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"" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 3 261 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 
}0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 364 26 " Math 2331 Practice Exam
 3" }}{PARA 0 "" 0 "" {TEXT 259 2 "  " }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 263 1 "1" }{TEXT 307 7 ". If   " }{XPPEDIT 305 1 "u=<3/5,-4/5>" 
"/%\"uG-%-anglebracketG6$*&\"\"$\"\"\"\"\"&!\"\",$*&\"\"%F)F*F+F+" }
{TEXT 280 2 ", " }{XPPEDIT 306 1 "v=<4/5,-3/5>" "/%\"vG-%-anglebracket
G6$*&\"\"%\"\"\"\"\"&!\"\",$*&\"\"$F)F*F+F+" }{TEXT 281 6 ",     " }
{XPPEDIT 303 1 "D[u](f)(P)=-13" "/--&%\"DG6#%\"uG6#%\"fG6#%\"PG,$\"#8!
\"\"" }{TEXT 304 7 ", and  " }{XPPEDIT 283 1 "D[v](f)(P)= -8" "/--&%\"
DG6#%\"vG6#%\"fG6#%\"PG,$\"\")!\"\"" }{TEXT 282 15 "  then what is " }
{XPPEDIT 367 1 "diff(f(x,y),x)" "-%%diffG6$-%\"fG6$%\"xG%\"yGF(" }
{TEXT 365 6 "   at " }{XPPEDIT 368 1 "P" "I\"PG6\"" }{TEXT 366 2 " ?" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 372 74 "a)  -4
           b)  -3             c)   -2         d)  -1           e)  0" 
}{TEXT 369 1 " " }{TEXT 374 20 "                    " }}{PARA 0 "" 0 "
" {TEXT 370 36 "f)   1            g)   2            " }{TEXT -1 1 " " 
}{TEXT 371 39 "h)   3           i)  4             j)  " }{TEXT -1 1 " \+
" }{TEXT 373 1 "5" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 274 
4 "2.  " }{TEXT -1 6 "If    " }{XPPEDIT 19 1 "v= <2/9,b,-4/9>" "/%\"vG
-%-anglebracketG6%*&\"\"#\"\"\"\"\"*!\"\"%\"bG,$*&\"\"%F)F*F+F+" }
{TEXT -1 43 "  is a unit vector and if the gradient of  " }{XPPEDIT 
19 1 "F" "I\"FG6\"" }{TEXT -1 6 "  at  " }{XPPEDIT 19 1 "P" "I\"PG6\"
" }{TEXT -1 7 "  is   " }{XPPEDIT 19 1 "<13,0,2>" "-%-anglebracketG6%
\"#8\"\"!\"\"#" }{TEXT -1 15 " then what is  " }{XPPEDIT 19 1 "D[v](F)
(P)" "--&%\"DG6#%\"vG6#%\"FG6#%\"PG" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 284 4 "a)  " }{XPPEDIT 285 
1 "0" "\"\"!" }{TEXT 286 9 "     b)  " }{XPPEDIT 287 1 "1" "\"\"\"" }
{TEXT 288 10 "      c)  " }{XPPEDIT 289 1 "-1" ",$\"\"\"!\"\"" }{TEXT 
290 7 "    d) " }{XPPEDIT 291 1 "2" "\"\"#" }{TEXT 292 10 "      e)  \+
" }{XPPEDIT 293 1 "-2" ",$\"\"#!\"\"" }{TEXT 294 11 "        f) " }
{XPPEDIT 295 1 "3" "\"\"$" }{TEXT 296 11 "       g)  " }{XPPEDIT 297 
1 "-3" ",$\"\"$!\"\"" }{TEXT 298 8 "    h)  " }{XPPEDIT 299 1 "4" "\"
\"%" }{TEXT 300 8 "     i) " }{XPPEDIT 301 1 "-4" ",$\"\"%!\"\"" }
{TEXT 302 76 "                          \nj) Insufficient information \+
to determine answer  " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
261 9 "3. Let   " }{TEXT 309 2 "  " }{XPPEDIT 312 1 "F(x,y)=x^2+2*y^3
" "/-%\"FG6$%\"xG%\"yG,&*$F&\"\"#\"\"\"*&F*F+*$F'\"\"$F+F+" }{TEXT 
310 43 ".  What is the greatest rate of change of  " }{XPPEDIT 19 1 "F
" "I\"FG6\"" }{TEXT 375 18 " (per unit change " }{XPPEDIT 19 1 "x" "I
\"xG6\"" }{TEXT 376 8 " of and " }{XPPEDIT 19 1 "y" "I\"yG6\"" }{TEXT 
377 15 " ) at the point" }{TEXT 311 2 "  " }{XPPEDIT 313 1 "`(`*1,1*`)
`" "6$*&%\"(G\"\"\"F%F%*&F%F%%\")GF%" }{TEXT 308 1 " " }{TEXT 314 2 "?
 " }{TEXT 315 2 "  " }}{PARA 3 "" 0 "" {TEXT 273 3 "a) " }{TEXT 388 1 
" " }{XPPEDIT 389 1 "sqrt(5)/2" "*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }
{TEXT 387 13 "          b) " }{XPPEDIT 390 1 "sqrt(5)" "-%%sqrtG6#\"\"
&" }{TEXT 386 13 "         c)  " }{XPPEDIT 391 1 "2*sqrt(5)" "*&\"\"#
\"\"\"-%%sqrtG6#\"\"&F$" }{TEXT 385 14 "           d) " }{XPPEDIT 392 
1 "3*sqrt(5)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"&F$" }{TEXT 384 11 "       e
)  " }{XPPEDIT 393 1 "4*sqrt(5)" "*&\"\"%\"\"\"-%%sqrtG6#\"\"&F$" }
{TEXT 383 14 "       \nf)    " }{XPPEDIT 394 1 "sqrt(10)/2" "*&-%%sqrt
G6#\"#5\"\"\"\"\"#!\"\"" }{TEXT 382 11 "      g)   " }{XPPEDIT 395 1 "
sqrt(10)" "-%%sqrtG6#\"#5" }{TEXT 381 11 "       h)  " }{XPPEDIT 396 
1 "2*sqrt(10)" "*&\"\"#\"\"\"-%%sqrtG6#\"#5F$" }{TEXT 378 11 "      i)
   " }{XPPEDIT 397 1 "3*sqrt(10)" "*&\"\"$\"\"\"-%%sqrtG6#\"#5F$" }
{TEXT 379 10 "     j)   " }{XPPEDIT 398 1 "4*sqrt(10)" "*&\"\"%\"\"\"-
%%sqrtG6#\"#5F$" }{TEXT 380 1 " " }{TEXT 316 7 "      \n" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 1 " " }
{TEXT 319 18 "4. The function   " }{XPPEDIT 318 1 "f(x,y)=x*y-x^2-y^2-
2*x-2*y+4" "/-%\"fG6$%\"xG%\"yG,.*&F&\"\"\"F'F*F**$F&\"\"#!\"\"*$F'F,F
-*&F,F*F&F*F-*&F,F*F'F*F-\"\"%F*" }{TEXT 317 163 "  has one critical p
oint P = (a,b). Choose the ordered list [a,what] where a is the abscis
sa of the critical point P = (a,b) and \"what\" describes the behavior
 of  " }{XPPEDIT 400 1 "f" "I\"fG6\"" }{TEXT 399 10 "   at P.\n " }}
{PARA 0 "" 0 "" {TEXT 266 2 "a)" }{TEXT 256 24 " [0,local minimum]    \+
  " }{TEXT 320 2 "b)" }{TEXT 276 56 " [-1,local minimum]        c) [-2
,local minimum]        " }{TEXT 321 7 "  \nd)  " }{TEXT 401 18 "[0,loc
al maximum] " }{TEXT 402 8 "     e) " }{TEXT 277 22 "[-1,local maximum
]    " }{TEXT 260 4 "    " }{TEXT 405 3 " f)" }{TEXT 406 1 " " }{TEXT 
403 18 "[-2,local maximum]" }{TEXT 404 1 " " }{TEXT 257 12 "       \ng
)  " }{TEXT 407 16 "[0,saddle point]" }{TEXT 408 16 "             h) \+
" }{TEXT 409 17 "[-1,saddle point]" }{TEXT 410 19 "                i) \+
" }{TEXT 413 17 "[-2,saddle point]" }{TEXT 258 6 "    \n " }{TEXT 414 
3 "j) " }{TEXT 415 1 " " }{TEXT 411 17 "[1,local minimum]" }{TEXT 412 
1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 267 2 " 5" }{TEXT -1 2 ". " }{TEXT 429 15 "The function   " }
{XPPEDIT 428 1 "f(x,y)=x^5*y+x*y^5+x*y" "/-%\"fG6$%\"xG%\"yG,(*&F&\"\"
&F'\"\"\"F+*&F&F+*$F'F*F+F+*&F&F+F'F+F+" }{TEXT 427 163 "  has one cri
tical point P = (a,b). Choose the ordered list [b,what] where b is the
 ordinate of the critical point P = (a,b) and \"what\" describes the b
ehavior of  " }{XPPEDIT 431 1 "f" "I\"fG6\"" }{TEXT 430 9 "   at P. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 436 2 "a)" }
{TEXT 432 24 " [0,local minimum]      " }{TEXT 439 2 "b)" }{TEXT 437 
56 " [-1,local minimum]        c) [-2,local minimum]        " }{TEXT 
440 7 "  \nd)  " }{TEXT 441 18 "[0,local maximum] " }{TEXT 442 8 "    \+
 e) " }{TEXT 438 22 "[-1,local maximum]    " }{TEXT 435 4 "    " }
{TEXT 445 3 " f)" }{TEXT 446 1 " " }{TEXT 443 18 "[-2,local maximum]" 
}{TEXT 444 1 " " }{TEXT 433 12 "       \ng)  " }{TEXT 447 16 "[0,saddl
e point]" }{TEXT 448 16 "             h) " }{TEXT 449 17 "[-1,saddle p
oint]" }{TEXT 450 19 "                i) " }{TEXT 453 17 "[-2,saddle p
oint]" }{TEXT 434 6 "    \n " }{TEXT 454 3 "j) " }{TEXT 455 1 " " }
{TEXT 451 17 "[1,local maximum]" }{TEXT 452 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 18 "6. Th
e function   " }{XPPEDIT 457 1 "f(x,y)=x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y
^4+y^2-2*y+1" "/-%\"fG6$%\"xG%\"yG,2*$F&\"\"%\"\"\"*(F*F+*$F&\"\"$F+F'
F+!\"\"*(\"\"'F+*$F&\"\"#F+F'F3F+*(F*F+F&F+F'F.F/*$F'F*F+*$F'F3F+*&F3F
+F'F+F/F+F+" }{TEXT 456 163 "  has one critical point P = (a,b). Choos
e the ordered list [b,what] where b is the ordinate of the critical po
int P = (a,b) and \"what\" describes the behavior of  " }{XPPEDIT 459 
1 "f" "I\"fG6\"" }{TEXT 458 9 "   at P. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 464 2 "a)" }{TEXT 460 24 " [0,local minim
um]      " }{TEXT 467 2 "b)" }{TEXT 465 54 " [1,local minimum]        \+
c) [2,local minimum]        " }{TEXT 468 7 "  \nd)  " }{TEXT 469 18 "[
0,local maximum] " }{TEXT 470 8 "     e) " }{TEXT 466 21 "[1,local max
imum]    " }{TEXT 463 4 "    " }{TEXT 473 3 " f)" }{TEXT 474 1 " " }
{TEXT 471 17 "[2,local maximum]" }{TEXT 472 1 " " }{TEXT 461 12 "     \+
  \ng)  " }{TEXT 475 16 "[0,saddle point]" }{TEXT 476 16 "            \+
 h) " }{TEXT 477 16 "[1,saddle point]" }{TEXT 478 19 "                \+
i) " }{TEXT 481 16 "[2,saddle point]" }{TEXT 462 6 "    \n " }{TEXT 
482 3 "j) " }{TEXT 483 1 " " }{TEXT 479 18 "[-1,local maximum]" }
{TEXT 480 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 611 16 "6. (Replacement
)" }{TEXT -1 1 " " }{TEXT 582 15 "The function   " }{XPPEDIT 584 1 "f(
x,y)=2*x^3-6*x*y+3*y^2" "/-%\"fG6$%\"xG%\"yG,(*&\"\"#\"\"\"*$F&\"\"$F+
F+*(\"\"'F+F&F+F'F+!\"\"*&F-F+*$F'F*F+F+" }{TEXT 583 176 "  has one cr
itical point P = (a,b)  with ab > 0. Choose the ordered list [b,what] \+
where b is the ordinate of the critical point P = (a,b) and \"what\" d
escribes the behavior of  " }{XPPEDIT 586 1 "f" "I\"fG6\"" }{TEXT 585 
9 "   at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
591 2 "a)" }{TEXT 587 24 " [0,local minimum]      " }{TEXT 594 2 "b)" 
}{TEXT 592 54 " [1,local minimum]        c) [2,local minimum]        \+
" }{TEXT 595 7 "  \nd)  " }{TEXT 596 18 "[0,local maximum] " }{TEXT 
597 8 "     e) " }{TEXT 593 21 "[1,local maximum]    " }{TEXT 590 4 " \+
   " }{TEXT 600 3 " f)" }{TEXT 601 1 " " }{TEXT 598 17 "[2,local maxim
um]" }{TEXT 599 1 " " }{TEXT 588 12 "       \ng)  " }{TEXT 602 16 "[0,
saddle point]" }{TEXT 603 16 "             h) " }{TEXT 604 16 "[1,sadd
le point]" }{TEXT 605 19 "                i) " }{TEXT 608 16 "[2,saddl
e point]" }{TEXT 589 6 "    \n " }{TEXT 609 3 "j) " }{TEXT 610 1 " " }
{TEXT 606 18 "[-1,local maximum]" }{TEXT 607 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 33 
"7.  What is the maximum value of " }{TEXT 328 1 " " }{XPPEDIT 322 1 "
f(x,y)=x^2/2+y^2/2" "/-%\"fG6$%\"xG%\"yG,&*&F&\"\"#F*!\"\"\"\"\"*&F'F*
F*F+F," }{TEXT 323 2 "  " }{TEXT 326 4 " if " }{TEXT 327 3 "   " }
{XPPEDIT 324 1 "x^2/2+y^2=1" "/,&*&%\"xG\"\"#F&!\"\"\"\"\"*$%\"yGF&F(F
(" }{TEXT 325 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 329 3 "a) " }{XPPEDIT 
19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 416 46 "            b)  1/2   \+
     c) 3/2          d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }
{TEXT 417 11 "       e)  " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%
%sqrtG6#F#F$" }{TEXT 418 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sqrt(3)" "*
&\"\"#\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 419 59 "      g) 1             h
) 2            i) 3            j) 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 "
 " }{TEXT 340 36 "8.  When subject to the conditions  " }{XPPEDIT 338 
1 "x^2+y^2=2" "/,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'F&" }{TEXT 339 1 " " }
{TEXT 335 5 " and " }{TEXT 336 1 " " }{XPPEDIT 330 1 "x+z=1" "/,&%\"xG
\"\"\"%\"zGF%F%" }{TEXT 331 1 " " }{TEXT 337 12 "the function" }{TEXT 
425 1 " " }{TEXT 333 1 " " }{XPPEDIT 334 0 "f(x,y,z)=x+y+z" "/-%\"fG6%
%\"xG%\"yG%\"zG,(F&\"\"\"F'F*F(F*" }{TEXT 332 40 "  has a maximum at  \+
(a,b,c).  What is c?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 420 3 "a) " }
{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 421 46 "            b
)  1/2        c) 3/2          d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6
#\"\"$" }{TEXT 422 11 "       e)  " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"
#\"\"\"-%%sqrtG6#F#F$" }{TEXT 423 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sq
rt(3)" "*&\"\"#\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 424 43 "      g) 1     \+
        h) 2            i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#
\"\"#\"\"\"F&!\"\"" }{TEXT 426 16 "            j)  " }{XPPEDIT 19 1 "s
qrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 268 15 "9. The vector  " }{XPPEDIT 341 1 "`<`*a,b,c*`>` " "6%*&%
\"<G\"\"\"%\"aGF%%\"bG*&%\"cGF%%\">GF%" }{TEXT 342 36 "  is a unit nor
mal to the surface   " }{XPPEDIT 343 1 "x^2+2*y^3+3*z^4=6" "/,(*$%\"xG
\"\"#\"\"\"*&F&F'*$%\"yG\"\"$F'F'*&F+F'*$%\"zG\"\"%F'F'\"\"'" }{TEXT 
344 54 " at the point (1,1,1). What is a possible value for c?" }
{TEXT 484 1 " " }}{PARA 258 "" 0 "" {TEXT 345 5 "\na)  " }{XPPEDIT 
346 1 "1/sqrt(26)" "*&\"\"\"F#-%%sqrtG6#\"#E!\"\"" }{TEXT 347 9 "     \+
b)  " }{XPPEDIT 348 1 "2/sqrt(26)" "*&\"\"#\"\"\"-%%sqrtG6#\"#E!\"\"" 
}{TEXT 349 10 "      c)  " }{XPPEDIT 350 1 "3/sqrt(26)" "*&\"\"$\"\"\"
-%%sqrtG6#\"#E!\"\"" }{TEXT 351 9 "      d) " }{XPPEDIT 485 1 "4/sqrt(
26)" "*&\"\"%\"\"\"-%%sqrtG6#\"#E!\"\"" }{TEXT 486 10 "       e) " }
{XPPEDIT 487 1 "5/sqrt(26)" "*&\"\"&\"\"\"-%%sqrtG6#\"#E!\"\"" }{TEXT 
488 12 "       \n f) " }{XPPEDIT 495 1 "3/sqrt(46)" "*&\"\"$\"\"\"-%%s
qrtG6#\"#Y!\"\"" }{TEXT 496 7 "     g)" }{XPPEDIT 493 1 "6/sqrt(46)" "
*&\"\"'\"\"\"-%%sqrtG6#\"#Y!\"\"" }{TEXT 494 12 "         h) " }
{XPPEDIT 491 1 "12/sqrt(46)" "*&\"#7\"\"\"-%%sqrtG6#\"#Y!\"\"" }{TEXT 
492 11 "       i)  " }{XPPEDIT 489 1 "13/sqrt(46)" "*&\"#8\"\"\"-%%sqr
tG6#\"#Y!\"\"" }{TEXT 490 11 "        j) " }{XPPEDIT 497 1 "15/sqrt(46
)" "*&\"#:\"\"\"-%%sqrtG6#\"#Y!\"\"" }{TEXT 498 3 "   " }{TEXT -1 1 " \+
" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 16 "10. Calculate   \+
" }{XPPEDIT 505 1 "int(int(`(`*2*x+3*x*y^2*`)`,x=0..1),y=-1..1)" "-%$i
ntG6$-F#6$,&*(%\"(G\"\"\"\"\"#F*%\"xGF*F***\"\"$F*F,F*%\"yGF+%\")GF*F*
/F,;\"\"!F*/F/;,$F*!\"\"F*" }{TEXT 506 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 502 74 "a)  -4           b)  -3       \+
      c)   -2         d)  -1           e)  0" }{TEXT 499 1 " " }{TEXT 
504 20 "                    " }}{PARA 0 "" 0 "" {TEXT 500 36 "f)   1  \+
          g)   2            " }{TEXT -1 1 " " }{TEXT 501 39 "h)   3   \+
        i)  4             j)  " }{TEXT -1 1 " " }{TEXT 503 1 "5" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT 278 1 " " }}{PARA 0 "" 0 "" 
{TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 270 17 "11.  Calculate   " }{XPPEDIT 507 1 "int(int(15/16*x
^2+x*y^2,A=R..``),``=``..``)" "-%$intG6$-F#6$,&*(\"#:\"\"\"\"#;!\"\"%
\"xG\"\"#F**&F-F**$%\"yGF.F*F*/%\"AG;%\"RG%!G/F6;F6F6" }{TEXT 508 47 "
  where R is the region in the upper half plane" }{TEXT 354 12 " bound
ed by " }{XPPEDIT 352 1 "y=4-x^2" "/%\"yG,&\"\"%\"\"\"*$%\"xG\"\"#!\"
\"" }{TEXT 353 5 ".  \n\n" }{TEXT 275 70 "a)  1           b)  2       \+
      c)   3         d)  4           e)  5" }{TEXT 355 21 "           \+
          " }}{PARA 0 "" 0 "" {TEXT 357 34 "f)  6            g)  7    \+
        " }{TEXT -1 3 " h)" }{TEXT 356 36 "  8           i) 9         \+
   j)  12" }{TEXT -1 2 "  " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }
{TEXT 271 135 "12. The triangular region in the first quadrant that is
 bounded by   y = 1-x   has mass density   1 + x + 2y.\n        What i
s its mass?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" 
{TEXT 515 70 "a)  1           b)  2             c)   3         d)  4  \+
         e)  5" }{TEXT 516 21 "                    \n" }{TEXT 518 34 "
f)  6            g)  7            " }{TEXT -1 3 " h)" }{TEXT 517 36 " \+
 8           i) 9            j)  12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 1 
" " }{TEXT 358 98 "13.  What is the abscissa of the center of mass of \+
the triangular region of the preceding problem?" }{TEXT 519 3 " \n\n" 
}{TEXT 359 152 "a) 1/6            b) 1/3          c) 1/2            d)
 2/3         e) 5/6  \nf)   1/12        g)   1/4         h)  5/12     \+
     i)  1/2         j)  7/12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 261 "" 0 "" {TEXT -1 33 "14.  Calculate  the integral of
  " }{XPPEDIT 19 1 "sqrt(x^2+y^2)/Pi" "*&-%%sqrtG6#,&*$%\"xG\"\"#\"\"
\"*$%\"yGF)F*F*%#PiG!\"\"" }{TEXT -1 42 "  over the region bounded by \+
y = x, x=0,  " }{XPPEDIT 19 1 "x^2+y^2=1" "/,&*$%\"xG\"\"#\"\"\"*$%\"y
GF&F'F'" }{TEXT -1 7 ", and  " }{XPPEDIT 19 1 "x^2+y^2=4" "/,&*$%\"xG
\"\"#\"\"\"*$%\"yGF&F'\"\"%" }{TEXT -1 1 "." }}{PARA 3 "" 0 "" {TEXT 
520 152 "a) 1/6            b) 1/3          c) 1/2            d) 2/3   \+
      e) 5/6  \nf)   1/12        g)   1/4         h)  5/12          i)
  1/2         j)  7/12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 362 14 "15. Integr
ate " }{TEXT 510 1 " " }{XPPEDIT 511 1 "12*x" "*&\"#7\"\"\"%\"xGF$" }
{TEXT 509 73 "  over the solid region in the first octant that is boun
ded by the plane " }{TEXT 513 1 " " }{XPPEDIT 514 1 "x+2*y+z=2" "/,(%
\"xG\"\"\"*&\"\"#F%%\"yGF%F%%\"zGF%F'" }{TEXT 512 1 "." }{TEXT 363 3 "
  \n" }{TEXT 361 142 "\na)  1           b)  2            c)  3        \+
   d)  4           e) 5 \nf)   8          g)   12         h)  15      \+
    i)  16          j)  20" }{TEXT 836 1 "\n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 837 2 "16" }{TEXT -1 2 ". " }
{TEXT 838 29 "The sum of iterated integrals" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 841 8 "        " }
{XPPEDIT 842 1 "Int(Int(f(x,y),y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..-1/s
qrt(2))+Int(Int(f(x,y),y= x..1/sqrt(2)),x=-1/sqrt(2)..1/sqrt(2))" ",&-
%$IntG6$-F$6$-%\"fG6$%\"xG%\"yG/F,;,$-%%sqrtG6#,&\"\"\"F4*$F+\"\"#!\"
\"F7-F16#,&F4F4*$F+F6F7/F+;,$F4F7,$*&F4F4-F16#F6F7F7F4-F$6$-F$6$-F)6$F
+F,/F,;F+*&F4F4-F16#F6F7/F+;,$*&F4F4-F16#F6F7F7*&F4F4-F16#F6F7F4" }}
{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 39 "can b
e written as the iterated integral" }}{PARA 267 "" 0 "" {TEXT -1 0 "" 
}}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 17 "   \+
              " }{XPPEDIT 19 1 "Int(Int(f(x,y),x= phi(y)..psi(y)),y=a.
.b)" "-%$IntG6$-F#6$-%\"fG6$%\"xG%\"yG/F*;-%$phiG6#F+-%$psiG6#F+/F+;%
\"aG%\"bG" }}{PARA 270 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" 
{TEXT -1 11 "What is    " }{XPPEDIT 19 1 "(-a*sqrt(2)+3*b*sqrt(2)+phi(
0)+psi(0))" ",**&%\"aG\"\"\"-%%sqrtG6#\"\"#F%!\"\"*(\"\"$F%%\"bGF%-F'6
#F)F%F%-%$phiG6#\"\"!F%-%$psiG6#F3F%" }{TEXT -1 1 "?" }}{PARA 272 "" 
0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 
"" {TEXT 839 144 "a)  1           b)  2            c)  3           d) \+
 4           e) 5 \nf)   6          g)   7            h)  8           \+
i)  9            j)  10" }{TEXT 840 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Solutions" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "1.  (j)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Let the gradient at P be \+
\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gf := vector( [a,b] ):
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Then
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "u := vector( [3/5,-4/5] ): v
 := vector( [4/5,-3/5] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "eqn1 := dotprod(gf,u) = -13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%eqn1G/,&%\"aG#\"\"$\"\"&%\"bG#!\"%F*!#8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "eqn2 := dotprod(gf,v) = -8;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/,&%\"aG#\"\"%\"\"&%\"bG#!\"$F*!\")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{eqn1,eqn2\},\{a,b\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG\"\"&/%\"bG\"#?" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 7 "2.  (d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v := vector( [2/9,b,-4/9] );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%#\"\"#\"\"*%\"
bG#!\"%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gf := vector( \+
[13,0,2] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gfG-%'VECTORG6#7%\"#
8\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(v,gf
);" }}{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 1 {PARA 4 "" 0 "" {TEXT -1 
7 "3.  (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "grad(x^2+2*y^3,[x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7$,$%\"xG\"\"#,$*$%\"yGF)\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(\{x=1,y=1\},\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$\"\"#\"\"'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "norm(\",2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,$*$\"#5#\"\"\"\"\"#F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 6 "4. (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := (x,y) -> x*y-x^2-y^2-2*
x-2*y+4; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%
)operatorG%&arrowGF),.*&9%\"\"\"9$F0F0*$F1\"\"#!\"\"*$F/F3F4F1!\"#F/F6
\"\"%F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 := diff
(f(x,y),x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,(%\"yG\"
\"\"%\"xG!\"#F*F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e
qn2 := diff(f(x,y),y) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G
/,(%\"xG\"\"\"%\"yG!\"#F*F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "solve(\{eqn1,eqn2\},\{x,y\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"xG!\"#/%\"yGF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "diff(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^
2; \n#positive => extremum" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "diff(f(x,y),x$2); #negativ
e => maximum" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "5. (g)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "f := (x,y) -> x^5*y+x*y^5+x*y; " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF
),(*&9$\"\"&9%\"\"\"F2*&F/F2F1F0F2*&F1F2F/F2F2F)F)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "eqn1 := diff(f(x,y),x) = 0;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%eqn1G/,(*&%\"xG\"\"%%\"yG\"\"\"\"\"&*$F*F,F+F*F
+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := diff(f(x,
y),y) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(*$%\"xG\"\"&
\"\"\"*&F(F*%\"yG\"\"%F)F(F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "solve(\{eqn1,eqn2\},\{x,y\});\n#Only one solution is \+
real-valued" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<$/%\"yG\"\"!/%\"xGF&<$
F$/F(-%'RootOfG6#,&-F,6#,&*$%#_ZG\"\"#\"\"\"F5F5!\"\"F2F5<$F'/F%F+<$/F
%-F,6#,&*$F3\"\"%\"\"'F5F5/F(F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "diff(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2; " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"%%\"yGF&\"$+%*$,(*$F%F&\"
\"&*$F'F&F,\"\"\"F.\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "subs(\{x=0,y=0\},\"); #negative => saddle point" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 6 "6. (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := (x
,y) -> x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4+y^2-2*y+1; " }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),2*$9$
\"\"%\"\"\"*&F/\"\"$9%F1!\"%*&F/\"\"#F4F7\"\"'*&F/F1F4F3F5*$F4F0F1*$F4
F7F1F4!\"#F1F1F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 \+
:= diff(f(x,y),x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,**
$%\"xG\"\"$\"\"%*&F(\"\"#%\"yG\"\"\"!#7*&F(F.F-F,\"#7*$F-F)!\"%\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := diff(f(x,y),y) = \+
0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,.*$%\"xG\"\"$!\"%*&F(
\"\"#%\"yG\"\"\"\"#7*&F(F.F-F,!#7*$F-F)\"\"%F-F,!\"#F.\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "solve(\{eqn1,eqn2\},\{x,y\})
; #Add eqn1 and eqn2 to get this" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$
/%\"yG\"\"\"/%\"xGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dif
f(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2; " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&*&,(*$%\"xG\"\"#\"#7*&F'\"\"\"%\"yGF+!#C*$F,F(F)
F+,*F&F)F*F-F.F)F(F+F+F+*$,(F&!#7F*\"#CF.F2F(!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 50 "subs(\{x=1,y=1\},\");  #Second Derivative T
est fails " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 
732 32 "Ad Hoc Brute Force Investigation" }{TEXT 733 51 ":  After some
 algebraic manipulation I noticed that" }{TEXT -1 2 " :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn3 :
= f(x,y) = (x-y)^4+(y-1)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3
G/,2*$%\"xG\"\"%\"\"\"*&F(\"\"$%\"yGF*!\"%*&F(\"\"#F-F0\"\"'*&F(F*F-F,
F.*$F-F)F**$F-F0F*F-!\"#F*F*,&*$,&F(F*F-!\"\"F)F**$,&F-F*F9F*F0F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 734 72 "T
he right side of eqn3 shows that  f(x,y)  has a global minimum at (1,1
)" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 20 "6. Replacement   (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := (x,y) -> 2*x^3-6*x*y+3*y^2; " 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%
&arrowGF),(*$9$\"\"$\"\"#*&9%\"\"\"F/F4!\"'*$F3F1F0F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn1 := diff(f(x,y),x) = 0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*$%\"xG\"\"#\"\"'%\"yG!\"'
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := diff(f(x,y
),y) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG!\"'%\"yG
\"\"'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "solve(\{eqn1,
eqn2\},\{x,y\});  \n#Only one solution (a,b) has ab > 0 " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$<$/%\"yG\"\"!/%\"xGF&<$/F%\"\"\"/F(F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "diff(f(x,y),x$2)*diff(f(x,y)
,y$2)-(diff(f(x,y),x,y))^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"x
G\"#s!#O\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(\{x=
1,y=1\},\");  #positive => extremum  " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"#O" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diff(f(x,y),x$2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\"#7" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "subs(\{x=1\},\"); #positive => local minimum
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 6 "7. (g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 55 "f := (x,y) -> x^2/2+y^2/2 :  phi := (x,y) -> x^2/2
+y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x,y
),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
eqn1G/%\"xG*&%'lambdaG\"\"\"F&F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*diff(phi(x,y),y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"yG,$*&%'lambdaG\"\"\"F&F*
\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x,y) =
 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"##\"\"\"F
)*$%\"yGF)F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{
eqn1,eqn2,eqn3\}, \{x,y,lambda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%
<%/%'lambdaG#\"\"\"\"\"#/%\"xG\"\"!/%\"yGF'<%F$F)/F-!\"\"<%/F-F+/F%F'/
F*-%'RootOfG6#,&*$%#_ZGF(F'!\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "f(0,1);\nf(0,-1); #Clearly same value as preceding\n
f(sqrt(2),0);\nf(-sqrt(2),0); #Clearly same value as preceding" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{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 92 "levelCurve[1] := implicitplot( f(x,y)=f(0,1), x = -
2..2,y=-3/2..3/2,color=pink,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 99 "levelCurve[2] := implicitplot( f(x,y)=f(sqrt(2),0),
 x = -2..2,y=-3/2..3/2,color=khaki,thickness=2):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 92 "levelCurve[3] := implicitplot( phi(x,y)=1, x \+
= -2..2,y=-3/2..3/2,color=maroon, thickness=3):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 102 "levelCurve[4] := contourplot(f(x,y), x = -2..
2,y=-3/2..3/2, contours=8,color=COLOR(RGB,0.75,0.7,0.7)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "display( seq(levelCurve[j], j = 1..
4) );" }}{PARA 13 "" 1 "" {INLPLOT "6--%'CURVESG6`q7$7$$!11++++++))!#;
$!1$*********\\7ZF*7$$!1,++++Dc!*F*$!1'************>%F*7$F-7$$!1M/8R<_
c\"*F*$!1t@l&p3E$RF*7$7$$!1*********\\i]*F*$!1'*************HF*F37$F97
$$!1o2Bp2Bp'*F*$!1Ap2Bp2[BF*7$7$$!1(********\\i!)*F*$!1'************z
\"F*F?7$7$$!1*********\\i!)*F*FH7$$!1/ieF[M5**F*$!1\\M5$z8Cn*!#<7$7$$!
1'********\\i&**F*$!1h************fFSFN7$FU7$FV$\"1-+++](=n#FS7$7$FV$
\"1R++++++gFSFen7$Fin7$$!1aG9dG9d)*F*$\"1Vr&G9dGR\"F*7$7$$!1'********
\\i!)*F*$\"1/++++++=F*F]o7$Fco7$$!1HE0@%otk*F*$\"1u%*y:j_NCF*7$7$$!1'*
*******\\i]*F*$\"1/++++++IF*Fio7$7$$!1(********\\i]*F*Fbp7$$!1?^>7&>7N
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