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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#Math 170 section 03
1 Maple lab (the only lab)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "#the purpose of the
 exercise is to show you some of what Maple and other similar software
 can do.  It is likely to be used in Math 275 and Math 333, though we \+
no longer use it as much in M170 or M175." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "#the \+
exercises should be turned in by the end of class for" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "#credit equivalent to a quiz." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "#math notation in Maple" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "#this will not be so different from your calculator
.  It is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "#important to n
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "#\"x times y\" is always x*y and \"
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{MPLTEXT 1 0 68 "#or 2x.  Maple uses ^ for exponentiation, just like y
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1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+2;" }}{PARA 11 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "(sin(x^2));" }}{PARA 8 "" 1 
"" {TEXT -1 31 "Error, missing operator or `;`\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 60 "# _always_ put parentheses around the argumen
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{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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$\"3]MLe9tOcPF1$\"3QvGetk9iEF-7$$\"31,++]Qk\\RF1$\"3xjY!\\&Q(=`#F-7$$
\"3![LL3dg6<%F1$\"3-\\J(**H9uR#F-7$$\"3%ymmmw(GpVF1$\"3c(y6y`-()G#F-7$
$\"3C++D\"oK0e%F1$\"3'\\)p%**>_J=#F-7$$\"35,+v=5s#y%F1$\"3?Xz^&)*f34#F
-7$$\"\"&F*$\"35+++++++?F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fhal-%+AXES
LABELSG6$Q!6\"F\\bl-%%VIEWG6$;F(F]alFabl" 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 55 "#derivatives can be taken of expressions and functi
ons." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "f:=x->arcsin(x^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%'arcsinG6#*$
)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dif
f(arcsin(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\",&
F&F&*$)F%\"\"%F&!\"\"#F+\"\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "D(x->x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)
operatorG%&arrowGF&,$9$\"\"#F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "#notice that we use D for functions and diff (with an
 argument indicating" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "#th
e variable) for expressions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(a^2+b^3,a);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"aG\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "diff(a^2+b^3,b);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*$)%\"bG\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "#it makes a difference which variable you pick if the
re is more than one!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "#Maple can solve equations:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "solve((x-1)/(x+1)=3,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solv
e(x^2+a*x+b=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&%\"aG#!\"\"\"\"
#*&#\"\"\"F'F*-%%sqrtG6#,&*$)F$F'F*F**&\"\"%F*%\"bGF*F&F*F*,&F$F%*&#F*
F'F**$F+F*F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#  we can
 combine derivatives and equation solving ability to " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "# determine critical points and int
ervals of increase and decrease" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "# for functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=x^3+6*x^2+2*x+1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,**$)%\"xG\"\"$\"\"\"F**&\"\"'F*
)F(\"\"#F*F**&F.F*F(F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "solve(diff(A,x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&!\"#
\"\"\"*&#F%\"\"$F%-%%sqrtG6#\"#IF%F%,&F$F%*&#F%F(F%*$F)F%F%!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(%);  #evalf gives nume
rical values." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!*U\"eU<!\"*$!+e=uDQ
F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sqrt(2);evalf(sqrt(2)
);  #notice the notation for square root." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+iN@99!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "solv
e(diff(A,x)<0,x);  #this gives the intervals on which A decreases." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$-%%OpenG6#,&!\"#\"\"\"*
&#F+\"\"$F+*$-%%sqrtG6#\"#IF+F+!\"\"-F'6#,&F*F+*&#F+F.F+F0F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "solve(diff(A,x)>0.0,x);  #t
his gives the intervals on which A increases.  Putting in the decimal \+
point makes it display approximations." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$-%*RealRangeG6$,$%)infinityG!\"\"-%%OpenG6#$!+e=uDQ!\"*-F$6$-F*6
#$!+;9eU<!#5F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "diff(A,x,
x);  #this is the second derivative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&%\"xG\"\"'\"#7\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "
solve(diff(A,x,x)>0,x);  #where is the  graph concave up?" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%*RealRangeG6$-%%OpenG6#!\"#%)infinityG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot(A,x=-7..3);  #I fooled \+
around with the limits to get both critical points in the picture." }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$
$!\"(\"\"!$!#iF*7$$!3YLLLe%G?y'!#<$!3ErY6^nT`[!#;7$$!3OmmT&esBf'F0$!3%
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ommmh5$\\IF0$\"3luL`!)fxLAF37$$!3M***\\(=[jLGF0$\"3#*=<>;;qv?F37$$!3g*
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mmTg=><AF0$\"3=`]O;t;;:F37$$!3VLL$e*e$\\+#F0$\"3Vxz!Q)e$\\I\"F37$$!3#G
L$3-;Y%y\"F0$\"3S8)GX#HY&3\"F37$$!3%)***\\73QDf\"F0$\"3Qx!f_wHI**)F07$
$!3_KL$3Ub_Q\"F0$\"3oN+0%)G([3(F07$$!3!*******\\@6r6F0$\"3#foW!)481G&F
07$$!30(****\\PZhh*!#=$\"308mc0+yNPF07$$!3a%***\\(=_\"*e(Fht$\"3[ayBXW
y+DF07$$!3*******\\i>&Q`Fht$\"3K5*HW)o8!\\\"F07$$!3ptmmmhA;LFht$\"3u#)
e#G1k7g*Fht7$$!3G&*****\\i*p:\"Fht$\"3Mlf<vOqt%)Fht7$$\"3/@mm\"zpe*z!#
>$\"3-TyZj*)y)>\"F07$$\"3l6++]#\\'QHFht$\"3W\"HL*)eY78#F07$$\"3]HL$e9S
8&\\Fht$\"3cHQ*f4+Ee$F07$$\"3C2+]i?=bqFht$\"3%H'>3Ezu[dF07$$\"33HLL$3s
?6*Fht$\"3'))4rP#*z2c)F07$$\"3a***\\7`Wl7\"F0$\"3awJ*>^S(H7F37$$\"3enm
mm*RRL\"F0$\"3[vW<;iyr;F37$$\"3%zmmTvJga\"F0$\"3YV&yYcpG@#F37$$\"3]MLe
9tOc<F0$\"3q&zh,$z(R%GF37$$\"31,++]Qk\\>F0$\"3(4t\">^nn6NF37$$\"3![LL3
dg6<#F0$\"3Am[PHn1'Q%F37$$\"3%ymmmw(GpBF0$\"3#pY8Pwx>F&F37$$\"3C++D\"o
K0e#F0$\"3rA%G\"H5,IjF37$$\"35,+v=5s#y#F0$\"3m+7'z2wuX(F37$$\"\"$F*$\"
#))F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELSG6$Q\"x6\"Q!Fg[
l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "# sometimes Maple can't solve an equation exactly:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "solve(cos(x)=x,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RootOfG6#,&%#_ZG\"\"\"-%$cosG6#F'!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "#Then use fsolve to get a numerical answer." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "fsolve(cos(x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# you m
ight want to give fsolve an interval in which to find" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "#a solution, if there is more than \+
one." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%
\"xG\"\"$\"\"\"F(*&\"\"'F()F&\"\"#F(F(*&F,F(F&F(F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(A=0,x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6%,(*$),&\"%/9\"\"\"*&\"#7F(-%%sqrtG6#\"%*o\"F(F(#F(\"\"$
F(#!\"\"\"\"'*&\"#?F(F&#F2F0F2\"\"#F2,*F$#F(F**&\"#5F(F&F6F(F7F2*(^##F
(F7F(-F,6#F0F(,&F$F1*&F5F(F&F6F(F(F(,*F$F9*&F;F(F&F6F(F7F2*(^##F2F7F(F
?F(FAF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#this is insan
ely complicated!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%$!+JI#)yc!\"*^$$!*&[)eg\"F%$!+=\"3p(Q!#5^$F'$\"+=\"3p
(QF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "#notice that Maple \+
also knows about complex solutions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(
A=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+II#)yc!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(x^3-10*x+1=0,x);  #Yecch!" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6%,&*$),&!$3\"\"\"\"*&^#\"#7F(-%%sqrtG6#
\"&>>\"F(F(#F(\"\"$F(#F(\"\"'*&\"#?F(F&#!\"\"F1F(,(F$#F7F+*&\"#5F(F&F6
F7*(^##F(\"\"#F(-F-6#F1F(,&F$F2*&F5F(F&F6F7F(F(,(F$F9*&F;F(F&F6F7*(^##
F7F?F(F@F(FBF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "evalf(%
);  #the solutions are all actually real!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%^$$\"+_!R56$!\"*$\"\"#!#5^$$!+a$R6@$F&$\"\"!F.^$$\"*-.5
+\"F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(x^3-10*x+
1=0,x=2..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_!R56$!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fsolve(x^3-10*x+1=0,x=-1..1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7I+,5!#5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "fsolve(x^3-10*x+1=0,x=-4..-2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$!+`$R6@$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "#all solutions can be approximated, but fsolve will u
sually only" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#give one at
 a time.  You could guess the right intervals to use" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "#by looking at a graph." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "#in the exercises, we will use solve (and perhaps fsolve) to" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#find critical points for c
urve sketching purposes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#Integrals" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A:=x^5+
3*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*$)%\"xG\"\"&\"\"\"F
**&\"\"$F*)F(\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "i
nt(x^5+3*x^2,x);  #this is how to get an antiderivative.  Maple does n
ot include a constant." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"
\"'\"\"\"#F(F'*$)F&\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 86 "Int(A,x=2..4);int(A,x=2..4);  #Int allows one to set up an integ
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$)%\"xG\"\"&\"\"\"F+*&\"\"$F+)F)\"\"#F+F+/F);F/\"\"%" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"$G(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "
with(student);  #this loads a package needed for the pictures of boxes
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*DoubleintG%$IntG%&
LimitG%(LineintG%(ProductG%$SumG%*TripleintG%*changevarG%/completesqua
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urve 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" "Curve 30" "Curve 31" "Curve 32" "C
urve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve
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urve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve
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" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "C
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{XPPMATH 20 "6#,$-%$SumG6$,&*$),&\"\"#\"\"\"*&#F,\"\"%F,%\"iGF,F,\"\"&
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "leftsum(A,x=2..4,100);  eval
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "plot(x
^5+x+1,x=-2..2);  #I give you a plot command to start with -- make the
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f:F37$$\"3_mmm1^rZ<F0$\"3!*ekhZJR0>F37$$\"34++]sI@K=F0$\"3w$oh'>..[BF3
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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "#finish by using fsolv
e to estimate the root." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#Exercise 2  Use Maple \+
to find all critical points, intervals of" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 70 "#increase and decrease, local maxima and minima, po
ints of inflection," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "#and
 intervals of concavity up and down for f(x) = (x^2-4)/(x^2-9)." }}}
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ts and asymptotes.  Choose a viewing window" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 63 "# for a plot which will show all important feature
s.  Write out" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "#comments \+
in English  explaining all features of the graph." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "A:=(x^2-4)/(x^2-9);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG*&,&*$)%\"xG\"\"#\"\"\"F+\"\"%!\"\"F+,&F'F+\"\"*F
-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "#Exercise 2 continu
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"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#in \+
both parts of exercise 2, remember that complex \"critical points\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#(things with I = sqrt(-1)
 in them) are not geometrically meaningful." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 60 "#you might want to use the command evalf(%) to get
 numerical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "#values for t
he critical points in part b; they are rather hard to read" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#in exact form." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
67 "#Exercise 3:  consider the integral from 2 to 4 of x^3+2*x.  Draw \+
a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "#variety of pictures o
f rectangles whose areas approximate this integral" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 70 "# and evaluate the corresponding sums.  Com
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132 "# integral and compare it with the sums already computed (use eva
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