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" }}{PARA 0 "" 0 "" {TEXT 227 24 "As you see the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 227 19 " is increasing on [" } {XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 227 30 ", 0] and decreasi ng on [0, 1]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 78 "Compute the lef t-hand sums and right-hand sums with 10, 100, 1000 subintervals" } {TEXT 227 2 ". " }{TEXT 209 35 "Do not forget the student package." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(student):with(plots): " }}{PARA 7 "" 1 "" {TEXT 229 49 "Warning, the name changecoords has b een redefined" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "for j fro m 1 to 3 do lhsum[j]:=evalf(leftsum(f(x), x=-1..1, 10^j)); rhsum[j]:=e valf(rightsum(f(x),x=-1..1,10^j)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"\"$\"+9W_=:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " >&I&rhsumG6\"6#\"\"\"$\"+9W_=:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">& I&lhsumG6\"6#\"\"#$\"+cU8p:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&r hsumG6\"6#\"\"#$\"+cU8p:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsu mG6\"6#\"\"$$\"+RPuq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6 \"6#\"\"$$\"+RPuq:!\"*" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 201 139 "Wha t do you notice? Can you explain what you noticed? You may want to gra ph some left-hand sums and right-hand sums to explain your answer." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "for j from 1 to 3 do lhsum [j]:=evalf(2*leftsum(f(x), x=-1..0, 10^j)); rhsum[j]:=evalf(2*rightsum (f(x),x=-1..0,10^j)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6 \"6#\"\"\"$\"+j\"fAX\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG 6\"6#\"\"\"$\"+j\"fAl\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsum G6\"6#\"\"#$\"+;&3-c\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG 6\"6#\"\"#$\"+;&3-e\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6 \"6#\"\"$$\"+Lxxp:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6 #\"\"$$\"+Lxxr:!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 13 "" 1 "" {TEXT 228 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 63 "As a result we do not get underestimates and overestimates for " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 227 29 " using the previous comma nds." }}{PARA 0 "" 0 "" {TEXT 210 42 "Find underestimates and overesti mates for " }{XPPEDIT 201 0 "Pi;" "6#%#PiG" }{TEXT 211 98 ", using lef t-hand sums and right-hand sums with 5, 50, 500, 5000 on the intervals [-1,0] and [0,1]" }{TEXT 227 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "for j from 0 to 3 do lhsum[j]:=evalf(4*leftsum(f(x), x=-1..0, 5*10^j)); rhsum[j]:=evalf(4*rightsum(f(x),x=-1..0,5*10^j)); \+ od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"!$\"+H)[qj#!\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"!$\"+H)[qV$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"\"$\"+6&o#)4$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"\"$\"+6&o#yJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"#$\"+xu[PJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"#$\"+xu[XJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"$$\"+F$*=TJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"$$\"+F$*)>9$!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), 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font_style_name = \"2D Input\", siz e = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mo(\"⁢\", form = \"\", fence = \"false \", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretch y = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize \+ = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[ 255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"0\"), T ypesetting:-mo(\"to\", form = \"\", fence = \"false\", separator = \"f alse\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", st retchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", min size = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground \+ = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"3 \"), Typesetting:-mo(\"do\", form = \"\", fence = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Input\", size = \"12\", fore ground = \"[255,0,0]\", background = \"[255,255,255]\")), Typesetting: -mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\" , depth = \"0.0 ex\", linebreak = \"increaseindentnewline\"), Typesett ing:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Ty pesetting:-msub(Typesetting:-mi(\"lhsum\"), Typesetting:-mrow(Typesett ing:-mi(\"j\")), subscriptshift = \"0\", placeholder = \"false\"), Typ esetting:-mo(\":=\", form = \"infix\", fence = \"false\", separator = \+ \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", s tretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mi nsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(T ypesetting:-mi(\"evalf\"), Typesetting:-mo(\"⁡\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\" , rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsi ze = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\") , Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \+ \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \+ \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), \+ Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting :-mn(\"2\"), Typesetting:-mo(\"⁢\", form = \"infix\", f ence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \" 0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", fo reground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting :-mrow(Typesetting:-mi(\"leftsum\"), Typesetting:-mo(\"⁡ \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"fals e\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", font_style_name = \"2D Inp ut\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,2 55,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\" , fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Inpu t\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow( Typesetting:-mi(\"f\"), Typesetting:-mo(\"⁡\", form = \" infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", r space = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize \+ = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \+ \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), T ypesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"t rue\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"t hinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Input\", size = \" 12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mrow(Typesetting:-mi(\"x\")), Typesetting:-mo(\")\", form = \+ \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinm athspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symme tric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", backgro und = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mo(\", \", form = \"infix\", fence = \"false\", separator = \"true\", lspace \+ = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symm etric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", backgro und = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Ty pesetting:-mo(\"=\", form = \"infix\", fence = \"false\", separator = \+ \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", s tretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mi nsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(T ypesetting:-mn(\"0\"), Typesetting:-mo(\"..\", form = \"postfix\", fen ce = \"false\", separator = \"false\", lspace = \"mediummathspace\", r space = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize \+ = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \+ \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), T ypesetting:-mn(\"1\")), Typesetting:-mi(\"\")), Typesetting:-mo(\",\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \+ \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmet ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", font_style_nam e = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", backgroun d = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mn(\"5\"), Type setting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[25 5,0,0]\", background = \"[255,255,255]\"), Typesetting:-msup(Typesetti ng:-mn(\"10\"), Typesetting:-mi(\"j\"), superscriptshift = \"0\"), Typ esetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mo(\")\", fo rm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \" thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", bac kground = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi (\"\")), Typesetting:-mi(\"\")), Typesetting:-mo(\")\", form = \"postf ix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspac e\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \+ \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), T ypesetting:-mo(\";\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"thickmathspace\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.3 em\", depth = \"0.0 ex\", linebreak = \"fir stprocnewline\"), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-msub(Typesetting:-mi( \"rhsum\"), Typesetting:-mrow(Typesetting:-mi(\"j\")), subscriptshift \+ = \"0\", placeholder = \"false\"), Typesetting:-mo(\":=\", form = \"in fix\", fence = \"false\", separator = \"false\", lspace = \"thickmaths pace\", rspace = \"thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"evalf\"), Type setting:-mo(\"⁡\", form = \"infix\", fence = \"false\", \+ separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesettin g:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false \", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesettin g:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"2\"), Typesetting:-mo (\"⁢\", form = \"infix\", fence = \"false\", separator \+ = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", large op = \"false\", movablelimits = \"false\", accent = \"false\", font_st yle_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"rig htsum\"), Typesetting:-mo(\"⁡\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em \", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Input\", size = \"12\", fore ground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separ ator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace \", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\" , minsize = \"1\", largeop = \"false\", movablelimits = \"false\", acc ent = \"false\", font_style_name = \"2D Input\", size = \"12\", foregr ound = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Ty pesetting:-mo(\"⁡\", form = \"infix\", fence = \"false\" , separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy \+ = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[25 5,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetti ng:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false \", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesettin g:-mi(\"x\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"tr ue\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"ve rythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mo(\",\", form = \"infix\", fenc e = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"very 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largeop = \"false\", movablelimits = \"false\", accent = \"fal se\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[2 55,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1\")), T ypesetting:-mi(\"\")), Typesetting:-mo(\",\", form = \"infix\", fence \+ = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"veryth ickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \+ \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), T ypesetting:-mrow(Typesetting:-mn(\"5\"), Typesetting:-mo(\"&InvisibleT imes;\", form = \"infix\", fence = \"false\", separator = \"false\", l space = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mn(\"10\"), Typeset ting:-mi(\"j\"), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Ty pesetting:-mi(\"\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,25 5]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-m i(\"\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythi nmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \" infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", font_style_name = \"2D Input\", size = \"12 \", foreground = \"[255,0,0]\", background = \"[255,255,255]\")), Type setting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Ty pesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0 ex\", width = \+ \"0.0 em\", depth = \"0.0 ex\", linebreak = \"decreaseindentnewline\") , Typesetting:-mo(\"end do\", form = \"\", fence = \"false\", separato r = \"false\", lspace = \"mediummathspace\", rspace = \"0em\", stretch y = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize \+ = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[ 255,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), \+ Typesetting:-mo(\";\", form = \"infix\", fence = \"false\", separator \+ = \"true\", lspace = \"0em\", rspace = \"thickmathspace\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%F+-F#6+F+-F#6--I#moGF $63Q$forF'/%%formGF./%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF' /%'rspaceGQ0mediummathspaceF'/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ )infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentG F=/%0font_style_nameGQ)2D~InputF'/%%sizeGQ#12F'/%+foregroundGQ*[255,0, 0]F'/%+backgroundGQ.[255,255,255]F'-F663Q1⁢F'F9F;F>F@/F DFBFFFHFJFMFPFRFTFVFYFfnFinF\\o-F,6#Q\"jF'F\\o-F663Q%fromF'F9F;F>/FAFE FCFFFHFJFMFPFRFTFVFYFfnFinF\\o-I#mnGF$6#Q\"0F'-F663Q#toF'F9F;F>FfoFCFF FHFJFMFPFRFTFVFYFfnFin-Fho6#Q\"3F'-F663Q#doF'F9F;F>FfoFCFFFHFJFMFPFRFT FVFYFfnFinF+-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.3~emF'/%&de pthGFip/%*linebreakGQ6increaseindentnewlineF'F+-F#6'F+-F#6'F+-F#6'F+-I 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r-F#6&F_sFbs-F#6%Fgs-F#6%F+-F#6&FftFit-F#6&-F,6#Q)rightsumF'FbsF`uF+F+ F+F`vF+F+F+F+-Fep6&Fgp/F[qQ'0.0~emF'F]q/F`qQ6decreaseindentnewlineF'-F 663Q'end~doF'F9F;F>FfoF_oFFFHFJFMFPFRFTFVFYFfnFinF+Ffx" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "for j fr om 0 to 3 do lhsum[j]:=evalf(4*leftsum(f(x), x=0..1, 5*10^j)); rhsum[j ]:=evalf(4*rightsum(f(x),x=0..1,5*10^j)); od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"! $\"+H)[qV$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"!$ \"+H)[qj#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"\"$ \"+6&o#yJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"\"$ \"+6&o#)4$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"#$ \"+xu[XJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"#$\"+ xu[PJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&lhsumG6\"6#\"\"$$\"+F$* )>9$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&rhsumG6\"6#\"\"$$\"+F$*= TJ!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 45 "We can actually predict beforehand how large " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 227 73 " should be so that our und erestimates and overestimates are within, say, " }{XPPEDIT 18 0 "epsil on;" "6#%(epsilonG" }{TEXT 227 31 ". Recall that on the interval [" } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT 227 2 ", " }{XPPEDIT 18 0 "b;" "6# %\"bG" }{TEXT 227 1 "]" }}{PARA 0 "" 0 "" {TEXT 227 20 "the difference |LHS(" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 227 6 ")-RHS(" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 227 14 ")|=|f(b)-f(a)|" }{XPPEDIT 18 0 "Del ta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 227 50 " for a \+ monotone function. If we split [-1,0] into " }{XPPEDIT 18 0 "n;" "6#% \"nG" }{TEXT 227 37 " subintervals the error is less than " }{XPPEDIT 18 0 "(f(0)-f(-1))/n = 1/n;" "6#/*&,&-%\"fG6#\"\"!\"\"\"-F'6#,$F*!\"\" F.F*%\"nGF.*&F*F*F/F." }{TEXT 227 1 "." }}{PARA 0 "" 0 "" {TEXT 227 6 "since " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT 227 7 "=(b-a)/" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 227 42 ". On the interval [0,1], if we split into " }{XPPEDIT 18 0 "n;" "6 #%\"nG" }{TEXT 227 42 " subintervals, we have an error less than " } {XPPEDIT 18 0 "(f(0)-f(1))/n = 1/n;" "6#/*&,&-%\"fG6#\"\"!\"\"\"-F'6#F *!\"\"F*%\"nGF-*&F*F*F.F-" }{TEXT 227 58 ". So in the whole interval \+ [-1,1] the error is less than " }{XPPEDIT 18 0 "2/n;" "6#*&\"\"#\"\"\" %\"nG!\"\"" }{TEXT 227 12 " and, since " }{XPPEDIT 18 0 "Pi;" "6#%#PiG " }{TEXT 227 48 " is twice the integral, the error in estimating " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 227 14 " is less than " } {XPPEDIT 18 0 "4/n;" "6#*&\"\"%\"\"\"%\"nG!\"\"" }{TEXT 227 67 ". If w e want the error to be, say less than 0.001, we need to make " } {XPPEDIT 18 0 "4/n < Float(1, -3);" "6#2*&\"\"%\"\"\"%\"nG!\"\"-%&Floa tG6$F&!\"$" }{TEXT 227 14 ", which gives " }{XPPEDIT 18 0 "4/Float(1, \+ -3) < n;" "6#2*&\"\"%\"\"\"-%&FloatG6$F&!\"$!\"\"%\"nG" }{TEXT 227 100 ", i.e., n>4000. This explains why with 5000 subintervals we had g otten the first 3 decimals correct." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "4/0.001;" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+++++S!\" '" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 230 30 "How large do you need to \+ take " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 230 26 ", so that you can \+ compute " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 230 35 " with an error less than 0.0000001?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "4/ .0000001" }}{PARA 7 "" 1 "" {TEXT 229 55 "Warning, inserted missing se micolon at end of statement" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+++++S !\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 137 "Unfortunately, this num ber of subintervals is too large to work out with Maple. So we resort \+ to the trapezoid rule and the midpoint rule." }}{PARA 0 "" 0 "" {TEXT 212 13 "Compute TRAP(" }{XPPEDIT 201 0 "n;" "6#%\"nG" }{TEXT 213 7 "), MID(" }{XPPEDIT 201 0 "n;" "6#%\"nG" }{TEXT 214 8 "), with " } {XPPEDIT 201 0 "n;" "6#%\"nG" }{TEXT 215 113 "=10, 100, 1000 on the wh ole interval [-1,1]. Which ones give underestimates and which ones giv e overestimates of " }{XPPEDIT 201 0 "Pi;" "6#%#PiG" }{TEXT 216 1 "?" }{TEXT 227 2 " " }{TEXT 217 34 "Why? What is the relation of TRAP(" } {XPPEDIT 201 0 "n;" "6#%\"nG" }{TEXT 218 11 ") with LHS(" }{XPPEDIT 201 0 "n;" "6#%\"nG" }{TEXT 219 2 ")?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "for j from 1 to 3 do midsum[j]:=evalf(2*middlesum(f( x), x=-1..1, 10^j)); trapsum[j]:=evalf(2*sum(((f(-1+2*k/10^j)+f(-1+2*( k+1)/10^j))/2)*2/10^j,k=0..2*10^j-1 )); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'midsumG6\"6#\"\"\"$\"+Cy)><$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I(trapsumG6\"6#\"\"\",&$\"+H)[q.$!\"*F'*&$\"+X&)owmF+F' ^#F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'midsumG6\"6#\"\"#$\"+_bc UJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I(trapsumG6\"6#\"\"#,&$\"+7& o#QJ!\"*\"\"\"*&$\"+,3%4s'F+F,^#F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I'midsumG6\"6#\"\"$$\"+dMiTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " >&I(trapsumG6\"6#\"\"$,&$\"+#[([TJ!\"*\"\"\"*&$\"+FB[AnF+F,^#F,F,F," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "for j from 1 to 3 do n:=1 0^j; dx:=2/n; evalf(sum( (f(-1+k*dx) + f(-1+(k+1)*dx))*dx/2, k=0..n-1) ); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+9W_=:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6\"\"$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+cU8p:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6\" \"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+TPuq:!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 59 "To compute more accurately, we can use Simpson's rule: SIM(" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 227 4 ")= " }{XPPEDIT 18 0 "(TRA P(n)+2*MID(n))/3;" "6#*&,&-%%TRAPG6#%\"nG\"\"\"*&\"\"#F)-%$MIDGF'F)F)F )\"\"$!\"\"" }{TEXT 227 3 " " }{TEXT 227 46 ". There is a Maple comm and for Simpson's rule:" }}{PARA 0 "" 0 "" {TEXT 227 65 "simpson(f(x), x=lowerlimit..upperlimit, 2*number of subintervals)" }}{PARA 200 "" 0 "" {TEXT 230 8 "Compute " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 230 27 " using Simpson's rule with " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 230 92 "=10, 100, 1000,10000 subintervals. Verify your answers using t he formula for Simpson's rule." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "for j from 1 to 3 do n:=10^j; dx:=2/n; trap[j]:=evalf(2*sum( \+ (f(-1+k*dx) + f(-1+(k+1)*dx))*dx/2, k=0..n-1)); mid[j]:=evalf(2*middle sum(f(x), x=-1..1, 10^j)); simp[j]:=(trap[j]+2*mid[j])/3; od;" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">& I%trapG6\"6#\"\"\"$\"+H)[q.$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I$ midG6\"6#\"\"\"$\"+Cy)><$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I%sim pG6\"6#\"\"\"$\"+f\"3q7$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6 \"\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"#]" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I%trapG6\"6#\"\"#$\"+3&o#QJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I$midG6\"6#\"\"#$\"+_bcUJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I%simpG6\"6#\"\"#$\"+/K8TJ!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I\"nG6\"\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#dxG6\"#\"\"\"\"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">& I%trapG6\"6#\"\"$$\"+yu[TJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I$mi dG6\"6#\"\"$$\"+dMiTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I%simpG6 \"6#\"\"$$\"+J\"y:9$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "for j from 1 to 3 do simpp[j]:=evalf(2*simpson(f(x), x=-1..1, 2*10^j) ); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&simppG6\"6#\"\"\"$\"+e\"3q 7$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&simppG6\"6#\"\"#$\"+0K8TJ! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I&simppG6\"6#\"\"$$\"+I\"y:9$! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fp:=D(f); fpp:=D(fp) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#fpG6\"f*6#I\"xGF$F$6$I)operator GF$I&arrowGF$F$,$*&F'\"\"\"-I%sqrtG6$%*protectedGI(_syslibGF$6#,&F-F-* $)F'\"\"#F-!\"\"F8F8F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$fppG6\" f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,&*$-I%sqrtG6$%*protectedGI(_s yslibGF$6#,&\"\"\"F4*$)F'\"\"#F4!\"\"F8F8*&F6F4)F-\"\"$F8F8F$F$F$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 220 33 "Topic 2: Substitutions with Maple " }{TEXT 227 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 156 "All the in tegrals in this section can be evaluated by hand. Although Maple can h elp us verify our answers, you should still learn the substitution me thod. " }}{PARA 0 "" 0 "" {TEXT 227 86 "You are encouraged to evaluate the integrals by hand, as a test of your understanding." }}{PARA 0 "" 0 "" {TEXT 227 30 "Let us introduce the integral " }{XPPEDIT 19 1 "in t((x^2+1)^4*x, x);" "6#-%$intG6$*&),&*$)%\"xG\"\"#\"\"\"F-F-F-\"\"%F-F +F-F+" }{TEXT 227 29 ": The command is Int(f(x), x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E:=Int((x^2+1)^4*x, x);" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I\"EG6\"-I$IntGI(_syslibGF$6$*&),&*$)I\"xGF$\"\"#\"\" \"F0F0F0\"\"%F0F.F0F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 67 "If we k now which variable to change, we introduce the substitution " } {XPPEDIT 18 0 "u = g(x);" "6#/%\"uG-%\"gG6#%\"xG" }{TEXT 227 61 " the \+ following way: changevar(u=g(x), E,u). This substitutes " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT 227 5 " for " }{XPPEDIT 18 0 "g(x);" "6#-%\"g G6#%\"xG" }{TEXT 227 0 "" }}{PARA 0 "" 0 "" {TEXT 227 18 "in the integ ral E." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:=changevar(u=x^ 2+1, E, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"FG6\"-I$IntGI(_syslib GF$6$,$*&#\"\"\"\"\"#F,)I\"uGF$\"\"%F,F,F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 54 "To compute this integral we can use the value command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "V:=value(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"VG6\",$*&#\"\"\"\"#5F()I\"uGF$\"\"&F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 227 111 "Notice that Maple does not inclu de constants of integration. We always want to switch to the original \+ variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 227 38 ". This is done with the subs command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S:=subs(u=x^2+1, V);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"SG6\",$*&# \"\"\"\"#5F(),&*$)I\"xGF$\"\"#F(F(F(F(\"\"&F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 111 "We do not need to do all these substitutions and c hanges, we can ask Maple to evaluate the integral E directly:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&#\"\"\"\"#5F%)I\"xG6\"F&F%F%*&#F%\"\"#F%)F(\"\")F%F% *$)F(\"\"'F%F%*$)F(\"\"%F%F%*&F+F%)F(F,F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 123 "The problem is that this answer does not see to be the same as the one we got before. The reason is that we need to expand " }{XPPEDIT 18 0 "(x^2+1)^5;" "6#*$),&*$)%\"xG\"\"#\"\"\"F*F*F*\"\"&F*" }{TEXT 227 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "expand( S, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&#\"\"\"\"#5F%)I\"xG6\"F&F% F%*&#F%\"\"#F%)F(\"\")F%F%*$)F(\"\"'F%F%*$)F(\"\"%F%F%*&F+F%)F(F,F%F%F $F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 57 "We still notice that ther e is a difference: the constant " }{XPPEDIT 18 0 "1/10;" "6#*&\"\"\"F$ \"#5!\"\"" }{TEXT 227 105 ". This is so, because in integration Maple \+ does not care about constants. The most general antiderivative" }} {PARA 0 "" 0 "" {TEXT 227 38 "is any of the previous expressions +C." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 124 "One problem is that Maple doe s not tell us which substitution to use. So the choice is ours and thi s is where our work lies." }}{PARA 200 "" 0 "" {TEXT 230 18 "Try to su bstitute " }{XPPEDIT 18 0 "u = x^2+1;" "6#/%\"uG,&*$)%\"xG\"\"#\"\"\"F *F*F*" }{TEXT 230 17 " in the integral " }{XPPEDIT 19 1 "int((x^2+1)^4 , x);" "6#-%$intG6$*$),&*$)%\"xG\"\"#\"\"\"F-F-F-\"\"%F-F+" }{TEXT 230 21 ". What do you notice?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E1:=Int((x^2+1)^4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E1G6\" -I$IntGI(_syslibGF$6$*$),&*$)I\"xGF$\"\"#\"\"\"F0F0F0\"\"%F0F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F1:=changevar(u=x^2+1, E1, u );" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#F1G6\"-I$IntGI(_syslibGF$6$,$*( #\"\"\"\"\"#F,)I\"uGF$\"\"%F,),&F,!\"\"F/F,F+F3F,F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 99 "In fac t one does not need substitution in this integral to compute it. We ca n expand the integrand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " g:=expand((x^2+1)^4, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"gG6\",,* $)I\"xGF$\"\")\"\"\"F**&\"\"%F*)F(\"\"'F*F**&F.F*)F(F,F*F**&F,F*)F(\" \"#F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "E2:=Int(g, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E2G6\"-I$IntGI(_syslibGF$6$,,*$ )I\"xGF$\"\")\"\"\"F.*&\"\"%F.)F,\"\"'F.F.*&F2F.)F,F0F.F.*&F0F.)F,\"\" #F.F.F.F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "V2:=value(E2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#V2G6\",,*&#\"\"\"\"\"*F()I\"xGF $F)F(F(*&#\"\"%\"\"(F()F+F/F(F(*&#\"\"'\"\"&F()F+F4F(F(*&#F.\"\"$F()F+ F8F(F(F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "value(Int((x^ 2+1)^4, x));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&#\"\"\"\"\"*F%)I\"xG 6\"F&F%F%*&#\"\"%\"\"(F%)F(F-F%F%*&#\"\"'\"\"&F%)F(F2F%F%*&#F,\"\"$F%) F(F6F%F%F(F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 47 "The last command verified our previous answer. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 221 22 "Evaluate the integral " }{XPPEDIT 201 1 "int(cos^2*x*sin*x, x);" " 6#-%$intG6$**)%$cosG\"\"#\"\"\"%\"xGF*%$sinGF*F+F*F+" }{TEXT 222 21 " \+ using a substitution" }{TEXT 227 2 ". " }{TEXT 223 41 "Check your answ er with the value command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E3:= Int((cos(x))^2*sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I #E3G6\"-I$IntGI(_syslibGF$6$*&)-I$cosG6$%*protectedGF'6#I\"xGF$\"\"#\" \"\"-I$sinGF-F/F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F3:= changevar(u=cos(x), E3, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#F3G6\" -I$IntGI(_syslibGF$6$,$*$)I\"uGF$\"\"#\"\"\"!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "V3:=value(F3); V3p:=value(E3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#V3G6\",$*&#\"\"\"\"\"$F()I\"uGF$F)F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$V3pG6\",$*&#\"\"\"\"\"$F()-I$cosG6$ %*protectedGI(_syslibGF$6#I\"xGF$F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "S3:=subs(u=cos(x), V3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#S3G6\",$*&#\"\"\"\"\"$F()-I$cosG6$%*protectedGI(_sysli bGF$6#I\"xGF$F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT 231 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 22 "Evaluate the integral " }{XPPEDIT 201 1 "int(e^(-x)*tan(e^(-x)), x );" "6#-%$intG6$*&)%\"eG,$%\"xG!\"\"\"\"\"-%$tanG6#F'F,F*" }{TEXT 225 21 " using a substitution" }{TEXT 227 2 ". " }{TEXT 226 41 "Check your answer with the value command." }}{PARA 11 "" 1 "" {TEXT 231 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "E4:= Int(exp(-x)*tan(exp(-x) ),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E4G6\"-I$IntGI(_syslibGF$6$* &-I$expG6$%*protectedGF'6#,$I\"xGF$!\"\"\"\"\"-I$tanGF,6#F*F2F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F4:=changevar(u=exp(-x), E4, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#F4G6\"-I$IntGI(_syslibGF$6$,$- I$tanG6$%*protectedGF'6#I\"uGF$!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "V4:=value(F4);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#V4 G6\"-I#lnG6$%*protectedGI(_syslibGF$6#-I$cosGF'6#I\"uGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S4:=subs(u=exp(-x),V4);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#S4G6\"-I#lnG6$%*protectedGI(_syslibGF$6#-I$co sGF'6#-I$expGF'6#,$I\"xGF$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(E4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I#lnG6$%*protecte dGI(_syslibG6\"6#-I$cosGF$6#-I$expGF$6#,$I\"xGF'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }