{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 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 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 49 "High School Modul es > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 32 "Intervals and Number Line Graphs" }}{PARA 0 "" 0 "" {TEXT -1 131 "\nThe number line graphs of finite and infinite interval s. This worksheet also covers interval notation, and more complicated \+ sets.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Co de Resource section first. Although there will be no output immediatel y, these definitions are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 5 "Code " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1289 "intfin := proc( a, b, leq, req )\n #description \"plot a finite interval on a number li ne\";\nlocal r, M, v1, v2, wid, bar, Axesplot, leftpt, rightpt, pt_col or,\n leftend, rightend, LT, RT, IT, textpos, lt, rt;\n \nr := b- a ;\nM := evalf( 1.2 * max( b-a, abs(a), abs(b) ) );\nv1 := [a, 0];\nv 2 := [b, 0];\nwid := ceil( M/15) ;\n\nbar := arrow( v1, v2,\n shape = double_arrow, color = red , difference = true,\n width = wid, he ad_width = 0, head_length = 0):\n\n\nif( leq ) then lt := `[`; else lt := `(`; fi;\nif( req ) then rt := `]`; else rt := `)`; fi;\n\ntextpos := max(wid, 2) ;\nLT := textplot( [a, textpos, a] ):\nRT := textplot( [b, textpos, b] ):\nIT := textplot( [(a+b)/2, textpos + wid, cat(lt,a ,`,`,b,rt)], color = green ):\n\nleftend := min( a - 2*wid, 0);\nrigh tend := max( b + 2*wid, 0);\n\nAxesplot := plot( 0, x = leftend..right end , \n y = (-wid)..(2*wid), axes = none, thickne ss = 2,\n scaling = constrained, tickmarks = [1,1] ):\n\n\nif( leq ) then pt_color := red; else pt_color := white; fi;\n leftpt := plottools[disk]([ a, 0], wid/2, color=pt_color):\nif( req ) \+ then pt_color := red; else pt_color := white; fi;\nrightpt := plottool s[disk]([ b, 0], wid/2, color=pt_color ):\n\ndisplay( leftpt, rightpt , bar, Axesplot, LT, RT, IT );\n\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1201 "#_____________________________________________ _____________\nintinfr := proc(a, leq )\n# description \"plot \+ an infinite interval going to +inf\";\nlocal bb, r, M, v1, v2, wid, bar, Axesplot, textpos,\n leftpt, rightpt, pt_color, leftend, rightend, LT, RT, IT,lt;\n\nbb := max( 3*abs(a), 5) ; \n\nr := bb-a; \nM := evalf(1.2* max( abs(r), abs(a)));\nv1 := [a, 0];\nv2 := [bb, 0] ;\nwid := ceil( M/15);\nbar := arrow( v1, v2,\n shape = double_arro w, color = red , difference = true,\n width = wid, head_width = 1.8* wid, head_length = 1.2*wid):\n\nif( leq ) then lt := `[`; else lt := ` (`; fi;\n\ntextpos := max(wid, 3) ;\nLT := textplot( [a, textpos, a] ) :\nRT := textplot( [bb, textpos, `Infinity`] ): \nIT := textplot( [(a+ bb)/2, textpos + wid, cat(lt,a,`, infinty)`)], color = green ):\n\nlef tend := min( 0, a*1.5);\nrightend := max( 0, bb);\n\nAxesplot := plot( 0, x = leftend..rightend , \n y = (-wid)..(3*wid) , axes = none, thickness = 2,\n scaling = constrai ned, tickmarks = [1,1] ):\n\n\nif( leq ) then pt_color := red; else pt _color := white; fi;\nleftpt := plottools[disk]([ a, 0], wid/2, color= pt_color):\n\ndisplay( leftpt, bar, Axesplot, LT, RT, IT );\n\nend p roc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1211 "#______________ ____________________________________________\nintinfl := proc( b, req \+ )\n# description \"plot a infinite interval going to -inf\";\n local aa, bb, r, M, v1, v2, wid, bar, Axesplot,textpos, \n \+ leftpt, rightpt, pt_color, leftend, rightend, LT, RT, IT, rt;\n\naa := min( -3*abs(b), -5); \n\nr := b-aa;\nM := evalf(1.2*max( abs(r), abs( b) ));\nv1 := [aa, 0];\nv2 := [b, 0];\nwid := ceil( M/15);\nbar := arr ow( v2, v1,\n shape = double_arrow, color = red , difference = true ,\n width = wid, head_width = 1.8*wid, head_length = 1.2*wid):\n\nte xtpos := max(wid, 3) ;\nif( req ) then rt := `]`; else rt := `)`; fi; \n\n\nLT := textplot( [aa, textpos, `-Infinity`] ):\nRT := textplot( [ b, textpos, b] ):\nIT := textplot( [(aa+b)/2, textpos + wid, cat(`(-in finity,`,b,rt)], color = green ):\n\nleftend := min( 0, aa);\nrightend := max( 0, b*1.5);\n\nAxesplot := plot( 0, x = leftend..rightend , \n y = (-wid)..(3*wid), axes = none, thickness = 2, \n scaling = constrained, tickmarks = [1,1] ):\n\n \nif( req ) then pt_color := red; else pt_color := white; fi;\nrightpt := plottools[disk]([ b, 0], wid/2, color=pt_color):\n\ndisplay( righ tpt, bar, Axesplot, LT, RT , IT);\n\nend proc:\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 900 "#__________________________________________ ________________\nintall := proc()\n #description \"plot a in finite interval of all real numbers \";\nlocal a, b, v1, v2, wid, b ar1, bar2, LT, RT, Axesplot ;\na := -25; b := 25;\n\nv1 := [a, 0];\nv2 := [b, 0];\nwid := ceil( (b-a)/15);\nbar1 := arrow( v1,\n shape = \+ double_arrow, color = red , difference = false,\n width = wid, head_ width = 1.8*wid, head_length = 1.2*wid):\n\nbar2 := arrow( v2,\n \+ shape = double_arrow, color = red , difference = false,\n \+ width = wid, head_width = 1.8*wid, head_length = 1.2*wid):\n\nLT := t extplot( [a, max( wid, 6), `-Infinity`] ):\nRT := textplot( [b, max( w id, 6), `Infinity`] ): \n\n\nAxesplot := plot( 0, x = a..b , \n \+ y = (-wid)..(3*wid), axes = none, thickness = 2,\n \+ scaling = constrained, tickmarks = [1,1] ):\n\ndisplay( bar1, bar2, Axesplot, LT, RT );\nend proc:\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 623 "#____________________________________________ ______________\n#_____________________________________________________ _____\nintv := proc( expr1, expr2 )\n # description \"plot a i nfinite interval going to -inf\";\nlocal a, b, leq, req;\n\nif type ( expr1, `<=`) then leq := true; else leq := false; fi;\nif type( expr 2, `<=`) then req := true; else req := false; fi;\na := lhs( expr1); \+ b := rhs( expr2);\n\nif( abs(a) <> infinity) \n then if( abs(b) <> infinity) then intfin( a,b, leq, req );\n else intinfr( a, leq) ; fi;\n else if( abs(b) <> infinity) then intinfl( b, req);\n \+ else intall(); fi;\n\nfi;\n\nend proc:" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Finite Intervals" }}{PARA 0 "" 0 "" {TEXT -1 181 "Finite intervals may include one, both, or nei ther endpoint - depending on whether the inequalities are \"less than \", or \"less than an equal\". This example is open on both endpoints. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "intv( 3 < x, x < 17 );" }}}{PARA 0 "" 0 "" {TEXT -1 47 "\nWhil e this example is closed on both endpoints" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "intv( 10 <= x, x <= 200 );" }}}{PARA 0 "" 0 "" {TEXT -1 89 "\n\nIts also possible to have half-open intervals (or ha lf-closed if you're a pessimist). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "intv( -30 <= x, x < 7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "intv( -140 < x,x <= -110 );" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Infinite Intervals" }}{PARA 0 "" 0 "" {TEXT -1 123 " There are also infinite intervals. These occur when there is only a si ngle inequality, and no bound in the other direction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "intv( -4 < x, x < infinity );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "intv( -infinity < x, x < 10 \+ );" }}}{PARA 0 "" 0 "" {TEXT -1 61 "\nIts also possible to have infini te sets with closed endpoint" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "intv( 12 <= x, x < infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "intv( -infinity < x, x <= 100 );" }}}{PARA 0 "" 0 "" {TEXT -1 85 "\nAnd all real numbers can be represented as an interval \+ too....a really big interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "intv( -infinity < x, x < infinity );" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 33 "Unions and More Complicated Sets " }}{PARA 0 "" 0 "" {TEXT -1 40 "\nYou can also form more complicate sets." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display( intv( -infinity " 0 "" {MPLTEXT 1 0 33 "solve( (x-3) *(x-1)*(x+4) > 0, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "di splay( intv( -4 < x, x < 1 ), intv( 3 < x, x < infinity ) );" }}} {PARA 0 "" 0 "" {TEXT -1 90 "\nHere is another example. Note Maple's m ethod of representating intervals as \"real range\"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "solve( ((x-1)*(x+7))/ (x^2 - 4) >= 0, x) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "display( intv( -inf inity <= x, x < -7 ), \n intv( -2 < x, x <= 1 ), \n \+ intv( 2 < x, x < infinity ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 \+ 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 \+ 1" 11 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }