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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "1.  Evaluate the following
, if they exist.  If they are divergent, state clearly why. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sum(2^k * 5^(-k-1), k = 2..i
nfinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#v" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "limit((x^2+3*x-4)/(2*x^2+x-3), x=1)
;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "limit((x^2+3*x-4)
/(2*x^2+x-3), x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"
\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(By l'Hopital's rule)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "limit((x^2+3*x-4)/(2*x^2+x
-3), x=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 124 "L'Hopital's rule is NOT valid here, because th
e denominator  does not tend to 0.  It is 0  because the numerator ten
ds to 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "2.  Determine whether
 the following converge.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "int(((3+x)/(3*x^2))^(1/2), x=2..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We
ll, that's divergent.  But why?  Because the integrand is greater than
 (1/3x)^(1/2), which cannot be integrated:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "int((1/3*x)^(1/2),x=0..R);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&%\"RG#\"\"$\"\"#F'#\"\"\"F(#F(\"\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 25 "which gets big as R -> °." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sum(1/(sin(k) + sqrt(k)),k=2..infin
ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*$,&-%$sinG6#%\"kG\"
\"\"*$F+#F,\"\"#F,!\"\"/F+;F/%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 83 "Maple is telling us she doesn't know the exact answer.  B
ut as k -> °, we know that" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "limit(sqrt(k)/(sin(k) + sqrt
(k)),k=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "So this series converges if and on
ly if " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sum(1/(sqrt(k)),k
=2..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*$%\"kG#!\"
\"\"\"#/F';F*%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Here
, even if Maple doesn't say so, we know that the series diverges.  It \+
is our friend the p-series with p=1/2." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 245 "Problems 3 and 4 are concern
ed with estimating the integral of 1/(4+x^2).   No credit will be give
n for an accurate estimate of this integral, only for the approximatio
n,.\n\n\n3.  (10 points)  In this problem we use Taylor's polynomial a
nd series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "taylor(1/(4+x
^2),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG#\"\"\"\"\"%\"\"!#
!\"\"\"#;\"\"##F&\"#k\"\"%-%\"OG6#F&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "int(1/4-(1/16)*x^2+(1/64)*x^4,x=0..1/2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"%jP\"&?2$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'*[$
\\A\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "For what positive val
ues of x is the Taylor series convergent for int(1/(4+t^2),t=0..x);" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Answer: \+
 SAME AS FOR int(1/(4+x^2), except possibly for the ends of the interv
al.  Since this is " }}{PARA 0 "" 0 "" {TEXT -1 93 "the sum of a geome
tric series in powers  ((x/2)^2)^n, we have convergence for -2 < x < 2
.  We" }}{PARA 0 "" 0 "" {TEXT -1 98 "actually do a little better, for
 it converges when x=2,-2 by the alternating series test.  Answer:" }}
{PARA 0 "" 0 "" {TEXT -1 11 "-2 ² x ² 2." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "f := x -> 1/(4+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(*$,&\"\"%\"\"\"*$9$\"\"#F/!
\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(student[t
rapezoid](f(x),x=0..2,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*eqR
\"R!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(student[sim
pson](f(x),x=0..2,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+mmm;R!#5
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Just for fun:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(f(x),x=0..2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=3*p
#R!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "29 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }