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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 
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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 "" 
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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 9 "Math 2331" }{TEXT 463 10 "
 Fall 2002" }}{PARA 18 "" 0 "" {TEXT 462 6 "Exam 3" }}{PARA 0 "" 0 "" 
{TEXT 259 2 "  " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 1 "1" }{TEXT 
303 7 ". If   " }{XPPEDIT 301 1 "u=<5/13,12/13>" "/%\"uG-%-anglebracke
tG6$*&\"\"&\"\"\"\"#8!\"\"*&\"#7F)F*F+" }{TEXT 278 2 ", " }{XPPEDIT 
302 1 "v=<4/5,3/5>" "/%\"vG-%-anglebracketG6$*&\"\"%\"\"\"\"\"&!\"\"*&
\"\"$F)F*F+" }{TEXT 279 6 ",     " }{XPPEDIT 299 1 "D[u](f)(P)=3" "/--
&%\"DG6#%\"uG6#%\"fG6#%\"PG\"\"$" }{TEXT 300 7 ", and  " }{XPPEDIT 
281 1 "D[v](f)(P)= 18/5" "/--&%\"DG6#%\"vG6#%\"fG6#%\"PG*&\"#=\"\"\"\"
\"&!\"\"" }{TEXT 280 15 "  then what is " }{XPPEDIT 350 1 "diff(f(x,y)
,x)" "-%%diffG6$-%\"fG6$%\"xG%\"yGF(" }{TEXT 348 6 "   at " }{XPPEDIT 
351 1 "P" "I\"PG6\"" }{TEXT 349 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 355 74 "a)  -4           b)  -3             \+
c)   -2         d)  -1           e)  0" }{TEXT 352 1 " " }{TEXT 357 
20 "                    " }}{PARA 0 "" 0 "" {TEXT 353 36 "f)   1      \+
      g)   2            " }{TEXT -1 1 " " }{TEXT 354 39 "h)   3       \+
    i)  4             j)  " }{TEXT -1 1 " " }{TEXT 356 1 "5" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 347 9 "Solution:" }{TEXT 464 4 " (h)" }
{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linal
g):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u := vector([5/13,12
/13]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v := vector([4/5,
3/5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gradient_f := vec
tor([a,b]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn1 := dotp
rod(u,gradient_f)=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&%\"
aG#\"\"&\"#8%\"bG#\"#7F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "eqn2 := dotprod(v,gradient_f)=18/5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/,&%\"aG#\"\"%\"\"&%\"bG#\"\"$F*#\"#=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 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 273 
4 "2.  " }{TEXT -1 6 "If    " }{XPPEDIT 19 1 "v= <cos(alpha),sin(alpha
)>" "/%\"vG-%-anglebracketG6$-%$cosG6#%&alphaG-%$sinG6#F*" }{TEXT -1 
6 ", if  " }{XPPEDIT 19 1 "r(t)=<1,2>+t*v" "/-%\"rG6#%\"tG,&-%-anglebr
acketG6$\"\"\"\"\"#F+*&F&F+%\"vGF+F+" }{TEXT -1 10 ", and if  " }
{XPPEDIT 19 1 "f(x,y)=x^2+3*y" "/-%\"fG6$%\"xG%\"yG,&*$F&\"\"#\"\"\"*&
\"\"$F+F'F+F+" }{TEXT -1 27 "   then for what value of  " }{XPPEDIT 
19 1 "alpha" "I&alphaG6\"" }{TEXT -1 8 " does   " }{XPPEDIT 19 1 "t ->
 f(r(t)" ":6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"fG6#-%\"rG6#F$F)F)" }
{TEXT -1 54 "  have the greatest instantaneous rate of change at   " }
{XPPEDIT 19 1 "t=0" "/%\"tG\"\"!" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 282 4 "a)  " }{XPPEDIT 283 1 "
arcsin(1/2)" "-%'arcsinG6#*&\"\"\"F&\"\"#!\"\"" }{TEXT 284 8 "    b)  \+
" }{XPPEDIT 285 1 "arcsin(1/3)" "-%'arcsinG6#*&\"\"\"F&\"\"$!\"\"" }
{TEXT 286 12 "        c)  " }{XPPEDIT 287 1 "arcsin(2/3)" "-%'arcsinG6
#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 288 12 "         d) " }{XPPEDIT 289 
1 "arcsin(2/9)" "-%'arcsinG6#*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT 290 12 " \+
       e)  " }{XPPEDIT 291 1 "arcsin(3/5)" "-%'arcsinG6#*&\"\"$\"\"\"
\"\"&!\"\"" }{TEXT 292 13 "        \nf)  " }{XPPEDIT 293 1 "arcsin(4/5
)" "-%'arcsinG6#*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 294 9 "     g)  " }
{XPPEDIT 295 1 "arcsin(1/sqrt(13)" "-%'arcsinG6#*&\"\"\"F&-%%sqrtG6#\"
#8!\"\"" }{TEXT 296 7 "   h)  " }{XPPEDIT 297 1 "arcsin(2/sqrt(13))" "
-%'arcsinG6#*&\"\"#\"\"\"-%%sqrtG6#\"#8!\"\"" }{TEXT 298 7 "    i) " }
{XPPEDIT 468 1 "arcsin(3/sqrt(13))" "-%'arcsinG6#*&\"\"$\"\"\"-%%sqrtG
6#\"#8!\"\"" }{TEXT 469 8 "    j)  " }{TEXT -1 1 " " }{XPPEDIT 467 1 "
arcsin(5/sqrt(13))" "-%'arcsinG6#*&\"\"&\"\"\"-%%sqrtG6#\"#8!\"\"" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 465 9 "Solution:" }{TEXT 466 4 " (i)" }{TEXT -1 1 "\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := (x,y) -> x^2 + 3*y:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "grad(f(x,y),[x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$,$%\"xG\"\"#\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=1,\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$\"\"#\"\"$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "map(z->z/sqrt(2^2+3^2),\");" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'VECTORG6#7$,$*$\"#8#\"\"\"\"\"##F,F),$F(#\"\"$F)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solve(sin(alpha)=3/13*13^
(1/2),alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arcsinG6#,$*$\"#8#
\"\"\"\"\"##\"\"$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 45 "3. The greate
st directional derivative  of   " }{XPPEDIT 470 1 "f" "I\"fG6\"" }
{TEXT 358 14 "  at the point" }{TEXT 305 2 "  " }{XPPEDIT 306 1 "P" "I
\"PG6\"" }{TEXT 304 1 " " }{TEXT 307 13 " is  2.  At  " }{XPPEDIT 472 
1 "P" "I\"PG6\"" }{TEXT 471 21 "  the gradient  of   " }{XPPEDIT 474 
1 "f" "I\"fG6\"" }{TEXT 473 22 "  makes an angle of   " }{XPPEDIT 478 
1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 477 27 "    with the unit ve
ctor   " }{XPPEDIT 476 1 "u" "I\"uG6\"" }{TEXT 475 41 " .  What is the
 directional derivative   " }{XPPEDIT 480 1 "D[u](f)(P)" "--&%\"DG6#%
\"uG6#%\"fG6#%\"PG" }{TEXT 479 3 "  ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 481 3 "a) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqr
tG6#\"\"#" }{TEXT 482 46 "            b)  1/2        c) 3/2          d
) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 483 11 "       e
)  " }{XPPEDIT 19 1 "2*sqrt(2)/3" "*(\"\"#\"\"\"-%%sqrtG6#F#F$\"\"$!\"
\"" }{TEXT 484 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sqrt(3)/3" "*(\"\"#\"
\"\"-%%sqrtG6#\"\"$F$F(!\"\"" }{TEXT 485 43 "      g) 1             h)
 2            i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"
\"F&!\"\"" }{TEXT 486 16 "            j)  " }{XPPEDIT 19 1 "sqrt(3)/2
" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 493 9 "Solutio
n:" }{TEXT 494 4 " (g)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "2*cos(Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
262 1 " " }{TEXT 310 18 "4. The function   " }{XPPEDIT 309 1 "f(x,y)=2
*x^3-24*x*y+16*y^3" "/-%\"fG6$%\"xG%\"yG,(*&\"\"#\"\"\"*$F&\"\"$F+F+*(
\"#CF+F&F+F'F+!\"\"*&\"#;F+*$F'F-F+F+" }{TEXT 308 185 "  has one criti
cal point P = (a,b) for which  a > 0. Choose the ordered list   [a, wh
at]   where a is the abscissa of the critical point P = (a,b) and \"wh
at\" describes the behavior of  " }{XPPEDIT 360 1 "f" "I\"fG6\"" }
{TEXT 359 10 "   at P.\n " }}{PARA 0 "" 0 "" {TEXT 266 2 "a)" }{TEXT 
256 24 " [1,local minimum]      " }{TEXT 311 2 "b)" }{TEXT 275 54 " [2
,local minimum]        c) [3,local minimum]        " }{TEXT 312 7 "  \+
\nd)  " }{TEXT 361 18 "[1,local maximum] " }{TEXT 362 8 "     e) " }
{TEXT 276 21 "[2,local maximum]    " }{TEXT 260 4 "    " }{TEXT 365 3 
" f)" }{TEXT 366 1 " " }{TEXT 363 17 "[3,local maximum]" }{TEXT 364 1 
" " }{TEXT 257 12 "       \ng)  " }{TEXT 367 16 "[1,saddle point]" }
{TEXT 368 16 "             h) " }{TEXT 369 16 "[2,saddle point]" }
{TEXT 370 19 "                i) " }{TEXT 373 16 "[3,saddle point]" }
{TEXT 258 6 "    \n " }{TEXT 374 3 "j) " }{TEXT 375 1 " " }{TEXT 371 
17 "[4,local minimum]" }{TEXT 372 2 " \n" }}{PARA 0 "" 0 "" {TEXT 487 
9 "Solution:" }{TEXT 488 4 " (b)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 34 "f := (x,y) -> 2*x^3-24*x*y+16*y^3:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y),x) = 0 ,
 diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<
$/%\"xG\"\"!/%\"yGF&<$/F(\"\"\"/F%\"\"#<$/F(-%'RootOfG6#,(*$%#_ZGF-F+F
5F+F+F+/F%,&!\"#F+F0F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 81 "The  last solution involves complex numbe
rs and is not relevant for our purposes." }}{PARA 0 "" 0 "" {MPLTEXT 
0 21 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$\"\"#\"\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(f(x,y),x$2)*diff(f(x
,y),y$2)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&
%\"xG\"\"\"%\"yGF&\"%_6!$w&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "subs(\{x=2,y=1\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%G<" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(\{x=2,y=1\},diff(f(x
,y),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 267 2 " 5" }{TEXT -1 2 ". " }{TEXT 384 15 "The function   \+
" }{XPPEDIT 383 1 "f(x,y)=x^2*y-x*y^2-3*x*y" "/-%\"fG6$%\"xG%\"yG,(*&F
&\"\"#F'\"\"\"F+*&F&F+*$F'F*F+!\"\"*(\"\"$F+F&F+F'F+F." }{TEXT 382 
178 "  has one critical point P = (a,b) with ab  <  0.  Choose the ord
ered list [b,what] where b is the ordinate of the critical point P = (
a,b) and \"what\" describes the behavior of  " }{XPPEDIT 386 1 "f" "I
\"fG6\"" }{TEXT 385 8 "  at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 391 2 "a)" }{TEXT 387 24 " [1,local minimum]    \+
  " }{TEXT 394 2 "b)" }{TEXT 392 56 " [-1,local minimum]        c) [-2
,local minimum]        " }{TEXT 395 7 "  \nd)  " }{TEXT 396 18 "[1,loc
al maximum] " }{TEXT 397 8 "     e) " }{TEXT 393 22 "[-1,local maximum
]    " }{TEXT 390 4 "    " }{TEXT 400 3 " f)" }{TEXT 401 1 " " }{TEXT 
398 18 "[-2,local maximum]" }{TEXT 399 1 " " }{TEXT 388 12 "       \ng
)  " }{TEXT 402 16 "[1,saddle point]" }{TEXT 403 16 "             h) \+
" }{TEXT 404 17 "[-1,saddle point]" }{TEXT 405 19 "                i) \+
" }{TEXT 408 17 "[-2,saddle point]" }{TEXT 389 6 "    \n " }{TEXT 409 
3 "j) " }{TEXT 410 1 " " }{TEXT 406 17 "[2,local maximum]" }{TEXT 407 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 489 9 "S
olution:" }{TEXT 490 4 " (e)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "f := (x,y) -> x^2*y-x*y^2-3*x*y:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y),x) = 0 , diff(f
(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&<$/%\"xG
\"\"!/%\"yGF&<$/F%\"\"$F'<$/F(!\"$F$<$/F(!\"\"/F%\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "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'!\"#!\"$F&F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "subs(\{x=1,y=-1\},\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sub
s(\{x=1,y=-1\},diff(f(x,y),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!
\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 18 "6. The functi
on   " }{XPPEDIT 412 1 "f(x,y)=x^3+2*x^2*y+y^2+x+2" "/-%\"fG6$%\"xG%\"
yG,,*$F&\"\"$\"\"\"*(\"\"#F+*$F&F-F+F'F+F+*$F'F-F+F&F+F-F+" }{TEXT 
411 165 "  has one critical point P = (a,b). Choose the ordered list [
a,what] where  a  is the abscissa of the critical point P = (a,b) and \+
\"what\" describes the behavior of  " }{XPPEDIT 414 1 "f" "I\"fG6\"" }
{TEXT 413 9 "   at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 419 2 "a)" }{TEXT 415 24 " [0,local minimum]      " }{TEXT 
422 2 "b)" }{TEXT 420 54 " [1,local minimum]        c) [2,local minimu
m]        " }{TEXT 423 7 "  \nd)  " }{TEXT 424 18 "[0,local maximum] \+
" }{TEXT 425 8 "     e) " }{TEXT 421 21 "[1,local maximum]    " }
{TEXT 418 4 "    " }{TEXT 428 3 " f)" }{TEXT 429 1 " " }{TEXT 426 17 "
[2,local maximum]" }{TEXT 427 1 " " }{TEXT 416 12 "       \ng)  " }
{TEXT 430 16 "[0,saddle point]" }{TEXT 431 16 "             h) " }
{TEXT 432 16 "[1,saddle point]" }{TEXT 433 19 "                i) " }
{TEXT 436 16 "[2,saddle point]" }{TEXT 417 6 "    \n " }{TEXT 437 3 "j
) " }{TEXT 438 1 " " }{TEXT 434 18 "[-1,local maximum]" }{TEXT 435 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 491 9 "Solu
tion:" }{TEXT 492 4 " (h)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "f := (x,y) -> x^3+2*x^2*y+y^2+x+2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y),x) = 0 , diff(f
(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yG
!\"\"/%\"xG\"\"\"<$/F(-%'RootOfG6#,(*$%#_ZG\"\"#\"\"%F1F)F)F)/F%,&#F)F
3F)F,F6" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 45 "Only the first is
 a solution in real numbers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "diff(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"#7%\"yG\"\")*$F$\"\"#!#;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(\{y = -1, x = 1\},\");
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 1 " " }{TEXT 265 33 "7.  What is the maximum value of " }{TEXT 319 
1 " " }{XPPEDIT 313 1 "f(x,y)=x^2+y" "/-%\"fG6$%\"xG%\"yG,&*$F&\"\"#\"
\"\"F'F+" }{TEXT 314 2 "  " }{TEXT 317 4 " if " }{TEXT 318 3 "   " }
{XPPEDIT 315 1 "x^2/2+y^2=1" "/,&*&%\"xG\"\"#F&!\"\"\"\"\"*$%\"yGF&F(F
(" }{TEXT 316 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 320 158 "a)  1         \+
   b)  9/8            c) 5/4              d) 11/8        e) 4/3 \n f) \+
 13/8       g) 7/4             h) 15/8            i) 2              j)
 17/8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 495 9 "Solution:" }{TEXT 496 4 " (j)" }
{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := (x,y)
 -> x^2 + y:  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/,$%\"xG\"\"#*&%'lambdaG\"\"
\"F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,
y),y) = lambda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%eqn2G/\"\"\",$*&%'lambdaGF&%\"yGF&\"\"#" }}}{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%#F0F(F)<%/F%F(/F*,$-%'RootOfG6#,&*$%#_ZGF(F(!#
:F'F&/F-#F'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f(0,1),
 \nf(0,-1), \nf(sqrt(15/2)/2,1/4), \nf(-sqrt(15/2)/2,1/4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&\"\"\"!\"\"#\"#<\"\")F%" }}}{EXCHG {PARA 0 "
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F\\hl$!1+++++O29F*Fd_]l7$Fj_]l7$$!1hjjjjjvkF-$!1jjjjjV#R\"F*7$7$$!1+++
+++sjF-FZF^`]l7$Fd`]l7$$!19=====IfF-$!1=====)pK\"F*7$7$Feil$!1+++++Oj7
F*Fh`]l7$F^a]l7$$!1Sr&G9dGQ&F-$!1'G9dG9<E\"F*7$7$$!1***********\\O&F-F
hpFba]l7$Fha]l7$$!1bz*[C71x%F-$!1/-^v(QH?\"F*7$7$Fajl$!1+++++;[6F*F\\b
]l7$Fbb]l7$$!1!)[`R\"eD:%F-$!17l/'=WZ9\"F*7$7$$!1jmmmmm'3%F-F[tFfb]l7$
F\\c]l7$$!1H-$4s$)[X$F-$!1xp!zi6X4\"F*7$7$Fa[m$!1+++++wh5F*F`c]l7$Ffc]
l7$$!1Ly$y$y$yr#F-$!1;i@;i@[5F*7$7$$!1\"***********H@F-F_vFjc]l7$F`d]l
7$$!1@q-Fq-2>F-$!1(H(H(H(H45F*7$7$Ff\\m$!1+++++;/5F*Fdd]l7$Fjd]l7$$!1c
***********f*F3$!1************R)*F-7$7$F8$!1+++++g`(*F-F^e]l7$Fde]l7$$
\"1]+++++O:F3Fee]l7$7$F[blFee]lFhe]l7$F\\f]l7$$\"1J%ot%*y:f\"F-$!1F%ot
%*y:***F-7$7$FPF[e]lF^f]l7$7$Fbo$!1,++++wh5F*7$$\"1%***********H@F-F_v
7$Fif]lFdf]l7$7$F^p$!1,++++;[6F*7$$\"1immmmm'3%F-F[t7$Fag]lFff]l7$7$Ff
qF_a]l7$$\"1)**********\\O&F-Fhp7$Fgg]lF^g]l7$7$FerF[`]l7$$\"1++++++sj
F-FZ7$7$$\"1,+++++sjF-FZFfg]l7$7$$\"1NLLLLLVsF-F(F\\h]lF^dr-F$6%7&7$\"
\"!$\"\"\"F\\i]l7$F\\i]l$FfdrF\\i]l7$$\"1:Hw$R1$p8F*$\"1+++++++DF-7$$!
1:Hw$R1$p8F*Fdi]l-%'SYMBOLG6#%(DIAMONDG-%&STYLEG6#%&POINTG-%+AXESLABEL
SG6$%\"xG%\"yG" 2 514 443 443 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 
1 0 0 0 237 224 0 0 0 0 0 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 331 36 "8.  When subject to the \+
conditions  " }{XPPEDIT 329 1 "x^2+y^2+z^2=1" "/,(*$%\"xG\"\"#\"\"\"*$
%\"yGF&F'*$%\"zGF&F'F'" }{TEXT 330 1 " " }{TEXT 326 5 " and " }{TEXT 
327 1 " " }{XPPEDIT 321 1 "x+ z=1" "/,&%\"xG\"\"\"%\"zGF%F%" }{TEXT 
322 1 " " }{TEXT 328 12 "the function" }{TEXT 380 1 " " }{TEXT 324 1 "
 " }{XPPEDIT 325 0 "f(x,y,z)=x+y+z" "/-%\"fG6%%\"xG%\"yG%\"zG,(F&\"\"
\"F'F*F(F*" }{TEXT 323 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 376 31 "a) 1/2        b)  3/2   \+
    c) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 499 14 "   \+
        d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 377 11 
"       e)  " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$
" }{TEXT 378 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"#\"\"\"
-%%sqrtG6#\"\"$F$" }{TEXT 379 43 "      g) 1             h) 2         \+
   i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }
{TEXT 381 16 "            j)  " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG
6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 497 9 "Solution:" }{TEXT 498 5 " (a)\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 78 "f := (x,y,z) -> x+y+z:  \nphi := (x,y,z) -> x
^2+y^2+z^2:\npsi := (x,y,z) -> x+z:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "eqn1 := diff(f(x,y,z),x) = lambda*diff(phi(x,y,z),x)+
mu*diff(psi(x,y,z),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"
\"\",&*&%'lambdaGF&%\"xGF&\"\"#%#muGF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "eqn2 := diff(f(x,y,z),y) = lambda*diff(phi(x,y,z),y)+
mu*diff(psi(x,y,z),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"
\"\",$*&%'lambdaGF&%\"yGF&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 75 "eqn3 := diff(f(x,y,z),z) = lambda*diff(phi(x,y,z),z)+mu*diff(p
si(x,y,z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/\"\"\",&*&%'
lambdaGF&%\"zGF&\"\"#%#muGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "eqn4 := phi(x,y,z) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq
n4G/,(*$%\"xG\"\"#\"\"\"*$%\"yGF)F**$%\"zGF)F*F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "eqn5 := psi(x,y,z) = 1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn5G/,&%\"xG\"\"\"%\"zGF(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "solve( \{eqn1,eqn2,eqn3,eqn4,eqn5\}, \{x,y,z,l
ambda,mu\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"xG#\"\"\"\"\"#/%
\"zGF&/%\"yG-%'RootOfG6#,&!\"\"F'*$%#_ZGF(F(/%'lambdaGF-/%#muG,&F'F'F-
F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 15 "9. The vector  " }{XPPEDIT 
332 1 "`<`*a,2,-1*`>` " "6%*&%\"<G\"\"\"%\"aGF%\"\"#,$*&F%F%%\">GF%!\"
\"" }{TEXT 333 30 "  is tangent to the surface   " }{XPPEDIT 334 1 "x^
2+2*y^3+3*z^2=6" "/,(*$%\"xG\"\"#\"\"\"*&F&F'*$%\"yG\"\"$F'F'*&F+F'*$%
\"zGF&F'F'\"\"'" }{TEXT 335 36 " at the point (1,1,1).  What is  a ?" 
}{TEXT 439 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 504 72 "a)  -5           b)  -3             c)   -3        d)  -
2           e)  " }{TEXT 501 3 "-1 " }{TEXT 505 20 "                  \+
  " }}{PARA 0 "" 0 "" {TEXT 502 37 "f)   0            g)   1          \+
   " }{TEXT -1 1 " " }{TEXT 503 37 "h)   2           i)  3           j
)  " }{TEXT -1 1 " " }{TEXT 506 1 "4" }{TEXT -1 2 "  " }}{PARA 0 "" 0 
"" {TEXT 500 1 " " }}{PARA 0 "" 0 "" {TEXT 507 9 "Solution:" }{TEXT 
508 5 " (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := (x,y,z
) -> x^2+2*y^3+3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6%%\"x
G%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"\"*$9%\"\"$F1*$9&F
1F5F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(F(x,y,z),[x
,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$%\"xG\"\"#,
$*$%\"yGF)\"\"',$%\"zGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"v := subs(\{x=1,y=1,z=1\}, \" );  #Normal to surface" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"#\"\"'F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "solve(dotprod(\",[a,2,-1])=0,a); \n#Tange
nt vector is perpendicular to Normal vector" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 16 "10. Calculate   " }
{XPPEDIT 446 1 "int(int(`(`*x*y+3*x^2*`)`,x=0..1),y=-1..1)" "-%$intG6$
-F#6$,&*(%\"(G\"\"\"%\"xGF*%\"yGF*F**(\"\"$F**$F+\"\"#F*%\")GF*F*/F+;
\"\"!F*/F,;,$F*!\"\"F*" }{TEXT 447 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 443 74 "a)  -4           b)  -3            \+
 c)   -2         d)  -1           e)  0" }{TEXT 440 1 " " }{TEXT 445 
20 "                    " }}{PARA 0 "" 0 "" {TEXT 441 36 "f)   1      \+
      g)   2            " }{TEXT -1 1 " " }{TEXT 442 39 "h)   3       \+
    i)  4             j)  " }{TEXT -1 1 " " }{TEXT 444 1 "5" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT 277 1 " " }}{PARA 0 "" 0 "" {TEXT 
509 9 "Solution:" }{TEXT 510 5 " (g)\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "int(int(x*y+3*x^2, x = 0 .. 1),y = -1 .. 1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 270 40 "11.  Calculate the double i
ntegral of   " }{XPPEDIT 448 1 "5*(x-y)" "*&\"\"&\"\"\",&%\"xGF$%\"yG!
\"\"F$" }{TEXT 449 83 "  over the region in the first quadrant of the \+
xy-plane that is bounded above by   " }{XPPEDIT 336 1 "y=2*x-x^2" "/%
\"yG,&*&\"\"#\"\"\"%\"xGF'F'*$F(F&!\"\"" }{TEXT 337 5 ".  \n\n" }
{TEXT 274 70 "a)  1           b)  2             c)   3         d)  4  \+
         e)  5" }{TEXT 338 21 "                     " }}{PARA 0 "" 0 "
" {TEXT 340 34 "f)  6            g)  7            " }{TEXT -1 3 " h)" 
}{TEXT 339 36 "  8           i) 9            j)  12" }{TEXT -1 2 "  " 
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 511 9 "Solutio
n:" }{TEXT 512 5 " (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
plot(2*x-x^2,x=0..2);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7S7$
\"\"!F(7$$\"1LLLL3VfV!#<$\"1$\\gZH:)G&)F,7$$\"1nmm\"H[D:)F,$\"1w`o9c/k
:!#;7$$\"1LLLe0$=C\"F4$\"1>c5.oWHBF47$$\"1LLL3RBr;F4$\"1c..Rb;jIF47$$
\"1mm;zjf)4#F4$\"1*=o?3#ycPF47$$\"1LL$e4;[\\#F4$\"1fPYc9AnVF47$$\"1++]
i'y]!HF4$\"1@(o974i'\\F47$$\"1LL$ezs$HLF4$\"1gJIqKF]bF47$$\"1++]7iI_PF
4$\"1!RjPBKm4'F47$$\"1nmm;_M(=%F4$\"17#zpV/8i'F47$$\"1LLL3y_qXF4$\"1O=
*><$3_qF47$$\"1+++]1!>+&F4$\"1dHv)G+>](F47$$\"1+++]Z/NaF4$\"1W(\\lN=h
\"zF47$$\"1+++]$fC&eF4$\"1e(fcl!zz#)F47$$\"1LL$ez6:B'F4$\"1pc]l'\\)z&)
F47$$\"1mmm;=C#o'F4$\"1qCpj![#**))F47$$\"1mmmm#pS1(F4$\"1zo!H2J!Q\"*F4
7$$\"1++]i`A3vF4$\"1u6f:f5z$*F47$$\"1mmmm(y8!zF4$\"1[-!>*)y&f&*F47$$\"
1++]i.tK$)F4$\"1iqe&>@?s*F47$$\"1++](3zMu)F4$\"1<]k>b6U)*F47$$\"1nmm\"
H_?<*F4$\"1flAf-XJ**F47$$\"1nm;zihl&*F4$\"1*\\2$y58\")**F47$$\"1LLL3#G
,***F4$\"0$Hxa-******!#:7$$\"1LLezw5V5F]s$\"11%4'zsT\")**F47$$\"1++v$Q
#\\\"3\"F]s$\"1Gu!R\"**eL**F47$$\"1LL$e\"*[H7\"F]s$\"1zS&4kN)[)*F47$$
\"1+++qvxl6F]s$\"1^*[G(z<D(*F47$$\"1++]_qn27F]s$\"1C\"\\'=Cqo&*F47$$\"
1++Dcp@[7F]s$\"1d)*HEM)QQ*F47$$\"1++]2'HKH\"F]s$\"1hR&G(R;S\"*F47$$\"1
nmmwanL8F]s$\"1IFring')))F47$$\"1+++v+'oP\"F]s$\"1X*4(Q[wz&)F47$$\"1LL
eR<*fT\"F]s$\"1BV)fs3&p#)F47$$\"1+++&)Hxe9F]s$\"1+>M#[t_*yF47$$\"1mm\"
H!o-*\\\"F]s$\"12xq*\\A(4vF47$$\"1++DTO5T:F]s$\"1:*fU\\o?2(F47$$\"1nmm
T9C#e\"F]s$\"1<#fg.\\*4mF47$$\"1++D1*3`i\"F]s$\"1*GWwr())*3'F47$$\"1LL
L$*zym;F]s$\"1n]Y>x$Rb&F47$$\"1LL$3N1#4<F]s$\"1qmP>NEq\\F47$$\"1nm\"HY
t7v\"F]s$\"1)*><R=)eN%F47$$\"1+++q(G**y\"F]s$\"1tGE$QD,w$F47$$\"1nm;9@
BM=F]s$\"1Q,$pzn0/$F47$$\"1LLL`v&Q(=F]s$\"11G#[wHPO#F47$$\"1++DOl5;>F]
s$\"1t-SU\")[2;F47$$\"1++v.Uac>F]s$\"1cyAF'=B])F,7$$\"\"#F(F(-%'COLOUR
G6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(Fbz%(DEFAU
LTG" 2 374 374 374 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 318 32208 0 0 0 0 0 0 }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "int(int(5*x-5*y, y = 0 .. 2*
x-x^2),x = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 271 136 "12. The triangular region in the first quadrant that
 is bounded by   y = 2 - 2x   has mass density   8/3 + 2y.\n        Wh
at is its mass?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" 
{TEXT 456 70 "a)  1           b)  2             c)   3         d)  4  \+
         e)  5" }{TEXT 457 21 "                    \n" }{TEXT 459 34 "
f)  6            g)  7            " }{TEXT -1 3 " h)" }{TEXT 458 36 " \+
 8           i) 9            j)  12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 513 9 "Solution:" }{TEXT 514 5 " (d)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(int(8/3+2*y,y = 0 .. 2-2
*x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 1 " " }{TEXT 
341 98 "13.  What is the ordinate of the center of mass of the triangu
lar region of the preceding problem?" }{TEXT 460 3 " \n\n" }{TEXT 342 
154 "a) 2/9           b) 1/3          c) 4/9            d) 5/9        \+
   e) 2/3 \nf)   7/9          g)   8/9         h)  1             i)  1
0/9         j)  11/9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 515 9 "Solution:" }{TEXT 516 5 " (f)\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 86 "Int(Int(y*(8/3+2*y),y = 0 .. 2-2*x),x=0..1)/Int(
Int((8/3+2*y),y = 0 .. 2-2*x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&-%$IntG6$-F%6$*&%\"yG\"\"\",&#\"\")\"\"$F+F*\"\"#F+/F*;\"\"!,&
F0F+%\"xG!\"#/F5;F3F+F+-F%6$-F%6$F,F1F7!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"
\"(\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 "arctan
(y/x)" "-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 47 "  over the \+
region bounded by y = 2x, x=0, and  " }{XPPEDIT 19 1 "x^2+y^2=4" "/,&*
$%\"xG\"\"#\"\"\"*$%\"yGF&F'\"\"%" }{TEXT -1 1 "." }}{PARA 3 "" 0 "" 
{TEXT 461 152 "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 548 9 "Solution:" }{TEXT 549 5 " (f)\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 56 "p1 := plot(2*x,x=0..2/sqrt(5),thickness=2,colo
r=MAROON):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p2 := plot( x
,x=0..sqrt(2),thickness=2,color=MAROON):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "p3 := plot(sqrt(4-x^2),x= sqrt(4/5)..sqrt(2),scaling=
constrained,thickness=2,color=MAROON):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "display(p1,p2,p3);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%
'CURVESG6%7S7$\"\"!F(7$$\"1/XuPnf\\>!#<$\"14!*[vM>**QF,7$$\"1)))=[VIfk
$F,$\"1wxjp3'=H(F,7$$\"1b?=\"4NOb&F,$\"16kB=qs56!#;7$$\"14RM__)RZ(F,$
\"1#yo/0(z%\\\"F97$$\"1fESC$3_Q*F,$\"1K0)[mTq(=F97$$\"1z$yln:d6\"F9$\"
1en:`8VJAF97$$\"1ku:u1>*H\"F9$\"1G\\J[8Q)f#F97$$\"1]u6z2%*)[\"F9$\"1**
[Be:)y(HF97$$\"1VV3`B3y;F9$\"1'oohqkhN$F97$$\"1hRO5xjs=F9$\"1Bzs?aFXPF
97$$\"1QX&\\<-S/#F9$\"1w!4*\\V+)3%F97$$\"1x/_uz\"pB#F9$\"1b4/\\f$QZ%F9
7$$\"1.W*[!fiICF9$\"11))y4=Dh[F97$$\"1Rj,*Q*H<EF9$\"1zE.y()fM_F97$$\"1
Q%Qiz;oy#F9$\"1xoZ#fLOd&F97$$\"1w,]*Q*Q))HF9$\"1_.+z(yn(fF97$$\"1bMJ;y
9fJF9$\"15piKcH=jF97$$\"1Q(Q4Y!ydLF9$\"1xu(=#4c:nF97$$\"1$\\Q%3SgLNF9$
\"1()p(o,3s1(F97$$\"1<*ylI5ls$F9$\"1My:81-`uF97$$\"1#Gu2s--\"RF9$\"1l&
[:W0/#yF97$$\"1quQ%[m=5%F9$\"1T\\xoHt.#)F97$$\"1>gL]O(yF%F9$\"1P?n+tub
&)F97$$\"1OPbc6snWF9$\"1su58BWN*)F97$$\"1_V%)f$>\\m%F9$\"1/()o>(Q)H$*F
97$$\"1D&3b(4eO[F9$\"1^q,^>;t'*F97$$\"1o%\\KA!)>-&F9$\"1%*)\\Y/'R/5!#:
7$$\"11`\\(y:N@&F9$\"1h!*\\dJqU5F\\t7$$\"1]8spf*3S&F9$\"1qU%R>z,3\"F\\
t7$$\"1/X$4$f>#e&F9$\"1,p='=Rk6\"F\\t7$$\"1<X1F')\\$y&F9$\"1.HTD(*pc6F
\\t7$$\"1Rk$G0yV'fF9$\"1)Gn0hvG>\"F\\t7$$\"1d$)yZa]dhF9$\"1rwb*3,:B\"F
\\t7$$\"18'*)>d2DL'F9$\"1BzR9:]m7F\\t7$$\"1]*fy6JQ_'F9$\"1!*>dBiw/8F\\
t7$$\"1^8Lk;&Qq'F9$\"1qi'GLq2M\"F\\t7$$\"1]m&f+D?*oF9$\"1I8>,]Sy8F\\t7
$$\"1x2NU))*f2(F9$\"1b,Zo(*>:9F\\t7$$\"1ewC*R-'osF9$\"1K&\\)z/s`9F\\t7
$$\"1*3:g^-TX(F9$\"1=I?.0#3\\\"F\\t7$$\"1,`yxJ!Qk(F9$\"1gqbN1wG:F\\t7$
$\"1f#)HAI$>$yF9$\"1_'fWg'Qm:F\\t7$$\"1be*4\"[![+)F9$\"1r\"*>i4'4g\"F
\\t7$$\"1ZYT*QNH?)F9$\"1HH)y2(eS;F\\t7$$\"1)=%oSd9!Q)F9$\"1Qo8[\"Hgn\"
F\\t7$$\"1V-HO*)3p&)F9$\"1\\!esy<Qr\"F\\t7$$\"1mP4$oJ*\\()F9$\"1`(=mL'
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#fGD;F\\t7$$\"1F')z\"p*\\v6F\\t$\"1bK)[V&3=;F\\t7$$\"1-u(>'RF'=\"F\\t$
\"11v$)>J?5;F\\t7$$\"1s^LP[S(>\"F\\t$\"1I/[(3V>g\"F\\t7$$\"13(3tA%H37F
\\t$\"1q>82cu$f\"F\\t7$$\"10,.s-$)=7F\\t$\"1$>'>5Eq&e\"F\\t7$$\"1nnM`(
G0B\"F\\t$\"1]L-!oTmd\"F\\t7$$\"1j27f./T7F\\t$\"16,Cd4Qo:F\\t7$$\"1L=%
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1<r/\\Bbt7F\\t$\"1]f]2u4U:F\\t7$$\"1\"**4?/9SG\"F\\t$\"1-xH:vRL:F\\t7$
$\"1,gbD&\\\\H\"F\\t$\"1fpz%ytT_\"F\\t7$$\"1Qmaf4k08F\\t$\"1D$)G3`-::F
\\t7$$\"1.V5+R$oJ\"F\\t$\"10qcsnI0:F\\t7$$\"1uQ<KShF8F\\t$\"1,8N-y!e\\
\"F\\t7$$\"1&)RAu#Q'Q8F\\t$\"1$fCm0]f[\"F\\t7$$\"1,R)*H7d\\8F\\t$\"14o
^1w-w9F\\t7$$\"1/W(3[<'f8F\\t$\"1#o,y))ynY\"F\\t7$$\"1_P6=;8r8F\\t$\"1
()*>0K@gX\"F\\t7$$\"1];s***H9Q\"F\\t$\"1))R$4=aiW\"F\\t7$$\"1t<ZA-T#R
\"F\\t$\"1y[^>eoN9F\\t7$$\"1%\\)o'p>HS\"F\\t$\"1/cNZzTD9F\\t7$Ffdl$\"1
!>YFc8UT\"F\\tFfzFa[l-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%
!G-%%VIEWG6$;F($\"+iN@99!\"*%(DEFAULTG" 2 375 375 375 2 0 1 0 2 9 0 4 
1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 
0 0 0 0 3840 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
Int(Int(1,y = x .. 2*x),x=0..2/sqrt(5));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$IntG6$-F$6$\"\"\"/%\"yG;%\"xG,$F,\"\"#/F,;\"\"!,$*$\"\"&#F(F.
#F.F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 54 "Int(Int(1,y = x .. sqrt(4-x^2)),x=2/sqrt(5)..sqrt(
2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$\"\"\"/%\"yG;%\"
xG*$,&\"\"%F(*$F,\"\"#!\"\"#F(F1/F,;,$*$\"\"&F3#F1F8*$F1F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%#PiG#\"\"\"\"\"##!\"#\"\"&F&-%'arcsinG6#,$*$F*F%#F&F
*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 95 "Int(Int(1,y = x .. 2*x),x=0..2/sqrt(5)) +Int
(Int(1,y = x .. sqrt(4-x^2)),x=2/sqrt(5)..sqrt(2));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&-%$IntG6$-F%6$\"\"\"/%\"yG;%\"xG,$F-\"\"#/F-;\"\"!,
$*$\"\"&#F)F/#F/F5F)-F%6$-F%6$F)/F+;F-*$,&\"\"%F)*$F-F/!\"\"F6/F-;F3*$
F/F6F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)36]V'!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "Int(Int( r, r = 0 .. 2),theta=Pi/4..Pi/3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$%\"rG/F(;\"\"!\"\"#/%&th
etaG;,$%#PiG#\"\"\"\"\"%,$F1#F3\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "value(\"); evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%#PiG#\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ex)fB
&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(\");  #Verifi
cation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!Ry2P\"!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(Int(arctan(y/x),x = y .. sqrt(1
-x^2)),y=0..1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F
$6$-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"/F.;F,*$,&F-F-*$F.\"\"#F/#F-F5/F
,;\"\"!,$*$F5F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(
\");" }}{PARA 8 "" 1 "" {TEXT -1 50 "Error, (in evalf/int) unable to h
andle singularity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 345 14 "15. Integrate " }{TEXT 451 1 " " }{XPPEDIT 452 1 "1
2*x" "*&\"#7\"\"\"%\"xGF$" }{TEXT 450 73 "  over the solid region in t
he first octant that is bounded by the plane " }{TEXT 454 1 " " }
{XPPEDIT 455 1 "x+2*y+z=2" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%F%%\"zGF%F'
" }{TEXT 453 1 "." }{TEXT 346 3 "  \n" }{TEXT 344 140 "\na)  1        \+
   b)  2            c)  3           d)  4           e) 5 \nf)   8     \+
     g)   12         h)  15          i)  16          j)  " }{TEXT 343 
3 "20\n" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 520 14 "16. Integrate " }{TEXT 523 1 " " }{XPPEDIT 524 1 "12*x" 
"*&\"#7\"\"\"%\"xGF$" }{TEXT 522 73 "  over the solid region in the fi
rst octant that is bounded by the plane " }{TEXT 526 1 " " }{XPPEDIT 
527 1 "x+2*y+z=2" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%F%%\"zGF%F'" }{TEXT 
525 1 "." }{TEXT 521 3 "  \n" }{TEXT 519 140 "\na)  1           b)  2 \+
           c)  3           d)  4           e) 5 \nf)   8          g)  \+
 12         h)  15          i)  16          j)  " }{TEXT 518 3 "20\n" 
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 530 14 "17. Integrate " }{TEXT 533 1 " \+
" }{XPPEDIT 534 1 "12*x" "*&\"#7\"\"\"%\"xGF$" }{TEXT 532 73 "  over t
he solid region in the first octant that is bounded by the plane " }
{TEXT 536 1 " " }{XPPEDIT 537 1 "x+2*y+z=2" "/,(%\"xG\"\"\"*&\"\"#F%%
\"yGF%F%%\"zGF%F'" }{TEXT 535 1 "." }{TEXT 531 3 "  \n" }{TEXT 529 
140 "\na)  1           b)  2            c)  3           d)  4         \+
  e) 5 \nf)   8          g)   12         h)  15          i)  16       \+
   j)  " }{TEXT 528 3 "20\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 540 14 "18. Integrate " }{TEXT 543 1 " " }
{XPPEDIT 544 1 "12*x" "*&\"#7\"\"\"%\"xGF$" }{TEXT 542 73 "  over the \+
solid region in the first octant that is bounded by the plane " }
{TEXT 546 1 " " }{XPPEDIT 547 1 "x+2*y+z=2" "/,(%\"xG\"\"\"*&\"\"#F%%
\"yGF%F%%\"zGF%F'" }{TEXT 545 1 "." }{TEXT 541 3 "  \n" }{TEXT 539 
140 "\na)  1           b)  2            c)  3           d)  4         \+
  e) 5 \nf)   8          g)   12         h)  15          i)  16       \+
   j)  " }{TEXT 538 3 "20\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 264 "" 0 "" {TEXT 552 63 "19. This problem and the next concer
n the conversion of \n\n     " }}{PARA 3 "" 0 "" {TEXT 553 59 "into a \+
single integral in polar coordinates of the form .  " }{TEXT 570 1 "\n
" }{TEXT 551 140 "\na)  1           b)  2            c)  3           d
)  4           e) 5 \nf)   8          g)   12         h)  15          \+
i)  16          j)  " }{TEXT 550 3 "20\n" }}{PARA 3 "" 0 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 562 14 "20. Integrate " }{TEXT 
565 1 " " }{XPPEDIT 566 1 "12*x" "*&\"#7\"\"\"%\"xGF$" }{TEXT 564 73 "
  over the solid region in the first octant that is bounded by the pla
ne " }{TEXT 568 1 " " }{XPPEDIT 569 1 "x+2*y+z=2" "/,(%\"xG\"\"\"*&\"
\"#F%%\"yGF%F%%\"zGF%F'" }{TEXT 567 1 "." }{TEXT 563 3 "  \n" }{TEXT 
561 140 "\na)  1           b)  2            c)  3           d)  4     \+
      e) 5 \nf)   8          g)   12         h)  15          i)  16   \+
       j)  " }{TEXT 560 3 "20\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}}
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