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2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle45" -1 249 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle46" -1 250 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }} {SECT 0 {PARA 245 "" 0 "" {TEXT 461 38 " Worksheet 2, Math 152 Spr ing 2006" }}{PARA 246 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 462 20 "HYPERBOLIC FUNCTIONS" }}{PARA 247 "" 0 "" {TEXT -1 0 "" }} {PARA 247 "" 0 "" {TEXT 463 2 " " }{TEXT 464 5 "Goals" }{TEXT 465 61 ": In this worksheet we examine the hyperbolic trigonometric " }} {PARA 247 "" 0 "" {TEXT 465 42 " functions and some of their propert ies." }{TEXT 466 4 " " }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 248 "" 0 "" {TEXT 467 1 " " }{TEXT 468 28 " It is true for any functio n" }{TEXT 469 3 " f" }{TEXT 468 15 " with Domain (" }{TEXT 469 2 " f " }{TEXT 468 9 " ) = ( " }{XPPEDIT 329 0 "-a;" "6#,$%\"aG!\"\"" } {TEXT 469 1 " " }{TEXT 469 1 " " }{TEXT 469 1 " " }{TEXT 469 1 " " } {TEXT 469 1 " " }{TEXT 469 4 " , a" }{TEXT 467 1 " " }{TEXT 468 10 ") \+ where " }{TEXT 469 1 "a" }{TEXT 468 10 " > 0 or " }{TEXT 469 1 "a " }{TEXT 468 6 " = +" }{XPPEDIT 324 0 "infinity;" "6#%)infinityG" } {TEXT 468 1 " " }{TEXT 468 1 " " }{TEXT 468 1 " " }{TEXT 468 1 " " } {TEXT 468 1 " " }{TEXT 468 3 " , " }}{PARA 248 "" 0 "" {TEXT 468 22 " \+ that we may express " }{TEXT 469 4 " f " }{TEXT 468 53 "as the sum o f one even and one odd function. That is," }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 27 " \+ " }{TEXT 471 3 " f " }{TEXT 470 2 "( " }{TEXT 471 1 "x" }{TEXT 470 7 " ) = " }{XPPEDIT 303 1 "Epsilon;" "6#%(EpsilonG" }{TEXT 472 1 " " } {TEXT 473 1 " " }{TEXT 473 1 " " }{TEXT 473 1 " " }{TEXT 473 1 " " } {TEXT 470 3 " ( " }{TEXT 471 1 "x" }{TEXT 470 7 " ) + " }{XPPEDIT 302 1 "Phi;" "6#%$PhiG" }{TEXT 472 1 " " }{TEXT 473 1 " " }{TEXT 473 1 " " }{TEXT 473 1 " " }{TEXT 473 1 " " }{TEXT 473 3 " ( " }{TEXT 474 1 "x" }{TEXT 473 2 " )" }{TEXT 466 5 " " }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 466 2 " " }{TEXT 473 8 " wher e " }{XPPEDIT 355 1 "Epsilon;" "6#%(EpsilonG" }{TEXT 475 1 " " } {TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " } {TEXT 473 1 " " }{TEXT 470 2 "( " }{TEXT 471 1 "x" }{TEXT 470 7 " ) = " }{XPPEDIT 292 0 "(f(x)+f(-x))/2;" "6#*&,&-%\"fG6#%\"xG\"\"\"-F&6#, $F(!\"\"F)F)\"\"#F-" }{TEXT 476 1 " " }{TEXT 476 1 " " }{TEXT 476 1 " \+ " }{TEXT 476 1 " " }{TEXT 476 1 " " }{TEXT 470 16 " is even and " } {XPPEDIT 353 1 "Phi;" "6#%$PhiG" }{TEXT 477 1 " " }{TEXT 478 1 " " } {TEXT 478 1 " " }{TEXT 478 1 " " }{TEXT 478 1 " " }{TEXT 473 3 " ( " } {TEXT 474 1 "x" }{TEXT 473 2 " )" }{TEXT 466 2 " " }{TEXT 473 1 "=" } {TEXT 466 2 " " }{XPPEDIT 355 0 "(f(x)-f(-x))/2;" "6#*&,&-%\"fG6#%\"x G\"\"\"-F&6#,$F(!\"\"F-F)\"\"#F-" }{TEXT 462 1 " " }{TEXT 462 1 " " } {TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 470 11 " is odd." }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 25 " Note: " }{TEXT 473 1 " " }{XPPEDIT 355 1 "E psilon;" "6#%(EpsilonG" }{TEXT 475 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 473 1 " " }{TEXT 470 2 "( " }{XPPEDIT 18 0 "-x;" "6#,$%\"xG!\"\"" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 7 " ) = " }{XPPEDIT 355 1 "Epsilon;" "6#%(EpsilonG" }{TEXT 475 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 473 1 " " }{TEXT 470 2 "( " }{TEXT 471 1 "x" }{TEXT 470 12 " ) and " }{XPPEDIT 355 1 "Phi;" "6#%$PhiG" }{TEXT 475 1 " " } {TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " } {TEXT 473 3 " ( " }{XPPEDIT 18 0 "-x;" "6#,$%\"xG!\"\"" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " \+ " }{TEXT 473 2 " )" }{TEXT 466 1 " " }{TEXT 473 3 " = " }{TEXT 479 1 " " }{XPPEDIT 355 1 "-Phi;" "6#,$%$PhiG!\"\"" }{TEXT 475 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 462 1 " " }{TEXT 473 3 " ( " }{TEXT 474 1 "x" }{TEXT 473 4 " ) ." }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 17 " If we express " } {TEXT 471 1 "f" }{TEXT 470 11 " ( x ) = " }{XPPEDIT 18 0 "exp(x);" " 6#-%$expG6#%\"xG" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " } {TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 26 " as such a sum, we fi nd " }}{PARA 247 "" 0 "" {TEXT 470 30 " \+ " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 5 " = " }{XPPEDIT 18 0 "[(exp(x)+exp(-x))/2];" "6#7#*&,&-%$expG6 #%\"xG\"\"\"-F'6#,$F)!\"\"F*F*\"\"#F." }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 6 " \+ + " }{XPPEDIT 18 0 "[(exp(x)-exp(-x))/2];" "6#7#*&,&-%$expG6#%\"xG\" \"\"-F'6#,$F)!\"\"F.F*\"\"#F." }{TEXT 470 1 " " }{TEXT 470 1 " " } {TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }}{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 27 " We call the even par t the" }{TEXT 480 18 " hyperbolic cosine" }{TEXT 470 22 " and the odd \+ part the " }{TEXT 480 15 "hyperbolic sine" }{TEXT 470 10 " and write" }}{PARA 247 "" 0 "" {TEXT 470 21 " cosh ( " }{TEXT 471 1 "x" }{TEXT 470 7 " ) = " }{XPPEDIT 18 0 "(exp(x)+exp(-x))/2;" "6#*&, &-%$expG6#%\"xG\"\"\"-F&6#,$F(!\"\"F)F)\"\"#F-" }{TEXT 470 1 " " } {TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " } {TEXT 470 16 " , sinh( " }{TEXT 471 1 "x" }{TEXT 470 7 " ) = \+ " }{XPPEDIT 18 0 "(exp(x)-exp(-x))/2" "6#*&,&-%$expG6#%\"xG\"\"\"-F&6# ,$F(!\"\"F-F)\"\"#F-" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 3 " ." }}{PARA 247 " " 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 47 " We may now p lot the two curves in the plane." }}{PARA 247 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 21 "restart; with(plots):" } }}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 123 "P:=plot([cosh(x),sinh (x)],x=-3..3,y=-4..4,thickness=2, color=[magenta,navy],scaling=constra ined,xtickmarks=3,ytickmarks=3):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 83 "t1:=textplot([-0.7,3.4,`y = cosh ( x )`],font=[TIME S,BOLDITALIC,10],color=magenta):" }}{PARA 247 "> " 0 "" {MPLTEXT 1 481 75 "t2:=textplot([2,1,`y = sinh ( x )`],font=[TIMES,BOLDITALIC,10] ,color=blue):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 17 "displa y(P,t1,t2);" }}}{PARA 247 "" 0 "" {TEXT 470 2 " " }}{PARA 247 "" 0 " " {TEXT 470 39 " Because it is easily verified that " }{TEXT 482 1 " " }{XPPEDIT 316 0 " cosh^2*t-sinh^2*t=1" "6#/,&*&%%coshG\"\"#%\"tG \"\"\"F)*&%%sinhGF'F(F)!\"\"F)" }{TEXT 482 1 " " }{TEXT 482 1 " " } {TEXT 482 1 " " }{TEXT 482 1 " " }{TEXT 482 1 " " }{TEXT 470 15 " \+ and " }{TEXT 482 1 " " }{TEXT 483 7 "cosh t " }{TEXT 482 1 " " } {TEXT 484 1 ">" }{TEXT 482 3 " 1" }{TEXT 470 10 " for all " }{TEXT 471 1 "t" }{TEXT 470 4 " , " }}{PARA 247 "" 0 "" {TEXT 470 34 " it follows that the point " }{XPPEDIT 18 0 "P[t];" "6#&%\"PG6#%\"tG " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 8 " = ( " }{TEXT 471 6 "cosh t" }{TEXT 470 4 " , " }{TEXT 471 6 "sinh t" }{TEXT 470 47 " ) must always lie \+ on the right branch of the " }}{PARA 247 "" 0 "" {TEXT 470 17 " hyp erbola " }{XPPEDIT 18 0 "x^2-y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF& !\"\"" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 8 " = 1 ." }}{PARA 247 "" 0 "" {TEXT 470 47 " Thus the designation: hyperbolic functions." }} {PARA 247 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 38 "restart; with(plots): with(plottools):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 94 "H:=plot([sqrt(x^2-1),-sqrt(x^2-1)],x=1..4 ,thickness=2,color=maroon,xtickmarks=3,ytickmarks=1):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 67 "e1 := ellipse([cosh(2),sinh(2)] , 0.1, 0.1, filled=true, color=red):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 68 "e2 := ellipse([cosh(2),sinh(-2)], 0.1, 0.1, filled= true, color=red):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 67 "e3 := ellipse([cosh(0),sinh(0)], 0.1, 0.1, filled=true, color=red):" }}} {EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 79 "T:=textplot([1.2,2,`x^2 \+ - y^2 = 1`],font=[TIMES,BOLDITALIC,10],color=maroon):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 100 "T1:=textplot([2.4,3.7,`( cosh \+ ( t ), sinh ( t ) ) , t > 0 `],font=[TIMES,BOLDITALIC,10],color=blue): " }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 96 "T2:=textplot([2.6,- 3.7,`( cosh ( - t ), sinh ( - t ) )`],font=[TIMES,BOLDITALIC,10],color =blue):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 93 "T3:=textplot ([0,0.3,`( cosh 0, sinh 0 = ( 1 , 0 )`],font=[TIMES,BOLDITALIC,10],c olor=blue):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 54 "T4:=text plot([4,-0.3,`x`],font=[TIMES,BOLDITALIC,10]):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 56 "T5:=textplot([-0.3,3.8,`y`],font=[TIMES,BOL DITALIC,10]):" }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 68 "L:=lin e([cosh(2),sinh(2)],[cosh(2),sinh(-2)],linestyle=2,color=red):" }}} {EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 481 39 "display(H,e1,e2,e3,T,T1, T2,T3,T4,T5,L);" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG } {EXCHG }{PARA 247 "" 0 "" {TEXT 470 1 " " }}{PARA 247 "" 0 "" {TEXT 470 97 " In analogy with the circular trigonometric functions, we d efine the remaining four hyperbolic" }}{PARA 247 "" 0 "" {TEXT 470 17 " functions as:" }}{PARA 247 "" 0 "" {TEXT 470 20 " t anh " }{TEXT 471 1 "x" }{TEXT 470 5 " = " }{XPPEDIT 18 0 "sinh*x/(co sh*x);" "6#*(%%sinhG\"\"\"%\"xGF%*&%%coshGF%F&F%!\"\"" }{TEXT 470 1 " \+ " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 10 " , sech " }{TEXT 471 1 "x" }{TEXT 470 5 " = " } {XPPEDIT 18 0 "1/(cosh*x);" "6#*&\"\"\"F$*&%%coshGF$%\"xGF$!\"\"" } {TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " } {TEXT 470 1 " " }{TEXT 470 11 " , coth " }{TEXT 471 1 "x" }{TEXT 470 5 " = " }{XPPEDIT 18 0 "cosh*x/(sinh*x);" "6#*(%%coshG\"\"\"%\"x GF%*&%%sinhGF%F&F%!\"\"" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 12 " , csch " } {TEXT 471 1 "x" }{TEXT 470 5 " = " }{XPPEDIT 18 0 "1/(sinh*x);" "6#* &\"\"\"F$*&%%sinhGF$%\"xGF$!\"\"" }{TEXT 470 1 " " }{TEXT 470 1 " " } {TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 3 " ." } }{PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 482 8 " \+ " }{TEXT 485 168 " \+ _ __ ____________ " }}{PARA 247 "" 0 " " {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 480 11 " Exercises" }} {PARA 247 "" 0 "" {TEXT -1 0 "" }}{PARA 247 "" 0 "" {TEXT 470 1 " " } {TEXT 480 9 " 1. ( a )" }{TEXT 470 6 " Use " }{TEXT 471 5 "Maple" } {TEXT 470 57 " to find the derivatives of sinh(x), cosh(x) and coth(x) ." }}{PARA 247 "" 0 "" {TEXT 470 45 " Please use command ' diff ' " }}{EXCHG }{PARA 247 "" 0 "" {TEXT 470 5 " " }{TEXT 480 7 " ( b ) " }{TEXT 470 93 " Prove the results from part ( a ) dire ctly from the definitions of the hyperbolic functions." }}{PARA 247 " " 0 "" {TEXT 470 15 " " }}{PARA 247 "" 0 "" {TEXT 470 2 " " }{TEXT 480 3 "2. " }{TEXT 470 1 " " }{TEXT 486 87 "It is well kno wn that a homogeneous hanging cable influenced only by the downward fo rce" }}{PARA 247 "" 0 "" {TEXT 486 97 " of gravity which is susp ended at an equal height above the ground at both of its enpoints " }} {PARA 247 "" 0 "" {TEXT 486 13 " ( at " }{TEXT 487 1 "x" } {TEXT 486 5 " = " }{TEXT 484 1 "+" }{TEXT 486 1 " " }{TEXT 487 3 "a \+ " }{TEXT 486 17 ") has the shape:" }}{PARA 247 "" 0 "" {TEXT 486 41 " " }{TEXT 487 3 " y" }{TEXT 486 5 " = " }{TEXT 487 2 "d " }{TEXT 486 5 "cosh " }{XPPEDIT 352 0 " x/d;" "6#*&%\"xG\"\"\"%\"dG!\"\"" }{TEXT 486 1 " " }{TEXT 486 1 " " } {TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 30 " . \+ Such a curve is called a " }{TEXT 482 8 "catenary" }{TEXT 486 1 "." }} {PARA 247 "" 0 "" {TEXT 486 47 " Consequently, a curve having t he form " }{TEXT 487 1 "y" }{TEXT 486 5 " = " }{TEXT 487 8 "c - b " }{TEXT 486 7 " cosh " }{XPPEDIT 330 0 "x/b;" "6#*&%\"xG\"\"\"%\"bG !\"\"" }{TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 1 " " }{TEXT 486 22 " is often called an " }{TEXT 482 17 "inverted catenary" }{TEXT 486 1 "." }}{EXCHG }{EXCHG }{PARA 247 "" 0 "" {TEXT 470 98 " Suppose that a farmer built the larg e hayshed shown below having length 120 feet and width" }}{PARA 247 " " 0 "" {TEXT 470 95 " 36 feet. Its cross sections have the shap e of the inverted catenary with the equation: " }}{PARA 247 "" 0 "" {TEXT 470 39 " " }{TEXT 471 1 "y " }{TEXT 470 5 " = " }{XPPEDIT 18 0 "37-24*cosh(x/24);" "6#,&\"#P\" \"\"*&\"#CF%-%%coshG6#*&%\"xGF%F'!\"\"F%F-" }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 1 " " }{TEXT 470 3 " . 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