,&*&%%coshG\"\"#%\"xG\"\"\" F)*&%%sinhGF'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^2*x-sinh^2*x = 1;" "6#/,&*&%%coshG\"\"#%\"xG\"\"\" F)*&%%sinhGF'F(F)!\"\"F)" }{TEXT -1 21 " means that, for any " }{TEXT 298 1 "t" }{TEXT -1 11 ", the point" }{XPPEDIT 18 0 "``(cosh*t,sinh*t) ;" "6#-%!G6$*&%%coshG\"\"\"%\"tGF(*&%%sinhGF(F)F(" }{TEXT -1 13 " lies on the " }{TEXT 263 21 "rectangular hyperbola" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^2-y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "This explains why the fu nctions are called hyperbolic." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 370 342 342 {PLOTDATA 2 "6.-%'CURVESG6%7X7$$\"3W;.TqJJWT!#<$ 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However, in the case of the hyper bolic functions, we do not have this geometrical interpretation of the parameter " }{TEXT 299 1 "t" }{TEXT -1 51 ", because the line joining the origin to the point " }{XPPEDIT 18 0 "`` ( cosh*t, sinh*t )" "6#- %!G6$*&%%coshG\"\"\"%\"tGF(*&%%sinhGF(F)F(" }{TEXT -1 27 " does not ma ke an angle of " }{TEXT 300 1 "t" }{TEXT -1 10 " with the " }{TEXT 301 1 "x" }{TEXT -1 20 " axis ( except when " }{XPPEDIT 18 0 "t = 0" " 6#/%\"tG\"\"!" }{TEXT -1 4 " ). " }}{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 hyperbolic 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#*&%%sinhG\"\"\"%\"xGF'*&%%cos hGF'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 "[cosh*x] = sinh*x;" "6#/7#*&%%coshG\"\"\"%\"xGF'*&%%sin hGF'F(F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 54 "These formulas are easy to check from th e definitions " }{XPPEDIT 18 0 "sinh*x = (exp(x)-exp(x))/2;" "6#/*&%%s inhG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#F'!\"\"F&\"\"#F/" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*x = (exp(x)+exp(-x))/2;" "6#/*&%%coshG\"\" \"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F&F&\"\"#F0" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 102 "Then we can use the differentiation rul es to obtain 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^2*x;" "6 #/7#*&%%tanhG\"\"\"%\"xGF'*&%%sechG\"\"#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 "[sech*x] = -sech*x*tanh*x;" "6# /7#*&%%sechG\"\"\"%\"xGF',$**F&F'F(F'%%tanhGF'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 "[csch*x] = -csch*x*c oth*x;" "6#/7#*&%%cschG\"\"\"%\"xGF',$**F&F'F(F'%%cothGF'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] = -csch^2*x;" "6#/7#*&%%cothG\"\"\"%\"xGF',$*&%%cschG\"\"#F(F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 283 16 "____ ____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "Diff(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(csch(x),x)=diff(csch(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$-%%sechG6#%\"xGF*,$*&F' \"\"\"-%%tanhGF)F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$ -%%cschG6#%\"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 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 306 11 "Definitions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sinh*x = (exp(x)-exp(-x))/2;" "6#/*&%%sinhG \"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F0F&\"\"#F0" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "cosh*x = (exp(x)+exp(-x))/2;" "6#/*&%%coshG\"\" \"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F&F&\"\"#F0" }{TEXT -1 6 " , \+ " }{XPPEDIT 18 0 "tanh*x = sinh*x/(cosh*x);" "6#/*&%%tanhG\"\"\"%\"x GF&*(%%sinhGF&F'F&*&%%coshGF&F'F&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " sech*x = 1/(cosh*x);" "6#/*&%%sechG\"\"\"%\"xGF&*&F&F&*&%%coshGF&F'F&! \"\"" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "csch*x = 1/(sinh*x);" "6#/ *&%%cschG\"\"\"%\"xGF&*&F&F&*&%%sinhGF&F'F&!\"\"" }{TEXT -1 7 ", \+ " }{XPPEDIT 18 0 "coth*x = cosh*x/(sinh*x);" "6#/*&%%cothG\"\"\"%\"xGF &*(%%coshGF&F'F&*&%%sinhGF&F'F&!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 15 "_______________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 307 16 "Basic identities" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([cosh(-x) = cosh*x, ``],[sinh(-x) = -sinh*x, \+ ``],[tanh(-x) = -tanh*x, ``]);" "6#-%*PIECEWISEG6%7$/-%%coshG6#,$%\"xG !\"\"*&F)\"\"\"F,F/%!G7$/-%%sinhG6#,$F,F-,$*&F4F/F,F/F-F07$/-%%tanhG6# ,$F,F-,$*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 316 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Prove the identity " }{XPPEDIT 18 0 "sinh*3*x = 3*sinh*x+4*sinh^3*x;" "6#/*(%%sinhG\"\"\"\"\"$F&%\"xG F&,&*(F'F&F%F&F(F&F&*(\"\"%F&*$F%F'F&F(F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 8 "Solution" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 318 8 "Method I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 39 "Use the identities already considered. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*3*x=sinh(x+2*x)" "6#/*(%%sinhG\"\"\"\"\" $F&%\"xGF&-F%6#,&F(F&*&\"\"#F&F(F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sinh*x*cosh*2*x+cosh*x*sinh*2*x " "6#/%!G,&*,%%sinhG\"\"\"%\"xGF(%%coshGF(\"\"#F(F)F(F(*,F*F(F)F(F'F(F +F(F)F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sinh*x*(2*sinh^2*x+1)+cosh*x*``(2*sinh*x*cosh*x); " "6#/%!G,&*(%%sinhG\"\"\"%\"xGF(,&*(\"\"#F(*$F'F,F(F)F(F(F(F(F(F(*(%% coshGF(F)F(-F$6#*,F,F(F'F(F)F(F/F(F)F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*sinh^3*x+sinh*x+2*si nh*x*cosh^2*x;" "6#/%!G,(*(\"\"#\"\"\"*$%%sinhG\"\"$F(%\"xGF(F(*&F*F(F ,F(F(*,F'F(F*F(F,F(%%coshGF'F,F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*sinh^3*x+sinh*x+2*sinh*x*(1+sinh ^2*x)" "6#/%!G,(*(\"\"#\"\"\"*$%%sinhG\"\"$F(%\"xGF(F(*&F*F(F,F(F(**F' F(F*F(F,F(,&F(F(*&F*F'F,F(F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*sinh^3*x+sinh*x+2*sinh*x+2*sinh^ 3*x;" "6#/%!G,**(\"\"#\"\"\"*$%%sinhG\"\"$F(%\"xGF(F(*&F*F(F,F(F(*(F'F (F*F(F,F(F(*(F'F(*$F*F+F(F,F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=3*sinh*x+4*sinh^3*x" "6#/%!G,&*(\"\" $\"\"\"%%sinhGF(%\"xGF(F(*(\"\"%F(*$F)F'F(F*F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 319 9 "Method II " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 22 "Use the definition o f " }{XPPEDIT 18 0 "sinh*x" "6#*&%%sinhG\"\"\"%\"xGF%" }{TEXT -1 13 " \+ in terms of " }{XPPEDIT 18 0 "exp*x" "6#*&%$expG\"\"\"%\"xGF%" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 ". " }{XPPEDIT 18 0 "3*sinh*x +4*sinh^3*x = 3/2*(exp(x)-exp(-x))+(exp(x)-exp(-x))^3/2;" "6#/,&*(\"\" $\"\"\"%%sinhGF'%\"xGF'F'*(\"\"%F'*$F(F&F'F)F'F',&*(F&F'\"\"#!\"\",&-% $expG6#F)F'-F36#,$F)F0F0F'F'*&,&-F36#F)F'-F36#,$F)F0F0F&F/F0F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3/2*(exp(x)-exp(-x))+(exp(3*x)-3*exp(x)+3*exp(-x)-exp(-3*x))/2;" "6#/ %!G,&*(\"\"$\"\"\"\"\"#!\"\",&-%$expG6#%\"xGF(-F-6#,$F/F*F*F(F(*&,*-F- 6#*&F'F(F/F(F(*&F'F(-F-6#F/F(F**&F'F(-F-6#,$F/F*F(F(-F-6#,$*&F'F(F/F(F *F*F(F)F*F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(exp(3*x)-exp(-3*x))/2" "6#/%!G*&,&-%$expG6#*&\"\"$ \"\"\"%\"xGF,F,-F(6#,$*&F+F,F-F,!\"\"F2F,\"\"#F2" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sinh*3*x" "6#/%!G* (%%sinhG\"\"\"\"\"$F'%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 23 "We find the derivative " }{XPPEDIT 18 0 "`f '`(x) " "6#-%$f~'G6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = cosh(sqrt(x))/sqrt(x);" "6#/-%\"fG6#%\"xG*&-%%coshG6#-%%sqrtG6 #F'\"\"\"-F-6#F'!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=x^(-1/2)*co sh(x^(1/2))" "6#/-%\"fG6#%\"xG*&)F',$*&\"\"\"F,\"\"#!\"\"F.F,-%%coshG6 #)F'*&F,F,F-F.F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so tha t " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=-1/2" "6#/-%$f~'G6#%\"xG,$*&\"\"\"F*\"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-3/2)*cosh(x^(1/2))+x^(-1/2)*sinh(x^(1/2))*``(1/2)*x^(-1/2)" "6#,&*&)%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F,F*-%%coshG6#)F&*&F*F*F+F,F*F** *)F&,$*&F*F*F+F,F,F*-%%sinhG6#)F&*&F*F*F+F,F*-%!G6#*&F*F*F+F,F*)F&,$*& F*F*F+F,F,F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-cosh(sqrt(x))/(2*x*sqrt(x))+sinh(sqrt(x))/(2*x)" "6 #/%!G,&*&-%%coshG6#-%%sqrtG6#%\"xG\"\"\"*(\"\"#F.F-F.-F+6#F-F.!\"\"F3* &-%%sinhG6#-F+6#F-F.*&F0F.F-F.F3F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(s qrt(x)*sinh(sqrt(x))-cosh(sqrt(x)))/(2*x*sqrt(x))" "6#/%!G*&,&*&-%%sqr tG6#%\"xG\"\"\"-%%sinhG6#-F)6#F+F,F,-%%coshG6#-F)6#F+!\"\"F,*(\"\"#F,F +F,-F)6#F+F,F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f := x -> cosh(sqrt(x))/sqrt (x):\n'f(x)'=f(x);\nDiff(f(x),x)=diff(f(x),x);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%%coshG6#*$F'#\"\" \"\"\"#F.F'#!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*&-% %coshG6#*$%\"xG#\"\"\"\"\"#F.F,#!\"\"F/F,,&*&F-F.*&-%%sinhGF*F.F,F1F.F .*&#F.F/F.*&F(F.F,#!\"$F/F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G, $*&#\"\"\"\"\"#F(*&%\"xG#!\"$F),&*&-%%sinhG6#*$F+F'F(F+F'F(-%%coshGF2! \"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = exp(sech*x)*tanh*x;" "6#/-%\" fG6#%\"xG*(-%$expG6#*&%%sechG\"\"\"F'F.F.%%tanhGF.F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = exp(se ch*x)*(-sech*x*tanh*x)*tanh*x+exp(sech*x)*sech^2*x;" "6#/-%$f~'G6#%\"x G,&**-%$expG6#*&%%sechG\"\"\"F'F/F/,$**F.F/F'F/%%tanhGF/F'F/!\"\"F/F2F /F'F/F/*(-F+6#*&F.F/F'F/F/*$F.\"\"#F/F'F/F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(sech*x)*sech*x*(se ch*x-tanh^2*x);" "6#/%!G**-%$expG6#*&%%sechG\"\"\"%\"xGF+F+F*F+F,F+,&* &F*F+F,F+F+*&%%tanhG\"\"#F,F+!\"\"F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(sech*x)*sech*x*(sech^2*x+se ch*x-1)" "6#/%!G**-%$expG6#*&%%sechG\"\"\"%\"xGF+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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> exp(sech (x))*tanh(x):\n'f(x)'=f(x);\nDiff(f(x),x)=diff(f(x),x);\n``=factor(rhs (%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%$expG6#-%%s echGF&\"\"\"-%%tanhGF&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6 $*&-%$expG6#-%%sechG6#%\"xG\"\"\"-%%tanhGF-F/F.,&*(F(F/F+F/)F0\"\"#F/! \"\"*&F(F/,&F/F/*$F4F/F6F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ *&-%$expG6#-%%sechG6#%\"xG\"\"\",(*&F*F.)-%%tanhGF,\"\"#F.F.F.!\"\"*$F 1F.F.F.F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "exp(sech(x))*sech(x)*(sech(x)^2+sech(x)-1);\nint(%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#-%%sechG6#%\"xG\"\" \"F'F+,(*$)F'\"\"#F+F+F'F+F+!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*(,&-%$expG6#,$*&\"\"#\"\"\"%\"xGF+F+F+F+!\"\"F+,&F%F+F+F+F--F&6#,$*& F*F+,&-F&6#F,F+-F&6#,$F,F-F+F-F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = arctan (sinh*x);" "6#/-%\"fG6#%\"xG-%'arctanG6#*&%%sinhG\"\"\"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) \+ = ``(1/(1+sinh^2*x))*cosh*x;" "6#/-%$f~'G6#%\"xG*(-%!G6#*&\"\"\"F-,&F- F-*&%%sinhG\"\"#F'F-F-!\"\"F-%%coshGF-F'F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cosh*x/(cosh^2*x)" "6#/%!G*(%%coshG\"\"\"%\"xGF'*&F&\"\"#F(F'!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(cosh*x)" "6#/%!G*&\"\"\"F&*& %%coshGF&%\"xGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=sech*x" "6#/ %!G*&%%sechG\"\"\"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f := x -> arctan(s inh(x)):\n'f(x)'=f(x);\nDiff(f(x),x)=diff(f(x),x);\n``=simplify(rhs(%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%'arctanG6#-%%sin hGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%'arctanG6#-%%sinh G6#%\"xGF-*&-%%coshGF,\"\"\",&F1F1*$)F*\"\"#F1F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&-%%coshG6#%\"xG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The following plot s hows the graph of " }{XPPEDIT 18 0 "f(x) = arctan(sinh*x);" "6#/-%\"f G6#%\"xG-%'arctanG6#*&%%sinhG\"\"\"F'F-" }{TEXT -1 4 " in " }{TEXT 320 3 "red" }{TEXT -1 33 " and the graph of the derivative " } {XPPEDIT 18 0 "`f '`(x) = sech*x;" "6#/-%$f~'G6#%\"xG*&%%sechG\"\"\"F' F*" }{TEXT -1 5 " in " }{TEXT 256 4 "blue" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "p1 \+ := plot([arctan(sinh(x)),sech(x)],x=-4..4,y,color=[red,blue],thickness =2):\np2 := plot([-Pi/2,Pi/2],x=-4..4,color=black,linestyle=2):\nplots [display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 617 209 209 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$!\"%\"\"!$!3F:F-7$$!35LLLo!)*Qn$F-$!3?9\"R'4D0?:F-7$$!3nmmmwxE.NF-$ !34EQ7jnh5:F-7$$!3YmmmOk]JLF-$!3?(R?_G[$*\\\"F-7$$!3_LLL[9cgJF-$!3QC\\ %)RK/'[\"F-7$$!3smmmhN2-IF-$!39qCO\"45:Z\"F-7$$!3!******\\`oz$GF-$!3;2 %p@(3%QX\"F-7$$!3!omm;)3DoEF-$!3!p;lVMrAV\"F-7$$!3?+++:v2*\\#F-$!3JnUp pI%oS\"F-7$$!3BLLL8>1DBF-$!3ow>4\">heP\"F-7$$!3kmmmw))yr@F-$!3/6`w8'GQ M\"F-7$$!3;+++S(R#**>F-$!3j`Q<$=e:I\"F-7$$!30++++@)f#=F-$!3arzQ^-U^7F- 7$$!3-+++gi,f;F-$!3]F+'z&Hk%>\"F-7$$!3qmmm\"G&R2:F-$!3Hx&fnah[8\"F-7$$ !3XLLLtK5F8F-$!3K6mH)*zC_5F-7$$!3eLLL$HsV<\"F-$!3;a\\JOi'Qr*!#=7$$!3+- ++]&)4n**F]q$!3<'[Js=Yjj)F]q7$$!37PLLL\\[%R)F]q$!35#ok*zM8`vF]q7$$!3G) *****\\&y!pmF]q$!3!RG'f$4^MA'F]q7$$!3Y******\\O3E]F]q$!3tn#G(3$Gp#[F]q 7$$!3NKLLL3z6LF]q$!35\"R^lQgGD$F]q7$$!3sLLL$)[`P(******z-6j'F]q$\"3[8bC\\6c#>'F]q7$$\"3q\"******4#32$)F ]q$\"3CHgsq'3$*[(F]q7$$\"3r$*****\\#y'G**F]q$\"3R!)G=v)[8h)F]q7$$\"3G* *****H%=H<\"F-$\"3t^Q-n#fcq*F]q7$$\"35mmm1>qM8F-$\"3wSPsA;+c5F-7$$\"3% )*******HSu]\"F-$\"34h/sb0)[8\"F-7$$\"3'HLL$ep'Rm\"F-$\"3e\"*H=$Rdk>\" F-7$$\"3')******R>4N=F-$\"3YyP')o\"oUD\"F-7$$\"3#emm;@2h*>F-$\"3r^]Z!o B2I\"F-7$$\"3]*****\\c9W;#F-$\"3XD%e7Cj@M\"F-7$$\"3Lmmmmd'*GBF-$\"3iV* 4D$fhw8F-7$$\"3j*****\\iN7]#F-$\"3\"3\"f0T\\>29F-7$$\"3aLLLt>:nEF-$\"3 z[^)=_>@V\"F-7$$\"35LLL.a#o$GF-$\"3Qo@_0uq`9F-7$$\"3ammm^Q40IF-$\"3!>y 8F-4=Z\"F-7$$\"3y******z]rfJF-$\"3EdC9j:(f[\"F-7$$\"3gmmmc%GpL$F-$\"3? 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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 10 "Example 5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x^(cosh*x);" "6#/-%\"fG6#%\"xG)F'*&%%coshG\"\"\"F'F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(f(x)) = ln(x^(cosh*x));" "6#/-%#lnG6#-%\"fG6#%\"xG-F%6#)F**&%%coshG \"\"\"F*F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=cosh*x*ln*x" "6#/%!G**%%coshG\"\"\"%\"xGF'%#lnGF'F(F '" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)/f(x)=sinh*x* ln*x+cosh*x/x" "6#/*&-%$f~'G6#%\"xG\"\"\"-%\"fG6#F(!\"\",&**%%sinhGF)F (F)%#lnGF)F(F)F)*(%%coshGF)F(F)F(F-F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=f(x)*(sinh*x*ln*x+cosh*x/x)" "6#/-%$f~'G6#%\"xG*&-%\"fG 6#F'\"\"\",&**%%sinhGF,F'F,%#lnGF,F'F,F,*(%%coshGF,F'F,F'!\"\"F,F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x^(cosh*x)*(sinh*x*ln*x+cosh*x/x)" " 6#/%!G*&)%\"xG*&%%coshG\"\"\"F'F*F*,&**%%sinhGF*F'F*%#lnGF*F'F*F**(F)F *F'F*F'!\"\"F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f := x -> x^cosh(x):\n'f(x)' =f(x);\nDiff(f(x),x)=diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"xG)F'-%%coshGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Dif fG6$)%\"xG-%%coshG6#F(F(*&F'\"\"\",&*&-%%sinhGF+F--%#lnGF+F-F-*&F)F-F( !\"\"F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 6 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 321 8 "Question" }{TEXT -1 8 ": Find " }{XPPEDIT 18 0 "Int(sinh *x/sqrt(4-cosh^2*x),x);" "6#-%$IntG6$*(%%sinhG\"\"\"%\"xGF(-%%sqrtG6#, &\"\"%F(*&%%coshG\"\"#F)F(!\"\"F2F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 8 "Solution" }{TEXT -1 2 " : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sinh*x/sqrt (4-cosh^2*x),x);" "6#-%$IntG6$*(%%sinhG\"\"\"%\"xGF(-%%sqrtG6#,&\"\"%F (*&%%coshG\"\"#F)F(!\"\"F2F)" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 " PIECEWISE([u=cosh*x,``],[du=sinh*x*dx,``])" "6#-%*PIECEWISEG6$7$/%\"uG *&%%coshG\"\"\"%\"xGF+%!G7$/%#duG*(%%sinhGF+F,F+%#dxGF+F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(1/sqrt (4-u^2),u)" "6#/%!G-%$IntG6$*&\"\"\"F)-%%sqrtG6#,&\"\"%F)*$%\"uG\"\"#! \"\"F2F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=arcsin(u/2)+c" "6#/%!G,&-%'a rcsinG6#*&%\"uG\"\"\"\"\"#!\"\"F+%\"cGF+" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=arcsin(cosh*x/2)+c" "6#/%!G,&-%'arcsinG6#*(%%coshG\"\"\"%\"xGF+ \"\"#!\"\"F+%\"cGF+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Int(sinh(x)/sqrt(4-cosh(x)^2),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%%sinhG6#%\"xG\"\"\",&\"\"%F+*$)-%% coshGF)\"\"#F+!\"\"#F3F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-% 'arcsinG6#,$*&#\"\"\"\"\"#F,-%%coshG6#%\"xGF,F,F,%\"cGF," }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 32 "Prove the following identiti es: " }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "cosh(x+y)=c osh*x*cosh*y+sinh*x*sinh*y" "6#/-%%coshG6#,&%\"xG\"\"\"%\"yGF),&**F%F) F(F)F%F)F*F)F)**%%sinhGF)F(F)F.F)F*F)F)" }{TEXT -1 8 " (b) " } {XPPEDIT 18 0 "2*sinh*2*x*cosh*x= sinh*3*x+sinh*x" "6#/*.\"\"#\"\"\"%% sinhGF&F%F&%\"xGF&%%coshGF&F(F&,&*(F'F&\"\"$F&F(F&F&*&F'F&F(F&F&" } {TEXT -1 9 " (c) " }{XPPEDIT 18 0 "cosh*3*x=4*cosh^3*x-3*cosh*x" " 6#/*(%%coshG\"\"\"\"\"$F&%\"xGF&,&*(\"\"%F&*$F%F'F&F(F&F&*(F'F&F%F&F(F &!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "______________ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 40 "Check the derivative s of the functions " }{XPPEDIT 18 0 "tanh*x, sech*x, csch*x" "6%*&%%t anhG\"\"\"%\"xGF%*&%%sechGF%F&F%*&%%cschGF%F&F%" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "coth*x" "6#*&%%cothG\"\"\"%\"xGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 20 "Find the derivative " }{XPPEDIT 18 0 "`f \+ '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 33 " in each of the following case s. " }}{PARA 0 "" 0 "" {TEXT -1 7 " (a) " }{XPPEDIT 18 0 "f(x)=x^3*s inh*3*x" "6#/-%\"fG6#%\"xG**F'\"\"$%%sinhG\"\"\"F)F+F'F+" }{TEXT -1 12 " (b) " }{XPPEDIT 18 0 "f(x)=ln(sinh*x)" "6#/-%\"fG6#%\"xG-% #lnG6#*&%%sinhG\"\"\"F'F-" }{TEXT -1 11 " (c) " }{XPPEDIT 18 0 " f(x)=2*sqrt(x)*tanh(sqrt(x))" "6#/-%\"fG6#%\"xG*(\"\"#\"\"\"-%%sqrtG6# F'F*-%%tanhG6#-F,6#F'F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " (d) " }{XPPEDIT 18 0 "f(x)=ln((1+cosh(x))/(1-cosh(x)))" "6#/-%\"f G6#%\"xG-%#lnG6#*&,&\"\"\"F--%%coshG6#F'F-F-,&F-F--F/6#F'!\"\"F4" } {TEXT -1 8 " (e) " }{XPPEDIT 18 0 "f(x)=(coth*x)^x" "6#/-%\"fG6#%\" xG)*&%%cothG\"\"\"F'F+F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "3*x^2*(sinh*3*x+x*cosh*3*x);" "6#*( \"\"$\"\"\"*$%\"xG\"\"#F%,&*(%%sinhGF%F$F%F'F%F%**F'F%%%coshGF%F$F%F'F %F%F%" }{TEXT -1 9 " (b) " }{XPPEDIT 18 0 "coth*x;" "6#*&%%cothG\" \"\"%\"xGF%" }{TEXT -1 9 " (c) " }{XPPEDIT 18 0 "tanh(sqrt(x))/sqr t(x)+sech^2*sqrt(x)" "6#,&*&-%%tanhG6#-%%sqrtG6#%\"xG\"\"\"-F)6#F+!\" \"F,*&%%sechG\"\"#-F)6#F+F,F," }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "-2/(sinh*x)" "6#,$*&\"\"#\"\"\"*&%% sinhGF&%\"xGF&!\"\"F*" }{TEXT -1 9 " (e) " }{XPPEDIT 18 0 "(coth*x )^x*(ln(coth*x)-x*sech*x*csch*x)" "6#*&)*&%%cothG\"\"\"%\"xGF'F(F',&-% #lnG6#*&F&F'F(F'F'*,F(F'%%sechGF'F(F'%%cschGF'F(F'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 " _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Q4. Find the following indefinite integrals." }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) \+ " }{XPPEDIT 18 0 "Int(tanh*x,x)" "6#-%$IntG6$*&%%tanhG\"\"\"%\"xGF(F) " }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "Int(tanh^2*x,x)" "6#-%$Int G6$*&%%tanhG\"\"#%\"xG\"\"\"F)" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 " Int(sinh*x*sqrt(1+cosh*x),x)" "6#-%$IntG6$*(%%sinhG\"\"\"%\"xGF(-%%sqr tG6#,&F(F(*&%%coshGF(F)F(F(F(F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " (d) " }{XPPEDIT 18 0 "Int(cosh^3*x,x)" "6#-%$IntG6$*&%%c oshG\"\"$%\"xG\"\"\"F)" }{TEXT -1 8 " (e) " }{XPPEDIT 18 0 "Int(cos h^2*x,x)" "6#-%$IntG6$*&%%coshG\"\"#%\"xG\"\"\"F)" }{TEXT -1 9 " (f ) " }{XPPEDIT 18 0 "Int(x*sinh*x,x)" "6#-%$IntG6$*(%\"xG\"\"\"%%sinhG F(F'F(F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "Ans" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "ln(cosh*x)+c" "6#,&-%#lnG6#*&%%coshG\"\"\"%\"xGF)F)%\" cGF)" }{TEXT -1 9 " (b) " }{XPPEDIT 18 0 "x - tanh*x+c" "6#,(%\"xG \"\"\"*&%%tanhGF%F$F%!\"\"%\"cGF%" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1+cosh*x)^(3/2)+c;" "6#,&),&\"\"\"F&*&%%coshGF&%\"xGF&F&*&\"\"$F&\" \"#!\"\"F&%\"cGF&" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "(d) " }{XPPEDIT 18 0 "sinh*x+sinh^3*x/3+c" "6#,(*&%%sinhG\"\"\"%\"xGF&F &*(F%\"\"$F'F&F)!\"\"F&%\"cGF&" }{TEXT -1 9 " (e) " }{XPPEDIT 18 0 "x/2+sinh*2*x/4+c" "6#,(*&%\"xG\"\"\"\"\"#!\"\"F&**%%sinhGF&F'F&F%F& \"\"%F(F&%\"cGF&" }{TEXT -1 7 " (f) " }{XPPEDIT 18 0 "x*cosh*x-sinh* x+c" "6#,(*(%\"xG\"\"\"%%coshGF&F%F&F&*&%%sinhGF&F%F&!\"\"%\"cGF&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 23 "Code for first picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 501 "aa := evalf(ln((1+sqrt(2)))): bb := evalf( sqrt(2)):\np1 := plot([cosh(x),sinh(x)],x=-1.5..1.5,color=[red,blue],t hickness=2):\np2 := plot([[[aa,0],[aa,bb]],[[0,1],[aa,1]],\n [[0, bb],[aa,bb]]],color=black,linestyle=2):\nt1 := plots[textplot]([-.7,1. 8,`y = cosh x`],color=red):\nt2 := plots[textplot]([-.7,-1.4,`y = sinh x`],color=blue):\nt3 := plots[textplot]([[.91,1.62,`A`],[.95,.94,`B`] ,[1.5,-.1,`x`],\n [-.08,2.4,`y`]],color=black):\nplots[display]([p 1,p2,t1,t2,t3],tickmarks=[3,5],labels=[``,``]);" }}}{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 0 529 "f := x ->sqrt(x^2-1):\np1 := plot([cosh(t),sinh(t),t=-2.1..2.1],color=red,thi ckness=2):\np2 := plot([-cosh(t),sinh(t),t=-2.1..2.1],color=red,thickn ess=2):\np3 := plot([x,-x],x=-3..3,color=black,linestyle=2):\npt := [1 .5,f(1.5)]:\np4 := plot([[pt]$3],style=point,symbol=[circle,diamond,cr oss],\n color=COLOR(RGB,.4,0,.8)):\nt1 := plots[textplot]([2.4 ,1.1,`(cosh t,sinh t)`],color=COLOR(RGB,.4,0,.8)):\nt2 := plots[textpl ot]([[3.3,-.2,`x`],[-.2,3.3,`y`]],color=black):\nplots[display]([p1,p2 ,p3,p4,t1,t2],view=[-3.3..3.3,-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 0 673 "f := x->s qrt(1-x^2):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness= 2):\npt := [.55,f(.55)]:\np2 := plot([[0,0],pt],color=COLOR(RGB,.4,0,. 8)):\np3 := plot([[pt]$3],style=point,\n symbol=[circle,diamond ,cross],color=COLOR(RGB,.4,0,.8)):\np4 := plot([.25*cos(t),.25*sin(t), t=0..arccos(.55)],\n color=COLOR(RGB,.4,0,.8)):\nt1 := plots [textplot]([.83,.93,`(cos t,sin t)`],color=COLOR(RGB,.4,0,.8)):\nt2 := plots[textplot]([.15,.1,`t`],color=COLOR(RGB,.4,0,.8)):\nt3 := plots[ textplot]([[1.15,-.08,`x`],[-.08,1.15,`y`]],color=COLOR(RGB,.01,.01,.0 1)):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],tickmarks=[3,3],scaling=co nstrained,\n view=[-1.15..1.15,-1.15..1.15]);" }}}{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 }