{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Module 4 : Trigonometry" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 22 "403: Inverse Fu nctions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________________ _______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 19 "A. The Inverse Idea" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____ ______________________________________________________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here are two functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f \+ := x -> (37*x - 53) / 89;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6# %\"xG6\"6$%)operatorG%&arrowGF(,&9$#\"#P\"#*)#\"#`F0!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g := x -> (89*x + 53 ) / 37; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&ar rowGF(,&9$#\"#*)\"#P#\"#`F0\"\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "If we evaluate f(x) at x = -8, the n evaluate g(x) " }{TEXT 256 14 "at that result" }{TEXT -1 65 ", notic e that the result is the number, -8, that we started with." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f(-8); g(%); simplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!$\\$\"#*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 261 "There was noth ing special about x = -8. Any number will work. In each case, you will get back the number you originally started with. In other words, howe ver f(x) transforms a number, g(x) does the opposite transformation wh ich returns it to its original value." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(33); g(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%o6\"#*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#L" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(0); g(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6##!#`\"#*)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f(1 19/47); g(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%7>\"%$=%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"$>\"\"#Z" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Even if we do it the other way, the same thing happens." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g(100); f(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"%`*)\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$+\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g(-987); f(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!&!z()\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$()*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Composing the functions together, we can see this even more immediately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "f( g( 39 )); g( f ( -222 ));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$A#" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "What we have found, is that in general, the composition of f(x) and g(x) is e quivalent to doing nothing at all." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f(g(x)); 'f(g(x))' = simpl ify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#-%\"gG6#%\"xGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g(f(x)); 'g(f(x))' = simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#-% \"fG6#%\"xGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Lets examine this from a numeric point of view." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ar ray( [[ k, f(k), g(k), f(g(k)), g(f(k)) ] $ k = 0..8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7+7'\"\"!#!#`\"#*)#\"#`\"#PF(F(7 '\"\"\"#!#;F+#\"$U\"F.F0F07'\"\"##\"#@F+#\"$J#F.F6F67'\"\"$#\"#eF+#\"$ ?$F.F " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-2)/(x -7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG %&arrowGF(*&,&9$\"\"\"\"\"#!\"\"F/,&F.F/\"\"(F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(a) = f(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"aG\"\"\"\"\"#!\"\"F',&F&F'\"\"(F)F)*&,&%\"bGF'F (F)F',&F.F'F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve( %,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"bG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "This equation f(a) = f(b ) can be solved and has a single unique solution. That means the value s of a and b are the same. Thus the function IS one-to-one. Lets try a nother example to see what happens when the function is not one-to-one ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := x -> x^4 - 8*x^2 + 16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"%\" \"\"F1*&\"\")F1)F/\"\"#F1!\"\"\"#;F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "f(a) = f(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( *$)%\"aG\"\"%\"\"\"F)*&\"\")F))F'\"\"#F)!\"\"\"#;F),(*$)%\"bGF(F)F)*&F +F))F3F-F)F.F/F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(% ||a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 259 "Note that there is not a single answer, but rather 4 answers! If there is a single solution, t he functioin is one-to-one and has an inverse function. However, if th ere are multiple solutions, then the function is not one-to-one and no inverse function exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 272 "When looking at t he graph of a function, it is easy to see if its one-to-one. If every \+ horizontal line crosses the function only once, then the funciton is o ne-to-ont. However, if there is even one horizontal line which crosses in two or more places, it is not one-to-one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; wit h(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " f := x -> 2*x*sqrt(abs(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&9$\"\"\"-%%sqrtG6#-%$absG6#F.F/\" \"#F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "If you try to solve f(a) = f(b) as we did above, even Maple ha s some trouble. Lets look at the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "display(plot( \{ k/2 $ k = -9..9\}, x = -Pi..2*Pi, color = coral),\n plot( \{ f(x) \} , x = -2..2, thickness = 3, color = green));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "68-%'CURVESG6$7S7$$!3*****4tk#fTJ!# <$!\"%\"\"!7$$!3S_J_4$fh$HF*F+7$$!3I/4wgGTdFF*F+7$$!37p-i%y$RcDF*F+7$$ !3mEnbBA/aBF*F+7$$!3u\"zSTS_E:#F*F+7$$!3//4&=EQf'>F*F+7$$!3kC\\T#e1Ex \"F*F+7$$!3Q4%yHoiEd\"F*F+7$$!3yaJ;1+Ot8F*F+7$$!3/&pXcq_$o6F*F+7$$!3sk 6iP;#y()*!#=F+7$$!3$QLxh,D]%yFOF+7$$!3#Rx)\\O:)Q!eFOF+7$$!3%f+R\"z:'o$ QFOF+7$$!3++n(R,>10#FOF+7$$\"3IXD63lhRt!#?F+7$$\"3U;GFHcrs=FOF+7$$\"3= )okm\">vlRFOF+7$$\"3pIqN#oV%=eFOF+7$$\"3w'=q^=S6&yFOF+7$$\"3')G\")pYzu 'y*FOF+7$$\"3)eUE(e^j!=\"F*F+7$$\"3sEG8=y4m8F*F+7$$\"3kLwV^V9m:F*F+7$$ \"39$R([xk$Rx\"F*F+7$$\"3'o9cS8?[&>F*F+7$$\"3'Q5?'Q%z,:#F*F+7$$\"3=yj: ;Z+_BF*F+7$$\"3I>&y`P^%\\DF*F+7$$\"31-Spq6\\SFF*F+7$$\"3O&R)*>H3E&HF*F 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Finding Inverses" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 264 "We have seen how to test to see if a fun ction has an inverse or not, but how do we actually find the inverse f or a function which has one? Just like we would do by hand, we interch ange x and y, and solve for y. We then turn this solution into a funct ion using the " }{TEXT 257 7 "unapply" }{TEXT 258 1 " " }{TEXT -1 8 "c ommand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-2)/(x-7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"\"\"#!\"\"F/ ,&F.F/\"\"(F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "so lve( x = f(y), y );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"( \"\"#!\"\"\"\"\",&F%F)F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := unapply (%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6 #%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"(\"\"#!\"\"\"\"\",&F.F2F2F1F 1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Lets verify that f(x) and g(x) are inverses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f(g(x)): ' f(g(x))' = simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#-% \"gG6#%\"xGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g(f(x)): \+ 'g(f(x))' = simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6# -%\"fG6#%\"xGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "In both cases we got 'x', which means that f undoes what \+ g does and vice-versa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "D. Graphing The Inverse" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________ __________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 287 "There is an intersting relationship between the graph \+ of f(x) and graph of its inverse. They are mirror images of each other through the line y = x. for every point (x,y) on the graph of f(x), t here corresponds a point(y,x) on the graph of the inverse. Lets see ho w this works in action." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> x^3; g:= x \+ -> (x)^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%) operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$#\"\"\"\" \"$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a := 0 : b \+ := 1.8:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "c := min(f(a), f (b)): d := max(f(a), f(b)): m:=min( a,c ): M := max( b,d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "display( plot([f(x), g(x), \+ x], x= m..M, y = m..M, thickness = 3, color = [red,blue, green], lines tyle = [1,1,3] ),\n plot ( [ [[a,c],[a,d],[b,d], [b,c], [a,c]],[ [c,a],[d,a], [d,b], [c,b],[c,a]] ], style = line, color = [coral, viol et], linestyle = 2) 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*)yu\"3i#F*$\"3[6)p&**p_v\\FI7$$\"3-i-HnWIXFF*$\"3_^@G2'o*fQFI7$$\"3u= RWhN.yGF*$\"3H4EwtQ=0EFI7$$\"3ss(*RSA20IF*$\"3%eLwJMn4O\"FI7$$\"3!)*** \\/l#fTJF*$\"3pawpOMzRJFibo-F\\u6&F^uFbuFbuF_u-%+AXESLABELSG6%Q\"x6\"Q !6\"%(DEFAULTG-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFbepF[ep" 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" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "The restricted version of the \+ sine is one-to-one and therefore has an inverse. In mathematics, we de signate the inverse of the sine as the arcsine or sin-1(x). In Maple, \+ we use " }{TEXT 259 9 "arcsin(x)" }{TEXT -1 91 ". Lets take a peek at \+ the sine and arcsin, and see how they are reflections of one another. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> sin(x); g := x -> arcsin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG% 'arcsinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a := -Pi/2: b := Pi/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "c := min(f(a), f(b)): d := max(f(a),f(b)): m := min(a,c): M := max (b,d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "display( plot([f(x), g(x), \+ x], x= m..M, y = m..M, thickness = 3, color = [red,blue, green], lines tyle = [1,1,3] ),\n plot ( [ [[a,c],[a,d],[b,d], [b,c], [a,c]],[ [c,a],[d,a], [d,b], [c,b],[c,a]] ], style = line, color = [coral, viol et], linestyle = 2, scaling = constrained) );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6&7S7$$!3+++lBjzq:!#<$! \"\"\"\"!7$$!3WNzQW&=B]\"F*$!3m`D4FJcw**!#=7$$!3.42![ROFW\"F*$!3U3C*F37$$!3,%4``jnW6\"F*$!3'>>2jDjn(*)F37$$!3)yYu)o'>y/\"F*$!3;GB9]H Oj')F37$$!3EWfpKW&Q\")*F3$!3\"p*GLjHo7$)F37$$!3(fe1Vw'\\I\"*F3$!3'yIzL \"zr8zF37$$!333*H!e\\fG&)F3$!3$4mG5]X;`(F37$$!35`u@%3'*4&yF3$!3#)**zF% Rc*oqF37$$!3%ewEV#\\hqrF3$!3CPT#\\TEfF3$!3E(yNitD)zbF37$$!3!=qRrNA:@&F3$!3'[fQ.S(zy\\F37$ $!3Z[5T-#\\F3$!3!e^J'*R[4'>F37$$!3)*RWU;s`+8F3$!3Y97m2T(oH\"F37$$!3.gSQ< NGBo!#>$!3%p\"pQ+-*z\"oFjr7$$!3y]Py0ul]:!#?$!3W[zM%yc1b\"F`s7$$\"3O=#e 5YQ8x'Fjr$\"3w$p_/5lhw'Fjr7$$\"3I^(**zOz+G\"F3$\"3!)4c_Fjew7F37$$\"3dN v()\\qFJ>F3$\"3-&4Sv&QH>>F37$$\"3y0j*\\(z-/EF3$\"3CoO$\\!zpuDF37$$\"3_ \"oMd]$=iKF3$\"3!4\"))*Q>JY?$F37$$\"3SH4XBG)*)*QF3$\"3>l%fdLV4!QF37$$ \"3=DLY%*)Rgg%F3$\"3&[Dw?K#*[W%F37$$\"3U6;S?@OT_F3$\"3\"*eoz8Ol/]F37$$ \"3KMYS.Uq>fF3$\"3U?'z@Ws*zbF37$$\"3g!oi?&HQMlF3$\"3Y(GhB%**>zgF37$$\" 3Jr$zA=*Q1sF3$\"3!ogrtz[')f'F37$$\"3gHJCuYpQyF3$\"3U]!e4/]-1(F37$$\"3) fG\"*Q5O'*\\)F3$\"3=L,8\")Qc7vF37$$\"3#=wm/;Fe9*F3$\"3>!)Qd?13BzF37$$ \"37E5$3JHB#)*F3$\"3G\"fd*>4R<$)F37$$\"31h>eG\")QZ5F*$\"3))[j\\nn?h')F 37$$\"3PK#)fG(=S6\"F*$\"3_3O')>Uyu*)F37$$\"330`g$f(4!=\"F*$\"3o5rJQ=VY #*F37$$\"3&oOhy?<3C\"F*$\"35a:'o4\"\\g%*F37$$\"3XnP+Q(3/J\"F*$\"3qln3S E!Hm*F37$$\"3/%yp@B_EP\"F*$\"3MurX7gL/)*F37$$\"3cNK@zn,R9F*$\"3?l,yOjH 8**F37$$\"3*p@f'=h`-:F*$\"3c3edku_>fF3$!3Y4l?,`z MjF37$$!3Z+++gB_6_F3$!3y!>'>`Y+#[&F37$$!3))*******>\\l]'H$F3$!3Qk^s&>N$fLF3 7$$!3<+++?5%*=EF3$!3Q9tA`G%)\\EF37$$!3%*******H%QP(>F3$!3#4`,9S$y')>F3 7$$!35+++?s`+8F3$!3XBg#ohJUI\"F37$$!3c++++NGBoFjr$!3WDpis\"*eGoFjr7$$! 3+++++ql]:F`s$!3ohgV@wl]:F`s7$$\"35++++&Q8x'Fjr$\"3%3K4FvBlx'Fjr7$$\"3 ))******p$z+G\"F3$\"3)RAJcI,OG\"F37$$\"36+++]qFJ>F3$\"3wLO\\1()[V>F37$ $\"3>+++qz-/EF3$\"3rWOz%p$RMEF37$$\"3=+++5N=iKF3$\"33!oan+.IK$F37$$\"3 %)******>G)*)*QF3$\"3Bg?#3I6_+%F37$$\"3\"********))Rgg%F3$\"3!)\\:x$[b ny%F37$$\"3X+++?@OT_F3$\"3$R!)*)HD0q^&F37$$\"3(********>/(>fF3$\"3!>0v ;k9]L'F37$$\"3Z******\\HQMlF3$\"3z6B'40x67(F37$$\"3<+++!=*Q1sF3$\"3%pt FsCMs/)F37$$\"3U+++qYpQyF3$\"3#fF))p1L(3!*F37$$\"3!)*******4O'*\\)F3$ \"3*3RXz@;f,\"F*7$$\"3i******fr#e9*F3$\"3C5#*G^[Za6F*7$$\"3o*****\\ByS [*F3$\"3C%***f!yv\"[7F*7$$\"3w******4$HB#)*F3$\"3oS+z)46?Q\"F*7$$\"3.+ ++2Lx.**F3$\"3mWCSrs&>V\"F*7$$\"3Q+++0t@&)**F3$\"3\")GKwgbT;:F*7$%%FAI LGFggl-F\\[l6&F^[lFb[lFb[lF_[lFc[lFf[l-F$6&7S7$F(F(7$F/F/7$F5F57$F:F:7 $F?F?7$FDFD7$FIFI7$FNFN7$FSFS7$FXFX7$FgnFgn7$F\\oF\\o7$FaoFao7$FfoFfo7 $F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_q7$FdqFdq7$FiqFiq7$F^rF^r7$FcrFc r7$FhrFhr7$F^sF^s7$FdsFds7$FisFis7$F^tF^t7$FctFct7$FhtFht7$F]uF]u7$Fbu Fbu7$FguFgu7$F\\vF\\v7$FavFav7$FfvFfv7$F[wF[w7$F`wF`w7$FewFew7$FjwFjw7 $F_xF_x7$FdxFdx7$FixFix7$F^yF^y7$FcyFcy7$FhyFhy7$F]zF]z7$FbzFbz7$FgzFg z-F\\[l6&F^[lFb[lF_[lFb[l-Fd[lFh[lFf[l-F$6&7'7$$!3c'*[zEjzq:F*F+7$Fe[m Fiz7$$\"3c'*[zEjzq:F*Fiz7$Fi[mF+Fd[m-F\\[l6&F^[lF_[l$\")AR!)\\Fa[lFb[l -%&STYLEG6#%%LINEG-Fd[l6#\"\"#-F$6&7'7$F+Fe[m7$FizFe[m7$FizFi[m7$F+Fi[ mFj\\m-F\\[l6&F^[l$\")#R!)4$Fa[l$\")t8V=Fa[lF`]mF`\\mFd\\m-%+AXESLABEL SG6%Q\"x6\"Q\"yFh]m%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!+ Fjzq:!\"*$\"+Fjzq:Fe^mFb^m" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "As befor e, the yellow box and violet boxes (and the graphs inside them) are mi rror images through the green line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 204 "Lets take a look at the cosine and arcc osine. Note the restricted range of the cosine is different. This make s both the restricted cosine and the arccosine appear much different t han the sine and arcsine." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := x -> cos(x); g := x -> arccos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$cosG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG%'arccosG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a := 0: b:= Pi:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "c := min(f(a),f(b)): d:= max(f(a),f(b)): m := mi n(a,c): M := max(b,d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "display( plot( f(x), x= a..b, y = c..a, thickness = 3, color = re d), \n plot( [g(x), x], x= m..M, y = m..M, thickness = [3,1], co lor = [blue,green], linestyle = [1,4] ),\n plot ( [ [[a,c],[a,d ],[b,d], [b,c], [a,c]],[[c,a],[d,a], [d,b], [c,b],[c,a]] ], style = li ne, color = [coral, violet], linestyle = 2, scaling = constrained) ); 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with(plots):" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> ta n(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$tanG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := x -> arctan(x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gG%'arctanG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := -Pi/2 + .1: b := Pi/2 - .1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "c := min(f(a),f(b)): d := max(f(a),f(b) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "m := min (a, c): M \+ := max( b, d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "display( plot( f(x), x= a..b, y = m..M, thickness = 3, color = red, discont = \+ true), \n plot( [g(x), x], x= m..M, y = m..M, thickness = 3, col or = [blue,green], linestyle = [1,4] ),\n plot ( [ [[a,c],[a,d] ,[b,d], [b,c], [a,c]],[[c,a],[d,a], [d,b], [c,b],[c,a]] ], style = lin e, color = [coral, violet], linestyle = 2, scaling = constrained) );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6%7e n7$$!3!****\\NK'zq9!#<$!3Uwdo(4Wm'**F*7$$!3)p!RWOnwa9F*$!3c.%eI))z(z&) F*7$$!309yL\\rtQ9F*$!3E1k%))y\"HGvF*7$$!37@g)4?f$QgF*7$$!33yI1fMyy8F*$!3yo4XN@#Q9&F*7$$!3?G0+V*) )3N\"F*$!3YPR&3.HQZ%F*7$$!3'*GJeT$[\")G\"F*$!3Q-JDO![KW$F*7$$!3bBJta=* \\A\"F*$!3gP\"zg#*4cx#F*7$$!3KK1kRb8i6F*$!3ym24[eD4BF*7$$!3q'\\\"Q*ofQ 5\"F*$!37dSapAm$)>F*7$$!3sWr!>U=N/\"F*$!3*e'*H6TFur\"F*7$$!3qZ4mnR86)* !#=$!3KNTwjEb%\\\"F*7$$!3Ku6K`]3*=*Fao$!3CV#>9+%H58F*7$$!3=()>R&GJ#\\& )Fao$!3wK^\"*G!*p\\6F*7$$!3%GH?$Qxk&)zFao$!3g'o[)GhoE5F*7$$!3KVwr[h=^t Fao$!3i$Q;!y-yT!*Fao7$$!3wq-())R>Tr'Fao$!35/i2u\"\\b%zFao7$$!3FwTzRv=+ hFao$!3g2m3xxY*)pFao7$$!36u)HgfyEa&Fao$!3'>b(eUz#**='Fao7$$!31$fu%Qluz [Fao$!3>$3_Z(Q*yI&Fao7$$!3\"37%yGh:=VFao$!3*)yCgQN?3YFao7$$!3kL,&f(H*[ m$Fao$!3WmxZBzHQQFao7$$!3mL1^%RWm3$Fao$!3OhJ2>`b)=$Fao7$$!3=S:?`S@_CFa o$!3lP,Q#HzD]#Fao7$$!3w!po&Qj3[=Fao$!3?FDA*z<%p=Fao7$$!3!>J3bWUx@\"Fao $!30#z&R_wzB7Fao7$$!3'HM;hz**))Q'!#>$!3IOy9eng(R'F]t7$$!31]%=!*[R>X\"! #?$!37w3J40%>X\"Fct7$$\"3Q+Qzk;ESjF]t$\"3U[Zwp5x[jF]t7$$\"3Ed#oT)pe)> \"Fao$\"3tfrKO)fV?\"Fao7$$\"3\"4D@R8G$3=Fao$\"3$)y\"[\"*\\+$G=Fao7$$\" 3ZE0$[S]#QCFao$\"3UvM,)GYx[#Fao7$$\"3'Q+FIX1X0$Fao$\"3)*y=Ssg=`JFao7$$ \"357F&p'ew]OFao$\"3'F ao7$$\"3dcL177R=hFao$\"3O5n^x*)f;qFao7$$\"3)Qf?o>;wu'Fao$\"3b-#Hf6R.+) Fao7$$\"3mPu#3(ymRtFao$\"3#3gIA9n3-*Fao7$$\"3s!=+@YK&ezFao$\"3iQ)e5(=8 @5F*7$$\"3')\\y@=dej&)Fao$\"3=Cq/kt.`6F*7$$\"3u:d&RS?q>*Fao$\"3S)Rox<_ CJ\"F*7$$\"3T'[>=H$42)*Fao$\"3%4`FR#oC$\\\"F*7$$\"3>yRW$4)4V5F*$\"3'Qr (Q(\\odr\"F*7$$\"3c5&QsCq\\5\"F*$\"3gb#3;)[:*)>F*7$$\"3!RP#yIV#=;\"F*$ \"36NVk#['G2BF*7$$\"3JxO]Eb)pA\"F*$\"3)ym(y[%fIz#F*7$$\"3#\\1\\nZm_G\" F*$\"3To%*eIGc1MF*7$$\"3wCm\\DhSZ8F*$\"3x8xyp(fY\"F*7$$!3))4w9ig&*G()F*$!3c<'Q\"eBtc9F*7$$!353\" )z6])4I)F*$!3#R$e_Lk!4X\"F*7$$!3Ia]9L![](yF*$!3p1*o>(*)[W9F*7$$!3#z?0- $*\\,[(F*$!3#G-4W;(*yV\"F*7$$!3Y2=+=eDrqF*$!3b;,M6.JI9F*7$$!3]y>BUpP[m F*$!3!e?$3LG]@9F*7$$!3Yi)zVCaoA'F*$!3z>!)yH7c69F*7$$!3cVL\"zLmKz&F*$!3 QQZ71l')*R\"F*7$$!3\"p\"3*Q(=O6aF*$!3?=.nk91)Q\"F*7$$!3EUxA,zU\")\\F*$ !3e5wA&*Qos8F*7$$!3K2dOy&G(\\XF*$!3m:8CeJWa8F*7$$!3SH)[))G1P8%F*$!397 \\1,FWL8F*7$$!3;**)>V>=fv$F*$!3jo&HhE'e58F*7$$!3(z6;@;4#F*$!3$eVIdHB[7\"F*7$$!3%*fEUj$3Nl-\"F*7$$\"3)G?%3PL%)p?F*$\"3'==qw!pt?6F*7$$\"3; )4.F9!*QZ#F*$\"3IHC*p)el'=\"F*7$$\"3_a$[AB:D#HF*$\"3=\"y9)eI6T7F*7$$\" 3%yp6'G[iDLF*$\"3%yiErp,(y7F*7$$\"3)=6]ZOIgv$F*$\"3mBV-\"p$f58F*7$$\"3 C'HK8vTg9%F*$\"3'R%z&4xATL\"F*7$$\"3'HJtB@FCd%F*$\"31$e%)=A%[b8F*7$$\" 3g9p@-Fit\\F*$\"3gk/W$3\"Qs8F*7$$\"3-0$H$)e()HR&F*$\"3q1+B5FX(Q\"F*7$$ \"3hhlU3([Z,S\"F*7$$\"3f!zGJ_JAB'F*$\"3&pVlGJ'p69F*7$$ \"33G!z]$)Qck'F*$\"3UVHJKAW@9F*7$$\"37A#*G@vSoqF*$\"3A1TBVWDI9F*7$$\"3 !GHE$pan([(F*$\"3#**[<23 " 0 "" {MPLTEXT 1 0 51 "sin(Pi): % = evalf(%); arcsin(Pi) : % = evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%'arcsinG6#%#PiG^$$\"+Fjzq:!\"*$!+ti_6=F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "In this \+ example, it makes no sense to compute the arcsin for a number larger t han 1, so Maple returns a complex number as a solution (why its a comp lex number is beyond the scope of our discussioin here)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "On the other hand , it makes sense to compute the sine of 1 radian, but its not somethin g that comes up very often. However, the inverse sine of 1 is a famili ar number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "sin(1): % = evalf(%); arcsin(1 ) : % : evalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#\"\"\"$\"+[)4ZT)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }}}}{MARK "2 0" 22 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }