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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT -1 21 "Symbolic Riemann Sums" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 521 "T
his worksheet will allow you to investigate left, right and midpoint R
iemann sums as well as the sums obtained from the trapezoid and Simpso
n rules. Each sum will be simplified to a closed-form formula in terms
of the number of partitions, n. You can then investigate such issues \+
as what happens to the sum in the limit as n goes to infinity, or how \+
the errors for the different sums change as n changes. The worksheet w
ill also plot the approximations as functions of n to illustrate the c
onvergence rates to the limit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "Larry Riddle, Agnes Scott College, Januar
y 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 61 "Be sure to execute these first two commands before you st
art." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: with(stude
nt):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 219 "To investigate your own function, change the example in the ne
xt line. The only thing you will need to change in the rest of the wor
ksheet is the lower and upper limits for the integral and each Riemann
sum calculation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 40 "f := x^4-3*x^3-3*x^2+30;\na := 0;\nb := 2;"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot([f,0], x=a..b);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ans := Int(f, x=a..b): % = v
alue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LS := leftsum(f
, x=a..b, n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "LEFT(n) :=
expand(value(LS));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "RS :
= rightsum(f, x=a..b, n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
30 "RIGHT(n) := expand(value(RS));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 32 "TRAP(n) := (LEFT(n)+RIGHT(n))/2;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "MS := middlesum(f, x=a..b, n);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "MID(n) := expand(value(MS));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "SIMP(n) := expand((2*MID(n) \+
+ TRAP(n))/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([LEF
T(n), RIGHT(n), SIMP(n)], n=5..100);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 42 "plot([MID(n), TRAP(n), SIMP(n)], n=5..40);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 1 0" 0 }
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