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1 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 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "#jan 23, 2001, sfb" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "# 7.6" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "#v'= what 1, x, x^2, x^3, xe^x^2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "#u x^3*exp(x^2) etc" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "intparts(Int(x^3*exp(x^2),x),x^3*exp(x^2));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "intparts(Int(x^3*exp(x^2),x)
,x^2*exp(x^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "intparts
(Int(x^3*exp(x^2),x),x*exp(x^2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "intparts(Int(x^3*exp(x^2),x),exp(x^2));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "intparts(Int(x^3*exp(x^2),x),x^2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "#7.6" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 88 "f:=x^3*exp(x);a:=3;b:=6;answer:=int(f,x=a..
b);error1:=array[1..10];error2:=array[1..10];" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "d:=2;Digits:=10;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "for i from 1 to 10 do error1[i]:=value(leftsum(f,x=a
..b,d^i)-answer); error2[i]:=value(rightsum(f,x=a..b,d^i)-answer); od:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "for i from 1 to 10 do e
valf([i,error1[i],error2[i],error1[i]/error2[i]]); od;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "for i from 1 to 9 do evalf([i,error
1[i]/error1[i+1],error2[i]/error2[i+1]]);od;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 125 "for i from 1 to 10 do error1[i]:=value(trapezoi
d(f,x=a..b,d^i)-answer); error2[i]:=value(middlesum(f,x=a..b,d^i)-answ
er); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "for i from 1 to
 10 do evalf([i,error1[i],error2[i],error1[i]/error2[i]]); od;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "for i from 1 to 9 do evalf([
i,error1[i]/error1[i+1],error2[i]/error2[i+1]]);od;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 73 "for i from 1 to 10 do error1[i]:=value(si
mpson(f,x=a..b,d^i)-answer); od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 
"for i from 1 to 10 do evalf([i,error1[i]]); od;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 57 "for i from 1 to 9 do evalf([i,error1[i]/error
1[i+1]]);od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "# 7.7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# Tails and spikes" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# Tails (aka integrating to \+
infinity)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for i from 1 t
o 10 do evalf(int(1/x^2,x=1..100^i)); od;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "for i from 1 to 10 do evalf(int(1/x^(1/2),x=1..100^
i)); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([1/x^(1/2)
,1/x^2],x=1..100,legend=[\"sqrt\",\"square\"]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "plot(1/x^2,x=100..10000);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "plot(1/x^(1/2),x=100..10000);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=int(1/x^2,x=1..L);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(a,L=infinity);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "int(1/x^2,x=1..infinity);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "b:=int(1/sqrt(x),x=1..L);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "limit(b,L=infinity);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "int(1/sqrt(x),x=1..infinity);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "int(sin(x),x=0..infinity);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "#spikes (aka points where the function is
 unbounded)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(1/sqrt(
x),x=0..1/100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "c:=int(1
/sqrt(x),x=delta..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "li
mit(c,delta=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(1/sq
rt(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(1/sqr
t(x),x=-1..1);# Maple lies!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "int(1/x,x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "in
t(1/x,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(1/x,x
=-1..0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(1/x,x=s..t)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ln(-1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "#7.8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "#(for large x: comparison theorems.)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x^2 - 1 < x^2 ;x^2 < x^2 + 1;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " 1/(x^2+1) < 1/x^2; 1/x^2 < \+
1/(x^2-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Int(1/(x^2+1)
,x=1..L) < Int(1/x^2,x=1..L); Int(1/x^2,x=1..L)<Int(1/(x^2-1),x=1..L);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Int(1/x^2,x=1..infinity
)< infinity;Int(1/(x^2+1),x=1..infinity)< infinity;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 118 "1/x<1/(x-1);Int(1/x,x=1..L)<Int(1/(x-1),
x=1..L); Int(1/x,x=1..infinity)=infinity; Int(1/(x-1),x=1..infinity)=i
nfinity;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# How do you ge
t the other one?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x^2-1 >
 x^2/2;1/(x^2-1)<2/x^2;Int(1/(x^2-1),x=1..L)<2*Int(1/x^2,x=1..L);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "x+1<2*x;1/(x+1)>1/(2*x);(1/2
)*Int(1/x,x=1..L)<Int(1/(x+1),x=1..L);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "# These are for large x" }}}}{MARK "59" 0 }{VIEWOPTS 
1 1 0 2 1 1805 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }