[ver] 4 [sty] finalbkb.sty [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Lexmark Optra R [port] LPT1: [lang] 1 [desc] 833996180 16 821915075 224 18 0 0 0 0 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] [ParaNum] 1 [tag] Number List 6 [fnt] Helv 200 0 16384 [algn] 1 1 3960 0 0 [spc] 33 273 1 0 72 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 1 <*:>. 360 1 1 0 20 0 0 [nfmt] 272 1 2 . , $ Number List 0 0 [tag] Practice 12 [fnt] Times New Roman 240 0 49152 [algn] 1 1 3960 288 0 [spc] 33 288 1 0 576 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 5 <*:>. 360 1 1 0 2 0 0 [nfmt] 272 1 2 . , $ Number List 0 0 [tag] Bullet 0 [fnt] Helvetica 240 0 16384 [algn] 24 1 3960 144 288 [spc] 34 417 1 0 1440 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*:>. 360 1 1 0 16 0 0 [nfmt] 272 1 2 . , $ Bullet 0 3960 [tag] Box1 0 [fnt] Helvetica 200 0 16384 [algn] 2 1 0 0 0 [spc] 33 244 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Box1 0 0 [tag] bullet 2 0 [fnt] Helv 240 0 16384 [algn] 24 1 576 288 288 [spc] 33 273 1 72 72 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*1> 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ bullet 2 0 576 [tag] Table Text 0 [fnt] Helv 240 0 16384 [algn] 8 1 0 0 0 [spc] 34 417 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Table Text 0 0 [frm] 0 512 1440 12340 3700 13376 0 1 1 1 0 0 0 0 0 0 0 16777215 133 [frmname] Frame128 [frmlay] 13376 2260 1 72 0 1 12340 0 0 2 0 0 0 0 1 1512 3690 0 [txt] @Box@<:s>See Appendix 2 @Box@<:s><+!> @Box@<+!>factor(<-!> > [frm] 1 25362944 1421 2693 10787 6679 6 1 1 5 0 0 0 0 0 0 0 16777215 137 [frmlay] 6679 9366 1 288 72 1 2765 288 0 2 0 2 8 2 1 1709 10499 0 [txt] @Box@<+C><:f240,BArial,0,0,0>Nottu Swift<:f><:f240,BArial,0,0,0> got a job with a new car dealer. His boss told him to go out to the lot and take an inventory by counting the vehicles<:f><:f240,BArial,0,0,0> in the lot. Nottu (who<:f><:f240,BArial,0,0,0> just learne d about variables<:f><:f240,BArial,0,0,0>) took a paper and pencil and went<:f><:f240,BArial,0,0,0> to<:f><:f240,BArial,0,0,0> the lot. Soon<:f><:f240,BArial,0,0,0> he returned and<:f><:f240,BArial,0,0,0> gave his boss this note:<:f> @Box@<+C><:s><:f240,BArial,0,0,0> @Box@<+C><:s><:f240,BArial,0,0,0>How may vehicles were in<:f><:f240,BArial,0,0,0> the lot?<:f> @Box@<+C><:s> @Box@<+C><:s> @Box@ > [frm] 1 42140160 1934 4023 10274 5253 1 1 1 3 0 0 0 0 13 0 0 16777215 138 [frmname] Frame5 [frmlay] 5253 8340 1 72 72 1 4095 72 0 2 0 2 8 2 1 2028 10180 0 [txt] <:s> For each vehicle I wrote V v + v + v + v + v + v + v + v + v + v + v + v + v + v + v + v + v + v + v + v > [frm] 8 66048 1457 2462 10782 4565 6 1 3 5 0 0 0 0 0 12779519 0 16777215 145 [frmlay] 4565 9325 1 144 0 1 2462 144 0 2 0 1 1 0 1 1601 10638 0 [txt] > [frm] 18 66048 1437 8558 10803 13381 1 1 3 3 0 0 0 0 0 12779519 0 16777215 231 [frmlay] 13381 9366 1 216 144 1 8702 216 0 2 0 1 1 0 1 1653 10592 0 [txt] <+@> <+@><:S+-1><:f,2Times New Roman,>The explorations at the end of this paper are taken from <+@><:S+-1><:f,2Times New Roman,> <+B><:S+-1><-!><+"><+!><:f,2Times New Roman,>Discovering Mathematics with a TI-92 Beginning Algebra<-!> <-"> <+B><:S+-1><:f,2Times New Roman,>and <+B><:S+-1><+!><+"><:f,2Times New Roman,>Discovering Mathematics with a TI-92 Functions<-!><-"> <+@><:S+-1><:f,2Times New Roman,> <+B><:S+-1><:f,2Times New Roman,>by Lin McMullin. <+@><:S+-1><:f,2Times New Roman,> <+@><:S+-1><:f,2Times New Roman,>Published by D & S Marketing Systems, Inc. 1325 East 17th Street Brooklyn, New York 11230 Phone: 800 - 633 - 8383. D & S Marketing System, 1996 <+@><:S+-1><:f,2Times New Roman,> <+@><:S+-1><:f,2Times New Roman,>For face-to face teaching purposes, classroom teachers my reproduce these explorations.<:f> <-!><-!> > [frm] 4 537395328 7861 1440 8121 1775 0 1 3 0 0 0 0 0 0 12779519 0 16777215 136 0 6421 260 236 [frmname] Frame136 [frmlay] 1775 260 1 0 0 1 1440 0 0 2 0 1 1 0 1 7861 8121 0 [isd] .X136 .tex 1 1 0 0 260 65201 100 0 0 .tex 0 65300 0 [frm] 2 537395392 5808 1440 7184 1775 0 1 3 0 0 0 0 0 0 0 0 16777215 27 1 4368 1376 236 [frmname] Frame3 [frmlay] 1775 1376 1 0 0 1 1440 0 0 2 0 27 0 512 1 5808 7040 0 [isd] .X27 .tex 1 1 0 0 1376 65201 100 0 0 .tex 0 65300 0 [frm] 1 537395328 5688 9319 8408 9607 0 1 3 0 0 0 0 0 0 0 0 16777215 9 2 4248 2720 206 [frmname] Frame9 [frmlay] 9607 2720 1 0 0 1 9319 0 0 2 0 2 8 2 1 5688 8408 0 [isd] .X9 .tex 1 1 0 0 2720 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5688 10183 6656 10471 0 1 3 0 0 0 0 0 0 0 0 16777215 10 3 4248 968 206 [frmname] Frame10 [frmlay] 10471 968 1 0 0 1 10183 0 0 2 0 1 1 0 1 5688 6656 0 [isd] .X10 .tex 1 1 0 0 968 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5688 11047 7382 11335 0 1 3 0 0 0 0 0 0 0 0 16777215 11 4 4248 1694 206 [frmname] Frame11 [frmlay] 11335 1694 1 0 0 1 11047 0 0 2 0 2 8 2 1 5688 7382 0 [isd] .X11 .tex 1 1 0 0 1694 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5688 11911 10232 12199 0 1 3 0 0 0 0 0 0 0 0 16777215 12 5 4248 4544 206 [frmname] Frame12 [frmlay] 12199 4544 1 0 0 1 11911 0 0 2 0 57 126 22 1 5688 10232 0 [isd] .X12 .tex 1 1 0 0 4544 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5688 12775 7040 13063 0 1 3 0 0 0 0 0 0 0 0 16777215 13 6 4248 1352 206 [frmname] Frame13 [frmlay] 13063 1352 1 0 0 1 12775 0 0 2 0 125 38 26 1 5688 7040 0 [isd] .X13 .tex 1 1 0 0 1352 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 11115 7310 11403 0 1 3 0 0 0 0 0 0 0 0 16777215 14 7 4386 1484 206 [frmname] Frame14 [frmlay] 11403 1484 1 0 0 1 11115 0 0 2 0 26 191 30 1 5826 7310 0 [isd] .X14 .tex 1 1 0 0 1484 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 11979 7538 12267 0 1 3 0 0 0 0 0 0 0 0 16777215 15 8 4386 1712 206 [frmname] Frame15 [frmlay] 12267 1712 1 0 0 1 11979 0 0 2 0 114 32 28261 1 5826 7538 0 [isd] .X15 .tex 1 1 0 0 1712 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 12843 7196 13131 0 1 3 0 0 0 0 0 0 0 0 16777215 16 9 4386 1370 206 [frmname] Frame16 [frmlay] 13131 1370 1 0 0 1 12843 0 0 2 0 0 0 0 1 5826 7196 0 [isd] .X16 .tex 1 1 0 0 1370 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 13707 7742 13995 0 1 3 0 0 0 0 0 0 0 0 16777215 17 10 4386 1916 206 [frmname] Frame17 [frmlay] 13995 1916 1 0 0 1 13707 0 0 2 0 0 0 0 1 5826 7742 0 [isd] .X17 .tex 1 1 0 0 1916 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 2351 7316 2639 0 1 3 0 0 0 0 0 0 0 0 16777215 18 11 4386 1490 206 [frmname] Frame18 [frmlay] 2639 1490 1 0 0 1 2351 0 0 2 0 0 0 0 1 5826 7316 0 [isd] .X18 .tex 1 1 0 0 1490 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5826 5135 6734 5423 0 1 3 0 0 0 0 0 0 0 0 16777215 19 12 4386 908 206 [frmname] Frame19 [frmlay] 5423 908 1 0 0 1 5135 0 0 2 0 0 0 0 1 5826 6734 0 [isd] .X19 .tex 1 1 0 0 908 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5765 5999 7019 6287 0 1 3 0 0 0 0 0 0 0 0 16777215 20 13 4325 1254 206 [frmname] Frame20 [frmlay] 6287 1254 1 0 0 1 5999 0 0 2 0 0 0 0 1 5765 7019 0 [isd] .X20 .tex 1 1 0 0 1254 65248 100 0 0 .tex 0 65330 0 [frm] 4 537395328 5826 6863 6706 7151 0 1 3 0 0 0 0 0 0 0 0 16777215 21 14 4386 880 206 [frmname] Frame21 [frmlay] 7151 880 1 0 0 1 6863 0 0 2 0 0 0 0 1 5826 6706 0 [isd] .X21 .tex 1 1 0 0 880 65248 100 0 0 .tex 0 65330 0 [frm] 4 537395328 5826 7727 6706 8015 0 1 3 0 0 0 0 0 0 0 0 16777215 22 15 4386 880 206 [frmname] Frame22 [frmlay] 8015 880 1 0 0 1 7727 0 0 2 0 0 0 0 1 5826 6706 0 [isd] .X22 .tex 1 1 0 0 880 65248 100 0 0 .tex 0 65330 0 [frm] 4 537395328 5822 8591 6582 8879 0 1 3 0 0 0 0 0 0 0 0 16777215 23 16 4382 760 206 [frmname] Frame23 [frmlay] 8879 760 1 0 0 1 8591 0 0 2 0 51 0 204 1 5822 6582 0 [isd] .X23 .tex 1 1 0 0 760 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 7050 10766 7988 11054 0 1 3 0 0 0 0 0 0 0 0 16777215 24 17 5610 938 206 [frmname] Frame24 [frmlay] 11054 938 1 0 0 1 10766 0 0 2 0 51 153 102 1 7050 7988 0 [isd] .X24 .tex 1 1 0 0 938 65248 100 0 0 .tex 0 65330 0 [frm] 4 537395392 5769 6796 8481 7265 0 1 3 0 0 0 0 0 0 0 0 16777215 230 18 4329 2712 298 [frmname] Frame22 [frmlay] 7265 2712 1 0 0 1 6796 0 0 2 0 10 9 12553 1 5769 8337 0 [isd] .X230 .tex 1 1 0 0 2712 65062 100 0 0 .tex 0 65238 0 [frm] 4 537395392 5769 5912 9153 6196 0 1 3 0 0 0 0 0 0 0 0 16777215 229 19 4329 3384 226 [frmname] Frame19 [frmlay] 6196 3384 1 0 0 1 5912 0 0 2 0 48 0 161 1 5769 9009 0 [isd] .X229 .tex 1 1 0 0 3384 65248 100 0 0 .tex 0 65310 0 [frm] 4 537395392 5769 5028 7671 5312 0 1 3 0 0 0 0 0 0 0 0 16777215 228 20 4329 1902 226 [frmname] Frame18 [frmlay] 5312 1902 1 0 0 1 5028 0 0 2 0 144 36 240 1 5769 7527 0 [isd] .X228 .tex 1 1 0 0 1902 65248 100 0 0 .tex 0 65310 0 [frm] 4 537395392 5769 4144 7911 4428 0 1 3 0 0 0 0 0 0 0 0 16777215 227 21 4329 2142 226 [frmname] Frame17 [frmlay] 4428 2142 1 0 0 1 4144 0 0 2 0 255 255 0 1 5769 7767 0 [isd] .X227 .tex 1 1 0 0 2142 65248 100 0 0 .tex 0 65310 0 [frm] 4 537395392 5769 3260 7015 3544 0 1 3 0 0 0 0 0 0 0 0 16777215 226 22 4329 1246 226 [frmname] Frame16 [frmlay] 3544 1246 1 0 0 1 3260 0 0 2 0 0 0 0 1 5769 6871 0 [isd] .X226 .tex 1 1 0 0 1246 65248 100 0 0 .tex 0 65310 0 [frm] 4 537395392 5769 2376 7005 2660 0 1 3 0 0 0 0 0 0 0 0 16777215 225 23 4329 1236 226 [frmname] Frame15 [frmlay] 2660 1236 1 0 0 1 2376 0 0 2 0 0 0 0 1 5769 6861 0 [isd] .X225 .tex 1 1 0 0 1236 65248 100 0 0 .tex 0 65310 0 [frm] 4 537395392 5764 12963 7570 13247 0 1 3 0 0 0 0 0 0 0 0 16777215 224 24 4324 1806 226 [frmname] Frame14 [frmlay] 13247 1806 1 0 0 1 12963 0 0 2 0 10 9 13833 1 5764 7426 0 [isd] .X224 .tex 1 1 0 0 1806 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 12103 7976 12387 0 1 3 0 0 0 0 0 0 0 0 16777215 223 25 4248 2288 226 [frmname] Frame13 [frmlay] 12387 2288 1 0 0 1 12103 0 0 2 0 9 9 14386 1 5688 7832 0 [isd] .X223 .tex 1 1 0 0 2288 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 11243 7494 11527 0 1 3 0 0 0 0 0 0 0 0 16777215 222 26 4248 1806 226 [frmname] Frame12 [frmlay] 11527 1806 1 0 0 1 11243 0 0 2 0 49 52 12340 1 5688 7350 0 [isd] .X222 .tex 1 1 0 0 1806 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 10383 6520 10667 0 1 3 0 0 0 0 0 0 0 0 16777215 221 27 4248 832 226 [frmname] Frame11 [frmlay] 10667 832 1 0 0 1 10383 0 0 2 0 1 0 2 1 5688 6376 0 [isd] .X221 .tex 1 1 0 0 832 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 9523 6520 9807 0 1 3 0 0 0 0 0 0 0 0 16777215 220 28 4248 832 226 [frmname] Frame10 [frmlay] 9807 832 1 0 0 1 9523 0 0 2 0 1 0 2 1 5688 6376 0 [isd] .X220 .tex 1 1 0 0 832 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 6749 7022 7033 0 1 3 0 0 0 0 0 0 0 0 16777215 219 29 4248 1334 226 [frmname] Frame9 [frmlay] 7033 1334 1 0 0 1 6749 0 0 2 0 1 1 0 1 5688 6878 0 [isd] .X219 .tex 1 1 0 0 1334 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 5889 7032 6173 0 1 3 0 0 0 0 0 0 0 0 16777215 218 30 4248 1344 226 [frmname] Frame8 [frmlay] 6173 1344 1 0 0 1 5889 0 0 2 0 1 1 0 1 5688 6888 0 [isd] .X218 .tex 1 1 0 0 1344 65248 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 4995 6512 5313 0 1 3 0 0 0 0 0 0 0 0 16777215 217 31 4248 824 226 [frmname] Frame7 [frmlay] 5313 824 1 0 0 1 4995 0 0 2 0 1 1 0 1 5688 6368 0 [isd] .X217 .tex 1 1 0 0 824 65201 100 0 0 .tex 0 65310 0 [frm] 3 537395392 5688 4135 6298 4419 0 1 3 0 0 0 0 0 0 0 0 16777215 216 32 4248 610 226 [frmname] Frame5 [frmlay] 4419 610 1 0 0 1 4135 0 0 2 0 1 1 0 1 5688 6154 0 [isd] .X216 .tex 1 1 0 0 610 65248 100 0 0 .tex 0 65310 0 [frm] 1 721408 1440 2305 10800 5757 6 1 3 6 0 0 0 0 0 12779519 4194434 16777215 215 33 0 9360 3452 [frmlay] 5757 9360 1 288 144 1 2449 288 0 2 0 1 1 0 1 1728 10512 0 [txt] <:s><:f360,BBrushScript,>Warm-up: <:f200,,>Solve this equation and explain in words the <:f><:f200,,>steps<:f><:f200,,> you do as you solve it:<:f> <:s>4x + 2(x-3) + 7 = 3x - (x-2) +1 S<:f200,,>teps:<:f> > [frm] 3 537395328 5910 7841 7650 8147 0 1 3 0 0 0 0 0 0 12779519 0 16777215 214 34 4470 1740 248 [frmname] Frame8 [frmlay] 8147 1740 1 0 0 1 7841 0 0 2 0 1 1 0 1 5910 7650 0 [isd] .X214 .tex 1 1 0 0 1740 65204 100 0 0 .tex 0 65288 0 [frm] 2 537395392 5688 7609 6830 7915 0 1 3 0 0 0 0 0 0 12779519 0 16777215 213 35 4248 1142 248 [frmname] Frame3 [frmlay] 7915 1142 1 0 0 1 7609 0 0 2 0 108 121 29286 1 5688 6830 0 [isd] .X213 .tex 1 1 0 0 1142 65204 100 0 0 .tex 0 65288 0 [frm] 2 537395392 5764 13847 7248 14153 0 1 3 0 0 0 0 0 0 12779519 0 16777215 212 36 4324 1484 248 [frmname] Frame4 [frmlay] 14153 1484 1 0 0 1 13847 0 0 2 0 9 9 8241 1 5764 7248 0 [isd] .X212 .tex 1 1 0 0 1484 65204 100 0 0 .tex 0 65288 0 [frm] 2 537526272 5760 5184 10800 8064 0 1 3 2 0 0 0 0 0 12779519 0 16777215 211 37 4320 5040 2880 [frmname] Frame11 [frmlay] 8064 5040 1 0 0 1 5184 0 0 2 0 1 1 0 1 5760 10401 0 [isd] .X211 .bmp 18 1 0 0 5000 62832 100 0 48 4 3 0 0 16 18447 24 18447 32 18447 40 18447 48 18447 56 18447 7638 61404 0 62 0 0 0 0 0 0 46 0 8 18447 8 18439 24 18439 32 18439 40 18439 16 18439 48 18439 0 0 0 0 0 0 .bmp 0 0 0 [frm] 2 537526336 5760 9792 10800 12903 0 1 1 1 0 0 0 0 0 0 0 16777215 210 38 4320 5040 1530 [frmname] Frame12 [frmlay] 12903 5040 1 0 0 1 9792 0 0 2 0 1 1 0 1 5760 9720 0 [isd] .X210 .bmp 24 1 0 0 5011 62829 100 0 48 4 3 0 0 16 18447 24 18447 32 18447 40 18447 48 18447 56 18447 7638 61404 0 62 0 65521 65388 0 0 0 47 0 8 18447 8 18439 24 18439 32 18439 40 18439 16 18439 48 18439 0 0 0 0 0 0 .bmp 65521 65388 0 [frm] 3 537395328 6019 3216 6167 3504 0 1 3 0 0 0 0 0 0 12779519 0 16777215 209 39 4579 148 206 [frmname] Frame30 [frmlay] 3504 148 1 0 0 1 3216 0 0 2 0 1 1 0 1 6019 6167 0 [isd] .X209 .tex 1 1 0 0 148 65248 100 0 0 .tex 0 65330 0 [frm] 5 537395328 8270 8171 8432 8459 0 1 3 0 0 0 0 0 0 12779519 0 16777215 208 40 6830 162 206 [frmname] Frame31 [frmlay] 8459 162 1 0 0 1 8171 0 0 2 0 20 0 65535 1 8270 8432 0 [isd] .X208 .tex 1 1 0 0 162 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 4924 2829 6354 3117 0 1 3 0 0 0 0 0 0 0 0 16777215 207 41 3484 1430 206 [frmname] Frame5 [frmlay] 3117 1430 1 0 0 1 2829 0 0 2 0 1 1 0 1 4924 6210 0 [isd] .X207 .tex 1 1 0 0 1430 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 4559 3593 5891 3886 0 1 3 0 0 0 0 0 0 0 0 16777215 206 42 3119 1332 211 [frmname] Frame6 [frmlay] 3886 1332 1 0 0 1 3593 0 0 2 0 1 1 0 1 4559 5747 0 [isd] .X206 .tex 1 1 0 0 1332 65243 100 0 0 .tex 0 65325 0 [frm] 1 537395328 5755 9293 7185 9581 0 1 3 0 0 0 0 0 0 0 0 16777215 205 43 4315 1430 206 [frmname] Frame7 [frmlay] 9581 1430 1 0 0 1 9293 0 0 2 0 1 1 0 1 5755 7041 0 [isd] .X205 .tex 1 1 0 0 1430 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5750 10157 7420 10445 0 1 3 0 0 0 0 0 0 0 0 16777215 204 44 4310 1670 206 [frmname] Frame8 [frmlay] 10445 1670 1 0 0 1 10157 0 0 2 0 1 1 0 1 5750 7276 0 [isd] .X204 .tex 1 1 0 0 1670 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5755 11021 6817 11314 0 1 3 0 0 0 0 0 0 0 0 16777215 203 45 4315 1062 211 [frmname] Frame9 [frmlay] 11314 1062 1 0 0 1 11021 0 0 2 0 2 8 2 1 5755 6673 0 [isd] .X203 .tex 1 1 0 0 1062 65243 100 0 0 .tex 0 65325 0 [frm] 1 537395328 5755 11890 7185 12178 0 1 3 0 0 0 0 0 0 0 0 16777215 202 46 4315 1430 206 [frmname] Frame10 [frmlay] 12178 1430 1 0 0 1 11890 0 0 2 0 141 134 29742 1 5755 7041 0 [isd] .X202 .tex 1 1 0 0 1430 65248 100 0 0 .tex 0 65330 0 [frm] 1 537395328 5755 12754 8027 13047 0 1 3 0 0 0 0 0 0 0 0 16777215 201 47 4315 2272 211 [frmname] Frame11 [frmlay] 13047 2272 1 0 0 1 12754 0 0 2 0 0 0 76 1 5755 7883 0 [isd] .X201 .tex 1 1 0 0 2272 65243 100 0 0 .tex 0 65325 0 [frm] 2 537395328 5813 2916 7145 3209 0 1 3 0 0 0 0 0 0 0 0 16777215 200 48 4373 1332 211 [frmname] Frame12 [frmlay] 3209 1332 1 0 0 1 2916 0 0 2 0 120 0 0 1 5813 7001 0 [isd] .X200 .tex 1 1 0 0 1332 65243 100 0 0 .tex 0 65325 0 [frm] 2 537395328 5813 3785 7025 4078 0 1 3 0 0 0 0 0 0 0 0 16777215 199 49 4373 1212 211 [frmname] Frame13 [frmlay] 4078 1212 1 0 0 1 3785 0 0 2 0 146 7 0 1 5813 6881 0 [isd] .X199 .tex 1 1 0 0 1212 65243 100 0 0 .tex 0 65325 0 [frm] 2 537395328 5813 4654 6697 4947 0 1 3 0 0 0 0 0 0 0 0 16777215 198 50 4373 884 211 [frmname] Frame14 [frmlay] 4947 884 1 0 0 1 4654 0 0 2 0 0 0 0 1 5813 6553 0 [isd] .X198 .tex 1 1 0 0 884 65243 100 0 0 .tex 0 65325 0 [frm] 2 537395328 5769 5523 6533 5816 0 1 3 0 0 0 0 0 0 0 0 16777215 197 51 4329 764 211 [frmname] Frame15 [frmlay] 5816 764 1 0 0 1 5523 0 0 2 0 112 0 0 1 5769 6389 0 [isd] .X197 .tex 1 1 0 0 764 65243 100 0 0 .tex 0 65325 0 [frm] 2 537395328 5769 6392 8553 6685 0 1 3 0 0 0 0 0 0 0 0 16777215 196 52 4329 2784 211 [frmname] Frame16 [frmlay] 6685 2784 1 0 0 1 6392 0 0 2 0 0 0 65535 1 5769 8409 0 [isd] .X196 .tex 1 1 0 0 2784 65243 100 0 0 .tex 0 65325 0 [frm] 3 537395328 5400 8234 8146 8527 0 1 3 0 0 0 0 0 0 0 0 16777215 195 53 3960 2746 211 [frmname] Frame20 [frmlay] 8527 2746 1 0 0 1 8234 0 0 2 0 1 1 0 1 5400 8002 0 [isd] .X195 .tex 1 1 0 0 2746 65243 100 0 0 .tex 0 65325 0 [frm] 3 537395328 5400 9055 8232 9343 0 1 3 0 0 0 0 0 0 0 0 16777215 194 54 3960 2832 206 [frmname] Frame21 [frmlay] 9343 2832 1 0 0 1 9055 0 0 2 0 1 1 0 1 5400 8088 0 [isd] .X194 .tex 1 1 0 0 2832 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5400 10591 7030 10879 0 1 3 0 0 0 0 0 0 0 0 16777215 193 55 3960 1630 206 [frmname] Frame22 [frmlay] 10879 1630 1 0 0 1 10591 0 0 2 0 1 1 0 1 5400 6886 0 [isd] .X193 .tex 1 1 0 0 1630 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5400 11023 7030 11311 0 1 3 0 0 0 0 0 0 0 0 16777215 192 56 3960 1630 206 [frmname] Frame23 [frmlay] 11311 1630 1 0 0 1 11023 0 0 2 0 2 8 2 1 5400 6886 0 [isd] .X192 .tex 1 1 0 0 1630 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5400 11455 7160 11743 0 1 3 0 0 0 0 0 0 0 0 16777215 191 57 3960 1760 206 [frmname] Frame24 [frmlay] 11743 1760 1 0 0 1 11455 0 0 2 0 0 0 65535 1 5400 7016 0 [isd] .X191 .tex 1 1 0 0 1760 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5400 11887 7160 12175 0 1 3 0 0 0 0 0 0 0 0 16777215 190 58 3960 1760 206 [frmname] Frame25 [frmlay] 12175 1760 1 0 0 1 11887 0 0 2 0 121 32 26963 1 5400 7016 0 [isd] .X190 .tex 1 1 0 0 1760 65248 100 0 0 .tex 0 65330 0 [frm] 3 537395328 5400 12607 8232 12895 0 1 3 0 0 0 0 0 0 0 0 16777215 189 59 3960 2832 206 [frmname] Frame26 [frmlay] 12895 2832 1 0 0 1 12607 0 0 2 0 101 110 8308 1 5400 8088 0 [isd] .X189 .tex 1 1 0 0 2832 65248 100 0 0 .tex 0 65330 0 [frm] 4 537395328 5755 2887 7145 3180 0 1 3 0 0 0 0 0 0 0 0 16777215 188 60 4315 1390 211 [frmname] Frame27 [frmlay] 3180 1390 1 0 0 1 2887 0 0 2 0 110 32 26982 1 5755 7001 0 [isd] .X188 .tex 1 1 0 0 1390 65243 100 0 0 .tex 0 65325 0 [frm] 4 537395328 5755 3756 7265 4049 0 1 3 0 0 0 0 0 0 0 0 16777215 187 61 4315 1510 211 [frmname] Frame28 [frmlay] 4049 1510 1 0 0 1 3756 0 0 2 0 111 32 26740 1 5755 7121 0 [isd] .X187 .tex 1 1 0 0 1510 65243 100 0 0 .tex 0 65325 0 [frm] 4 537395328 5755 4625 8741 4918 0 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0 16777215 148 100 4262 780 226 [frmname] Frame165 [frmlay] 14252 780 1 0 0 1 13893 0 0 2 0 0 0 6656 1 5702 6194 0 [isd] .X148 .tex 1 1 0 0 780 65201 100 0 0 .tex 0 65310 0 [frm] 26 537395328 9853 2671 10501 2955 0 1 3 0 0 0 0 0 0 12779519 0 16777215 147 101 8413 648 226 [frmname] Frame166 [frmlay] 2955 648 1 0 0 1 2671 0 0 2 0 240 0 0 1 9853 10213 0 [isd] .X147 .tex 1 1 0 0 648 65248 100 0 0 .tex 0 65310 0 [frm] 26 537395328 7289 2964 8129 3248 0 1 3 0 0 0 0 0 0 12779519 0 16777215 146 102 5849 840 226 [frmname] Frame167 [frmlay] 3248 840 1 0 0 1 2964 0 0 2 0 70 105 27758 1 7289 7841 0 [isd] .X146 .tex 1 1 0 0 840 65248 100 0 0 .tex 0 65310 0 [lay] Standard 513 [rght] 15840 12240 1 1440 1440 1 1440 1440 0 1 0 1 0 2 1 1440 10800 12 1 720 1 1440 1 2160 1 2880 1 3600 1 4320 1 5040 1 5760 1 6480 1 7200 1 7920 1 8640 [hrght] [lyfrm] 1 11200 0 0 12240 1440 0 1 3 1 0 0 0 0 0 0 0 0 1 [frmlay] 1440 12240 1 1440 72 1 792 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] @Header@Discovering Mathematics on the TI-92 > [frght] [lyfrm] 1 13248 0 14400 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14472 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] @Footer@<+@> <+!><:P11,6,><-!> > [elay] [l1] 0 [pg] 18 14 53 50 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 32 0 7 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 51 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 69 0 11 0 0 1 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 84 0 2 0 1 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 102 0 7 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 120 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 141 59 14 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 162 0 0 0 5 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 180 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 198 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 212 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 231 105 10 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 244 300 52 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 261 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 278 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 297 0 17 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 308 0 50 1025 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 [edoc] @Subhead@<+A><:s>Exploration 1: Adding the Quick Way <+A><:s> @Body Single@<+@><:s> @Body Single@<+@><:s> @Body Single@<+@><:s>In this section you will learn how to simplify addition and subtraction expressions by exploring how the TI-92 does the simplification. @Body Single@<+@><:s> @Body Single@<+@>Type the following on the entry line and push<+!> <[>ENTER]<-!> . Record the TI-92s answers below and look for a pattern: @Body Single@<+@><:s> @Practice@<+@><:s><:A2> @Practice@<+@><:s><:A3> @Practice@<+@><:s><:A4> @Practice@<+@><:s><:A5> @Practice@<+@><:s><:A6> @Practice@<+@><:s>If you see a pattern go on to the next questions, if not make up some similar problems of your own and check the answers on the TI-92. Record your work and answers below. @Practice@<+@><:s><:#864,9360>Complete the sentence: <+!>When adding a bunch of variables which are all the same the answer can be found by ...<-!> <+A><:s><:#447,9360> <+A><:s><:#447,9360> <+A><:s><:#447,9360> @Practice@<:s><:#576,9360>Explain in your own words what an expressions like <+!>5x, <-!><+!>3b or 8y <-!> mean. @Body Single@<:s><:#240,9360> <:s><:#447,9360> <:s><:#447,9360> @Practice@<:s><:#576,9360>Complete this sentence: <+!>Multiplying a number by a variable is the same as adding ...<-!> @Body Single@<:s><:#240,9360><+!> @Body Single@<:s><:#240,9360><+!> @Body Single@<:s><:#240,9360><+!> @Body Single@<:s><:#480,9360>Try the following on the TI-92 and look for a pattern. They are similar to your answers above: @Body Single@<:s><:#240,9360> @Practice@<:s> <:A7> @Practice@<:s> <:A8> @Practice@<:s> <:A9> @Practice@<:s> <:A10> @Practice@<:s> <:A1> enter <:A0> as (1 / 2) a @Practice@<:s> <:A11> @Practice@<:s><:#576,9360> Express the pattern in problems 10 - 15 in your own words. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360>Here are some subtraction problems. They have a similar pattern @Practice@<:s> <:A12> @Practice@<:s> <:A13> @Practice@<:s> <:A14>. @Practice@<:s> <:A15> @Practice@<:s> <:A16> @Practice@<:s><:#288,9360> Explain how the pattern changes with subtraction. <:s><:#447,9360> @Practice@<:s> Explain why <:A17> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> @Subhead@<:s><:#346,9360>Exploration 1: Step-by-step <:s><:#447,9360> <+A><:s><:A33> <:s><:#447,9360> <:s><:#1593,9360><:f200,,>As you may expect the TI-92 can solve equations very easily. The purpose of this exploration is to help you understand <+!><+">how<-"><-!> linear <:f><:f200,,>equations are solved. You have already learned the steps in solving <:f><:f200,,> a linear equation in one variable<:f><:f200,,> like the one above,<:f><:f200,,>. T<:f><:f200,,>his exploration will give you a chance to practice what you have learned on some very complicated equations, without concerning yourself with the ar ithmetic. <:f> @Body Single@<:s><:#240,9360><-"> @Body Single@<:s><:#480,9360>Your steps in solving the equation above probably went something like this: <:s><:#439,9360><+!>Procedure<-!> @Body Single@<:s><:#480,9360><+#><+!>Step 1<-#>: <-!>On each side of the equal sign: remove parentheses and collect terms @Body Single@<:s><:#240,9360>To do this on the TI-92 type<+!>: @Body Single@<:s><:#240,9360><+!> expand( <-!>4x + 2(x-3) + 7 = 3x - (x+2) +1<+!>) <-!><-"> @Body Single@<:s><:#480,9360>and push <+!><[>Enter]<-!>. The <+!>expand(<-!> operation removes parentheses and the TI-92 automatically collects like terms. @Body Single@<:s><:#240,9360>You should see the answer @Body Single@<:s><:#240,9360> 6x + 1 = 2x -1 @Body Single@<:s><:#480,9360><+#><+!>Step 2<-#>:<-!> Next subtract 2x from both sides to get all the terms with the variable on one side (subtracting 6x also works) @Body Single@<:s><:#240,9360>To do this on the TI-92: @Body Single@<:#720,9360>Push the subtraction key, then 2 x. The entry line will show ans(1)-2x indicating that 2x will be subtracted from the last answer. Push <+!><[>Enter]<-!> and you will see @Body Single@<:s><:#240,9360> 4x +1 = -1 @Body Single@<:s><:#240,9360><+#><+!>Step 3<-#>: <-!>Subtract 1 from both sides To do this on the TI-92: @Body Single@<:#480,9360>Push the subtraction key then 1 (you will see ans(1) - 1 on the entry line) Push<+!> <[>Enter] <-!>the result is @Body Single@<:s><:#240,9360> 4x = - 2 @Body Single@<:s><:#240,9360><+#><+!>Step 4<-#> <-!> Divide both sides by 4. To do this on the TI-92: @Body Single@Push <+!> <[> <:A39>] <-!>4 (You will see ans(1) / 4 on the entry line<:f,2Symbol,>)<:f> @Body Single@<:s><:#240,9360>Push<+!> <:f,2Symbol,> <:f><-!><+!><[><-!><+!>Enter]<-!> you will see @Body Single@<:s><:#240,9360> x = -1/2 the answer. @Body Single@<:s><:#240,9360><+#> @Body Single@<:s><:#240,9360>The screen looks like this when the problem is done: @Body Single@<:s><:A37> @Body Single@<:s><:#1440,9360>You can even check your answer on the TI-92. Retrieve the original equation type the<+!> <-!><+!>|<-!> (with bar) after it. Then retrieve the answer and Push<+!> <[>Enter]<-!>. This substitutes -1/2 for the variable and determines if the resulting equation (now all number) is true or false. Since it is true the TI-92 returns True. The screen will look like this: @Body Single@<:s><:A38> <:s><:#439,9360><+!> <:s><:#439,9360><+!> <:s><:#439,9360><+!> <:s><:#439,9360><+!>Practice<-!> @Body Single@<:s><:#960,9360>Here are some equations for you to practice on. <-">Often you will start on the second, third or fourth step or skip steps depending on the equation. The steps are the same as you learned to do with paper and pencil; the TI-92 does the work. @Body Single@<:s><:#240,9360>Solve these equation by doing the appropriate steps on the TI-92: @Body Single@<:s><:#240,9360>Group 1 - one step equations (start at step 4 by dividing): @Body Single@<:s><:#240,9360> @Practice@<:s><:A32> @Practice@<:s><:A31> @Practice@<:s><:A30> @Practice@<:s><:A29> @Practice@<:s><:A35> @Body Single@<:s><:#480,9360>Group 2 - one step equations (Start at step 3 by adding or subtracting): @Body Single@<:s><:#240,9360> @Practice@<:s><:A28> @Practice@<:s><:A27> @Practice@<:s><:A26> @Practice@<:s><:A25> @Practice@<:s> <:A24> @Practice@<:s> <:A36> @Body Single@<:s><:#480,9360>Group 3 Equations requiring several steps (Start at step 1, use <+!>expand( <-!>): @Body Single@<:s><:#240,9360> @Practice@<:s> <:A23> @Practice@<:s> <:A22> @Practice@<:s> <:A21> @Practice@<:s> <:A20> @Practice@<:s> <:A19> @Practice@<:s> <:A18> @Practice@<:s> <:A34> (Hint: enter the equation and then square both side using <+!><[> <:A40> ]<-!> 2 @Practice@<:s><:#864,9360> Make up some complicated equations. Try them out on your friends see if they can do them without a calculator. <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> @Subhead@<:s><:#389,9360>Exploration 4: The <+!><:f,BHelvetica,>Expand(<-!><:f,BBrushScript,> <:f,BHelvetica,> <:f>and<:f,BHelvetica,> <:f><:f,BBrushScript,> <:f,BHelvetica,>Factor<:f>( Operations <:s><:#447,9360> <:s><:#447,9360> <:s> Write <+">without<-"> parentheses: <:A41> <:s><:#447,9360> <:s> Write <+">with<-"> parentheses: <:A42> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:#1788,9360>In this exploration you will explore the relationship between two of the TI-92s built in operations: <+!>factor(<-!> and <+!>expand(<-!> .With these two utilities you will investigate the relationship between the multiplied form of two or more binomials and the factored form of the same expression. <:s><:#447,9360> <:#439,9360><+!>How to <-!><+!>unexpand<-!> @Body Single@<:#480,9360>On the Home Screen use <+!>expand<-!>( to multiply the binomials below. Then use <+!> factor( <-!>to unexpand your answer. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s> <:A43> @Practice@<:s> <:A44> @Practice@<:s> <:A45> @Practice@<:s> <:A46> @Practice@<:s> <:A47> @Practice@<:s><:f200,,> <:f> Was the factored answers 5 what you expected? What happened? <:s><:#468,9360><+!>Using <+">factor(<-!><-"> @Body Single@<:s><:#480,9360>Do these the other way: first use <+!>factor( <-!>, write your answer and then <+!>expand<-!><+!>(<-!> the answer: @Body Single@<:s><:#240,9360> @Practice@<:s> <:A48> @Practice@<:s> <:A49> @Practice@<:s> <:A50> @Practice@<:s> <:A51> @Practice@<:s> <:A52> @Practice@<:s><:#1152,9360> Polynomials may be written in an expanded or factored form. Write a brief paragraph explaining the relationship between the <+!>expand( <-!>and the <+!>factor(<-!> commands when used with polynomials. <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> @Header@<:s><:#240,9360> @Subhead@<+A><:s><:#346,9360><:f,BArial,>Exploration 5 Using factor( to Solve Equations<:f> <:s><:#447,9360> <:#2682,9360>One important use of the factored form of polynomials is to help understand the solution of polynomial equations. This investigation will show you how factoring is used to solve polynomial equations. The TI-92 has several built in ways of solving polynomial equations which are explored in the next section. This exploration is to help you understand what a solution is rather than just how to find it. <:s><:#439,9360><+!>Why factor?<-!> @Body Single@<:s><:#480,9360>This rather simple idea about numbers is the basis for solving polynomial equations: @Body Single@<:s><:#480,9360><+!>If the product of two or more numbers is zero then one of the numbers<-!><+!> must be zero.<-!> @Body Single@<:s><:#960,9360>The TI-92 (or any other computer) cannot factor every polynomial, because there are polynomials which do not factor. Nevertheless it is easy to see where the solution of a polynomial equation comes from if we look at those which do factor: @Body Single@<:s><:#240,9360>Lets start with a large equation like @Body Single@<:s><:A53> @Body Single@<:s><:#240,9360>Use the TI-92 to factor the left side: @Body Single@<:s><:A54> @Body Single@<:s><:#960,9360>Since the four numbers on the left multiply to zero, one of the four must be zero. You can easily solve each of the resulting linear equations. You have traded in your big equation for four very simple ones. @Body Single@<:s><:A55> @Body Single@<:s><:A56> @Body Single@<:s><:A57> @Body Single@<:s><:A58> @Body Single@<:s>We see that <:A70> and only these numbers will make <:A59>, therefore together they must be the solution set of the original equation. Substitute the four numbers, one at a time, into the fourth degree polynomial see what the value is. @Body Single@<:s><:#240,9360> <:s><:#439,9360><+!>Practice<-!> @Body Single@<:s><:#480,9360>Use the TI-92 to solve these equation by factoring them. Write the factors and the solution set. @Body Single@<:s><:#240,9360> @Practice@<:s> <:A60> @Practice@<:s> <:A61> @Practice@<:s> <:A62> @Practice@<:s> <:A63> Begin by using the TI-92 to subtract <:A64> from each side so that the resulting expression will be equal to zero. Why is this necessary? @Practice@<:s> <:A65> @Practice@<:s> <:A66> @Body Single@<:s><:#720,9360>The next three examples are somewhat different than the first six. Make a list of any differences you can find in the equations themselves and in the steps on the way to the answer. @Practice@<:s> <:A67> @Practice@<:s> <:A68> @Practice@<:s> <:A69> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> @Title@<:s><:#432,9360>Chapter 1 @Subhead@<:s><:#346,9360>Exploration 3 Limits <:#3576,9360>The limit of a function is a simple but extremely important tool for analyzing functions. The limit of a function at a point is the number the y-values of a function are getting closer to as the x-value gets closer the specified point. Think of it as if you were walking along the graph approaching the place where the x-coordinate is some number: what y-value are you getting closer to? Thats the limit. What happens when x equals the number is never in question (the value either exist and you can find it or it is undefined), but often what happens near that value is interesting and important. The concept of limit and the associated vocabulary help describe these situations. <:s><:#447,9360> <:s><:#439,9360><+!>Finding Limits<-!> @Body Single@<:s><:#1680,9360>In this exploration you will be asked to investigate the limit of various functions. In each case you may use the TI-92 to find the limit for you. You will also look at the graph of the function in the neighborhood of the limit. As you will see each kind of limit means something different on the graph of the function. You also may need to use the table to look at values of parts of a functions (for example the numerator and denominator). <:s><:#439,9360><+!>The value of a functions<-!> @Body Single@<:s><:#720,9360>As you will see the value of a function may be very different from the limit of a functions at the same point. Often there will be a limit but <+">no<-"> value or a value but <+">no<-"> limit. @Body Single@Once you have defined a function in the Equation Editor or on the Home Screen you may find its value at <:A80> by typing y1(a) or f(a). @Body Single@<:#720,9360>The <+!>value<-!> operation found on the Graph Screen under <+!><[>F5 Math] <[>1:Value] <-!>may also be used to find values. The expression must be entered in the Equation Editor. @Body Single@<:s><:#960,9360>The <+!>value<-!> operation prompts you for an x value and returns the y-value of the selected function. The down up or down cursor may by used to find the values of any other functions which are graphed for the same <+">x<-"> value. @Body Single@<:s><:#240,9360>Values may also be found by setting up a table. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> <:s><:#439,9360><+!>The limit( operation<-!> @Body Single@<:s><:#240,9360>Limits may be found using the <+!>limit(<-!> operation. @Body Single@<:#480,9360>The <+!>limit(<-!> operation is found on the Home Screen under <+!><[> F3 Calc ] <[> 3:limit( ]<-!> or it may be typed on the entry line. @Body Single@<:s><:#240,9360>The syntax is @Body Single@<:s><:#240,9360><+!>limit(<-!> <+">expression <+!>,<-!> variable <+!>,<-!> number <-"><[><+!>,<-!> <+">direction<-">]<+"> <+!>)<-!><-"><-!><-!> @Body Single@<+@><:s><:#240,9360>The usual mathematical notation is @Body Single@<+@><:s> <:A79><+">expression<-"> @Body Single@<+@><:s> for example <:A78> @Body Single@<:s><:#720,9360>The <+!>limit(<-!> operation gives the number which the <+">expression<-"> approaches as the <+!>variable<-!> approaches the <+">number<-">. The <+">direction<-"> is used for <+">one-sided limit <-">and will be explained below. @Body Single@<:s><:#1200,9360>It is important to realize that the <+!>limit(<-!> operation tells you what y-value the coordinates on the graph are approaching as x approaches the <+">number<-">. What happens at the number may be very different from the limit. What happens when x = the <+">number <-"> is never in question: you may find it using <+!>value<-!>. @Body Single@<:s><:#240,9360>Use <+!>value<-!> and <+!>limit(<-!> to answer these questions. <:s><:#447,9360> @Practice@<:s> Find <:A77> and find the value of the expression when <+">x<-"> = 3. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@For the expression <:A76> find the value and the limit at <+">x<-"> = 2. Draw the examine the graph using ZoomDec with <+!>xres<-!> = 1. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s>For the expression <:A81> find the value and limit at x = -2. Draw and examine the graph. Explain what is happening. @Practice@<:s>For the expression <:A82> find the value and limit at x = -2. Draw and examine the graph. Explain what is happening. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s>For the expression <:A83> find the value and limit as <:A84>. Draw and examine the graph. Explain what is happening. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> <:s><:#439,9360><+!>One-sided limits<-!> @Body Single@<:s> <:A85> and <:A86> @Body Single@<:s><:#240,9360> @Body Single@<:s>Sometimes it is necessary to consider what happens as x approaches its value from one side at a time. If you approach x from the right side writing <:A75> the value it is as if you were walking <+">left<-"> on the graph approaching a from the right side. In the same way you may approach <:f240,2Times New Roman,><+">a<-"><:f> from the left side by writing <:A74>. The syntax is shown in the next problem using the fifth parameter for <[><+">direction<-">] . @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s> For the function <:A73> . <:A87> <:A88> <:A89> For the second and third limits you must enter the <+">direction<-"> number. The syntax is for the second limit is <+!>limit(<-!> 2 / (x - 2) , x, 2, 1) and for the third is <+!>limit(<-!> 2 / (x - 2) , x, 2, -1). The pretty print should look like the expressions written above. Any positive number may be used for from the right and any negative number for from the left. The <+">direction<-"> number does not have to be actually left or right of the <+">number<-">. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s><:#864,9360>Graph of the function in the last problem using Dot style. Explain what the three <+">limits<-"> tell you about the behavior of the <+">graph<-"> at and near <+">x<-"> = 4. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s><:#576,9360>Using the graph from 35, explain what the graph tells you about the three limits at <+">x<-"> = 4. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s>Define and graph <:A90> and examine the value, the limit, the two one-sided limits and the graph at <+">x = <-">0. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s>Define and graph <:A91>and examine the value, the limit, the two one-sided limits and the graph at <+">x = <-">5. <:s><:#447,9360> <:s><:#447,9360> @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> <:s><:#439,9360><+!>Infinite Limits and Undefined Limits<-!> @Body Single@<:s><:#720,9360>Limits may be equal to infinity this means that the y-value becomes infinitely large in absolute value at the value being considered. @Body Single@<:s><:#720,9360>Limits may also be undefined. In this case the two corresponding one-sided limits may be different as you saw in problem 38 above. @Body Single@<:s><:#240,9360>The next two problems explore these ideas: @Body Single@<:s><:#240,9360> @Practice@<:s> Examine the limit as <:A92> for this function. <:A72> Consider the graph and the one-sided limits. Write an explanation of what you learned. @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:#1440,9360>Make a table of values for the function in problem 39. Under <+!>TblSet<-!> choose <+!>Independent ASK<-!>. Use decimals approaching <+">x<-"> = 4 such as 3.9, 3.99, 3.999, 3.9999 and also 4.1, 4.01, 4.001. 4.0001. What do you observe? @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Body Single@<:s><:#240,9360> @Practice@<:s> Examine the limit as <:A93> for this function. <:A71> Consider the graph and the one-sided limits. Write an explanation of what you learned. How and why does this differ from 40. above? <:s><:#447,9360> <:s><:#447,9360> <:s><:#447,9360> <:s><:#439,9360><+!>Limits at Infinity @Body Single@<:s>You may also use the<+!> limit(<-!> operation to determine what happens as <+">x<-"> gets larger or smaller without bound. This is equivalent to asking what happens as you walk along the graph a long way to the right or left. These are called limits at infinity (as <:A101>) or at negative infinity (as <:A102>). @Body Single@<:s><:#480,9360>If the limit is finite the <+">y<-">-coordinates of the graph get closer to the limiting value. The graph flattens out as it goes right or left. @Body Single@<:s><:#480,9360>If these limits are infinite then the graph goes up (or down) as it goes off the screen to the left or right. @Body Single@<:s><:#480,9360>If the limit does not exist the function may go up and down forever as it moves left or right. @Body Single@<:s><:#240,9360>The problems below illustrate these situations. <:s><:#439,9360><+!>Practice @Body Single@<:s><:#240,9360>For each problem below @Body Single@<:s><:#240,9360>(a) Find the indicated limit @Body Single@<:s><:#240,9360>(b) Discuss what the <+">limit<-"> tells you about the <+">graph<-">. @Body Single@<:s><:#240,9360>(c) Discuss what the <+">graph<-"> tells you about the <+">limit<-">. @Body Single@<:s><:#240,9360> @Practice@<:s><:A94> @Practice@<:s><:A95> @Practice@<:s><:A96> @Practice@<:s><:A97> @Practice@<:s><:A98> @Practice@<:s><:A99> @Practice@<:A100><-!><-#><-!><-#><-!><-#> <+B> <+B> <+B><:f,2Times New Roman,>Lin McMullin <+B><:f,2Times New Roman,>Mathematics Department Representative <+B><:f,2Times New Roman,>Burnt Hills Ballston Lake High School <+B><:f,2Times New Roman,>88 Lake Hill Road <+B><:f,2Times New Roman,>Burnt Hills, New York 12027 <+B><:f,2Times New Roman,> <+B><:f,2Times New Roman,>School Phone 518 - 399 - 9141 (ext. 221) <+B><:f,2Times New Roman,><:f><:f,2Times New Roman,> <+B><:f,2Times New Roman,>e-mail; Lmcmullin@aol.com<:f> > Times New Roman,18,12,0,0,0,0,0 $\frac 12a$Times New Roman,18,12,0,0,0,0,0 $\frac 12a+\frac 13a+\frac 14a=$Times New Roman,18,12,0,0,0,0,0 $a+a+a+a+a+a+a+a=$Times New Roman,18,12,0,0,0,0,0 $x+x+x=$Times New Roman,18,12,0,0,0,0,0 $b+b+b+b+b=$Times New Roman,18,12,0,0,0,0,0 $w+w+w+w+w+w+w+w+w+w+w+w=$Times New Roman,18,12,0,0,0,0,0 $g+g+g+g=$Times New Roman,18,12,0,0,0,0,0 $3w+5w+2w=$Times New Roman,18,12,0,0,0,0,0 $6b+8b+4b+b=$Times New Roman,18,12,0,0,0,0,0 $5p+8p+4p=$Times New Roman,18,12,0,0,0,0,0 $2.3a+5.7a+3.1a=$Times New Roman,18,12,0,0,0,0,0 $5g+8g+15g=$Times New Roman,18,12,0,0,0,0,0 $6a-3a=$Times New Roman,18,12,0,0,0,0,0 $20m-12m=$Times New Roman,18,12,0,0,0,0,0 $6y-3y=$Times New Roman,18,12,0,0,0,0,0 $6y-2y=$Times New Roman,18,12,0,0,0,0,0 $6y-y=$Times New Roman,18,12,0,0,0,0,0 $6y-y\neq 6$Times New Roman,18,12,0,0,0,0,0 $$\sqrt{15}(x-12.6)=\frac 27x+\sqrt{12}x$$Times New Roman,18,12,0,0,0,0,0 $4.5x-(2.3x-12)=4(23.23+5.6x)$Times New Roman,18,12,0,0,0,0,0 $5.3(3.82x+5.3)=0$Times New Roman,18,12,0,0,0,0,0 $4.31(x-3.24)=6.943$Times New Roman,18,12,0,0,0,0,0 $-2(x-5)=7$Times New Roman,18,12,0,0,0,0,0 $3(x-5)=15$Times New Roman,18,12,0,0,0,0,0 $9.432+x=-5.765$Times New Roman,18,12,0,0,0,0,0 $x-8.8832=4,567,894$Times New Roman,18,12,0,0,0,0,0 $x+3.456=-9.874$Times New Roman,18,12,0,0,0,0,0 $x-5=7$Times New Roman,18,12,0,0,0,0,0 $x+5=7$Times New Roman,18,12,0,0,0,0,0 $8.81x=37.66$Times New Roman,18,12,0,0,0,0,0 $3.71x=-9.43$Times New Roman,18,12,0,0,0,0,0 $4x=-\frac 3{17}$Times New Roman,18,12,0,0,0,0,0 $4x=8$Times New Roman,18,12,0,0,0,0,0 $\sqrt{3x+5}=\sqrt{x+6}$Times New Roman,18,12,0,0,0,0,0 $\sqrt{17}x=5.4$Times New Roman,18,12,0,0,0,0,0 $\sqrt{43.2}+x=12$BMО>(є„€џџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяnЊпџџџџџџЊŸQБџџџџџџqlўЙпџџџџџџяnЊŸџџџџџџЊЏUЕџџџџџџuKўОЏџџџџџџя`jŠџџџџџџ˜ЏЕџџџџџџ5 јнЏџџџџџџяdЊ_џџџџџџЊЏUЕџџџџџџu+њюЏџџџџџџяnкпџџџџџџŸЕџџџџџџlћщпџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџя@/@џџџџџџџџџџџџџџџџџџџџџяOYŸŒпџџџџџџџџџџџџџџџџџџџџџяY™™ƒŸџџџџџџџџџџџџџџџџџџџџџяY™ƒ ŸџџџџџџџџџџџџџџџџџџџџџяO™Œ †пџџџџџџџџџџџџџџџџџџџџџяA™™ŒƒŸџџџџџџџџџџџџџџџџџџџџџяO Ÿџџџџџџџџџџџџџџџџџџџџџя@ЧŸџџџџџџџџџџџџџџџџџџџџџя@CŸџџџџџџџџџџџџџџџџџџџџџя@‚ Ÿџџџџџџџџџџџџџџџџџџџџџя @џџџџџџџџџџџџџџџџџџџџџя 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ппѕ§ўџџўћџћџџџџџџџџџџџџџї§џ§яaоћ§ўќўћџћџџџџџџџџџџџџџƒўќўяaооѕ№~џџўћСћџџџџџџџџџџџџџЗНџџo aп_ю§ўќŽћџћџџџџџџџџџџџџџзћМЏaпŸџ§ќџџќћџѓџџџџџџџџџџџџџчџџџћЏ япџџўџџўїџћџџџџџџџџџџџџџїџџџќoїџџџџџџџяџџџџџџџџџџџџџџџџџџџџя џџџџџџџџџџџџџџџџџџџџџџџџџџџџџя џџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяїџџџџџџџџџџяџџџџџџџџџџџџџџџџџяяџџџџџџџџџџїџџџџџџџџџџџџџџџџџяпюџќўюџќ{џрўяџџџџџџпюџќџќoояѕ§ўџџѕџўћџїџ_џџџџџџпѕ§ўџџўяaояћ§ўќПћџўћџћџПџџџџџўћ§ўќўяaоѕ№~џџоѕ№~ћС§я_џџџџџўоѕ№~џџўяaоџю§ўќяюџўћџўўяџџџџџџ_ю§ўќŽяaпџ§ќџўяџџќћџюџџџџџџџџŸџ§ќџџќя яŸџџўџџџџўїџёџџџџџџџџпџџўџџўя їџџџџџџџџџџяџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџюџќўюџќoџџџџџџџџџџџџџџџџџџўяѕ§ўџџѕџўяџџџџџџџџџџџџџџџџџџўяћ§ўќПћџўяџџџџџџџџџџџџџџџџџџўѕ№~џџоѕ№~яџџџџџџџџџџџџџџџџџџўџю§ўќяюџўяџџџџџџџџџџџџџџџџџџџџ§ќџўяџџќяџџџџџџџџџџџџџџџџџџџŸџџўџџџџўяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяџџџџџПџџџџўџџџпџџџџџџџяџџ§џџїяџўџџџџџџџ§џџџяџџџџџџџпџџўџџћяёКќ.ТўџwџСћнџјїџяџјџwџНпџў=я@/_џџўџџџџџїџџўџџџџџџџўџџџџџџџџЏ_јўџќџџїџџўџџџџў?џўџџџџџџџџЏ_ќџўЛБ†ќ68cŽЧ›БП_зcŽЧpПќ?‚ЏЏ^ќџўЛМ/КћЕзwvЛkЏП_~wvЛЛюПћПпџЏ\xЛЛ Кќ5јw~Л{ П_}еwvПИ0Пќ?яџЏX8ƒЛЎšoЕџw~ЛyЎ›C5еwvПЛО›џ їџЏ_ќџўЛМ1ІœuјїŽК:qЇ]NїvПМqЇќƒџЏ_ќџўЛПџПџѕпїўЛ{џџ]џџїvЛПџџџџџџЏ_ўЮЧ?џПџі?чўЧ{џџCџџуŽЧ?џџџџџџЏ_џ~ЖџџџџџїџџўџџџџџџџўџџџџџџџџЏ[њ6џџqџџїї?ўџпџ§Яџўџџџю?џџџЏ[пш6џџwпџїїнўџпwџ§їџўџџџюПџџџЏYп 6џџ;џїѓИўџЬcџќЯџўџџџц?џџџЏ[ž€6џџ}џїїа~џнAџ§пџўџџџюџџџџЏhи џџџыё?§ХџўПќGџ§џџу?џџџowџџћПџџџџнџџћПџџ§пџџџћПџџџџџџўяxР>РрРяџџџџџџџџџџџџџџџџџџџџџџџџџџџџџяTimes New Roman,18,12,0,0,0,0,0 $\div $Times New Roman,18,12,0,0,0,0,0 $\wedge $Times New Roman,18,12,0,0,0,0,0 $(x+8)(x-2)=$Times New Roman,18,12,0,0,0,0,0 $x^2+6x-16=$Times New Roman,18,12,0,0,0,0,0 $(x-7)(x-3)=$Times New Roman,18,12,0,0,0,0,0 $(3x+5)(5x+6)=$Times New Roman,18,12,0,0,0,0,0 $(2a-7)^2=$Times New Roman,18,12,0,0,0,0,0 $(x-3)(x+3)=$Times New Roman,18,12,0,0,0,0,0 $(x-9)(x-4)(x^2-25)=$Times New Roman,18,12,0,0,0,0,0 $x^2-3x-40=$Times New Roman,18,12,0,0,0,0,0 $x^2-3x-4=$Times New Roman,18,12,0,0,0,0,0 $x^2-49=$Times New Roman,18,12,0,0,0,0,0 $x^3+8=$Times New Roman,18,12,0,0,0,0,0 $x^5+x^4-27x^3-25x^2+50x=$Times New Roman,18,12,0,0,0,0,0 $x^4+5x^3-7x^2-29x+30=0$Times New Roman,18,12,0,0,0,0,0 $(x-2)(x-1)(x+3)(x+5)=0$Times New Roman,18,12,0,0,0,0,0 $x-2=0$ or $x=2$Times New Roman,18,12,0,0,0,0,0 $x-1=0$ or $x=1$Times New Roman,18,12,0,0,0,0,0 $x+3=0$ or $x=-3$Times New Roman,18,12,0,0,0,0,0 $x+5=0$ or $x=-5$Times New Roman,18,12,0,0,0,0,0 $(x-2)(x-1)(x+3)(x+5)=0$Times New Roman,18,12,0,0,0,0,0 $x^2+3x-4=0$Times New Roman,18,12,0,0,0,0,0 $x^2+2x-15=0$Times New Roman,18,12,0,0,0,0,0 $x^4+2x^3-31x^2-32x+240=0$Times New Roman,18,12,0,0,0,0,0 $x^2-8x=20$Times New Roman,18,12,0,0,0,0,0 $20$Times New Roman,18,12,0,0,0,0,0 $12x^2-x-35=0$Times New Roman,18,12,0,0,0,0,0 $x^5+15x^4+85x^3+225x^2+274x+120=0$Times New Roman,18,12,0,0,0,0,0 $x^2+4=0$Times New Roman,18,12,0,0,0,0,0 $x^3-64=0$Times New Roman,18,12,0,0,0,0,0 $x^2-x-5=0$Times New Roman,18,12,0,0,0,0,0 $2,1,-3$ and $5$Times New Roman,18,12,0,0,0,0,0 $$f(x)=\frac 2{(x-4)}$$Times New Roman,18,12,0,0,0,0,0 $$f(x)=\frac 2{(x-4)^2}$$Times New Roman,18,12,0,0,0,0,0 $$f(x)=\frac{x^2-4x}{\sqrt{x^2-8x+16}}$$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow a-$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow a+$Times New Roman,18,12,0,0,0,0,0 $$\frac{\sin (x-2)}{x-2}$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow 3}(x^2-2x)$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow 2}(2x+4)$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow number}$$Times New Roman,18,12,0,0,0,0,0 $x=a$Times New Roman,18,12,0,0,0,0,0 $$\frac{3x^2}{(x+2)^2}$$Times New Roman,18,12,0,0,0,0,0 $$\frac{3x}{x+2}$$Times New Roman,18,12,0,0,0,0,0 $$\frac{3x}{x+2}$$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow \infty $Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow a+}f(x)$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow a\,-}f(x)$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow 4}f(x)=\_\_\_\_\_\_\_$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow 4+}f(x)=\_\_\_\_\_\_\_$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow 4-}f(x)=\_\_\_\_\_\_\_$$Times New Roman,18,12,0,0,0,0,0 $$g(x)=\frac{\sin (x)}x$$Times New Roman,18,12,0,0,0,0,0 $$h(x)=\frac 1{x-5}$$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow 4$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow 4$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \infty }\frac{4x}{x-4}$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \;-\;\infty }\frac{2x^2}{(x-3)^2}$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \infty }x^2$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \;-\;\infty }\frac 1x$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \infty }\sin (x)$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \infty }3^x$$Times New Roman,18,12,0,0,0,0,0 $$\lim _{x\rightarrow \;-\;\infty }3^x$$Times New Roman,18,12,0,0,0,0,0 $x\rightarrow \infty $Times New Roman,18,12,0,0,0,0,0 $x\rightarrow -\ \infty $ [Embedded] 136 .tex 70538 44 0 0 27 .tex 70582 65 0 0 9 .tex 70647 51 0 0 10 .tex 70698 41 0 0 11 .tex 70739 45 0 0 12 .tex 70784 59 0 0 13 .tex 70843 43 0 0 14 .tex 70886 44 0 0 15 .tex 70930 46 0 0 16 .tex 70976 44 0 0 17 .tex 71020 50 0 0 18 .tex 71070 45 0 0 19 .tex 71115 41 0 0 20 .tex 71156 43 0 0 21 .tex 71199 41 0 0 22 .tex 71240 41 0 0 23 .tex 71281 40 0 0 24 .tex 71321 45 0 0 230 .tex 71366 75 0 0 229 .tex 71441 63 0 0 228 .tex 71504 51 0 0 227 .tex 71555 53 0 0 226 .tex 71608 44 0 0 225 .tex 71652 44 0 0 224 .tex 71696 49 0 0 223 .tex 71745 53 0 0 222 .tex 71798 49 0 0 221 .tex 71847 40 0 0 220 .tex 71887 40 0 0 219 .tex 71927 46 0 0 218 .tex 71973 46 0 0 217 .tex 72019 50 0 0 216 .tex 72069 39 0 0 214 .tex 72108 57 0 0 213 .tex 72165 49 0 0 212 .tex 72214 51 0 0 211 .bmp 72265 4286 0 0 210 .bmp 76551 4286 0 0 209 .tex 80837 40 0 0 208 .tex 80877 42 0 0 207 .tex 80919 46 0 0 206 .tex 80965 45 0 0 205 .tex 81010 46 0 0 204 .tex 81056 48 0 0 203 .tex 81104 44 0 0 202 .tex 81148 46 0 0 201 .tex 81194 54 0 0 200 .tex 81248 45 0 0 199 .tex 81293 44 0 0 198 .tex 81337 42 0 0 197 .tex 81379 41 0 0 196 .tex 81420 59 0 0 195 .tex 81479 57 0 0 194 .tex 81536 57 0 0 193 .tex 81593 49 0 0 192 .tex 81642 49 0 0 191 .tex 81691 50 0 0 190 .tex 81741 50 0 0 189 .tex 81791 57 0 0 188 .tex 81848 45 0 0 187 .tex 81893 46 0 0 186 .tex 81939 59 0 0 185 .tex 81998 44 0 0 184 .tex 82042 37 0 0 183 .tex 82079 47 0 0 182 .tex 82126 68 0 0 181 .tex 82194 42 0 0 180 .tex 82236 43 0 0 179 .tex 82279 44 0 0 178 .tex 82323 49 0 0 177 .tex 82372 56 0 0 176 .tex 82428 58 0 0 175 .tex 82486 73 0 0 174 .tex 82559 50 0 0 173 .tex 82609 50 0 0 172 .tex 82659 59 0 0 171 .tex 82718 67 0 0 170 .tex 82785 65 0 0 169 .tex 82850 64 0 0 168 .tex 82914 38 0 0 167 .tex 82952 57 0 0 166 .tex 83009 51 0 0 165 .tex 83060 51 0 0 164 .tex 83111 55 0 0 163 .tex 83166 64 0 0 162 .tex 83230 66 0 0 161 .tex 83296 78 0 0 160 .tex 83374 79 0 0 159 .tex 83453 79 0 0 158 .tex 83532 58 0 0 157 .tex 83590 54 0 0 156 .tex 83644 49 0 0 155 .tex 83693 49 0 0 154 .tex 83742 79 0 0 153 .tex 83821 90 0 0 152 .tex 83911 68 0 0 151 .tex 83979 78 0 0 150 .tex 84057 73 0 0 149 .tex 84130 68 0 0 148 .tex 84198 73 0 0 147 .tex 84271 55 0 0 146 .tex 84326 58 0 0 00084386