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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 "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 
0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 264 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 "" 3 265 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 -1 9 "Math 2331" }{TEXT 462 10 "
 Fall 2002" }}{PARA 18 "" 0 "" {TEXT 461 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 349 1 "diff(f(x,y)
,x)" "-%%diffG6$-%\"fG6$%\"xG%\"yGF(" }{TEXT 347 6 "   at " }{XPPEDIT 
350 1 "P" "I\"PG6\"" }{TEXT 348 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 354 74 "a)  -4           b)  -3             \+
c)   -2         d)  -1           e)  0" }{TEXT 351 1 " " }{TEXT 356 
20 "                    " }}{PARA 0 "" 0 "" {TEXT 352 36 "f)   1      \+
      g)   2            " }{TEXT -1 1 " " }{TEXT 353 39 "h)   3       \+
    i)  4             j)  " }{TEXT -1 1 " " }{TEXT 355 1 "5" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 346 9 "Solution:" }{TEXT 463 4 " (h)" }
{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linal
g):" }}{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 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 := vector([a,b
]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn1 := dotprod(u,gr
adient_f)=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&%\"aG#\"\"&
\"#8%\"bG#\"#7F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eq
n2 := dotprod(v,gradient_f)=18/5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%eqn2G/,&%\"aG#\"\"%\"\"&%\"bG#\"\"$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-%-an
glebracketG6$-%$cosG6#%&alphaG-%$sinG6#F*" }{TEXT -1 6 ", if  " }
{XPPEDIT 19 1 "r(t)=<1,2>+t*v" "/-%\"rG6#%\"tG,&-%-anglebracketG6$\"\"
\"\"\"#F+*&F&F+%\"vGF+F+" }{TEXT -1 10 ", and if  " }{XPPEDIT 19 1 "f(
x,y)=x^2+3*y" "/-%\"fG6$%\"xG%\"yG,&*$F&\"\"#\"\"\"*&\"\"$F+F'F+F+" }
{TEXT -1 27 "   then for what value of  " }{XPPEDIT 19 1 "alpha" "I&al
phaG6\"" }{TEXT -1 8 " does   " }{XPPEDIT 19 1 "t -> f(r(t)" ":6#%\"tG
7\"6$%)operatorG%&arrowG6\"-%\"fG6#-%\"rG6#F$F)F)" }{TEXT -1 54 "  hav
e the greatest instantaneous rate of change at   " }{XPPEDIT 19 1 "t=0
" "/%\"tG\"\"!" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT 282 4 "a)  " }{XPPEDIT 283 1 "arcsin(1/2)" "-%
'arcsinG6#*&\"\"\"F&\"\"#!\"\"" }{TEXT 284 8 "    b)  " }{XPPEDIT 285 
1 "arcsin(1/3)" "-%'arcsinG6#*&\"\"\"F&\"\"$!\"\"" }{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)" "-%'arcsin
G6#*&\"\"%\"\"\"\"\"&!\"\"" }{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 467 1 "
arcsin(3/sqrt(13))" "-%'arcsinG6#*&\"\"$\"\"\"-%%sqrtG6#\"#8!\"\"" }
{TEXT 468 8 "    j)  " }{TEXT -1 1 " " }{XPPEDIT 466 1 "arcsin(5/sqrt(
13))" "-%'arcsinG6#*&\"\"&\"\"\"-%%sqrtG6#\"#8!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 464 9 "Solution:" }{TEXT 465 4 " (i)" }{TEXT -1 1 "\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := (x,y) -> x^2 + 3*y:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "grad(f(x,y),[x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$,$%\"xG\"\"#\"\"$" }}}
{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 469 1 "f" "I\"fG6\"" }
{TEXT 357 14 "  at the point" }{TEXT 305 2 "  " }{XPPEDIT 306 1 "P" "I
\"PG6\"" }{TEXT 304 1 " " }{TEXT 307 13 " is  2.  At  " }{XPPEDIT 471 
1 "P" "I\"PG6\"" }{TEXT 470 21 "  the gradient  of   " }{XPPEDIT 473 
1 "f" "I\"fG6\"" }{TEXT 472 22 "  makes an angle of   " }{XPPEDIT 477 
1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 476 27 "    with the unit ve
ctor   " }{XPPEDIT 475 1 "u" "I\"uG6\"" }{TEXT 474 41 " .  What is the
 directional derivative   " }{XPPEDIT 479 1 "D[u](f)(P)" "--&%\"DG6#%
\"uG6#%\"fG6#%\"PG" }{TEXT 478 3 "  ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 480 3 "a) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqr
tG6#\"\"#" }{TEXT 481 46 "            b)  1/2        c) 3/2          d
) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 482 11 "       e
)  " }{XPPEDIT 19 1 "2*sqrt(2)/3" "*(\"\"#\"\"\"-%%sqrtG6#F#F$\"\"$!\"
\"" }{TEXT 483 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sqrt(3)/3" "*(\"\"#\"
\"\"-%%sqrtG6#\"\"$F$F(!\"\"" }{TEXT 484 43 "      g) 1             h)
 2            i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"
\"F&!\"\"" }{TEXT 485 16 "            j)  " }{XPPEDIT 19 1 "sqrt(3)/2
" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 492 9 "Solutio
n:" }{TEXT 493 4 " (g)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "2*cos(Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
262 1 " " }{TEXT 310 18 "4. The function   " }{XPPEDIT 309 1 "f(x,y)=2
*x^3-24*x*y+16*y^3" "/-%\"fG6$%\"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 359 1 "f" "I\"fG6\"" }
{TEXT 358 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 360 18 "[1,local maximum] " }{TEXT 361 8 "     e) " }
{TEXT 276 21 "[2,local maximum]    " }{TEXT 260 4 "    " }{TEXT 364 3 
" f)" }{TEXT 365 1 " " }{TEXT 362 17 "[3,local maximum]" }{TEXT 363 1 
" " }{TEXT 257 12 "       \ng)  " }{TEXT 366 16 "[1,saddle point]" }
{TEXT 367 16 "             h) " }{TEXT 368 16 "[2,saddle point]" }
{TEXT 369 19 "                i) " }{TEXT 372 16 "[3,saddle point]" }
{TEXT 258 6 "    \n " }{TEXT 373 3 "j) " }{TEXT 374 1 " " }{TEXT 370 
17 "[4,local minimum]" }{TEXT 371 2 " \n" }}{PARA 0 "" 0 "" {TEXT 486 
9 "Solution:" }{TEXT 487 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%<
$/%\"yG\"\"!/%\"xGF&<$/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 47 "subs(\{x=2,y=1\},\"); # positive => local extremum" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"%G<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "subs(\{x=2,y=1\},diff(f(x,y),x$2)); # positive => local min" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
267 2 " 5" }{TEXT -1 2 ". " }{TEXT 383 15 "The function   " }{XPPEDIT 
382 1 "f(x,y)=x^2*y-x*y^2-3*x*y" "/-%\"fG6$%\"xG%\"yG,(*&F&\"\"#F'\"\"
\"F+*&F&F+*$F'F*F+!\"\"*(\"\"$F+F&F+F'F+F." }{TEXT 381 178 "  has one \+
critical point P = (a,b) with ab  <  0.  Choose the ordered list [b,wh
at] where b is the ordinate of the critical point P = (a,b) and \"what
\" describes the behavior of  " }{XPPEDIT 385 1 "f" "I\"fG6\"" }{TEXT 
384 8 "  at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 390 2 "a)" }{TEXT 386 24 " [1,local minimum]      " }{TEXT 393 
2 "b)" }{TEXT 391 56 " [-1,local minimum]        c) [-2,local minimum]
        " }{TEXT 394 7 "  \nd)  " }{TEXT 395 18 "[1,local maximum] " }
{TEXT 396 8 "     e) " }{TEXT 392 22 "[-1,local maximum]    " }{TEXT 
389 4 "    " }{TEXT 399 3 " f)" }{TEXT 400 1 " " }{TEXT 397 18 "[-2,lo
cal maximum]" }{TEXT 398 1 " " }{TEXT 387 12 "       \ng)  " }{TEXT 
401 16 "[1,saddle point]" }{TEXT 402 16 "             h) " }{TEXT 403 
17 "[-1,saddle point]" }{TEXT 404 19 "                i) " }{TEXT 407 
17 "[-2,saddle point]" }{TEXT 388 6 "    \n " }{TEXT 408 3 "j) " }
{TEXT 409 1 " " }{TEXT 405 17 "[2,local maximum]" }{TEXT 406 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 488 9 "Solution:
" }{TEXT 489 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&<$/%\"yG\"\"!/%\"
xGF&<$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$2)-(diff(f(x,y)
,x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"%\"yGF&!\"
%*$,(F%\"\"#F'!\"#!\"$F&F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "subs(\{x=1,y=-1\},\"); # positive => local extremum" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "subs(\{x=1,y=-1\},diff(f(x,y),x$2));  # negative => local max
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 1 " " }{TEXT 264 18 "6. The function   " }{XPPEDIT 411 1 "f(x,y)=x^
3+2*x^2*y+y^2+x+2" "/-%\"fG6$%\"xG%\"yG,,*$F&\"\"$\"\"\"*(\"\"#F+*$F&F
-F+F'F+F+*$F'F-F+F&F+F-F+" }{TEXT 410 165 "  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 413 1 "f" "I\"fG6\"" }{TEXT 412 9 "   at P. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 418 2 "a)" }{TEXT 414 24 
" [0,local minimum]      " }{TEXT 421 2 "b)" }{TEXT 419 54 " [1,local \+
minimum]        c) [2,local minimum]        " }{TEXT 422 7 "  \nd)  " 
}{TEXT 423 18 "[0,local maximum] " }{TEXT 424 8 "     e) " }{TEXT 420 
21 "[1,local maximum]    " }{TEXT 417 4 "    " }{TEXT 427 3 " f)" }
{TEXT 428 1 " " }{TEXT 425 17 "[2,local maximum]" }{TEXT 426 1 " " }
{TEXT 415 12 "       \ng)  " }{TEXT 429 16 "[0,saddle point]" }{TEXT 
430 16 "             h) " }{TEXT 431 16 "[1,saddle point]" }{TEXT 432 
19 "                i) " }{TEXT 435 16 "[2,saddle point]" }{TEXT 416 
6 "    \n " }{TEXT 436 3 "j) " }{TEXT 437 1 " " }{TEXT 433 18 "[-1,loc
al maximum]" }{TEXT 434 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 490 9 "Solution:" }{TEXT 491 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\"\"\"<$/F%,&#F)\"\"%F)-%'RootOfG6#,
(*$%#_ZG\"\"#F.F4F)F)F)F-/F(F/" }}}{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$\"\"#!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(\{y =
 -1, x = 1\},\"); # negative => saddle point" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 
33 "7.  What is the maximum value of " }{TEXT 319 1 " " }{XPPEDIT 313 
1 "f(x,y)=x^2+y" "/-%\"fG6$%\"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 494 9 "Solution:" }{TEXT 495 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/,$%\"xG\"\"#*&%'lambdaG\"\"\"F'F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lam
bda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"
\",$*&%'lambdaGF&%\"yGF&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "eqn3 := phi(x,y) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3
G/,&*$%\"xG\"\"##\"\"\"F)*$%\"yGF)F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,eqn3\}, \{x,y,lambda\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%<%/%\"xG\"\"!/%\"yG\"\"\"/%'lambdaG#F)\"\"#<
%/F(!\"\"F$/F+#F0F-<%/F%,$-%'RootOfG6#,&*$%#_ZGF-F-!#:F)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 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 109 "levelCurve[0] := implicitplot( f(x,y)=f(sqrt(15/2)/2,1/4), x = \+
-3/2..3/2,y=-3/2..3/2,color=pink,thickness=2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 96 "levelCurve[1] := implicitplot( f(x,y)=f(0,1), \+
x = -3/2..3/2,y=-3/2..3/2,color=plum,thickness=2):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 103 "levelCurve[2] := implicitplot( f(x,y)=f(0,
-1), x = -3/2..3/2,y=-3/2..3/2,color=aquamarine,thickness=2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "levelCurve[3] := implicitplo
t( phi(x,y)=1, x = -3/2..3/2,y=-3/2..3/2,color=maroon, thickness=3):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "levelCurve[4] := contour
plot(f(x,y), x = -3/2.. 3/2, y= -3/2..3/2, contours=10,color=COLOR(RGB
,0.75,0.7,0.7)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "pointP
lot := plot([ [0,1],[0,-1],[sqrt(15/2)/2,1/4],[-sqrt(15/2)/2,1/4]],sty
le=POINT,symbol=DIAMOND, color=CLUE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "display(seq(levelCurve[j],j=0..4),pointPlot);" }}
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+!eD\"F*F(7$$\"1+++++!GD\"F*$!1**********z#\\\"F*7$7$$\"1+++++!e?\"F*F
ZF_[\\l7$Fe[\\l7$$\"19dG9d3n6F*$!18dG9d3(G\"F*7$7$$\"1+++++!e:\"F*FhpF
i[\\l7$F_\\\\lFgi[lF^dr-F$6fn7$7$$!1yxxxxxD5F*F(7$$!13')3n$\\R-\"F*$!1
#R6Hj]g\\\"F*7$7$F_v$!1*********>v[\"F*Fi\\\\l7$F_]\\l7$$!14y\">#3yJ)*
F-$!1>#3y\">#oT\"F*7$7$$!1-+++++S'*F-FZFc]\\l7$Fi]\\l7$$!1>#3y\">#3U*F
-$!1y\">#3y\"zL\"F*7$7$$!1(**********\\,*F-FhpF]^\\l7$7$$!1***********
\\,*F-Fhp7$$!1F')p8I')4!*F-$!1P,j)p8!f7F*7$7$Ffx$!1+++++7d7F*Fj^\\l7$7
$Fc]yFa_\\l7$$!1joiu])Hc)F-$!18t`#\\,P=\"F*7$7$$!1Tr&G9dGI)F-F[tFe_\\l
7$F[`\\l7$$!16qf!)QA:\")F-$!1)HS>hx%36F*7$7$F`gl$!1+++++_b5F*F_`\\l7$F
e`\\l7$$!1$>3&HiUawF-$!1!=\\qPdX.\"F*7$7$$!1ILLLLL`vF-F_vFi`\\l7$F_a\\
l7$$!1o'>3&HiirF-$!1D.=\\qPP'*F-7$7$$!1'***********>nF-FfxFca\\l7$7$$!
1(***********>nF-Ffx7$$!1V68s'>3n'F-$!1\\)oyK!=H*)F-7$7$F\\hl$!1&*****
****>F))F-F`b\\l7$Ffb\\l7$$!1144444LhF-$!1(34444pE)F-7$7$$!1'*********
*Ru&F-F`glFjb\\l7$F`c\\l7$$!1fjjjjj(e&F-$!1KOOOOO7wF-7$7$Feil$!1'*****
****>(Q(F-Fdc\\l7$Fjc\\l7$$!1r(QpMn$)*\\F-$!1@71`Ej,qF-7$7$$!1'*******
****zXF-FdcmF^d\\l7$7$$!1&***********zXF-F\\hl7$$!1(ez*[C7'Q%F-$!1//-^
v(QT'F-7$7$Fajl$!1'*********>NiF-F[e\\l7$7$Fajl$!1(*********>NiF-7$$!1
e6l/'=Wr$F-$!1L)[`R\"e&)eF-7$7$$!1$***********RIF-FeilFhe\\l7$F^f\\l7$
$!11l/'=Wn,$F-$!1'[`R\"eD$Q&F-7$7$Fa[m$!1)*********>r`F-Fbf\\l7$Fhf\\l
7$$!1V'['['['3AF-$!1\\8N^8N\"*\\F-7$7$Ff\\m$!1**********>&z%F-F\\g\\l7
$Fbg\\l7$$!1$***********>8F-$!1************zYF-7$7$F8$!1**********>2XF
-Ffg\\l7$F\\h\\l7$$!1E*********z#HF3F]h\\l7$7$FD$!1,++++?2XF-F`h\\l7$7
$F[blFeh\\l7$$\"1+eJE0@/5F-$!1#z:j_5Ug%F-7$7$FP$!1-++++?&z%F-Fih\\l7$F
_i\\l7$$\"1-ah%Q:Y%HF-$!1%R:YQ:YM&F-7$7$Fbo$!1.++++?r`F-Fci\\l7$7$F^p$
!1.++++?NiF-7$$\"1%***********RIF-Feil7$F`j\\lFii\\l7$7$Ffq$!1/++++?(Q
(F-7$$\"1'***********zXF-F\\hl7$7$Fij\\lFh_pF]j\\l7$7$Fer$!10++++?F))F
-7$$\"1(**********Ru&F-F`gl7$7$$\"1)**********Ru&F-F`gl7$Fjq$!10++++?(
Q(F-7$7$Ffbt$!1,++++_b5F*7$$\"1JLLLLL`vF-F_v7$7$$\"1ILLLLL`vF-F_v7$$\"
1bsssss#*pF-$!1[sssss#R*F-7$7$$\"1(***********>nF-Fc]yFf\\]l7$F\\]]lF^
[]l7$7$Fit$!1,++++7d7F*7$$\"1Tr&G9dGI)F-F[t7$Fd]]lF\\\\]l7$7$Feu$!1,++
++_([\"F*7$$\"1(***********R'*F-FZ7$F\\^]l7$$\"1\"3EyM/8.*F-$!13EyM/8j
7F*7$7$$\"1++++++:!*F-FhpF`^]l7$Ff^]lFa]]l7$7$$\"1xxxxxxD5F*F(Fi]]lF^d
r-F$6D7$7$$!1LLLLLLVsF-F(7$$!1Qv9J@nzpF-$!1Y_)oyK?Y\"F*7$7$F\\hl$!1+++
++O29F*Fd_]l7$Fj_]l7$$!1hjjjjjvkF-$!1jjjjjV#R\"F*7$7$$!1++++++sjF-FZF^
`]l7$Fd`]l7$$!19=====IfF-$!1=====)pK\"F*7$7$Feil$!1+++++Oj7F*Fh`]l7$F^
a]l7$$!1Sr&G9dGQ&F-$!1'G9dG9<E\"F*7$7$$!1***********\\O&F-FhpFba]l7$Fh
a]l7$$!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$$!1jmmmmm'3%F-F[tFfb]l7$F\\c]l7$$!1
H-$4s$)[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]l7$$!1@q-Fq-
2>F-$!1(H(H(H(H45F*7$7$Ff\\m$!1+++++;/5F*Fdd]l7$Fjd]l7$$!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$$\"1%***********H@F-F_v7$Fif]lFdf]
l7$7$F^p$!1,++++;[6F*7$$\"1immmmm'3%F-F[t7$Fag]lFff]l7$7$FfqF_a]l7$$\"
1)**********\\O&F-Fhp7$Fgg]lF^g]l7$7$FerF[`]l7$$\"1++++++sjF-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-%+AXESLABELSG6$%\"xG%
\"yG" 2 514 443 443 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 
10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 
237 224 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 1 " " }{TEXT 331 36 "8.  When subject to the conditions \+
 " }{XPPEDIT 329 1 "x^2+y^2+z^2=1" "/,(*$%\"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 379 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 375 31 "a) 1/2        b)  3/2   \+
    c) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 498 14 "   \+
        d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 376 11 
"       e)  " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$
" }{TEXT 377 9 " \n\n f)   " }{XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"#\"\"\"
-%%sqrtG6#\"\"$F$" }{TEXT 378 43 "      g) 1             h) 2         \+
   i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }
{TEXT 380 16 "            j)  " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG
6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 496 9 "Solution:" }{TEXT 497 5 " (a)\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 78 "f := (x,y,z) -> x+y+z:  \nphi := (x,y,z) -> x
^2+y^2+z^2:\npsi := (x,y,z) -> x+z:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "eqn1 := diff(f(x,y,z),x) = lambda*diff(phi(x,y,z),x)+
mu*diff(psi(x,y,z),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"
\"\",&*&%'lambdaGF&%\"xGF&\"\"#%#muGF&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "eqn2 := diff(f(x,y,z),y) = lambda*diff(phi(x,y,z),y)+
mu*diff(psi(x,y,z),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"
\"\",$*&%'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**$%\"zGF)F*F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "eqn5 := psi(x,y,z) = 1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn5G/,&%\"xG\"\"\"%\"zGF(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "solve( \{eqn1,eqn2,eqn3,eqn4,eqn5\}, \{x,y,z,l
ambda,mu\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"zG#\"\"\"\"\"#/%
\"xGF&/%#muG,&F'F'-%'RootOfG6#,&!\"\"F'*$%#_ZGF(F(F2/%\"yGF./%'lambdaG
F." }}}{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'*$%
\"zGF&F'F'\"\"'" }{TEXT 335 36 " at the point (1,1,1).  What is  a ?" 
}{TEXT 438 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 503 72 "a)  -5           b)  -4             c)   -3        d)  -
2           e)  " }{TEXT 500 3 "-1 " }{TEXT 504 20 "                  \+
  " }}{PARA 0 "" 0 "" {TEXT 501 37 "f)   0            g)   1          \+
   " }{TEXT -1 1 " " }{TEXT 502 37 "h)   2           i)  3           j
)  " }{TEXT -1 1 " " }{TEXT 505 1 "4" }{TEXT -1 2 "  " }}{PARA 0 "" 0 
"" {TEXT 499 1 " " }}{PARA 0 "" 0 "" {TEXT 506 9 "Solution:" }{TEXT 
507 5 " (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := (x,y,z
) -> x^2+2*y^3+3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6%%\"x
G%\"yG%\"zG6\"6$%)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 445 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 446 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 442 74 "a)  -4           b)  -3            \+
 c)   -2         d)  -1           e)  0" }{TEXT 439 1 " " }{TEXT 444 
20 "                    " }}{PARA 0 "" 0 "" {TEXT 440 36 "f)   1      \+
      g)   2            " }{TEXT -1 1 " " }{TEXT 441 39 "h)   3       \+
    i)  4             j)  " }{TEXT -1 1 " " }{TEXT 443 1 "5" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT 277 1 " " }}{PARA 0 "" 0 "" {TEXT 
508 9 "Solution:" }{TEXT 509 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 447 1 "5*(x-y)" "*&\"\"&\"\"\",&%\"xGF$%\"yG!
\"\"F$" }{TEXT 448 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 510 9 "Solutio
n:" }{TEXT 511 5 " (d)\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 455 70 "a)  1           b)  2             c)   3         d)  4  \+
         e)  5" }{TEXT 456 21 "                    \n" }{TEXT 458 34 "
f)  6            g)  7            " }{TEXT -1 3 " h)" }{TEXT 457 36 " \+
 8           i) 9            j)  12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 512 9 "Solution:" }{TEXT 513 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 459 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 514 9 "Solution:" }{TEXT 515 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 29 "  over the \+
region bounded by " }{XPPEDIT 19 1 "y=sqrt(3)*x" "/%\"yG*&-%%sqrtG6#\"
\"$\"\"\"%\"xGF)" }{TEXT -1 4 " ,  " }{XPPEDIT 19 1 "y=0" "/%\"yG\"\"!
" }{TEXT -1 4 " ,  " }{XPPEDIT 19 1 "x^2+y^2=1" "/,&*$%\"xG\"\"#\"\"\"
*$%\"yGF&F'F'" }{TEXT -1 8 " , and  " }{XPPEDIT 19 1 "x^2+y^2=4" "/,&*
$%\"xG\"\"#\"\"\"*$%\"yGF&F'\"\"%" }{TEXT -1 1 "." }}{PARA 265 "" 0 "
" {TEXT 460 4 "a)  " }{XPPEDIT 19 1 "Pi/12" "*&%#PiG\"\"\"\"#7!\"\"" }
{TEXT 539 14 "           b) " }{XPPEDIT 19 1 "Pi/6" "*&%#PiG\"\"\"\"\"
'!\"\"" }{TEXT 541 18 "               c) " }{XPPEDIT 19 1 "Pi/4" "*&%#
PiG\"\"\"\"\"%!\"\"" }{TEXT 542 17 "              d) " }{XPPEDIT 19 1 
"Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 543 14 "           e) " }
{XPPEDIT 19 1 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 544 10 "     \nf
)  " }{XPPEDIT 19 1 "Pi^2/12" "*&%#PiG\"\"#\"#7!\"\"" }{TEXT 540 16 " \+
          g)   " }{XPPEDIT 19 1 "Pi^2/6" "*&%#PiG\"\"#\"\"'!\"\"" }
{TEXT 545 13 "         h)  " }{XPPEDIT 19 1 "Pi^2/4" "*&%#PiG\"\"#\"\"
%!\"\"" }{TEXT 546 15 "           i)  " }{XPPEDIT 19 1 "Pi^2/3" "*&%#P
iG\"\"#\"\"$!\"\"" }{TEXT 547 12 "        j)  " }{XPPEDIT 19 1 "Pi^2/2
" "*&%#PiG\"\"#F$!\"\"" }{TEXT 548 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 528 9 "Solution:" }{TEXT 529 5 " (f)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p0 := plot(sqrt(3)*x,x=0..1/2,thick
ness=1,color=gray):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p1 :
= plot(sqrt(3)*x,x=1/2..1,thickness=3,color=MAROON):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "p2 := plot( 0,x=1..2,thickness=3,color=MA
ROON):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "p3 := plot(sqrt(4
-x^2),x= 1..2,scaling=constrained,thickness=3,color=MAROON):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "p4 := plot(sqrt(1-x^2),x= 1/
2..1,scaling=constrained,thickness=3,color=MAROON):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "display(p0,p1,p2,p3,p4);" }}{PARA 13 "" 
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vr7F[^lF(7$$\"1++]nHi#H\"F[^lF(7$$\"1n;z*ev:J\"F[^lF(7$$\"1LL$347TL\"F
[^lF(7$$\"1LLLjM?`8F[^lF(7$$\"1+]7o7Tv8F[^lF(7$$\"1LLLQ*o]R\"F[^lF(7$$
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$\"1MbC%*>G><F[^l7$Ff[m$\"1?9aWz'yq\"F[^l7$Fi[m$\"1K'='zio%p\"F[^l7$F
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EMTA$*eqFH7$Fial$\"1OBRs4PapFH7$F^bl$\"14c]_m#>%oFH7$Fcbl$\"1#)*R(zP^N
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\"1ZBDv'>e$eFH7$F[el$\"1di&>Qdfn&FH7$F`el$\"1mH3m#=k_&FH7$Feel$\"1*[N8
mY+O&FH7$Fjel$\"1sZ**\\Rw-_FH7$F_fl$\"1DP;&)H+B]FH7$Fdfl$\"1_]x[RiX[FH
7$Fifl$\"15)[l\">m]YFH7$F^gl$\"10**f=/S\\WFH7$Fcgl$\"1)H&R5dwDUFH7$Fhg
l$\"1%4%[#\\Ie*RFH7$F]hl$\"1p@C<1:VPFH7$Fbhl$\"1]\"pXkj7Z$FH7$Fghl$\"1
&42\\G&3)>$FH7$F\\il$\"1<x@.O(*[GFH7$Fail$\"1#\\k\"*=?:\\#FH7$Ffil$\"1
XjW$*HKP?FH7$$\"1+++DM\"3%)*FH$\"1#yEqH'=x<FH7$F[jl$\"1H.f%oH+Z\"FH7$$
\"1^7.#Q?&=**FH$\"1#>z^*=&RF\"FH7$$\"1+voa-oX**FH$\"1P%\\8H')3/\"FH7$$
\"1\\PMF,%G(**FH$\"1&=hJ*p=ltF,Fa[mFcjlF[[m-%(SCALINGG6#%,CONSTRAINEDG
-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(F`dm%(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 22660 5000 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "Int(Int(theta*r,r=1..2),theta=0..Pi/3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%&thetaG\"\"\"%\"rGF*/F+;F*\"\"#
/F);\"\"!,$%#PiG#F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%#PiG\"\"##\"\"\"\"#
7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT 344 14 "15. Integrate " }{TEXT 450 1 " \+
" }{XPPEDIT 451 1 "12*x*y*z" "**\"#7\"\"\"%\"xGF$%\"yGF$%\"zGF$" }
{TEXT 449 74 "   over the solid region in the first octant that is bou
nded by the plane " }{TEXT 453 1 " " }{XPPEDIT 454 1 "x+y+z=1" "/,(%\"
xG\"\"\"%\"yGF%%\"zGF%F%" }{TEXT 452 1 "." }{TEXT 345 3 "  \n" }{TEXT 
343 169 "\na)  1/10           b) 1/48           c)   1/25           d)
  1/60         e) 1/96 \nf)   2/15          g)   3/25          h)   5/
48           i)  3/64          j)  1/120" }{TEXT 573 1 "\n" }}{PARA 0 
"" 0 "" {TEXT 571 9 "Solution:" }{TEXT 572 5 " (d)\n" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(Int(In
t(12*x*y*z,z=0..1-x-y),y=0..1-x),x=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$,$*(%\"xG\"\"\"%\"yGF-%\"zGF-\"#7/F/
;\"\"!,(F-F-F,!\"\"F.F5/F.;F3,&F-F-F,F5/F,;F3F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
\"\"\"\"#g" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 518 17 "16. Calculate    " }
{TEXT 521 1 " " }{XPPEDIT 522 1 "Pi*Int(Int( sin(Pi*y^2) ,y=x..1),x=0.
.1)" "*&%#PiG\"\"\"-%$IntG6$-F&6$-%$sinG6#*&F#F$*$%\"yG\"\"#F$/F/;%\"x
GF$/F3;\"\"!F$F$" }{TEXT 520 3 "  ." }{TEXT 519 3 "  \n" }{TEXT 517 
140 "\na)  1           b)  2            c)  3           d)  4         \+
  e) 5 \nf)   8          g)   12         h)  15          i)  16       \+
   j)  " }{TEXT 516 3 "20\n" }}{PARA 3 "" 0 "" {TEXT 574 9 "Solution:
" }{TEXT 575 4 " (a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Pi*i
nt(int(sin(Pi*y^2),y = x .. 1),x = 0 .. 1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Pi
*int(int(sin(Pi*y^2),x = 0 .. y),y = 0 .. 1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 523 3 "17." }{TEXT 524 1 " " }{TEXT 601 38 " A s
olid is bounded by the surfaces   " }{XPPEDIT 602 1 "z=16-2*x^2-y^2" "
/%\"zG,(\"#;\"\"\"*&\"\"#F&*$%\"xGF(F&!\"\"*$%\"yGF(F+" }{TEXT 599 9 "
   and   " }{XPPEDIT 603 1 "z=1+x^2+2*y^2" "/%\"zG,(\"\"\"F%*$%\"xG\"
\"#F%*&F(F%*$%\"yGF(F%F%" }{TEXT 600 1 "." }{TEXT 604 1 "\n" }{TEXT 
609 21 "Its mass density is  " }{XPPEDIT 610 1 "delta(x,y,z" "-%&delta
G6%%\"xG%\"yG%\"zG" }{TEXT 605 36 ".   Its mass is therefore \n\n     \+
   " }{XPPEDIT 611 1 "Int(Int(Int (delta(x,y,z),z=phi(x,y)..psi(x,y)  \+
  ), y = mu(x) .. nu(x) ),  x = a .. b)" "-%$IntG6$-F#6$-F#6$-%&deltaG
6%%\"xG%\"yG%\"zG/F.;-%$phiG6$F,F--%$psiG6$F,F-/F-;-%#muG6#F,-%#nuG6#F
,/F,;%\"aG%\"bG" }{TEXT 606 17 "  .\n\n   What is  " }{XPPEDIT 612 1 "
nu(1)" "-%#nuG6#\"\"\"" }{TEXT 608 1 "?" }}{PARA 3 "" 0 "" {TEXT 607 
22 "\na)  1           b)   " }{XPPEDIT 618 1 "sqrt(2)" "-%%sqrtG6#\"\"
#" }{TEXT 613 17 "            c)   " }{XPPEDIT 619 1 "sqrt(3)" "-%%sqr
tG6#\"\"$" }{TEXT 614 32 "           d)  2            e)  " }{XPPEDIT 
620 1 "2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 615 23 " \nf)  \+
 3          g)   " }{XPPEDIT 621 1 "2*sqrt(3)" "*&\"\"#\"\"\"-%%sqrtG6
#\"\"$F$" }{TEXT 616 54 "          h)   4               i)  5         \+
    j)   " }{XPPEDIT 622 1 "3*sqrt(3)" "*&\"\"$\"\"\"-%%sqrtG6#F#F$" }
{TEXT 617 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT 623 9 "Solution:" }{TEXT 624 4 " (d)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 527 62 "18.   Find the surface area of that portion of the parab
oloid " }{XPPEDIT 577 1 "z=4-x^2-y^2" "/%\"zG,(\"\"%\"\"\"*$%\"xG\"\"#
!\"\"*$%\"yGF)F*" }{TEXT 576 30 " that\nlies above the xy-plane." }}
{PARA 3 "" 0 "" {TEXT 526 5 "\na)  " }{XPPEDIT 587 1 "Pi/3" "*&%#PiG\"
\"\"\"\"$!\"\"" }{TEXT 585 50 "                                       \+
     b)    " }{XPPEDIT 588 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 
586 16 "           \n\nc) " }{XPPEDIT 589 1 "Pi/3*(3*sqrt(3)-1)" "*(%#
PiG\"\"\"\"\"$!\"\",&*&F%F$-%%sqrtG6#F%F$F$F$F&F$" }{TEXT 584 19 "    \+
            d) " }{XPPEDIT 590 1 "Pi/6*(3*sqrt(3)-1)" "*(%#PiG\"\"\"\"
\"'!\"\",&*&\"\"$F$-%%sqrtG6#F)F$F$F$F&F$" }{TEXT 583 16 "            \+
\ne) " }{XPPEDIT 591 1 "Pi/3*(5*sqrt(5)-1)" "*(%#PiG\"\"\"\"\"$!\"\",&
*&\"\"&F$-%%sqrtG6#F)F$F$F$F&F$" }{TEXT 582 20 "                 f) " 
}{XPPEDIT 592 1 "Pi/6*(5*sqrt(5)-1)" "*(%#PiG\"\"\"\"\"'!\"\",&*&\"\"&
F$-%%sqrtG6#F)F$F$F$F&F$" }{TEXT 581 16 "          \ng)   " }{XPPEDIT 
593 1 "Pi/3*(15*sqrt(15)-1)" "*(%#PiG\"\"\"\"\"$!\"\",&*&\"#:F$-%%sqrt
G6#F)F$F$F$F&F$" }{TEXT 580 13 "         h)  " }{XPPEDIT 594 1 "Pi/6*(
15*sqrt(15)-1)" "*(%#PiG\"\"\"\"\"'!\"\",&*&\"#:F$-%%sqrtG6#F)F$F$F$F&
F$" }{TEXT 579 7 "\n  i)  " }{XPPEDIT 595 1 "Pi/3*(17*sqrt(17)-1)" "*(
%#PiG\"\"\"\"\"$!\"\",&*&\"#<F$-%%sqrtG6#F)F$F$F$F&F$" }{TEXT 578 16 "
           j)   " }{XPPEDIT 596 1 "Pi/6*(17*sqrt(17)-1)" "*(%#PiG\"\"
\"\"\"'!\"\",&*&\"#<F$-%%sqrtG6#F)F$F$F$F&F$" }{TEXT 525 1 "\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 597 9 "Solution:
" }{TEXT 598 4 " (j)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "Int(Int(r*sqrt(4*r^2+1),r=0..2),theta=0..
2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%\"rG\"\"\",
&*$F)\"\"#\"\"%F*F*#F*F-/F);\"\"!F-/%&thetaG;F2,$%#PiGF-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&%#PiG\"\"\"\"#<#F&\"\"##F'\"\"'F%#!\"\"F+" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 264 "" 0 "" {TEXT 531 66 "19. This problem concerns the \+
conversion of \n\n                    " }{XPPEDIT 19 1 "Int(Int(f[1](x
,y),y = x .. 2*x),x = 0 .. 2/sqrt( 5))+Int(Int(f[1](x,y),y = x .. (4-x
^2)^(1/2)),x = 2/sqrt(5) .. sqrt(2))" ",&-%$IntG6$-F$6$-&%\"fG6#\"\"\"
6$%\"xG%\"yG/F/;F.*&\"\"#F,F.F,/F.;\"\"!*&F3F,-%%sqrtG6#\"\"&!\"\"F,-F
$6$-F$6$-&F*6#F,6$F.F//F/;F.),&\"\"%F,*$F.F3F<*&F,F,F3F</F.;*&F3F,-F96
#F;F<-F96#F3F," }}{PARA 3 "" 0 "" {TEXT 532 70 "into a single integral
 in polar coordinates of the form\n\n             " }{XPPEDIT 19 1 "In
t(    Int(f[2](r,theta), r=a(theta)..b(theta) ),theta=alpha..beta  )" 
"-%$IntG6$-F#6$-&%\"fG6#\"\"#6$%\"rG%&thetaG/F-;-%\"aG6#F.-%\"bG6#F./F
.;%&alphaG%%betaG" }{TEXT 549 3 ".  " }{TEXT 538 2 "\n\n" }{TEXT 552 
26 "To what does the sum      " }{XPPEDIT 551 1 "tan(alpha)+tan(beta)+
a(Pi/6)+b(Pi/6)" ",*-%$tanG6#%&alphaG\"\"\"-F$6#%%betaGF'-%\"aG6#*&%#P
iGF'\"\"'!\"\"F'-%\"bG6#*&F/F'F0F1F'" }{TEXT 550 1 " " }{TEXT 553 13 "
    evaluate?" }{TEXT 554 1 "\n" }{TEXT 530 93 "\na)  1           b)  \+
  2             c)   3             d)   4                 e)    5 \nf)
   " }{XPPEDIT 562 1 "Pi" "I#PiG%*protectedG" }{TEXT 557 16 "         \+
 g)    " }{XPPEDIT 563 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 558 14 "
         h)   " }{XPPEDIT 564 1 "3*Pi" "*&\"\"$\"\"\"%#PiGF$" }{TEXT 
559 14 "         i)   " }{XPPEDIT 565 1 "Pi+2" ",&%#PiG\"\"\"\"\"#F$" 
}{TEXT 560 14 "          j)  " }{XPPEDIT 566 1 "2*Pi+1" ",&*&\"\"#\"\"
\"%#PiGF%F%F%F%" }{TEXT 561 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 555 9 "Solution:
" }{TEXT 556 5 " (e)\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{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$$\"1mP4$oJ*\\()F9$\"1`(=mL'
)*\\<F\\t7$$\"1+++7>FW*)F9$\"1++S#Qa))y\"F\\t-%'COLOURG6&%$RGBG$\")vio
b!\")$\")!\\DP\"F\\[l$\")%yg>%F\\[l-%*THICKNESSG6#\"\"#-F$6%7SF'7$$\"1
\"[]N5$e#3$F,Fi[l7$$\"1muYz@skdF,F\\\\l7$$\"1g^]'3o5y)F,F_\\l7$$\"1L'>
#H3u\"=\"F9Fb\\l7$$\"1(z!HI<$R[\"F9Fe\\l7$$\"1'p6(y85k<F9Fh\\l7$$\"1g@
f@3?a?F9F[]l7$$\"1z1T!3AUN#F9F^]l7$$\"1xrCn6G`EF9Fa]l7$$\"19H4(>+4'HF9
Fd]l7$$\"1o:-17&=B$F9Fg]l7$$\"13L]ny(o`$F9Fj]l7$$\"1\\Mw(*p:VQF9F]^l7$
$\"1_=\">p8$QTF9F`^l7$$\"1Q_rYUM1WF9Fc^l7$$\"1fn%3]e]s%F9Ff^l7$$\"1(p8
*z70&*\\F9Fi^l7$$\"1L**4nq64`F9F\\_l7$$\"1;%f^]=re&F9F__l7$$\"1Q$oN9I@
*eF9Fb_l7$$\"1UHI_Ld#='F9Fe_l7$$\"1!4C6P?c['F9Fh_l7$$\"1RFWN@\"Rw'F9F[
`l7$$\"1!)=<*R(3kqF9F^`l7$$\"1-bYN^)eP(F9Fa`l7$$\"1#z%\\\")fIZwF9Fd`l7
$$\"1$[B5$zWSzF9Fg`l7$$\"1@q%)[AHV#)F9Fj`l7$$\"1H%33Lm&R&)F9F]al7$$\"1
`bBRnAE))F9F`al7$$\"1%=K&[U^W\"*F9Fcal7$$\"1rS/K(40V*F9Ffal7$$\"1'o1_&
4(et*F9Fial7$$\"1h$**3Od7+\"F\\tF\\bl7$$\"16Jjp#3:.\"F\\tF_bl7$$\"1EJC
<?(*f5F\\tFbbl7$$\"1m;&\\$[s*3\"F\\tFebl7$$\"1;<e_O\")=6F\\tFhbl7$$\"1
o!H)[pE\\6F\\tF[cl7$$\"1$QvD4(fy6F\\tF^cl7$$\"1:D!3S\"f37F\\tFacl7$$\"
1FP,TtLQ7F\\tFdcl7$$\"1*)Rxq2nl7F\\tFgcl7$$\"1;_&e'z*pH\"F\\tFjcl7$$\"
1p+f#Q<]K\"F\\tF]dl7$$\"1(z+\\#>*[N\"F\\tF`dl7$$\"1ryztc[$Q\"F\\tFcdl7
$$\"1+++iN@99F\\tFfdlFfzFa[l-F$6%7S7$Fbz$\"1!z**=Qa))y\"F\\t7$$\"1)f!e
[0dd!*F9$\"1PQ&yXWJy\"F\\t7$$\"1e[Y'3^h:*F9$\"1&\\8Gj-\"y<F\\t7$$\"16B
`6_,n#*F9$\"1>WgM*\\Bx\"F\\t7$$\"1T_)f@9'y$*F9$\"1xq48*pkw\"F\\t7$$\"1
J0&)4Go*[*F9$\"1Yd1.\"G0w\"F\\t7$$\"1=L89wl#f*F9$\"1')4:<&Q\\v\"F\\t7$
$\"1(pM%f?G*p*F9$\"1A%fi6o!\\<F\\t7$$\"1HKH8Kb4)*F9$\"1T]$)Hv!Hu\"F\\t
7$$\"1NG;E2Z>**F9$\"1[!oLFvmt\"F\\t7$$\"1&*G()RMD.5F\\t$\"1:eja$o,t\"F
\\t7$$\"1.nI%47K,\"F\\t$\"1f(y[JbVs\"F\\t7$$\"1$G)\\!=BW-\"F\\t$\"1A=
\"ze=xr\"F\\t7$$\"10p[+.oN5F\\t$\"1])ejza4r\"F\\t7$$\"1^&*[@'Gl/\"F\\t
$\"1J%)o#)3M/<F\\t7$$\"1!oZi$*zj0\"F\\t$\"1;+ljCD)p\"F\\t7$$\"1eYL-T4o
5F\\t$\"1h$*=:&44p\"F\\t7$$\"1C+cPv,y5F\\t$\"1V&G`I+Yo\"F\\t7$$\"1?h\"
=&3c*3\"F\\t$\"1u#e8Ydrn\"F\\t7$$\"1#4s3ky(*4\"F\\t$\"1+'*[k]Zq;F\\t7$
$\"1U*)*[<*)46\"F\\t$\"1L!fTISIm\"F\\t7$$\"1mGNaUm@6F\\t$\"1H;K#3fel\"
F\\t7$$\"1iO()zD!G8\"F\\t$\"1A;9x\"f#[;F\\t7$$\"1s1rR5.V6F\\t$\"1]-R8D
=T;F\\t7$$\"1:=Y:Q1a6F\\t$\"1*>+o?VMj\"F\\t7$$\"1:@woP_l6F\\t$\"1#[$)p
#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=%
>ujAD\"F\\t$\"1h&)=fVVf:F\\t7$$\"1+/!\\zLCE\"F\\t$\"1+B[BF@^:F\\t7$$\"
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 0 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 24 "The required integral is" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "Int(Int(r*f[1](r,theta),r=0..2),theta= Pi/4 .. arctan
(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%\"rG\"\"\"-&
%\"fG6#F*6$F)%&thetaGF*/F);\"\"!\"\"#/F0;,$%#PiG#F*\"\"%-%'arctanG6#F4
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+
required sum is therefore " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "tan(Pi/4) + tan(arctan(2)) + 0 + 2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 535 66 "20. This problem concerns t
he fubination of \n\n                    " }{XPPEDIT 630 1 "Int(Int(f(
x,y),y = 0 .. x+1),x = -1 .. 0)+Int(Int(f(x,y),y = 0 .. 1-x/2),x =0 ..
 2)" ",&-%$IntG6$-F$6$-%\"fG6$%\"xG%\"yG/F,;\"\"!,&F+\"\"\"F1F1/F+;,$F
1!\"\"F/F1-F$6$-F$6$-F)6$F+F,/F,;F/,&F1F1*&F+F1\"\"#F5F5/F+;F/F@F1" }
{TEXT 567 62 " \n\ninto a single iterated integral  of the form\n\n   \+
          " }{XPPEDIT 628 1 "Int(    Int(f(x,y), x=phi(y) .. psi(y)) ,
y= a .. b  )" "-%$IntG6$-F#6$-%\"fG6$%\"xG%\"yG/F*;-%$phiG6#F+-%$psiG6
#F+/F+;%\"aG%\"bG" }{TEXT 537 1 "." }{TEXT 536 3 "  \n" }{TEXT 534 10 
"\nWhat is  " }{XPPEDIT 626 1 "5*phi(3)+2*psi(3)" ",&*&\"\"&\"\"\"-%$p
hiG6#\"\"$F%F%*&\"\"#F%-%$psiG6#F)F%F%" }{TEXT 568 147 " ?\n\n\na)  1 \+
          b)  2            c)  3            d)  4           e) 5 \nf) \+
  6           g)   8           h)  12          i)  16          j)  " }
{TEXT 533 3 "20\n" }}{PARA 0 "" 0 "" {TEXT 569 9 "Solution:" }{TEXT 
570 5 " (b)\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "5*(y-1) + 2*(2*(1-y));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%\"yG\"\"\"!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "subs( y = 3 , \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"#" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 0 5" 3 }{VIEWOPTS 
1 1 0 1 1 1803 }