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{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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Inverse hyperbolic functions " }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 20 "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 45 "The inverse of the hyperbolic sine function, " } {XPPEDIT 18 0 "arcsinh*x" "6#*&%(arcsinhG\"\"\"%\"xGF%" }{TEXT -1 21 " , and its derivative " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(sinh(x),x=-3..3,y=-5..5,title=`y = sinh x`,thickness=2);" } }{PARA 13 "" 1 "" {GLPLOT2D 333 336 336 {PLOTDATA 2 "6'-%'CURVESG6$7S7 $$!\"$\"\"!$!3E!*4u#\\(y,5!#;7$$!3!******\\2<#pG!#<$!3sp&RK!fB$y)F17$$ !3#)***\\7bBav#F1$!3gX*))zWf?$yF17$$!36++]K3XFEF1$!3Ky'G2(H4$)oF17$$!3 %)****\\F)H')\\#F1$!3Kg)[K'y!=/'F17$$!3#****\\i3@/P#F1$!3,UK;$>GUI&F17 $$!3;++Dr^b^AF1$!3y8)edYL')p%F17$$!3$****\\7Sw%G@F1$!3GM$H.g=:9%F17$$! 3*****\\7;)=,?F1$!3K1G5kILJOF17$$!3/++DO\"3V(=F1$!3yk&*)*GSU\"=$F17$$! 3#******\\V'zV5iViY&R=F1 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$\"3P+++!)RO+?F1$\"3DW267+BGOF17$$\"30++]_!>w7#F1$\"3\"H[OeNny8%F17$$ \"3O++v)Q?QD#F1$\"3HIr\"yLF&4ZF17$$\"3G+++5jypBF1$\"3\"[jL\"\\H!3I&F17 $$\"3<++]Ujp-DF1$\"37#HBrChn1'F17$$\"3++++gEd@EF1$\"3S-;>soKUoF17$$\"3 9++v3'>$[FF1$\"3Qp\\&*>k;wxF17$$\"37++D6EjpGF1$\"3_Q.Z$**4py)F17$$\"\" $F*$\"3E!*4u#\\(y,5F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%*THICKNESS G6#\"\"#-%&TITLEG6#%+y~=~sinh~xG-%+AXESLABELSG6$Q\"x6\"Q\"yF_\\l-%%VIE WG6$;F(Fgz;$!\"&F*$\"\"&F*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 77 "Since the function sinh is one-to-on e, it has an inverse function denoted by " }{XPPEDIT 18 0 "sinh^(-1)" "6#)%%sinhG,$\"\"\"!\"\"" }{TEXT -1 28 " or arcsinh and defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 385 56 56 {PLOTDATA 2 "6 '-%'CURVESG6%7'7$$\"\"!F)F(7$$\"\")F)F(7$F+$\"\"#F)7$F(F.F'-%'COLOURG6 &%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#F/-%%TEXTG6%7$$\"\"%F)$\"\"\"F )QHy~=~arcsinh~x~~exactly~when~~x~=~sinh~y6\"-F26&F4F)F)F)-%*AXESSTYLE G6#%%NONEG-%+AXESLABELSG6$Q!FDFN-%%VIEWG6$%(DEFAULTGFR" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "It is possible to expres s " }{XPPEDIT 18 0 "arcsinh*x;" "6#*&%(arcsinhG\"\"\"%\"xGF%" }{TEXT -1 44 " in terms of the natural logarithm function." }}{PARA 0 "" 0 " " {TEXT -1 16 "First note that " }{XPPEDIT 18 0 "x = sinh*y;" "6#/%\"x G*&%%sinhG\"\"\"%\"yGF'" }{TEXT -1 18 " is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=(exp(y)-exp(-y))/2" "6#/% \"xG*&,&-%$expG6#%\"yG\"\"\"-F(6#,$F*!\"\"F/F+\"\"#F/" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x= (exp(2*y)-1)/(2*exp(y))" "6#/%\"xG*&,&-%$e xpG6#*&\"\"#\"\"\"%\"yGF,F,F,!\"\"F,*&F+F,-F(6#F-F,F." }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x*exp(y) = exp(2*y)-1" "6#/*(\"\"#\"\"\"%\" xGF&-%$expG6#%\"yGF&,&-F)6#*&F%F&F+F&F&F&!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(2*y)-2*x*exp(y)-1=0" "6#/,(-%$expG6#*&\"\"#\"\"\"% \"yGF*F**(F)F*%\"xGF*-F&6#F+F*!\"\"F*F0\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "u=exp(y)" "6#/%\"uG-%$ expG6#%\"yG" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u^2-2*x*u-1=0" "6#/,(*$%\"uG\"\"#\"\"\"*(F'F(%\"xGF( F&F(!\"\"F(F+\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Thi s quadratic equation can be solved for " }{XPPEDIT 18 0 "u=exp(y)" "6# /%\"uG-%$expG6#%\"yG" }{TEXT -1 34 " by completing the square to give \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u^2-2*x*u+x^2 = x ^2+1;" "6#/,(*$%\"uG\"\"#\"\"\"*(F'F(%\"xGF(F&F(!\"\"*$F*F'F(,&*$F*F'F (F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(u-x)^2 = x^2+1;" "6# /*$,&%\"uG\"\"\"%\"xG!\"\"\"\"#,&*$F(F*F'F'F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "u-x = ``;" "6#/,&%\"uG\"\"\"%\"xG!\"\"%!G" }{TEXT 268 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2+1);" "6#-%%sqrtG6#,&*$% \"xG\"\"#\"\"\"F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u = x;" "6# /%\"uG%\"xG" }{TEXT -1 1 " " }{TEXT 269 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(x^2+1);" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F*F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "u=exp(y)" "6#/%\"uG-%$expG6#%\"yG " }{TEXT -1 52 " cannot be negative, this rules out the possibility " }{XPPEDIT 18 0 "u = x-sqrt(x^2+1);" "6#/%\"uG,&%\"xG\"\"\"-%%sqrtG6#,& *$F&\"\"#F'F'F'!\"\"" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u =exp(y) " "6#/%\"uG-%$expG6#%\"yG" } {XPPEDIT 18 0 "`` = x+sqrt(x^2+1);" "6#/%!G,&%\"xG\"\"\"-%%sqrtG6#,&*$ F&\"\"#F'F'F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "We con clude that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln (x+sqrt(x^2+1));" "6#/%\"yG-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F)\"\"#F *F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh*x = ln(x+sqrt(x^2+1 ));" "6#/*&%(arcsinhG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sqrtG6#,&*$F'\"\"#F &F&F&F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }{XPPEDIT 18 0 "y=a rcsinh*x" "6#/%\"yG*&%(arcsinhG\"\"\"%\"xGF'" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "y=ln(x+sqrt(x^2+1))" "6#/%\"yG-%#lnG6#,&%\"xG\"\"\"-%%s qrtG6#,&*$F)\"\"#F*F*F*F*" }{TEXT -1 20 " appear to coincide." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "f := x -> arcsinh(x): g := x -> ln(x+sqrt(x^2+1)):\n'f(x)'=f(x); \+ 'g(x)'=g(x);\nplot([f(x),g(x)],x=-4..4,y,thickness=[1,2],legend=[`f(x) `,`g(x)`],tickmarks=[6,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG-%(arcsinhGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG -%#lnG6#,&F'\"\"\"*$,&*$)F'\"\"#F,F,F,F,#F,F1F," }}{PARA 13 "" 1 "" {GLPLOT2D 491 294 294 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$!\"%\"\"!$!3A,6E ZDr%4#!#<7$$!3ommmmFiDQF-$!3[(3D)\\&G:0#F-7$$!35LLLo!)*Qn$F-$!3*f>R_%z U7?F-7$$!3nmmmwxE.NF-$!3mD?q6yhm>F-7$$!3YmmmOk]JLF-$!3#z>%=&Qr$=>F-7$$ !3_LLL[9cgJF-$!3iD52ON.o=F-7$$!3smmmhN2-IF-$!3&zxMJ(>5>=F-7$$!3!****** \\`oz$GF-$!3QHBg%)G#fw\"F-7$$!3!omm;)3DoEF-$!3&*pNDYv'zq\"F-7$$!3?+++: v2*\\#F-$!3-\"yT&*[))ok\"F-7$$!3BLLL8>1DBF-$!3_A!4ACz^\"F-7$$!3;+++S(R#**>F-$!3#[0k[U&HV9F-7$$!30++++@)f #=F-$!3y8tp/'*)HO\"F-7$$!3-+++gi,f;F-$!3u!e(=1H&)z7F-7$$!3qmmm\"G&R2:F -$!3qav3$Re))>\"F-7$$!3XLLLtK5F8F-$!3I#>!\\b'o[4\"F-7$$!3eLLL$HsV<\"F- $!3o]l6xri%***!#=7$$!3+-++]&)4n**F]q$!3Ehh8&)=X!z)F]q7$$!37PLLL\\[%R)F ]q$!3!Q3(HI(oz3*G(******z-6j'F]q$\"3Zwnwkf$=A'F]q7$ $\"3q\"******4#32$)F]q$\"3!>/jYf\"okvF]q7$$\"3r$*****\\#y'G**F]q$\"3uy Q*yu8Kw)F]q7$$\"3G******H%=H<\"F-$\"3OQ\"[\\6)>&)**F]q7$$\"35mmm1>qM8F -$\"3+5?/jJV*4\"F-7$$\"3%)*******HSu]\"F-$\"3;s-]zK)))>\"F-7$$\"3'HLL$ ep'Rm\"F-$\"3O>,bYeS#G\"F-7$$\"3')******R>4N=F-$\"3)3w2'3qNn8F-7$$\"3# emm;@2h*>F-$\"3r9Co5K*=W\"F-7$$\"3]*****\\c9W;#F-$\"3_([6+kN[^\"F-7$$ \"3Lmmmmd'*GBF-$\"3=`I;wrw\"e\"F-7$$\"3j*****\\iN7]#F-$\"3A*oP#[**oZ;F -7$$\"3aLLLt>:nEF-$\"3U#R(*\\#=e2>F-7$$\"3/LLL8-V&\\$F-$\"3YD;S'HkW'>F-7 $$\"3=+++XhUkOF-$\"3a*))[RHP*4?F-7$$\"3=+++:oR_%zU7?F-7$F9$!35E?q6yhm>F-7$F>$!39)>%=&Qr$=>F-7$FC$! 3=D52ON.o=F-7$FH$!31zZ8t>5>=F-7$FM$!3sGBg%)G#fw\"F-7$FR$!3RoNDYv'zq\"F -7$FW$!39!yT&*[))ok\"F-7$Ffn$!3=B!4ACz^\"F-7$F` o$!3rbS'[U&HV9F-7$Feo$!3c8tp/'*)HO\"F-7$Fjo$!3_!e(=1H&)z7F-F^pFcp7$Fip $!3e\\l6xri%***F]q7$F_q$!39gh8&)=X!z)F]q7$Fdq$!3\"\\3(HI(o z3*G " 0 "" {MPLTEXT 1 0 23 "convert(arcsinh(x),ln); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#, &*$)F'\"\"#F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "We \+ can find the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG !\"\"" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "y = arcsinh*x;" "6#/%\"yG* &%(arcsinhG\"\"\"%\"xGF'" }{TEXT -1 47 " by implicit differentiation f rom the equation " }{XPPEDIT 18 0 "x = sinh*y;" "6#/%\"xG*&%%sinhG\"\" \"%\"yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differentiat ing both sides of the equation " }{XPPEDIT 18 0 "x = sinh*y;" "6#/%\"x G*&%%sinhG\"\"\"%\"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 303 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1 = cosh*y;" "6#/\"\"\"*&%%coshGF$%\"yGF$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 1/(cosh*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F& F&*&%%coshGF&%\"yGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the identity " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cosh^2*y-sinh^2*y = 1;" "6#/,&*&%%coshG\"\"#%\"yG\"\"\"F)*&%%sin hGF'F(F)!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*y;" "6#*&%%c oshG\"\"\"%\"yGF%" }{TEXT -1 3 " = " }{TEXT 267 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(sinh^2*y+1);" "6#-%%sqrtG6#,&*&%%sinhG\"\"#%\"yG \"\"\"F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cosh*y;" "6#*&%%co shG\"\"\"%\"yGF%" }{TEXT -1 28 " is never negative, we have " } {XPPEDIT 18 0 "cosh*y = sqrt(sinh^2*y+1);" "6#/*&%%coshG\"\"\"%\"yGF&- %%sqrtG6#,&*&%%sinhG\"\"#F'F&F&F&F&" }{TEXT -1 5 ", so " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(sinh^2*y+1);" "6# /*&%#dyG\"\"\"%#dxG!\"\"*&F&F&-%%sqrtG6#,&*&%%sinhG\"\"#%\"yGF&F&F&F&F (" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(x^2+1);" "6# /*&%#dyG\"\"\"%#dxG!\"\"*&F&F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&F&F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(arcsinh(x),x)=diff(arcsinh(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%(arcsinhG6#%\"xGF**&\"\"\"F,*$- %%sqrtG6#,&*$)F*\"\"#F,F,F,F,F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([arcsinh(x),1/sqrt(1+x^2)],x=-3..3,y,color=[red,blue],thi ckness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-% 'CURVESG6$7S7$$!\"$\"\"!$!3$o1K#fkW==!#<7$$!3!******\\2<#pGF-$!3gi7aot Dw\"F-7$$!3/+++v l[p8F-$!3[Wc*yd<,7\"F-7$$!3\"******\\>iUC\"F-$!3j/cq6O+W5F-7$$!3-++Dhk aI6F-$!3+g$R&4VR2(*!#=7$$!3s******\\XF`**Fcp$!3K5(eDOd1y)Fcp7$$!3u**** ***>#z2))Fcp$!377Uev!=^%zFcp7$$!3S++]7RKvuFcp$!3Ga*GNC><\"pFcp7$$!3s,+ ++P'eH'Fcp$!38*\\f#)4*GSfFcp7$$!3q)***\\7*3=+&Fcp$!3f$4&eKht8[Fcp7$$!3 [)***\\PFcpPFcp$!3E`RN2tb&o$Fcp7$$!3;)****\\7VQ[#Fcp$!3g(**Q-<()*eCFcp 7$$!32)***\\i6:.8Fcp$!3u)pn$32\\*H\"Fcp7$$!3Wb+++v`hH!#?$!3EH)))3*4#))Fcp7$$\"36+++D-eI6F-$\"3(4Nc]+=wq*F cp7$$\"3u***\\(=_(zC\"F-$\"3#***\\*o^Fj/\"F-7$$\"3M+++b*=jP\"F-$\"35'[ o\\?ST7\"F-7$$\"3g***\\(3/3(\\\"F-$\"3E$\\XviUJ>\"F-7$$\"33++vB4JB;F-$ \"3S7TtNPFh7F-7$$\"3u*****\\KCnu\"F-$\"3$3)3YH9FC8F-7$$\"3s***\\(=n#f( =F-$\"3QPSw7# F-$\"3U!4%)Q%RH*\\\"F-7$$\"3O++v)Q?QD#F-$\"3a*yQYY3$[FF-$\"3:scRpJqN$\"30H!o#[Am:PFcp7$FC$\"3E!=$3bW$p )QFcp7$FH$\"3+mFpTB&Fcp7$F `o$\"3+X=_[?Y[bFcp7$Feo$\"3kY48e@<(*eFcp7$Fjo$\"33Zx_J&oWE'Fcp7$F_p$\" 3T7k(4ir`i'Fcp7$Fep$\"37QXD0qg(3(Fcp7$Fjp$\"3J\"f%Q[WB/vFcp7$F_q$\"3>G 1WUoZ4!)Fcp7$Fdq$\"3sJnVeM\\i%)Fcp7$Fiq$\"3vQ#ypoCO%*)Fcp7$F^r$\"3'f2$ 3@)fsN*Fcp7$Fcr$\"3Z3:]&G.^q*Fcp7$$!35)**\\P9(\\$*=Fcp$\"3Dm/,cUTD)*Fc p7$Fhr$\"3e3C'=Bch\"**Fcp7$$!3%p)*\\i:sw%)*!#>$\"3=&=4\"=:'=&**Fcp7$$! 39$***\\(oKQm'Fe`l$\"3ewecU.(y(**Fcp7$$!3O***\\(=K**zMFe`l$\"3YR%GyJ]R ***Fcp7$F]s$\"3o[mwYh&*****Fcp7$$\"3l%**\\ilg4,$Fe`l$\"3Q16xP,Z&***Fcp 7$$\"3%[***\\i]2=jFe`l$\"31$pR,_+,)**Fcp7$$\"3/&**\\(o%*=D'*Fe`l$\"3u0 ])*fs*R&**Fcp7$Fcs$\"3*QB2BD7u\"**Fcp7$$\"3K******\\4+p=Fcp$\"3GO!*HV$ )yH)*Fcp7$Fhs$\"3e/-awn\"Rr*Fcp7$F]t$\"3SiRO=\\8#Q*Fcp7$Fbt$\"3e4`3hW! Q&*)Fcp7$Fgt$\"39g!4D([[([)Fcp7$F\\u$\"3E_\")>(HY0-)Fcp7$Fau$\"3(HZ8$z PH3vFcp7$Ffu$\"3/@&)\\l(Ru1(Fcp7$F[v$\"35Jh851EDmFcp7$F`v$\"3-\\CR@Z7` iFcp7$Fev$\"3Kedk?&G!yeFcp7$Fjv$\"3Arl+1W[abFcp7$F_w$\"3NSmkgz#\\C&Fcp 7$Fdw$\"3C\")3`u%)Ro\\Fcp7$Fiw$\"3Kk2UU@2/ZFcp7$F^x$\"3#[\\>2$\\[rWFcp 7$Fcx$\"3!pt\"*HvxOD%Fcp7$Fhx$\"3!3F$3qMjbSFcp7$F]y$\"335#oFE=y)QFcp7$ Fby$\"3MU1wBmX5PFcp7$Fgy$\"3gr)>ad;Sc$Fcp7$F\\z$\"39#GQh!fF>MFcp7$Faz$ \"3')4I<,ho!H$Fcp7$FfzFf[l-F[[l6&F][lFa[lFa[lF^[l-%*THICKNESSG6#\"\"#- %+AXESLABELSG6$Q\"x6\"Q\"yFigl-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 259 6 "legend" }{TEXT -1 2 " \+ " }{TEXT 277 5 "_____" }{TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = arcsin h*x;" "6#/-%\"fG6#%\"xG*&%(arcsinhG\"\"\"F'F*" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 19 " " }{TEXT 278 5 "____ _" }{TEXT -1 6 " f '(" }{TEXT 281 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = 1/sqrt(x^2+1);" "6#/%!G*&\"\"\"F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&F &!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "An alternative method of finding the de rivative " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arcsinh*x];" "6#7#*&%(arcsinhG\"\"\"%\"xGF& " }{TEXT -1 24 " is to use the formula " }{XPPEDIT 18 0 "arcsinh*x = \+ ln(x+sqrt(x^2+1));" "6#/*&%(arcsinhG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sqrt G6#,&*$F'\"\"#F&F&F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([ln(x+sqrt(x ^2+1))],x) = 1/(x+sqrt(x^2+1));" "6#/-%%DiffG6$7#-%#lnG6#,&%\"xG\"\"\" -%%sqrtG6#,&*$F,\"\"#F-F-F-F-F,*&F-F-,&F,F--F/6#,&*$F,F3F-F-F-F-!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x+sqrt(x^2+1)],x);" "6#-%%Diff G6$7#,&%\"xG\"\"\"-%%sqrtG6#,&*$F(\"\"#F)F)F)F)F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2+1));" "6#/%!G*&\"\"\"F&,&%\"xGF&-%%s qrtG6#,&*$F(\"\"#F&F&F&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+` `(1/2)*(x^2+1)^(-1/2)*2*x);" "6#-%!G6#,&\"\"\"F'**-F$6#*&F'F'\"\"#!\" \"F'),&*$%\"xGF,F'F'F',$*&F'F'F,F-F-F'F,F'F1F'F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2+1) );" "6#/%!G*&\"\"\"F&,&%\"xGF&-%%sqrtG6#,&*$F(\"\"#F&F&F&F&!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+x/sqrt(x^2+1));" "6#-%!G6#,&\"\"\" F'*&%\"xGF'-%%sqrtG6#,&*$F)\"\"#F'F'F'!\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2+1));" "6#/%!G*&\"\"\"F&,&%\"xGF&-%%s qrtG6#,&*$F(\"\"#F&F&F&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``((x+ sqrt(x^2+1))/sqrt(x^2+1));" "6#-%!G6#*&,&%\"xG\"\"\"-%%sqrtG6#,&*$F(\" \"#F)F)F)F)F)-F+6#,&*$F(F/F)F)F)!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqrt(x^2+1);" "6#/%!G*&\"\" \"F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Diff(ln (x+sqrt(x^2+1)),x);\nvalue(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%#lnG6#,&%\"xG\"\"\"*$-%%sqrtG6#,&*$)F*\"\"# F+F+F+F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&,&*$)% \"xG\"\"#F%F%F%F%#!\"\"F+F*F%F%F%,&F*F%*$-%%sqrtG6#F'F%F%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$-%%sqrtG6#,&*$)%\"xG\"\"#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 47 "The invers e of the hyperbolic cosine function, " }{XPPEDIT 18 0 "arccosh*x" "6#* &%(arccoshG\"\"\"%\"xGF%" }{TEXT -1 21 ", and its derivative " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "p1 := plot(cosh(x ),x=-3..0,y,color=red,thickness=2):\np2 := plot(cosh(x),x=0..3,color=b lue,thickness=2):\np3 := plot([[0,0],[3,0]],color=blue,thickness=2):\n t1 := plots[textplot]([2.7,3,`y = cosh x`],color=blue):\nplots[display ]([p1,p2,p3,t1],view=[-3..3,0..6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 380 257 257 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$!\"$\"\"!$\"3`wxd*>mn+\"!#;7$$ !3&*****\\P&3Y$H!#<$\"3bO\"fK4WOV*F17$$!3!***\\iv%*3^F17$$!31++]-\\r\\AF1$\"3!\\*)>]GI_z%F 17$$!3!*****\\(GVZ=#F1$\"3'fiT(elV+XF17$$!31++](4J@7#F1$\"31x!=P\\VVB% F17$$!3!***\\iIKFl?F1$\"3Cmnxq\"=r+%F17$$!3)*****\\FPm(*>F1$\"3'4F)\\; Et`PF17$$!3********4'*QS>F1$\"3k`p+N`b_NF17$$!3-+]i&>mP(=F1$\"3-o_+![h JL$F17$$!34+++&=$z9=F1$\"3A3yk:*Q8:$F17$$!3%***\\iX/4]!)z#F17$$!3\"****\\i:#>C;F1$\"3i[ m0pWpNEF17$$!3!***\\7ev:l:F1$\"3Y$Q0%3=C'\\#F17$$!3.++vo2[,:F1$\"3[S#4 f;lbN#F17$$!3-+]i![Q`V\"F1$\"3xdn.k2b>AF17$$!3/+]PC9wx8F1$\"3;Sfh6z24@ F17$$!3%****\\iiwbJ\"F1$\"3sN/;w%4w*>F17$$!35+++XOL^7F1$\"3/_-b[>c!*=F 17$$!33++D@W[)=\"F1$\"3[\\VdO%Fgu$\"3Kbz/$)*[m4\"F17$$!3E)**\\i0)*3t$Fg u$\"3'[u>.24/2\"F17$$!3e)*****\\%o5:$Fgu$\"3_c%QTJe+0\"F17$$!39****\\( G=l[#Fgu$\"3NyTWsM2J5F17$$!3+++++n8#*=Fgu$\"3p,#4tPaz,\"F17$$!3G***\\i &>Se7Fgu$\"3`[;tJ$Gz+\"F17$$!3V$***\\P%p$=l!#>$\"3=FBL4_7-5F17$$F*F*$ \"\"\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")FgzFgz-%*THICKNESSG6#\"\"#-F$6 %7SFfz7$$\"3s******\\i9RlFcz$\"3$Rm[OyQ@+\"F17$$\"3/++vVA)GA\"Fgu$\"3Z .alFl[25F17$$\"3+++]Peui=Fgu$\"35IVgL$*R<5F17$$\"3A++]i3&o]#Fgu$\"3\"H HhJS'eJ5F17$$\"3%)***\\(oX*y9$Fgu$\"3Ir=c#pc*\\5F17$$\"3z***\\P9CAu$Fg u$\"3g/)GM>U32\"F17$$\"3!)***\\P*zhdVFgu$\"3qHJ99hX'4\"F17$$\"31++v$>f S*\\Fgu$\"3MH4b\"f;t7\"F17$$\"3$)***\\(=$f%GcFgu$\"3K=g0AQii6F17$$\"3Q +++Dy,\"G'Fgu$\"34rkS\"zEQ?\"F17$$\"33++]7F17$$\"33++vVy!eP\"F1$\"3I.6igUX0@F17$$\"34+](=WU[V\"F1 $\"39)*)\\$G\"o&=AF17$$\"3)****\\7B>&)\\\"F1$\"3C;0lA#f#\\BF17$$\"3)** *\\P>:mk:F1$\"3WQ\\@#e2^\\#F17$$\"3'***\\iv&QAi\"F1$\"3:ikjZa$4j#F17$$ \"31++vtLU%o\"F1$\"3]jIT(*oW(y#F17$$\"3!******\\Nm'[F1$\"3#=dl3L(p]NF17$$\"3z*****\\@80+#F1$\"33V7z Xv0kPF17$$\"31++]7,Hl?F1$\"3`2-nyO=2SF17$$\"3()**\\P4w)R7#F1$\"3vsGXQ& *)>C%F17$$\"3;++]x%f\")=#F1$\"3f?K<^GX:XF17$$\"3!)**\\P/-a[AF1$\"3-8. \"=\\C(*y%F17$$\"3/+](=Yb;J#F1$\"3Ch\"*zym5&4&F17$$\"3')****\\i@OtBF1$ \"34L.3rIH8aF17$$\"3')**\\PfL'zV#F1$\"3)zl-SI=&odF17$$\"3>+++!*>=+DF1$ \"3?(3Nzl!RLhF17$$\"3-++DE&4Qc#F1$\"3So%)3@25JlF17$$\"3=+]P%>5pi#F1$\" 3_#)=Eh^j^pF17$$\"39+++bJ*[o#F1$\"3I*>Sf*=VitF17$$\"33++Dr\"[8v#F1$\"3 EVQC$R)yjyF17$$\"3++++Ijy5GF1$\"3KEr5^4dT$)F17$$\"31+]P/)fT(GF1$\"3![5 &=c0]$)))F17$$\"31+]i0j\"[$HF1$\"3Y6,CkKfN%*F17$$\"\"$F*F+-F[[l6&F][lF gzFgzF^[lFa[l-F$6%7$7$FgzFgz7$FdjlFgzFfjlFa[l-%%TEXTG6%7$$\"#F!\"\"Fdj lQ+y~=~cosh~x6\"Ffjl-%+AXESLABELSG6%Q\"xFe[mQ\"yFe[m-%%FONTG6#%(DEFAUL TG-%%VIEWG6$;F(Fdjl;Fgz$\"\"'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 37 "The function cosh is one-to-one when " }{TEXT 260 13 "restricte d to" }{TEXT -1 14 " the domain [ " }{XPPEDIT 18 0 "0,infinity;" "6$\" \"!%)infinityG" }{TEXT -1 51 " ), and its inverse with this domain is \+ denoted by " }{XPPEDIT 18 0 "cosh^(-1);" "6#)%%coshG,$\"\"\"!\"\"" } {TEXT -1 12 " or arccosh." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 504 70 70 {PLOTDATA 2 "6'-%'CURVESG6%7'7$$\"\"!F)F(7$$\"\")F )F(7$F+$\"\"#F)7$F(F.F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNES SG6#F/-%%TEXTG6%7$$\"\"%F)$\"\"\"F)Qiny~=~arccosh~x~~exactly~when~~x~= ~cosh~y,~and~y~is~not~negative6\"-F26&F4F)F)F)-%*AXESSTYLEG6#%%NONEG-% +AXESLABELSG6$Q!FDFN-%%VIEWG6$%(DEFAULTGFR" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 26 "It is possible to express " } {XPPEDIT 18 0 "arccosh*x;" "6#*&%(arccoshG\"\"\"%\"xGF%" }{TEXT -1 44 " in terms of the natural logarithm function." }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }{XPPEDIT 18 0 "x = cosh*y;" "6#/%\"xG* &%%coshG\"\"\"%\"yGF'" }{TEXT -1 18 " is equivalent to " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (exp(y)+exp(-y))/2;" "6#/%\" xG*&,&-%$expG6#%\"yG\"\"\"-F(6#,$F*!\"\"F+F+\"\"#F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (exp(2*y)+1)/(2*exp(y));" "6#/%\"xG*&,&-% $expG6#*&\"\"#\"\"\"%\"yGF,F,F,F,F,*&F+F,-F(6#F-F,!\"\"" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x*exp(y) = exp(2*y)+1;" "6#/*(\"\"#\"\"\"% \"xGF&-%$expG6#%\"yGF&,&-F)6#*&F%F&F+F&F&F&F&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(2*y)-2*x*exp(y)+1 = 0;" "6#/,(-%$expG6#*&\"\"#\"\" \"%\"yGF*F**(F)F*%\"xGF*-F&6#F+F*!\"\"F*F*\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "u=exp(y)" "6#/% \"uG-%$expG6#%\"yG" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u^2-2*x*u+1 = 0;" "6#/,(*$%\"uG\"\"#\"\"\"*(F 'F(%\"xGF(F&F(!\"\"F(F(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "This quadratic equation can be solved for " }{XPPEDIT 18 0 "u=e xp(y)" "6#/%\"uG-%$expG6#%\"yG" }{TEXT -1 34 " by completing the squar e to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u^2-2*x *u+x^2 = x^2-1;" "6#/,(*$%\"uG\"\"#\"\"\"*(F'F(%\"xGF(F&F(!\"\"*$F*F'F (,&*$F*F'F(F(F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(u-x)^2 = x^2-1 ;" "6#/*$,&%\"uG\"\"\"%\"xG!\"\"\"\"#,&*$F(F*F'F'F)" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "u-x = ``;" "6#/,&%\"uG\"\"\"%\"xG!\"\"%!G" }{TEXT 264 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2-1);" "6#-%%sqrtG6#,&*$% \"xG\"\"#\"\"\"F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 1 " " }{TEXT 265 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(x^2-1);" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F*!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "u=exp(y)" "6#/%\"uG-%$expG6#%\"yG " }{TEXT -1 52 " cannot be negative, this rules out the possibility " }{XPPEDIT 18 0 "u = x-sqrt(x^2-1);" "6#/%\"uG,&%\"xG\"\"\"-%%sqrtG6#,& *$F&\"\"#F'F'!\"\"F." }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u =exp(y) " "6#/%\"uG-%$expG6#%\"yG" } {XPPEDIT 18 0 "`` = x+sqrt(x^2-1);" "6#/%!G,&%\"xG\"\"\"-%%sqrtG6#,&*$ F&\"\"#F'F'!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "We \+ conclude that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln(x+sqrt(x^2-1));" "6#/%\"yG-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F)\" \"#F*F*!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccosh*x = ln(x+sqr t(x^2-1));" "6#/*&%(arccoshG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sqrtG6#,&*$F '\"\"#F&F&!\"\"F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 262 15 "_______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " } {XPPEDIT 18 0 "y = arccosh*x;" "6#/%\"yG*&%(arccoshG\"\"\"%\"xGF'" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "y = ln(x+sqrt(x^2-1));" "6#/%\"yG- %#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F)\"\"#F*F*!\"\"F*" }{TEXT -1 20 " a ppear to coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 159 "f := x -> arccosh(x): g := x -> ln(x+sqrt(x ^2-1)):\n'f(x)'=f(x); 'g(x)'=g(x);\nplot([f(x),g(x)],x=0..4,y,thicknes s=[1,2],legend=[`f(x)`,`g(x)`],tickmarks=[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(arccoshGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%#lnG6#,&F'\"\"\"*$,&*$)F'\"\"#F,F,F,!\" \"#F,F1F," }}{PARA 13 "" 1 "" {GLPLOT2D 491 294 294 {PLOTDATA 2 "6'-%' CURVESG6&7J7$$\"3#*******H,Q+5!#<$\"3ev'[L$R>dF!#>7$$\"30+++Nt.A5F*$\" 3WUv^%Qgb4#!#=7$$\"3&*******RXpV5F*$\"3w[>Wa[]XHF37$$\"33+++X#)36d(=9%F37$$\"3)****** *p=\\q6F*$\"3c)eJPEZ%fdF37$$\"3)*******eBIY7F*$\"3/gJ5\"*e'>)oF37$$\"3 \"******HO[kL\"F*$\"3G`d0[=))))zF37$$\"31+++`Q\"GT\"F*$\"3C`]-+\\s*z)F 37$$\"33+++s]k,:F*$\"3QE-Ysf$*Q'*F37$$\"3++++`dF!e\"F*$\"3ST6<9N,J5F*7 $$\"3/+++sgam;F*$\"3ab[bP=_)4\"F*7$$\"3)******p\"ep[F*$\"3;cSs&R!Gl7 F*7$$\"3)******>kD!)*>F*$\"3G**G()\\s\"eJ\"F*7$$\"35+++f`@'3#F*$\"3azW !oXi`O\"F*7$$\"3?+++nZ)H;#F*$\"3*z_qF@CjS\"F*7$$\"3#)*****>$y*eC#F*$\" 3YKDo_p`[9F*7$$\"3')******R^bJBF*$\"3S^r&QW_,\\\"F*7$$\"3.+++0TN:CF*$ \"3e$GF*$\"3&oU?yT&Q,=F*7$$\"3 y*****>\"yh]KF*$\"3I75I`eWZ=F*7$$\"38+++()fdLLF*$\"3m3J_]l*Q(=F*7$$\"3 \"******>q7%=MF*$\"3BjJj.%3-!>F*7$$\"3&******f#pa-NF*$\"3PY>=YjgD>F*7$ $\"3!*******Rv&)zNF*$\"3cvJ*eXl$[>F*7$$\"3%******zAk%oOF*$\"3k'3RzM.Q( >F*7$$\"3))*****p5:xu$F*$\"3YR]k8\"**f*>F*7$$\"30+++sI@KQF*$\"3w%)R.iX 6>?F*7$$\"30+++2%)38RF*$\"3#pw*fDBtS?F*7$$\"\"%\"\"!$\"3ygb*)oqVj?F*-% 'COLOURG6&%$RGBG$\"#5!\"\"$FjwFjwFdx-%*THICKNESSG6#\"\"\"-%'LEGENDG6#% %f(x)G-F$6&7JF'F.F4F97$F?$\"3R?#)36d(=9%F3FC7$FI$\"3#*eJ5\"*e'>)oF3FMF RFWFfnF[oF`oFeoFjoF_pFdpFipF^qFcqFhqF]rFbrFgrF\\sFasFfsF[tF`tFetFjtF_u FduFiuF^vFcvFhvF]wFbwFgw-F^x6&F`xFdxFaxFdx-Ffx6#\"\"#-Fjx6#%%g(x)G-%+A XESLABELSG6$Q\"x6\"Q\"yFbz-%*AXESTICKSG6$Fiw\"\"$-%%VIEWG6$;FdxFhw%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x) " "g(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The domain of the function " }{XPPEDIT 18 0 "f(x) = arccosh*x;" "6 #/-%\"fG6#%\"xG*&%(arccoshG\"\"\"F'F*" }{TEXT -1 14 " is the set \{ " }{TEXT 310 1 "x" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "x>=1" "6#1\"\"\"%\" xG" }{TEXT -1 6 " \} = [" }{XPPEDIT 18 0 "-1,infinity" "6$,$\"\"\"!\" \"%)infinityG" }{TEXT -1 30 "), and the range is the set \{ " }{TEXT 311 1 "y" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "y>=0" "6#1\"\"!%\"yG" } {TEXT -1 5 "\} = [" }{XPPEDIT 18 0 "0,infinity" "6$\"\"!%)infinityG" } {TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 14 "convert(..,ln)" } {TEXT -1 26 " can make this conversion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(arccosh(x),ln); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&%\"xG\"\"\"*&,&F'F(F(!\" \"#F(\"\"#,&F'F(F(F(F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can find the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dx G!\"\"" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "y = arccosh*x;" "6#/%\"yG *&%(arccoshG\"\"\"%\"xGF'" }{TEXT -1 47 " by implicit differentiation \+ from the equation " }{XPPEDIT 18 0 "x = cosh*y;" "6#/%\"xG*&%%coshG\" \"\"%\"yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differenti ating both sides of the equation " }{XPPEDIT 18 0 "x = cosh*y;" "6#/% \"xG*&%%coshG\"\"\"%\"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 304 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1 = sinh*y;" "6#/\"\"\"*&%%sinhGF$%\"yGF$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 1/(sinh*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F& F&*&%%sinhGF&%\"yGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the identity " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cosh^2*y-sinh^2*y = 1;" "6#/,&*&%%coshG\"\"#%\"yG\"\"\"F)*&%%sin hGF'F(F)!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*y;" "6#*&%%s inhG\"\"\"%\"yGF%" }{TEXT -1 3 " = " }{TEXT 263 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(cosh^2*y-1);" "6#-%%sqrtG6#,&*&%%coshG\"\"#%\"yG \"\"\"F+F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\" !%\"yG" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "0 <= sinh*y;" "6 #1\"\"!*&%%sinhG\"\"\"%\"yGF'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "sinh*y = sqrt(cosh^2*y-1);" "6#/*&%%sinhG\"\"\"%\"yGF&-%%sqrtG6#,&* &%%coshG\"\"#F'F&F&F&!\"\"" }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(cosh^2*y-1);" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&F&F&-%%sqrtG6#,&*&%%coshG\"\"#%\"yGF&F&F&F(F(" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/sqrt(x^2-1);" "6#/*&%# dyG\"\"\"%#dxG!\"\"*&F&F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(arccosh(x),x)=diff(arccosh(x),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%(arccoshG6#%\"xGF**&\"\"\"F,*&-%%sq rtG6#,&F*F,F,!\"\"F,-F/6#,&F*F,F,F,F,F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot([arccosh(x),1/sqrt(x^2-1)],x=0..3,y=0..4,color=[ red,blue],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 439 336 336 {PLOTDATA 2 "6'-%'CURVESG6$7G7$$\"33+++vMA+5!#<$\"3#)*[^G$)zS6#!#>7$$ \"3(******>FOB+\"F*$\"3Czz003FMoF-7$$\"3\"******>![l;5F*$\"3icV8)GoD#= !#=7$$\"3*******4Lt4.\"F*$\"3ky5fV$F87$$\"3&******R!QBE6F*$\"3g.,I*H1 K(\\F87$$\"3!******\\\"o?&=\"F*$\"3O07:#pRf*fF87$$\"3/+++a&4*\\7F*$\"3 %o1>xtl-$pF87$$\"32+++j=_68F*$\"35BLh0\"e:q(F87$$\"35+++Wy!eP\"F*$\"3( y8v<,[(=%)F87$$\"36+++UC%[V\"F*$\"3e;A*yb'3&)\\\" F*$\"3ymKT;.)4h*F87$$\"3%*******=:mk:F*$\"3\\j*Q$yj9=5F*7$$\"3*******f dQAi\"F*$\"3k4s0yNdk5F*7$$\"33+++uLU%o\"F*$\"3i()=n*R?=6\"F*7$$\"3!*** ***\\Nm'[`!Gbp0?\"F*7$$ \"3'******RVDB(=F*$\"3cR6Md/*)R7F*7$$\"3$******4TW)R>F*$\"3o#o]J))3:G \"F*7$$\"3z*****\\@80+#F*$\"3%H'o1^TD<8F*7$$\"3-+++7,Hl?F*$\"3iN'3JaeQ N\"F*7$$\"3$)******3w)R7#F*$\"3i**Q(RDadQ\"F*7$$\"3?+++y%f\")=#F*$\"3s xxf/SN>9F*7$$\"3A+++/-a[AF*$\"3QlxAf*\\)\\9F*7$$\"30+++ial6BF*$\"3'H;Z *)ea1[\"F*7$$\"3#)*****>;iLP#F*$\"3IZ=jNry4:F*7$$\"3$)******eL'zV#F*$ \"3\"o\"*HMe:$R:F*7$$\"3>+++!*>=+DF*$\"3')o'y/myoc\"F*7$$\"3++++E&4Qc# F*$\"3EiU>>RB%f\"F*7$$\"39+++%>5pi#F*$\"3Sm]0!)He?;F*7$$\"39+++bJ*[o#F *$\"3AwubS.:W;F*7$$\"31+++r\"[8v#F*$\"3y%>gS:X/n\"F*7$$\"3++++Ijy5GF*$ \"3EgO]H2N$p\"F*7$$\"3/+++/)fT(GF*$\"3b$[`w))prr\"F*7$$\"35+++1j\"[$HF *$\"3,?hrYYTRdq25F*$\"3EQP.&4l(R!)F*7$$\"30+++Pb\\45F*$\"3em%Q7 H-$RsF*7$$\"3.+++q^285F*$\"39[%4>ooP;'F*7$F4$\"3axd\"Q./lX&F*7$$\"3\"* *****f19Q-\"F*$\"3Iei(*H+5bXF*7$F:$\"3m)o/Pmqq)RF*7$F?$\"3=)[rau\"e&G$ F*7$FD$\"3Eu4X'z]R&GF*7$$\"3)******p4AH4\"F*$\"3+]k(e/(enAF*7$FI$\"38z #=&4$>-$>F*7$$\"3)*******4.sb6F*$\"3&e\"oiO<'fs\"F*7$FN$\"3[C33*>,>d\" F*7$FS$\"38y@g#R,OL\"F*7$FX$\"39/(z:)zVy6F*7$Fgn$\"3;p&)4.kIe5F*7$F\\o $\"3SB\"o%>e[=(*F87$Fao$\"3=vzZ\"z*>g*)F87$Ffo$\"3Fre%H$e!)4$)F87$F[p$ \"3M))QCmRiGyF87$F`p$\"3!Hj\\id[vP(F87$Fep$\"3![%4()>f*4(pF87$Fjp$\"36 A/4rcN?mF87$F_q$\"36I$yf]![okXUSl\\F87$Fbs$\"3/>gu713)z%F87$Fgs$\"3'o+9oXlfk%F87$F\\t$ \"3GKl%3\\Wv\\%F87$Fat$\"3+'oblhzRO%F87$Fft$\"39W\\^`+&fB%F87$F[u$\"3? >?<'e0n6%F87$F`u$\"3i*High(H8SF87$Feu$\"320&Q')G'R,RF87$Fju$\"3!*p1k`: z1QF87$F_v$\"3g>gW:]96PF87$Fdv$\"3&*\\YfGzCCOF87$Fiv$\"3'ytKf!R`NNF8-F _w6&FawFewFewFbw-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yF[bl-%% VIEWG6$;FewFiv;Few$\"\"%F[w" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 259 6 "legend" }{TEXT -1 2 " " }{TEXT 275 5 "_ ____" }{TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = arccosh*x;" "6#/-%\"fG6# %\"xG*&%(arccoshG\"\"\"F'F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 14 " " }{TEXT 276 5 "_____" }{TEXT -1 6 " f '( " }{TEXT 282 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` = 1/sqrt(x^2-1); " "6#/%!G*&\"\"\"F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&!\"\"F." }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "An alternative method of finding the derivative " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[arccosh*x];" "6#7#*&%(arccoshG\"\"\"%\"xGF&" }{TEXT -1 24 " is to use the formula " }{XPPEDIT 18 0 "arccosh*x = ln(x+sqrt (x^2-1));" "6#/*&%(arccoshG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sqrtG6#,&*$F' \"\"#F&F&!\"\"F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([ln(x+sqrt(x^2-1 ))],x) = 1/(x+sqrt(x^2-1));" "6#/-%%DiffG6$7#-%#lnG6#,&%\"xG\"\"\"-%%s qrtG6#,&*$F,\"\"#F-F-!\"\"F-F,*&F-F-,&F,F--F/6#,&*$F,F3F-F-F4F-F4" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x+sqrt(x^2-1)],x);" "6#-%%DiffG6$ 7#,&%\"xG\"\"\"-%%sqrtG6#,&*$F(\"\"#F)F)!\"\"F)F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2-1) );" "6#/%!G*&\"\"\"F&,&%\"xGF&-%%sqrtG6#,&*$F(\"\"#F&F&!\"\"F&F/" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+``(1/2)*(x^2-1)^(-1/2)*2*x);" "6#- %!G6#,&\"\"\"F'**-F$6#*&F'F'\"\"#!\"\"F'),&*$%\"xGF,F'F'F-,$*&F'F'F,F- F-F'F,F'F1F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2-1));" "6#/%!G*&\"\"\"F&,&%\"xGF&-%%s qrtG6#,&*$F(\"\"#F&F&!\"\"F&F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1+x /sqrt(x^2-1));" "6#-%!G6#,&\"\"\"F'*&%\"xGF'-%%sqrtG6#,&*$F)\"\"#F'F'! \"\"F0F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(x+sqrt(x^2-1));" "6#/%! G*&\"\"\"F&,&%\"xGF&-%%sqrtG6#,&*$F(\"\"#F&F&!\"\"F&F/" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``((x+sqrt(x^2-1))/sqrt(x^2-1));" "6#-%!G6#*&,&%\"xG \"\"\"-%%sqrtG6#,&*$F(\"\"#F)F)!\"\"F)F)-F+6#,&*$F(F/F)F)F0F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqr t(x^2-1);" "6#/%!G*&\"\"\"F&-%%sqrtG6#,&*$%\"xG\"\"#F&F&!\"\"F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Diff(ln(x+sqrt(x^2-1)),x);\nvalue(%);\nsimplify( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%#lnG6#,&%\"xG\"\"\" *$-%%sqrtG6#,&!\"\"F+*$)F*\"\"#F+F+F+F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&,&!\"\"F%*$)%\"xG\"\"#F%F%#F(F,F+F%F%F%,& F+F%*$-%%sqrtG6#F'F%F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$ *$-%%sqrtG6#,&!\"\"F$*$)%\"xG\"\"#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 48 "The inverse of the hyperbolic tangent fun ction, " }{XPPEDIT 18 0 "arctanh*x" "6#*&%(arctanhG\"\"\"%\"xGF%" } {TEXT -1 21 ", and its derivative " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 132 "plot([tanh(x),1,-1],x=-3..3,y=-1.2..1.2,lin estyle=[1,3$2],\n color=[red,black$2],ytickmarks=3,title=`y = tanh x` ,thickness=[2,1$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 545 196 196 {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$$ 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XESLABELSG6$Q\"x6\"Q\"yFbcl-%*AXESTICKSG6$%(DEFAULTGFgz-%%VIEWG6$;F(Ff z;$!#7Fi_l$\"#7Fi_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 77 "Since the functi on tanh is one-to-one, it has an inverse function denoted by " } {XPPEDIT 18 0 "tanh^(-1);" "6#)%%tanhG,$\"\"\"!\"\"" }{TEXT -1 28 " or arctanh and defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 346 64 64 {PLOTDATA 2 "6'-%'CURVESG6%7'7$$\"\"!F)F(7$$\"\")F )F(7$F+$\"\"#F)7$F(F.F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNES SG6#F/-%%TEXTG6%7$$\"\"%F)$\"\"\"F)QHy~=~arctanh~x~~exactly~when~~x~=~ tanh~y6\"-F26&F4F)F)F)-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FDFN-%%V IEWG6$%(DEFAULTGFR" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "It is possible to express " }{XPPEDIT 18 0 "arctanh*x;" " 6#*&%(arctanhG\"\"\"%\"xGF%" }{TEXT -1 44 " in terms of the natural lo garithm function." }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " } {XPPEDIT 18 0 "x = tanh*y;" "6#/%\"xG*&%%tanhG\"\"\"%\"yGF'" }{TEXT -1 18 " is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = (exp(y)-exp(-y))/(exp(y)+exp(-y));" "6#/%\"xG*&,&-% $expG6#%\"yG\"\"\"-F(6#,$F*!\"\"F/F+,&-F(6#F*F+-F(6#,$F*F/F+F/" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (exp(2*y)-1)/(exp(2*y)+1);" "6#/%\"xG*&,&-%$expG6#*&\"\"#\"\"\"%\"yGF,F,F,!\"\"F,,&-F(6#*&F+F,F-F, F,F,F,F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(2*y)+x = exp(2* y)-1;" "6#/,&*&%\"xG\"\"\"-%$expG6#*&\"\"#F'%\"yGF'F'F'F&F',&-F)6#*&F, F'F-F'F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+x = exp(2*y)-x*exp( 2*y);" "6#/,&\"\"\"F%%\"xGF%,&-%$expG6#*&\"\"#F%%\"yGF%F%*&F&F%-F)6#*& F,F%F-F%F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+x=exp(2*y)*(1-x )" "6#/,&\"\"\"F%%\"xGF%*&-%$expG6#*&\"\"#F%%\"yGF%F%,&F%F%F&!\"\"F%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*y) = (1+x)/(1-x)" "6#/-%$expG6# *&\"\"#\"\"\"%\"yGF)*&,&F)F)%\"xGF)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 "2*y=ln((1+x)/(1-x))" "6#/*&\"\"#\"\"\"%\"yGF& -%#lnG6#*&,&F&F&%\"xGF&F&,&F&F&F-!\"\"F/" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F(,&F(F( F)!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctanh*x = 1/2;" "6# /*&%(arctanhG\"\"\"%\"xGF&*&F&F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+ " }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 13 "__ ___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }{XPPEDIT 18 0 "y = arctanh*x; " "6#/%\"yG*&%(arctanhG\"\"\"%\"xGF'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F(,&F(F( F)!\"\"F+" }{TEXT -1 20 " appear to coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "f := x -> a rctanh(x): g := x -> ln((1+x)/(1-x))/2:\n'f(x)'=f(x); 'g(x)'=g(x);\npl ot([f(x),g(x)],x=-1.2..1.2,y,thickness=[1,2],legend=[`f(x)`,`g(x)`],ti ckmarks=[4,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(ar ctanhGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,$*&#\"\"\" \"\"#F+-%#lnG6#*&,&F+F+F'F+F+,&F+F+F'!\"\"F3F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 254 377 377 {PLOTDATA 2 "6'-%'CURVESG6&7hn7$$!3g******4$>X** *!#=$!3)*z'>vB\"*45%!#<7$$!3%)******>K\\y**F*$!3*)RFM+<-QJF-7$$!3Y******H5WY**F*$!3!RQC5O6+'HF-7$$!3p*** ***R\\TI**F*$!3h.n'=1A!)*F*$!3D)f1/yhJI#F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 14 "convert(..,ln)" } {TEXT -1 26 " can make this conversion." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG" } {XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "convert(arct anh(x),ln);\nassume(x_>-1,x_<1):\nsubs(x_=x,combine(subs(x=x_,%),ln)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&%\"xG\"\"\"F)F)#F)\"\" #*&#F)F+F)-F%6#,&F)F)F(!\"\"F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% #lnG6#*&,&%\"xG\"\"\"F)F)#F)\"\"#,&F)F)F(!\"\"#F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 27 "We can find the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 6 " for " } {XPPEDIT 18 0 "y = arctanh*x;" "6#/%\"yG*&%(arctanhG\"\"\"%\"xGF'" } {TEXT -1 47 " by implicit differentiation from the equation " } {XPPEDIT 18 0 "x = tanh*y;" "6#/%\"xG*&%%tanhG\"\"\"%\"yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differentiating both sides of \+ the equation " }{XPPEDIT 18 0 "x = tanh*y;" "6#/%\"xG*&%%tanhG\"\"\"% \"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 305 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = sech^ 2*y;" "6#/\"\"\"*&%%sechG\"\"#%\"yGF$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx = 1/(sech^2*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&%%sechG\"\"#% \"yGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the ide ntity " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sech^2*y = 1-tanh^2*y;" "6#/*&%%sechG\"\"#%\"yG\"\"\",&F(F(*&%%tanhGF&F'F(!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(1-tanh^2*y);" "6#/*&% #dyG\"\"\"%#dxG!\"\"*&F&F&,&F&F&*&%%tanhG\"\"#%\"yGF&F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(1-x^2);" "6#/*&%#dyG\"\"\"%# dxG!\"\"*&F&F&,&F&F&*$%\"xG\"\"#F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(arct anh(x),x)=diff(arctanh(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%D iffG6$-%(arctanhG6#%\"xGF**&\"\"\"F,,&F,F,*$)F*\"\"#F,!\"\"F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "p1 := plot([arctanh(x),1/(1-x^2)],x=-0.99..0.99,y=-2.5..3.7,color =[red,blue],thickness=2):\np2 := plots[implicitplot](\{x=-1,x=1\},x=-1 .1..1.1,y=-2.5..3.7,\n color=black,linestyle=3):\nplots[display]([ p1,p2],xtickmarks=3,view=[-1.1..1.1,-2.5..3.7]);" }}{PARA 13 "" 1 "" {GLPLOT2D 242 402 402 {PLOTDATA 2 "6)-%'CURVESG6%7en7$$!3!************ ***)*!#=$!3qXAO7ClYE!#<7$$!3y**\\(o3/@z*F*$!3w?pV3/+yAF-7$$!3m***\\P<3 Uo*F*$!3u)e`%zVCm?F-7$$!3_**\\igAJw&*F*$!3%yR5!=%Rl\">F-7$$!3Q****\\Zj To%*F*$!3Dki0\"4W.!=F-7$$!3O*\\7L.d1G*F*$!3OO\"y%)pfUk\"F-7$$!3M**\\7> x*G4*F*$!3FKY\"GC3M_\"F-7$$!3O++DZxeq')F*$!3V(y!Gai5@8F-7$$!3a***\\2Vy aC)F*$!3AFnBT0'3<\"F-7$$!3A+]i%e*QAyF*$!3!Q>Bko860\"F-7$$!3C+]7l?8IuF* $!3)zv`+jHRF*7$$!3)*)***\\,1e%G$ F*$!34z8+A'))4T$F*7$$!3u*****fUrl!HF*$!3%3BT,'*QG*HF*7$$!3'***\\7\"*o& oY#F*$!3HYm7Sl!)=DF*7$$!3_+++@]jx?F*$!3g?p5O\"G$3@F*7$$!3y)*\\7Tpf];F* $!3gI?kfo$em\"F*7$$!3G**\\P.d&RC\"F*$!3M-r.uBV]7F*7$$!39#***\\7Bo'>)!# >$!37\"*yz,L6:#)Fhs7$$!3l&)*\\i$))R+VFhs$!3v\\Q__F0.VFhs7$$!3eQ,+]P2t( *!#@$!390'>:'o2t(*Fct7$$\"3K2+vy-mnUFhs$\"3')R\\Z**RDqUFhs7$$\"39(**\\ 7*fun!)Fhs$\"3ku(R>f=`3)Fhs7$$\"3J,+vmU><7F*$\"3z6FQO%fKA\"F*7$$\"3**) ****H%z>T;F*$\"3g*y1V:whl\"F*7$$\"3A++v>G+c?F*$\"3cmB6<1t&3#F*7$$\"3_, ](o'yMdCF*$\"3A6C![)Qo3DF*7$$\"3P++D9J(H!HF*$\"3UE'=Y04*))HF*7$$\"3@++ +>sQ.LF*$\"3?5S=(=%3KMF*7$$\"3******\\UZ\"4t$F*$\"3s_gpXp4?RF*7$$\"3=* *\\(=A=$=TF*$\"3/d?%e!>:yVF*7$$\"3+.+]^D&=a%F*$\"3EAkB)=2'**[F*7$$\"3! ***\\()[`OS\\F*$\"3D;oG#yhQT&F*7$$\"3U,]P[g#pN&F*$\"3skdF*$\"3'>\"oz-@$3d'F*7$$\"3A+](='F*$\"3Z)>YT$ysMsF*7$$ \"3>.++M6?,mF*$\"3u$Rt&yZEIzF*7$$\"3U,+DtG9@qF*$\"3ty_V'y#e9()F*7$$\"3 %G+vGG2wV(F*$\"3w8lUu'Q%)e*F*7$$\"3X-++B[H?yF*$\"3]$oH$pTd]5F-7$$\"33- +DIz*)e#)F*$\"3i<:W;r1v6F-7$$\"3M,++y(*=^')F*$\"3%)f%[h_VLJ\"F-7$$\"3; -]()3ZXp!*F*$\"3k(=ovF-7$$\"3E+Dc 3Q*[o*F*$\"3oz*obb[t1#F-7$$\"33]7G/pW#z*F*$\"3SQ=DUU$)yAF-7$$\"3!***** **********)*F*$\"3qXAO7ClYEF--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj] lFi]l-%*THICKNESSG6#\"\"#-F$6%7ho7$F($\"3(fpS\"Gc7D]!#;7$$!3iu$f3,8l)) *F*$\"3\\K=_hS$4V%Fe^l7$$!3K\\(=<-EI()*F*$\"3CG')yXW(H'RFe^l7$$!3/C\"y D.R&f)*F*$\"3!>%=c3Y)[e$Fe^l7$$!3')*\\PM/_g%)*F*$\"3#eqQuwWIF$Fe^l7$$! 3P]i:l!y!>)*F*$\"3%eptW5^))y#Fe^l7$F/$\"3tl3w#G7.V#Fe^l7$$!3=\\Pf3,8l( *F*$\"3q$*QY*GNT:#Fe^l7$$!3s*\\7.8c\"Q(*F*$\"3W/Y5=l'[$>Fe^l7$$!3C]7._ @=6(*F*$\"3#*eKii7cc$\"3Khd)=d$oi'*F-7$FC$\"3:*y2;i9,@(F-7$FH$\"339*e'f]$Rx&F-7$$!3')* \\(=LFu\")))F*$\"3`.nT'3\\gt%F-7$FM$\"3v:pLb9')GSF-7$$!3%*******)3L!e% )F*$\"3\\;&=?]'\\8NF-7$FR$\"3&[rIgA?Q7$F-7$FW$\"3/5F-7$F`o$\"3%RFt%*y>Mx\"F-7$Feo$\"30[5krVh >;F-7$Fjo$\"3W]Xj([%4&\\\"F-7$F_p$\"3ZGv2k?K19F-7$Fdp$\"32+g1%\\?UK\"F -7$Fip$\"3'fq%fdAmc7F-7$F^q$\"3FXYHsry-7F-7$Fcq$\"3oTdS.^ph6F-7$Fhq$\" 3ePt2Q8$47\"F-7$F]r$\"3Or*4nIxA4\"F-7$Fbr$\"3YC0:tpzk5F-7$Fgr$\"3]_Q11 I6X5F-7$F\\s$\"3(3ysew2!G5F-7$Fas$\"3;B(oTZZap'4HsF-7$F \\z$\"3p1IX$zP!QAF-7$Faz$\"3==+wyrYuDF-7$Ffz$\"3gpD@AMeXJF-7$F[[l$\"3I !RcF8]](RF-7$$\"3u,vVVAKg))F*$\"3E4ZEf=J_YF-7$F`[l$\"3ZWq()=IRNcF-7$Fe [l$\"3Z9c@3#Q_5(F-7$Fj[l$\"3Wd.D@:*po*F-7$F_\\l$\"3,[*GToD&37Fe^l7$Fd \\l$\"3B[(47&o;7;Fe^l7$$\"3W(=#\\#3#y6(*F*$\"3og:6-S;gFe^l7$$\"3!Hc^.jebw*F*$\"3yydp?f-e@Fe^l7$Fi\\l$\"3 wpzW)G%GMCFe^l7$$\"3EP4@y^L>)*F*$\"3E`AyDQy#z#Fe^l7$$\"3cD19_MAY)*F*$ \"3_c#Q@![mwKFe^l7$$\"3qpa5*en'f)*F*$\"3aQdIVa9)e$Fe^l7$$\"3u7.2E<6t)* F*$\"3]X/$=1Mc'RFe^l7$$\"3wb^.jeb'))*F*$\"3B\"3JFC)fKWFe^l7$F^]lFc^l-F c]l6&Fe]lFi]lFi]lFf]lF[^l-F$6V7$7$$\"\"\"Fj]l$!3++++++++DF-7$Fa_m$!3/= =====mCF-7$7$Fa_m$!3y***********>D#F-Fe_m7$Fi_m7$Fa_m$!3\"y\"======AF- 7$7$Fa_m$!3c***********R+#F-F]`m7$Fa`m7$Fa_m$!3#y\"=====q>F-7$7$Fa_m$! 3c***********fv\"F-Fe`m7$Fi`m7$Fa_m$!3\"y\"=====A,\"F-F]bm7$Fab m7$Fa_m$!3?y\"=====y*F*7$7$Fa_m$!3o&***********RwF*Febm7$Fibm7$Fa_m$!3 5x\"=====I(F*7$7$Fa_m$!3q&***********f^F*F]cm7$Facm7$Fa_m$!3nx\"=====# [F*7$7$Fa_m$!3s&***********zEF*Fecm7$Ficm7$Fa_m$!3Ux\"=====M#F*7$7$Fa_ m$!3Qd************>FhsF]dm7$Fadm7$Fa_m$\"3IC#=====Q\"Fhs7$7$Fa_m$\"3E/ +++++!G#F*Fedm7$Fidm7$Fa_m$\"3FA======EF*7$7$Fa_m$\"3A/+++++gZF*F]em7$ Faem7$Fa_m$\"3q@=====)4&F*7$7$Fa_m$\"3@/+++++SsF*Feem7$Fiem7$Fa_m$\"3! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Diff(1/2*ln((1+x)/(1-x)) ,x);\nvalue(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Di ffG6$,$*&#\"\"\"\"\"#F)-%#lnG6#*&,&%\"xGF)F)F)F),&F)F)F0!\"\"F2F)F)F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\",&*&\"\"\"F),&F)F)% \"xGF&F&F)*&,&F+F)F)F)F)F*!\"#F)F)F-F&F*F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&*$)%\"xG\"\"#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 47 "The inverse of the hyperbolic s ecant function, " }{XPPEDIT 18 0 "arcsech*x" "6#*&%(arcsechG\"\"\"%\"x GF%" }{TEXT -1 21 ", and its derivative " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 274 "p1 := plot(sech(x),x=-4..0,y,color=red,thic kness=2):\np2 := plot(sech(x),x=0..4,color=blue,thickness=2):\np3 := p lot([[0,0],[4,0]],color=blue,thickness=2):\nt1 := plots[textplot]([1.7 ,.8,`y = sech x`],color=blue):\nplots[display]([p1,p2,p3,t1],view=[-3. .3,0..1.2],ytickmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 483 187 187 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"%\"\"!$\"3+`'otM**=m$!#>7$$!3cLLL$Q 6G\"R!#<$\"3kO@5[@D&*RF-7$$!3bmm;M!\\p$QF1$\"32(y%>:B&)4VF-7$$!37LLL)) Qj^PF1$\"3+0'H!fIG$p%F-7$$!3ALLL=KvlOF1$\"32iuY(GJO6&F-7$$!3wmm;C2G!e$ F1$\"3I0@b7'\\#pbF-7$$!39LL$3yO5]$F1$\"3J,))>?JtFgF-7$$!3&*****\\nU)*= MF1$\"3(zWh')z?@a'F-7$$!3iLL$3WDTL$F1$\"3.KDDSC5?rF-7$$!3))****\\d(Q& \\KF1$\"3eNIv>'**HF1$\"3i'>&)f_]l$**F-7$$!3.+++]5*H\"H F1$\"3O(4,=tlI3\"!#=7$$!3z******H\"3&HGF1$\"3S2@(QrNn<\"Fjo7$$!3OLL$3k (p`FF1$\"3i9Ksd;po7Fjo7$$!3%pmmmj^Nm#F1$\"3yS%e28lsQ\"Fjo7$$!3!ommm9'= (e#F1$\"3%oG`;tgh\\\"Fjo7$$!3K++]F\\N)\\#F1$\"3cA'*[S6OL;Fjo7$$!33nmmY Us>CF1$\"3UWTd'\\f\\w\"Fjo7$$!3\"*****\\FRXLBF1$\"367Y%HCW6#>Fjo7$$!3? ++]#=/8D#F1$\"3o07\\gl;#3#Fjo7$$!3%omm;a*el@F1$\"3W``2%erQE#Fjo7$$!3om m;Wn(o3#F1$\"3$=WyXDfQW#Fjo7$$!3PLLLeV(>+#F1$\"3!*)z;einHl#Fjo7$$!3mLL $3k%y8>F1$\"3U[5H!\\zv)GFjo7$$!3?++]K_,P=F1$\"3d\"=o')=1q5$Fjo7$$!3aLL Lo@5ad(es$f%Fjo7$$!3&pmmm/\\EL\"F1$\"3q!*o[cWOK\\Fjo7$$!3 3+++])ziC\"F1$\"3_J]cR$[@J&Fjo7$$!3_LL$3_;!o6F1$\"33o\")ez&)>rcFjo7$$! 31+++ISX#3\"F1$\"3sLgw))\\xxgFjo7$$!34nm;%RY>+\"F1$\"3'48Uv>Q4Z'Fjo7$$ !3W-++vr#z<*Fjo$\"35$oQ-6^!*)oFjo7$$!3Qommm6(>%Fjo$\"35)y))3Z4&z\"*Fjo7$$\"3Qmmm\">K'*)\\Fjo$ \"3)G!of\")\\Vs))Fjo7$$\"3P*****\\Kd,\"eFjo$\"3n=6/`3&3_)Fjo7$$\"3-mmm \"fX(emFjo$\"3))3iC(3H+8)Fjo7$$\"3.*****\\U7Y](Fjo$\"3%ya;^!Qj@xFjo7$$ \"3'QLLLV!pu$)Fjo$\"37xdq.iW!H(Fjo7$$\"3xmmm;c0T\"*Fjo$\"3=$ep-Plu!pFj o7$$\"3#*******H,Q+5F1$\"3u*f/Nmm'ykFjo7$$\"3)*******\\*3q3\"F1$\"3)\\ &RFdw!e0'Fjo7$$\"3)*******p=\\q6F1$\"3'o=))\\VU'fcFjo7$$\"3mmm;fBIY7F1 $\"3mPDGMq/7`Fjo7$$\"3GLLLj$[kL\"F1$\"3R(\\K)\\73;\\Fjo7$$\"3?LLL`Q\"G T\"F1$\"3I;yE6Fp'f%Fjo7$$\"3!*****\\s]k,:F1$\"3,b6E'>MYC%Fjo7$$\"39LLL `dF!e\"F1$\"3eQp!\\?T3&RFjo7$$\"33++]sgam;F1$\"3I(e)3([/yk$Fjo7$$\"3/+ +]F1$\"3O3s([Y3%*)GFjo7$$\"3immmTc-)*>F1$\"3g?(>![e3jEFjo7$$\"3Mm m;f`@'3#F1$\"3M-D$o/FaW#Fjo7$$\"3y****\\nZ)H;#F1$\"3_mJRdAipAFjo7$$\"3 YmmmJy*eC#F1$\"3SEW-$f/K4#Fjo7$$\"3')******R^bJBF1$\"3Op_yYtsC>Fjo7$$ \"3f*****\\5a`T#F1$\"3qID[dtcs$GF1$ \"375g3Am%Q<\"Fjo7$$\"3$*******pfa([?&**F-7$$\"3w****\\#G2A3$F1$\"3ehQi\"zPB:*F-7$$\"3;LL L$)G[kJF1$\"3S&[O!)[`@V)F-7$$\"3#)****\\7yh]KF1$\"3P]Ka7@VQxF-7$$\"3xm mm')fdLLF1$\"3)fdO^u0S7(F-7$$\"3bmmm,FT=MF1$\"3+$G%f+J&ea'F-7$$\"3FLL$ e#pa-NF1$\"3,T*)*3u`'=gF-7$$\"3!*******Rv&)zNF1$\"3I&z6'oKgrbF-7$$\"3I LLLGUYoOF1$\"3O)*))z&*\\!)*4&F-7$$\"3_mmm1^rZPF1$\"3_74Ho1p6ZF-7$$\"34 ++]sI@KQF1$\"3*=i!RuGHIVF-7$$\"34++]2%)38RF1$\"3uu:=.k9%*RF-7$$\"\"%F* F+-Fjz6&F\\[lFfzFfzF][lF`[l-F$6%7$7$FfzFfz7$FcjlFfzFejlF`[l-%%TEXTG6%7 $$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "The function sec h is one-to-one when " }{TEXT 260 13 "restricted to" }{TEXT -1 14 " th e domain [ " }{XPPEDIT 18 0 "0,infinity;" "6$\"\"!%)infinityG" }{TEXT -1 51 " ), and its inverse with this domain is denoted by " }{XPPEDIT 18 0 "sech^(-1);" "6#)%%sechG,$\"\"\"!\"\"" }{TEXT -1 12 " or arcsech. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 480 62 62 {PLOTDATA 2 "6'-%'CURVESG6%7'7$$\"\"!F)F(7$$\"\")F)F(7$F+$\"\"#F)7$F(F.F'-%'COLOUR G6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#F/-%%TEXTG6%7$$\"\"%F)$\"\" \"F)Qiny~=~arcsech~x~~exactly~when~~x~=~sech~y,~and~y~is~not~negative6 \"-F26&F4F)F)F)-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FDFN-%%VIEWG6$% (DEFAULTGFR" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 " Curve 1" "Curve 2" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "It is possible to express " }{XPPEDIT 18 0 "arcsech*x;" "6#*&%(arcsechG\"\"\"%\"xGF%" }{TEXT -1 44 " in term s of the natural logarithm function." }}{PARA 0 "" 0 "" {TEXT -1 16 "F irst note that " }{XPPEDIT 18 0 "x = sech*y;" "6#/%\"xG*&%%sechG\"\"\" %\"yGF'" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 0 "1/x = cosh* y;" "6#/*&\"\"\"F%%\"xG!\"\"*&%%coshGF%%\"yGF%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "u = 1/x" "6#/%\" uG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arccosh*u;" "6#/%\"yG*&%(arccoshG\" \"\"%\"uGF'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "ln(u+sqrt(u^2-1));" "6# -%#lnG6#,&%\"uG\"\"\"-%%sqrtG6#,&*$F'\"\"#F(F(!\"\"F(" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= ln(1/x+sqrt(1/(x^2)-1))" "6#/%!G-%#lnG6#,&*&\"\"\"F *%\"xG!\"\"F*-%%sqrtG6#,&*&F*F**$F+\"\"#F,F*F*F,F*" }{XPPEDIT 18 0 "`` = ln(1/x+sqrt((1-x^2)/x^2))" "6#/%!G-%#lnG6#,&*&\"\"\"F*%\"xG!\"\"F*-% %sqrtG6#*&,&F*F**$F+\"\"#F,F**$F+F3F,F*" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "x>0" "6#2\"\"!%\"xG" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 11 ", we have " } {XPPEDIT 18 0 "sqrt(x^2)=x" "6#/-%%sqrtG6#*$%\"xG\"\"#F(" }{TEXT -1 39 ", so the last expression simplifies to " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 " " }{XPPEDIT 18 0 " ``= ln(1/x+sqrt(1-x^2)/x)" "6#/%!G-%#lnG6#,&*&\"\"\"F*%\"xG!\"\"F**&-% %sqrtG6#,&F*F**$F+\"\"#F,F*F+F,F*" }{XPPEDIT 18 0 "``= ln((1+sqrt(1-x^ 2))/x)" "6#/%!G-%#lnG6#*&,&\"\"\"F*-%%sqrtG6#,&F*F**$%\"xG\"\"#!\"\"F* F*F0F2" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 13 "Thus we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsech*x = ln((1+sqrt(1-x^2))/x);" "6#/*&%(arcsechG\" \"\"%\"xGF&-%#lnG6#*&,&F&F&-%%sqrtG6#,&F&F&*$F'\"\"#!\"\"F&F&F'F3" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 16 "_____ ___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of \+ " }{XPPEDIT 18 0 "y = arcsech*x;" "6#/%\"yG*&%(arcsechG\"\"\"%\"xGF'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = ln((1+sqrt(1-x^2))/x);" "6#/% \"yG-%#lnG6#*&,&\"\"\"F*-%%sqrtG6#,&F*F**$%\"xG\"\"#!\"\"F*F*F0F2" } {TEXT -1 20 " appear to coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := x -> arcsech(x): g := \+ x -> ln((1+sqrt(1-x^2))/x):\n'f(x)'=f(x); 'g(x)'=g(x);\nplot([f(x),g(x )],x=0..1.05,y=0..5,thickness=[1,2],legend=[`f(x)`,`g(x)`],tickmarks=[ 3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(arcsechGF& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%#lnG6#*&,&\"\"\"F- *$,&F-F-*$)F'\"\"#F-!\"\"#F-F2F-F-F'F3" }}{PARA 13 "" 1 "" {GLPLOT2D 230 356 356 {PLOTDATA 2 "6'-%'CURVESG6&7^o7$$\"3i******47>_r!#@$\"3/+( e>log$z!#<7$$\"3$******>CQ/V\"!#?$\"3w$fWx3@HC(F-7$$\"3))*****ROdc9#F1 $\"3M(yX(R`XPoF-7$$\"3')*****\\[w3'GF1$\"3iUS@sBx\\lF-7$$\"3w*****zs98 H%F1$\"3^#*=F1ZIWhF-7$$\"3p******pHv@dF1$\"3G(>*4`!>m&eF-7$$\"3%3++]XH Ee)F1$\"3'y(\\\"QrV6X&F-7$$\"3%******Rf]V9\"!#>$\"3u@\"*=lFP$\"3sQEW6%QCU$F-7$ $\"3S+++DUzYwFP$\"3iEKk4eciKF-7$$\"3_+++>!yRx)FP$\"3eO=(>#pfCJF-7$$\"3 0+++*4j<5\"!#=$\"3Y([Vn%>x&*GF-7$$\"34+++]%y(48Fhp$\"3`n3;Fhp$\"3\\/SFsC(yI#F-7$$\"3++++RiN)>#Fhp$\"3#\\@>)*R:d >#F-7$$\"3/+++*4F&*R#Fhp$\"3xf4eV^u0@F-7$$\"36+++Ty*fi#Fhp$\"3yV7xczc7 ?F-7$$\"37+++%\\)R`GFhp$\"3H9kN]c@E>F-7$$\"39+++f6asIFhp$\"3#f&e6xPu[= F-7$$\"3E+++$pV:F$Fhp$\"3!)\\y>i,d#y\"F-7$$\"3=+++ap<3NFhp$\"3(zG;AwT$ 3&QA[TFhp$\"38?iL#)p&p_\"F-7$$\"33+++SMouVFhp$\" 35s\\=![,#o9F-7$$\"3#******4_E.f%Fhp$\"3FD\"y]6uVT\"F-7$$\"34+++`uK:[F hp$\"3Et4Fm5:g8F-7$$\"3E+++Z&[>-&Fhp$\"3 !RJb'Fhp$\"3!\\&))p?)HJ&)*Fhp7$$\"3[******QaX*y'Fhp$\"3%HrLD7SvP*Fhp7$ $\"3*)*****>D'z,qFhp$\"3!)*[70z(G_*)Fhp7$$\"3@+++%R:&GsFhp$\"33!Go7yM( )\\)Fhp7$$\"3a*****HjcRV(Fhp$\"3!4ShIqFm3)Fhp7$$\"3Y*****4]I4?(Fhp7$$\"3u*****p6%z !4)Fhp$\"3*o`w<9U9u'Fhp7$$\"3!)******ovw1$)Fhp$\"3q'onmk!=\"G'Fhp7$$\" 3?+++e<(G`)Fhp$\"35+&G]5sJy&Fhp7$$\"3W+++lpj]()Fhp$\"3CPz&y$)\\?G&Fhp7 $$\"3a+++ULLt*)Fhp$\"3r\")QZ#f/\"RZFhp7$$\"3#*******zc=%>*Fhp$\"3a\")) 4Y)yucTFhp7$$\"3A+++Ug7(R*Fhp$\"3!\\'HZIK;jNFhp7$$\"30+++*f=(H'*Fhp$\" 3)pf[+%\\MkFFhp7$$\"3(******\\PNPt*Fhp$\"3)yB%H8+rLBFhp7$$\"3A+++b@vP) *Fhp$\"3GXlokKo8=Fhp7$$\"3=+++]R?$*)*Fhp$\"3M;3ZQv-o9Fhp7$$\"3w******R dl[**Fhp$\"30(4\\7cGb,\"Fhp7$$\"3c*******o=D'**Fhp$\"3C4ZtW*\\;n)FP7$$ \"33+++N;Qw**Fhp$\"3#*)z-[:s'zoFP7$$\"3a+++5JJ$)**Fhp$\"350'z!>a-\"y&F P7$$\"3[******zXC!***Fhp$\"3(3f/!HU!*=WFP7$$\"3%******\\0wr***Fhp$\"3S K1El'3oP#FP7$%*undefinedGF\\`l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fe`lFd `l-%*THICKNESSG6#\"\"\"-%'LEGENDG6#%%f(x)G-F$6&7^oF'F.F4F9F>FCFHFMFSFX FgnF\\oFaoFfo7$F\\p$\"31FKk4eciKF-F`pFep7$F\\q$\"33n3;)*R:d>#F-FdrFir7$F_s$\"329kN]c@E>F-Fcs7$Fis$\"3e\\y>i,d#y \"F-7$F^t$\"3>)G;AwT$3a -\"y&FP7$Fb_l$\"3R#y/!HU!*=WFP7$Fg_l$\"39C0El'3oP#FPF[`l-F^`l6&F``lFd` lFa`lFd`l-Fg`l6#\"\"#-F[al6#%%g(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yF[gl-%*A XESTICKSG6$\"\"$\"\"%-%%VIEWG6$;Fd`l$\"$0\"!\"#;Fd`l$\"\"&Fe`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "g(x)" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The domai n of the function " }{XPPEDIT 18 0 "f(x) = arcsech*x;" "6#/-%\"fG6#%\" xG*&%(arcsechG\"\"\"F'F*" }{TEXT -1 14 " is the set \{ " }{TEXT 313 1 "x" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "0 < x;" "6#2\"\"!%\"xG" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 6 " \} = (" } {XPPEDIT 18 0 "0,1;" "6$\"\"!\"\"\"" }{TEXT -1 30 "], and the range is the set \{ " }{TEXT 314 1 "y" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "y>=0 " "6#1\"\"!%\"yG" }{TEXT -1 6 " \} = [" }{XPPEDIT 18 0 "0,infinity" "6 $\"\"!%)infinityG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 14 "conve rt(..,ln)" }{TEXT -1 26 " can make this conversion." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(ar csech(x),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&*&\"\"\"F(% \"xG!\"\"F(*&-%%sqrtG6#,&F'F(F(F*F(-F-6#,&F'F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can find the derivative " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 6 " for \+ " }{XPPEDIT 18 0 "y = arcsech*x;" "6#/%\"yG*&%(arcsechG\"\"\"%\"xGF'" }{TEXT -1 47 " by implicit differentiation from the equation " } {XPPEDIT 18 0 "x = sech*y;" "6#/%\"xG*&%%sechG\"\"\"%\"yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differentiating both sides of \+ the equation " }{XPPEDIT 18 0 "x = sech*y;" "6#/%\"xG*&%%sechG\"\"\"% \"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 306 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = -sech *y*tanh*y;" "6#/\"\"\",$**%%sechGF$%\"yGF$%%tanhGF$F(F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/(sech*y*tanh*y);" "6#/*&%#dyG\"\"\"%# dxG!\"\",$*&F&F&**%%sechGF&%\"yGF&%%tanhGF&F-F&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The identity " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1-tanh^2*y = sech^2*y;" "6#/,&\"\"\"F% *&%%tanhG\"\"#%\"yGF%!\"\"*&%%sechGF(F)F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "tanh*y;" "6#*&%%tanhG\"\"\"%\"yGF%" }{TEXT -1 3 " = " } {TEXT 285 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1-sech^2*y);" "6# -%%sqrtG6#,&\"\"\"F'*&%%sechG\"\"#%\"yGF'!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " } {XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "0 <= tanh*y;" "6#1\"\"!*&%%tanhG\"\"\"%\"yGF'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "tanh*y = sqrt(1-sech^2*y);" "6#/*&% %tanhG\"\"\"%\"yGF&-%%sqrtG6#,&F&F&*&%%sechG\"\"#F'F&!\"\"" }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = - 1/(sech*y*sqrt(1-sech^2*y));" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&*(%%s echGF&%\"yGF&-%%sqrtG6#,&F&F&*&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 "dy/dx = -1/(x*sqrt(1-x^2));" "6#/*&%#dyG\"\"\"%# dxG!\"\",$*&F&F&*&%\"xGF&-%%sqrtG6#,&F&F&*$F,\"\"#F(F&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Diff(arcsech(x),x)=diff(arcsech(x),x);\nassume(x_>0,x _<=1):\nsubs(x_=x,simplify(subs(x=x_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%(arcsechG6#%\"xGF*,$*&\"\"\"F-*()F*\"\"#F- -%%sqrtG6#,&*&F-F-F*!\"\"F-F-F6F--F26#,&F5F-F-F-F-F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%(arcsechG6#%\"xGF*,$*&\"\"\"F-*&-%%sq rtG6#,&F-F-*$)F*\"\"#F-!\"\"F-F*F-F6F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "p1 := plot([arcsech(x),-1/(x*sqrt(1-x^2))],x=0..1.1, y=-6..6,color=[red,blue],thickness=2):\np2 := plots[implicitplot](x=1, x=.9..1.1,y=-6..2,color=black,linestyle=4):\nplots[display]([p1,p2],xt ickmarks=3,view=[0..1.1,-6..6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 282 428 428 {PLOTDATA 2 "6(-%'CURVESG6%7in7$$\"3o******\\(F-7$$\"37+++D:$yC#F 1$\"3#=4D:@N4z'F-7$$\"3))*******p3r*HF1$\"3^T'*ec@D.lF-7$$\"3C+++]Im& 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Diff(ln((1+sqrt(1-x^2))/x),x);\nvalue(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%#lnG6#*&,&\"\"\"F+*$-%%sqrtG6#,& F+F+*$)%\"xG\"\"#F+!\"\"F+F+F+F3F5F3" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*(,&*&\"\"\"F&*$-%%sqrtG6#,&F&F&*$)%\"xG\"\"#F&!\"\"F&F0F0*&,&F&F&*$ -F)6#F+F&F&F&F.!\"#F0F&F2F0F.F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* &\"\"\"F%*&%\"xGF%-%%sqrtG6#,&F%F%*$)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 49 "The inverse of the hyperbolic c osecant function, " }{XPPEDIT 18 0 "arccsch*x" "6#*&%(arccschG\"\"\"% \"xGF%" }{TEXT -1 21 ", and its derivative " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(csch(x),x=-3..3,y=-3..3,discont=true ,title=`y = csch x`,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 434 340 340 {PLOTDATA 2 "6&-%'CURVESG6&7gn7$$!\"$\"\"!$!3AtA)o'p:#)**!#>7$ $!3*GyIw`3Y$H!#<$!3!>)))*)*4Ug1\"!#=7$$!3Ew&pex6x(GF1$!30d%)R&fS)G6F47 $$!3;\\Di;as8GF1$!3D#o3m#R&R?\"F47$$!3?q8D9\\J\\FF1$!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!=#y;k<%=l\"F47$$!3* =p]#pS:PCF1$!3Ea/O.[khF47$$!3Sq0+/\\r\\AF1$!3(o+h*HmGK@F47$$!3S808*GVZ=#F1$ !3q%QG+&*y*yAF47$$!3%ytb#*4J@7#F1$!3gHzle\"*QICF47$$!3Q`W\\KKFl?F1$!3! 4-(G!R(4xDF47$$!3ksY]HPm(*>F1$!3mU;=pi*Qw#F47$$!3y?#>@h*QS>F1$!3ZJvsm7 \\LHF47$$!3ynu(y>mP(=F1$!3I(yU[oK]9$F47$$!3t8/P(=$z9=F1$!3)pQ$*p&o>YLF 47$$!3=>[7[/4]C;F1$!3_xQCLPn+TF47$$!3%[o%*4cd^c\"F1$!3w-wU hxP\"F1$!3%\\&))f*\\;_Q&F47$$!3pY)='Hmd:8F1$!3q18yG.s#y&F47$$!3)HL(\\[ OL^7F1$!39\"pTI83FB'F47$$!3IJI([U%[)=\"F1$!3$\\dU4!RAF(yF47$$!3Ko-,!*y'[** *F4$!3z%3<$f!=\\^)F47$$!3q-eI;*)4Z$*F4$!3s6QW@=-'G*F47$$!3Ua(H([R7g()F 4$!35(o:+CDw+\"F17$$!3\"p=j(o_S=\")F4$!3&)*=%[J\"*=16F17$$!3!p!3A,!)f9 vF4$!3wNUrtlG87F17$$!3G5J[FaW$)oF4$!3C4a1%RwSM\"F17$$!3uWsYA%yjE'F4$!3 %3wptfsf\\\"F17$$!37p#4]Xm.i&F4$!3eu6f49\"*)o\"F17$$!3MRO+],=)*\\F4$!3 ,$)3qY2z>>F17$$!3#>>w()y/>O%F4$!3F>r,U*f9A#F17$$!3;.#)y3\")*3t$F4$!3s$ *e]LL8>EF17$$!3]iyp.&o5:$F4$!3okcf[4h@JF17$$!3Okp_U$=l[#F4$!3;q`qFEa!) 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" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 289 22 "______________________ " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }{XPPEDIT 18 0 "y = arcsech*x;" "6#/%\"y G*&%(arcsechG\"\"\"%\"xGF'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=ln( 1/x+sqrt(1+x^2)/abs(x))" "6#/%\"yG-%#lnG6#,&*&\"\"\"F*%\"xG!\"\"F**&-% %sqrtG6#,&F*F**$F+\"\"#F*F*-%$absG6#F+F,F*" }{TEXT -1 20 " appear to c oincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := x -> arccsch(x):\ng := x -> ln(1/x+sqrt(1+x^2)/ abs(x)):\n'f(x)'=f(x); 'g(x)'=g(x);\nplot([f(x),g(x)],x=-3..3,y=-3..3, thickness=[1,2],legend=[`f(x)`,`g(x)`],discont=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(arccschGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%#lnG6#,&*&\"\"\"F-F'!\"\"F-*&,&F-F-*$)F '\"\"#F-F-#F-F3-%$absGF&F.F-" }}{PARA 13 "" 1 "" {GLPLOT2D 353 322 322 {PLOTDATA 2 "6&-%'CURVESG6'7gn7$$!\"$\"\"!$!3I%esB],XF$!#=7$$!3*Gy Iw`3Y$H!#<$!3LTA))pr)[M$F-7$$!3Ew&pex6x(GF1$!35m[!f,%f3MF-7$$!3;\\Di;a s8GF1$!3M8aV*\\ZJ[$F-7$$!3?q8D9\\J\\FF1$!3-Nl\\y!*\\hNF-7$$!3yyXvV0@&o #F1$!3=@::%R()Hk$F-7$$!3sWMP'exdi#F1$!34x:4e4(=s$F-7$$!3[Bl\\,#QUc#F1$ !3W'e59/ar!QF-7$$!3p6Qi\"3%f+DF1$!3#zi=u%3Z**QF-7$$!3*=p]#pS:PCF1$!3W* 3oI3^f*RF-7$$!3m.iv=#)*=P#F1$!3a<7#>`z,5%F-7$$!3ie67I3U9BF1$!3Y[o()*yw k>%F-7$$!3Sq0+/\\r\\AF1$!3E>oH-lH5VF-7$$!3S808*GVZ=#F1$!3\"*4GxW@zIWF- 7$$!3%ytb#*4J@7#F1$!3Mu<*HGsKb%F-7$$!3Q`W\\KKFl?F1$!3gsrV<5LqYF-7$$!3k sY]HPm(*>F1$!3F%3=!3yM<[F-7$$!3y?#>@h*QS>F1$!3/cOrUA3\\\\F-7$$!3ynu(y> mP(=F1$!39:57_YN6^F-7$$!3t8/P(=$z9=F1$!3#\\pI_/,QE&F-7$$!3=>[7[/4]C;F1$ !3?%*o6+\"HB#eF-7$$!3%[o%*4cd^c\"F1$!3&)Rq<-A0>gF-7$$!3lQqur2[,:F1$!3[ -4.Z)zfC'F-7$$!30BVv$[Q`V\"F1$!3wh:'H@3'*\\'F-7$$!3=x%>wUhxP\"F1$!3c(R J\"e%oot'F-7$$!3pY)='Hmd:8F1$!3='*ef)zeA,(F-7$$!3)HL(\\[OL^7F1$!3Tt^M! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 14 "convert(..,ln)" }{TEXT -1 55 " \+ can make this conversion, but gives the first formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert (arccsch(x),ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&*&\"\"\" F(%\"xG!\"\"F(*$-%%sqrtG6#,&F(F(*&F(F(*$)F)\"\"#F(F*F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can find the derivative " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 6 " for \+ " }{XPPEDIT 18 0 "y = arccsch*x;" "6#/%\"yG*&%(arccschG\"\"\"%\"xGF'" }{TEXT -1 47 " by implicit differentiation from the equation " } {XPPEDIT 18 0 "x = csch*y;" "6#/%\"xG*&%%cschG\"\"\"%\"yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differentiating both sides of \+ the equation " }{XPPEDIT 18 0 "x = csch*y;" "6#/%\"xG*&%%cschG\"\"\"% \"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 307 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = -csch *y*coth*y;" "6#/\"\"\",$**%%cschGF$%\"yGF$%%cothGF$F(F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/(csch*y*coth*y);" "6#/*&%#dyG\"\"\"%# dxG!\"\",$*&F&F&**%%cschGF&%\"yGF&%%cothGF&F-F&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The identity " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "coth^2*y-1 = csch^2*x;" "6#/,&*&%%cothG \"\"#%\"yG\"\"\"F)F)!\"\"*&%%cschGF'%\"xGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "coth*y;" "6#*&%%cothG\"\"\"%\"yGF%" }{TEXT -1 3 " = " } {TEXT 290 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+csch^2*y);" "6# -%%sqrtG6#,&\"\"\"F'*&%%cschG\"\"#%\"yGF'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "coth*x)" "6#*&%%cothG\"\"\"%\"xGF%" }{TEXT -1 22 " has \+ the same sign as " }{XPPEDIT 18 0 "sinh*x" "6#*&%%sinhG\"\"\"%\"xGF%" }{TEXT -1 17 " it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = -1/(abs(csch*y)*sqrt(1+csch^2*y));" "6#/*&%#dyG \"\"\"%#dxG!\"\",$*&F&F&*&-%$absG6#*&%%cschGF&%\"yGF&F&-%%sqrtG6#,&F&F &*&F0\"\"#F1F&F&F&F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/(abs(x)*sqrt(1+x^2));" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&*&-%$abs G6#%\"xGF&-%%sqrtG6#,&F&F&*$F/\"\"#F&F&F(F(" }{TEXT -1 2 ". 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"Since the func tion coth is one-to-one, it has an inverse function denoted by " } {XPPEDIT 18 0 "coth^(-1);" "6#)%%cothG,$\"\"\"!\"\"" }{TEXT -1 28 " or arccoth and defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 393 63 63 {PLOTDATA 2 "6'-%'CURVESG6%7'7$$\"\"!F)F(7$$\"\")F )F(7$F+$\"\"#F)7$F(F.F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNES SG6#F/-%%TEXTG6%7$$\"\"%F)$\"\"\"F)QHy~=~arccoth~x~~exactly~when~~x~=~ coth~y6\"-F26&F4F)F)F)-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FDFN-%%V IEWG6$%(DEFAULTGFR" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "It is possible to express " }{XPPEDIT 18 0 "arccoth*x;" " 6#*&%(arccothG\"\"\"%\"xGF%" }{TEXT -1 44 " in terms of the natural lo garithm function." }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " } {XPPEDIT 18 0 "x = coth*y;" "6#/%\"xG*&%%cothG\"\"\"%\"yGF'" }{TEXT -1 18 " is equivalent to " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = (exp(y)+exp(-y))/(exp(y)-exp(-y));" "6#/%\"xG*&,&-% $expG6#%\"yG\"\"\"-F(6#,$F*!\"\"F+F+,&-F(6#F*F+-F(6#,$F*F/F/F/" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (exp(2*y)+1)/(exp(2*y)-1);" "6#/%\"xG*&,&-%$expG6#*&\"\"#\"\"\"%\"yGF,F,F,F,F,,&-F(6#*&F+F,F-F,F,F ,!\"\"F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(2*y)-x = exp(2* y)+1;" "6#/,&*&%\"xG\"\"\"-%$expG6#*&\"\"#F'%\"yGF'F'F'F&!\"\",&-F)6#* &F,F'F-F'F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(2*y)-exp(2*y) = x+1;" "6#/,&*&%\"xG\"\"\"-%$expG6#*&\"\"#F'%\"yGF'F'F'-F)6#*&F,F'F-F' !\"\",&F&F'F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*y)*(x-1) = \+ x+1;" "6#/*&-%$expG6#*&\"\"#\"\"\"%\"yGF*F*,&%\"xGF*F*!\"\"F*,&F-F*F*F *" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*y) = (x+1)/(x-1);" "6#/-%$ex pG6#*&\"\"#\"\"\"%\"yGF)*&,&%\"xGF)F)F)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 "2*y = ln((x+1)/(x-1));" "6#/*&\"\"#\"\" \"%\"yGF&-%#lnG6#*&,&%\"xGF&F&F&F&,&F-F&F&!\"\"F/" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = 1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln((x+1)/(x-1));" "6#-%#lnG6#*&,&%\"xG\"\"\"F)F)F),& F(F)F)!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccoth*x = 1/2; " "6#/*&%(arccothG\"\"\"%\"xGF&*&F&F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln((x+1)/(x-1));" "6#-%#lnG6#*&,&%\"xG\"\"\"F)F)F),&F(F )F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 294 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{XPPEDIT 18 0 "x>1 or x <-1" "6#52\"\"\"%\"xG2F&,$F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }{XPPEDIT 18 0 "y = arccoth*x;" "6#/%\"y G*&%(arccothG\"\"\"%\"xGF'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 1 /2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln ((x+1)/(x-1));" "6#-%#lnG6#*&,&%\"xG\"\"\"F)F)F),&F(F)F)!\"\"F+" } {TEXT -1 20 " appear to coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "f := x -> arccoth(x):\ng := x -> ln((x+1)/(x-1))/2:\n'f(x)'=f(x); 'g(x)'=g(x);\nplot([f(x),g(x)], x=-4..4,y=-4..4,thickness=[1,2],legend=[`f(x)`,`g(x)`],discont=true); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(arccothGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,$*&#\"\"\"\"\"#F+-%#lnG 6#*&,&F'F+F+F+F+,&F'F+F+!\"\"F3F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 353 322 322 {PLOTDATA 2 "6&-%'CURVESG6'7gn7$$!\"%\"\"!$!3g`*H)=\"GTb#!#=7$ $!3NxVnP&3Y$R!#<$!3SbQ#[R'\\)f#F-7$$!3s,6&fx6x(QF1$!3FNH4xPSQEF-7$$!30 Knu;as8QF1$!33gES9'4[o#F-7$$!3S$\\=W\"\\J\\PF1$!3`A3%f+NF1$!3%H&HC\\`SQHF-7$$!3SAfipS:PMF1$! 3Q(*oy45\"f*HF-7$$!3/Q\\<>#)*=P$F1$!3w#))3[fLv0$F-7$$!3!3@y0$3U9LF1$!3 ^%[&f!4=S6$F-7$$!31g2]/\\r\\KF1$!3C!Hv:+U-=$F-7$$!3wLF-7$$!3q/w6LKFlIF1$!3sU959@4'Q $F-7$$!3c'*Ga$F-7 $$!33!HG')>mP(GF1$!3=Q/h\")RQJOF-7$$!3Y^0;)=$z9GF1$!3s2U*))ouXr$F-7$$! 3=#4e*[/4]FF1$!30%\\u%QCc5QF-7$$!3S\"Rs.9y%)o#F1$!3^;cf^;$p!RF-7$$!3r? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The \+ Maple procedure " }{TEXT 0 14 "convert(..,ln)" }{TEXT -1 31 " can make this conversion when " }{XPPEDIT 18 0 "x>1" "6#2\"\"\"%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "convert(arccoth(x),ln);\nassume(x_>1):\nsubs(x_=x,com bine(subs(x=x_,%),ln));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\" \"\"#F&-%#lnG6#,&%\"xGF&F&F&F&F&*&#F&F'F&-F)6#,&F,F&F&!\"\"F&F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#*&,&%\"xG\"\"\"F)F)#F)\"\"#,&F (F)F)!\"\"#F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can f ind the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\" " }{TEXT -1 6 " for " }{XPPEDIT 18 0 "y = arccoth*x;" "6#/%\"yG*&%(ar ccothG\"\"\"%\"xGF'" }{TEXT -1 47 " by implicit differentiation from t he equation " }{XPPEDIT 18 0 "x = coth*y;" "6#/%\"xG*&%%cothG\"\"\"%\" yGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Differentiating b oth sides of the equation " }{XPPEDIT 18 0 "x = coth*y;" "6#/%\"xG*&%% cothG\"\"\"%\"yGF'" }{TEXT -1 17 " with respect to " }{TEXT 308 1 "x" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = -csch^2*y;" "6#/\"\"\",$*&%%cschG\"\"#%\"yGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " ," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/(csch^2*y);" "6#/*&%#dyG\"\"\"%#dxG!\"\" ,$*&F&F&*&%%cschG\"\"#%\"yGF&F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the identity " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "csch^2*y = coth^2*y-1;" "6#/*&%%cschG\"\"#%\"yG\"\"\", &*&%%cothGF&F'F(F(F(!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/(coth^2*y-1);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&,&*&%%cothG\" \"#%\"yGF&F&F&F(F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "t hat is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1 /(1-x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&,&F&F&*$%\"xG\"\"#F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(arccoth(x),x)=diff(arccoth(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%(arccothG6#%\"xGF*,$*&\"\"\"F-, &*$)F*\"\"#F-F-F-!\"\"F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "p1 := plot(arccoth(x),x=-5..5,y=-3 ..3,color=red,thickness=2):\np2 := plot(1/(1-x^2),x=1.01..5,color=blue ,thickness=2):\np3 := plot(1/(1-x^2),x=-5..-1.01,color=blue,thickness= 2):\np4 := plots[implicitplot](\{x=-1,x=1\},x=-1.1..1.1,y=-3..3,\n \+ color=black,linestyle=3):\nplots[display]([p1,p2,p2,p3,p4],view=[-5.. 5,-3..3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 449 306 306 {PLOTDATA 2 "6*-% 'CURVESG6%7gn7$$!\"&\"\"!$!3P@3aSbKF?!#=7$$!35+++e%G?y%!#<$!3e)R^INqC7 #F-7$$!3w*****fesBf%F1$!3QR6<(z`H@#F-7$$!3!******4s%3zVF1$!3))\\)fjwmX K#F-7$$!3?+++YIQkTF1$!3-jA;+x8\\CF-7$$!31+++6=q]RF1$!3w'p'>7:V(e#F-7$$ !3'******>&>f_PF1$!3\"*Q)RxtN2t#F-7$$!3!*******o1YZNF1$!31#[Nvnft*GF-7 $$!3#)*****>g8`L$F1$!3oxKG7)QK4$F-7$$!3:+++%*o%Q7$F1$!3gHB!*p%F-7$$!3#* *****\\KqP2#F1$!3!RdI#>Ske_F-7$$!3))*****H5WU)=F1$!3d&e!f!)[U6fF-7$$!3 1+++#4z)e;F1$!3Ct`=m$*fvpF-7$$!3#******pOlzY\"F1$!3]'f?`IvPJ)F-7$$!3/+ ++Vj#pN\"F1$!3(y*3$)o5#zV*F-7$$!3%*******=t)eC\"F1$!37!=GJH\"*f5\"F17$ $!3-+++WJu'>\"F1$!3Co!zk0;k?\"F17$$!31+++o*)fZ6F1$!3(y$*p)3M!)Q8F17$$! 31+++#za%)4\"F1$!39O+fI&z'H:F17$$!3%******ph5$\\5F1$!3-Nm?F17$$!3*******>9^B-\"F1$!3%H+UYJpDD#F17$$ !31+++u7h:5F1$!3fCyIAAMICF17$$!30+++09()35F1$!3K?l@&eZ7r#F17$$!3/+++O: 8-5F1$!3/<)4nSiDU$F17$%*undefinedGFft7$$\"3.+++#H^c+\"F1$\"3Nf$H F17$$\"3%******f3)>75F1$\"3]PrBi>&Gb#F17$$\"3*******Rn\"HD5F1$\"3'puCv (o\\\">#F17$$\"3.+++i_QQ5F1$\"3cYemD)>h)>F17$$\"3'******>qAP1\"F1$\"32 5!z'RH')Q9g!*3\"F1$\"3Ws`LG)yvd\"F17$$\"30+++#e(R96F1 $\"3W`>nQEUe9F17$$\"3!******4-N(R6F1$\"35to\"F1$\"3Xt7sGCL@7F17$$\"3++++!y%3T7F1$\"3[$fuUK$y96F17$$\"3%******zS; ON\"F1$\"3Q(4XV%zZx%*F-7$$\"3))*****f.[hY\"F1$\"30T&>iFZ&H$)F-7$$\"31+ ++#Qx$o;F1$\"3y#HWrVk=#pF-7$$\"3%******RP+V)=F1$\"3#)ySp^U?6fF-7$$\"3# ******ppe*z?F1$\"3!Gi@(\\p'*R_F-7$$\"3=+++C\\'QH#F1$\"33yE-+\"R@n%F-7$ $\"3#******H,M^\\#F1$\"3-mTkx$zdC%F-7$$\"3=+++0#=bq#F1$\"3k,)y7fq(zQF- 7$$\"3\"******p?27\"HF1$\"3Uq)H\\$\\b!e$F-7$$\"3))******HXaEJF1$\"30UY 2E+r9LF-7$$\"3*)*****\\'*RRL$F1$\"3/Yzg'>'f%4$F-7$$\"3%)*****HvJga$F1$ \"3a]*\\Ap$f)*GF-7$$\"3;+++8tOcPF1$\"3G`A[FH&ys#F-7$$\"35+++\\Qk\\RF1$ \"3)>X@>&f:)e#F-7$$\"3U+++p0;rTF1$\"3rN!)))ov*\\W#F-7$$\"3;+++lxGpVF1$ \"31#=!*eXp*HBF-7$$\"39+++!oK0e%F1$\"3cxC\"yDj)=AF-7$$\"33+++<5s#y%F1$ \"3]DAJ)p`@7#F-7$$\"\"&F*$\"3P@3aSbKF?F--%'COLOURG6&%$RGBG$\"*++++\"! \")$F*F*F_^l-%*THICKNESSG6#\"\"#-F$6%7hn7$$\"3,++++++55F1$!3)3X4\"yV7v \\!#;7$$\"3Dc,mKyr75F1$!3(>`R\"eak1RF\\_l7$$\"3r7.KlcV:5F1$!3EQfyNGW9K F\\_l7$$\"3&*o/)z\\`\"=5F1$!39w([.*\\^HFF\\_l7$$\"3>D1kI8(3-\"F1$!3?h* \\k()))3P#F\\_l7$$\"3mP4'f*pIE5F1$!3weU3s'ff(=F\\_l7$$\"39]7GhEuJ5F1$! 3IxY6>yb]:F\\_l7$$\"34v=#>*RhU5F1$!3)=[v`5X)[6F\\_l7$$\"3/+DcA`[`5F1$! 3^wOG%fq[5*F17$$\"3'*\\P%Q)zAv5F1$!3#*3L9oa_0kF17$$\"33+]7X1(p4\"F1$!3 TM*)*[Qgx\"\\F17$$\"30v=x\"*p![8\"F1$!3AH:7F1$!3a5!R.8ox4#F17$$\"3%**\\(Q'>XxD\"F1$ !3[lu7u/W=j&*e\"F1$!3 wg'\\*4v-]lF-7$$\"3+](oF()4Un\"F1$!3C(*3cJsPYbF-7$$\"3-]PR*3&eeF1$!33H]'= o)y7PF-7$$\"3=+]nz\"zy+#F1$!3oOFFr7h)H$F-7$$\"3)***\\iF9H%4#F1$!3[ZER= zG`HF-7$$\"35+]KSccx@F1$!3!ym`g=;Dn#F-7$$\"3)*\\(oKg'=`AF1$!3_1(=SBuGX #F-7$$\"3!***\\UUs5VBF1$!3\\k'\\CF1$!3E`\\&Q8811 #F-7$$\"3**\\(=)f4*y]#F1$!3_+DIe<`!*=F-7$$\"3%****\\R1Dje#F1$!37=sVtUv dnO#G;F-7$$\"3!*\\i&z2CVv#F1$!3=A4u%)GI=: F-7$$\"38+v=KW#)RGF1$!3ufi(ou2bT\"F-7$$\"3B]PpZ/M=HF1$!34D'f\"*4p.L\"F -7$$\"3\"**\\ivdII+$F1$!3'*\\B6[Q;Z7F-7$$\"3!*\\(o2#)**44$F1$!3C-vnra+ p6F-7$$\"3s\\7e0tdnJF1$!3\"42,1,&)p5\"F-7$$\"3O+v3(3$G]KF1$!3_4H'\\V]b /\"F-7$$\"3t***\\@DEdL$F1$!3U4-pzZ_u)*!#>7$$\"3#**\\P(>dJ>MF1$!3UM[ft? .`$*F`[m7$$\"3t\\(=x#G>+NF1$!3K=o`$[Ay)))F`[m7$$\"3?+D'p1$***e$F1$!3a! 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3GcPnGU7f?F-7$$!3J++0CuJRBF1$!3WR4#z&H$fB#F-7$$!3'***\\P]T;`AF1$!3U]tx 6^$HX#F-7$$!3=]7`zk4v@F1$!3-)[K_Z9-o#F-7$$!3Y**\\#\\*yu*3#F1$!3odt;h@' *pHF-7$$!35]7=G:W4?F1$!3;:m@tgz\"H$F-7$$!3y\\iqN#)\\D>F1$!3WzjPd[Q$p$F -7$$!3****\\(QKGM%=F1$!3;KWDa`upTF-7$$!3#*\\7.K(3vv\"F1$!3AeTp#4`ty%F- 7$$!3U****H`zvu;F1$!3:\"*\\ETltSbF-7$$!3!)*\\(3IL8!f\"F1$!3aVjJO!fAa'F -7$$!3Z\\7[T%4i]\"F1$!3kT'[Qy*G#)yF-7$$!3_***\\Q5#4H9F1$!3+<'olqFTf*F- 7$$!3f*\\PA$pqS8F1$!3'4(y-Mi#RD\"F17$$!3%*****4\"=a;E\"F1$!3#H$e$G3U)* o\"F17$$!3)[iS1#3^>7F1$!3me3Xvr^_?F17$$!3#)\\7=guOx6F1$!3eMJ*pEr$*e#F1 7$$!3()***\\o)3.P6F1$!3@(\\6*GH%[T$F17$$!3!*\\(=NJ%p'4\"F1$!3*yL;CFkC$ \\F17$$!3Qi!R^t?]2\"F1$!3E\"G'30E'QU'F17$$!3%[Pfn:ZL0\"F1$!3H3cMwe1H\" *F17$$!33J&pvO5D/\"F1$!3g&**yn)Rq^6F\\_l7$$!3K(oz$yNnJ5F1$!3UjtliD*Rb \"F\\_l7$$!3blZy$=bi-\"F1$!3sS9mo(4(z=F\\_l7$$!3xV)*=*yO3-\"F1$!3oa&Hc sf[P#F\\_l7$$!3*GQ#*=fF\"=5F1$!3sT9>?3XLFF\\_l7$$!3+A\\f%R=a,\"F1$!3W \\0%3.r!=KF\\_l7$$!37huH(>4F,\"F1$!3I+-BJmJ4RF\\_l7$$!3,++++++55F1Fj^l FcamF`^l-F$6V7$7$$!\"\"F*$!\"$F*7$Fbdn$!3]ssssss#z#F17$7$Fbdn$!3!)**** ********fFF1Ffdn7$Fjdn7$Fbdn$!3Issssss_DF17$7$Fbdn$!3d************>DF1 F^en7$Fben7$Fbdn$!33ssssss7BF17$7$Fbdn$!3O************zAF1Ffen7$Fjen7$ Fbdn$!3'=FFFFFF2#F17$7$Fbdn$!3:************R?F1F^fn7$Fbfn7$Fbdn$!3(=FF FFFF$=F17$7$Fbdn$!3;*************z\"F1Fffn7$7$Fbdn$!3#*)************z \"F17$Fbdn$!35ssssss#f\"F17$7$Fbdn$!3;************f:F1Fagn7$Fegn7$Fbdn $!3)=FFFFFFN\"F17$7$Fbdn$!3<************>8F1Fign7$F]hn7$Fbdn$!3!>FFFFF F6\"F17$7$Fbdn$!3=************z5F1Fahn7$Fehn7$Fbdn$!33>FFFFFF()F-7$7$F bdn$!3#>************R)F-Fihn7$F]in7$Fbdn$!3G?FFFFFFjF-7$7$Fbdn$!3+#*** **********fF-Fain7$Fein7$Fbdn$!3\")>FFFFFFRF-7$7$Fbdn$!33#************ f$F-Fiin7$F]jn7$Fbdn$!3!*>FFFFFF:F-7$7$Fbdn$!3=#************>\"F-Fajn7 $7$Fbdn$!3/#************>\"F-7$Fbdn$\"37+GFFFFF()F`[m7$7$Fbdn$\"3s2+++ +++7F-F\\[o7$F`[o7$Fbdn$\"3QzssssssKF-7$7$Fbdn$\"3i2++++++OF-Fd[o7$7$F bdn$\"332++++++OF-7$Fbdn$\"3Q!GFFFFFn&F-7$7$Fbdn$\"3a2++++++gF-F_\\o7$ Fc\\o7$Fbdn$\"3=zssssss!)F-7$7$Fbdn$\"3Y2++++++%)F-Fg\\o7$F[]o7$Fbdn$ \"3-GFFFFFZ5F17$7$Fbdn$\"3u++++++!3\"F1F_]o7$Fc]o7$Fbdn$\"3-GFFFFF(G\" F17$7$Fbdn$\"3t++++++?8F1Fg]o7$F[^o7$Fbdn$\"3+GFFFFFF:F17$7$Fbdn$\"3s+ +++++g:F1F_^o7$Fc^o7$Fbdn$\"3yFFFFFFn " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "An alternative method of finding the de rivative " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arccoth*x];" "6#7#*&%(arccothG\"\"\"%\"xGF& " }{TEXT -1 24 " is to use the formula " }{XPPEDIT 18 0 "arccoth*x = \+ 1/2;" "6#/*&%(arccothG\"\"\"%\"xGF&*&F&F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln((x+1)/(x-1));" "6#-%#lnG6#*&,&%\"xG\"\"\"F)F)F),&F(F )F)!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG !\"\"" }{TEXT -1 1 " " }{TEXT 297 2 "[ " }{XPPEDIT 18 0 "1/2" "6#*&\" \"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((x+1)/(x-1));" "6 #-%#lnG6#*&,&%\"xG\"\"\"F)F)F),&F(F)F)!\"\"F+" }{TEXT 298 2 " ]" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {XPPEDIT 18 0 "``((x-1)/(x+1));" "6#-%!G6#*&,&%\"xG\"\"\"F)!\"\"F),&F( F)F)F)F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[(x+1)/(x-1)];" "6#7#*&,&%\"xG\" \"\"F'F'F',&F&F'F'!\"\"F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {XPPEDIT 18 0 "``((x-1)/(x+1))*``((``(x-1)-(x+1))/((x-1)^2));" "6#*&-% !G6#*&,&%\"xG\"\"\"F*!\"\"F*,&F)F*F*F*F+F*-F%6#*&,&-F%6#,&F)F*F*F+F*,& F)F*F*F*F+F**$,&F)F*F*F+\"\"#F+F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``((x-1)/(x+1))*``((-2)/((x-1)^2));" "6#*&-%!G6#*&,&% \"xG\"\"\"F*!\"\"F*,&F)F*F*F*F+F*-F%6#*&,$\"\"#F+F**$,&F)F*F*F+F1F+F* " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = (-1)/((x+1)*(x-1));" "6#/%!G*&,$\"\"\"!\"\"F'*&,&%\"xGF'F'F'F',&F+ F'F'F(F'F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 1/(1-x^2)" "6#/%!G*&\"\"\"F&,&F&F&*$%\"xG\"\"#!\"\" F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Diff(1/2*ln((x+1)/(x-1)),x);\nvalue(%);\nsi mplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$-%#lnG6#*&,& %\"xG\"\"\"F-F-F-,&F,F-F-!\"\"F/#F-\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&*&\"\"\"F',&%\"xGF'F'!\"\"F*F'*&,&F)F'F'F'F'F(!\" #F*F'F,F*F(F'#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%, &*$)%\"xG\"\"#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 8 "Summary " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arcsinh*x;" "6 #/%\"yG*&%(arcsinhG\"\"\"%\"xGF'" }{TEXT -1 16 " exactly when " } {XPPEDIT 18 0 "x = sinh*y;" "6#/%\"xG*&%%sinhG\"\"\"%\"yGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh*x \+ = ln(x+sqrt(x^2+1));" "6#/*&%(arcsinhG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sq rtG6#,&*$F'\"\"#F&F&F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arcsinh*x] = 1/sqrt(x^2+1);" "6#/7#*&%(arcsinhG \"\"\"%\"xGF'*&F'F'-%%sqrtG6#,&*$F(\"\"#F'F'F'!\"\"" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 309 18 "__________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arccosh*x;" "6#/%\"yG*&%(arccoshG\" \"\"%\"xGF'" }{TEXT -1 16 " exactly when " }{XPPEDIT 18 0 "x = cosh* y;" "6#/%\"xG*&%%coshG\"\"\"%\"yGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y >= 0" "6#1\"\"!%\"yG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccosh*x = ln(x+sqrt(x^2-1));" "6#/*&%(arcco shG\"\"\"%\"xGF&-%#lnG6#,&F'F&-%%sqrtG6#,&*$F'\"\"#F&F&!\"\"F&" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arccosh *x] = 1/sqrt(x^2-1);" "6#/7#*&%(arccoshG\"\"\"%\"xGF'*&F'F'-%%sqrtG6#, &*$F(\"\"#F'F'!\"\"F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y \+ = arctanh*x;" "6#/%\"yG*&%(arctanhG\"\"\"%\"xGF'" }{TEXT -1 16 " exac tly when " }{XPPEDIT 18 0 "x = tanh*y;" "6#/%\"xG*&%%tanhG\"\"\"%\"yG F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctanh*x = 1/2;" "6#/*&%(arctanhG\"\"\"%\"xGF&*&F&F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F (%\"xGF(F(,&F(F(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 "[arctanh*x] = 1/(1-x^2);" "6#/7#*&%(arctanhG\"\" \"%\"xGF'*&F'F',&F'F'*$F(\"\"#!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 18 "__________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = arcsech*x;" "6#/%\"yG*&%(arcsechG\"\"\"%\"xGF'" } {TEXT -1 16 " exactly when " }{XPPEDIT 18 0 "x = sech*y;" "6#/%\"xG* &%%sechG\"\"\"%\"yGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y >= 0" "6# 1\"\"!%\"yG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arcsech*x = ln((1+sqrt(1-x^2))/x);" "6#/*&%(arcsechG\" \"\"%\"xGF&-%#lnG6#*&,&F&F&-%%sqrtG6#,&F&F&*$F'\"\"#!\"\"F&F&F'F3" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arcsech *x] = -1/(x*sqrt(1-x^2));" "6#/7#*&%(arcsechG\"\"\"%\"xGF',$*&F'F'*&F( F'-%%sqrtG6#,&F'F'*$F(\"\"#!\"\"F'F2F2" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 300 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arccsch*x;" "6#/%\"yG*&%(arccschG\"\"\"%\"xGF'" } {TEXT -1 16 " exactly when " }{XPPEDIT 18 0 "x = csch*y;" "6#/%\"xG* &%%cschG\"\"\"%\"yGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccsch*x = ln(1/x+sqrt(1+x^2)/abs(x))" "6#/*&%(ar ccschG\"\"\"%\"xGF&-%#lnG6#,&*&F&F&F'!\"\"F&*&-%%sqrtG6#,&F&F&*$F'\"\" #F&F&-%$absG6#F'F-F&" }{XPPEDIT 18 0 "``=PIECEWISE([ln((1+sqrt(1+x^2)) /x), 0 < x],[ln((1-sqrt(1+x^2))/x), x < 0])" "6#/%!G-%*PIECEWISEG6$7$- %#lnG6#*&,&\"\"\"F.-%%sqrtG6#,&F.F.*$%\"xG\"\"#F.F.F.F4!\"\"2\"\"!F47$ -F*6#*&,&F.F.-F06#,&F.F.*$F4F5F.F6F.F4F62F4F8" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arccsch*x] = -1/(abs(x) *sqrt(1+x^2));" "6#/7#*&%(arccschG\"\"\"%\"xGF',$*&F'F'*&-%$absG6#F(F' -%%sqrtG6#,&F'F'*$F(\"\"#F'F'!\"\"F5" }{TEXT -1 3 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 301 18 "__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arccoth*x" "6#/%\"yG*&%(arccothG\"\"\"%\"xGF'" } {TEXT -1 16 " exactly when " }{XPPEDIT 18 0 "x = coth*y" "6#/%\"xG*& %%cothG\"\"\"%\"yGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccoth*x = 1/2;" "6#/*&%(arccothG\"\"\"%\"xGF&*&F& F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((x+1)/(x-1));" "6#-%# lnG6#*&,&%\"xG\"\"\"F)F)F),&F(F)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 "[arccoth*x] = 1/(1-x^2);" "6#/7 #*&%(arccothG\"\"\"%\"xGF'*&F'F',&F'F'*$F(\"\"#!\"\"F-" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 302 18 "__________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctanh*x" "6#*&%(arctanhG\"\"\"%\"xGF% " }{TEXT -1 18 " is defined when " }{XPPEDIT 18 0 "abs(x)<1" "6#2-%$a bsG6#%\"xG\"\"\"" }{TEXT -1 9 ", while " }{XPPEDIT 18 0 "arccoth*x" " 6#*&%(arccothG\"\"\"%\"xGF%" }{TEXT -1 18 " is defined when " } {XPPEDIT 18 0 "abs(x)>1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arccosh*x" "6#*&%(arcc oshG\"\"\"%\"xGF%" }{TEXT -1 17 " is defined when " }{XPPEDIT 18 0 "x> =1" "6#1\"\"\"%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "arcsech*x" "6 #*&%(arcsechG\"\"\"%\"xGF%" }{TEXT -1 17 " is defined when " } {XPPEDIT 18 0 "00" "6 #0%\"xG\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 21 "Find the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG " }{TEXT -1 33 " in each of the following cases. " }}{PARA 0 "" 0 "" {TEXT -1 9 " (a) " }{XPPEDIT 18 0 "f(x)=(x-1)*arctanh*x" "6#/-%\"f G6#%\"xG*(,&F'\"\"\"F*!\"\"F*%(arctanhGF*F'F*" }{TEXT -1 9 " (b) \+ " }{XPPEDIT 18 0 "f(x)=x*arcsech*x+arcsin*x" "6#/-%\"fG6#%\"xG,&*(F'\" \"\"%(arcsechGF*F'F*F**&%'arcsinGF*F'F*F*" }{TEXT -1 9 " (c) " } {XPPEDIT 18 0 "f(x)=arcsinh(tan*x)" "6#/-%\"fG6#%\"xG-%(arcsinhG6#*&%$ tanG\"\"\"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 "arctanh*x-1/(x+1);" "6#,&*&%(arctanhG\"\"\"% \"xGF&F&*&F&F&,&F'F&F&F&!\"\"F*" }{TEXT -1 9 " (b) " }{XPPEDIT 18 0 "arcsech*x" "6#*&%(arcsechG\"\"\"%\"xGF%" }{TEXT -1 9 " (c) " } {XPPEDIT 18 0 "abs(sec*x)" "6#-%$absG6#*&%$secG\"\"\"%\"xGF(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Diff((x-1)*arctanh(x),x);\n``=value(%);\n``=map(simpl ify,rhs(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,&%\"xG \"\"\"F)!\"\"F)-%(arctanhG6#F(F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G,&-%(arctanhG6#%\"xG\"\"\"*&,&F)F*F*!\"\"F*,&F*F**$)F)\"\"#F*F-F-F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%(arctanhG6#%\"xG\"\"\"*&F *F*,&F)F*F*F*!\"\"F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "assume(x>0,x<=1):\ninterface(showassumed =0):\nDiff(x*arcsech(x)+arcsin(x),x);\n``=simplify(value(%));\nx := 'x ': interface(showassumed=1):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Dif fG6$,&*&%#x|irG\"\"\"-%(arcsechG6#F(F)F)-%'arcsinGF,F)F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G-%(arcsechG6#%#x|irG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Diff(arcsinh (tan(x)),x);\nvalue(%);\nsimplify(%,\{1+tan(x)^2=sec(x)^2\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%(arcsinhG6#-%$tanG6#%\"xGF ," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&\"\"\"F%*$)-%$tanG6#%\"xG\"\" #F%F%#F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$*$)-%$secG6#%\"xG\"\"# \"\"\"#F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 40 "Use integration by p arts to show that " }{XPPEDIT 18 0 "Int(arctanh*x,x) = x*arctanh*x+1 /2" "6#/-%$IntG6$*&%(arctanhG\"\"\"%\"xGF)F*,&*(F*F)F(F)F*F)F)*&F)F)\" \"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1-x^2)+c" "6#,&-%#lnG6# ,&\"\"\"F(*$%\"xG\"\"#!\"\"F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(arctanh*x,x)" "6#-%$IntG6$*&%(arctanhG\"\"\"%\"xGF(F)" } {TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=arctanh*x,v=x],[ du/dx=1/(1-x^2),dv/dx=1])" "6#-%*PIECEWISEG6$7$/%\"uG*&%(arctanhG\"\" \"%\"xGF+/%\"vGF,7$/*&%#duGF+%#dxG!\"\"*&F+F+,&F+F+*$F,\"\"#F4F4/*&%#d vGF+F3F4F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$ 6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{XPPEDIT 18 0 "``=u*v-Int(v*``(du/dx),x )" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&%#duGF(%#dxG! \"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x*arctanh*x-Int(x/(1-x^2),x)" "6#/%!G,&*(%\"xG\"\"\" %(arctanhGF(F'F(F(-%$IntG6$*&F'F(,&F(F(*$F'\"\"#!\"\"F1F'F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x*arcta nh*x-1/2" "6#/%!G,&*(%\"xG\"\"\"%(arctanhGF(F'F(F(*&F(F(\"\"#!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1-x^2)+c" "6#,&-%#lnG6#,&\"\"\"F(*$ %\"xG\"\"#!\"\"F(%\"cGF(" }{TEXT -1 2 ". " }}{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 0 "" 0 " " {TEXT -1 23 "Code for first picture " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "p1 := plot([[0,0],[8,0] ,[8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots[textplot]([4, 1,`y = arcsinh x exactly when x = sinh y`],\n color=black):\nplots[ display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 21 "Code for 2nd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "p1 := plot([[0,0],[8,0],[ 8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots[textplot]([4,1, `y = arccosh x exactly when x = cosh y, and y is not negative`],colo r=black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 3rd picture " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "p1 := plo t([[0,0],[8,0],[8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots [textplot]([4,1,`y = arctanh x exactly when x = tanh y`],color=black ):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 21 "Code for 4th picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "p1 := plot([[0,0],[8 ,0],[8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots[textplot]( [4,1,`y = arcsech x exactly when x = sech y, and y is not negative`] ,color=black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 5th picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "p1 \+ := plot([[0,0],[8,0],[8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots[textplot]([4,1,`y = arccsch x exactly when x = csch y`],color =black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 6th picture " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "p1 := plo t([[0,0],[8,0],[8,2],[0,2],[0,0]],color=red,thickness=2):\nt1 := plots [textplot]([4,1,`y = arccoth x exactly when x = coth y`],\n color=b lack):\nplots[display]([p1,t1],axes=none);" }}}{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 }