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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 18 " Lab 8    Math 151" }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "
  #1 Enter the function " }{XPPEDIT 18 0 "f(x) = x*(1-x);" "6#/-%\"fG6
#%\"xG*&F'\"\"\",&F)F)F'!\"\"F)" }{TEXT -1 44 " and plot its graph on \+
the interval  [0,1] ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 67 "  #2 Integrate the above function on the interval  [0,
1] using the " }{TEXT 256 3 "int" }{TEXT -1 48 " command. The answer, \+
of course, should be  1/6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 134 "  #3 In order to access routines which show what
 some of the Riemann sums corresponding to this integral look like, ty
pe the command  " }{TEXT 257 16 "with(student) . " }{TEXT -1 10 "Then \+
type " }{TEXT 259 22 "middlebox(f(x),x=0..1)" }{TEXT -1 215 ". This di
splays in green the vertical strips whose total area equals the value \+
of the Riemann sum when the interval [0,1] is divided into four equal \+
parts and function evaluations occur at midpoints of subintervals." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "#4 Type " 
}{TEXT 258 22 "middlesum(f(x),x=0..1)" }{TEXT -1 209 " in order to see
 what the corresponding Riemann sum looks like.  Copy the resulting su
m into the argument of evalf( ) to obtain the numerical value of the R
iemann sum. How does it compare with the exact value?" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "#5 Using a partition \+
of the interval [0,1] into four equal subintervals obviously leads to \+
a Riemann sum which gives only a crude approximation of the exact valu
e of the interval. To see what happens with ten subintervals, use the \+
commands " }{TEXT -1 101 "middlebox(f(x),x=0..1,10 ) , etc. to get the
 graph, the Riemann sum and the value of the Riemann sum." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 313 "#6 Leftbox, lefts
um, and rightbox, rightsum,  refer to the cases in which the function \+
evaluations in the Riemann sums are taken at the left-hand and right-h
and endpoints of the subintervals respectively. Using ten subintervals
, do the graphs and numerical valuations of the Riemann sums for both \+
of these cases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 276 "#7 The numbers which you obtain in #6 should turn out to
 be equal. Can you see why by inspecting the corresponding graphs? Is \+
this number more or less accurate than the middlesum case? Explain the
 difference in accuracy by comparing the middlebox graph with the left
box graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 20 "#8 For the integral " }{XPPEDIT 18 0 "int(x*cos(x^2),x = 0 .. s
qrt(4*pi));" "6#-%$intG6$*&%\"xG\"\"\"-%$cosG6#*$F'\"\"#F(/F';\"\"!-%%
sqrtG6#*&\"\"%F(%#piGF(" }{TEXT -1 173 " , plot the middlebox graph an
d find the value of the middlesum for ten subintervals and for forty s
ubintervals. What should the exact value be? (Maple note: use Pi, not \+
pi)" }}}{MARK "0 0" 6 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }