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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "Maple and Calculus Operat
ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 346 "
Although our initial Intro to Maple included discussion of a number of
 Maple functions related to calculus, you may not have noticed that fa
ct if you didn't look carefully at what Maple can do. This handout is,
 in the first instance, just a summary of Maple's calculus-related fun
ctionality. As always, I'm only scratching the surface. The Maple " }
{TEXT 292 4 "help" }{TEXT -1 14 " command (say " }{TEXT 290 6 "?diff;
" }{TEXT -1 16 " to get help on " }{TEXT 291 4 "diff" }{TEXT -1 92 ", \+
for instance) offers a lot more information on any of the commands use
d in this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "
" {TEXT -1 14 "Basic Commands" }}{PARA 4 "" 0 "" {TEXT -1 6 "Limits" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Maple's \+
command for limits is, logically enough, " }{TEXT 256 5 "limit" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(sin(
x)/x, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(1/x, x=0);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "limit(1/x^2, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinity
G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(sin(1/x), x=0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!\"\"\"\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 127 "It is useful to plot all the functions above, and
 to make sure you understand what Maple is telling you with all these \+
answers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
414 "Our investigation of derivatives could have been speeded up subst
antially had we been willing to use Maple uncritically as a black box \+
for taking limits. Here, for instance, are calculations of the derivat
ives of the sine, exponential, and cube root functions. It would be us
eful review to try to remember how we obtained these results in class.
 Are you sure Maple doesn't have bugs that affect these calculations?
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit((sin(x+h)-sin(x))
/h, h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#%\"xG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit((exp(x+h)-exp(x))/h, h
=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%\"xG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g := x -> x^(1/3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$#\"
\"\"\"\"$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(g(x)-
g(a))/(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"xG#\"\"\"\"
\"$F)F)*$)%\"aGF(F)!\"\"F),&F'F)F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "limit(%, x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*
&\"\"\"F%*$)%\"aG#\"\"#\"\"$F%!\"\"#F%F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 159 "For limits, as for most calculus functions, Maple has an
 \"inert\" form of the limit command that writes the limit without eva
luating it. This command is called " }{TEXT 257 5 "Limit" }{TEXT -1 
65 ". It's often useful in writing down formulas in Maple worksheets.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Limit(x^x, x=0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$)%\"xGF'/F'\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit(x^x, x=0) = limit(x^x,
 x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)%\"xGF(/F(\"\"!
\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Thinks about why the va
lue of this limit wasn't obvious. (What's " }{XPPEDIT 18 0 "x^0;" "6#*
$%\"xG\"\"!" }{TEXT -1 9 "? What's " }{XPPEDIT 18 0 "0^x;" "6#)\"\"!%
\"xG" }{TEXT -1 14 "? What should " }{XPPEDIT 18 0 "0^0;" "6#*$\"\"!F$
" }{TEXT -1 18 " be, if anything?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 4 "" 0 "" {TEXT -1 11 "Derivatives" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Maple has two functions f
or computing derivatives." }}{PARA 0 "" 0 "" {TEXT -1 7 "    If " }
{TEXT 258 1 "f" }{TEXT -1 24 " is an expression (like " }{XPPEDIT 18 
0 "x^2*sin(x);" "6#*&%\"xG\"\"#-%$sinG6#F$\"\"\"" }{TEXT -1 8 "), then
 " }{TEXT 259 10 "diff(f, x)" }{TEXT -1 22 " is the derivative of " }
{TEXT 260 1 "f" }{TEXT -1 30 " with respect to the variable " }{TEXT 
261 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "    If " }
{TEXT 262 1 "f" }{TEXT -1 21 " is a function (like " }{TEXT 263 3 "sin
" }{TEXT -1 36 ", or like the user-defined function " }{TEXT 264 1 "g
" }{TEXT -1 14 " above), then " }{TEXT 265 4 "D(f)" }{TEXT -1 22 " is \+
the derivative of " }{TEXT 266 1 "f" }{TEXT -1 22 " as a function. Usi
ng " }{TEXT 267 1 "D" }{TEXT -1 95 " often results in much less typing
, and it is the easy way to get the derivative as a function." }}}
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'F'F'F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^x;
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rrowGF()9$F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fprim
e := D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fprimeGf*6#%\"xG6\"6$
%)operatorG%&arrowGF(*&)9$F.\"\"\",&-%#lnG6#F.F/F/F/F/F(F(F(" }}}
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133.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "D(sin)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG
6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "D(D(sin))(x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#%\"xG!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D(D(D(sin)))(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$-%$cosG6#%\"xG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "(D@@4)(sin)(x) = D(D(D(D(sin))))(x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$sinG6#%\"xGF$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 107 "These operators can also be used with undefined function
s to get general differentiation rules. Right now, " }{TEXT 268 2 "f \+
" }{TEXT -1 4 "and " }{TEXT 269 2 " g" }{TEXT -1 36 " have value, but \+
we can remove those" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG#\"\"\"\"\"$F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g := 'g';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g
(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6#%\"xG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f := 'f'; f(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%
\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Here are Maple versions o
f the product rule, the chain rule, and an extended chain rule for " }
{XPPEDIT 18 0 "f(g(h(x)));" "6#-%\"fG6#-%\"gG6#-%\"hG6#%\"xG" }{TEXT 
-1 92 ". Play with things like this, and make sure the notation makes \+
sense. If you need help, try " }{TEXT 271 2 "?D" }{TEXT -1 4 " or " }
{TEXT 270 5 "?diff" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "D(f*g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%\"DG6
#%\"fG\"\"\"%\"gGF)F)*&F(F)-F&6#F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "D(f@g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"@G6$-
%\"DG6#%\"fG%\"gG\"\"\"-F(6#F+F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "diff(f(g(h(x))), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*(--%\"DG6#%\"fG6#-%\"gG6#-%\"hG6#%\"xG\"\"\"--F&6#F+F,F1-%%diffG6$
F-F0F1" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "Integrals" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Maple does both inde
finite and definite integrals with " }{TEXT 272 3 "int" }{TEXT -1 20 "
. The inert form of " }{TEXT 273 3 "int" }{TEXT -1 23 " is, logically \+
enough, " }{TEXT 274 3 "Int" }{TEXT -1 65 ". As with limits, the inert
 forms are useful in writing formulas." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(sin(x), x=0..Pi);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "int(x/(1+x^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,$-%#lnG6#,&\"\"\"F(*$)%\"xG\"\"#F(F(#F(F," }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "int(x/(1+x^4), x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$-%'arctanG6#*$)%\"xG\"\"#\"\"\"#F+F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "int(x/(1+x^3), x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(-%#lnG6#,&%\"xG\"\"\"F)F)#!\"\"\"\"$*&#F)\"\"'F)-F%6#,(*$)F(\"
\"#F)F)F(F+F)F)F)F)*(#F)F,F)-%%sqrtG6#F,F)-%'arctanG6#,$*&,&F(F5F)F+F)
F8F)F7F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Int(x^3*exp(x
),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"$\"\"\"-
%$expG6#F(F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(x^3*e
xp(x),x) = int(x^3*exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&)%\"xG\"\"$\"\"\"-%$expG6#F)F+F),*F'F+*(F*F+)F)\"\"#F+F,F+!\"\"
*(\"\"'F+F)F+F,F+F+*&F5F+F,F+F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "Int(x^n,x) = int(x^n,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%$IntG6$)%\"xG%\"nGF(*&)F(,&F)\"\"\"F-F-F-F,!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The values of iner
t forms can also be taken using the command " }{TEXT 280 5 "value" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "Int(ln(x)^n/x, x) = value(Int(ln(x)^n/x, x));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%#lnG6#%\"xG%\"nG\"\"\"
F,!\"\"F,*&)F),&F-F.F.F.F.F2F/" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 4 "
Sums" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "S
ums are done with " }{TEXT 275 3 "sum" }{TEXT -1 5 " and " }{TEXT 276 
3 "Sum" }{TEXT -1 34 ", just as integrals are done with " }{TEXT 277 
4 "int " }{TEXT -1 4 "and " }{TEXT 278 3 "Int" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "sum(k, k=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#b" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sum(k, k=1..n);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,(*$),&%\"nG\"\"\"F(F(\"\"#F(#F(F)*&#F(F)F(F'F(
!\"\"#F(F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"nG\"\"\",&F%F&F&F&F&#F&\"\"
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sum(k^2, k=1..n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$),&%\"nG\"\"\"F(F(\"\"$F(#F(F)*&#F
(\"\"#F(*$)F&F-F(F(!\"\"*&#F(\"\"'F(F'F(F(F2F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$*(%\"nG\"\"\",&F%F&F&F&F&,&F%\"\"#F&F&F&#F&\"\"'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "Sum(k^3, k=1..n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$SumG6$*$)%\"kG\"\"$\"\"\"/F(;F*%\"nG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum(k^3, k=1..n) = factor(sum(k^3, \+
k=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"kG\"\"$\"
\"\"/F);F+%\"nG,$*&)F.\"\"#F+),&F.F+F+F+F2F+#F+\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 119 "The range of sums Maple understands is pretty \+
impressive. Here's an identity that should look beautiful and surprisi
ng:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Sum(1/n^2, n=1..infi
nity) = sum(1/n^2, n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%$SumG6$*&\"\"\"F(*$)%\"nG\"\"#F(!\"\"/F+;F(%)infinityG,$*$)%#PiGF,F
(#F(\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "If Maple can do all
 this, then Maple ought to be able to handle the nasty work we encount
ered with Riemann sums. Here, for instance, is Maple's treatment of a \+
computation of" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "int(t^3,t = 0 .. x)
;" "6#-%$intG6$*$%\"tG\"\"$/F';\"\"!%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 
19 "using Riemann sums." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
f := x -> x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$
%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "RSum := Sum(f(k*x/n)*(1/n), k=1..n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%RSumG-%$SumG6$*(%\"kG\"\"$%\"xGF*%\"nG!\"%/F)
;\"\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "RSum = value(R
Sum);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(%\"kG\"\"$%\"xGF)
%\"nG!\"%/F(;\"\"\"F+,(*(F*F)F+F,,&F+F/F/F/\"\"%#F/F3*&#F/\"\"#F/*(F2F
)F*F)F+F,F/!\"\"**F4F/F*F)F+F,F2F7F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "RSum = factor(value(RSum));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$*(%\"kG\"\"$%\"xGF)%\"nG!\"%/F(;\"\"\"F+,$*(F
*F),&F+F/F/F/\"\"#F+!\"##F/\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "This works out the Riemann sum for " }{TEXT 279 1 "n" }{TEXT -1 
54 " slices. Now we want to take the limit of this sum as " }{XPPEDIT 
18 0 "proc (n) options operator, arrow; infinity end proc" "6#f*6#%\"n
G7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "Int(t^3, t=0..x) = Limit(RS
um, n=infinity);\n\nLimit(RSum, n=infinity) = limit(value(RSum), n=inf
inity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)%\"tG\"\"$\"\"
\"/F);\"\"!%\"xG-%&LimitG6$-%$SumG6$*(%\"kGF*F/F*%\"nG!\"%/F7;F+F8/F8%
)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%$SumG6$*(%
\"kG\"\"$%\"xGF,%\"nG!\"%/F+;\"\"\"F./F.%)infinityG,$*$)F-F,F2#F2\"\"%
" }}}{EXCHG {PAGEBK }{PARA 3 "" 0 "" {TEXT -1 29 "The student calculus
 package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
105 "Maple also has a whole library of functions useful for the calcul
us student and contained in the package " }{TEXT 281 7 "student" }
{TEXT -1 62 ". Let's load this package and see what functions it conta
ins. " }{TEXT 284 35 "You need to load the package using " }{TEXT 285 
13 "with(student)" }{TEXT 286 34 " before you can use the functions." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*DoubleintG%$IntG%&LimitG%(Li
neintG%(ProductG%$SumG%*TripleintG%*changevarG%/completesquareG%)dista
nceG%'equateG%*integrandG%*interceptG%)intpartsG%(leftboxG%(leftsumG%)
makeprocG%*middleboxG%*middlesumG%)midpointG%(powsubsG%)rightboxG%)rig
htsumG%,showtangentG%(simpsonG%&slopeG%(summandG%*trapezoidG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "It's worth asking for help about \+
most of these functions, just to see what they do. Here's a quick teas
er:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Th
e midpoint approximation to the integral of " }{XPPEDIT 18 0 "y = Zeta
(x);" "6#/%\"yG-%%ZetaG6#%\"xG" }{TEXT -1 9 " between " }{XPPEDIT 18 
0 "x = -20;" "6#/%\"xG,$\"#?!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"x = -19;" "6#/%\"xG,$\"#>!\"\"" }{TEXT -1 15 " is shown below" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "middlebox(Zeta(x), x=-20..-1
9, 10);" }}{PARA 13 "" 1 "" {GLPLOT2D 363 272 272 {PLOTDATA 2 "6/-%)PO
LYGONSG6$7&7$$!#?\"\"!$F*F*7$F($\"57hi'\\#erDGi!#>7$$!5++++++++!*>!#=F
-7$F1F+-%&COLORG6&%$RGBG$\"\"(!\"\"$\"\"*F;F9-F$6$7&F47$F1$\"5l0%)=i*p
[sk\"F37$$!5++++++++!)>F3FB7$FEF+F5-F$6$7&FG7$FE$\"5`=L$=Mfs9S#F37$$!5
++++++++q>F3FL7$FOF+F5-F$6$7&FQ7$FO$\"5_<lSDfuV<HF37$$!5++++++++g>F3FV
7$FYF+F5-F$6$7&Fen7$FY$\"5O1(zq=Se\"GKF37$$!5++++++++]>F3Fjn7$F]oF+F5-
F$6$7&F_o7$F]o$\"52,.IX0eSmLF37$$!5++++++++S>F3Fdo7$FgoF+F5-F$6$7&Fio7
$Fgo$\"58_I:'R*=ojLF37$$!5++++++++I>F3F^p7$FapF+F5-F$6$7&Fcp7$Fap$\"5b
axucM:Y\\KF37$$!5++++++++?>F3Fhp7$F[qF+F5-F$6$7&F]q7$F[q$\"5LvB`]xPm]I
F37$$!5++++++++5>F3Fbq7$FeqF+F5-F$6$7&Fgq7$Feq$\"5uH*yX)41K\"z#F37$$F/
F*F\\r7$F_rF+F5-%'CURVESG6&7SF'7$$!5LLLLe%G?y*>F3$\"5OHu0\"Hya%4GF/7$$
!5nmmT&esBf*>F3$\"5vX04c'[c]8&F/7$$!5LLL$3s%3z$*>F3$\"5sVGJ]Qgi?wF/7$$
!5LLL$e/$Qk\"*>F3$\"5I+3C:njZ')**F/7$$!5nmmT5=q]*)>F3$\"5h_!z'R3!>3A\"
F37$$!5LLL3_>f_()>F3$\"54Uq6$eq0_T\"F37$$!5+++vo1YZ&)>F3$\"5h-`'o/Y!)
\\g\"F37$$!5LLL3-OJN$)>F3$\"5OQ,BW+\\@*y\"F37$$!5+++v$*o%Q7)>F3$\"5%>1
N-G#o(4'>F37$$!5nmmm\"RFj!z>F3$\"5vQ@6^.UfD@F37$$!5LLL$e4OZr(>F3$\"5%R
H0(y#[=2E#F37$$!5++++v'\\!*\\(>F3$\"5'*ztK/T32-CF37$$!5++++DwZ#G(>F3$
\"57&GY$)3O\\G`#F37$$!5++++D.xtq>F3$\"5v**)pP&*R:'[EF37$$!5LLL3-TC%)o>
F3$\"5'R!=>/<1FXFF37$$!5nmmm\"4z)em>F3$\"55<NI^[2.]GF37$$!5nmmmm`'zY'>
F3$\"5q9hVHnnRIHF37$$!5+++v=t)eC'>F3$\"5%p*)HY!4]`9IF37$$!5nmmm;1J\\g>
F3$\"53t:%fs$=!43$F37$$!5+++v=[jLe>F3$\"5,Pe0]!Ql_9$F37$$!5+++Dc/EGc>F
3$\"5****HU@]vi)>$F37$$!5nmm;aQ(RT&>F3$\"5m$[BD8DsjC$F37$$!5nmmTg=><_>
F3$\"5**G*p!p!**\\LG$F37$$!5LLL$e*e$\\+&>F3$\"5]Wo1i%zahJ$F37$$!5LLL3-
;Y%y%>F3$\"5VZ&)*)\\?pwULF37$$!5+++D\"3QDf%>F3$\"5:0XO<V0+gLF37$$!5LLL
$3Ub_Q%>F3$\"5G.Ay1#RwEP$F37$$!5++++]@6rT>F3$\"5tvxC,[$z&zLF37$$!5+++]
PZhhR>F3$\"5x<6f[wT_!Q$F37$$!5+++v=_\"*eP>F3$\"5&49xZuHHiP$F37$$!5+++]
i>&Q`$>F3$\"5'fd2l(4pvlLF37$$!5nmmm;EiJL>F3$\"5d&R/5Q\"y[^LF37$$!5++++
D'*p:J>F3$\"5O\"=KtPbZ9L$F37$$!5LLL3-8/?H>F3$\"5i0]Ft#oH#4LF37$$!5++++
v]81F>F3$\"5(>XPtN<r2G$F37$$!5nmmT&)f'[]#>F3$\"5!)R%)oG1GC]KF37$$!5+++
v$z\"[%H#>F3$\"5KothVpun9KF37$$!5nmmm\"z#z)3#>F3$\"5430vt#)o^wJF37$$!5
+++voaXt=>F3$\"5mjPn.F5@LJF37$$!5LLLLL+1m;>F3$\"5\"R%H\\T]!y%)3$F37$$!
5LLL$eCoRX\">F3$\"5Pdw^q^H))RIF37$$!5nmmT&oKOC\">F3$\"5'R%=)e_nc!*)HF3
7$$!5++++]hN]5>F3$\"5n&\\&oyRr@SHF37$$!5nmm;H%R)G3>F3$\"5;EWTjx8$>)GF3
7$$!5LLLLLArI1>F3$\"5/$z!4d&4')y#GF37$$!5+++v=tY>/>F3$\"5U+KA*[@b%oFF3
7$$!5+++D\")*ys@!>F3$\"5S4C87S9**4FF37$F_r$\"57777777iXEF3-%'COLOURG6&
F8$\"*++++\"!\")F+F+-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-%+AXESLABEL
SG6$Q\"x6\"Q!Febl-%%VIEWG6$;F(F_r%(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" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The sum of the areas of t
he reactangles is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "middlesum(Zeta(x), x=-20..-19, 10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$SumG6$-%%ZetaG6#,&#!$*R\"#?\"\"\"
*&#F.\"#5F.%\"iGF.F./F2;\"\"!\"\"*F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "Numerically, this sum is approximately" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "% = evalf(value(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$-%$SumG6$-%%ZetaG6#,&#!$*R\"#?\"\"\"*&#F/\"#5F/%\"iG
F/F//F3;\"\"!\"\"*F1$\"5q)pJK\"\\w'Qm#!#=" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 52 "By comparison, a more exact value of the integral is" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Int(Zeta(x), x=-20..-19) = e
valf(int(Zeta(x), x=-20..-19));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
$IntG6$-%%ZetaG6#%\"xG/F*;!#?!#>$\"5a&*4ZqD#3rl#!#=" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 47 "The error in the midpoint sum is therefore only
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "middlesum(Zeta(x), x=-
20..-19, 10) -\nInt(Zeta(x), x=-20..-19) = \nevalf(value(middlesum(Zet
a(x), x=-20..-19, 10)))-\nevalf(int(Zeta(x), x=-20..-19));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&-%$SumG6$-%%ZetaG6#,&#!$*R\"#?\"\"\"*&#F/
\"#5F/%\"iGF/F//F3;\"\"!\"\"*F1-%$IntG6$-F)6#%\"xG/F=;!#?!#>!\"\"$\"2;
.2wUB%fn!#=" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 27 "Integration by Sub
stitition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
95 "One of the neatest uses of the student package is that it lets one
 explore possible choices of " }{TEXT 282 1 "u" }{TEXT -1 56 " when do
ing substitution. The command for doing this is " }{TEXT 283 9 "change
var" }{TEXT -1 91 ", which takes as arguments (1) the substitution you
 want to do, written either in the form " }{XPPEDIT 18 0 "u = f(x);" "
6#/%\"uG-%\"fG6#%\"xG" }{TEXT -1 16 " or in the form " }{XPPEDIT 18 0 
"x = f(u);" "6#/%\"xG-%\"fG6#%\"uG" }{TEXT -1 160 ", (2) the inert int
egral to be computed, and (3) the name of the new variable. For instan
ce, here are a bunch of different substitutions applied to the integra
l" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "puzzle := Int(x^2/(1+x
^6),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'puzzleG-%$IntG6$*&%\"xG
\"\"#,&\"\"\"F,*$)F)\"\"'F,F,!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "changevar(u=x^2, puzzle, u);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&%\"uG#\"\"\"\"\"#,&F*F)*&F*F))F'\"\"$F)F)!\"
\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "changevar(u=x^6, pu
zzle, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"F(*&-%
%sqrtG6#%\"uGF(,&F(F(F-F(F(!\"\"#F(\"\"'F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "changevar(u=1+x^6, puzzle, u);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"F(*&-%%sqrtG6#,&!\"\"F(%\"uGF(F(F/F(
F.#F(\"\"'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "changevar(u
=x^3, puzzle, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"
F',&\"\"$F'*&F)F')%\"uG\"\"#F'F'!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 197 "The last of these has an obvious integral using the arct
an function, so apparently that's the right substitution to have used.
 But isn't it nice to be able to prospect for a good choice so quickly
?" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 20 "Integration by Parts" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "The anal
ogous command letting you explore possible integrations by part is int
parts, whose arguments are (1) the integral you are working on, and (2
) your choice of " }{TEXT 287 1 "u" }{TEXT -1 61 " (the rest of the in
tegral will automatically be taken to be " }{TEXT 288 2 "v'" }{TEXT 
-1 87 "). For instance, here are some possible ways to do integration \+
by parts on the integral" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"puzzle2 := Int(x^3*exp(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
(puzzle2G-%$IntG6$*&-%$expG6#*$)%\"xG\"\"#\"\"\"F0)F.\"\"$F0F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "intparts(puzzle2, x^3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**^##!\"\"\"\"#\"\"\")%\"xG\"\"$F)-%
%sqrtG6#%#PiGF)-%$erfG6#*&^#F)F)F+F)F)F)-%$IntG6$**^##!\"$F(F))F+F(F)F
-F)F1F)F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "intparts(puz
zle2, exp(x^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"xG\"\"%\"
\"\"-%$expG6#*$)F&\"\"#F(F(#F(F'-%$IntG6$,$*&)F&\"\"&F(F)F(#F(F.F&!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "intparts(puzzle2, x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\",&*&F%F&-%$expG6#*$
)F%\"\"#F&F&#F&F.*(^##F&\"\"%F&-%%sqrtG6#%#PiGF&-%$erfG6#*&^#F&F&F%F&F
&F&F&F&-%$IntG6$F'F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "intparts(puzzle2, x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%
\"xG\"\"#\"\"\"-%$expG6#*$F%F(F(#F(F'-%$IntG6$*&F&F(F)F(F&!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Bingo! The last of these finally \+
gave us an integral that doesn't involve weird functions like erf, and
 that we can do with the substitution " }{XPPEDIT 18 0 "u = x^2;" "6#/
%\"uG*$%\"xG\"\"#" }{TEXT -1 129 ". (Actually, I would have done this \+
substitution first, and then used parts; but doing parts first shows o
ff better the power of " }{TEXT 289 8 "intparts" }{TEXT -1 23 " as a p
rospecting tool." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 17 "Problems for \+
you." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 249 "
Here are a few problems you might want to do in order to reinforce the
 ideas in this handout, and to get comfortable using these commands. Y
ou need not turn these in, but I would be happy to read them for extra
 credit if you do want to turn them in." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 32 "1. Use Maple to explore the sums" }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "Sum(1/(n^s),n = 1 .. infinity);" "6#-
%$SumG6$*&\"\"\"F')%\"nG%\"sG!\"\"/F);F'%)infinityG" }}{PARA 0 "" 0 "
" {TEXT -1 45 "for various values of s. What do you observe?" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "2. As efficient
ly as possible, compute the 10th derivative of " }{XPPEDIT 18 0 "sec(x
);" "6#-%$secG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 46 "3. What happens when you use the subs
titution " }{XPPEDIT 18 0 "u = arcsin(x);" "6#/%\"uG-%'arcsinG6#%\"xG
" }{TEXT -1 16 " on the integral" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "I
nt(sqrt(1-x^2),x);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#!\"\"F
," }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 16 "What if you use " }
{XPPEDIT 18 0 "u = arctan(x);" "6#/%\"uG-%'arctanG6#%\"xG" }{TEXT -1 
16 " on the integral" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(sqrt(1+x^
2),x);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#F*F," }{TEXT -1 1 
"?" }}{PARA 0 "" 0 "" {TEXT -1 141 "Can you understand what's going on
 by doing these integrals by hand? Can you find other integral where t
hese substitutions work? Be creative!" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 33 "4. Use Maple to explore sums like" }}
{PARA 260 "" 0 "" {XPPEDIT 18 0 "Sum(k^s,k = 1 .. n);" "6#-%$SumG6$)%
\"kG%\"sG/F';\"\"\"%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 94 "for various v
alues of s, and see if you observe anything. Also, use Maple to look a
t sums like" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "Sum(k*(k+1),k = 1 .. n
);" "6#-%$SumG6$*&%\"kG\"\"\",&F'F(F(F(F(/F';F(%\"nG" }{TEXT -1 5 ",  \+
  " }{XPPEDIT 18 0 "Sum(k*(k+1)*(k+2),k = 1 .. n);" "6#-%$SumG6$*(%\"k
G\"\"\",&F'F(F(F(F(,&F'F(\"\"#F(F(/F';F(%\"nG" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "Sum(k*(k+1)*(k+2)*(k+3),k = 1 .. n);" "6#-%$SumG6$**%\"
kG\"\"\",&F'F(F(F(F(,&F'F(\"\"#F(F(,&F'F(\"\"$F(F(/F';F(%\"nG" }{TEXT 
-1 7 ", . . ." }}{PARA 0 "" 0 "" {TEXT -1 121 "See if you notice anyth
ing here. In both parts of this problem, it would be useful to factor \+
the expressions you compute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 25 "5. Let f(x) be a parabola" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "f := x -> a*x^2 + b*x + c;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"
\"\")9$\"\"#F/F/*&%\"bGF/F1F/F/%\"cGF/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 10 "What must " }{TEXT 293 1 "a" }{TEXT -1 2 ", " }{TEXT 
294 1 "b" }{TEXT -1 6 ", and " }{TEXT 295 1 "c" }{TEXT -1 17 " be in o
rder for " }{TEXT 296 1 "f" }{TEXT -1 86 " to have the same value, the
 same first derivative, and the same second derivative as " }{XPPEDIT 
18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x
 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 137 "? (COmpute the derivatives of bo
th functions at 0, set them equal to one another, and solve the result
ing system of equations.) Plot sin(" }{TEXT 297 1 "x" }{TEXT -1 30 ") \+
together with this parabola." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 26 "Now take a generic cubic, " }{XPPEDIT 18 0 "f(x
) = a*x^3+b*x^2+c*x+d;" "6#/-%\"fG6#%\"xG,**&%\"aG\"\"\"*$F'\"\"$F+F+*
&%\"bGF+*$F'\"\"#F+F+*&%\"cGF+F'F+F+%\"dGF+" }{TEXT -1 127 ", and try \+
to make its value and its first, second, and third derivatives equal t
o the value and the first 3 derivatives of sin(" }{TEXT 298 1 "x" }
{TEXT -1 49 ") at 0. Again, plot this cubic together with sin(" }
{TEXT 299 1 "x" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 137 "Keep doing this with polynomials of incr
easing degree. You should be getting curves that approximate the sine \+
function better and better." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 26 "6. Ask Maple about the sum" }}{PARA 262 "" 0 "
" {XPPEDIT 18 0 "Sum((-1)^n*x^(2*n+1)/(2*n+1)!,n = 0 .. infinity);" "6
#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF))%\"xG,&*&\"\"#F)F+F)F)F)F)F)-%*facto
rialG6#,&*&F0F)F+F)F)F)F)F*/F+;\"\"!%)infinityG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 39 "What has this got to do with Problem 5?" 
}}}}{MARK "66 0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }