{VERSION 6 0 "Windows XP" "6.0" }
{USTYLETAB {PSTYLE "Normal256" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 
0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "He
ading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0
 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 5" -1 201 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3
 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 202 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 
-1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0
 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bul
let Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Out
put" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1
 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 
-1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 
2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1
 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnosti
c" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 
0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal259" -1 203 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0
 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal260" -1 204 1 {CSTYLE "" -1 -1 "Time
s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }
{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2
 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3"
 -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1
 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 
2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item"
 -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 
1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 
-1 1 }{PSTYLE "Ordered List 4" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 
0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "
Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 
2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output"
 -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 
}1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 
2 -1 1 }{PSTYLE "Normal258" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0
 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warni
ng" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0
 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1
 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 
0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1
 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE
 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 
1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE
 "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 
2 0 2 2 -1 1 }{PSTYLE "Normal257" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE 
"Ordered List 2" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{CSTYLE "Help Variable" 
-1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 
200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic \+
Small201203204" -1 201 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "2D Math Italic Small" -1 202 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 
0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2
 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2
 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2
 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 
1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 
2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2
 2 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2
 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 
2 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2
 2 1 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2
 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 203 "Courier" 1 12 
200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "T
imes" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1
 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 
"Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "T
imes" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2"
 -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" 
-1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic
 Small201" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "He
lp Italic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Ou
tput Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "H
elp Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "
Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "
2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "H
elp Maple Name" -1 35 "Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2
 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2
 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 
1 2 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2
 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2
 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2
 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 
0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small201203" -1 205 "
Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "T
imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 
"Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 206 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 207 "Times" 1 12 0 0 0 1
 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 208 "Times" 1 12 0 0 0 1 2 1 2 2 2
 2 0 0 0 1 }{CSTYLE "" -1 209 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1
 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 
2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 210 "Times" 1 
12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 211 "Times" 1 12 0 0 0 1
 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Times" 1 12 0 0 0 1 2 1 2 2 2
 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1
 }{CSTYLE "" -1 214 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE 
"" -1 215 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }}
{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 216 12 "MAT 156 F'07" }}{PARA 
208 "" 0 "" {TEXT 217 5 "Lab 3" }}{PARA 207 "" 0 "" {TEXT 218 34 "More
 on Riemann Sums and Integrals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 59 "You should not do thi
s lab until you have completed Lab 2. " }}{PARA 0 "" 0 "" {TEXT 219 0 
"" }}{PARA 0 "" 0 "" {TEXT 219 53 "You can add color to your leftbox c
ommand as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(
student): with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "l
eftbox(x^2,x=0..4,5,shading=yellow);" }}{PARA 13 "" 1 "" {TEXT 220 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 69 "Now I am going to combine tw
o leftsum pictures with different colors." }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 44 "lb10:=leftbox(x^2,x=0..4,10,shading=yellow):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lb5:=leftbox(x^2,x=0..4,5,sh
ading=white):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(lb
5,lb10);" }}{PARA 13 "" 1 "" {TEXT 220 0 "" }{TEXT 220 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 219 344 "You can easily see the difference betwe
en the two left sums now.  (Warning:  Maple seems tempermental about t
hese commands.  For example writing display(lb10,lb5) instead of displ
ay(lb5,lb10) matters.  So be patient and experiment wtih the commands.
)  The problem is that the first plot covers the second and the second
 does not show through.  " }}{PARA 0 "" 0 "" {TEXT 219 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 206 20 "Assignment Part 1:  " }{TEXT 219 82 " (D
o not forget the plots  and student packages).   Using sin(x) and x=0.
.Pi/4.   " }}{PARA 0 "" 0 "" {TEXT 219 117 "A)    Graph the left-hands
 sums with 5 and 10 subintervals simultaneously. Explain why LHS(5) is
 less than LHS(10).  " }}{PARA 0 "" 0 "" {TEXT 219 52 "B)    Graph and
 explain why RHS(5)>RHS(10)>RHS(20). " }}{PARA 0 "" 0 "" {TEXT 219 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 20 "Assignment Part 2:  " }{TEXT
 219 27 "Using sin(x) and x=0..Pi/2." }}{PARA 0 "" 0 "" {TEXT 219 108 
" Write commands to compute the left-hand sums and the right-hand sums
 with 10, 20, 40, 80, 160 subintervals." }}{PARA 0 "" 0 "" {TEXT 219 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 178 "We see that indeed the l
eft-hand sums are increasing and the right-hand sums are decreasing. A
t the same time the right-hand sums are always larger than all the lef
t-hand sums.\n" }{TEXT 219 108 "As you see we have not gotten great ac
curacy.   To compute more accurately it will be nice to increase the\n
" }{TEXT 219 126 "number of subintervals. As it is tiring to write the
 same commands over and over, it is important to introduce loops in Ma
ple." }}}{EXCHG {PARA 203 "" 0 "" {TEXT 221 2 "\n" }{TEXT 208 40 "Loop
s and convergence of Riemann sums.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
219 422 "The following commands produce a table of the left-hand sums \+
 with 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120 subintervals.
 Notice that every time we multiply the number of subintervals by 2, s
o at the first step (j=0) we have 5*2^0 subintervals, at the second st
ep (j=1) we have 5*2^1=10 subintervals, at the third step (j=2) we hav
e 5*2^2=20 subintervals etc. This gives the formula 5*2^\{j\} in the c
ommands. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for j from 0 to 10 do n:= 5*2^(j): \+
evalf(leftsum(x^2, x=0..4, n)):od;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 219 248 "We see that
 the first two decimal places have stabilized. However, as you have no
ticed before, the left-hand sums are increasing. To be certain that th
e second decimal place is indeed 2 we need to give overestimates, whic
h are the right-hand sums. " }}{PARA 0 "" 0 "" {TEXT 219 2 "\n" }{TEXT
 209 19 "Assignment Part 3: " }{TEXT 219 197 "  Write commands that pr
oduce the right-hand sums with the same number of subintervals, using \+
a loop. All you have to do is copy the command for the left-hand sums \+
and change leftsum into rightsum." }}{PARA 0 "" 0 "" {TEXT 219 0 "" }}
}{EXCHG {PARA 210 "" 0 "" {TEXT 200 30 "Computing Integrals with Maple
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(x^2,x=0..4); " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 210 19 "Assignment Part 4: " }{TEXT 219 
9 "  Study  " }{XPPEDIT 18 0 "int(1/x, x = 1 .. 2);" "6#-%$intG6$*&\"
\"\"F'%\"xG!\"\"/F(;F'\"\"#" }{TEXT 219 12 " as follows:" }}{PARA 0 ""
 0 "" {TEXT 211 3 "A) " }{TEXT 219 120 " Compute the left-hand sums an
d right-hand sums with 2, 4, 8, 16, 32, 64, 128, 256, 516, 1024 subint
ervals. Use a loop. " }}{PARA 0 "" 0 "" {TEXT 212 2 "B)" }{TEXT 219 
47 "  How many decimal digits are you certain of?\n" }{TEXT 213 3 "C) \+
" }{TEXT 219 64 "  Which are overestimates and which are underestimate
s and why? " }}{PARA 0 "" 0 "" {TEXT 214 3 "D) " }{TEXT 219 150 "  Do \+
the left-hand sums form an increasing or decreasing sequence of number
s, as the number of subintervals increase? Show this in appropriate gr
aphs." }}{PARA 0 "" 0 "" {TEXT 215 3 "E) " }{TEXT 219 30 " Compute the
 exact integral.\n" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }