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0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "T
imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 
2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 3 260 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 "" 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 "" 3 263 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 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 418 33 "Math  233  Spring 2004 F
inal Exam" }}{PARA 0 "" 0 "" {TEXT 256 2 "  " }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT 258 1 "1" }{TEXT 267 2 ". " }{TEXT 450 43 "A line that is per
pendicular to the plane  " }{XPPEDIT 455 1 "4*x+3*y+2*z=13" "/,(*&\"\"
%\"\"\"%\"xGF&F&*&\"\"$F&%\"yGF&F&*&\"\"#F&%\"zGF&F&\"#8" }{TEXT 452 
33 "   has symmetric equations \n\n    " }{TEXT 460 2 "  " }{TEXT 657 
5 "     " }{XPPEDIT 461 1 "(4*x+1)/a = (3*y-2)/b" "/*&,&*&\"\"%\"\"\"%
\"xGF'F'F'F'F'%\"aG!\"\"*&,&*&\"\"$F'%\"yGF'F'\"\"#F*F'%\"bGF*" }
{TEXT 458 2 "  " }{XPPEDIT 462 1 "` `=4*z" "/%\"~G*&\"\"%\"\"\"%\"zGF&
" }{TEXT 459 1 "\n" }{TEXT 463 14 "\n     What is " }{TEXT 451 6 "  a?
\n\n" }{TEXT 457 142 "a)  1            b)  2           c)  3         d
)  4          e)   5  \nf)   6           g)   7          h)  8        \+
  i)   9          j)   10" }{TEXT 456 1 "\n" }}{PARA 3 "" 0 "" {TEXT 
453 8 "Solution" }{TEXT 454 6 ": (b)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 8 "2. If   " }{XPPEDIT 19 1 "`
<`*14 , -2*`>`=`<`*a,b*`>`+`<`*c,d*`>`" "6&*&%\"<G\"\"\"\"#9F%/,$*&\"
\"#F%%\">GF%!\"\"*&F$F%%\"aGF%,&*&%\"bGF%F+F%F%*&F$F%%\"cGF%F%*&%\"dGF
%F+F%" }{TEXT -1 9 "   where " }{XPPEDIT 19 1 "`<`*a,b*`>`" "6$*&%\"<G
\"\"\"%\"aGF%*&%\"bGF%%\">GF%" }{TEXT -1 18 "   is parallel to " }
{XPPEDIT 19 1 "`<`*3,1*`>`" "6$*&%\"<G\"\"\"\"\"$F%*&F%F%%\">GF%" }
{TEXT -1 5 " and " }{XPPEDIT 19 1 "`<`*c,d*`>`" "6$*&%\"<G\"\"\"%\"cGF
%*&%\"dGF%%\">GF%" }{TEXT -1 28 "  is perpendicular \n    to  " }
{XPPEDIT 19 1 "`<`*3,1*`>`" "6$*&%\"<G\"\"\"\"\"$F%*&F%F%%\">GF%" }
{TEXT -1 16 ",  then what is " }{TEXT 471 1 "b" }{TEXT -1 1 "?" }
{TEXT 265 2 "\n\n" }{TEXT 470 142 "a)  1            b)  2           c)
  3         d)  4          e)   5  \nf)   6           g)   7          \+
h)  8          i)   9          j)   10" }{TEXT 469 3 "\n \n" }}{PARA 
3 "" 0 "" {TEXT 419 8 "Solution" }{TEXT 420 6 ": (d)\n" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT 257 32 "3. A curve is parameterized by  " }{XPPEDIT 464 
1 "r(t)=`<`*t,t^3/3,t^2*`>`" "6%/-%\"rG6#%\"tG*&%\"<G\"\"\"F'F**&F'\"
\"$F,!\"\"*&F'\"\"#%\">GF*" }{TEXT 465 51 " .  Calculate its principal
 unit normal vector \n   " }{XPPEDIT 468 1 "N=`<`*a,b,c*`>`" "6%/%\"NG
*&%\"<G\"\"\"%\"aGF'%\"bG*&%\"cGF'%\">GF'" }{TEXT 472 40 "   at the po
int (1, 1/3, 1).  What is   " }{XPPEDIT 467 1 "a" "I\"aG6\"" }{TEXT 
466 3 "?\n " }{TEXT 424 1 "\n" }{TEXT 425 7 "a)     " }{XPPEDIT 540 1 
"1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT 526 20 "                b)  " }
{XPPEDIT 541 1 "-1/3" ",$*&\"\"\"F$\"\"$!\"\"F&" }{TEXT 527 17 "      \+
      c)   " }{XPPEDIT 542 1 "1/sqrt(2)" "*&\"\"\"F#-%%sqrtG6#\"\"#!\"
\"" }{TEXT 528 17 "            d)   " }{XPPEDIT 543 1 "-1/sqrt(2)" ",$
*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"F)" }{TEXT 531 1 " " }{TEXT 529 2 "  " }
{TEXT 530 10 "       e) " }{TEXT 532 3 "   " }{TEXT 533 1 " " }
{XPPEDIT 544 1 "1/sqrt(3)" "*&\"\"\"F#-%%sqrtG6#\"\"$!\"\"" }{TEXT 
534 8 "      \n " }{TEXT 422 5 "f)   " }{TEXT 521 1 " " }{XPPEDIT 545 
1 "-1/sqrt(3)" ",$*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"F)" }{TEXT 522 7 "    \+
   " }{TEXT 523 7 "   g)  " }{XPPEDIT 546 1 "1/sqrt(6)" "*&\"\"\"F#-%%
sqrtG6#\"\"'!\"\"" }{TEXT 524 8 "        " }{TEXT 525 3 "   " }{TEXT 
520 1 " " }{TEXT 423 3 "h) " }{XPPEDIT 547 1 "-1/sqrt(6)" ",$*&\"\"\"F
$-%%sqrtG6#\"\"'!\"\"F)" }{TEXT 535 7 "       " }{TEXT 536 12 "     i)
     " }{XPPEDIT 548 1 "2/sqrt(6)" "*&\"\"#\"\"\"-%%sqrtG6#\"\"'!\"\"
" }{TEXT 537 6 "      " }{TEXT 538 11 "      j)   " }{XPPEDIT 549 1 "-
2/sqrt(6)" ",$*&\"\"#\"\"\"-%%sqrtG6#\"\"'!\"\"F*" }{TEXT 539 6 "     \+
 " }}{PARA 3 "" 0 "" {TEXT 421 1 " " }}{PARA 3 "" 0 "" {TEXT 426 8 "So
lution" }{TEXT 427 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 473 17 "4. Suppose that \+
 " }{XPPEDIT 474 1 "x^2 +y*z-z^3=2*x*z-9" "/,(*$%\"xG\"\"#\"\"\"*&%\"y
GF'%\"zGF'F'*$F*\"\"$!\"\",&*(F&F'F%F'F*F'F'\"\"*F-" }{TEXT 475 16 " .
  Calculate   " }{XPPEDIT 476 1 "diff( z ,x)" "-%%diffG6$%\"zG%\"xG" }
{TEXT 477 8 "   at   " }{XPPEDIT 478 1 "``(3,1,2)" "-%!G6%\"\"$\"\"\"
\"\"#" }{TEXT 479 202 ".\n\na)  3/8               b)  4/15            \+
 c)  2/9             d)  3/7             e)  5/13               \nf)  \+
 2/17             g)  2/11             h) 3/19            i)  4/17    \+
         j)  8/21 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT 480 8 "Solution" }{TEXT 481 6 ": (f)\n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn := x^2 +y*z-z^
3=2*x*z-9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*$%\"xG\"\"#\"
\"\"*&%\"yGF*%\"zGF*F**$F-\"\"$!\"\",&*&F(F*F-F*F)!\"*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := (x,y,z) -> lhs(eqn)-rhs(eqn);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6%%\"xG%\"yG%\"zG6\"6$%)ope
ratorG%&arrowGF*,&-%$lhsG6#%$eqnG\"\"\"-%$rhsGF1!\"\"F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expr := - diff(F(x,y,z),x)/diff(F(x
,y,z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,$*&,&%\"xG\"\"#%
\"zG!\"#\"\"\",(%\"yGF,*$F*F)!\"$F(F+!\"\"F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "subs(\{x=3,y=1,z=2\},expr);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 
"" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 482 
2 "5." }{TEXT -1 1 " " }{TEXT 283 15 "The function   " }{XPPEDIT 282 
1 "f(x,y)=3*x^2*y- x*y^2- 9*x*y" "/-%\"fG6$%\"xG%\"yG,(*(\"\"$\"\"\"*$
F&\"\"#F+F'F+F+*&F&F+*$F'F-F+!\"\"*(\"\"*F+F&F+F'F+F0" }{TEXT 281 133 
"  has exactly one critical point P = (a,b) that does not lie on a coo
rdinate axis.  Choose the ordered list [b,what] where b is the  " }
{TEXT 655 1 "y" }{TEXT 656 82 "-coordinate of the critical point P = (
a,b) and \"what\" describes the behavior of  " }{XPPEDIT 285 1 "f" "I
\"fG6\"" }{TEXT 284 12 "  at P. \n\na)" }{TEXT 286 145 " [-1,local min
imum]         b) [-2,local minimum]          c) [-3,local minimum]    \+
      \nd)  [-1,local maximum]       e) [-2,local maximum]    " }
{TEXT 289 4 "    " }{TEXT 292 3 " f)" }{TEXT 293 1 " " }{TEXT 290 18 "
[-3,local maximum]" }{TEXT 291 1 " " }{TEXT 287 95 "       \ng)  [-1,s
addle point]            h) [-2,saddle point]              i) [-3,saddl
e point]" }{TEXT 288 5 "    \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 3 "" 0 "" {TEXT 439 8 "Solution" }{TEXT 440 6 ": (f)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := (x,y) -> 3*x^2*y - x*y^
2- 9*x*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%
)operatorG%&arrowGF),(*&9$\"\"#9%\"\"\"\"\"$*&F/F2F1F0!\"\"*&F/F2F1F2!
\"*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x
,y),x) = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&<$/%\"yG\"\"!/%\"xGF&<$/F(\"\"$F$<$/F%!\"*F'<$/F(\"\"\"
/F%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(f(x,y),x$2)
*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&%\"xG\"\"\"%\"yGF&!#7*$,(F%\"\"'F'!\"#!\"*F&\"\"#!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=1,y=-3\},\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "subs(\{x=1,y=-3\},diff(f(x,y),x$2));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#!#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "
f(1,-3);  f(.99,-3.1); f(.99,-2.9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'<#**)!\"&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"'B')*)!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 
" " }{TEXT 444 8 "6. Let  " }{XPPEDIT 511 1 "u" "I\"uG6\"" }{TEXT 508 
43 "  be the upward unit normal of the plane   " }{XPPEDIT 510 1 "x-2*
y+2*z=7" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%!\"\"*&F'F%%\"zGF%F%\"\"(" }
{TEXT 509 46 ".  Calculate the directional derivative of    " }
{XPPEDIT 517 1 "F(x,y,z) = x^3+3*y*z+z^2" "/-%\"FG6%%\"xG%\"yG%\"zG,(*
$F&\"\"$\"\"\"*(F+F,F'F,F(F,F,*$F(\"\"#F," }{TEXT 512 25 "   in the di
rection of   " }{XPPEDIT 519 1 "u" "I\"uG6\"" }{TEXT 518 28 "  at the \+
point  ( 2, 2, 3). " }{TEXT 513 2 "\n\n" }{TEXT 515 142 "a)  1        \+
    b)  2           c)  3         d)  4          e)   5  \nf)   6     \+
      g)   7          h)  8          i)   9          j)   10" }{TEXT 
514 1 "\n" }{TEXT 516 7 "      \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 3 "" 0 "" {TEXT 445 8 "Solution" }{TEXT 446 6 ": (f)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "F := (x,y,z) -> x^3+3*y*z+z^2;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"FG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*
,(*$9$\"\"$\"\"\"*&9%F29&F2F1*$F5\"\"#F2F*F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "u := [1/3,-2/3,2/3];  v := grad(F(x,y,z),[x,y,z]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%#\"\"\"\"\"$#!\"#F(#\"\"
#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%,$*$%\"xG\"
\"#\"\"$,$%\"zGF-,&%\"yGF-F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "dp := dotprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dpG,
(*$%\"xG\"\"#\"\"\"%\"zG#!\"#\"\"$%\"yGF(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "subs(\{x=2,y=2,z=3\}, dp );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 8 "7.  Let " }{TEXT 488 1 " \+
" }{TEXT 485 1 "(" }{TEXT 483 7 " a , b " }{TEXT 484 48 ")   be the po
int in the first quadrant at which " }{TEXT 272 1 " " }{XPPEDIT 268 1 
"f(x,y)=x*y+1" "/-%\"fG6$%\"xG%\"yG,&*&F&\"\"\"F'F*F*F*F*" }{TEXT 269 
1 " " }{TEXT 270 46 " has a maximum value subject to the constraint" }
{TEXT 271 3 "   " }{XPPEDIT 486 1 "x^2+2*y^2=1" "/,&*$%\"xG\"\"#\"\"\"
*&F&F'*$%\"yGF&F'F'F'" }{TEXT 487 3 ".  " }{TEXT 428 10 "Calculate " }
{TEXT 489 1 "a" }{TEXT 490 6 "  and " }{TEXT 491 1 "b" }{TEXT 492 25 "
.  What is the value of  " }{XPPEDIT 494 1 "a^2+b^2" ",&*$%\"aG\"\"#\"
\"\"*$%\"bGF%F&" }{TEXT 493 4 " ?\n\n" }{TEXT 273 4 "a)  " }{XPPEDIT 
497 1 "1/8" "*&\"\"\"F#\"\")!\"\"" }{TEXT 429 15 "            b) " }
{XPPEDIT 498 1 "1/4" "*&\"\"\"F#\"\"%!\"\"" }{TEXT 430 22 "           \+
      c)   " }{XPPEDIT 499 1 "3/8" "*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 
431 26 "                      d)  " }{XPPEDIT 500 1 "1/2" "*&\"\"\"F#
\"\"#!\"\"" }{TEXT 432 19 "               e)  " }{XPPEDIT 501 1 "5/8" 
"*&\"\"&\"\"\"\"\")!\"\"" }{TEXT 433 11 "     \n f)  " }{XPPEDIT 502 
1 "3/4" "*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 434 15 "            g) " }
{XPPEDIT 503 1 "7/8" "*&\"\"(\"\"\"\"\")!\"\"" }{TEXT 435 20 "        \+
         h) " }{XPPEDIT 504 1 "3/16" "*&\"\"$\"\"\"\"#;!\"\"" }{TEXT 
436 26 "                      i)  " }{XPPEDIT 505 1 "5/16" "*&\"\"&\"
\"\"\"#;!\"\"" }{TEXT 495 18 "              j)  " }{XPPEDIT 506 1 "7/1
6" "*&\"\"(\"\"\"\"#;!\"\"" }{TEXT 496 3 "   " }{TEXT 507 3 "   " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 437 8 "Solution
" }{TEXT 438 6 ": (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f
 := (x,y) -> x*y+1:  phi := (x,y) -> x^2 + 2*y^2:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x,y),x) = lambda*diff(phi(x,y
),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"yG,$*&%'lambdaG\"
\"\"%\"xGF*\"\"#" }}}{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/%\"xG,$*&%'lambdaG\"\"\"%\"yGF*\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x,y) = 1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"#\"\"\"*$%\"yGF)
F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,
eqn3\}, \{x,y,lambda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG-%
'RootOfG6#,&!\"\"\"\"\"*$%#_ZG\"\"#F./%\"yG#F+F./%'lambdaG,$F&F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c := 1/sqrt(2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 98 "levelCurve[1] := implicitplot( f(x,y)=f(c,1/2)
, x = -3/2..3/2,y=-3/2..3/2,color=pink,thickness=2):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 94 "levelCurve[2] := implicitplot( x^2+2*y^2=
1, x = -3/2..3/2,y=-3/2..3/2,color=plum,thickness=2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "levelCurve[3] := contourplot(f(x,y
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0.7)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(seq(level
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45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -1 -1 
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{TEXT -1 1 " " }{TEXT 447 33 "8.  Calculate  the integral of   " }
{XPPEDIT 554 1 "x+2*y" ",&%\"xG\"\"\"*&\"\"#F$%\"yGF$F$" }{TEXT 550 
96 "  over the region in the first quadrant that is bounded by the coo
rdinate axes and the circle   " }{XPPEDIT 555 1 "x^2+y^2=4" "/,&*$%\"x
G\"\"#\"\"\"*$%\"yGF&F'\"\"%" }{TEXT 551 3 "  ." }{TEXT 552 1 "\n" }
{TEXT 553 145 "\na)  1            b)  2           c)  3         d)  4 \+
         e)   5  \nf)   6            g)   7          h)  8          i)
   9          j)   10\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 
0 "" {TEXT 448 8 "Solution" }{TEXT 449 6 ": (h)\n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int(int( (r*
cos(t)+2*r*sin(t))*r,r=0..2),t=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 
15 "9. The vector  " }{XPPEDIT 274 1 "`<`*a,-3,-1*`>` " "6%*&%\"<G\"\"
\"%\"aGF%,$\"\"$!\"\",$*&F%F%%\">GF%F)" }{TEXT 275 30 "  is tangent to
 the surface   " }{XPPEDIT 276 1 "x^2+2*y^3-3*z^2=3" "/,(*$%\"xG\"\"#
\"\"\"*&F&F'*$%\"yG\"\"$F'F'*&F+F'*$%\"zGF&F'!\"\"F+" }{TEXT 277 36 " \+
at the point (2,1,1).  What is  a ?" }{TEXT 294 2 " \n" }{TEXT 556 
144 "\na)  1            b)  2           c)  3         d)  4          e
)   5  \nf)   6           g)   7          h)  8          i)   9       \+
   j)   10\n" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 298 1 " " }{TEXT 441 8 "Solution" }{TEXT 442 5 ": (c)" }{TEXT 
443 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := (x,y,z) ->
 x^2+2*y^3-3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6%%\"xG%\"
yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"\"*$9%\"\"$F1*$9&F1!\"
$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(F(x,y,z),[x,y
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$%\"yGF)\"\"',$%\"zG!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"v := subs(\{x=2,y=1,z=1\}, \" );  #Normal to surface" }}{PARA 11 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "solve(dotprod(v,[a,-3,-1])=0,a); \n
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{PARA 3 "" 0 "" {TEXT 621 27 "10. Calculate\n     \n       " }{BITMAP 
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:B:vYyA::::::::ra>:<::rYvyy=::y]ykYkywI<:;ZYlyyIZyL:rYvyy=B:vQxyy;::Z:
B:xiy=::::J:vy:jywY::Z:J:ryxI::J:B:vYyA::::::::ri>:<:::y=yA::xCy]::xC;
:vyy=ri:ry:xEryxI:::<Z:Zy=::::J:vy:jywY::Z:J:ryxI::J:B:vYyA:::::::::::
:::::::;Z:jywY:>:<:yqy;jIsYry:::<Z:Zy=::::J:vyy[y=::B::yA::J::xI::::::
::::::::::::vI<:y_yyY:;Z:jywY:ZykyryyA::<Z:Zy=::::jyrYvyy=::rYvIxyy;::
vIxAyyy::::::::::::::::::jIC:xyIsyyAxE;Z:jIky;ZYL:rYvQxyy;Z:B:xiy=::::
jyrYvyy=::rYvIxyy;::vIxAyyy::::::::::::::::::jYC:xY:ryxyyB:>::jywYvMJ:
:ryxI:<Z:Zy=::::jyravyy=::rY:yA::jy:xI:::::::::::::::::Zy<yI:Zyuy:::::
:::B:<:xI:::::y=ry:::xAjy;::vIZy=::::::::::::::::::::::::::::B:Zy=::::
jyrivyy=::rY:yA::jy:xI::::::::::::::::::::::::::::Z::xI:::::y^ylIC:xyy
ryyA:xCy?>:Zy=:y^y<<:jy;::::::::::::::::::::::::::Z:B:xiy=:::::<ZyA::x
iy=:>:<::yqy;:<:;:Zyuy:::::::::::::::::::::::::::B:<:xI:::::ri>:yI:vMx
yy;:vQJ:rary::xEB:vMvY:::::::::::::::::::::::::::B:<Zyuy::::::::::::::
::::::::::::::::::::::::::::::Z::xI:::::::::::::::::::::::::::::::::::
:::::::::B:Zy=::::::::::::::::::::::::::::::::::::::::::::<:ry::::::::
::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::::::::::::::::
:::::::::::::::B:Zy=::::::::::::::::::::::::::::::::::::::::::::<:ry::
::::::::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::::::::::
:::::::::::::::::::::B:Zy=::::::::::::::::::::::::::::::::::::::::::::
<:ry::::::::::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::::
:::::::::::::::::::::::::::B:3:" }{TEXT 636 7 "\nwhere " }{TEXT 634 1 
"U" }{TEXT 635 65 "  is the solid half-ball that is described by the i
nequalities   " }{XPPEDIT 639 1 "x^2+y^2+z^2<=1" "1,(*$%\"xG\"\"#\"\"
\"*$%\"yGF&F'*$%\"zGF&F'F'" }{TEXT 638 9 "   and   " }{XPPEDIT 640 1 "
z>=0" "1\"\"!%\"zG" }{TEXT 637 7 ". \na)  " }{XPPEDIT 645 1 "0" "\"\"!
" }{TEXT 622 15 "           b)  " }{XPPEDIT 646 1 "Pi/12" "*&%#PiG\"\"
\"\"#7!\"\"" }{TEXT 623 19 "               c)  " }{XPPEDIT 647 1 "Pi/8
" "*&%#PiG\"\"\"\"\")!\"\"" }{TEXT 624 19 "               d)  " }
{XPPEDIT 648 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 625 18 "       \+
        e) " }{XPPEDIT 649 1 "Pi/4" "*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 
626 11 "     \nf)   " }{XPPEDIT 650 1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"
" }{TEXT 627 19 "              g)   " }{XPPEDIT 651 1 "Pi/2" "*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT 628 17 "             h)  " }{XPPEDIT 652 1 "P
i" "I#PiG%*protectedG" }{TEXT 629 18 "              i)  " }{XPPEDIT 
653 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 630 18 "              j)  \+
" }{XPPEDIT 654 1 "3*Pi" "*&\"\"$\"\"\"%#PiGF$" }{TEXT 631 5 "    \n" 
}}{PARA 3 "" 0 "" {TEXT 632 8 "Solution" }{TEXT 633 6 ": (e)\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "int(int(int(rho*cos(phi)*rho^2*sin(phi),rho=0..1),phi=0..Pi/2),the
ta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 
16 "11. Calculate   " }{XPPEDIT 295 1 "int( y*dx+2*x*dy+(z-x*y+3),z=C.
.``)" "-%$intG6$,(*&%\"yG\"\"\"%#dxGF(F(*(\"\"#F(%\"xGF(%#dyGF(F(,(%\"
zGF(*&F,F(F'F(!\"\"\"\"$F(F(/F/;%\"CG%!G" }{TEXT 296 11 ".   where  " 
}{TEXT 343 1 "C" }{TEXT 344 42 "   is the oriented curve parameterized
 by " }{XPPEDIT 346 1 "r(t)=[3*t,t^2,t^3]" "/-%\"rG6#%\"tG7%*&\"\"$\"
\"\"F&F**$F&\"\"#*$F&F)" }{TEXT 345 7 ",      " }{XPPEDIT 594 1 "0 <=t
" "1\"\"!%\"tG" }{TEXT 593 1 " " }{XPPEDIT 596 1 "``<=1" "1%!G\"\"\"" 
}{TEXT 595 1 " " }{TEXT 597 3 ". \n" }{TEXT 598 152 "\na)  1          \+
  b)   2           c)   3          d)  4          e)   5  \nf)   6    \+
        g)   7           h)   8          i)   9          j)   10\n  " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 341 8 "Solutio
n" }{TEXT 342 6 ": (g)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
r := t -> [3*t,t^2,t^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%
\"tG6\"6$%)operatorG%&arrowGF(7%,$9$\"\"$*$F.\"\"#*$F.F/F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Int( r(t)[2]*diff(r(t)[1],t
)+ 2*r(t)[1]*diff(r(t)[2],t)+(r(t)[3]-r(t)[1]*r(t)[2]+3)*diff(r(t)[3],
t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$%\"tG\"\"#
\"#:*&,&*$F(\"\"$!\"#F.\"\"\"F0F(F)F./F(;\"\"!F0" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 358 38 "12. The vector field  \n\n \+
             " }{TEXT -1 3 "   " }{BITMAP 238 29 29 1 "?TM_TNB:n@>?nWf
=F:k:;::::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:yay=
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::J:Zy=::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::J:Zy=::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::J:Zy=::::::::::::::::::::::xoixIxuYyyyyI:::::::::::::::::::
:::::ZYxmyyqyly;:::::::::::::::::vQxCyyy::::::::jyxqy;xyy;::::::::::::
::::::vI:vY::::::::y^y\\y=:y^:J::yvaxyy;:>:xiy=:::::::::::::::::::::jI
sIysyy]ylIsyyA::::::::::::::::::::::::jIsavyy=::::::::::::::::ravIxyy;
::::::::vIxAy?vyy=::::::::::::::::::y]ykY;xI:::::::ZYlIsyyAJ:B:vQ<:;ZY
\\y=:;Zyuy:::::::::::::::::::::ZykyryyA:xCxI::::::::::::::::::::::::ZY
lIsyyA:::::::::::::::jyrYvyy=:::::::::::y>yA:::::::::::::::::::jIky;::
:::::vMxCyyyjyB:jy=:;:vYyA:;:xI:::::::::::::::::::::y?jy;:jyryyA::::::
::::::::::::::::::J:rivyy=::::::::::::::ri>:xiy=:::::::::::Zy[y=::::::
:::::::::::::rYry::::::::<:y_yyY:B:>:jywY:>:ry>:<:::ryxI::::::::::::::
::J:rYy]y=:ri>:<:y_yyY:::::::::::;Zyuy:::::::y]y\\y=:::y]::yA;Z:::xiy=
:::Z::yA:::::B:vIxyy;ZyK:B:::Zyly:ryyAJ:rYvyy=:y_:J:rivyy=:::::B:vYyA:
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:::::::<:;Zyuy::::::::;:xI::<:y_yyY:::::;Zyuy:>:<:ryxI:jIC:jy;::::::::
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:::xCy^yyYriZ:::jYsyyA::::::::::::::::::::jYsaryjyB:vIxyy;:::::::Z:jyw
YZYlIsyyA::<:;:vYyA:;:xI;Z::::::jywY:::::::::::::::Zy<vY:::::::;:xI:::
:::::::::xE;Z::::xAxiYlYsyyA::xC;ZYlyyI:::::::::::::vQxCxIvQxA;Z:jyriv
yy=:::::::::::::::::::::rivI<:;ZykyyI::::::::Z:jywYvMxEyyy:::xC;ZYlyyI
:>:xiy=::::::::::::::::::::::::::::::J:Zy=::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::Z::yA::::::::;:xI::::::
:::::::::::::::::::::::;:xIJ:Zy=::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::Z::yAB:jy;:::::::J:Zy=:::::::::::::
::::::::::::::::<:;:vYyA::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::;Z::xiy=::::::::>:ry:::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::>:ry:::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::>:ry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:ry:::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::>:1:" }{TEXT -1 1 "\n" }
{TEXT 362 39 "is the gradient field of some function " }{TEXT 363 2 "f
(" }{TEXT 365 3 "x,y" }{TEXT 366 1 ")" }{TEXT 364 13 ".  Calculate " }
{TEXT 361 1 " " }{TEXT 359 15 "f(2,1) - f(0,1)" }{TEXT 360 3 ".\n\n" }
{TEXT 367 79 "a)  2              b)  4             c)   6            d
)  8          e)  10   " }{TEXT 372 4 "  \nf" }{TEXT 368 34 ")  12    \+
        g)  14           " }{TEXT 557 1 " " }{TEXT 371 80 "h)  16     \+
      i) 18          j)  Cannot be calculated from given information.
" }{TEXT -1 2 "  " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT 369 8 "Solution" }{TEXT 370 6 ": (h)\n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "restart: with(linalg):" }}{PARA 7 "" 1 "" {TEXT 
-1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 
33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "F := vector([y^2+7,2*x*y+3*y^2]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7$,&*$%\"yG\"\"#\"\"\"\"\"(F-,&*&%
\"xGF-F+F-F,F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "curl
(vector([y^2+7,2*x*y+3*y^2,0]),[x,y,z]); #F is conservative" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "eqn := u(x,y) = int(y^2+7,x) + g(y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6$%\"xG%\"yG,&*&,&*$F*\"
\"#\"\"\"\"\"(F0F0F)F0F0-%\"gG6#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d
iffG6$-%\"uG6$%\"xG%\"yGF+,&*&F*\"\"\"F+F.\"\"#-F%6$-%\"gG6#F+F+F." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F[2] = rhs(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\"%\"yGF'\"\"#*$F(F)\"\"$,&F%F
)-%%diffG6$-%\"gG6#F(F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "diff(g(y),y) = solve(\",diff(g(y),y));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"gG6#%\"yGF*,$*$F*\"\"#\"\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := subs(g(y)=y^3+C,eqn);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6$%\"xG%\"yG,(*&,&*$F*
\"\"#\"\"\"\"\"(F0F0F)F0F0*$F*\"\"$F0%\"CGF0" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "subs( \{x=2,y=1\}, rhs(eqn2)) - subs(\{x=0,y=1\}
, rhs(eqn2));" }}{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 263 17 "13.  Calculate \+
  " }{XPPEDIT 347 1 "int( (y+z)*dx+(x+1)*dy+(2*z+x),z=C..``)" "-%$intG
6$,(*&,&%\"yG\"\"\"%\"zGF)F)%#dxGF)F)*&,&%\"xGF)F)F)F)%#dyGF)F),&*&\"
\"#F)F*F)F)F.F)F)/F*;%\"CG%!G" }{TEXT 348 11 "    where  " }{TEXT 349 
1 "C" }{TEXT 350 42 "   is the oriented curve parameterized by " }
{XPPEDIT 352 1 "r(t)=[sqrt(t),cos(Pi*sqrt(t)),t*ln(t^2-t+1)]" "/-%\"rG
6#%\"tG7%-%%sqrtG6#F&-%$cosG6#*&%#PiG\"\"\"-F)6#F&F0*&F&F0-%#lnG6#,(*$
F&\"\"#F0F&!\"\"F0F0F0" }{TEXT 351 8 ",       " }{XPPEDIT 353 1 "0 <=t
" "1\"\"!%\"tG" }{TEXT 278 1 " " }{XPPEDIT 559 1 "``<=1" "1%!G\"\"\"" 
}{TEXT 558 1 " " }{TEXT 560 6 ".   \n\n" }{TEXT 266 72 "a)  1         \+
  b)  -1             c)   2          d)  -2           e)  " }{TEXT 
356 6 "3    \n" }{TEXT 279 35 "f)  -3            g)  4            " }
{TEXT 561 1 " " }{TEXT 357 38 "h)  -4           i) 5             j)  \+
" }{TEXT 562 2 "-5" }{TEXT -1 2 "  " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }
}{PARA 3 "" 0 "" {TEXT 354 8 "Solution" }{TEXT 355 6 ": (f)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r := (t) -> [sqrt(t), cos(Pi
*sqrt(t)),t*ln(t^2-t+1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#
%\"tG6\"6$%)operatorG%&arrowGF(7%-%%sqrtG6#9$-%$cosG6#*&%#PiG\"\"\"F-F
6*&F0F6-%#lnG6#,(*$F0\"\"#F6F0!\"\"F6F6F6F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "P := r(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"PG7%\"\"!\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Q :=
 r(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG7%\"\"\"!\"\"\"\"!" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F :=
 vector([y+z,x+1,2*z+x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'
VECTORG6#7%,&%\"yG\"\"\"%\"zGF+,&%\"xGF+F+F+,&F,\"\"#F.F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "curl(F,[x,y,z]); #F is conservative
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := u(x,y,z) = int(y+z,x)
 + g(y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6%%\"xG%\"
yG%\"zG,&*&,&F*\"\"\"F+F/F/F)F/F/-%\"gG6$F*F+F/" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%%diffG6$-%\"uG6%%\"xG%\"yG%\"zGF+,&F*\"\"\"-F%6$-%\"gG6$F+F,F+F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x + 1 = rhs(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"F&F&,&F%F&-%%diffG6$-%\"
gG6$%\"yG%\"zGF.F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "diff(
g(y,z),y) = solve(\",diff(g(y,z),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%%diffG6$-%\"gG6$%\"yG%\"zGF*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "eqn2 := subs(g(y,z)=y+h(z),eqn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/-%\"uG6%%\"xG%\"yG%\"zG,(*&,&F*\"\"\"F+F/F/F)F
/F/F*F/-%\"hG6#F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2*z+
x = diff(rhs(eqn2),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"zG\"\"
#%\"xG\"\"\",&F'F(-%%diffG6$-%\"hG6#F%F%F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "diff(h(z),z) = solve( \" , diff(h(z),z));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"hG6#%\"zGF*,$F*\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn3 := subs(h(z)=z^2 + C, e
qn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"uG6%%\"xG%\"yG%
\"zG,**&,&F*\"\"\"F+F/F/F)F/F/F*F/*$F+\"\"#F/%\"CGF/" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "evalm( grad(rhs(eqn3),[x,y,z]) - F );  #C
heck" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs( \{x=1,y=-1,z=0\}, rhs(
eqn3)) - subs(\{x=0,y=1,z=0\}, rhs(eqn3));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 264 49 "14.  What  is the surface area of the part of    " 
}{XPPEDIT 566 1 "z=1+x^2/2+y^2/2" "/%\"zG,(\"\"\"F%*&%\"xG\"\"#F(!\"\"
F%*&%\"yGF(F(F)F%" }{TEXT 563 35 "   that lies over the unit disk    \+
" }{XPPEDIT 565 1 "x^2+y^2<=1" "1,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'F'" }
{TEXT 564 10 "  ? \n\na)  " }{XPPEDIT 582 1 "sqrt(2)/3*Pi" "*(-%%sqrtG
6#\"\"#\"\"\"\"\"$!\"\"%#PiGF'" }{TEXT 568 22 "                  b)  \+
" }{XPPEDIT 583 1 "(sqrt(2)-1)/3*Pi" "*(,&-%%sqrtG6#\"\"#\"\"\"F(!\"\"
F(\"\"$F)%#PiGF(" }{TEXT 569 11 "        c) " }{XPPEDIT 584 1 "(2-sqrt
(2))/3*Pi" "*(,&\"\"#\"\"\"-%%sqrtG6#F$!\"\"F%\"\"$F)%#PiGF%" }{TEXT 
572 12 "         d) " }{XPPEDIT 585 1 "4*sqrt(2)/3*Pi" "**\"\"%\"\"\"-
%%sqrtG6#\"\"#F$\"\"$!\"\"%#PiGF$" }{TEXT 573 20 "                e)  \+
" }{XPPEDIT 586 1 "2*(2*sqrt(2)-1)*Pi/3" "**\"\"#\"\"\",&*&F#F$-%%sqrt
G6#F#F$F$F$!\"\"F$%#PiGF$\"\"$F*" }{TEXT 570 14 "             \n" }
{TEXT 567 4 "f)  " }{XPPEDIT 587 1 "(4*(sqrt(2)-1))/3*Pi" "**\"\"%\"\"
\",&-%%sqrtG6#\"\"#F$F$!\"\"F$\"\"$F*%#PiGF$" }{TEXT 578 3 "   " }
{TEXT 579 6 "  g)  " }{XPPEDIT 588 1 "2*(4-sqrt(2))/3*Pi" "**\"\"#\"\"
\",&\"\"%F$-%%sqrtG6#F#!\"\"F$\"\"$F*%#PiGF$" }{TEXT 580 3 "   " }
{TEXT 581 8 "   h)   " }{TEXT 642 2 "  " }{XPPEDIT 643 1 "(4-sqrt(2))*
Pi/3" "*(,&\"\"%\"\"\"-%%sqrtG6#\"\"#!\"\"F%%#PiGF%\"\"$F*" }{TEXT 
641 2 "  " }{TEXT 644 6 "   i) " }{TEXT 589 1 " " }{XPPEDIT 590 1 "2*(
2-sqrt(2) )*Pi/3" "**\"\"#\"\"\",&F#F$-%%sqrtG6#F#!\"\"F$%#PiGF$\"\"$F
)" }{TEXT 574 2 "  " }{TEXT 575 7 "   j)  " }{TEXT 591 2 "  " }
{XPPEDIT 592 1 "4*(2*sqrt(2)-1)*Pi/3" "**\"\"%\"\"\",&*&\"\"#F$-%%sqrt
G6#F'F$F$F$!\"\"F$%#PiGF$\"\"$F+" }{TEXT 571 2 "  " }}{PARA 262 "" 0 "
" {TEXT 297 5 "     " }}{PARA 3 "" 0 "" {TEXT 576 8 "Solution" }{TEXT 
577 6 ": (e)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "2*Pi*int(r*(1+r^2)^(1/2),r=0..1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\",&*$\"\"##F&F)#F)\"\"$#!\"\"F,F&F
&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "answers := [sqrt(2)
/3*Pi,(sqrt(2)-1)/3*Pi,(2-sqrt(2))/3*Pi,4*sqrt(2)/3*Pi,2*(2*sqrt(2)-1)
*Pi/3,(4*(sqrt(2)-1))/3*Pi,2*(4-sqrt(2))/3*Pi,(4-sqrt(2))*Pi/3,2*(2-sq
rt(2) )*Pi/3,4*(2*sqrt(2)-1)*Pi/3]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "for i from 1 to 9 do\nfor j from i+1 to 10 do\nif (a
nswers[i]=answers[j]) then print(`repeat`);\nfi;\nod;\nod;" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 41 "int(int(sqrt(1+r^2)*r,r=0..1),t=0..2*Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-F$6$*&,&\"\"\"F**$%\"rG\"\"#
F*#F*F-F,F*/F,;\"\"!F*/%\"tG;F1,$%#PiGF-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "int(r*sqrt(1+r^2),r=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$intG6$*&,&\"\"\"F(*$%\"rG\"\"#F(#F(F+F*F(/F*;\"\"!F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(Int(r*sqrt(1+r^2)
,r=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#3dZ4'!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "testeq(diff((1+r^2)^(3/2)/3,
r) = sqrt(1+r^2)*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "subs(r=1,(1+r^2)^(3/2)/3) - \+
subs(r=0,(1+r^2)^(3/2)/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"
##\"\"\"F%#F%\"\"$#!\"\"F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "evalf( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"3dZ4'!#5" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 280 15 "15.  Calculate " }{XPPEDIT 388 1 "Int(-5*y*dx+11*x,
y=C..``)" "-%$IntG6$,&*(\"\"&\"\"\"%\"yGF(%#dxGF(!\"\"*&\"#6F(%\"xGF(F
(/F);%\"CG%!G" }{TEXT 387 8 "  where " }{TEXT 385 1 "C" }{TEXT 386 48 
" is the unit circle traversed counter-clockwise." }{TEXT 384 2 " \n" 
}{TEXT 373 4 "a)  " }{XPPEDIT 389 1 "0" "\"\"!" }{TEXT 374 22 "       \+
           b)  " }{XPPEDIT 390 1 "Pi" "I#PiG%*protectedG" }{TEXT 375 
23 "                  c)   " }{XPPEDIT 391 1 "2*Pi" "*&\"\"#\"\"\"%#Pi
GF$" }{TEXT 376 20 "                d)  " }{XPPEDIT 392 1 "3*Pi" "*&\"
\"$\"\"\"%#PiGF$" }{TEXT 377 22 "                 e)   " }{XPPEDIT 
393 1 "4*Pi" "*&\"\"%\"\"\"%#PiGF$" }{TEXT 378 11 "     \nf)   " }
{XPPEDIT 394 1 "6*Pi" "*&\"\"'\"\"\"%#PiGF$" }{TEXT 379 18 "          \+
   g)   " }{XPPEDIT 395 1 "8*Pi" "*&\"\")\"\"\"%#PiGF$" }{TEXT 380 19 
"               h)  " }{XPPEDIT 396 1 "12*Pi" "*&\"#7\"\"\"%#PiGF$" }
{TEXT 381 19 "               i)  " }{XPPEDIT 397 1 "16*Pi" "*&\"#;\"\"
\"%#PiGF$" }{TEXT 382 22 "                 j)   " }{XPPEDIT 398 1 "18*
Pi" "*&\"#=\"\"\"%#PiGF$" }{TEXT 383 5 "    \n" }}{PARA 3 "" 0 "" 
{TEXT 399 8 "Solution" }{TEXT 400 6 ": (i)\n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "r := t -> [cos(t),sin(t)]:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "Int( -5*r(t)[2]*diff(r(t)[1],t) + 11*r(t)[1]*d
iff(r(t)[2],t),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6
$,&*$-%$sinG6#%\"tG\"\"#\"\"&*$-%$cosGF*F,\"#6/F+;\"\"!,$%#PiGF," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "More easily by Green's Theorem:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(Int( r*(diff(11*x,x)-diff(-5*y,
y)),r=0..1),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG
6$-F$6$,$%\"rG\"#;/F);\"\"!\"\"\"/%&thetaG;F-,$%#PiG\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 261 "" 0 "" {TEXT -1 9 "16. Let  " }{TEXT 403 1 "G" }{TEXT -1 
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402 1 "S" }{TEXT -1 6 "  of  " }{TEXT 404 1 "G" }{TEXT -1 59 "  is an \+
orientable closed surface with outward unit normal " }{TEXT 401 1 "n" 
}{TEXT -1 46 ". Vector calculus tells us that\n\n             " }
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:::5:" }{TEXT 405 16 " \nthen what is  " }{TEXT 407 2 "f " }{TEXT 409 
13 "( 0 , 1 , 2 )" }{TEXT 410 1 " " }{TEXT 408 3 "?\n\n" }{TEXT 412 
71 "a)  0           b)  1             c)   2          d)  3           \+
e)  4" }{TEXT 414 3 "   " }{TEXT 413 35 "\nf)  5            g)  6     \+
       " }{TEXT -1 1 " " }{TEXT 415 38 "h)  7           i) 8          \+
   j)  9" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "
" 0 "" {TEXT 416 8 "Solution" }{TEXT 417 6 ": (e)\n" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linal
g):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diverge([x*y,2*y*z,x
^2+x^2*y^3-y*z],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"zG\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(\{x=1,y=2,z=3\}
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{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 20 "17. Suppose tha
t    " }{XPPEDIT 333 1 "f(x,y,z)=3*x^2 - 2*y^2 + 3*z^3" "/-%\"fG6%%\"x
G%\"yG%\"zG,(*&\"\"$\"\"\"*$F&\"\"#F,F,*&F.F,*$F'F.F,!\"\"*&F+F,*$F(F+
F,F," }{TEXT 332 14 " .  Calculate " }{TEXT 336 1 " " }{TEXT 334 11 "d
iv(grad(f(" }{TEXT 337 5 "x,y,z" }{TEXT 338 23 ")) at the point (3,2,1
)" }{TEXT 335 2 ". " }{TEXT 301 2 " \n" }{TEXT 299 142 "\na)  1       \+
    b)  2            c)  3           d)  4           e) 5 \nf)   8    \+
      g)   12         h)  15          i)  16          j)  20" }{TEXT 
411 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 
339 8 "Solution" }{TEXT 340 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "f := (x,y,z) -> 3*x^2 - 2*y^2 + 3*z^3;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG
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{MPLTEXT 1 0 23 "grad(f(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(\",[x,y,z]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"%\"zG\"#=" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 302 38 "18. Calcul
ate the surface integral of " }{TEXT 305 1 " " }{XPPEDIT 306 1 "f(x,y,
z)=2*x+z" "/-%\"fG6%%\"xG%\"yG%\"zG,&*&\"\"#\"\"\"F&F,F,F(F," }{TEXT 
304 31 "  over that part of  the plane " }{TEXT 307 1 " " }{XPPEDIT 
308 1 "2*x+2*y+z=2" "/,(*&\"\"#\"\"\"%\"xGF&F&*&F%F&%\"yGF&F&%\"zGF&F%
" }{TEXT 303 4 "   \n" }{TEXT 328 30 "that lies in the first octant." 
}{TEXT 329 1 "\n" }{TEXT 600 1 "\n" }{TEXT 599 144 "\na)  1           \+
b)  2            c)  3           d)  4          e)  5 \nf)   6        \+
   g)  7           h)  8            i)  9           j)  10" }{TEXT 
601 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 
330 8 "Solution" }{TEXT 331 6 ": (b)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ve
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1 "" {XPPMATH 20 "6#>%-vectorLengthG:6#%\"vG6\"6$%)operatorG%&arrowGF(
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:= map(z->diff(z,x),r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rxG
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0 31 "ry := map(z->diff(z,y),r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#ryG-%'VECTORG6#7%\"\"!\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "crossprod(rx,ry);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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37 "surfaceAreaFactor := vectorLength(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%2surfaceAreaFactorG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "f := (x,y,z) -> 2*x + z;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&9$
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{PARA 3 "" 0 "" {TEXT 310 9 "19. Let  " }{TEXT 602 3 "U  " }{TEXT 603 
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the surface of that part of the cube that lies above the  " }{XPPEDIT 
605 1 "xy" "I#xyG6\"" }{TEXT 604 27 " plane.  (In other words,  " }
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ttom.)   Let  " }{TEXT 319 1 "n" }{TEXT 320 46 "  denote the outward-p
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610 4 "  \n\n" }{TEXT 311 1 "a" }{TEXT 309 154 ")  1             b)  -
1             c)  2            d)  -2           e)  3 \nf)   -3       \+
    g)   4             h)  -4            i)  5             j)  " }
{TEXT 321 5 "-5\n  " }}{PARA 3 "" 0 "" {TEXT 322 8 "Solution" }{TEXT 
323 6 ": (a)\n" }}{PARA 0 "" 0 "" {TEXT -1 91 "Using Stokes's Theorem \+
the required surface integral can be calculated by the line integral" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }
{TEXT 324 5 "     " }{XPPEDIT 325 1 "Int(x*y*z^4*dx+2*x*y*dy+y^3,z)" "
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yGF(F(*$F)\"\"$F(F*" }}{PARA 0 "" 0 "" {TEXT -1 86 "\nintegrated count
erclockwise over the boundary of the square in the xy plane. Since   \+
" }{TEXT 326 10 "z = dz = 0" }{TEXT -1 24 "  on all four sides and " }
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0 "" {MPLTEXT 1 0 16 "int(2*y,y=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 
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{TEXT -1 0 "" }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 312 11 "20.  Let   " 
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ngth one and vertices at ( 0, 0, 0 )  and  ( 1, 1, 1 )  (as in the pre
ceding problem). In this problem, let  " }{TEXT 613 1 "S" }{TEXT 614 
51 "  be the entire surface of the cube. \n\n            " }{BITMAP 
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