{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 }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 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Module 9 : Integral Calculus" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "902 : Int eresting Integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 13 "P U R P O S E" }}{PARA 0 "" 0 "" {TEXT -1 165 "The purpos e of this project is to learn how to perform definite and indefinite i ntegration using Maple, and to explore various properties and theorems of integrals.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "_________________________________________________________ _________________________" }}{PARA 4 "" 0 "" {TEXT -1 35 "A. Definite \+ and Indefinite Integral" }}{PARA 0 "" 0 "" {TEXT -1 82 "______________ ____________________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "We define a function to use as a guinea p ig." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "\011f := x -> .01*x^4 + 3 + sqrt(x) + 6/x^2 + x*sin(3 *x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operato rG%&arrowGF(,,*$)9$\"\"%\"\"\"$F1!\"#\"\"$F1-%%sqrtG6#F/F1*&\"\"'F1*$) F/\"\"#F1!\"\"F1*&F/F1-%$sinG6#,$F/F4F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "and graph it while markin g off an area to compute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot( \{ f(x), [[1,0],[1,f(1)]], \+ \011[[6,0],[6,f(6)]] \}, x = 0..7, y = 5..20);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6$7]q7$$\"3M+++su7oZ!#? $\"3Jees+)G\"RE!#77$$\"3N*****\\%\\DO&*F*$\"3['\\)3l]0)f'!#87$$\"3#*** ***>CQ/V\"!#>$\"3/6si_KkKHF37$$\"3()******))4D2>F7$\"3'yP.vB](\\;F37$$ \"3++++OP1%Q#F7$\"3QZCfa[&f0\"F37$$\"3%)*****R[w3'GF7$\"3&*Hdbr'**RL(! #97$$\"3K+++J#*oPLF7$\"3];HAP05*Q&FI7$$\"3u*****z(>]9QF7$\"3)3P+%e/zET FI7$$\"3')*****\\s98H%F7$\"3eFa@y%e8E$FI7$$\"31+++tu7oZF7$\"3IO@'F7$\"3?hD\\Vp&[c\"FI7$$\"3k+++i%y`n'F7$\"3Sa,ft- v\\8FI7$$\"31+++47>_rF7$\"3C>m`d^@w6FI7$$\"3S*****\\&R+HwF7$\"3G[vT97> M5FI7$$\"3H+++.n\"e5)F7$\"3s$GI!>'p[;*!#:7$$\"3s******\\%HEe)F7$\"3)Ra 5Pq7&y\")F[q7$$\"3a+++(>U%f!*F7$\"3`%[$fwTxVtF[q7$$\"3()*****H%\\DO&*F 7$\"31*\\E.[-6j'F[q7$$\"32+++pnI,5!#=$\"3MJxt!45y,'F[q7$$\"3/+++W!))*[ 5F^r$\"39jOQabC'[&F[q7$$\"3/+++=$pm4\"F^r$\"3\"y&eyd#4D-&F[q7$$\"3.+++ $f]V9\"F^r$\"3h3%>ceNbh%F[q7$$\"3)******z'=.#>\"F^r$\"3qZ?)4$oUcUF[q7$ $\"3-+++UJrR7F^r$\"3>7sA\"yiz$RF[q7$$\"37+++]BaOF[ q7$$\"34+++#pv]L\"F^r$\"3DO4F:5P+MF[q7$$\"31+++npv#Q\"F^r$\"3aa\"zvcBB <$F[q7$$\"3.+++U#Q/V\"F^r$\"3K#ee\\c3n'HF[q7$$\"3!******f^>\"y9F^r$\"3 7'3ftwy1y#F[q7$$\"3))*****4z+e_\"F^r$\"30YX.PZ#=h#F[q7$$\"31+++9!)Gn:F ^r$\"3#=[FG=wsZ#F[q7$$\"3*)*****fBv(3;F^r$\"3o<&e&Q*3IN#F[q7$$\"3++++e CE];F^r$\"3pI%o#fA+QAF[q7$$\"33+++!o\\F[q7$$\"3%******zc)pd=F^r$\"3'4@n35(*Qx\"F[ q7$$\"3()*****>,t1%>F^r$\"3tFgZ@`eG;F[q7$$\"3()*****pXZO-#F^r$\"3R2!Qb E,3]\"F[q7$$\"32+++,>i1@F^r$\"3Q\\'z!Q(QyQ\"F[q7$$\"33+++Yjf*=#F^r$\"3 SLVggV\\(G\"F[q7$$\"3-+++N_abBF^r$\"3K#3/%4$Qx6\"F[q7$$\"3')*****H7%\\ @DF^r$\"3'*fhX([aX!)*!#;7$$\"3!)*****>,Vuo#F^r$\"35&eWr-'yy')Fdy7$$\"3 u*****4!>R`GF^r$\"3rY7^E/JWxFdy7$$\"36+++jckEKF^r$\"3E'4>s&\\RYhFdy7$$ \"3++++E%**)*f$F^r$\"3^n_ocLl@]Fdy7$$\"3#*******)=`J(RF^r$\"3.6\\r\"=@ 3?%Fdy7$$\"3!******H&pSYVF^r$\"31[Wt=9'Re$Fdy7$$\"3[+++:G'y4&F^r$\"3pW u6x89JFFdy7$$\"3]*****zn=$\\eF^r$\"3%)*)Hm*oWx=#Fdy7$$\"3Y+++,I?(f'F^r $\"3cSaS7&Q0#=Fdy7$$\"3S+++Dt3XtF^r$\"3-n@B\\5Od:Fdy7$$\"3')*****HLc=t )F^r$\"3lrgkm.]C7Fdy7$$\"34+++Kvx;5!#<$\"3Th%)GD)H`\"**F_]l7$$\"37+++y /Gl6F_]l$\"3aDup_xN7\")F_]l7$$\"3(******R<2LJ\"F_]l$\"3Gk9O\"GzQr'F_]l 7$$\"3#******\\#3dl9F_]l$\"3s\\W;y#>ql&F_]l7$$\"3!******Ht%o*f\"F_]l$ \"3)GT@%=PE\"3&F_]l7$$\"31+++F_m]F_]l$\"3Y=YHx(H:8&F_]l7$$\"35+++s2O[?F_]l$\"3)=xVtiw_v&F_]l7$$\"3/+++ G\"H5=#F_]l$\"3cgPi0K$\\_'F_]l7$$\"36+++NYyQBF_]l$\"3_/Y!3U'f!\\(F_]l7 $$\"3')*****HCCCZ#F_]l$\"3?vZ(f9&)z;)F_]l7$$\"3.+++w)yyi#F_]l$\"31=Z#4 \"=`$f)F_]l7$$\"3#******zc#[lFF_]l$\"3s!*zLF!p:`)F_]l7$$\"3')*****fibk \"HF_]l$\"3/%[mGGk+'zF_]l7$$\"31+++!o<-1$F_]l$\"35!\\Q\\GKn+(F_]l7$$\" 3$)*****4I=-@$F_]l$\"3o)y4@7/(zdF_]l7$$\"3!)*****pplzM$F_]l$\"3-$f.ZoQ &yYF_]l7$$\"3')*****>([a'\\$F_]l$\"3qH@V#y\")oz$F_]l7$$\"31+++yo(3l$F_ ]l$\"3)o2?5oh**[$F_]l7$$\"35+++ULA&y$F_]l$\"3O\"R_v%)oZ(QF_]l7$$\"3#)* ****\\?@.$RF_]l$\"3S'3HN$e*e+&F_]l7$$\"3!******R\\@-3%F_]l$\"3)e'\\w! \\-k%oF_]l7$$\"3K+++$opoA%F_]l$\"3gu$z7Euc1*F_]l7$$\"3/+++Y$f(oVF_]l$ \"30V$yB!*3$H6Fdy7$$\"3s*****\\i.j_%F_]l$\"3!3)QpMSoX8Fdy7$$\"3G+++nT' ym%F_]l$\"33#Qj>Ew4[\"Fdy7$$\"3&)*****4E5!>[F_]l$\"3U>(\\HV=@a\"Fdy7$$ \"3++++(3rf&\\F_]l$\"3.U)f>d#Q>:Fdy7$$\"3)******fW0d5&F_]l$\"3g.W(\\+S IU\"Fdy7$$\"3;+++4QfY_F_]l$\"3OkF_]l$\"3W_[&y_aB_#Fdy7$$\"3; +++N9]elF_]l$\"3-dK@=*=C!HFdy7$$\"3d*****\\(GP1nF_]l$\"3 " 0 "" {MPLTEXT 1 0 42 "Int( f(x), x ); va lue(%); int( f(x), x );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ ,,*$)%\"xG\"\"%\"\"\"$F+!\"#\"\"$F+*$-%%sqrtG6#F)F+F+*&\"\"'F+*$)F)\" \"#F+!\"\"F+*&F)F+-%$sinG6#,$F)F.F+F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"&\"\"\"$\"+++++?!#7*&$\"\"$\"\"!F(F&F(F(*&$\"+nmmm m!#5F()F&#F.\"\"#F(F(*&$\"\"'F/F(F&!\"\"F:*&$\"+66666F3F(-%$sinG6#,$F& F-F(F(*($\"+LLLLLF3F(F&F(-%$cosGF@F(F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"&\"\"\"$\"+++++?!#7*&$\"\"$\"\"!F(F&F(F(*&$\"+nmmm m!#5F()F&#F.\"\"#F(F(*&$\"\"'F/F(F&!\"\"F:*&$\"+66666F3F(-%$sinG6#,$F& F-F(F(*($\"+LLLLLF3F(F&F(-%$cosGF@F(F:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "We can also compute the definite in tegral which represents area under the curve. Again, the Int command j ust sets it up, and the int command gets the numerical answer." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "\011Int( f(x), x = 1..6); value(%); \n\011\011\011int( f(x), x = 1 ..6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,,*$)%\"xG\"\"%\" \"\"$F+!\"#\"\"$F+*$-%%sqrtG6#F)F+F+*&\"\"'F+*$)F)\"\"#F+!\"\"F+*&F)F+ -%$sinG6#,$F)F.F+F+/F);F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OQ: $H%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OQ:$H%!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "_________________________________________________________ _________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "B. Linearity Properties" }}{PARA 0 "" 0 "" {TEXT -1 82 "__________________________ ________________________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 301 "A process that takes T(a\341f(x)) + b\341g(x)) to a \341T(f(x)) + b\341T(g(x)) is called a linear transformation. The deri vative is an example of a linear transformation. The integral is anoth er example. \n\nLets define some functions, and explore the propositio n that the integral of a sum is the sum of integrals :" }}{PARA 0 "" 0 "" {METAFILE 205 35 35 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> x^3 + 10*x + \+ 3;\n\011\011" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)o peratorG%&arrowGF(,(*$)9$\"\"$\"\"\"F1*&\"#5F1F/F1F1F0F1F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g := x -> sin(10*x);\n\011 \011" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG %&arrowGF(-%$sinG6#,$9$\"#5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int( c*f(x), x = 0..14) : % = value(%);\n\011\011" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"cG\"\"\",(*$)%\"xG\"\"$ F)F)*&\"#5F)F-F)F)F.F)F)/F-;\"\"!\"#9,$F(\"&E1\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "\011c*Int( f(x), x = 0..14) : % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"cG\"\"\"-%$IntG6$,(*$)%\"xG\" \"$F&F&*&\"#5F&F-F&F&F.F&/F-;\"\"!\"#9F&,$F%\"&E1\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The fact that the resu lts are the same is an indication (but does not formally prove) that t his proposition is valid.\n\nLets explore the proposition " }}{PARA 0 "" 0 "" {METAFILE 130 35 35 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " Int( f(x) + g(x), x) : % = value(%);\n\011\011" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,**$)%\"xG\"\"$\"\"\"F,*&\"# 5F,F*F,F,F+F,-%$sinG6#,$F*F.F,F*,**$)F*\"\"%F,#F,F6*&\"\"&F,)F*\"\"#F, F,*&F+F,F*F,F,*&#F,F.F,-%$cosGF1F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " Int( f(x), x ) + Int( g(x), x ) : % = value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$IntG6$,(*$)%\"xG\"\"$\"\"\" F-*&\"#5F-F+F-F-F,F-F+F--F&6$-%$sinG6#,$F+F/F+F-,**$)F+\"\"%F-#F-F9*& \"\"&F-)F+\"\"#F-F-*&F,F-F+F-F-*&#F-F/F--%$cosGF4F-!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The fact that the results are the same is an indication (but does not formally prove) t hat this proposition is valid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "_______________________________________________ ___________________________________" }}{PARA 4 "" 0 "" {TEXT -1 25 "C. Upper and Lower Bounds" }}{PARA 0 "" 0 "" {TEXT -1 82 "______________ ____________________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 449 "An upper bound for a function is a const ant M which the function never exceeds in a given interval : f(x) M. \+ Similarly, a lower bound, m, for a function is a value which is never \+ larger than the value of the function : m f(x) over some interval. Th ere are many functions which are difficult or impossible to integrate. However, using the upper and lower bound of a function, we can get up per and lower bounds for an integral : \n\011\011\011\011\011\011\011 \011\011\n\011\011\011\011\011\011\011\011\011" }{METAFILE 169 35 35 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }{TEXT -1 151 "\n\nIn the student pac kage of commands, there are maximize and minimize commands which find \+ the maximum and minimum value of a function over an interval." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "\011restart; with(student):\n\011\011" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "g := x -> 1/3*x^3 - 7*x^2 + 35*x + 30; \n\011" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrow GF(,**$)9$\"\"$\"\"\"#F1F0*&\"\"(F1)F/\"\"#F1!\"\"*&\"#NF1F/F1F1\"#IF1 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "minimize( g(x), x =0..14); m := evalf(%); \011\n\011\011\011maximize( g(x),x=0..14); M \+ := evalf(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"$v#\"\"\"*&#F% \"\"$F%),&\"\"(F%*$-%%sqrtG6#\"#9F%F%F(F%F%*&F+F%)F*\"\"#F%!\"\"*&\"#N F%F-F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"*t>69\"!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"$v#\"\"\"*&#F%\"\"$F%),&\"\"(F%*$- %%sqrtG6#\"#9F%!\"\"F(F%F%*&F+F%)F*\"\"#F%F1*&\"#NF%F-F%F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG$\"*!paD\")!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Here is snapshot of the s ituation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "\011plot( \{ g(x), m, M\}, x=0..14, y=0..90 );" }} {PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'CURVESG6$7S7$ $\"\"!F)$\"3^++++paD\")!#;7$$\"3?LLL$e,;0$!#=F*7$$\"3Unm;/Qy1dF0F*7$$ \"3KLLL3R\"Gp)F0F*7$$\"3YLL$etj)p6!#$F:F*7$$\"3i+++b/L,NF:F*7$$\"38+++D8`/QF: F*7$$\"3I+++X:s'4%F:F*7$$\"3]LL3d#e?O%F:F*7$$\"3-nmmr#pvn%F:F*7$$\"3Kn mm'[[[%\\F:F*7$$\"3?++v`xvb_F:F*7$$\"3cmmmO^'4`&F:F*7$$\"3u++v`7\"H$eF :F*7$$\"35,+Dh`V?hF:F*7$$\"3Rnm;/mV?kF:F*7$$\"3EnmT&RJfp'F:F*7$$\"3?LL $eu*3$*pF:F*7$$\"3_LL3dPv,tF:F*7$$\"3Q++D'oY/d(F:F*7$$\"3#RLL3TU1'yF:F *7$$\"3'********)HWg\")F:F*7$$\"3w++]n$RPX)F:F*7$$\"37,+v$p=vt)F:F*7$$ \"3k****\\_sg_!*F:F*7$$\"3olmmO$GdL*F:F*7$$\"3t,++D0-Q'*F:F*7$$\"3qKL3 x@%>\"**F:F*7$$\"3?++]*3T6-\"F,F*7$$\"3km;/i(=$\\5F,F*7$$\"31+]()[Dxy5 F,F*7$$\"3qmm;4!pv5\"F,F*7$$\"3%***\\PMirP6F,F*7$$\"3eLLL&f^n;\"F,F*7$ $\"3SLLeXWW'>\"F,F*7$$\"3(omTSU\"*eA\"F,F*7$$\"3?+++R,&HD\"F,F*7$$\"3y mm\"*zC'RG\"F,F*7$$\"3RLLL(G+69\"F,7$F.Fdu7$F2Fdu7$F5Fdu7$F8Fdu7$Fz VF0$\"3/!R'=;bF,WF,7$F2$\"3Q$Qn&GyfvZF,7$$\"3Q++Dc))z*>(F0$\"3!*=U*>V5 &p^F,7$F5$\"3w-))yDbUNbF,7$$\"3YLLLjDd>5F:$\"3-1uXrL;weF,7$F8$\"3=3Z47 Z))*='F,7$F<$\"3/o\"\\`VCmt'F,7$F?$\"3+8&y'\\u'\\:(F,7$FB$\"3W#*)3]_:I ](F,7$FE$\"3UMr\\lC&ox(F,7$FH$\"3#=@;O#4#y'zF,7$$\"3aL$3-+y)yFF:$\"3Lw +(HBde.)F,7$FK$\"33g7[e3K%3)F,7$$\"3c++vebDlIF:$\"3+-i8fqN6\")F,7$FN$ \"3;4r2s(QU7)F,7$$\"35nmTg*\\.N$F:$\"3Nzu97aSA\")F,7$FQ$\"3d!\\\"e^BP$p!)F,7$FT$\"3RS!Q5Kc$>!)F,7$FW$\"3= $fR[?'>#)yF,7$FZ$\"3nb'[\"y.c9xF,7$Fgn$\"3it;@Lx=nuF,7$Fjn$\"3k:HK]U?@ sF,7$F]o$\"39YhkhTQ)*oF,7$F`o$\"33>M[hkI%e'F,7$Fco$\"3]B+\")3!oU@'F,7$ Ffo$\"3P$[[>zT?%eF,7$Fio$\"3(pXx/.L#QaF,7$F\\p$\"3GcuXjA4e]F,7$F_p$\"3 mvu;yw+VYF,7$Fbp$\"3WG,o\\Rz6UF,7$Fep$\"3h&p&z'f&*3%QF,7$Fhp$\"3w&))QI P$o\\MF,7$F[q$\"3Wb6]XE!31$F,7$F^q$\"3'z[UMG20q#F,7$Faq$\"3+z5f#zdcP#F ,7$Fdq$\"3a>D;06&z/#F,7$Fgq$\"3i/-DPd2)y\"F,7$Fjq$\"3k0ch<30_:F,7$F]r$ \"3%[)QIQdmz8F,7$F`r$\"3hMWt>7NT7F,7$$\"3^L3xD*H_.\"F,$\"3k#*He_c(e>\" F,7$Fcr$\"3=I\"p))R3P;\"F,7$$\"3WL$ealXS1\"F,$\"34Aoo#G<\\9\"F,7$Ffr$ \"3@f*Q?6<>9\"F,7$$\"3QL3-z2<$4\"F,$\"3S!p4$QJ'[:\"F,7$Fir$\"3sA)\\_t5 T=\"F,7$$\"3KL3x@EkA6F,$\"3G\\ci'RYGB\"F,7$F\\s$\"3Md(p\\5)y+8F,7$F_s$ \"3K\\`!R@:$)[\"F,7$Fbs$\"3m0/Z,&>:w\"F,7$Fes$\"3+LN%R^-*=@F,7$Fhs$\"3 (G.!*Q'feFDF,7$F[t$\"3msPU=D!e4$F,7$$\"33+]i$QJyH\"F,$\"3OCH%pg0fQ$F,7 $F^t$\"39357\\o+*p$F,7$$\"3!o;a8V([E8F,$\"3oK#GHIr(eSF,7$Fat$\"33]%4&e K$fW%F,7$$\"3=+++fxUb8F,$\"3*Gg;)eWvU[F,7$Fdt$\"3G,&3xuLeE&F,7$$\"31+D Jr/z%Q\"F,$\"33i8Xl*3/v&F,7$Fgt$\"3)GmmmmmmE'F,-Fjt6&F\\uF]uF]uF(-%+AX ESLABELSG6$Q\"x6\"Q\"yFdgl-%%VIEWG6$;F(Fgt;F($\"#!*F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 237 "Th e horizontal lines indicate the minimum and maximum values. The value \+ m(b-a) represents the area of the smaller rectangle which fits under t he curve, and the value M(b-a) represents the larger rectangle which t he curves fits within. \n" }}{PARA 0 "" 0 "" {TEXT -1 22 "Lets now ve rify that " }}{PARA 0 "" 0 "" {METAFILE 174 34 34 1 "adOIDKMdD\\f;Z;Z DZ::@:::<:2:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Int( m, x=0..14): % = value (%);\n\011\011\011Int( g(x),x=0..14): % = evalf(value(%));\n\011\011 \011Int( M, x = 0..14): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$$\"*t>69\"!\"(/%\"xG;\"\"!\"#9$\"+Awc(f\"F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$,**$)%\"xG\"\"$\"\"\"#F,F+*&\"\"(F,)F* \"\"#F,!\"\"*&\"#NF,F*F,F,\"#IF,/F*;\"\"!\"#9$\"+nmm'['!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$$\"*!paD\")!\"(/%\"xG;\"\"!\"#9$\" +mldP6!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Here you can see the min and max. Thus we have that m g(x) M for the interval. What can we say about the integral of g(x) if were \+ not able to compute it directly?\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "_________ ______________________________________________________________________ ___" }}{PARA 4 "" 0 "" {TEXT -1 35 "D. The Absolute Value and Integral s" }}{PARA 0 "" 0 "" {TEXT -1 82 "____________________________________ ______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Lets explore the proposit ion that " }{METAFILE 122 30 30 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" } {TEXT -1 120 "and see why its true.\n\nLet define a function as an ex ample, and then compute each integral and see which one is larger." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := x -> x^3 - 10*x^2 + 20*x + 10;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"$\" \"\"F1*&\"#5F1)F/\"\"#F1!\"\"*&\"#?F1F/F1F1F3F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Here is " }{METAFILE 53 35 35 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Int( abs( f(x) ), x = -3..8): % = evalf( value(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$absG6#,**$)%\"xG\"\"$\"\"\"F/*&\"#5F/)F-\"\"#F/!\" \"*&\"#?F/F-F/F/F1F//F-;!\"$\"\")$\"+nm;H8!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "and " }{METAFILE 51 35 35 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " abs( Int( f(x) , x = -3..8) ): % = evalf( value(%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$I ntG6$,**$)%\"xG\"\"$\"\"\"F-*&\"#5F-)F+\"\"#F-!\"\"*&\"#?F-F+F-F-F/F-/ F+;!\"$\"\")F2$\"+nm;H8!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The answers ar e different. If we look at the plot of each function we can see some \+ clues as to why there is a difference." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot( f(x), x = -3..8, y = -15..25);\n\011\011" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$!\"$\"\"!$!$n\"F*7$$!33LL3_c6!)G!#<$!3 Gd!\\kCPWa\"!#:7$$!3imm;/8BgFF0$!3b_@**pDBC9F37$$!3#)*\\iSd?fl#F0$!3?& \\I,)=#RK\"F37$$!3YL$eR%)4;b#F0$!3c!GzpB@vA\"F37$$!3smm\"H>$*pJ#F0$!3? 6i+y6jC5F37$$!3gmmT]8#33#F0$!3O$3K8'yT#R)!#;7$$!3[L$e9*>xX=F0$!3&yw^K` \\ss'FM7$$!3!om\"HZ6&yi\"F0$!3n%Q#HBe'pL&FM7$$!34+]iNn?-9F0$!3_>2d2oHY SFM7$$!3ym;Hi\\%)o6F0$!3%[^T#**edjGFM7$$!3+,+DJeJi$*!#=$!3%HMm3@c5$=FM 7$$!3-LLL38gppF_o$!3UcDIEwGN\"*F07$$!3Glm;aq4i[F_o$!3e)[:'*zKJ?#F07$$! 3o(****\\UY&*[#F_o$\"3aQ8(e>$p&Q%F07$$!3_S++](QD2\"!#>$\"3-!>YonSVy*F0 7$$\"3=*****\\UE&)=#F_o$\"3)\\:tH.d3R\"FM7$$\"3eHL3x[JtUF_o$\"3CchNpV& )z;FM7$$\"3qlmm\"**HBv'F_o$\"3,C!Q<%HJD>FM7$$\"3=kmmm4Q_))F_o$\"3]'Qu! y5?c?FM7$$\"3y**\\P\\R_H6F0$\"3@)>SE'4LF@FM7$$\"3wlmm@$edM\"F0$\"3>^6V CoFM7$$\"3'om;/wGY/#F0$\"3oE:***=4Nw\"FM7$$\"3%pmTN&*)3hAF0$\"3AuH5P ;kl:FM7$$\"3yKLe90d%\\#F0$\"3QG+E&y+'=8FM7$$\"3mK$3xB#4PFF0$\"3ntwX530 L5FM7$$\"3)***\\i5\"3#[HF0$\"3A8g^NKZqwF07$$\"3ULL3P!>i<$F0$\"3uYm!\\_ rNo%F07$$\"3&*)****\\jw6buk\\\"F07$$\"3O**\\PfK>lQF0$!3e&)p_@KU[VF07$$\"3Z***\\7%Gw7TF0$!3 5>K0Z$*GEtF07$$\"3*emm;7:_L%F0$!3e#>1oDv,w*F07$$\"3Y****\\7/tsXF0$!3&> ^Y(fd)G?\"FM7$$\"3%GL3xcazy%F0$!3%*[-ow!\\CP\"FM7$$\"3$4++vT^K-&F0$!3p ?W]PTN6:FM7$$\"3il;/;ukW_F0$!3KIdXcK$4f\"FM7$$\"3++](o-qgZ&F0$!3#41q** *Q#Rh\"FM7$$\"3vlm;HzK-dF0$!3a?rkf$)))p:FM7$$\"3g)*\\P%)*)>RfF0$!3))>n RyeqX9FM7$$\"3;MLLjRLnhF0$!3I')H/fPEV7FM7$$\"38LLeH\\j+kF0$!3Y)y'pQ&[l W*F07$$\"3rm;/YS+KmF0$!372Hn,R4'\\&F07$$\"3\"3++]B3Y%oF0$!3'ocE'4BLVW,G B\"FM7$$\"3P**\\P\\feQvF0$\"311q:$R()))3#FM7$$\"3o+]i?J*4w(F0$\"34#e.7 #HxNIFM7$$\"\")F*$\"#UF*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F]\\l-%+AXESL ABELSG6$Q\"x6\"Q\"yFb\\l-%%VIEWG6$;F(Fb[l;$F3F*$\"#DF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot( abs(f(x)), x = -3..8, y = -15..25); \n\n" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6%-%'CURVE SG6$7io7$$!\"$\"\"!$\"$n\"F*7$$!33LL3_c6!)G!#<$\"3Gd!\\kCPWa\"!#:7$$!3 imm;/8BgFF0$\"3b_@**pDBC9F37$$!3#)*\\iSd?fl#F0$\"3?&\\I,)=#RK\"F37$$!3 YL$eR%)4;b#F0$\"3c!GzpB@vA\"F37$$!34+vV=:IMCF0$\"3b4*4F6&pB6F37$$!3smm \"H>$*pJ#F0$\"3?6i+y6jC5F37$$!3mmmmrs!*)>#F0$\"3')Qy7z=A'H*!#;7$$!3gmm T]8#33#F0$\"3O$3K8'yT#R)FR7$$!3%**\\P4n'Hj>F0$\"3!eC,8]')y`(FR7$$!3[L$ e9*>xX=F0$\"3&yw^K`\\ss'FR7$$!39+]Pp:\"ot\"F0$\"3O\"Ha>G\\S,'FR7$$!3!o m\"HZ6&yi\"F0$\"3n%Q#HBe'pL&FR7$$!34+]iNn?-9F0$\"3_>2d2oHYSFR7$$!3ym;H i\\%)o6F0$\"3%[^T#**edjGFR7$$!3+,+DJeJi$*!#=$\"3%HMm3@c5$=FR7$$!3-LLL3 8gppFcp$\"3UcDIEwGN\"*F07$$!3:***\\7=\\e\"fFcp$\"3k$p(='4k%QbF07$$!3Gl m;aq4i[Fcp$\"3e)[:'*zKJ?#F07$$!3#=$3_D#Gbc%Fcp$\"3i)>A#Q`i58F07$$!3O)* \\(oRf*oUFcp$\"3/9#))>/y6Q%Fcp7$$!3l\"3_D)\\n?TFcp$\"36g1E!>-^J*!#?7$$ !3#\\;H#o0RsRFcp$\"3M?`)G2ja9%Fcp7$$!3?[i!R:1T#QFcp$\"3)o$zTt)f[L)Fcp7 $$!3[JLeR<#en$Fcp$\"3KhP2,N_Z7F07$$!3ek;H#3%o#3$Fcp$\"33GM([:V]&GF07$$ !3o(****\\UY&*[#Fcp$\"3aQ8(e>$p&Q%F07$$!3%3+++:+%)H\"Fcp$\"3;i%4j]ECB( F07$$!3_S++](QD2\"!#>$\"3-!>YonSVy*F07$$\"3=*****\\UE&)=#Fcp$\"3)\\:tH .d3R\"FR7$$\"3eHL3x[JtUFcp$\"3CchNpV&)z;FR7$$\"3qlmm\"**HBv'Fcp$\"3,C! 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The fundamental Theorem" }} {PARA 0 "" 0 "" {TEXT -1 82 "_________________________________________ _________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 49 "One version of the Fundamental Theorem is that if" }{METAFILE 82 46 46 1 "adOIDKMdD\\f;Z;ZDZ::@:::<:2:" }{TEXT -1 96 " , then F\325(x) \+ = f(x). We begin be defining f(x), then define F(x) by way of an integ ral of f(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> x^2 - 10*x + 30;\n\011\011" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$ \"\"#\"\"\"F1*&\"#5F1F/F1!\"\"\"#IF1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "F := x -> int( f(t), t = 0..x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-% \"fG6#%\"tG/F2;\"\"!9$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 108 "Locate f(x) by noting the Y-intercept fo 30. The other graph is F(x) which is measuring the area under f(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot( \{ f(x), F(x) \}, x = 0..10);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"#IF)7$ $\"3emmm;arz@!#=$\"3s()4j<'zny#!#;7$$\"3[LL$e9ui2%F/$\"3(elGjf))*3EF27 $$\"3nmmm\"z_\"4iF/$\"31&fB#*HQwT#F27$$\"39ommT&phN)F/$\"3!4TU_h3UB#F2 7$$\"3KLLe*=)H\\5!#<$\"3!G&H[zW!31#F27$$\"3smm\"z/3uC\"FE$\"3QgS)ej%>3 >F27$$\"3!****\\7LRDX\"FE$\"3y>Gj>xWe%eZ\"F27$$\"3oLLL3En$4#FE$\"3x'>b2*Q nW8F27$$\"3#pmmT!RE&G#FE$\"3eS?+2#zpB\"F27$$\"3D+++D.&4]#FE$\"3[g<\"y# \\_C6F27$$\"3;+++vB_Nu8<&F E7$$\"3]LLeR\"3Gy%FE$\"3u\\7B/BY2aFE$\"3aG9B:_-m^FE7$ $\"3Znm;zXu9cFE$\"3C+[h(*3\"zP&FE7$$\"34+++]y))GeFE$\"3gAw(y1bqo&FE7$$ \"3H++]i_QQgFE$\"3O'>xL&RCygFE7$$\"3b++D\"y%3TiFE$\"3Uc.DM9HSlFE7$$\"3 +++]P![hY'FE$\"3U0^'y1!f\\rFE7$$\"3iKLL$Qx$omFE$\"37t\"=K4$[$y(FE7$$\" 3Y+++v.I%)oFE$\"3uN^A.ze]&)FE7$$\"3?mm\"zpe*zqFE$\"3\\\">R]=GiK*FE7$$ \"3;,++D\\'QH(FE$\"3CEXTH;=E5F27$$\"3%HL$e9S8&\\(FE$\"38uI2v$pD7\"F27$ $\"3s++D1#=bq(FE$\"3)>-Nk(G)>B\"F27$$\"3\"HLL$3s?6zFE$\"3u&>&)4u7vM\"F 27$$\"3a***\\7`Wl7)FE$\"3qG*)eq!GvZ\"F27$$\"3enmmm*RRL)FE$\"3MMP8qb^6; F27$$\"3%zmmTvJga)FE$\"3gKe:?TVdF2 7$$\"31,++]Qk\\*)FE$\"3'GG%=a'o*f?F27$$\"3![LL3dg6<*FE$\"3emuw]!e)RAF2 7$$\"3%ymmmw(Gp$*FE$\"3m*H%zev14CF27$$\"3C++D\"oK0e*FE$\"3]J**Rkz7)f#F 27$$\"35,+v=5s#y*FE$\"3;0$>V.Uuy#F27$$\"#5F)F*-%'COLOURG6&%$RGBG$Ffz! \"\"F(F(-F$6$7S7$F(F(7$F-$\"3Khm]V./0jFE7$F4$\"35,j/.*f?9\"F27$F9$\"3A 6<^Zu&zn\"F27$F>$\"35name@kmF27$Fbs$\"3s#\\t(R'pZx'F2 7$Fgs$\"3'QjL6f_E(oF27$F\\t$\"3kBs57Hy\")pF27$Fat$\"3n@[$\\k$4+rF27$Ff t$\"3B=Ag&Q!=BsF27$F[u$\"3G)f\\D8I4N(F27$F`u$\"3QFV1%f%z/vF27$Feu$\"3S %>()H._cl(F27$Fju$\"3?*e]ZKH=$yF27$F_v$\"3N=)z.)=f1!)F27$Fdv$\"3hRWY!* *Gf@)F27$Fiv$\"3IS#>?oJ?V)F27$F^w$\"3ih5Tpnbz')F27$Fcw$\"3BX,2r')pW*)F 27$Fhw$\"31AFP81q[#*F27$F]x$\"3%R/Z4ry)o&*F27$Fbx$\"3Gr]O$)H)f#**F27$F gx$\"3fq1A2K;J5!#:7$F\\y$\"3'\\sRD@E&p5F_cl7$Fay$\"3ES>)=FKr6\"F_cl7$F fy$\"3Q/gMRK " 0 "" {MPLTEXT 1 0 23 "\011F(x); diff( F(x), x);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"#F(F'*&\"\"&F ()F&\"\"#F(!\"\"*&\"#IF(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$ )%\"xG\"\"#\"\"\"F(*&\"#5F(F&F(!\"\"\"#IF(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "2 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }