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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 83 "Lab # 6 Antidifferentiati
on, Integral Curves, Direction Fields, Area Appoximations " }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 72 "1. Antidifferentiation: The  \"Int\" an
d \"int\" commands, Integral Curves. " }}{PARA 0 "" 0 "" {TEXT -1 101 
"The command for antidifferentiation (Indefinite Integration) is \"int
(function, variable)\".    Note:  " }{TEXT 261 56 "Maple does not put \+
the \"+C\" in the answer of integration" }{TEXT -1 79 ".  We start wit
h an easy one. What is an antiderivative of the function f(x) = " }
{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 15 "?  ie. What is " }
{XPPEDIT 18 0 "Int(x^2,x);" "6#-%$IntG6$*$%\"xG\"\"#F'" }{TEXT -1 4 " \+
 ? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Int(x^2" }{TEXT -1 0 "
" }{MPLTEXT 1 0 5 ", x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*
$)%\"xG\"\"#\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "This has
 done us little good.  It just rewrote the integral in mathematical no
tation.  The only use I can think of for the \"Int\" with a capital \"
I\" is that it produces mathematical notation good for cutting and pas
ting into text.  You can actually evalute this integral with the comma
nd \"value\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"$\"\"\"#F(F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Note:  This is \"an\" antiderivat
ive. Specifically, the one where the constant of integration equals ze
ro." }}{PARA 0 "" 0 "" {TEXT -1 71 "The \"int\" command with a small \+
\"i\" replaces this sequence of commands. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "int(x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$
)%\"xG\"\"$\"\"\"#F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Notice,
 you must put in the independent variable.  Here's what happens if you
 don't.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(x^2);" }}
{PARA 8 "" 1 "" {TEXT -1 52 "Error, (in int) wrong number (or type) of
 arguments\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 87 "How about a more difficult problem. What is the antider
ivative of the function f(x) =  " }{XPPEDIT 18 0 "(exp(x)+exp(-x))/(ex
p(x)-exp(-x));" "6#*&,&-%$expG6#%\"xG\"\"\"-F&6#,$F(!\"\"F)F),&-F&6#F(
F)-F&6#,$F(F-F-F-" }{TEXT -1 22 "   ?  If you let u =  " }{XPPEDIT 18 
0 "exp(x)-exp(-x);" "6#,&-%$expG6#%\"xG\"\"\"-F%6#,$F'!\"\"F," }{TEXT 
-1 57 " you will notice that you have an integral of the form   " }
{XPPEDIT 18 0 "Int(1/u,u);" "6#-%$IntG6$*&\"\"\"F'%\"uG!\"\"F(" }
{TEXT -1 73 "  which you should recognize as ln(|u|) + C.  Can Maple f
igure this out? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x \+
-> (exp(x) + exp(-x))/(exp(x) - exp(-x));  # this defines the integran
d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&
arrowGF(*&,&-%$expG6#9$\"\"\"-F/6#,$F1!\"\"F2F2,&F.F2F3F6F6F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "int(f(x),x);   # this finds \+
the antiderivative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&-%$exp
G6#%\"xG\"\"\"-F(6#,$F*!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 
"Maple got it. Again, the arbitrary constant is not included in the an
swer.    The actual answer is " }{XPPEDIT 18 0 "ln(exp(x)-exp(-x));" "
6#-%#lnG6#,&-%$expG6#%\"xG\"\"\"-F(6#,$F*!\"\"F/" }{TEXT -1 7 "  + C. \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 16 "Integral Curves:" }{TEXT -1 
215 " Suppose we want to plot some integral curves.  This amounts to p
lotting the antideritive of f(x) for various values of the arbitrary c
onstant C.  Here I define a function (the antiderivative of f(x))  of \+
x and C.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := (x,C) ->
 ln(exp(x) - exp(-x)) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*
6$%\"xG%\"CG6\"6$%)operatorG%&arrowGF),&-%#lnG6#,&-%$expG6#9$\"\"\"-F3
6#,$F5!\"\"F:F69%F6F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "and \+
plot this function over a range of x values for different values of C \+
by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([F(x,-1),F(x,0)
,F(x,1)],x=0..1,color=[red,green,blue]);" }}{PARA 13 "" 1 "" 
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\\#F]p$!3>ah\\v@bQ:F-7$$\"3))******e'HU\"GF]p$!3^f<gW2<;9F-7$$\"3:+++7
*309$F]p$!31F%[cIbKI\"F-7$$\"3-+++ce*yU$F]p$!3*)3m7e)pD@\"F-7$$\"3w***
**z[D9v$F]p$!3a)z(3#HU&=6F-7$$\"3')*****>c$GwSF]p$!3fw;z!3(HJ5F-7$$\"3
?+++7XM*Q%F]p$!3ib80\"3x$H&*F]p7$$\"3'******p%QjtYF]p$!3)>0o][U%f))F]p
7$$\"3.+++i8o6]F]p$!3h\"=&QMSO2\")F]p7$$\"31+++]>0)H&F]p$!3;mi?i2L.vF]
p7$$\"3\"******>-p6j&F]p$!3O4kB_#\\S$oF]p7$$\"3d*****\\2Mg#fF]p$!35HGS
ur4oiF]p7$$\"3M+++sxa\\iF]p$!3;AK@2lZscF]p7$$\"3U+++;$4wb'F]p$!3K0A'*e
5GF^F]p7$$\"3[+++>#R!zoF]p$!3EBBR'yZ)yXF]p7$$\"3Y+++4A@urF]p$!3i\\6%p*
pu\"4%F]p7$$\"3%)*****f:'f#\\(F]p$!3C9e<G4N#e$F]p7$$\"3*)*****pf2L#yF]
p$!3e[C+*oQ\"pIF]p7$$\"3x*****z(G>6\")F]p$!3!ee(3&zgVj#F]p7$$\"3M+++po
6A%)F]p$!3'\\O<)z!)=w@F]p7$$\"3Z*****\\xJLu)F]p$!3TL#))G`4Ur\"F]p7$$\"
3e*****R*ydd!*F]p$!3)eH%=e\\Xs7F]p7$$\"3%******><F;O*F]p$!3.vXb9,GQ&)F
J7$$\"3w*****f0A#*p*F]p$!3)z,4=$f&H)RFJ7$$\"3%******zgc-+\"F-$\"3#o14:
c;\"pL!#@7$$\"3,+++c]kK5F-$\"3#p')>v3/*[UFJ7$$\"3'******\\!Q*>1\"F-$\"
3A![9\"*ok!3!)FJ7$$\"35+++R(zS4\"F-$\"3XWLFSAr07F]p7$$\"36+++-,FC6F-$
\"33mOZ&4P8e\"F]p7$$\"3-+++Jx#e:\"F-$\"3o7[LBV$)o>F]p7$$\"3!******43\"
o'=\"F-$\"3*pvYW<wHM#F]p7$$\"3'*******z;)*=7F-$\"36a6HH'[+t#F]p7$$\"35
+++&*44]7F-$\"3M%R'*=U-()4$F]p7$$\"3++++jZ!>G\"F-$\"3gT<+(o/=Z$F]p7$$
\"32+++(4bMJ\"F-$\"3:vxtu/?QQF]p7$$\"36+++ylWU8F-$\"3Q8*Qx(Q#><%F]p7$$
\"33+++'3ucP\"F-$\"3#y(G;`\"o6b%F]p7$$\"3++++lJR09F-$\"3o)G7;rMw)[F]p7
$$\"3-+++-*zqV\"F-$\"3c_*H>nIPC&F]p7$$\"31+++`\"3uY\"F-$\"3e&\\+2sq@e&
F]p7$$\"3++++++++:F-$\"34$)HdqFWVfF]p-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"
!Fj^lFi^l-%+AXESLABELSG6$Q\"x6\"Q!F__l-%%VIEWG6$;Fi^l$\"#:Fh^l%(DEFAUL
TG" 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 "" {TEXT -1 108 "It sure looks like this goes \+
through the point (1,0).  How can you be sure? Evaluate F at x = 1 and
 C = C1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(1,C1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&-%$expG6#\"\"\"F+-F)6#!\"
\"F.F+$\"+?a'ea)!#5F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Is this \+
zero?  Have Maple evaluate this as a floating point number" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(F(1,C1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Bingo
. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 10 "Summary:  \+
" }}{PARA 0 "" 0 "" {TEXT -1 91 "1) Use the \"int(function, variable)
\" to evaluate indefinite integrals (antidifferentiate). " }}{PARA 0 "
" 0 "" {TEXT -1 88 "2) If you want to see the integral in mathematical
 notation use \"Int(function,variable)\"" }}{PARA 0 "" 0 "" {TEXT -1 
78 "3) Remember, that the constant of integration is excluded from Map
le's answer." }}{PARA 0 "" 0 "" {TEXT -1 76 "4) You can solve for the \+
constant of integration using Maple if necessary.  " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 44 "2. Area Approximations, Summation Techniques" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 442 "In this section you will learn h
ow to use Maple to visualize the approximate area under a positive fun
ction using boxes (rectangles really).   There is another command to f
ind the summed area of these boxes (again rectangles).  This command w
ill be used with the limit command to find the area under the curve.  \+
The maple commands that draw boxes and compute the areas are part of t
he \"student\" package which you must load in order to access." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "restart:    # this clears al
l variables" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(student
):  # this loads the student library" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "To approximate the area under the cuve " }{XPPEDIT 18 0 "f(x) =
 x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 102 " over the interval f
rom 1 to 3, several boxes can be used.   The following command draws t
he graph of " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#
" }{TEXT -1 41 " and six boxes using the left end points." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "leftbox(x^2, x = 1..3,6);" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6&7S7$$\"\"
\"\"\"!F(7$$\"3ALLL3VfV5!#<$\"3EGdQ!3*3*3\"F.7$$\"3smm\"H[D:3\"F.$\"3C
H\")>qtpp6F.7$$\"3XLL$e0$=C6F.$\"3krF-Vvyj7F.7$$\"3QLL$3RBr;\"F.$\"3%y
HI%4q<i8F.7$$\"3imm\"zjf)47F.$\"3iZ)fMMgPY\"F.7$$\"3WLLe4;[\\7F.$\"3gd
po#H/7c\"F.7$$\"3-++Dmy]!H\"F.$\"3+GJ&Gb5am\"F.7$$\"3>LLezs$HL\"F.$\"3
)o,.8z@nx\"F.7$$\"31++D@1Bv8F.$\"3?hOih#f7*=F.7$$\"3pmmm@Xt=9F.$\"3PX(
oHk2G,#F.7$$\"3MLL$3y_qX\"F.$\"3r\\T81G+B@F.7$$\"3'******\\1!>+:F.$\"3
+/Z7J-d]AF.7$$\"3*******\\Z/Na\"F.$\"3yD]MkgS#Q#F.7$$\"35+++NfC&e\"F.$
\"3kCSVuY+8DF.7$$\"3LLLez6:B;F.$\"3`mFy^(>Yj#F.7$$\"3_mmm\"=C#o;F.$\"3
->uH?>(Hy#F.7$$\"3gmmmEpS1<F.$\"3PyfP*fC=\"HF.7$$\"3%)***\\i`A3v\"F.$
\"33#)3W`&z`1$F.7$$\"3Ymmmwy8!z\"F.$\"3&4kwuh$f/KF.7$$\"3/++DOIFL=F.$
\"3!RHTa-!*3O$F.7$$\"3!****\\(3zMu=F.$\"3y(\\NI3!=8NF.7$$\"3emm;H_?<>F
.$\"3e5Su!*envOF.7$$\"3mmm\"zihl&>F.$\"3%p\"f$QSL\"GQF.7$$\"3KLL$3#G,*
*>F.$\"3CSg&yD_g*RF.7$$\"3<LLezw5V?F.$\"3'=Rs.**)GuTF.7$$\"3o***\\PQ#
\\\"3#F.$\"37c#4Oa5EL%F.7$$\"3BLL$e\"*[H7#F.$\"3CDzB*47p]%F.7$$\"3#***
****pvxl@F.$\"3'[5:F[#f!p%F.7$$\"3z****\\_qn2AF.$\"3u'3N\"oz$Q([F.7$$
\"3%)***\\i&p@[AF.$\"3)Q,qB[zW0&F.7$$\"3#)****\\2'HKH#F.$\"3W.YrK?!*e_
F.7$$\"3_mmmwanLBF.$\"3c$R&RI7/YaF.7$$\"3u*****\\2goP#F.$\"3G/!Hh\"QY
\\cF.7$$\"3CLLeR<*fT#F.$\"3F+\\t&3;q$eF.7$$\"3'******\\)HxeCF.$\"3I5ew
\"fkb/'F.7$$\"3Cmm\"H!o-*\\#F.$\"3y$*eph\\8XiF.7$$\"3))***\\7k.6a#F.$
\"3U3Sd:x?dkF.7$$\"3emmmT9C#e#F.$\"3/X21j3(zm'F.7$$\"3\"****\\i!*3`i#F
.$\"3=rbB`oC#*oF.7$$\"3;LLL$*zymEF.$\"3YDoQ,#e<6(F.7$$\"30LL$3N1#4FF.$
\"3zkcR^!*zRtF.7$$\"3kmm\"HYt7v#F.$\"3wm%\\xm0&pvF.7$$\"3%*******p(G**
y#F.$\"3!Hrt;a-Py(F.7$$\"3Umm;9@BMGF.$\"3M_O(pnrG.)F.7$$\"3/LLL`v&Q(GF
.$\"35^5&oBd!f#)F.7$$\"30++DOl5;HF.$\"3,s*f2LxO])F.7$$\"3/++v.UacHF.$
\"3<@xsGO:T()F.7$$\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*
F][l-%*THICKNESSG6#\"\"#-%&STYLEG6#%%LINEG-%)POLYGONSG6$7&7$F(F][lF'7$
$\"+LLLL8!\"*F(7$F\\\\lF][l-%&COLORG6&Fiz$\"\"(!\"\"$FezFe\\lFc\\l-Fg[
l6$7&F_\\l7$F\\\\l$\"+yxxx<F^\\l7$$\"+nmmm;F^\\lF[]l7$F^]lF][lF`\\l-Fg
[l6$7&F`]l7$F^]l$\"+yxxxFF^\\l7$$Fa[lF*Fe]l7$Fh]lF][lF`\\l-Fg[l6$7&Fi]
l7$Fh]l$\"\"%F*7$$\"+LLLLBF^\\lF^^l7$Fa^lF][lF`\\l-Fg[l6$7&Fc^l7$Fa^l$
\"+WWWWaF^\\l7$$\"+nmmmEF^\\lFh^l7$F[_lF][lF`\\l-Fg[l6$7&F]_l7$F[_l$\"
+6666rF^\\l7$FbzFb_l7$FbzF][lF`\\l-%+AXESLABELSG6$Q\"x6\"Q!Fj_l-%%VIEW
G6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "To find the comb
ined area of these boxes we use the \"leftsum\" command" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "leftsum(x^2,x=1..3,6);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$-%$SumG6$*$),&\"\"\"F**&#F*\"\"$F*%\"iGF*F*\"
\"#F*/F.;\"\"!\"\"&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eva
lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p.Pqt!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 114 "From the graph, you may assume that this
 value is smaller than the actual area under the curve.  Using rightbo
xes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rightbox(x^2,x=1..3
,6);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%)POLYG
ONSG6$7&7$$\"\"\"\"\"!$F*F*7$F($\"+yxxx<!\"*7$$\"+LLLL8F/F-7$F1F+-%&CO
LORG6&%$RGBG$\"\"(!\"\"$\"\"*F:F8-F$6$7&F37$F1$\"+yxxxFF/7$$\"+nmmm;F/
FA7$FDF+F4-F$6$7&FF7$FD$\"\"%F*7$$\"\"#F*FK7$FNF+F4-F$6$7&FP7$FN$\"+WW
WWaF/7$$\"+LLLLBF/FU7$FXF+F4-F$6$7&FZ7$FX$\"+6666rF/7$$\"+nmmmEF/Fin7$
F\\oF+F4-F$6$7&F^o7$F\\o$F<F*7$$\"\"$F*Fco7$FeoF+F4-%'CURVESG6&7S7$F(F
(7$$\"3ALLL3VfV5!#<$\"3EGdQ!3*3*3\"F`p7$$\"3smm\"H[D:3\"F`p$\"3CH\")>q
tpp6F`p7$$\"3XLL$e0$=C6F`p$\"3krF-Vvyj7F`p7$$\"3QLL$3RBr;\"F`p$\"3%yHI
%4q<i8F`p7$$\"3imm\"zjf)47F`p$\"3iZ)fMMgPY\"F`p7$$\"3WLLe4;[\\7F`p$\"3
gdpo#H/7c\"F`p7$$\"3-++Dmy]!H\"F`p$\"3+GJ&Gb5am\"F`p7$$\"3>LLezs$HL\"F
`p$\"3)o,.8z@nx\"F`p7$$\"31++D@1Bv8F`p$\"3?hOih#f7*=F`p7$$\"3pmmm@Xt=9
F`p$\"3PX(oHk2G,#F`p7$$\"3MLL$3y_qX\"F`p$\"3r\\T81G+B@F`p7$$\"3'******
\\1!>+:F`p$\"3+/Z7J-d]AF`p7$$\"3*******\\Z/Na\"F`p$\"3yD]MkgS#Q#F`p7$$
\"35+++NfC&e\"F`p$\"3kCSVuY+8DF`p7$$\"3LLLez6:B;F`p$\"3`mFy^(>Yj#F`p7$
$\"3_mmm\"=C#o;F`p$\"3->uH?>(Hy#F`p7$$\"3gmmmEpS1<F`p$\"3PyfP*fC=\"HF`
p7$$\"3%)***\\i`A3v\"F`p$\"33#)3W`&z`1$F`p7$$\"3Ymmmwy8!z\"F`p$\"3&4kw
uh$f/KF`p7$$\"3/++DOIFL=F`p$\"3!RHTa-!*3O$F`p7$$\"3!****\\(3zMu=F`p$\"
3y(\\NI3!=8NF`p7$$\"3emm;H_?<>F`p$\"3e5Su!*envOF`p7$$\"3mmm\"zihl&>F`p
$\"3%p\"f$QSL\"GQF`p7$$\"3KLL$3#G,**>F`p$\"3CSg&yD_g*RF`p7$$\"3<LLezw5
V?F`p$\"3'=Rs.**)GuTF`p7$$\"3o***\\PQ#\\\"3#F`p$\"37c#4Oa5EL%F`p7$$\"3
BLL$e\"*[H7#F`p$\"3CDzB*47p]%F`p7$$\"3#*******pvxl@F`p$\"3'[5:F[#f!p%F
`p7$$\"3z****\\_qn2AF`p$\"3u'3N\"oz$Q([F`p7$$\"3%)***\\i&p@[AF`p$\"3)Q
,qB[zW0&F`p7$$\"3#)****\\2'HKH#F`p$\"3W.YrK?!*e_F`p7$$\"3_mmmwanLBF`p$
\"3c$R&RI7/YaF`p7$$\"3u*****\\2goP#F`p$\"3G/!Hh\"QY\\cF`p7$$\"3CLLeR<*
fT#F`p$\"3F+\\t&3;q$eF`p7$$\"3'******\\)HxeCF`p$\"3I5ew\"fkb/'F`p7$$\"
3Cmm\"H!o-*\\#F`p$\"3y$*eph\\8XiF`p7$$\"3))***\\7k.6a#F`p$\"3U3Sd:x?dk
F`p7$$\"3emmmT9C#e#F`p$\"3/X21j3(zm'F`p7$$\"3\"****\\i!*3`i#F`p$\"3=rb
B`oC#*oF`p7$$\"3;LLL$*zymEF`p$\"3YDoQ,#e<6(F`p7$$\"30LL$3N1#4FF`p$\"3z
kcR^!*zRtF`p7$$\"3kmm\"HYt7v#F`p$\"3wm%\\xm0&pvF`p7$$\"3%*******p(G**y
#F`p$\"3!Hrt;a-Py(F`p7$$\"3Umm;9@BMGF`p$\"3M_O(pnrG.)F`p7$$\"3/LLL`v&Q
(GF`p$\"35^5&oBd!f#)F`p7$$\"30++DOl5;HF`p$\"3,s*f2LxO])F`p7$$\"3/++v.U
acHF`p$\"3<@xsGO:T()F`pFdo-%'COLOURG6&F7$\"*++++\"!\")F+F+-%*THICKNESS
G6#FO-%&STYLEG6#%%LINEG-%+AXESLABELSG6$Q\"x6\"Q!Fj_l-%%VIEWG6$;F(Feo%(
DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(rightsum(x^2,x=1..3,6)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/Pq.5!\")" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 168 "and we may safely assume this value is larger \+
than the actual area under the curve.  Therefore, it is safe to assume
 that the actual area is between 7.37 and 10.04.    " }}{PARA 0 "" 0 "
" {TEXT -1 41 "Question: How do we find the exact area ?" }}{PARA 0 "
" 0 "" {TEXT -1 94 "Answer: By taking the limit of either of the above
 as the number of boxes goes to infinity.   " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 66 "First lets try 50 boxes evaluating f at the right-hand \+
endpoints. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rightbox(x^2
, x=1..3,50);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6
W-%'CURVESG6&7S7$$\"\"\"\"\"!F(7$$\"3ALLL3VfV5!#<$\"3EGdQ!3*3*3\"F.7$$
\"3smm\"H[D:3\"F.$\"3CH\")>qtpp6F.7$$\"3XLL$e0$=C6F.$\"3krF-Vvyj7F.7$$
\"3QLL$3RBr;\"F.$\"3%yHI%4q<i8F.7$$\"3imm\"zjf)47F.$\"3iZ)fMMgPY\"F.7$
$\"3WLLe4;[\\7F.$\"3gdpo#H/7c\"F.7$$\"3-++Dmy]!H\"F.$\"3+GJ&Gb5am\"F.7
$$\"3>LLezs$HL\"F.$\"3)o,.8z@nx\"F.7$$\"31++D@1Bv8F.$\"3?hOih#f7*=F.7$
$\"3pmmm@Xt=9F.$\"3PX(oHk2G,#F.7$$\"3MLL$3y_qX\"F.$\"3r\\T81G+B@F.7$$
\"3'******\\1!>+:F.$\"3+/Z7J-d]AF.7$$\"3*******\\Z/Na\"F.$\"3yD]MkgS#Q
#F.7$$\"35+++NfC&e\"F.$\"3kCSVuY+8DF.7$$\"3LLLez6:B;F.$\"3`mFy^(>Yj#F.
7$$\"3_mmm\"=C#o;F.$\"3->uH?>(Hy#F.7$$\"3gmmmEpS1<F.$\"3PyfP*fC=\"HF.7
$$\"3%)***\\i`A3v\"F.$\"33#)3W`&z`1$F.7$$\"3Ymmmwy8!z\"F.$\"3&4kwuh$f/
KF.7$$\"3/++DOIFL=F.$\"3!RHTa-!*3O$F.7$$\"3!****\\(3zMu=F.$\"3y(\\NI3!
=8NF.7$$\"3emm;H_?<>F.$\"3e5Su!*envOF.7$$\"3mmm\"zihl&>F.$\"3%p\"f$QSL
\"GQF.7$$\"3KLL$3#G,**>F.$\"3CSg&yD_g*RF.7$$\"3<LLezw5V?F.$\"3'=Rs.**)
GuTF.7$$\"3o***\\PQ#\\\"3#F.$\"37c#4Oa5EL%F.7$$\"3BLL$e\"*[H7#F.$\"3CD
zB*47p]%F.7$$\"3#*******pvxl@F.$\"3'[5:F[#f!p%F.7$$\"3z****\\_qn2AF.$
\"3u'3N\"oz$Q([F.7$$\"3%)***\\i&p@[AF.$\"3)Q,qB[zW0&F.7$$\"3#)****\\2'
HKH#F.$\"3W.YrK?!*e_F.7$$\"3_mmmwanLBF.$\"3c$R&RI7/YaF.7$$\"3u*****\\2
goP#F.$\"3G/!Hh\"QY\\cF.7$$\"3CLLeR<*fT#F.$\"3F+\\t&3;q$eF.7$$\"3'****
**\\)HxeCF.$\"3I5ew\"fkb/'F.7$$\"3Cmm\"H!o-*\\#F.$\"3y$*eph\\8XiF.7$$
\"3))***\\7k.6a#F.$\"3U3Sd:x?dkF.7$$\"3emmmT9C#e#F.$\"3/X21j3(zm'F.7$$
\"3\"****\\i!*3`i#F.$\"3=rbB`oC#*oF.7$$\"3;LLL$*zymEF.$\"3YDoQ,#e<6(F.
7$$\"30LL$3N1#4FF.$\"3zkcR^!*zRtF.7$$\"3kmm\"HYt7v#F.$\"3wm%\\xm0&pvF.
7$$\"3%*******p(G**y#F.$\"3!Hrt;a-Py(F.7$$\"3Umm;9@BMGF.$\"3M_O(pnrG.)
F.7$$\"3/LLL`v&Q(GF.$\"35^5&oBd!f#)F.7$$\"30++DOl5;HF.$\"3,s*f2LxO])F.
7$$\"3/++v.UacHF.$\"3<@xsGO:T()F.7$$\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBG$
\"*++++\"!\")$F*F*F][l-%*THICKNESSG6#\"\"#-%&STYLEG6#%%LINEG-%)POLYGON
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ABELSG6$Q\"x6\"Q!F\\[n-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu
rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C
urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve
 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23
" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "C
urve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve
 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42
" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "C
urve 49" "Curve 50" "Curve 51" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "evalf(rightsum(x^2,x=1..3,50));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+++?F))!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "It seems
 as though this is getting closer to the exact area by inspection of t
he graph, but is still too big.  We create a function of the number of
 boxes by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "boxarea := n -
> rightsum(x^2, x=1..3,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(boxar
eaGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%)rightsumG6%*$)%\"xG\"\"#\"\"
\"/F1;F3\"\"$9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ev
alf(boxarea(100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++!ou)!\"*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This should be closer still.  T
o get the limit as \"n\" goes to infinity we use the \"limit\" command
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(boxarea(n),n=inf
inity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#E\"\"$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 81 "This is the exact area which we can check
 by the fundemental theorem of calculus." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 47 "We can see that our error in using 100 boxes is" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(boxarea(100) - Limit(b
oxarea(n),n=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")LL8!)!
\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "Not bad. Notice that usin
g right boxes to approximate the area under an increasing function wil
l always be greater than the exact area. If the function is decreasing
, do you expect the approximation with right boxes to be larger or sma
ller than the exact area?  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "# end of this section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 59 "3.
 Integral Curves and Direction Fields, Combining Graphs. " }}{PARA 0 "
" 0 "" {TEXT -1 104 "Here we will investigate the direction field and \+
integral curves for a differential equation of the form" }}{PARA 257 "
" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "dy/dx = f(x);" "6#/*&%#dyG\"\"
\"%#dxG!\"\"-%\"fG6#%\"xG" }{TEXT -1 24 "                   (1). " }}
{PARA 0 "" 0 "" {TEXT -1 251 "The command we need to plot direction fi
elds is in the \"DEtools\" library and the command we need to combine \+
graphs is in the \"plots\" library. Therefore, if you want to use thes
e commands you must first import these libraries with the \"with\" com
mand.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart; # this c
lears all variables and restarts" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "with(DEtools):  # we need the \"dfieldplot\" program \+
in this library.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "with(
plots):  # we need the \"display\" program in this library. " }}{PARA 
7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin
ed\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "In this example, f(x) in \+
equation (1) above, will be " }{XPPEDIT 18 0 "sqrt(x^2+1);" "6#-%%sqrt
G6#,&*$%\"xG\"\"#\"\"\"F*F*" }{TEXT -1 210 "  .  We're looking for fun
ctions that have this as its derivative. Plotting a direction field am
ounts to plotting little segments of lines that have slope f(x) over a
 range of x values.  First we define f(x).  " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "f := x -> sqrt(x^2 + 1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&*$
)9$\"\"#\"\"\"F4F4F4F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Th
e command \"dfieldplot(diff(y(x),x) = f(x), y(x), xrange,yrange)\" wil
l plot a direction field for f(x) over the x and y ranges.  For exampl
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dfieldplot(diff(y(x),
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\"3=sG[KSc]_F*7$Fdhw$\"3%)3r!4_$[abF*7$Fihw$\"3%)>W\"=U%eYeF*7$F^iw$\"
3mhhJxqg'>'F*7$Fciw$\"3lwY,eUcBlF*7$Fhiw$\"3%4,Nbf8n)oF*7$F]jw$\"3#G*[
Qb:V[sF*7$Fbjw$\"3q-'o)>(REl(F*FfjwFjjw-%+AXESLABELSG6%Q\"x6\"Q%y(x)F]
^y-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+++++L!\"*$\"+++++LFi^y;$!++++]KFi^
y$\"++++]AFi^y" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "See how nicely the integral curves follow the direction f
ield.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this se
ction" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Assignment" }}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 51 "Problem # 1: Antiddiffe
rentiation, Integral Curves." }}{PARA 0 "" 0 "" {TEXT -1 68 "Generate \+
a graph of at least three integral curves of the function  " }
{XPPEDIT 18 0 "f(x) = (1+x)*sin(x);" "6#/-%\"fG6#%\"xG*&,&\"\"\"F*F'F*
F*-%$sinG6#F'F*" }{TEXT -1 21 "  over the interval (" }{XPPEDIT 18 0 "
-Pi/2,Pi/2;" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*&F%F&F'F(" }{TEXT -1 3 ").
 " }}{PARA 0 "" 0 "" {TEXT -1 48 "Find the solution to the initial val
ue problem: " }{TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx = f(x);" "6#/*&%#
dyG\"\"\"%#dxG!\"\"-%\"fG6#%\"xG" }{TEXT -1 18 ",       y(0) = 3.2" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 0 "" 
}{TEXT 259 0 "" }{TEXT 260 31 "Problem #2: Area Approximations" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }{XPPEDIT 18 0 "f(x
) = exp(-x^2);" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 
91 "  .   Approximate the area under this curve defined by y = f(x) ov
er the interval [0,3] by " }}{PARA 0 "" 0 "" {TEXT -1 110 "a) using 20
 rectangles with the height determined by f evaluated at the right han
d endpoint of each interval. " }}{PARA 0 "" 0 "" {TEXT -1 104 "b) usin
g 20 rectangles with height determined by f evaluated at the left hand
 endpoint of each interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 163 "c) Give an upper bound and a lower bound of th
e actual area under the curve based on the function being increasing o
r decreasing and the results from (a) and (b). " }}{PARA 0 "" 0 "" 
{TEXT -1 203 "d) Using a limiting process to find the actual area unde
r the curve. Does this lie within the bounds from part (c).  You will \+
need to use the command \"evalf\" on the limit answer to get an actual
 number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 263 49 "Problem #3:  Integral Curves and Directio
n Fields" }}{PARA 0 "" 0 "" {TEXT -1 48 "Superimpose  3 integral curve
s of the function  " }{XPPEDIT 18 0 "f(x) = x*sin*x;" "6#/-%\"fG6#%\"x
G*(F'\"\"\"%$sinGF)F'F)" }{TEXT -1 100 "  on the direction field for t
his function.  Of these three, make one of them go through the point (
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 42 ",0).  Identify this poi
nt on your graph.  " }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }