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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 278 27 "Maple Help for Assignme
nt 1" }}{PARA 257 "" 0 "" {TEXT -1 32 "Math 152, Spring 2009\nJ. Gerla
ch" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 37 "This is not a complete maple
 tutorial" }{TEXT -1 106 ".  Those of you, who have not worked with ma
ple before, should carefully study the first four maple files " }
{TEXT 282 46 "(Fundamentals, Functions, Limits, Derivatives)" }{TEXT 
-1 75 " which are posted on  http://www.radford.edu/~jgerlach/Maple/Ma
plePage.htm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 36 "This help manual is written for the " }{TEXT 279 23 "Classic Ma
ple Worksheet" }{TEXT -1 156 ". Most of the steps will work in the reg
ular maple as well, but the drag-and-click environment of the regular \+
maple 11 may lead to unexpected complications." }}{PARA 0 "" 0 "" 
{TEXT -1 41 "\nGo though this worksheet by pressing the" }{TEXT 283 1 
" " }{TEXT 280 5 "enter" }{TEXT -1 5 " key." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 9 "Summation" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The c
ommand for summation is " }{TEXT 256 3 "sum" }{TEXT -1 19 ".  Its synt
ax is  s" }{TEXT 257 22 "um( expression, range)" }{TEXT -1 75 ".   For
 example, if you want to add the first 12 integers, you should enter" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sum( k, k=1..12);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 110 "The disadvantage  in this method is that
 you just see the result (78 in this case).  An alternative is to use \+
" }{TEXT 258 3 "Sum" }{TEXT -1 12 " along with " }{TEXT 259 5 "value" 
}{TEXT -1 34 ".  In our example this becomes    " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Sum( k , k=1..12);  value(%);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 260 3 "Sum" }{TEXT -1 25 " is the inert version of " }
{TEXT 261 3 "sum" }{TEXT -1 74 ".  Nothing is being calculated with Su
m, it just displays the expression. " }{TEXT 262 8 "value(%)" }{TEXT 
-1 49 " executes the summation.  \nLet's look at another " }{TEXT 264 
7 "example" }{TEXT -1 13 ":  Evaluate  " }{XPPEDIT 18 0 "Sum(1/(k^2),k
 = 1 .. 100);" "6#-%$SumG6$*&\"\"\"F'*$%\"kG\"\"#!\"\"/F);F'\"$+\"" }
{TEXT -1 2 "  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum(1/k^2,k=1..10
0);\nvalue(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Maple is a sym
bolic algebra system; it finds the common denominator.  A floating poi
nt representation is much more useful in this example" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "
Another " }{TEXT 263 7 "example" }{TEXT -1 11 ":  Compute " }{XPPEDIT 
18 0 "1/8+1/16+1/32;" "6#,(*&\"\"\"F%\"\")!\"\"F%*&F%F%\"#;F'F%*&F%F%
\"#KF'F%" }{TEXT -1 10 "   + .. + " }{XPPEDIT 18 0 "1/1024;" "6#*&\"\"
\"F$\"%C5!\"\"" }{TEXT -1 53 "  .\nOnce you realize that the terms are
 of the form  " }{XPPEDIT 18 0 "1/(2^k);" "6#*&\"\"\"F$)\"\"#%\"kG!\"
\"" }{TEXT -1 63 "  , where k ranges from 3 to 10, everything is strai
ghtforward:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Sum( 1/2^k, k=3..10)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 34 "The answer is fairly close to 1/4." }{MPLTEXT 1 0 0 "" 
}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 12 "Riemann Sums" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "I
t is " }{TEXT 284 10 "essential " }{TEXT -1 14 "to invoke the " }
{TEXT 285 15 "student package" }{TEXT -1 86 " for this part.  Commands
 such as leftsum or leftbox will otherwise not be executable." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart; with(student):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "All examples will use the function  " }
{XPPEDIT 18 0 "f(x) = x^4-7*x^2+8*x+1;" "6#/-%\"fG6#%\"xG,**$F'\"\"%\"
\"\"*&\"\"(F+*$F'\"\"#F+!\"\"*&\"\")F+F'F+F+F+F+" }{TEXT -1 73 "  on t
he interval [0,2].  So, let's define the function once and for all " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> x^4 - 7*x^2 + 8*x +1;" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 265 25 "display of the
 rectangles" }{TEXT -1 42 " for the Riemann sum can be obtained with \+
" }{TEXT 266 8 "rightbox" }{TEXT -1 2 ", " }{TEXT 267 7 "leftbox" }
{TEXT -1 4 " or " }{TEXT 268 9 "middlebox" }{TEXT -1 138 " (select one
).  For these commands you need to communicate the function, the main \+
interval and the number of subintervals.  The syntax is " }{TEXT 269 
53 "xxxbox( expression, interval, number of subintervals)" }{TEXT -1 
97 ". For example, if in our case we want to use left endpoints and 10
 subintervals, we need to enter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
leftbox( f(x), x=0..2, 10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The
 " }{TEXT 270 32 "calculation of the combined area" }{TEXT -1 38 " of \+
all green rectangles is done with " }{TEXT 271 8 "rightsum" }{TEXT -1 
2 ", " }{TEXT 272 7 "leftsum" }{TEXT -1 4 " or " }{TEXT 273 9 "middles
um" }{TEXT -1 24 ".  Again, the syntax is " }{TEXT 274 53 "xxxsum( exp
ression, interval, number of subintervals)" }{TEXT -1 92 ".  Unfortuna
tely, these commands just display the sums which need to be evaluated.
  Example:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "leftsum( f(x), x=0..2
, 10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "In order to get the act
ual sum, you need to follow this with " }{TEXT 275 8 "value(%)" }
{TEXT -1 9 " or with " }{TEXT 276 8 "evalf(%)" }{TEXT -1 1 ":" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "value(%); evalf(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 101 "A more direct way to achive the same res
ult is to enclose the leftsum command with the evalf command:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf( leftsum( f(x), x=0..2,10));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 8 "Example:" }{TEXT -1 25 "  We \+
study the function  " }{XPPEDIT 18 0 "f(x) = 1/x;" "6#/-%\"fG6#%\"xG*&
\"\"\"F)F'!\"\"" }{TEXT -1 19 "  on the interval  " }{XPPEDIT 18 0 "[1
, 5];" "6#7$\"\"\"\"\"&" }{TEXT -1 57 "  .  We know from class that th
e area under the curve is " }{XPPEDIT 18 0 "ln(5);" "6#-%#lnG6#\"\"&" 
}{TEXT -1 16 ".  Question: By " }}{PARA 0 "" 0 "" {TEXT -1 114 "how mu
ch is the error reduced when we use left endpoints and when we double \+
the number of intervals from 25 to 50?" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "f := x -> 1/x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "L
25 := evalf( leftsum( f(x), x=1..5,25)); # sum with 25 subintervals" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "L50 := evalf( leftsum( f(x
), x=1..5,50)); # sum with 50 subintervals" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "(L50 - ln(5.0))/(L25-ln(5.0));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 45 "Result:  The error is reduced by a factor of " }
{TEXT 286 10 "about 1/2 " }{TEXT -1 155 "when we double the number of \+
intervals (and thus the work required in the computations).  The exact
 value of this factor is not important, just eyeball it." }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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