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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 48 "MAT 156 F'08  Lab 5\nMid
point and Trapezoid Rules" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 26 "Top
ic 1 :  Midpoint Rule  " }{TEXT -1 314 "   In addition to the Left Han
d Rule and Right Hand Rules for calculating Riemann sums there is also
 the Midpoint Rule.  For this rule one simply picks the middle point o
f each subinterval instead of one of the endpoints.  The Maple command
s for this are very similar to the ones for left and right hand endpoi
nts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(student): with
(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoord
s has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f
:=x->sqrt(x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "middlebox(f(x), x=1..4, 5);" }}{PARA 13 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf(mid
dlesum(f(x), x=1..4, 5));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 17 "Ass
ignment Part 1" }{TEXT -1 182 ":    Write commands that show the boxes
 associated to the midpoint rule with 10, 20, 40 subintervals.   Write
 commands that evaluate the midpoint rule with 10, 20, 40 subintervals
.\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 18 "Assignment Part 2:" }
{TEXT -1 306 "   Write commands that show the boxes associated to  the
 left-hand sums, the right-hand sums and the midpoint sums with  10, 2
0, 40 subintervals.   Do not forget to introduce the plots package. Wh
at do you notice? Which are larger, the left-hand sums, right-hand sum
s, or midpoint sums? Can you explain it?" }}}{EXCHG {PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 18 "Assignment Part \+
3:" }{TEXT -1 202 "  Write commands that compute the left-hand sums an
d right-hand sum and midpoints sums numerically with 5, 10, 20, 40, 80
, 160, 320, 640, 1280, 2560 subintervals. You can use a loop. What do \+
you notice?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT 260 14 "Trapezoid Rule" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 282 "Unlike our previous rules, the trapezoid rule does \+
not use approzimation by rectangles.  As the name suggests trapezoids \+
are used.    This amounts to averaging the left hand value and right h
and value for each box.   There is a Maple command called trapezoid th
at computes the sums." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ev
alf(trapezoid(f(x),x=1..4,5));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 18 "Assignment Part 4:" }{TEXT -1 
297 "    Write commands that calculate the left hand, right hand, midp
oint and trapezoid  rules with 5, 10, 20, 40, 80, 160, 320, 640, 1280,
 2560 subintervals. You can use a loop. What do you notice? Which are \+
larger, smaller? Which are overestimates and which are underestimates \+
of the integral? Why?\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 18 "Assignment Part 5:" }{TEXT 
-1 59 "  Work all the commands introduced today for the integral \n" }
{XPPEDIT 18 0 "Int(1/x, x= 1..2)" "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;
F'\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 229 "with n=1, 2, 4, 8, 16, 32, 64
 subintervals. Order in increasing order the left-hand sums, right-han
d sums, midpoint sums and trapezoid sums. Explain you answer. Graph yo
ur sums  for n=1  and n= 2 to explain your answer.          " }}}
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