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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 405 4 "Math" }{TEXT 652 5 " 2331
" }{TEXT 651 18 " \nExam 2 Fall 2002" }}{PARA 0 "" 0 "" {TEXT 260 2 " \+
 " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 "1" }{TEXT 315 2 ". " }
{TEXT 293 55 "Which of the following vector-valued vector functions  \+
" }{TEXT 366 2 "  " }{XPPEDIT 314 1 "r(t)=`<`*x(t),y(t),z(t)*`>`" "6%/
-%\"rG6#%\"tG*&%\"<G\"\"\"-%\"xG6#F'F*-%\"yG6#F'*&-%\"zG6#F'F*%\">GF*
" }{TEXT 360 73 "    parameterizes a curve that lies on a sphere cente
red at the origin?\n " }}{PARA 268 "" 0 "" {TEXT -1 4 "a)  " }{TEXT 
409 2 "  " }{XPPEDIT 408 1 "`<`*cos(t),sin(t),t*`>`" "6%*&%\"<G\"\"\"-
%$cosG6#%\"tGF%-%$sinG6#F)*&F)F%%\">GF%" }{TEXT -1 8 "   \nb)  " }
{TEXT 411 2 "  " }{XPPEDIT 410 1 "`<`*t*cos(t), t*sin(t),t*`>`" "6%*(%
\"<G\"\"\"%\"tGF%-%$cosG6#F&F%*&F&F%-%$sinG6#F&F%*&F&F%%\">GF%" }
{TEXT -1 15 "       \nc)     " }{XPPEDIT 412 1 "`<`*cos(t), sin(t),(1-
cos(t)*`>`" "6%*&%\"<G\"\"\"-%$cosG6#%\"tGF%-%$sinG6#F),&F%F%*&-F'6#F)
F%%\">GF%!\"\"" }{TEXT -1 18 "           \nd)    " }{XPPEDIT 413 1 "`<
`*cos(t)^2, sin(t)^2, t*`>`" "6%*&%\"<G\"\"\"*$-%$cosG6#%\"tG\"\"#F%*$
-%$sinG6#F*F+*&F*F%%\">GF%" }{TEXT -1 9 "  \ne)    " }{XPPEDIT 418 1 "
`<`*cos(t)*sin(t), sin(t)^2, cos(t)*`>`" "6%*(%\"<G\"\"\"-%$cosG6#%\"t
GF%-%$sinG6#F)F%*$-F+6#F)\"\"#*&-F'6#F)F%%\">GF%" }{TEXT -1 9 "  \nf) \+
   " }{XPPEDIT 419 1 "`<`*1-t, 1+t, 1*`>`" "6%,&*&%\"<G\"\"\"F&F&F&%\"
tG!\"\",&F&F&F'F&*&F&F&%\">GF&" }{TEXT -1 1 " " }}{PARA 269 "" 0 "" 
{TEXT 294 5 "g)   " }{XPPEDIT 414 1 "`<`*1-t, 1+t, sqrt(1-t)*`>`" "6%,
&*&%\"<G\"\"\"F&F&F&%\"tG!\"\",&F&F&F'F&*&-%%sqrtG6#,&F&F&F'F(F&%\">GF
&" }{TEXT -1 1 "\n" }{TEXT 415 7 "h)     " }{XPPEDIT 416 1 "`<`*1-t, 1
+t, sqrt(1-t^2)*`>`" "6%,&*&%\"<G\"\"\"F&F&F&%\"tG!\"\",&F&F&F'F&*&-%%
sqrtG6#,&F&F&*$F'\"\"#F(F&%\">GF&" }{TEXT 417 17 "        \ni)      " 
}{XPPEDIT 420 1 "`<`*sqrt(t), sqrt(t), sqrt(1-t^2)*`>`" "6%*&%\"<G\"\"
\"-%%sqrtG6#%\"tGF%-F'6#F)*&-F'6#,&F%F%*$F)\"\"#!\"\"F%%\">GF%" }
{TEXT 421 6 "\nj)   " }{XPPEDIT 422 1 "`<`*sqrt(t), sqrt(t), sqrt(1-t)
*`>`" "6%*&%\"<G\"\"\"-%%sqrtG6#%\"tGF%-F'6#F)*&-F'6#,&F%F%F)!\"\"F%%
\">GF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 653 12 "Solu
tion:  e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "magnitude := r -> sqrt(sum(op(i,r)^2,i=1..nops(r)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeG:6#%\"rG6\"6$%)operat
orG%&arrowGF(-%%sqrtG6#-%$sumG6$*$-%#opG6$%\"iG9$\"\"#/F6;\"\"\"-%%nop
sG6#F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "simplify(magn
itude([cos(t), sin(t), t])); #a: not constant " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,&*$%\"tG\"\"#\"\"\"F(F(#F(F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 60 "simplify(magnitude([t*cos(t),t*sin(t),t])); #b
: not constant" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(\"\"##\"\"\"F$-%%c
sgnG6#%\"tGF&F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "simpli
fy(magnitude([cos(t), sin(t), 1-cos(t)])); #c: not constant" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*$,(*$-%$cosG6#%\"tG\"\"#\"\"\"F*F+F&!\"##F+
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(magnitude([c
os(t)^2, sin(t)^2, t])); #d: not constant" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,**$-%$cosG6#%\"tG\"\"%\"\"#*$F)F+\"\"\"F-F-*$F&F+!\"
##F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "simplify(magnitud
e([cos(t)*sin(t), sin(t)^2, cos(t)])); #e:  constant!  Let's continue \+
though..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "simplify(magnitude([1-t, 1+t, 1])); #f: n
ot constant" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&\"\"$\"\"\"*$%\"tG
\"\"#F)#F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "simplify(ma
gnitude([1-t, 1+t, sqrt(1-t)])); #g: not constant" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,(\"\"$\"\"\"%\"tG!\"\"*$F'\"\"#F*#F&F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(magnitude([1-t, 1+t, sqrt(
1-t^2)])); #h: not constant" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&\"
\"$\"\"\"*$%\"tG\"\"#F&#F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 70 "simplify(magnitude([sqrt(t), sqrt(t), sqrt(1-t^2)])); #i: not co
nstant" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(%\"tG\"\"#\"\"\"F'*$F%F&
!\"\"#F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "simplify(magn
itude([sqrt(t), sqrt(t), sqrt(1-t)])); #j: not constant" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*$,&\"\"\"F%%\"tGF%#F%\"\"#" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 275 4 "2.  " }{TEXT 
-1 3 "If " }{TEXT 369 1 " " }{XPPEDIT 367 1 "r(t)=`<`*t^t,t^3,(t*ln(t)
)*`>`" "6%/-%\"rG6#%\"tG*&%\"<G\"\"\")F'F'F**$F'\"\"$*(F'F*-%#lnG6#F'F
*%\">GF*" }{TEXT 368 55 "  then what is the dot product of\n\n        \+
            " }{XPPEDIT 424 1 "limit( (r(1+Delta*t)-r(1))/(Delta*t),De
lta*t=0)" "-%&limitG6$*&,&-%\"rG6#,&\"\"\"F+*&%&DeltaGF+%\"tGF+F+F+-F(
6#F+!\"\"F+*&F-F+F.F+F1/*&F-F+F.F+\"\"!" }{TEXT 425 5 "   \n\n" }
{TEXT -1 23 "with the unit vector   " }{TEXT 423 1 "k" }{TEXT -1 3 " ?
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 295 4 "a)
  " }{XPPEDIT 296 1 "0" "\"\"!" }{TEXT 297 10 "      b)  " }{XPPEDIT 
298 1 "1" "\"\"\"" }{TEXT 299 10 "      c)  " }{XPPEDIT 300 1 "2" "\"
\"#" }{TEXT 301 8 "     d) " }{XPPEDIT 302 1 "3" "\"\"$" }{TEXT 303 
10 "      e)  " }{XPPEDIT 304 1 "4" "\"\"%" }{TEXT 305 11 "        f) \+
" }{XPPEDIT 306 1 "6" "\"\"'" }{TEXT 307 11 "       g)  " }{XPPEDIT 
308 1 "9" "\"\"*" }{TEXT 309 9 "     h)  " }{XPPEDIT 310 1 "12" "\"#7
" }{TEXT 311 9 "      i) " }{XPPEDIT 312 1 "15" "\"#:" }{TEXT 313 10 "
      j)  " }{XPPEDIT 371 1 "18" "\"#=" }{TEXT -1 1 " " }{TEXT 370 3 "
   " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 654 12 "Solution:  b" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Limit Method" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new defin
ition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition
 for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r := t -> vec
tor( [t^t,t^3,t*ln(t)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#
%\"tG6\"6$%)operatorG%&arrowGF(-%'vectorG6#7%)9$F1*$F1\"\"$*&F1\"\"\"-
%#lnG6#F1F5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(1/Delta
)*(r(1+Delta)-r(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&DeltaG!\"
\",&-%'VECTORG6#7%),&\"\"\"F-F$F-F,*$F,\"\"$*&F,F--%#lnG6#F,F-F--F(6#7
%F-F-\"\"!F%F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dotprod(
\",vector([0,0,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%&DeltaG!\"
\",&\"\"\"F'F$F'F'-%#lnG6#F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "limit(\",Delta=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "Derivative Method" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "diff(t*ln(t), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&-%#lnG6#%\"tG\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "subs(t=1, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"
\"F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 40 "3. The position vector of a pa
rticle is " }{TEXT 318 2 "  " }{XPPEDIT 321 1 "r(t)=`<`*8*t,cos(Pi*t)^
2,sin(Pi*t)*ln(t)*`>`" "6%/-%\"rG6#%\"tG*(%\"<G\"\"\"\"\")F*F'F**$-%$c
osG6#*&%#PiGF*F'F*\"\"#*(-%$sinG6#*&F1F*F'F*F*-%#lnG6#F'F*%\">GF*" }
{TEXT 316 1 " " }{TEXT 319 22 ". What is its speed at" }{TEXT 320 2 " \+
 " }{XPPEDIT 322 1 "t=1" "/%\"tG\"\"\"" }{TEXT 317 1 " " }{TEXT 323 2 
"? " }{TEXT 324 2 "  " }}{PARA 264 "" 0 "" {TEXT 274 4 "a)  " }
{XPPEDIT 19 1 "6" "\"\"'" }{TEXT 381 14 "          b)  " }{XPPEDIT 19 
1 "2*sqrt(10" "*&\"\"#\"\"\"-%%sqrtG6#\"#5F$" }{TEXT 380 13 "         \+
c)  " }{XPPEDIT 19 1 "7" "\"\"(" }{TEXT 379 14 "          d)  " }
{XPPEDIT 19 1 "3*sqrt(6)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"'F$" }{TEXT 378 
17 "            e)   " }{XPPEDIT 19 1 "8" "\"\")" }{TEXT 377 15 "     \+
    \nf)   " }{XPPEDIT 19 1 "6*sqrt(2)" "*&\"\"'\"\"\"-%%sqrtG6#\"\"#F
$" }{TEXT 376 11 "      g)   " }{XPPEDIT 19 1 "9" "\"\"*" }{TEXT 375 
15 "            h) " }{XPPEDIT 19 1 "sqrt(86)" "-%%sqrtG6#\"#')" }
{TEXT 374 13 "         i)  " }{XPPEDIT 19 1 "10" "\"#5" }{TEXT 373 17 
"              j) " }{XPPEDIT 19 1 "sqrt(120)" "-%%sqrtG6#\"$?\"" }
{TEXT 372 1 " " }{TEXT 325 7 "      \n" }}{PARA 0 "" 0 "" {TEXT 655 
12 "Solution:  e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 45 "magnitude := r -> sqrt(r[1]^2+r[2]^2+r[3]^2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeG:6#%\"rG6\"6$%)operat
orG%&arrowGF(-%%sqrtG6#,(*$&9$6#\"\"\"\"\"#F4*$&F26#F5F5F4*$&F26#\"\"$
F5F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "x := t -> 8*t: \+
y := t -> cos(Pi*t)^2: z := t ->  sin(Pi*t)*ln(t):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t), [x(t),y(t),z(
t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%\"\"),$*(-%$cos
G6#*&%#PiG\"\"\"%\"tGF.F.-%$sinGF+F.F-F.!\"#,&*(F)F.F-F.-%#lnG6#F/F.F.
*&F0F.F/!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=1
, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"),$*(-%$cosG6#%#PiG\"
\"\"-%$sinGF)F+F*F+!\"#,&*(F'F+F*F+-%#lnG6#F+F+F+F,F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%\"\")\"\"!F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "magnitude(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 263 1 " " }{TEXT 330 36 "4. A space curv
e is parameterized by" }{TEXT 331 3 "   " }{XPPEDIT 329 1 "r(t) = `<`*
t^3,t^4,1*`>`" "6%/-%\"rG6#%\"tG*&%\"<G\"\"\"*$F'\"\"$F**$F'\"\"%*&F*F
*%\">GF*" }{TEXT 326 1 " " }{TEXT 328 65 ". What is the second compone
nt of its unit tangent      vector at" }{TEXT 426 2 "  " }{XPPEDIT 
427 1 "t=1" "/%\"tG\"\"\"" }{TEXT 327 1 "?" }}{PARA 0 "" 0 "" {TEXT 
268 4 "a)  " }{XPPEDIT 256 0 "1/5" "*&\"\"\"F#\"\"&!\"\"" }{TEXT 257 
13 "             " }{TEXT 332 3 "b) " }{TEXT 333 1 " " }{XPPEDIT 276 
0 "-1/5" ",$*&\"\"\"F$\"\"&!\"\"F&" }{TEXT 277 14 "           c) " }
{XPPEDIT 278 0 "3/5" "*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 279 10 "        \+
  " }{TEXT 334 5 "  d) " }{XPPEDIT 280 0 "-3/5" ",$*&\"\"$\"\"\"\"\"&!
\"\"F'" }{TEXT 281 5 "     " }{TEXT 335 6 "    e)" }{TEXT 336 2 "  " }
{XPPEDIT 361 0 "4/5" "*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 362 3 "   " }
{TEXT 284 2 "  " }}{PARA 262 "" 0 "" {TEXT 261 4 "f)  " }{TEXT 258 2 "
  " }{XPPEDIT 282 0 "-4/5" ",$*&\"\"%\"\"\"\"\"&!\"\"F'" }{TEXT 283 
18 "              g)  " }{XPPEDIT 285 0 "3/10" "*&\"\"$\"\"\"\"#5!\"\"
" }{TEXT 286 17 "             h)  " }{XPPEDIT 287 0 "-3/10" ",$*&\"\"$
\"\"\"\"#5!\"\"F'" }{TEXT 288 15 "           i)  " }{XPPEDIT 289 0 "1/
sqrt(5)" "*&\"\"\"F#-%%sqrtG6#\"\"&!\"\"" }{TEXT 259 16 "            j
)  " }{XPPEDIT 291 0 "-1/sqrt(5)" ",$*&\"\"\"F$-%%sqrtG6#\"\"&!\"\"F)
" }{TEXT 292 1 " " }{TEXT 290 2 "  " }}{PARA 3 "" 0 "" {TEXT 656 12 "S
olution:  e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "x := t -> t^3: y := t -> t^4: z := t -> 1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t)
, [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%
,$*$%\"tG\"\"#\"\"$,$*$F(F*\"\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "unitTangent := map( u -> u/magnitude(velocity) , velo
city);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,unitTangentG7%,$*&%\"tG\"
\"#,&*$F(\"\"%\"\"**$F(\"\"'\"#;#!\"\"F)\"\"$,$*&F(F3F*F1F,\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%,$*$\"#D#\"\"\"\"\"##\"\"$F&,$F%#\"\"%F&\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7%#\"\"$\"\"&#\"\"%F&\"\"!" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 1 " " }{TEXT 382 2 
"5." }{TEXT -1 1 " " }{TEXT 431 33 "A space curve is parameterized by
" }{TEXT 432 3 "   " }{XPPEDIT 430 1 "r(t) = `<`*2*t^2,4*t,t^2*`>`" "6
%/-%\"rG6#%\"tG*(%\"<G\"\"\"\"\"#F*F'F+*&\"\"%F*F'F**&F'F+%\">GF*" }
{TEXT 428 1 " " }{TEXT 429 151 ".   What is the first component of its
 principal unit normal?   (You may need to use some of your computatio
ns for the question that follows this one.)" }}{PARA 261 "" 0 "" 
{TEXT -1 4 "a)  " }{XPPEDIT 19 1 "sqrt(5)/5" "*&-%%sqrtG6#\"\"&\"\"\"F
&!\"\"" }{TEXT -1 16 "            b)  " }{XPPEDIT 19 1 "2*sqrt(5)/5" "
*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT -1 14 "           c) " }
{XPPEDIT 19 1 "3*sqrt(5)/5" "*(\"\"$\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }
{TEXT -1 13 "          d) " }{XPPEDIT 19 1 "sqrt(5)" "-%%sqrtG6#\"\"&
" }{TEXT -1 17 "             e)  " }{XPPEDIT 19 1 "sqrt(5)/15" "*&-%%s
qrtG6#\"\"&\"\"\"\"#:!\"\"" }}{PARA 265 "" 0 "" {TEXT -1 5 "f)   " }
{XPPEDIT 19 1 "2*sqrt(5)/15" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"#:!\"\"
" }{TEXT -1 12 "        g)  " }{XPPEDIT 19 1 "4*sqrt(5)/15" "*(\"\"%\"
\"\"-%%sqrtG6#\"\"&F$\"#:!\"\"" }{TEXT -1 16 "            h)  " }
{XPPEDIT 19 1 "sqrt(5)/10" "*&-%%sqrtG6#\"\"&\"\"\"\"#5!\"\"" }{TEXT 
-1 16 "            i)  " }{XPPEDIT 19 1 "3*sqrt(5)/10" "*(\"\"$\"\"\"-
%%sqrtG6#\"\"&F$\"#5!\"\"" }{TEXT -1 14 "          j)  " }{XPPEDIT 19 
1 "sqrt(5)/2" "*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
657 12 "Solution:  g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "x := t -> 2*t^2: y := t -> 4*t: z := t ->
 t^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u-
>diff(u,t), [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)v
elocityG7%,$%\"tG\"\"%F(,$F'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "unitTangent := map( u -> u/magnitude(velocity) , velo
city);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,unitTangentG7%,$*&%\"tG\"
\"\",&*$F(\"\"#\"\"&\"\"%F)#!\"\"F,F,,$*$F*F/F,F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "N := map(u->diff(u,t), unitTangent);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%,&*$,&*$%\"tG\"\"#\"\"&\"\"%\"
\"\"#!\"\"F+F+*&F*F+F(#!\"$F+!#5,$*&F(F2F*F.F4,&F'F.F1!\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$*$\"\"*#\"\"\"\"\"##\"\")\"#\"),$F%#!#5F,,$F%#\"\"%
F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "N := simplify(\");" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%#\"\")\"#F#!#5F(#\"\"%F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "principalUnitNormal :=  map(
 u -> u/magnitude(N), N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4princi
palUnitNormalG7%,$*$\"\"&#\"\"\"\"\"##\"\"%\"#:,$F'#!\"\"\"\"$,$F'#F+F
." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 265 87 "6. Referring to the curve of question  5, what is the
 curvature at the point for which " }{TEXT 434 1 " " }{XPPEDIT 435 1 "
t=1" "/%\"tG\"\"\"" }{TEXT 433 1 "?" }}{PARA 0 "" 0 "" {TEXT 266 8 "\n
a)     " }{XPPEDIT 19 1 "sqrt(5)/3" "*&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"
" }{TEXT -1 5 "     " }{TEXT 383 11 "      b)   " }{XPPEDIT 19 1 "2*sq
rt(5)/3" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"\"$!\"\"" }{TEXT -1 5 "     \+
" }{TEXT 384 14 "         c)   " }{XPPEDIT 19 1 "sqrt(5)/27" "*&-%%sqr
tG6#\"\"&\"\"\"\"#F!\"\"" }{TEXT -1 5 "     " }{TEXT 385 13 "         \+
d)  " }{XPPEDIT 19 1 "2*sqrt(5)/27" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"#
F!\"\"" }{TEXT -1 5 "     " }{TEXT 386 13 "        e)   " }{XPPEDIT 
19 1 "sqrt(5)/81" "*&-%%sqrtG6#\"\"&\"\"\"\"#\")!\"\"" }{TEXT -1 5 "  \+
   " }{TEXT 387 11 " \n\n f)     " }{XPPEDIT 19 1 "2*sqrt(5)/81" "*(\"
\"#\"\"\"-%%sqrtG6#\"\"&F$\"#\")!\"\"" }{TEXT 388 15 "         g)    \+
" }{XPPEDIT 19 1 "2*sqrt(5)/5" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"
" }{TEXT 389 18 "         h)       " }{XPPEDIT 19 1 "3*sqrt(5)/5" "*(
\"\"$\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 390 14 "          i)  " }
{XPPEDIT 19 1 "6*sqrt(5)/5" "*(\"\"'\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }
{TEXT 391 16 "            j)  " }{XPPEDIT 19 1 "12*sqrt(5)/5" "*(\"#7
\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 392 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
658 12 "Solution:  c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "velocity;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7%,$%\"tG\"\"%F&,$F%\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "acceleration := map(u->diff(u,t), velocity);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%-accelerationG7%\"\"%\"\"!\"\"#" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "magnitude(crossprod(velocity,acceleration))/magnitude
(velocity)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"&#\"\"\"\"\"#,&
*$%\"tGF'F$\"\"%F&#!\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "subs(t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"&#\"\"\"
\"\"#\"\"*F&#F'\"#\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "si
mplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"##F
'\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 267 61 "7.  What is the \+
length of the curve that is parameterized by " }{TEXT 437 1 " " }
{XPPEDIT 436 1 "r(t)=`<`*t^2,2*t,ln(t)*`>` " "6%/-%\"rG6#%\"tG*&%\"<G
\"\"\"*$F'\"\"#F**&F,F*F'F**&-%#lnG6#F'F*%\">GF*" }{TEXT 438 1 " " }
{TEXT 338 1 " " }{TEXT 339 5 "for  " }{XPPEDIT 439 1 "t" "I\"tG6\"" }
{TEXT 393 1 " " }{TEXT 440 18 " in the interval  " }{TEXT 342 2 "  " }
{XPPEDIT 340 1 "[1,2]" "7$\"\"\"\"\"#" }{TEXT 341 3 "  ?" }{TEXT 337 
1 "\n" }}{PARA 0 "" 0 "" {TEXT 343 5 "a)   " }{XPPEDIT 19 1 "sqrt(2)" 
"-%%sqrtG6#\"\"#" }{TEXT 394 21 "                b)   " }{XPPEDIT 19 
1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 395 14 "          c
)  " }{XPPEDIT 19 1 "3*sqrt(2)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"#F$" }
{TEXT 396 15 "           d)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6
#\"\"#\"\"\"F&!\"\"" }{TEXT 397 15 "           e)  " }{XPPEDIT 19 1 "3
*sqrt(2)/2" "*(\"\"$\"\"\"-%%sqrtG6#\"\"#F$F(!\"\"" }{TEXT 398 8 " \n
\n f)  " }{XPPEDIT 19 1 "sqrt(2) +ln(2)" ",&-%%sqrtG6#\"\"#\"\"\"-%#ln
G6#F&F'" }{TEXT 399 9 "      g) " }{XPPEDIT 19 1 "3 + ln(2)" ",&\"\"$
\"\"\"-%#lnG6#\"\"#F$" }{TEXT 400 9 "      h) " }{XPPEDIT 19 1 "2+ln(3
)" ",&\"\"#\"\"\"-%#lnG6#\"\"$F$" }{TEXT 401 10 "      i)  " }
{XPPEDIT 19 1 "3+ln(3)" ",&\"\"$\"\"\"-%#lnG6#F#F$" }{TEXT 402 9 "    \+
  j) " }{XPPEDIT 19 1 "2*ln(3)" "*&\"\"#\"\"\"-%#lnG6#\"\"$F$" }{TEXT 
403 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
659 12 "Solution:  g" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x := t -> \+
 t^2 :  y := t ->  2*t:  z := t -> ln(t):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t), [x(t),y(t),z(t)]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,$%\"tG\"\"#F(*$F'!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "arcLength := Int(magnit
ude(velocity),t=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*arcLength
G-%$IntG6$*$,(*$%\"tG\"\"#\"\"%F-\"\"\"*$F+!\"#F.#F.F,/F+;F.F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "value(arcLength);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&\"\"$\"\"\"-%#lnG6#\"\"#F%" }}}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 1 " " }{TEXT 348 6 "8. If " }{TEXT 346 1 " " }
{XPPEDIT 344 1 "r(t)=`<`*t ,2*t,t^2*`>` " "6%/-%\"rG6#%\"tG*&%\"<G\"\"
\"F'F**&\"\"#F*F'F**&F'F,%\">GF*" }{TEXT 345 1 " " }{TEXT 347 79 "  is
 the position vector of a particle then what is its tangential compone
nt,  " }{XPPEDIT 19 1 "a[T]" "&%\"aG6#%\"TG" }{TEXT 404 23 " ,  of acc
eleration at " }{TEXT 364 1 " " }{XPPEDIT 363 1 "t=1" "/%\"tG\"\"\"" }
{TEXT 365 3 "  ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 349 172 "a)  0             b)  1/6              \+
c) 1/3          d)  2/3             e)   1\n\nf)   4/3          g)  5/
3              h) 2             i)  7/3               j)  8/3    " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 660 12 "Solution:  f" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "x := t -> t:    y := t -> \+
2*t:    z := t -> t^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "v
elocity := map(u -> diff(u,t), [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)velocityG7%\"\"\"\"\"#,$%\"tGF'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "speed := magnitude(velocity);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&speedG*$,&\"\"&\"\"\"*$%\"tG\"\"#\"\"%#F(F+" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[T] = diff(speed, t);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG,$*&,&\"\"&\"\"\"*$%\"t
G\"\"#\"\"%#!\"\"F/F.F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "subs(t=1, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG,$
*$\"\"*#\"\"\"\"\"##\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG#
\"\"%\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 270 55 "9. At an instant of time a particle's acceleration is  \+
" }{XPPEDIT 406 1 "`<`*3,2,6*`>`" "6%*&%\"<G\"\"\"\"\"$F%\"\"#*&\"\"'F
%%\">GF%" }{TEXT 407 37 "  and its principal unit normal is\n  " }
{TEXT 445 1 " " }{XPPEDIT 444 1 "N=2/3*i-2/3*j+k/3" "/%\"NG,(*(\"\"#\"
\"\"\"\"$!\"\"%\"iGF'F'*(F&F'F(F)%\"jGF'F)*&%\"kGF'F(F)F'" }{TEXT 446 
50 ".    What is its normal component of acceleration " }{TEXT 442 1 "
 " }{XPPEDIT 443 1 "a[N]" "&%\"aG6#%\"NG" }{TEXT 441 3 "?\n " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 447 169 "a)  0        \+
     b)  1/3            c) 2/3            d) 1                e)  4/3
\n\nf)   5/3          g)  2              h) 7/3            i)  8/3    \+
          j)  3    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 661 12 "Solution:  i" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := [3,2,6]:  N := [2/3,-2/3,1/3]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(a,N);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"\")\"\"$" }}}{PARA 258 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 276 "" 0 "" 
{TEXT -1 4 "10. " }{TEXT 671 75 "A particle is moving along a space cu
rve in such a way that its speed  is  " }{XPPEDIT 672 1 "1+ t^2" ",&\"
\"\"F#*$%\"tG\"\"#F#" }{TEXT 673 142 "  at time  t.  The curvature of \+
the particle's path is   3   at time  t = 1.  What is the magnitude of
 its acceleration vector at that time?  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 674 5 "a)
   " }{XPPEDIT 19 1 "3*sqrt(3)" "*&\"\"$\"\"\"-%%sqrtG6#F#F$" }{TEXT 
676 18 "             b)   " }{XPPEDIT 19 1 "5*sqrt(3)" "*&\"\"&\"\"\"-
%%sqrtG6#\"\"$F$" }{TEXT 677 16 "            c)  " }{XPPEDIT 19 1 "10*
sqrt(3)" "*&\"#5\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 678 20 "              \+
  d)  " }{XPPEDIT 19 1 "2*sqrt(37)" "*&\"\"#\"\"\"-%%sqrtG6#\"#PF$" }
{TEXT 675 20 "                e)  " }{XPPEDIT 19 1 "3*sqrt(37)" "*&\"
\"$\"\"\"-%%sqrtG6#\"#PF$" }{TEXT 679 12 "   \n\nf)     " }{XPPEDIT 
19 1 "3*sqrt(10)" "*&\"\"$\"\"\"-%%sqrtG6#\"#5F$" }{TEXT 680 17 "     \+
       g)   " }{XPPEDIT 19 1 "4*sqrt(10)" "*&\"\"%\"\"\"-%%sqrtG6#\"#5
F$" }{TEXT 681 20 "              h)    " }{XPPEDIT 19 1 "3*sqrt(15)" "
*&\"\"$\"\"\"-%%sqrtG6#\"#:F$" }{TEXT 682 15 "          i)   " }
{XPPEDIT 19 1 "4*sqrt(15)" "*&\"\"%\"\"\"-%%sqrtG6#\"#:F$" }{TEXT 683 
19 "              j)   " }{XPPEDIT 19 1 "5*sqrt(15)" "*&\"\"&\"\"\"-%%
sqrtG6#\"#:F$" }{TEXT 684 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 685 12 "Solution:  d" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "v := 1+t^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"v
G,&\"\"\"F&*$%\"tG\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "a[T]  := diff(v,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%
\"TG,$%\"tG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[T] :=
 subs(t = 1, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\"TG\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a[N] := kappa*v^2;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\"NG*&%&kappaG\"\"\",&F*F*
*$%\"tG\"\"#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a[N] :=
 subs(\{t=1,kappa=3\}, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"a
G6#%\"NG\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sqrt(a[T]^2
 + a[N]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"#P#\"\"\"\"\"#F(
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 271 71 "11.  What surfac
e is parameterized  by the vector-valued function\n\n    " }{TEXT 457 
9 "         " }{XPPEDIT 458 1 "r(u,v)=u*i+u*cos(v)*j+u*sin(v)*k" "/-%
\"rG6$%\"uG%\"vG,(*&F&\"\"\"%\"iGF*F**(F&F*-%$cosG6#F'F*%\"jGF*F**(F&F
*-%$sinG6#F'F*%\"kGF*F*" }{TEXT 448 29 "  \n\nof the two real variable
s" }{TEXT 452 2 "  " }{XPPEDIT 453 1 "u>0" "2\"\"!%\"uG" }{TEXT 449 2 
"  " }{TEXT 455 4 " and" }{TEXT 456 1 " " }{XPPEDIT 454 1 "v>=0,` `*v<
2*Pi" "6$1\"\"!%\"vG2*&%\"~G\"\"\"F%F)*&\"\"#F)%#PiGF)" }{TEXT 450 1 "
 " }{TEXT 451 2 "?\n" }}{PARA 3 "" 0 "" {TEXT 351 133 "a)  cylinder   \+
                                        b)  cone     \nc) two cones wi
th common vertex      d) sphere      \ne) plane   " }{TEXT 352 17 "   \+
              " }{TEXT 461 2 "  " }{TEXT 459 4 "    " }{TEXT 463 168 "
                 f)   ellipsoid   \ng)  elliptic paraboloid           \+
               h) hyperbolic paraboloid             \ni)  hyperboloid \+
of one sheet                j)" }{TEXT 462 2 "  " }{TEXT 464 25 "hyper
boloid of two sheets" }{TEXT 460 2 "  " }{TEXT -1 3 "   " }}{PARA 3 "
" 0 "" {TEXT -1 15 "\nSolution:  b \n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "x := u:   y := u*cos(v):  z := u*sin(v):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "testeq( x^2 = y^2 + z^2 );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "x := 'x':  y := 'y':  z := 'z':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "implicitplot3d( x^2 = y^2 + z^2 , x = 0..2, y=-2..2, \+
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 " }{TEXT 353 9 "12.   Let" }{TEXT 466 2 "  " }{XPPEDIT 467 1 "f(x,y)=
x^2*y/(x^4+y^2)" "/-%\"fG6$%\"xG%\"yG*(F&\"\"#F'\"\"\",&*$F&\"\"%F**$F
'F)F*!\"\"" }{TEXT 465 49 "  for all points other than the origin.    \+
Let   " }{TEXT 469 1 " " }{XPPEDIT 470 1 "r(t)=`<`*t,2*t^2*`>`" "6$/-%
\"rG6#%\"tG*&%\"<G\"\"\"F'F**(\"\"#F**$F'F,F*%\">GF*" }{TEXT 468 2 ". \+
" }{TEXT 354 3 " \n\n" }{TEXT 472 13 "Calculate    " }{XPPEDIT 473 1 "
limit(f(r(t)),t=0)" "-%&limitG6$-%\"fG6#-%\"rG6#%\"tG/F+\"\"!" }{TEXT 
471 2 ".\n" }{TEXT 474 1 "\n" }}{PARA 0 "" 0 "" {TEXT 479 4 "a)  " }
{XPPEDIT 475 0 "1/2" "*&\"\"\"F#\"\"#!\"\"" }{TEXT 476 8 "        " }
{TEXT 488 3 "b) " }{TEXT 489 1 " " }{XPPEDIT 480 0 "1/3" "*&\"\"\"F#\"
\"$!\"\"" }{TEXT 481 8 "        " }{TEXT 499 1 "c" }{TEXT 500 2 ") " }
{XPPEDIT 482 0 "2/3" "*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 483 7 "       " 
}{TEXT 490 7 "    d) " }{XPPEDIT 484 0 "1" "\"\"\"" }{TEXT 485 5 "    \+
 " }{TEXT 491 8 "      e)" }{TEXT 492 2 "  " }{TEXT 486 2 "0 " }}
{PARA 270 "" 0 "" {TEXT 478 4 "f)  " }{XPPEDIT 493 0 "4/5" "*&\"\"%\"
\"\"\"\"&!\"\"" }{TEXT 494 16 "           g)   " }{XPPEDIT 495 0 "3/5
" "*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 496 17 "           h)    " }
{XPPEDIT 497 0 "2/5" "*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 498 15 "        \+
   i)  " }{XPPEDIT 487 0 "1/5" "*&\"\"\"F#\"\"&!\"\"" }{TEXT 477 31 " \+
          j)  Does not exist  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 275 "" 0 "" {TEXT -1 14 "Solution:  h \n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := (x,y) -
> x^2*y/(x^4+y^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r := \+
t -> [t,2*t^2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "limit(f(
r(t)[1],r(t)[2]),t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"&
" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 266 "" 0 "" 
{TEXT -1 10 "13. If    " }{XPPEDIT 19 1 "f(x,y)=x-2*y+ 3*y^3/x^2" "/-%
\"fG6$%\"xG%\"yG,(F&\"\"\"*&\"\"#F)F'F)!\"\"*(\"\"$F)*$F'F.F)*$F&F+F,F
)" }{TEXT -1 18 "   then calculate " }{XPPEDIT 19 1 "Diff(f(x,y),y)" "
-%%DiffG6$-%\"fG6$%\"xG%\"yGF)" }{TEXT -1 7 "  at   " }{XPPEDIT 19 1 "
``(3,2)" "-%!G6$\"\"$\"\"#" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT 
501 5 "\na)  " }{XPPEDIT 502 1 "0" "\"\"!" }{TEXT 503 9 "     b)  " }
{XPPEDIT 504 1 "1" "\"\"\"" }{TEXT 505 9 "     c)  " }{XPPEDIT 506 1 "
2" "\"\"#" }{TEXT 507 7 "    d) " }{XPPEDIT 508 1 "3" "\"\"$" }{TEXT 
509 9 "     e)  " }{XPPEDIT 510 1 "4" "\"\"%" }{TEXT 511 9 "      f) \+
" }{XPPEDIT 512 1 "6" "\"\"'" }{TEXT 513 10 "      g)  " }{XPPEDIT 
514 1 "9" "\"\"*" }{TEXT 515 9 "     h)  " }{XPPEDIT 516 1 "12" "\"#7
" }{TEXT 517 8 "     i) " }{XPPEDIT 518 1 "15" "\"#:" }{TEXT 519 9 "  \+
   j)  " }{XPPEDIT 521 1 "18" "\"#=" }{TEXT -1 1 " " }{TEXT 520 3 "   \+
" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 355 3 "   " }{TEXT 356 1 "\n
" }{TEXT -1 14 "Solution:  c \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diff(x-2
*y+3*y^3/x^2,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"#\"\"\"*&%\"y
G\"\"#%\"xGF$\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(
\{x=3,y=2\}, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT 359 13 "14.    If    " }{XPPEDIT 523 1 "f(x,y)=x + 2*y+ \+
 x^3*y^2" "/-%\"fG6$%\"xG%\"yG,(F&\"\"\"*&\"\"#F)F'F)F)*&F&\"\"$F'F+F)
" }{TEXT 524 11 "     then  " }{XPPEDIT 525 1 "Diff(f(x,y),x,x)*Diff(f
(x,y),y,y)-(Diff(f(x,y),x,y))^2=k*x^p*y^q" "/,&*&-%%DiffG6%-%\"fG6$%\"
xG%\"yGF+F+\"\"\"-F&6%-F)6$F+F,F,F,F-F-*$-F&6%-F)6$F+F,F+F,\"\"#!\"\"*
(%\"kGF-)F+%\"pGF-)F,%\"qGF-" }{TEXT 526 13 " .  \nWhat is " }{TEXT 
530 1 " " }{TEXT 529 1 " " }{XPPEDIT 528 1 "q-k/p" ",&%\"qG\"\"\"*&%\"
kGF$%\"pG!\"\"F(" }{TEXT 527 3 "  ?" }{TEXT 522 1 "\n" }{TEXT 358 139 
"\na) 0             b) 1          c)  2           d)  3         e)  4 \+
  \nf)  5            g)  6          h) 7            i) 8           j) \+
 9" }{TEXT 357 1 "\n" }}{PARA 0 "" 0 "" {TEXT 350 13 "Solution:  i " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "f := (x,y) -> x+2*y+x^3*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "diff(f(x,y),x,x)*diff(f(x,y),y,y)-(diff(f(x,y),x,y))^
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"%%\"yG\"\"#!#C" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "k := -24:  p := 4:  q := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "q-k/p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 267 "" 0 "" {TEXT -1 11 "15.   If \+
  " }{XPPEDIT 19 1 "x^2 +y^3+z^4=6+2*x*y*z" "/,(*$%\"xG\"\"#\"\"\"*$%
\"yG\"\"$F'*$%\"zG\"\"%F',&\"\"'F'**F&F'F%F'F)F'F,F'F'" }{TEXT -1 20 "
   then calculate   " }{XPPEDIT 19 1 "diff( z ,x)" "-%%diffG6$%\"zG%\"
xG" }{TEXT -1 10 "   at     " }{XPPEDIT 19 1 "``(3,2,1)" "-%!G6%\"\"$
\"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT -1 177 "a) 1/6             b) 1/5            c)  1/4
           d) 1/3            e) 1/2               \nf)  2/3           \+
 g) 3/4             h) 3/5            i) 4/5             j) 5/6 " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 662 13 "Solution
:  c " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "eqn1 := x^2+y^3+z(x,y)^4=6+2*x*y*z(x,y);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%eqn1G/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"$F**$-%
\"zG6$F(F,\"\"%F*,&\"\"'F**(F(F*F,F*F/F*F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "eqn2 := map(u -> diff(u,x),eqn1);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG\"\"#*&-%\"zG6$F'%\"yG\"\"$-%%diffG6
$F*F'\"\"\"\"\"%,&*&F-F2F*F2F(*(F'F2F-F2F/F2F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "eqn3 := diff(z(x,y),x) = solve(eqn2, diff(z(x,
y),x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%diffG6$-%\"zG6
$%\"xG%\"yGF,*&,&F,\"\"\"*&F-F0F)F0!\"\"F0,&*$F)\"\"$!\"#*&F,F0F-F0F0F
2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{x=3,y=2\}, rhs(
eqn3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"$\"\"\"-%\"zG6$F%\"
\"#!\"#F&,&*$F'F%F+\"\"'F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "subs(z(3,2)=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"
\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 273 15 "16. The pla
ne  " }{XPPEDIT 533 1 "x=2" "/%\"xG\"\"#" }{TEXT 532 2 "  " }{TEXT 
534 24 " intersects the graph of" }{TEXT 535 2 "  " }{XPPEDIT 536 1 "z
=5-x^2+x*y^3" "/%\"zG,(\"\"&\"\"\"*$%\"xG\"\"#!\"\"*&F(F&*$%\"yG\"\"$F
&F&" }{TEXT 531 2 "  " }{TEXT 537 50 " in a curve.    The tangent line
 to this curve at " }{XPPEDIT 539 1 "``(2,1,3)" "-%!G6%\"\"#\"\"\"\"\"
$" }{TEXT 538 1 " " }{TEXT 540 27 " passes through the point  " }
{XPPEDIT 541 1 "``(2,2,c)" "-%!G6%\"\"#F%%\"cG" }{TEXT 542 12 ".   Wha
t is " }{TEXT 544 1 " " }{XPPEDIT 545 1 "c" "I\"cG6\"" }{TEXT 543 5 "?
    " }}{PARA 258 "" 0 "" {TEXT 546 5 "\na)  " }{XPPEDIT 547 1 "0" "\"
\"!" }{TEXT 548 10 "      b)  " }{XPPEDIT 549 1 "1" "\"\"\"" }{TEXT 
550 10 "      c)  " }{XPPEDIT 551 1 "2" "\"\"#" }{TEXT 552 8 "     d) \+
" }{XPPEDIT 553 1 "3" "\"\"$" }{TEXT 554 10 "      e)  " }{XPPEDIT 
555 1 "4" "\"\"%" }{TEXT 556 11 "        f) " }{XPPEDIT 557 1 "5" "\"
\"&" }{TEXT 558 11 "       g)  " }{XPPEDIT 559 1 "6" "\"\"'" }{TEXT 
560 9 "     h)  " }{XPPEDIT 561 1 "7" "\"\"(" }{TEXT 562 9 "      i) \+
" }{XPPEDIT 563 1 "8" "\"\")" }{TEXT 564 10 "      j)  " }{XPPEDIT 
566 1 "9" "\"\"*" }{TEXT -1 1 " " }{TEXT 565 3 "   " }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 665 13 "Solutio
n:  j " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "r := t -> [2,1+t,3+t*subs(\{x=2,y=1\}, diff(5-x^2+x*y
^3,y))]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"tG6\"6$%)oper
atorG%&arrowGF(7%\"\"#,&9$\"\"\"F0F0,&\"\"$F0*&F/F0-%%subsG6$<$/%\"xGF
-/%\"yGF0-%%diffG6$,(\"\"&F0*$F9F-!\"\"*&F9F0F;F2F0F;F0F0F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%\"\"#,&%\"tG\"\"\"F'F',&\"\"$F'F&\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%\"\"#F$\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 271 "" 0 "" 
{TEXT -1 46 "17. The plane that is tangent to the graph of " }{TEXT 
568 2 "  " }{XPPEDIT 569 1 "z=5-x^2+x*y^3" "/%\"zG,(\"\"&\"\"\"*$%\"xG
\"\"#!\"\"*&F(F&*$%\"yG\"\"$F&F&" }{TEXT 567 1 " " }{TEXT -1 5 " at  \+
" }{XPPEDIT 571 1 "`(`*2,1,3*`)`" "6%*&%\"(G\"\"\"\"\"#F%F%*&\"\"$F%%
\")GF%" }{TEXT 570 1 " " }{TEXT -1 15 "  has  what    " }{XPPEDIT 19 
1 "z" "I\"zG6\"" }{TEXT -1 11 " intercept?" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 272 "" 0 "" {TEXT 572 4 "a)  " }{XPPEDIT 573 1 "0" "\"
\"!" }{TEXT 574 10 "      b)  " }{XPPEDIT 575 1 "1" "\"\"\"" }{TEXT 
576 10 "      c)  " }{XPPEDIT 577 1 "2" "\"\"#" }{TEXT 578 8 "     d) \+
" }{XPPEDIT 579 1 "3" "\"\"$" }{TEXT 580 10 "      e)  " }{XPPEDIT 
581 1 "4" "\"\"%" }{TEXT 582 11 "        f) " }{XPPEDIT 583 1 "5" "\"
\"&" }{TEXT 584 11 "       g)  " }{XPPEDIT 585 1 "6" "\"\"'" }{TEXT 
586 9 "     h)  " }{XPPEDIT 587 1 "7" "\"\"(" }{TEXT 588 9 "      i) \+
" }{XPPEDIT 589 1 "8" "\"\")" }{TEXT 590 10 "      j)  " }{XPPEDIT 
592 1 "9" "\"\"*" }{TEXT -1 1 " " }{TEXT 591 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 666 12 "Solution:  d" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := subs(\{x=2,y=1
\},diff(5-x^2+x*y^3,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B := subs(\{x=2,y=1\},d
iff(5-x^2+x*y^3,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\"\"'" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn := A*(x-2) + B*(y-1) -
 (z-3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"$\"\"$
\"\"\"%\"yG\"\"'%\"zG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "solve( subs(\{x=0,y=0\},eqn),z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT 593 8 "18.  Let" }{TEXT 599 2 "  " }{XPPEDIT 600 1 "f(x,y)=12
*x/sqrt(y)" "/-%\"fG6$%\"xG%\"yG*(\"#7\"\"\"F&F*-%%sqrtG6#F'!\"\"" }
{TEXT 594 7 ".   Let" }{TEXT 601 2 "  " }{XPPEDIT 602 1 "L(x,y)" "-%\"
LG6$%\"xG%\"yG" }{TEXT 595 33 "   be the linear approximation of" }
{TEXT 603 1 " " }{XPPEDIT 604 1 "f(x,y)" "-%\"fG6$%\"xG%\"yG" }{TEXT 
596 3 " at" }{TEXT 605 2 "  " }{XPPEDIT 606 1 "``(1,4)" "-%!G6$\"\"\"
\"\"%" }{TEXT 597 30 ". What is the approximation of" }{TEXT 667 2 "  \+
" }{XPPEDIT 668 1 "f(9/10,42/10)" "-%\"fG6$*&\"\"*\"\"\"\"#5!\"\"*&\"#
UF'F(F)" }{TEXT 669 21 "  that results when  " }{TEXT 607 1 " " }
{XPPEDIT 608 1 "L(x,y)" "-%\"LG6$%\"xG%\"yG" }{TEXT 598 11 "  is used \+
?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 273 "" 0 "" {TEXT -1 192 "a) 25/4             b) 101/16         \+
    c)  49/8          d) 99/16           e) 85/16              \nf)  5
1/8             g) 81/16               h) 23/4           i)  51/8     \+
        j) 21/4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 670 12 "Solution:  j" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f :=
 (x,y) -> 12*x/y^(1/2);  a := 1:  b := 4:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\"
\"9%#!\"\"\"\"#\"#7F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 
"L := (x,y) -> f(a,b) + subs(\{u=a,v=b\}, diff(f(u,v),u))*(x-a) + subs
(\{u=a,v=b\}, diff(f(u,v),v))*(y-b); " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"LG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(-%\"fG6$%\"aG%\"bG\"
\"\"*&-%%subsG6$<$/%\"uGF1/%\"vGF2-%%diffG6$-F/6$F:F<F:F3,&9$F3F1!\"\"
F3F3*&-F66$F8-F>6$F@F<F3,&9%F3F2FDF3F3F)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "L(9/10,42/10); simplify(\"); evalf(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"%#\"\"\"\"\"##\"#@\"\")" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6##\"#@\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"++++]_!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f(9/10,42/1
0); simplify(\"); evalf(\");  # actual" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&\"#@#\"\"\"\"\"#\"\"&F&#\"#=\"#N" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&\"#@#\"\"\"\"\"#\"\"&F&#\"#=\"#N" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+'Rg)p_!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 630 9 "19. Let  " }{XPPEDIT 631 1 "z=(y^
4)/(2*x-y)" "/%\"zG*&%\"yG\"\"%,&*&\"\"#\"\"\"%\"xGF*F*F%!\"\"F," }
{TEXT 632 3 ",  " }{XPPEDIT 633 1 "x=2*t+1/t" "/%\"xG,&*&\"\"#\"\"\"%
\"tGF'F'*&F'F'F(!\"\"F'" }{TEXT 634 7 ",  and " }{XPPEDIT 635 1 "y=3*t
-1/t" "/%\"yG,&*&\"\"$\"\"\"%\"tGF'F'*&F'F'F(!\"\"F*" }{TEXT -1 2 ". \+
" }{TEXT 686 1 " " }{TEXT 687 20 "Find  dz/dt  at t=1." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 609 4 "a)  " }{XPPEDIT 610 1 "2" "\"\"#" }{TEXT 611 9 "     b)  \+
" }{XPPEDIT 612 1 "6" "\"\"'" }{TEXT 613 9 "     c)  " }{XPPEDIT 614 
1 "12" "\"#7" }{TEXT 615 7 "    d) " }{XPPEDIT 616 1 "18" "\"#=" }
{TEXT 617 9 "     e)  " }{XPPEDIT 618 1 "22" "\"#A" }{TEXT 619 8 "    \+
 f) " }{XPPEDIT 620 1 "24" "\"#C" }{TEXT 621 9 "     g)  " }{XPPEDIT 
622 1 "28" "\"#G" }{TEXT 623 9 "     h)  " }{XPPEDIT 624 1 "34" "\"#M
" }{TEXT 625 9 "     i)  " }{XPPEDIT 628 1 "40" "\"#S" }{TEXT 629 9 " \+
    j)  " }{XPPEDIT 627 1 "42" "\"#U" }{TEXT -1 1 " " }{TEXT 626 3 "  \+
 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 663 12 "Solution:  h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 6 "Direct" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "z := y^4/(2*x-y);  subs(\{x = 2*t+1
/t, y=3*t-1/t \}, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&%\"yG
\"\"%,&%\"xG\"\"#F&!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"t
G\"\"$*$F%!\"\"F(\"\"%,&F%\"\"\"F'F&F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "diff(\",t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&
%\"tG\"\"$*$F&!\"\"F)F',&F&\"\"\"F(F'F),&F'F+*$F&!\"#F+F+\"\"%*(F%F/F*
F.,&F+F+F-!\"$F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(
t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Chain Rule" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "diff(y^4/(2*
x-y),x)*diff(2*t+1/t,t)+diff(y^4/(2*x-y),y)*diff(3*t-1/t,t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*(%\"yG\"\"%,&%\"xG\"\"#F%!\"\"!\"#,&F)\"
\"\"*$%\"tGF+F*F-F+*&,&*&F%\"\"$F'F*F&*&F%F&F'F+F-F-,&F3F-F.F-F-F-" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{x=3,y=2,t=1\}, \");
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 636 10 "20.  Let  " }{XPPEDIT 
637 1 "z=x^3-y^4 " "/%\"zG,&*$%\"xG\"\"$\"\"\"*$%\"yG\"\"%!\"\"" }
{TEXT 638 3 ",  " }{XPPEDIT 639 1 "x=2*t+ s^2" "/%\"xG,&*&\"\"#\"\"\"%
\"tGF'F'*$%\"sGF&F'" }{TEXT 640 7 ",  and " }{XPPEDIT 641 1 "y=3*t-s^3
" "/%\"yG,&*&\"\"$\"\"\"%\"tGF'F'*$%\"sGF&!\"\"" }{TEXT 642 13 ".   Ca
lculate" }{TEXT 646 1 " " }{XPPEDIT 647 1 "diff(z,s)" "-%%diffG6$%\"zG
%\"sG" }{TEXT 643 1 " " }{TEXT 648 8 " when   " }{XPPEDIT 649 1 "t=1" 
"/%\"tG\"\"\"" }{TEXT 644 8 "   and  " }{XPPEDIT 650 1 "s=2" "/%\"sG\"
\"#" }{TEXT 645 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 196 "a) -5568           \+
  b) -4326            c)  -3216          d) -2812            e) -1192 \+
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         i) 4424             j)  5284 " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 664 12 "Solu
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6 "Direct" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "subs(\{x=2*t+s^2,y=3*t-s^3\}, x^3-y^4);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*$,&%\"tG\"\"#*$%\"sGF'\"\"\"\"\"$F**$,&F&F+*$
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0 "" {MPLTEXT 1 0 19 "subs(\{t=1,s=2\}, \");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!%ob" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Chain Rule" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := \+
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*$%\"yG\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A := d
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" {XPPMATH 20 "6#>%\"AG,&*&%\"xG\"\"#%\"sG\"\"\"\"\"'*&%\"yG\"\"$F)F(
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A := subs(\",A);" }}{PARA 
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"> " 0 "" {MPLTEXT 1 0 15 "A := subs(\",A);" }}{PARA 11 "" 1 "" 
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" 0 "" {MPLTEXT 1 0 17 "A := subs(s=2,A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG!%ob" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "diff( (2*t+s^2)^3 - (3*t-s^3)^4, s);" }}{PARA 11 "" 1 "" {XPPMATH 
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13 }{VIEWOPTS 1 1 0 1 1 1803 }