{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 "Maple Out put" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Given the following (arbi trary) two functions. We want to show that for all points on y=f(x), t here is a point on y=g(x) (say c) so that the distance is the shortest between these two curves." }}{PARA 0 "" 0 "" {TEXT -1 159 "Key: If we start with one point from y=f(x), the slope connecting the desired tw o points should be negative reciprocal to the slope of the tangent lin e at x=c." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=x->sin(x)+5 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%& arrowGF(,&-%$sinG6#9$\"\"\"\"\"&F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=x->-cos(x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$cosG6#,&9$\"\"\"F2!\"\"F3 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "graph1:=plot(\{f( x),g(x)\},x=-5..5,y=-10..10,scaling=constrained,thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(f(x)-g(x)=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6$/,(-%$sinG6#%\"xG\"\"\"\" \"&F,-%$cosG6#,&F+F,F,!\"\"F,\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(x=c,diff(g(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,&%\"cG\"\"\"F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eq1:=((g(c)-f(x))/(c-x))*subs(x=c,diff(g(x),x))+1=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&*(,(-%$cosG6#,&%\"cG\"\" \"F.!\"\"F/-%$sinG6#%\"xGF/\"\"&F/F.,&F-F.F3F/F/-F1F+F.F.F.F.\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq2:=((g(c)-f(x))/(c-x))*di ff(f(x),x)+1=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&*(,(-%$co sG6#,&%\"cG\"\"\"F.!\"\"F/-%$sinG6#%\"xGF/\"\"&F/F.,&F-F.F3F/F/-F*F2F. F.F.F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(\{eq1 ,eq2\},\{c,x\},x=-2..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"cG$\"+Nr-S:!\"*/%\"xG$!+#>p2.\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fsolve(\{eq1,eq2\},\{c,x\},x=-2.5..-1.5,c=-2.5..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"cG$!+B2eF?!\"*/%\"xG$!+d#3[o\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(\{eq1,eq2\},\{c,x\},c =-2..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"cG$!+TeJ6&)!#5/%\"xG$\"+6z#>U$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Case 1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x1:=-1.030769192;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+#>p2.\"!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "y1:=f(x1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#y1G$\"+q_IUT!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eq3: =((g(c)-y1)/(c-x1))*subs(x=c,diff(g(x),x))+1=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,&*(,&-%$cosG6#,&%\"cG\"\"\"F.!\"\"F/$\"+q_IUT! \"*F/F.,&F-F.$\"+#>p2.\"F2F.F/-%$sinGF+F.F.F.F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c1:=fsolve(eq3,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G$\"+Nr-S:!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g(c1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++t%pd)!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "with(linalg):with(plots ):with(student):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protecte d names norm and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "X1:=vector([c1,g(c1 )]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X1GK%'vectorG6#7$$\"+Nr-S:! \"*!++t%pd)!#5Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Y1:=vector([x1,y1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y1GK%'v ectorG6#7$$!+#>p2.\"!\"*$\"+q_IUTF+Q(pprint16\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "Z1:= arrow(, , difference,wid th=0.1, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gra ph2:=showtangent(f(x), x = x1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "graph3:=showtangent(g(x),x=c1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(\{graph1,graph2,graph3,Z1\});" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6/-%'CURVESG6$7S7$$!#5\"\"!!3 K+++/m5\"z'!#<7$$!3!pmmm\"p0k&*F-$!3UAr^GC'pc'F-7$$!3uKL3P'F-7$$!3WmmmT%p\"e()F-$!3'=1wj\"zg_hF-7$$!3/nmm\"4m(G)F-!3-j (z?E=fF-7$$!3OLL$3i.9!zF-$!3+nlJJQ47dF-7$$!3fmm;/R=0vF-!3ZP#eGzt3b F-7$$!3k++]P8#\\4(F-$!3s&yfH\\LuH&F-7$$!3Kmm;/siqmF-!3s)e^Ity#z]F-7$$ !3Q****\$$ypZiF-!3UV^A)fB='[F-7!3jKLLyaE\"eF-!36iCUjU9QYF-7!3< mmm\">s%HaF-!3\\hqdQF6W%F-7!3]******\\*4)*\\F-!3LSIL>UF-7!3 o******\\_&\\c%F-!3_tQ#z)zi'*RF-7!3%)******\\1aZTF-!3!)Q1#QS5?yF- 7!3Imm;/#)[oPF-!3O)*Gswq6(eF-7!3%HLLL=exJF-!3'\\]<.1qNF-7! 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