{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 
0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }
{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 
272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }
{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 
278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 18 0 
0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 12 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 282 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
283 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 12 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 12 0 0 0 0 0 2 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
288 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 18 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 
300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 301 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" 18 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 18 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 309 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 
311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 1 12 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 316 "" 0 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 
-1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 
0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" 
-1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 
"Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 
}{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 
2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 
0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 259 0 "" }{TEXT 260 30 "Converg
ence and Divergence of " }{TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 18 
"Improper Integrals" }{TEXT 261 1 ":" }}{PARA 258 "" 0 "" {TEXT -1 0 "
" }{TEXT 264 0 "" }{TEXT 265 26 "Comparison Test Benchmarks" }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 257 18 "Calculus 2 Project" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objective:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Explore b
enchmarks for the comparison test graphically and algebraically." }}
{PARA 0 "" 0 "" {TEXT -1 67 "Understand how to choose a benchmark for \+
using the Comparison Test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 96 "Using Maple allows students to quickly and easily
 graph functions and do algebraic calculations." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 0 "" }{TEXT 309 12 " Background:" }}{PARA 0 "" 0 "" {TEXT 
-1 34 "Some generally used benchmarks are" }}{PARA 0 "" 0 "" {XPPEDIT 
18 0 "int(exp(-x),x = 0 .. infinity);" "6#-%$intG6$-%$expG6#,$%\"xG!\"
\"/F*;\"\"!%)infinityG" }{TEXT -1 23 " = 1                   " }
{XPPEDIT 18 0 "int(1/(x^p),x = 1 .. infinity);" "6#-%$intG6$*&\"\"\"F'
)%\"xG%\"pG!\"\"/F);F'%)infinityG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 
"1/(p-1);" "6#*&\"\"\"F$,&%\"pGF$F$!\"\"F'" }{TEXT -1 7 "   if  " }
{TEXT 312 1 "p" }{TEXT -1 28 " > 1                        " }{XPPEDIT 
18 0 "int(1/(x^p),x = 1 .. infinity);" "6#-%$intG6$*&\"\"\"F')%\"xG%\"
pG!\"\"/F);F'%)infinityG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "infinity
;" "6#%)infinityG" }{TEXT -1 8 "   if   " }{TEXT 311 1 "p" }{TEXT -1 
1 " " }{TEXT 310 1 "<" }{TEXT -1 2 " 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 266 15 "Solved Examp
le:" }}{PARA 0 "" 0 "" {TEXT -1 30 "a. State the Comparison Test. " }}
{PARA 0 "" 0 "" {TEXT -1 79 "b. Determine the proper benchmark for usi
ng the Comparison Test to find whether" }{XPPEDIT 18 0 "int(1/(x^6+1),
x = 1 .. infinity);" "6#-%$intG6$*&\"\"\"F',&*$%\"xG\"\"'F'F'F'!\"\"/F
*;F'%)infinityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "     co
nverges or diverges. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 97 "c. Show graphically that the benchmark you chose satis
fies the conditions of the Comparison Test." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "d. Find an upper bound for the \+
value of the improper integral." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" 
}{TEXT 267 0 "" }{TEXT 268 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 4 
"a.  " }{TEXT 273 9 "Theorem: " }{TEXT -1 16 " Comparison Test" }}
{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }{TEXT 291 1 "f" }{TEXT -1 7 "  and
  " }{TEXT 292 1 "g" }{TEXT -1 50 "  be continuous functions.  Suppose
 that for all  " }{TEXT 293 1 "x" }{TEXT -1 1 " " }{TEXT 269 1 ">" }
{TEXT -1 1 " " }{TEXT 294 1 "a" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 48 "                                              0 " }{TEXT 
270 1 "<" }{TEXT -1 2 "  " }{TEXT 295 1 "f" }{TEXT -1 1 "(" }{TEXT 
296 1 "x" }{TEXT -1 2 ") " }{TEXT 271 1 "<" }{TEXT -1 2 "  " }{TEXT 
297 1 "g" }{TEXT -1 1 "(" }{TEXT 298 1 "x" }{TEXT -1 2 ")." }}{PARA 0 
"" 0 "" {TEXT -1 5 "If   " }{XPPEDIT 300 0 "int(g(x),x = a .. infinity
);" "6#-%$intG6$-%\"gG6#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 28 "  con
verges, then so does   " }{XPPEDIT 299 0 "int(f(x),x = a .. infinity);
" "6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 10 " , and \+
   " }{XPPEDIT 301 0 "int(f(x),x = a .. infinity)" "6#-%$intG6$-%\"fG6
#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 1 " " }{TEXT 272 1 "<" }{TEXT 
-1 1 " " }{XPPEDIT 302 0 "int(g(x),x = a .. infinity)" "6#-%$intG6$-%
\"gG6#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 5 "If   " }{XPPEDIT 303 0 "int(f(x),x = a .. infinity)" "6#-%
$intG6$-%\"fG6#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 26 "  diverges, th
en so does  " }{XPPEDIT 304 0 "int(g(x),x = a .. infinity)" "6#-%$intG
6$-%\"gG6#%\"xG/F);%\"aG%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "b. " }}{PARA 0 "" 0 "" 
{TEXT -1 12 "BENCHMARK:  " }{XPPEDIT 18 0 "int(1/(x^6),x = 1 .. infini
ty);" "6#-%$intG6$*&\"\"\"F'*$%\"xG\"\"'!\"\"/F);F'%)infinityG" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot([1/(x^6+1), 1
/x^6], x = 0..4, y = 0..0.6, color = [red,green]);" }}{PARA 13 "" 1 "
" {GLPLOT2D 306 163 163 {PLOTDATA 2 "6&-%'CURVESG6$7en7$\"\"!$\"\"\"F(
7$$\"1mmmm;')=()!#<$\"0,Fqg&******!#:7$$\"1LLLe'40j\"!#;$\"1o5q(47)***
*F57$$\"1nmm;6m$[#F5$\"120cHLl(***F57$$\"1nmm;yYULF5$\"1g'GJ!\\2')**F5
7$$\"1LLLeF>(>%F5$\"1L0+1niX**F57$$\"1*****\\Z7Mf%F5$\"1!R*f,B%p!**F57
$$\"1mmm\">K'*)\\F5$\"1q>Xq!H![)*F57$$\"1KLLeZ*)*R&F5$\"1w\"3@ix!e(*F5
7$$\"1*****\\Kd,\"eF5$\"1d2fApaH'*F57$$\"1mmm\"fX(emF5$\"1.=&Hs9#)>*F5
7$$\"1LLL3!z;3(F5$\"1TG#\\nr*z))F57$$\"1*****\\U7Y](F5$\"11G>*z\"Q%[)F
57$$\"1mm;H9lRzF5$\"1ruQ2!)z'*zF57$$\"1MLLL/pu$)F5$\"1M13pG(\\V(F57$$
\"1+++DI(yv)F5$\"1#)fJ(fO2*oF57$$\"1nmm;c0T\"*F5$\"1Uf%)e4Y:jF57$$\"1L
LLe%GCd*F5$\"1roPPJu^cF57$$\"1+++I,Q+5F1$\"1Ip$f8*H%*\\F57$$\"1+++SXpV
5F1$\"1-5)[+%*>O%F57$$\"1+++]*3q3\"F1$\"1\"yrR*f/uPF57$$\"1+++5/vG6F1$
\"1(*H9w_EfKF57$$\"1+++q=\\q6F1$\"1X@R]J$)*z#F57$$\"1nm;fBIY7F1$\"1qPn
MhP1@F57$$\"1LLLj$[kL\"F1$\"1d[c1.,$\\\"F57$$\"1LLL`Q\"GT\"F1$\"1Zi:OF
*p6\"F57$$\"1++]s]k,:F1$\"1Q)fO2e>-)F.7$$\"1LLL`dF!e\"F1$\"19\")H9XeLg
F.7$$\"1++]sgam;F1$\"16+.UWZfWF.7$$\"1++]<ep[<F1$\"165$Hx#)*yLF.7$$\"1
LLLe/TM=F1$\"19&)\\aTBdDF.7$$\"1LL$eDBJ\">F1$\"1&*G$eg>))*>F.7$$\"1nmm
Tc-)*>F1$\"1iy@HSYZ:F.7$$\"1mm;f`@'3#F1$\"1\"3\\&H+U)>\"F.7$$\"1++]nZ)
H;#F1$\"1%[_#z4sq'*!#=7$$\"1mmmJy*eC#F1$\"1r_\"[#*\\>t(Fbv7$$\"1+++S^b
JBF1$\"1Dvx))zH'='Fbv7$$\"1+++0TN:CF1$\"1KIvEN56]Fbv7$$\"1++]7RV'\\#F1
$\"1$\\2[*[B9TFbv7$$\"1+++:#fke#F1$\"1\"yP]BK!HLFbv7$$\"1LLL`4NnEF1$\"
1!\\:g;[*oFFbv7$$\"1+++],s`FF1$\"1OHP*ef\")G#Fbv7$$\"1mm;zM)>$GF1$\"11
IO)e/Z$>Fbv7$$\"1+++qfa<HF1$\"1F%)\\f')y=;Fbv7$$\"1LL$eg`!)*HF1$\"1kEe
!Q+_P\"Fbv7$$\"1++]#G2A3$F1$\"1KX7a8*\\;\"Fbv7$$\"1LLL$)G[kJF1$\"1MBhN
RN[**!#>7$$\"1++]7yh]KF1$\"1dqS`l3p%)F_z7$$\"1nmm')fdLLF1$\"1e!)f53^\"
G(F_z7$$\"1nmm,FT=MF1$\"1B.(y+zHE'F_z7$$\"1LL$e#pa-NF1$\"1s>'>:(G8aF_z
7$$\"1+++Sv&)zNF1$\"1')3Ynz'*[ZF_z7$$\"1LLLGUYoOF1$\"1sJ32)[75%F_z7$$
\"1nmm1^rZPF1$\"1=HFl<\"yg$F_z7$$\"1++]sI@KQF1$\"1DIa_O?cJF_z7$$\"1++]
2%)38RF1$\"1%[e\"=*)e%y#F_z7$$\"\"%F($\"1!)e.\\.\"3W#F_z-%'COLOURG6&%$
RGBG$\"*++++\"!\")F(F(-F$6$7_o7$$\"1+++m;')=()F.$\"10!zmthjF#!\"*7$$\"
1,++2:$f&*)F.$\"1U*3\"*4=z$>Fa^l7$$\"1+++Z8+$>*F.$\"13-i$[Ynl\"Fa^l7$$
\"1+++)=r+V*F.$\"13%y$HY.A9Fa^l7$$\"1*****z-Trm*F.$\"1%\\P,A8_A\"Fa^l7
$$\"1,++p3@/**F.$\"1nnjY1Xf5Fa^l7$$\"1+++r!GT,\"F5$\"11Z)4'=q#>*!#57$$
\"1+++b]$y.\"F5$\"1)\\l!z#zD+)F``l7$$\"1+++R?ah5F5$\"1\"[9X&[Q))pF``l7
$$\"1+++3g&*36F5$\"1O&)f!yymP&F``l7$$\"1+++w*pj:\"F5$\"1V:N$GQB=%F``l7
$$\"1+++WRy.7F5$\"1/O(3Y6jG$F``l7$$\"1+++7z>^7F5$\"1<Zp2sT1EF``l7$$\"1
+++&y`3W\"F5$\"1k3h6\\e<6F``l7$$\"1+++e'40j\"F5$\"1R24qY\"=K&!#67$$\"1
+++\"f`rt\"F5$\"1[#*yiZ!*QOFibl7$$\"1+++BvzV=F5$\"1H4l6D>XDFibl7$$\"1+
++c9W]>F5$\"11^QCBO;=Fibl7$$\"1+++)Q&3d?F5$\"1xj(Q-L(>8Fibl7$$\"1+++`K
PqAF5$\"1oN$H'Hd,t!#77$$\"1+++<6m$[#F5$\"1*)fy\"yc.E%Fcdl7$$\"1+++oW18
HF5$\"1P`e/vVO;Fcdl7$$\"1+++<yYULF5$\"1_O&Qey7<(!#87$$\"1+++eF>(>%F5$
\"1O64j\"Q\"H=Fcel7$$\"1+++\">K'*)\\F5$\"1%R^v&e?!['!#97$$\"1+++Dt:5eF
5$\"1;sH+sR*f#F^fl7$$\"1+++\"fX(emF5$\"1R\"okr;s9\"F^fl7$$\"1+++DCh/vF
5$\"1-WCVx'zf&F17$$\"1+++L/pu$)F5$\"1CSXTUf)*GF17$$\"1+++;c0T\"*F5$\"1
\"GO$3P/9<F17$F^q$\"1tz,9DAx**F57$Fhq$\"1M+-%\\&zhgF57$Fbr$\"1:QRTqc))
QF57$$\"1+++fBIY7F1$\"1Ns**HEXoEF57$$\"1+++j$[kL\"F1$\"1^Ji<&R]v\"F57$
$\"1+++`Q\"GT\"F1$\"1-3Ei)[uD\"F57$$\"1+++s]k,:F1$\"1?GR)G,;s)F.7$$\"1
+++`dF!e\"F1$\"1)4an/,5U'F.7$$\"1+++sgam;F1$\"1OyPSginYF.7$$\"1+++<ep[
<F1$\"12NO04:(\\$F.7$$\"1+++e/TM=F1$\"1_O\"))zWVi#F.7$$\"1+++cK78>F1$
\"1)H@^F(eR?F.7$$\"1+++Uc-)*>F1$\"1B`+koyr:F.7$$\"1+++f`@'3#F1$\"1MU(=
McH@\"F.7$$\"1+++nZ)H;#F1$\"1.1K-r:l(*Fbv7$$\"1+++Ky*eC#F1$\"1)oC*3))>
#z(Fbv7$Fiv$\"1=&)o'\\1[A'Fbv7$F^w$\"18,n`6MO]Fbv7$$\"1+++7RV'\\#F1$\"
13tgd<BJTFbv7$Fhw$\"1sFu2<:SLFbv7$$\"1+++`4NnEF1$\"1)4@2`Omx#Fbv7$Fbx$
\"1+H@rsS$H#Fbv7$$\"1+++zM)>$GF1$\"1mroE\\XQ>Fbv7$F\\y$\"1%e)4&Q89i\"F
bv7$$\"1+++1O0)*HF1$\"1Mw`gT4x8Fbv7$$\"1+++#G2A3$F1$\"1L54V,Nm6Fbv7$$
\"1+++$)G[kJF1$\"17V<s2Ee**F_z7$$\"1+++7yh]KF1$\"1;@d$=liZ)F_z7$$\"1++
+()fdLLF1$\"1=Tp4n\"oG(F_z7$$\"1+++-FT=MF1$\"1`i*H&R!pE'F_z7$$\"1+++Ep
a-NF1$\"1jfM0\">iT&F_z7$Fe[l$\"1#pl'3VA^ZF_z7$$\"1+++GUYoOF1$\"1+3iB:$
H5%F_z7$$\"1+++2^rZPF1$\"1WNpjQ64OF_z7$$\"1+++sI@KQF1$\"1eDnJ,?dJF_z7$
$\"1+++2%)38RF1$\"1(e8)HXO&y#F_z7$F^]l$\"1+++]iSTCF_z-Fc]l6&Fe]lF(Ff]l
F(-%+AXESLABELSG6$Q\"x6\"Q\"yFcbm-%%VIEWG6$;F(F^]l;F($\"\"'!\"\"" 1 2 
0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Since  0 " }{TEXT 276 1 "<" }
{TEXT -1 3 "   " }{XPPEDIT 18 0 "1/(x^6+1);" "6#*&\"\"\"F$,&*$%\"xG\"
\"'F$F$F$!\"\"" }{TEXT -1 2 "  " }{TEXT 275 1 "<" }{TEXT -1 3 "   " }
{XPPEDIT 18 0 "1/(x^6);" "6#*&\"\"\"F$*$%\"xG\"\"'!\"\"" }{TEXT -1 9 "
     for " }{TEXT 305 1 "x" }{TEXT -1 1 " " }{TEXT 277 1 ">" }{TEXT 
-1 28 " 1 , then the integral of   " }{TEXT 306 1 "f" }{TEXT -1 1 "(" 
}{TEXT 307 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "1/(x^6+1);" "6#*&
\"\"\"F$,&*$%\"xG\"\"'F$F$F$!\"\"" }{TEXT -1 12 "  converges." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By the fo
rmula,   " }{XPPEDIT 18 0 "int(1/(x^6),x = 1 .. infinity);" "6#-%$intG
6$*&\"\"\"F'*$%\"xG\"\"'!\"\"/F);F'%)infinityG" }{TEXT -1 17 "   conve
rges to  " }{XPPEDIT 18 0 "1/(6-1);" "6#*&\"\"\"F$,&\"\"'F$F$!\"\"F'" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }
{TEXT -1 18 "   and therefore  " }{XPPEDIT 18 0 "1/5;" "6#*&\"\"\"F$\"
\"&!\"\"" }{TEXT -1 25 "  is an upper bound for  " }{XPPEDIT 18 0 "int
(1/(x^6+1),x = 1 .. infinity);" "6#-%$intG6$*&\"\"\"F',&*$%\"xG\"\"'F'
F'F'!\"\"/F*;F'%)infinityG" }{TEXT -1 2 " ." }}}}{PARA 0 "" 0 "" 
{TEXT -1 80 "_________________________________________________________
_______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "
" 0 "" {TEXT -1 0 "" }{TEXT 278 0 "" }{TEXT 279 10 "ASSIGNMENT" }}
{PARA 260 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 280 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
286 1 " " }{TEXT 282 80 "a) Determine the proper benchmark for using t
he Comparison Test to find whether " }{TEXT 287 0 "" }{TEXT 288 0 "" }
}{PARA 0 "" 0 "" {XPPEDIT 18 0 "int(1/(exp(x)+x),x = 0 .. infinity);" 
"6#-%$intG6$*&\"\"\"F',&-%$expG6#%\"xGF'F,F'!\"\"/F,;\"\"!%)infinityG
" }{TEXT -1 26 "   converges or diverges. " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "b) Show graphically that the be
nchmark you chose satisfies the conditions of the Comparison Test." }}
{PARA 0 "" 0 "" {TEXT -1 62 "c) Find an upper bound for the value of t
he improper integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 76 "______________________________________________________
______________________" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 281 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 
81 "a) Determine the proper benchmark for using the Comparison Test to
 find whether  " }{XPPEDIT 18 0 "int(1/(x+1),x = 1 .. infinity);" "6#-
%$intG6$*&\"\"\"F',&%\"xGF'F'F'!\"\"/F);F'%)infinityG" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 27 "    converges or diverges. " }}{PARA 0 
"" 0 "" {TEXT -1 97 "b) Show graphically that the benchmark you chose \+
satisfies the conditions of the Comparison Test." }}{PARA 0 "" 0 "" 
{TEXT -1 62 "c) Find an upper bound for the value of the improper inte
gral." }}{PARA 0 "" 0 "" {TEXT -1 19 "Hint:  Compare to  " }{XPPEDIT 
18 0 "1/(2*x);" "6#*&\"\"\"F$*&\"\"#F$%\"xGF$!\"\"" }{TEXT -1 2 " ." }
}{PARA 0 "" 0 "" {TEXT -1 78 "________________________________________
______________________________________" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 284 10 "Problem 3:" }}{PARA 3 "" 0 "" {TEXT 313 
1 " " }{TEXT -1 0 "" }{TEXT 285 81 "a) Determine the proper benchmark \+
for using the Comparison Test to find whether  " }{XPPEDIT 18 0 "int(1
/(x*sqrt(x^2+1)),x = 1 .. infinity);" "6#-%$intG6$*&\"\"\"F'*&%\"xGF'-
%%sqrtG6#,&*$F)\"\"#F'F'F'F'!\"\"/F);F'%)infinityG" }{TEXT -1 0 "" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "    converges or diverges
. " }}{PARA 0 "" 0 "" {TEXT -1 97 "b) Show graphically that the benchm
ark you chose satisfies the conditions of the Comparison Test." }}
{PARA 0 "" 0 "" {TEXT -1 62 "c) Find an upper bound for the value of t
he improper integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 82 "____________________________________________________
______________________________" }}{PARA 0 "" 0 "" {TEXT 314 0 "" }
{TEXT 316 0 "" }{TEXT -1 172 "MSIP Grant #P120A80089-98:  \"Three Urba
n Calculus Reform programs: Adopting the Best\" 1998-2001, MSEIP Grant
 #P120A010031: \"Four Colleges: Calculus + Enhancements\"  2001-04" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "15 1 2" 53 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }