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"" -1 311 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 1 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 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2 " 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "C ourier" 1 10 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 278 55 "Maple Interactive Problem Solver for Integrals and Area" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 43 "IA Wor ksheet 1 Exercises: Reversing D-Rules" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Reversing the PowCR" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 290 102 "Arrow expressions for IA Worksheet 1 Exercises 1-6.\nTo check your work, copy each one and paste below." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 127 "7*(5*x-2)^2:\n7*(5*x-2)^(-2):\nx *(x^2+2)^(1/2):\n-8*x*(x^2+2)^(-1/2):\n17*(4*x^3 - 4)*(x^4-4*x)^2:\n-7 *(4*x^3 - 4)*(x^4-4*x)^(-5/2):" }}}{PARA 3 "" 0 "" {TEXT 256 17 "FOUR \+ STEP PROCESS" }}{PARA 0 "" 0 "" {TEXT 257 7 "STEP 1." }{TEXT -1 60 " T ranslate the integral notation into a derivative equation." }}{PARA 0 "" 0 "" {TEXT 259 16 "TO CHANGE, ENTER" }{TEXT -1 43 " the new arrow e xpression for the integral " }{TEXT 261 4 "HERE" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 154 "f:=x->7*(5*x-2 )^2:\nInt(f(x),x)= 'h(x) + C';\n`This integral problem translates into the following`;\n` derivative equation:`;\n`D-Equation: `*'D(h(x))'=f (x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 7 "STEP 2." }{TEXT -1 48 " Guess h(x) by \"reversing\" the PowCR, and check." }}{PARA 0 " " 0 "" {TEXT 260 5 "ENTER" }{TEXT -1 26 " the arrow expression for " } {TEXT 262 15 "YOUR GUESS HERE" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{MPLTEXT 1 0 191 "h:=x->(5*x-2)^3:\n`D-Equation: `*'D (h(x))'=f(x);\n`Guess: `*'h(x)'=h(x);\n`Check: `*'D'(h(x))=D(h)(x);\n` Is the constant out in front equal to the constant`; \n`in the derivat ive equation above?`;" }}}{PARA 0 "" 0 "" {TEXT 263 7 "STEP 3." } {TEXT -1 65 " If necessary, adjust your guess h(x) with a constant, an d check." }}{PARA 0 "" 0 "" {TEXT 264 5 "ENTER" }{TEXT -1 24 " the con stant factor to " }{TEXT 265 23 "ADJUST YOUR GUESS HERE:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 152 "ConstantF actor:=7/15:\nh2:=unapply(ConstantFactor*h(x),x):\n`D-Equation: `*'D(h (x))'=f(x);\n`Adjusted Guess: `*'h(x)'=h2(x);\n`Check: `*'D'(h2(x))=D( h2)(x);" }}}{PARA 0 "" 0 "" {TEXT 266 7 "STEP 4." }{TEXT -1 52 " Write the final answer using the integral notation." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 29 "Int(f(x),x)= int(f(x),x) + C;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 19 "Reversing the ExpCR" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 291 103 "Arrow expressions for IA Worksheet 1 Exercises 7-10.\nT o check your work, copy each one and paste below." }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 71 "5*exp(-x):\n-(4/3)*x*exp(x^2): \n(x^2-1)*exp(x^3-3*x):\nx^(-2)*exp(x^(-1)):" }}}{PARA 3 "" 0 "" {TEXT 267 17 "FOUR STEP PROCESS" }}{PARA 0 "" 0 "" {TEXT 268 7 "STEP 1 ." }{TEXT -1 60 " Translate the integral notation into a derivative eq uation." }}{PARA 0 "" 0 "" {TEXT 270 16 "TO CHANGE, ENTER" }{TEXT -1 43 " the new arrow expression for the integral " }{TEXT 272 4 "HERE" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 152 "f:=x->5*exp(-x):\nInt(f(x),x)= 'h(x) + C';\n`This integral proble m translates into the following`;\n` derivative equation:`;\n`D-Equati on: `*'D(h(x))'=f(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 7 " STEP 2." }{TEXT -1 48 " Guess h(x) by \"reversing\" the ExpCR, and che ck." }}{PARA 0 "" 0 "" {TEXT 271 5 "ENTER" }{TEXT -1 26 " the arrow ex pression for " }{TEXT 273 15 "YOUR GUESS HERE" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 189 "h:=x->exp(-x): \n`D-Equation: `*'D(h(x))'=f(x);\n`Guess: `*'h(x)'=h(x);\n`Check: `*'D '(h(x))=D(h)(x);\n`Is the constant out in front equal to the constant` ; \n`in the derivative equation above?`;" }}}{PARA 0 "" 0 "" {TEXT 274 7 "STEP 3." }{TEXT -1 65 " If necessary, adjust your guess h(x) wi th a constant, and check." }}{PARA 0 "" 0 "" {TEXT 275 5 "ENTER" } {TEXT -1 24 " the constant factor to " }{TEXT 276 23 "ADJUST YOUR GUES S HERE:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 154 "ConstantFactor:=5/(-1):\nh2:=unapply(ConstantFactor* h(x),x):\n`D-Equation: `*'D(h(x))'=f(x);\n`Adjusted Guess: `*'h(x)'=h2 (x);\n`Check: `*'D'(h2(x))=D(h2)(x);" }}}{PARA 0 "" 0 "" {TEXT 277 7 " STEP 4." }{TEXT -1 52 " Write the final answer using the integral nota tion." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 29 "Int(f( x),x)= int(f(x),x) + C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Reversing the LnCR" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 53 "Arrow expressions for IA \+ Worksheet 1 Exercises 11-14." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 50 "To check your work, copy each one and paste below." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 69 "-1/(2*x+1):\n-3/(5*x-2 ):\n9*x/(2*x^2-5):\n(x^2-2*x+1)/(x^3-3*x^2+3*x-2):" }}}{PARA 3 "" 0 " " {TEXT 279 17 "FOUR STEP PROCESS" }}{PARA 0 "" 0 "" {TEXT 280 7 "STEP 1." }{TEXT -1 60 " Translate the integral notation into a derivative \+ equation." }}{PARA 0 "" 0 "" {TEXT 282 16 "TO CHANGE, ENTER" }{TEXT -1 43 " the new arrow expression for the integral " }{TEXT 284 4 "HERE " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 153 "f:=x->-1/(2*x+1):\nInt(f(x),x)= 'h(x) + C';\n`This integral pro blem translates into the following`;\n` derivative equation:`;\n`D-Equ ation: `*'D(h(x))'=f(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 281 7 "STEP 2." }{TEXT -1 47 " Guess h(x) by \"reversing\" the LnCR, and c heck." }}{PARA 0 "" 0 "" {TEXT 283 5 "ENTER" }{TEXT -1 26 " the arrow \+ expression for " }{TEXT 285 15 "YOUR GUESS HERE" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 191 "h:=x->ln(2*x+1 ):\n`D-Equation: `*'D(h(x))'=f(x);\n`Guess: `*'h(x)'=h(x);\n`Check: `* 'D'(h(x))=D(h)(x);\n`Is the constant out in front equal to the constan t`; \n`in the derivative equation above?`;" }}}{PARA 0 "" 0 "" {TEXT 286 7 "STEP 3." }{TEXT -1 65 " If necessary, adjust your guess h(x) wi th a constant, and check." }}{PARA 0 "" 0 "" {TEXT 287 5 "ENTER" } {TEXT -1 24 " the constant factor to " }{TEXT 288 23 "ADJUST YOUR GUES S HERE:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 152 "ConstantFactor:=-1/2:\nh2:=unapply(ConstantFactor*h( x),x):\n`D-Equation: `*'D(h(x))'=f(x);\n`Adjusted Guess: `*'h(x)'=h2(x );\n`Check: `*'D'(h2(x))=D(h2)(x);" }}}{PARA 0 "" 0 "" {TEXT 289 7 "ST EP 4." }{TEXT -1 52 " Write the final answer using the integral notati on." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 29 "Int(f(x) ,x)= int(f(x),x) + C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "IA Worksheet 2 Exercises: Com puting AUC and AOC" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 294 44 "Computing \+ AUC and AOC by Reversing the PowCR" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 295 51 "Arrow expressions for IA Worksheet 2 Exer cises 1-6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 50 "To check yo ur work, copy each one and paste below." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 196 "7*(5*x-2)^2:a:=0:b:=1:\n7*(5*x-2)^(-2 ):a:=.5:b:=.8:\nx*(x^2+2)^(1/2):a:=0:b:=5:\n-8*x*(x^2+2)^(-1/2):a:=-3: b:=0:\n17*(4*x^3 - 4)*(x^4-4*x)^2:a:=1:b:=1.5:\n-7*(4*x^3 - 4)*(x^4-4* x)^(-5/2):a:=-.4:b:=-.2:" }}}{PARA 0 "" 0 "" {TEXT 297 16 "TO CHANGE, \+ ENTER" }{TEXT -1 31 " the arrow expression for f(x) " }{TEXT 298 3 "AN D" }{TEXT -1 24 " the values for a and b." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 639 "f:=x->7*(5*x-2)^2:a:=0:b:=1:\nh:=unap ply(int(f(x),x),x):\n'f(x)'= f(x);\np[0]:=plots[display](\n [plo t(f(x),x=a..b,color=blue),\n plot(0,x=0..0)]):\nif evalf(int(f( x),x=a..b))<0 then \ntitletxt:=`Computing the Area Over the Curve (AOC )`:\nelse titletxt:=`Computing the Area Under the Curve (AUC)`:fi:\nme sh:=400:\nlen:=(b-a)/mesh:\nfor i from 0 to mesh do\nx[i]:=a + i*len: \np[i+1]:=plot(\n [[x[i],0],\n [x[i],f(x[i])]],\n \+ linestyle=1,color=green):\nod:\nShadedArea:=\n plots[display]( \n \+ [seq(p[j],j=0..mesh+1)],\n labels=[``,``],\n #scaling=constrained,\n titlefont=[HELVETICA,DEFAULT,14],\n title=titletxt):\nShadedArea;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 299 4 "NEXT" }{TEXT -1 41 ", f ind the integral of f(x), namely h(x)." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 56 "Digits:=6:\nInt(f(x),x) = int(f(x),x) \+ + 'C';\n'h(x)'=h(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 300 3 "N OW" }{TEXT -1 31 ", use h(x) to compute the area." }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 309 "Digits:=6:\n'h(x)'=h(x);\nbtxt :=convert(b,symbol):\nif evalf(int(f(x),x=a..b))<0 then\n AOC[a,b](f(x )) = abs(Int(f(x),x=a..btxt));\n``= abs('h'(b) - 'h'(a));\n``= abs(h(b ) - h(a));\n``= abs(evalf(h(b) - h(a)));\nelse\nAUC[a,b](f(x)) = Int(f (x),x=a..b);\n``= 'h'(b) - 'h'(a);\n``= h(b) - h(a);\n``= evalf(h(b) - h(a));\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 301 44 "Computing AUC and AOC b y Reversing the ExpCR" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 52 " Arrow expressions for IA Worksheet 2 Exercises 7-11." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 304 50 "To check your work, copy each one and p aste below." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 122 "5*exp(-x):a:=-3:b:=0:\n(-4/3)*x*exp(x^2):a:=0:b:=2: \n(x^2-1)*exp(x^3-3*x):a:=2.2:b:=2.3:\nx^(-2)*exp(x^(-1)):a:=-3.5:b:=- .25:" }}}{PARA 0 "" 0 "" {TEXT 305 16 "TO CHANGE, ENTER" }{TEXT -1 31 " the arrow expression for f(x) " }{TEXT 306 3 "AND" }{TEXT -1 24 " th e values for a and b." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 638 "f:=x->5*exp(-x):a:=-3:b:=0:\nh:=unapply(int(f(x),x), x):\n'f(x)'= f(x);\np[0]:=plots[display](\n [plot(f(x),x=a..b,co lor=blue),\n plot(0,x=0..0)]):\nif evalf(int(f(x),x=a..b))<0 th en \ntitletxt:=`Computing the Area Over the Curve (AOC)`:\nelse titlet xt:=`Computing the Area Under the Curve (AUC)`:fi:\nmesh:=400:\nlen:=( b-a)/mesh:\nfor i from 0 to mesh do\nx[i]:=a + i*len:\np[i+1]:=plot(\n [[x[i],0],\n [x[i],f(x[i])]],\n linestyle=1,c olor=green):\nod:\nShadedArea:=\n plots[display]( \n [seq(p[j],j=0..me sh+1)],\n labels=[``,``],\n #scaling=constrained,\n titlefont=[HELV ETICA,DEFAULT,14],\n title=titletxt):\nShadedArea;" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 307 4 "NEXT" }{TEXT -1 41 ", find the integral \+ of f(x), namely h(x)." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 56 "Digits:=6:\nInt(f(x),x) = int(f(x),x) + 'C';\n'h(x)'= h(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 308 3 "NOW" }{TEXT -1 31 ", use h(x) to compute the area." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 309 "Digits:=6:\n'h(x)'=h(x);\nbtxt:=convert(b,s ymbol):\nif evalf(int(f(x),x=a..b))<0 then\n AOC[a,b](f(x)) = abs(Int( f(x),x=a..btxt));\n``= abs('h'(b) - 'h'(a));\n``= abs(h(b) - h(a));\n` `= abs(evalf(h(b) - h(a)));\nelse\nAUC[a,b](f(x)) = Int(f(x),x=a..b); \n``= 'h'(b) - 'h'(a);\n``= h(b) - h(a);\n``= evalf(h(b) - h(a));\nfi; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 0 "" }{TEXT 302 43 "Computing AUC and AOC by Reversing the LnCR" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 309 53 "Arrow expres sions for IA Worksheet 2 Exercises 11-14." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 310 50 "To check your work, copy each one and paste below. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 123 "-1/(2*x+1) :a:=0:b:=2.5:\n-3/(5*x-2):a:=10:b:=100:\n9*x/(2*x^2-5):a:=1.6:b:=1.65: \n(x^2-2*x+1)/(x^3-3*x^2+3*x-2):a:=2.1:b:=3.3:" }}}{PARA 0 "" 0 "" {TEXT 311 16 "TO CHANGE, ENTER" }{TEXT -1 31 " the arrow expression fo r f(x) " }{TEXT 312 3 "AND" }{TEXT -1 24 " the values for a and b." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 640 "f:=x->-1/(2*x+ 1):a:=0:b:=2.5:\nh:=unapply(int(f(x),x),x):\n'f(x)'= f(x);\np[0]:=plot s[display](\n [plot(f(x),x=a..b,color=blue),\n plot(0,x=0 ..0)]):\nif evalf(int(f(x),x=a..b))<0 then \ntitletxt:=`Computing the \+ Area Over the Curve (AOC)`:\nelse titletxt:=`Computing the Area Under \+ the Curve (AUC)`:fi:\nmesh:=400:\nlen:=(b-a)/mesh:\nfor i from 0 to me sh do\nx[i]:=a + i*len:\np[i+1]:=plot(\n [[x[i],0],\n [ x[i],f(x[i])]],\n linestyle=1,color=green):\nod:\nShadedArea: =\n plots[display]( \n [seq(p[j],j=0..mesh+1)],\n labels=[``,``],\n \+ #scaling=constrained,\n titlefont=[HELVETICA,DEFAULT,14],\n title=ti tletxt):\nShadedArea;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 313 4 " NEXT" }{TEXT -1 41 ", find the integral of f(x), namely h(x)." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 56 "Digits:=6:\nInt (f(x),x) = int(f(x),x) + 'C';\n'h(x)'=h(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 314 3 "NOW" }{TEXT -1 31 ", use h(x) to compute the are a." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 309 "Digits:= 6:\n'h(x)'=h(x);\nbtxt:=convert(b,symbol):\nif evalf(int(f(x),x=a..b)) <0 then\n AOC[a,b](f(x)) = abs(Int(f(x),x=a..btxt));\n``= abs('h'(b) - 'h'(a));\n``= abs(h(b) - h(a));\n``= abs(evalf(h(b) - h(a)));\nelse\n AUC[a,b](f(x)) = Int(f(x),x=a..b);\n``= 'h'(b) - 'h'(a);\n``= h(b) - h (a);\n``= evalf(h(b) - h(a));\nfi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 }