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{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 "" 0 261 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 "" 0 262 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 "" 3 263 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 }
{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 18 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 8 "Math 233" }{TEXT 292 1 " \+
" }{TEXT -1 9 "Fall 2005" }}{PARA 18 "" 0 "" {TEXT -1 6 "Exam 3" }}
{PARA 0 "" 0 "" {TEXT 259 2 "  " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
4 "Load" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
24 "The following function, " }{TEXT 534 12 "discriminant" }{TEXT -1 
26 ", is called as follows:   " }{TEXT 535 19 "discriminant(f,x,y)" }
{TEXT -1 2 " :" }}{PARA 15 "" 0 "" {TEXT -1 35 "if  x  and  y  are var
iables, then " }{TEXT 536 19 "discriminant(f,x,y)" }{TEXT -1 53 " will
 calculate the discriminant of f(x,y) at (x,y) ;" }}{PARA 15 "" 0 "" 
{TEXT -1 80 "if  x  and  y  are real constants and if  (x,y) is a crit
ical point of  f, then " }{TEXT 537 19 "discriminant(f,x,y)" }{TEXT 
-1 53 " will calculate the discriminant of f(x,y) at (x,y) ;" }}{PARA 
15 "" 0 "" {TEXT -1 110 "if  x  and  y  are real constants and if  (x,
y) is not a critical point of  f, print \"Not a critical point.\" ;" }
}{PARA 15 "" 0 "" {TEXT 446 72 "To use these functions in MapleV R5 an
d subsequent releases, change each" }{TEXT -1 2 "  " }{TEXT 450 1 "`" 
}{TEXT -1 6 "  to  " }{TEXT 451 1 "\"" }{TEXT -1 2 "  " }{TEXT 447 7 "
in the " }{TEXT -1 1 " " }{TEXT 449 6 "printf" }{TEXT -1 1 " " }{TEXT 
448 11 " statements" }{TEXT -1 4 ".   " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 413 "discriminant := proc()
\nlocal x, y, u, v, f;\nf := args[1]:\nx := args[2]:\ny := args[3]:\ni
f not type(x, realcons) or not type(y, realcons) then\nsubs(\{u=x,v=y
\},diff(f(u,v),u$2)*diff(f(u,v),v$2)-(diff(f(u,v),u,v))^2);\nelif subs
(\{u=x,v=y\},diff(f(u,v),u))<>0 or subs(\{u=x,v=y\},diff(f(u,v),v))<>0
 then printf(`Not a critical point.\\n`); else subs(\{u=x,v=y\},diff(f
(u,v),u$2)*diff(f(u,v),v$2)-(diff(f(u,v),u,v))^2) fi;\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%-discriminantG:6\"6'%\"xG%\"yG%\"uG%\"vG%
\"fGF&F&C&>8(&9\"6#\"\"\">8$&F16#\"\"#>8%&F16#\"\"$@'43-%%typeG6$F5%)r
ealconsG-FB6$F:FD-%%subsG6$<$/8&F5/8'F:,&*&-%%diffG6$-F/6$FLFN-%\"$G6$
FLF8F3-FR6$FT-FW6$FNF8F3F3*$-FR6%FTFLFNF8!\"\"50-FH6$FJ-FR6$FTFL\"\"!0
-FH6$FJ-FR6$FTFNFao-%'printfG6#%7Not~a~critical~point.|+GFGF&F&" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "c
riticalPoints := proc()\nlocal f,x,y,cp,N,j,k,returnList,nextcp;\nf :=
 args[1]:\nreturnList := []:\ncp := [solve(\{diff(f(x,y),x) = 0 , diff
(f(x,y),y) = 0\}, \{x,y\} )];\nN := nops(cp);\nfor j from 1 to N do\ni
f lhs(cp[j][1]) = x then \nnextcp := [rhs(cp[j][1]),rhs(cp[j][2])];\ne
lse\nnextcp := [rhs(cp[j][2]),rhs(cp[j][1])];\nfi:\nreturnList := [op(
returnList),nextcp]:\nod:\nRETURN(returnList);\nend;\n  \n\n" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%/criticalPointsG:6\"6+%\"fG%\"xG%\"yG%#cpG
%\"NG%\"jG%\"kG%+returnListG%'nextcpGF&F&C(>8$&9\"6#\"\"\">8+7\">8'7#-
%&solveG6$<$/-%%diffG6$-F36$8%8&FH\"\"!/-FD6$FFFIFJ<$FHFI>8(-%%nopsG6#
F<?(8)F7F7FP%%trueGC$@%/-%$lhsG6#&&F<6#FUF6FH>8,7$-%$rhsGFfn-F^o6#&Fhn
6#\"\"#>F[o7$F_oF]o>F97$-%#opG6#F9F[o-%'RETURNGFjoF&F&" }}}{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 0 519 "discriminantTest \+
:= proc()\nlocal x, y, u, v, f, A, B;\nf := args[1]:\nx := args[2]:\ny
 := args[3]:\nif subs(\{u=x,v=y\},diff(f(u,v),u))<>0 or subs(\{u=x,v=y
\},diff(f(u,v),v))<>0 then printf(`Not a critical point.\\n`); \nelse \+
\nA := subs(\{u=x,v=y\},diff(f(u,v),u$2)*diff(f(u,v),v$2)-(diff(f(u,v)
,u,v))^2);\nB := subs(\{u=x,v=y\},diff(f(u,v),u$2)); \nfi;\nif A = 0 t
hen printf(`The test is not conclusive.\\n`);\nelif A < 0 then printf(
`Saddle point\\n`);\nelif B > 0 then printf(`Local minimum\\n`);\nelse
  printf(`Local maximum\\n`);\nfi:\nend;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%1discriminantTestG:6\"6)%\"xG%\"yG%\"uG%\"vG%\"fG%\"AG%\"BGF&F
&C'>8(&9\"6#\"\"\">8$&F36#\"\"#>8%&F36#\"\"$@%50-%%subsG6$<$/8&F7/8'F<
-%%diffG6$-F16$FHFJFH\"\"!0-FD6$FF-FL6$FNFJFP-%'printfG6#%7Not~a~criti
cal~point.|+GC$>8)-FD6$FF,&*&-FL6$FN-%\"$G6$FHF:F5-FL6$FN-F^o6$FJF:F5F
5*$-FL6%FNFHFJF:!\"\">8*-FD6$FFF[o@)/FfnFP-FW6#%=The~test~is~not~concl
usive.|+G2FfnFP-FW6#%.Saddle~point|+G2FPFio-FW6#%/Local~minimum|+G-FW6
#%/Local~maximum|+GF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 331 3 "1. " }{TEXT -1 1 " " }{TEXT 260 15 "The f
unction   " }{XPPEDIT 257 1 "f(x,y) = 3*x^2/2-x*y+2*y;" "/-%\"fG6$%\"x
G%\"yG,(*(\"\"$\"\"\"*$F&\"\"#F+F-!\"\"F+*&F&F+F'F+F.*&F-F+F'F+F+" }
{TEXT 256 56 "   has one critical point  P = (a,b) .  What is  a + b ?
" }{TEXT 340 2 "  " }}{PARA 0 "" 0 "" {TEXT 259 79 "a)  1             \+
b)  2              c)   3           d)  4              e)  6 " }{TEXT 
260 20 "                    " }}{PARA 0 "" 0 "" {TEXT 257 36 "f)   8  \+
          g)   9            " }{TEXT 332 1 " " }{TEXT 258 40 "h)   10 \+
         i)  12             j)  " }{TEXT 333 1 " " }{TEXT 261 2 "15" }
{TEXT 334 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 482 12 "Solution:  f" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f :=
 (x,y) -> 3*x^2/2-x*y+2*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6
$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*$9$\"\"##\"\"$F0*&F/\"\"\"9%F4
!\"\"F5F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn1 := di
ff( f(x,y), x ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&%\"
xG\"\"$%\"yG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "e
qn2 := diff( f(x,y), y ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq
n2G/,&%\"xG!\"\"\"\"#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "cp := solve( \{eqn1,eqn2\} , \{x,y\} );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#cpG<$/%\"xG\"\"#/%\"yG\"\"'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "rhs(cp[1]) + rhs(cp[2]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "criticalPoints(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"#\"
\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 335 34 "2.  What is the discriminant \+
of   " }{XPPEDIT 337 1 "f(x,y) = x^4/2+x*y+y^2-3*y;" "/-%\"fG6$%\"xG%
\"yG,**&F&\"\"%\"\"#!\"\"\"\"\"*&F&F-F'F-F-*$F'F+F-*&\"\"$F-F'F-F," }
{TEXT 257 2 "  " }{TEXT 336 28 "  at the point  ( -1 , 2 ) ?" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 258 77 "a)  2             b)  3              c)   4        \+
   d)  5            e)  6 " }{TEXT 259 20 "                    " }}
{PARA 0 "" 0 "" {TEXT 256 77 "f)   7            g)   8              h)
   9          i)  10             j)  " }{TEXT 338 1 " " }{TEXT 260 2 "
11" }{TEXT 339 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 483 12 "Solution:  j" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f \+
:= (x,y) -> x^4/2+x*y+y^2-3*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),**$9$\"\"%#\"\"\"\"\"#*&F/F2
9%F2F2*$F5F3F2F5!\"$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"criticalPoints(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$!\"\"\"\"#7
$,&-%'RootOfG6#,(*$%#_ZGF&\"\"\"F.!#5\"#FF/#F%F&\"\"$F/,$F)#F/\"\"%" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "discriminant(f,-1,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#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 330 3 "3. " }{TEXT 328 12 "The function
" }{TEXT 329 3 "   " }{XPPEDIT 263 1 "f(x,y) = x^3-6*x^2+12*x-8-3*y*x+
6*y+y^3;" "/-%\"fG6$%\"xG%\"yG,0*$F&\"\"$\"\"\"*&\"\"'F+*$F&\"\"#F+!\"
\"*&\"#7F+F&F+F+\"\")F0*(F*F+F'F+F&F+F0*&F-F+F'F+F+*$F'F*F+" }{TEXT 
262 60 "   has a critical point  P = ( 3 , b ).   \nSet  c = -1  if  \+
" }{XPPEDIT 266 1 "f" "I\"fG6\"" }{TEXT 265 29 "   las a local minimum
 at P, " }{TEXT 256 11 " set  c = 0" }{TEXT 341 1 " " }{TEXT 342 5 "  \+
if " }{TEXT 343 1 " " }{XPPEDIT 258 1 "f" "I\"fG6\"" }{TEXT 257 53 "  \+
 las a saddle point at P,  and set   c = +1   \nif  " }{XPPEDIT 258 1 
"f" "I\"fG6\"" }{TEXT 257 48 "   las a local maximum at P.    What is \+
 b + c ?" }}{PARA 3 "" 0 "" {TEXT 327 3 " a)" }{TEXT 530 3 " -4" }
{TEXT 533 4 "    " }{TEXT 531 101 "b) -3       c) -2       d) -1      \+
e)  0       f)  1       g)  2        h)  3       i) 4      j) 5   " }}
{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 532 13 "Solution:   e" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := (x,y) -> expand( (x
-2)^3-3*(x-2)*y+1*y^3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
f(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$%\"xG\"\"$\"\"\"*$F%\"
\"#!\"'F%\"#7!\")F'*&%\"yGF'F%F'!\"$F.\"\"'*$F.F&F'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "cp := criticalPoints(f);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#cpG7%7$\"\"#\"\"!7$\"\"$\"\"\"7$-%'RootOfG6#,(*
$%#_ZGF'F+F2!\"$F*F+,&F+F+F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "discriminantTest(f,3,1);" }}{PARA 6 "" 1 "" {TEXT -1 
13 "Local minimum" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "cp[2][
2]+(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 17 "4. The function  " }
{XPPEDIT 372 1 "f(x,y) = 2*x^3-6*x^2+6*x-2*y^3-4*x*y+4*y+3" "/-%\"fG6$
%\"xG%\"yG,0*&\"\"#\"\"\"*$F&\"\"$F+F+*&\"\"'F+*$F&F*F+!\"\"*&F/F+F&F+
F+*&F*F+*$F'F-F+F1*(\"\"%F+F&F+F'F+F1*&F6F+F'F+F+F-F+" }{TEXT 371 49 "
   has one saddle point (a,b). \nWhat is  f(a,b) ?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 375 2 "a)" }{TEXT 373 1 " " }
{TEXT 406 1 "-" }{TEXT 407 1 "4" }{TEXT -1 4 "    " }{TEXT 374 99 "b) \+
-3       c) -2        d) -1      e)  0        f)  1       g)  2       \+
 h)  3       i) 4     j) 5" }{TEXT 376 3 "   " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 9 "Solution:" }{TEXT 295 4 "   j" 
}{TEXT -1 2 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := (x,
y) -> 2*x^3-6*x^2+6*x-2*y^3-4*x*y+4*y+3;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),0*$9$\"\"$\"\"#*$F
/F1!\"'F/\"\"'*$9%F0!\"#*&F/\"\"\"F6F9!\"%F6\"\"%F0F9F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "criticalPoints(f);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%7$\"\"\"\"\"!7$#F%\"\"$#\"\"#F)7$,$-%'RootOfG6#,
(*$%#_ZGF+F%F3!\")\"#>F%F(,&#!\"&F)F%F.F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "discriminantTest(f,1,0);" }}{PARA 6 "" 1 "" {TEXT 
-1 12 "Saddle point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f(1,
0);  ### THE ANSWER ###" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "###\n\nHere are some details
 of finding the required critical point\n\n###" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "eqn1 := diff(f(x,y),x)=0;  \neqn2 := diff(f(x,
y),y)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,**$%\"xG\"\"#\"
\"'F(!#7F*\"\"\"%\"yG!\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e
qn2G/,(*$%\"yG\"\"#!\"'%\"xG!\"%\"\"%\"\"\"\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "eqn3 := x = solve(eqn2, x);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%eqn3G/%\"xG,&*$%\"yG\"\"##!\"$F*\"\"\"F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(eqn3,eqn1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,**$,&*$%\"yG\"\"##!\"$F+\"\"
\"F.F+\"\"'F)\"#=!\"'F.F*!\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "eqn5 := map(simplify, eqn4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn5G/,&*$%\"yG\"\"%#\"#F\"\"#F(!\"%\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# One obvious solution is  y
 = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eqn6 := y = 0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"yG\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "subs(eqn6, eqn3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"xG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "### Now verify that (1,0) is a saddle point" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 270 33 
"5.  What is the maximum value of " }{TEXT 383 1 " " }{XPPEDIT 377 1 "
f(x,y)=3*x+4*y" "/-%\"fG6$%\"xG%\"yG,&*&\"\"$\"\"\"F&F+F+*&\"\"%F+F'F+
F+" }{TEXT 378 2 "  " }{TEXT 381 4 " if " }{TEXT 382 3 "   " }
{XPPEDIT 379 1 "x^2+y^2=4" "/,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'\"\"%" }
{TEXT 380 3 "  ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 386 3 " a)" }{TEXT 384 1 " " }{TEXT 388 1 "6" }{TEXT -1 13 "    \+
         " }{TEXT 385 127 "b) 7             c) 8             d) 9     \+
    e)  10       \n f)  32/5       g)  36/5        h)  44/5       i) 4
8/5     j) 52/5" }{TEXT 387 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 484 9 "Solution:
" }{TEXT 293 5 "    e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "f := (x,y) -> 3*x+4*y:  phi := (x,y) -> x
^2 + y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f
(x,y),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%eqn1G/\"\"$,$*&%'lambdaG\"\"\"%\"xGF*\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*diff(phi(x,y),
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"%,$*&%'lambdaG\"\"
\"%\"yGF*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := p
hi(x,y) = 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"
#\"\"\"*$%\"yGF)F*\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "
solve( \{eqn1,eqn2,eqn3\}, \{x,y,lambda\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$<%/%\"yG#\"\")\"\"&/%\"xG#\"\"'F(/%'lambdaG#F(\"\"%<%/F
%#!\")F(/F.#!\"&F0/F*#!\"'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "f(6/5,8/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 87 "levelCurve[0] := implicitplot( f(x,y)=10, x = \+
-2..2, y=-2..2, color=pink, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 98 "constraintCurve := implicitplot( phi(x,y) = 4, x = \+
-2..2, y=-2..2, color=AQUAMARINE, thickness=2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 71 "pointPlot := plot([ [6/5,8/5]],style=POINT,sym
bol=DIAMOND, color=CLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "display(levelCurve[0],pointPlot,constraintCurve);" }}{PARA 13 "" 
1 "" {INLPLOT "6&-%'CURVESG617$7$$\"1)************>(!#;$\"1++++++g>!#:
7$$\"1wmmmmmmmF*$\"1**************>F-7$7$$\"1)************z)F*$\"1****
********R=F-F'7$7$$\"1++++++S5F-$\"1++++++?<F-F47$7$$\"1+++++++7F-$\"1
+++++++;F-7$$\"1NLLLLL$4\"F-$\"1************z;F-7$FEF:7$7$$\"1++++++g8
F-$\"1,+++++![\"F-7$$\"1ommmmm18F-$\"1++++++?:F-7$FQF@7$7$FTFMFL7$7$FH
$\"1,+++++S7F-FX7$7$F7$\"1,+++++?6F-7$$\"1LLLLLLL<F-FA7$F[oFZ7$7$F1$\"
1,++++++5F-7$$\"1nmmmmmY>F-F;7$FcoFhn-%'COLOURG6&%$RGBG$\")!\\DP(!\")$
\")J%yg&F]pF^p-%*THICKNESSG6#\"\"#-F$6%7#F@-%'SYMBOLG6#%(DIAMONDG-%&ST
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F-$!1/++++++sF*7$Fgq7$$!1+++++]x=F-$!12+++++DoF*7$7$$!1LLLLLL=>F-$!1/+
+++++cF*F]r7$Fcr7$$!1WWWWWWW>F-$!1gbbbbbbXF*7$7$$!1LLLLLLe>F-$!1/+++++
+SF*Fir7$F_s7$$!1++++++#)>F-$!1/+++++!e#F*7$7$$!1++++++&)>F-$!1/++++++
CF*Fes7$F[t7$$!1======)*>F-$!1A#======)!#<7$7$$!1LLLLLL)*>F-$!1N++++++
!)FftFat7$Fht7$Fit$\"1%HLLLLL$yFft7$7$Fit$\"1l************zFftF^u7$Fbu
7$$!1YQ:YQ:')>F-$\"1f%Q:YQ:E#F*7$7$F\\t$\"1(************R#F*Ffu7$F\\v7
$$!19dG9dGk>F-$\"1Rr&G9dGk$F*7$7$F`s$\"1(*************RF*F`v7$Ffv7$$!1
nmmmmmM>F-$\"1kmmmmmY\\F*7$7$Fdr$\"1'************f&F*Fjv7$7$Fdr$\"1(**
**********f&F*7$$!1+++++v)*=F-$\"1)*********\\(='F*7$7$FhqF(Fgw7$F]x7$
$!1HN#)eqkd=F-$\"1#HN#)eqkP(F*7$7$Fcq$\"1)************z(F*F_x7$7$$!1++
++++!o\"F-$!1Ur&G9dG3\"F-7$$!1FFFFFF2<F-$!1++++++S5F-7$F^y7$$!1*******
*****R<F-$!16++++++)*F*7$7$$!1aaaaaa%z\"F-$!1/++++++))F*Fdy7$7$F[z$!11
++++++))F*7$Fcq$!1*************z(F*7$Fex7$$!1+++++v3=F-$\"1)*********
\\([)F*7$7$F[zF5Fgz7$F][l7$$!1C)eqk<Tv\"F-$\"1N#)eqk<T&*F*7$7$F_yF;F_[
l7$Fe[l7$$!1nmmmmm'p\"F-$\"1nmmmmmc5F-7$7$Fjx$\"1Vr&G9dG3\"F-Fg[l7$7$$
!1++++++?:F-$!1**********\\(H\"F-7$$!1***********zf\"F-$!1+++++++7F-7$
Ff\\lFix7$F]\\l7$$!1`B)eqk<j\"F-$\"1`B)eqk<:\"F-7$7$$!1++++++)f\"F-FAF
]]l7$Fc]l7$$!1LLLLLLj:F-$\"1LLLLLLV7F-7$7$Fb\\l$\"1+++++](H\"F-Fg]l7$7
$$!1++++++g8F-$!1WWWWWWk9F-7$Fd^lFb^l7$Ff^lFa\\l7$F]^l7$$!1=THN#)e!\\
\"F-$\"1=THN#)eI8F-7$7$Fd^lFMFi^l7$F__l7$$!1AAAAAA79F-$\"1AAAAAA79F-7$
7$Fb^l$\"1XWWWWWk9F-Fa_l7$7$Fi\\lFg\\l7$Fd\\lFb\\l7$F\\`lFa^l7$Fg_l7$$
!1=THN#)eI8F-$\"1=THN#)e!\\\"F-7$7$$!1+++++](H\"F-FTF_`l7$Fe`l7$$!1LLL
LLLV7F-$\"1LLLLLLj:F-7$7$Fi\\l$\"1++++++)f\"F-Fi`l7$7$FayF_y7$F\\yFjx7
$FdalF[`l7$F_al7$$!1`B)eqk<:\"F-$\"1`B)eqk<j\"F-7$7$$!1Vr&G9dG3\"F-FHF
gal7$F]bl7$$!1nmmmmmc5F-$\"1mmmmmm'p\"F-7$7$Fay$\"1EFFFFF2<F-Fabl7$Fca
l7$FgyFey7$7$F]zF[zF[cl7$7$Fay$\"1FFFFFF2<F-7$$!1S#)eqk<T&*F*$\"1A)eqk
<Tv\"F-7$7$$!1.++++++))F*$\"1aaaaaa%z\"F-Fbcl7$7$FjqFhq7$FeqFcq7$7$Fdz
Fcq7$FazF[z7$Fhcl7$$!13++++]([)F*$\"1*********\\(3=F-7$7$$!19++++++yF*
F7Fddl7$Fjdl7$$!1,`B)eqkP(F*$\"1HN#)eqkd=F-7$7$Fjq$\"1***********\\'=F
-F^el7$F^dl7$F`rF^r7$7$FfrFdrFhel7$Fdel7$$!13++++](='F*$\"1*********\\
()*=F-7$7$Ffr$\"1LLLLLL=>F-F\\fl7$Fjel7$F\\sFjr7$7$FbsF`sFffl7$7$$!10+
+++++cF*Fcfl7$$!1wmmmmmY\\F*$\"1mmmmmmM>F-7$7$Fbs$\"1LLLLLLe>F-F]gl7$F
hfl7$FhsFfs7$7$F^tF\\tFggl7$Fcgl7$$!1_r&G9dGk$F*$\"19dG9dGk>F-7$7$F^t$
\"1++++++&)>F-F[hl7$Figl7$FdtFbt7$7$F[uFitFehl7$7$F^t$\"1***********\\
)>F-7$$!1s%Q:YQ:E#F*$\"1XQ:YQ:')>F-7$7$F[u$\"1LLLLLL)*>F-F\\il7$Fghl7$
F_uFit7$7$FcuFitFfil7$Fbil7$$!1KMLLLLLyFftFcil7$7$FcuFcilFjil7$Fhil7$F
iuFgu7$7$F]vF\\tF`jl7$F^jl7$$\"1w!======)Fft$\"1======)*>F-7$7$F]vFjhl
Fdjl7$Fbjl7$$\"1Sr&G9dGk$F*Fav7$7$FgvF`sF\\[m7$Fjjl7$$\"1))**********z
DF*$\"1++++++#)>F-7$7$FgvFdglFb[m7$F`[m7$F]wF[w7$7$FawFdrFj[m7$Fh[m7$$
\"1WbbbbbbXF*$\"1WWWWWWW>F-7$7$FewFcflF^\\m7$7$FewFdr7$FjwFhw7$7$F(Fhq
Fg\\m7$Fd\\m7$$\"1*)*********\\#oF*$\"1+++++]x=F-7$7$F($\"1++++++l=F-F
[]m7$Fi\\m7$$\"1\"HN#)eqkP(F*F`x7$7$FfxFcqFe]m7$Fi]m7$FjzFhz7$7$F5F[zF
[^m7$7$F5F[dl7$$\"16++++++yF*F77$7$$\"17++++++yF*F77$$\"1*************
>(F*Fb]m7$F]^m7$$\"1M#)eqk<T&*F*$!1B)eqk<Tv\"F-7$7$F;F_yF[_m7$F_^m7$$
\"1')***********z*F*$\"1++++++S<F-7$7$F;F`clFc_m7$Fa_m7$Fj[lFh[l7$7$F^
\\lFjxF[`m7$F]`m7$F`]lF^]l7$7$FAFd]lF_`m7$7$FAF`al7$$\"1Wr&G9dG3\"F-FH
7$Fd`mFi_m7$Fa`m7$Fj]lFh]l7$7$F^^lFb\\lFi`m7$F[am7$F\\_lFj^l7$7$FMFd^l
F]am7$7$FMFh_l7$$\"1,++++](H\"F-FT7$FbamFc`m7$F_am7$Fd_lFb_l7$7$Fh_lFb
^lFgam7$Fiam7$Fb`lF``l7$7$FTFf`lF[bm7$7$FTFcam7$Fh_lFM7$F`bmFaam7$F]bm
7$F\\alFj`l7$7$F`alFi\\lFcbm7$Febm7$FjalFhal7$7$FHF^blFgbm7$7$FHFe`m7$
F`alFA7$F\\cmF_bm7$Fibm7$FdblFbbl7$7$FhblFayF_cm7$7$F`clFay7$$\"1B)eqk
<Tv\"F-Fccl7$7$F[dlFiclFdcm7$Fhcm7$Fgdl$!12++++]([)F*7$7$F7F[elFjcm7$7
$F7Fa^m7$F[dlF57$Fadm7$Ff_mFd_m7$7$F`clF;Fcdm7$FedmF[cm7$F^dm7$FaelF_e
l7$7$FeelFjqFhdm7$Fjdm7$F_flF]fl7$7$FcflFfrF\\em7$7$FcflF[gl7$F`gl$!1v
mmmmmY\\F*7$7$FdglFbsFaem7$Feem7$F^hlF\\hl7$7$FbhlF^tFgem7$7$FjhlF^t7$
$\"1YQ:YQ:')>F-F]il7$7$FcilF[uF\\fm7$F`fm7$FcilF[jl7$7$FcilFcuFbfm7$Fd
fm7$FgjlFejl7$7$FjhlF]vFffm7$Fhfm7$Fe[m$\"1*)**********zDF*7$7$FdglFgv
Fjfm7$F^gm7$Fa\\mF_\\m7$7$FcflFewF`gm7$Fbgm7$F^]mF\\]m7$7$Fb]mF(Fdgm7$
7$Fb]mFh^m7$F7Fe^m-Fho6&Fjo$\")p:#R%F]p$\")`B)e)F]p$\")fqkdF]pF`p-%+AX
ESLABELSG6$%\"xG%\"yG" 2 375 375 375 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 0 144 
3240 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 1 " " }{TEXT 263 33 "6.  What is the maximum value of \+
" }{TEXT 277 1 " " }{XPPEDIT 271 1 "f(x,y)=x*y" "/-%\"fG6$%\"xG%\"yG*&
F&\"\"\"F'F)" }{TEXT 272 2 "  " }{TEXT 275 4 " if " }{TEXT 276 3 "   \+
" }{XPPEDIT 273 1 "x^2/4+y^2=2" "/,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*$%\"y
GF&F)F&" }{TEXT 274 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 344 37 "a)  1/2  \+
         b)  3/2         c) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"
#" }{TEXT 349 13 "          d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#
\"\"$" }{TEXT 345 16 "            e)  " }{XPPEDIT 19 1 "2*sqrt(2)" "*&
\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 346 9 " \n\n f)   " }{XPPEDIT 19 1 "
2*sqrt(3)" "*&\"\"#\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 347 44 "      g) 1 \+
            h) 2             i)  " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sq
rtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 348 16 "            j)  " }{XPPEDIT 
19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 296 9 "Solution:" }{TEXT 297 4 " (h)" }{TEXT -1 1 "\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := (x,y) -> x*y:  phi := (x,y) ->
 x^2/4 + y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := di
ff(f(x,y),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%eqn1G/%\"yG,$*&%'lambdaG\"\"\"%\"xGF*#F*\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*dif
f(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"xG,$*&%'
lambdaG\"\"\"%\"yGF*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "eqn3 := phi(x,y) = 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/
,&*$%\"xG\"\"##\"\"\"\"\"%*$%\"yGF)F+F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,eqn3\}, \{x,y,lambda\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&<%/%\"xG\"\"#/%'lambdaG\"\"\"/%\"yGF)<%F'/F+
!\"\"/F%!\"#<%/F(F.F*F/<%F$F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "f(2,1), \nf(-2,-1), \nf(-2,1), \nf(2,-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&\"\"#F#!\"#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "level
Curve[0] := implicitplot( f(x,y)=f(2,1), x = -3..3, y=-2..2, color=pin
k, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "levelCu
rve[1] := implicitplot( f(x,y)=f(2,-1), x = -3..3, y=-2..2,color=plum,
thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "constraint
Curve := implicitplot( phi(x,y) = 2, x = -3..3, y=-2..2, color=AQUAMAR
INE, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "point
Plot := plot([ [2,1],[2,-1],[-2,-1],[-2,1]],style=POINT,symbol=DIAMOND
, color=CLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(s
eq(levelCurve[j],j=0..1),pointPlot,constraintCurve);" }}{PARA 13 "" 1 
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FF*$!12'>!\\DPJsF-7$7$$\"1wxxxxxxFF*F5Fdam7$Fjam7$$\"1,++++]PHF*$!1KLL
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$\"12++++++%)F-F^fnFd_o7$7$$\"14++++++%)F-$\"1+++]P%)\\8F*7$$\"1.yxxxx
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Ffco7$$\"1#[?,`K\"Q:F*$!1))pu'o@a=\"F*7$7$F`w$!1cG9dGRy6F*Fjco7$7$F`w$
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$!1G9dG9dG6F*7$7$F\\x$!1cG9dGR)3\"F*F\\eo7$Fddo7$$\"1iIlK;3M;F*$\"1fz*
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FggoF\\fo7$F`go7$$\"1(=ridu%=@F*$!13z!=%Q;B$*F-7$7$$\"1![\"[\"[\"[6AF*
F[^nF\\ho7$Fbho7$$\"1f0J]z_XAF*$!1CPSNj=q&)F-7$7$Fdy$!1#*********\\P$)
F-Ffho7$7$Fdy$\"1#*********\\P$)F-7$$\"1#[\"[\"[\"[6AF*$\"1***********
**z)F-7$FcioFdgo7$F\\io7$$\"1fqk<THgBF*$!1eqk<THNxF-7$7$$\"1mmmmmmJCF*
F5Fjio7$F`jo7$$\"1BmPBmPoCF*$!1_T%e:We&oF-7$7$Ffz$!1()******\\(=P'F-Fd
jo7$7$Ffz$\"1()******\\(=P'F-7$$\"1nmmmmmJCF*F_\\l7$Fa[p7$Fdy$\"1$****
*****\\P$)F-7$Fjjo7$$\"1(Q*pA,YlDF*$!1uDm%[nI!fF-7$7$$\"1%[[[[[[f#F*Fa
hmFi[p7$F_\\p7$$\"1#3/-^v([EF*$!1S0O,M]e[F-7$7$$\"1@@@@@@6FF*F\\imFc\\
p7$Fi\\p7$$\"1nB*4xHJs#F*$!1QCGt%)>aPF-7$7$Fb[l$!1p*******\\P*HF-F]]p7
$7$Fb[l$\"1n*******\\P*HF-7$$\"1A@@@@@6FF*Fjim7$Fj]p7$$\"1eqk<THlEF*$
\"1?'>!\\DPJYF-7$7$$\"1&[[[[[[f#F*FhjmF^^p7$Fd^pF^[p7$Fc]p7$$\"1Vr&G9d
.y#F*$!1#G9dG9d`#F-7$7$$\"1*)))))))))Q'y#F*FhdmFi^p7$F__p7$$\"1EK!Hh^R
\"GF*$!1L[N>unf6F-7$7$$\"1XWWWW%>#GF*FdemFc_p7$Fi_p7$Fj_p$\"1T/Pq.PqQF
fem7$7$Fj_pF_fmF]`p7$Fa`p7$$\"1m&p3Ey4z#F*$\"1t&p3EyM>#F-7$7$$\"1!*)))
)))))Q'y#F*FjfmFc`p7$Fi`pFg]p-Fc\\l6&Fe\\l$\")p:#R%Fh\\l$\")`B)e)Fh\\l
$\")fqkdFh\\lF[]l-%+AXESLABELSG6$%\"xG%\"yG" 2 375 375 375 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 0 138 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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 
392 2 "7." }{TEXT -1 1 " " }{TEXT 389 12 "Calculate   " }{XPPEDIT 390 
1 "int(int(`(`*y - 2*x^2*`)`,x=-1..1),y=0..3)" "-%$intG6$-F#6$,&*&%\"(
G\"\"\"%\"yGF*F**(\"\"#F**$%\"xGF-F*%\")GF*!\"\"/F/;,$F*F1F*/F+;\"\"!
\"\"$" }{TEXT 391 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 395 73 "a)  -2           b)  -1             c)   1         \+
 d)  2            e)  " }{TEXT 396 21 "3                    " }}{PARA 
0 "" 0 "" {TEXT 393 36 "f)   4            g)   5            " }{TEXT 
-1 1 " " }{TEXT 394 43 "h)   6           i)  7             j)  8\n\n\n
" }{TEXT 538 11 "Answer  (g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(int(y-2*x^2, x = -1 .. 1
),y = 0 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT 397 106 "8. Let  a  and  b  be the constants and    f \+
 and  g   the functions of one variable such that\n\n          " }
{XPPEDIT 19 1 "int(int(psi(x,y),y=x^2..3*x-2),x=1..2)  = int(int(psi(x
,y),x=f(y)..g(y)),y=a..b)" "/-%$intG6$-F$6$-%$psiG6$%\"xG%\"yG/F,;*$F+
\"\"#,&*&\"\"$\"\"\"F+F4F4F0!\"\"/F+;F4F0-F$6$-F$6$-F)6$F+F,/F+;-%\"fG
6#F,-%\"gG6#F,/F,;%\"aG%\"bG" }{TEXT 404 33 "\n\nfor every continuous \+
function  " }{XPPEDIT 19 1 "psi" "I$psiG6\"" }{TEXT 405 13 ".   What i
s  " }{TEXT 413 1 " " }{XPPEDIT 414 1 "a+b+f(0)*g(9)" ",(%\"aG\"\"\"%
\"bGF$*&-%\"fG6#\"\"!F$-%\"gG6#\"\"*F$F$" }{TEXT 412 2 " ?" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 408 69 "a)  1           \+
b)  2           c)   3          d)  4           e)  5" }{TEXT 409 1 " \+
" }{TEXT 411 20 "                    " }}{PARA 0 "" 0 "" {TEXT 410 72 
"f)  6            g)  7           h)  8           i)  9            j) \+
 12" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 539 10 "Answer (g)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := \+
'f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "equivalent_integral
 := Int(Int(f(x,y),x=(y+2)/3..sqrt(y)),y=1..4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%4equivalent_integralG-%$IntG6$-F&6$-%\"fG6$%\"xG%\"yG
/F-;,&F.#\"\"\"\"\"$#\"\"#F4F3*$F.#F3F6/F.;F3\"\"%" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 60 "f := (x,y) -> 13*x^(3/2)+7*y^2; #make up we
ird test function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"
yG6\"6$%)operatorG%&arrowGF),&*$9$#\"\"$\"\"#\"#8*$9%F2\"\"(F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "one_way := int(int(f(x,y),y=
x^2..3*x-2),x=1..2);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(one_wayG,
&#\"&R=\"\"%g7\"\"\"*$\"\"##F)F+#\"$;%\"$:$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "other_way := int(int(f(x,y),x=(y+2)/3..sqrt(y)),y=
1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*other_wayG,(*$\"\"%#\"\"
\"F'#\"%k;\"#X#\"&R=\"\"%g7F)*$\"\"##F)F1#!%[7\"#N" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 46 "simplify(one_way - other_way);  #Should giv
e 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "f := y -> (y+2)/3;  g := sqrt;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"fG:6#%\"yG6\"6$%)operatorG%&arrowGF(,&9$#\"\"\"
\"\"$#\"\"#F0F/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG%%sqrtG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "answer := 1+4+f(0)*g(9);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG\"\"(" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 278 
17 "9.  Calculate    " }{TEXT 399 1 " " }{XPPEDIT 400 1 "int(int(Pi*si
n(Pi*x^2),x=y..1),y=0..1)" "-%$intG6$-F#6$*&%#PiG\"\"\"-%$sinG6#*&F(F)
*$%\"xG\"\"#F)F)/F/;%\"yGF)/F3;\"\"!F)" }{TEXT 398 2 " ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 402 69 "a)  1           b
)  2           c)   3          d)  4            e)  " }{TEXT 403 21 "5
                    " }}{PARA 0 "" 0 "" {TEXT 401 74 "f)   6          \+
 g)  7           h)   8          i)  9             j)  10\n" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 298 9 "Solution:" }{TEXT 299 5 " (a)\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 42 "int( int( Pi*sin(Pi*x^2), x=y..1),y=0..1);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "int( int( Pi*sin(Pi*x^2), y=0..x),x=0..1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 264 103 "10.  Let  a  and  b  be the con
stants and    f  and  g   the functions of one variable such that\n\n \+
    " }{TEXT 424 5 "     " }{XPPEDIT 425 1 "int(int(psi(x,y),x=-sqrt(y
)..sqrt(y)),y=0..1)  + int(int(psi(x,y),x=y-2..sqrt(y)),y=1..4)  = int
(int(psi(x,y),y=f(x)..g(x)),x=a..b)" "/,&-%$intG6$-F%6$-%$psiG6$%\"xG%
\"yG/F,;,$-%%sqrtG6#F-!\"\"-F26#F-/F-;\"\"!\"\"\"F:-F%6$-F%6$-F*6$F,F-
/F,;,&F-F:\"\"#F4-F26#F-/F-;F:\"\"%F:-F%6$-F%6$-F*6$F,F-/F-;-%\"fG6#F,
-%\"gG6#F,/F,;%\"aG%\"bG" }{TEXT 415 33 "\n\nfor every continuous func
tion  " }{XPPEDIT 19 1 "psi" "I$psiG6\"" }{TEXT 416 13 ".   What is  \+
" }{TEXT 422 1 " " }{XPPEDIT 423 1 "a+b+g(3)" ",(%\"aG\"\"\"%\"bGF$-%
\"gG6#\"\"$F$" }{TEXT 421 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 417 69 "a)  1           b)  2           c)   3  \+
        d)  4           e)  5" }{TEXT 418 1 " " }{TEXT 420 20 "       \+
             " }}{PARA 0 "" 0 "" {TEXT 419 72 "f)  6            g)  7 \+
          h)  8           i)  9            j)  12" }{TEXT -1 2 "  " }}
{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 1 " " }}
{PARA 0 "" 0 "" {TEXT 301 9 "Solution:" }{TEXT 302 5 " (f)\n" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := 'f':" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "Int(Int(f(x,y),y=x^2..x+2),x=-1..2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-%\"fG6$%\"xG%\"yG/F,;*$F+\"\"#,&F
+\"\"\"F0F2/F+;!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f
 := (x,y) -> (5*x+3)/(7+13);  #weird test function" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$#\"
\"\"\"\"%#\"\"$\"#?F0F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
86 "int(int(f(x,y),x=-sqrt(y)..sqrt(y)),y=0..1)  + int(int(f(x,y),x=y-
2..sqrt(y)),y=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#**\"#!)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "int(int(f(x,y),y=x^2..x+2)
,x=-1..2); #should be preceding result" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6##\"#**\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "-1+2+5;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 25 "11.  \+
The polar plot of   " }{TEXT 427 2 "  " }{XPPEDIT 428 1 "r=sin(3*theta
)" "/%\"rG-%$sinG6#*&\"\"$\"\"\"%&thetaGF)" }{TEXT 426 55 "  is shown.
   The area of one leaf is what multiple of " }{TEXT 430 1 " " }
{XPPEDIT 431 1 "Pi" "I#PiG%*protectedG" }{TEXT 429 3 " ?\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plots[polarplot](sin(3*theta),theta
=0..Pi,scaling=constrained,thickness=2,tickmarks=[0,0]);" }}{PARA 13 "
" 1 "" {INLPLOT "6&-%'CURVESG6$7jp7$\"\"!F(7$$\"1OtNi/wC5!#;$\"14.<fx.
5N!#=7$$\"1bn/LJ8N?F,$\"1>Md&Q'z&R\"!#<7$$\"1?hCA)H:*GF,$\"19&*=y^l]GF
57$$\"1UHA&3)H<PF,$\"1+6#pWjly%F57$$\"1aB91ml)f%F,$\"1(HCF%*f]\\(F57$$
\"1vkBNY*)=aF,$\"1-Mt`!e12\"F,7$$\"1>N'QHDA<'F,$\"1P:)>0?tV\"F,7$$\"1B
@7cQ\\VoF,$\"1Oc#o#=(*Q=F,7$$\"1+W\\L,h/zF,$\"1$e;l*fS/FF,7$$\"1_3-G[u
H&)F,$\"1Q>*))R,]_$F,7$$\"1?6<4\\46()F,$\"1V!*G'[BJ$RF,7$$\"1o:!4\"Ge%
z)F,$\"1\"R\"\\$yCrJ%F,7$$\"1S/Fx,Cz()F,$\"1E]4vl#)yYF,7$$\"1kJvryMj')
F,$\"1eP?>Eg%*\\F,7$$\"1xQ1')p8_%)F,$\"1O652QNa_F,7$$\"1<BHM)*=^\")F,$
\"1Ro&=8B2X&F,7$$\"1!o98(p]bxF,$\"1nHQvZtybF,7$$\"1#[vp_k9G(F,$\"1hzp`
b$\\i&F,7$$\"1f5$3w9r!oF,$\"1YvLiB]$f&F,7$$\"1S/VY0Z(G'F,$\"1]?:/\"H;
\\&F,7$$\"1x%[*R!GS*\\F,$\"1bbu**3,(*\\F,7$$\"1crt&))GSI%F,$\"1pFQY#f+
h%F,7$$\"1E_7-of.OF,$\"1A[X-I7LTF,7$$\"1$)eR&y%QIHF,$\"1h5m%)*3Af$F,7$
$\"1;wy/>\"*pAF,$\"1RX`F4owHF,7$$\"1Lf&on())*o\"F,$\"1YfhY][eBF,7$$\"1
t0MWu?O6F,$\"1K+!RV5)*o\"F,7$$\"1rXa.p&R@&F5$\"1$*=*oY[/Q)F57$$!180&yw
8Ul$F/$!1*G#>no>ljF/7$$!1B2(3!4?#e%F5$!11%RV#Qsm&)F57$$!1m=0\\-(\\G)F5
$!1%fz;0)Gn;F,7$$!1j=ls9-!>\"F,$!1AoxEwQ?EF,7$$!13Gd&>ZNZ\"F,$!1&*>$)*
z)\\qNF,7$$!14)Gyqixl\"F,$!17E$f')prR%F,7$$!1C]_*38!z<F,$!1.Q3:lu*>&F,
7$$!1mO$Gbs1%=F,$!1U'3uRL2/'F,7$$!1Em0YODI=F,$!19vJ\"p7!GoF,7$$!106,g(
3rv\"F,$!1%*>IK!3o^(F,7$$!1CJl_N6F;F,$!1c)QWe'[O\")F,7$$!1Mny2,=P9F,$!
1*zhU![/+()F,7$$!1f:BL6T*>\"F,$!1VB&yot.<*F,7$$!1&zR([p\\j%*F5$!1lCm:
\"yR^*F,7$$!1pe&*QjkvmF5$!1/9F!RR%o(*F,7$$!1&RT@d0C4&F5$!1vxZ;lYn)*F,7
$$!1-o^HFPpMF5$!1*QH'QA@R**F,7$$!17*yI#yH>=F5$!1*=#*ej0M)**F,7$$!1\\n#
Q!)R1b\"F/$!1A%RMxz)****F,7$$\"1\\qhRrqu:F5$!1+X'==wv)**F,7$$\"1E-r.yD
\"H$F5$!1A#ex**[`%**F,7$$\"11b$RYO,)\\F5$!1i&=,KqL()*F,7$$\"1]nDL&Qqi'
F5$!1l\"\\kSL>x*F,7$$\"1<!p#**)4CN*F5$!149WNjLE&*F,7$$\"1$\\xbm0O=\"F,
$!1Vv4+qy&>*F,7$$\"1$eV]<mgg\"F,$!1ktD*yLC@)F,7$$\"1^QG\\MxZ<F,$!1?sjN
k%[d(F,7$$\"1_G7(=)>G=F,$!1QxZrNChoF,7$$\"1RN#4e-B%=F,$!1P*)Q7/!**4'F,
7$$\"1ZcD2(H')y\"F,$!17t@)=LqG&F,7$$\"1N&p0fw(p;F,$!1V9ziQSiWF,7$$\"1!
G*R=B,%[\"F,$!10!=Jw(H6OF,7$$\"1TtY%Q:'*>\"F,$!1(Qp)\\b#)[EF,7$$\"1!fW
dHQ-N)F5$!1(\\>`*H$Go\"F,7$$\"1*Q))4V5AU%F5$!1$G?ni!oV#)F57$$!1l%Rv2m!
p!)!#>$\"126K1-(eR\"F/7$$!1R$3&\\LtjaF5$\"1Hx#))y-Ev)F57$$!1X=Viq_O6F,
$\"15$e#[7A!p\"F,7$$!1eaePwh3<F,$\"1qkt\\;pzBF,7$$!1cIi_T`3BF,$\"1f<&y
f`^,$F,7$$!1#zAvbMn)HF,$\"1o@TP#=2k$F,7$$!18V=F@PxOF,$\"1'p)))Q@S(=%F,
7$$!15f$*[&z!GVF,$\"1u#*>/k#\\i%F,7$$!1YVcRHSp\\F,$\"1>)=pg>Y)\\F,7$$!
1:2'*HGPNiF,$\"1(f@!4&Q!yaF,7$$!1T>nr.#ez'F,$\"1^QqtN)>f&F,7$$!1M#>M8u
UI(F,$\"15j-XqpCcF,7$$!1B*=&)=k1x(F,$\"1@:))G[hvbF,7$$!1N%3X,G*f\")F,$
\"1'49tEklW&F,7$$!1>l`6`u`%)F,$\"1a1VKj\"HD&F,7$$!1^G2k`?h')F,$\"1O4dA
'[$)*\\F,7$$!11[jElNy()F,$\"1;nsX)=Lo%F,7$$!1nt.D'e]z)F,$\"1.9Pp*eAK%F
,7$$!1t9@cYd4()F,$\"1EGHIJdGRF,7$$!1Aj%\\(4?@&)F,$\"1&*Q]&fV'4NF,7$$!1
Y:/-6*)3zF,$\"1izfJt!)3FF,7$$!1ijS9Nq.oF,$\"1**>fBcv7=F,7$$!1IZ@Q_D#='
F,$\"1!>2yCZFW\"F,7$$!1*H/$oc!3\\&F,$\"18Y#z@OC5\"F,7$$!1$H(Gr[$[o%F,$
\"1i&z3<h!*z(F57$$!1Jlvhe\"y\"QF,$\"16%=Y=F/1&F57$$!1_M#\\D**4%HF,$\"1
aoJ`BV^HF57$$!1%ej=\"zyG?F,$\"1`)G#)o4qQ\"F57$$!1#oxho>:-\"F,$\"1f-&\\
]6y[$F/7$$\"1@T5r0iI7!#C$\"1t([n'**3[]!#M-%'COLOURG6&%$RGBG$\"#5!\"\"F
(F(-%*AXESTICKSG6$F(F(-%(SCALINGG6#%,CONSTRAINEDG-%*THICKNESSG6#\"\"#
" 2 314 262 262 2 0 1 2 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 0 145 36 0 0 0 0 0 0 }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 81 "a)  1/24         \+
  b)  1/12             c)  1/8            d)  1/6            e) " }
{TEXT 286 23 "1/4                    " }}{PARA 0 "" 0 "" {TEXT 283 42 
"f)   1/3             g)   1/2             " }{TEXT -1 1 " " }{TEXT 
284 46 "h)   2/3           i)  3/8             j)  3/4" }}{PARA 0 "" 
0 "" {TEXT 269 1 " " }}{PARA 0 "" 0 "" {TEXT 303 9 "Solution:" }{TEXT 
304 5 " (b)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(sin(3*t
heta)^2 ,theta  = 0 .. Pi/3)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%
#PiG#\"\"\"\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT 266 17 "12.  Calculate   " }{XPPEDIT 433 1 "int(int((x^2+
y^2)^(3/2),x=y..sqrt(2-y^2)),y=0..1)" "-%$intG6$-F#6$),&*$%\"xG\"\"#\"
\"\"*$%\"yGF+F,*&\"\"$F,F+!\"\"/F*;F.-%%sqrtG6#,&F+F,*$F.F+F1/F.;\"\"!
F," }{TEXT 279 3 ".  " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 434 4 "a)  " }{XPPEDIT 19 1 "Pi*sqrt(2)" "*&%#PiG\"\"\"-%%sqr
tG6#\"\"#F$" }{TEXT 439 17 "             b)  " }{XPPEDIT 19 1 "Pi*sqrt
(2)/3" "*(%#PiG\"\"\"-%%sqrtG6#\"\"#F$\"\"$!\"\"" }{TEXT 440 15 "     \+
       c) " }{XPPEDIT 19 1 "Pi*sqrt(2)/4" "*(%#PiG\"\"\"-%%sqrtG6#\"\"
#F$\"\"%!\"\"" }{TEXT 438 18 "               d) " }{XPPEDIT 19 1 "Pi*s
qrt(2)/5" "*(%#PiG\"\"\"-%%sqrtG6#\"\"#F$\"\"&!\"\"" }{TEXT 435 16 "  \+
          e)  " }{XPPEDIT 19 1 "2*Pi*sqrt(2)/3" "**\"\"#\"\"\"%#PiGF$-
%%sqrtG6#F#F$\"\"$!\"\"" }{TEXT 436 9 " \n\n f)   " }{XPPEDIT 19 1 "2*
Pi*sqrt(2)/5" "**\"\"#\"\"\"%#PiGF$-%%sqrtG6#F#F$\"\"&!\"\"" }{TEXT 
441 12 "       g)   " }{XPPEDIT 19 1 "3*Pi*sqrt(2)/5" "**\"\"$\"\"\"%#
PiGF$-%%sqrtG6#\"\"#F$\"\"&!\"\"" }{TEXT 442 15 "          h)   " }
{XPPEDIT 19 1 "4*Pi*sqrt(2)/5" "**\"\"%\"\"\"%#PiGF$-%%sqrtG6#\"\"#F$
\"\"&!\"\"" }{TEXT 443 13 "         i)  " }{XPPEDIT 19 1 "Pi*sqrt(2)/2
" "*(%#PiG\"\"\"-%%sqrtG6#\"\"#F$F(!\"\"" }{TEXT 437 16 "            j
)  " }{XPPEDIT 19 1 "3*Pi*sqrt(2)/4" "**\"\"$\"\"\"%#PiGF$-%%sqrtG6#\"
\"#F$\"\"%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 444 9 "Solution:" }{TEXT 445 5 "
 (d)\n" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "unevaluated_answer  := Int(Int(r^3*
r,r=0..sqrt(2)),theta=0..Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3
unevaluated_answerG-%$IntG6$-F&6$*$%\"rG\"\"%/F+;\"\"!*$\"\"##\"\"\"F1
/%&thetaG;F/,$%#PiG#F3F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"answer := value(unevaluated_answer );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'answerG,$*&%#PiG\"\"\"\"\"##F(F)#F(\"\"&" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "evalf(answer); # first part of verification
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wew&)))!#5" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 89 "evalf(Int(Int((x^2+y^2)^(3/2),x=y..sqrt(2-y
^2)),y=0..1)); \n# second part of verification " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+wew&)))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 267 179 "13. The square \+
region in the first quadrant that is bounded by   x = 0, y = 0, x = 1,
 and y = 1   has \n  mass density   2 +  6y.   What  is the y-coordina
te of its center of mass?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 
261 "" 0 "" {TEXT 287 80 "a)  5/9           b)  7/9             c)   3
/5         d)  4/5           e)  2/3" }{TEXT 288 21 "                 \+
   \n" }{TEXT 290 38 "f)   5/6           g)  5/8            " }{TEXT 
-1 3 " h)" }{TEXT 289 42 "  7/8           i) 3/4            j)  7/10" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 9 "Solutio
n:" }{TEXT 306 5 " (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "
answer := Int(Int(y*(2+6*y),y = 0 .. 1),x=0..1)/Int(Int(2+6*y,y = 0 ..
 1),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG*&-%$IntG6$-
F'6$*&%\"yG\"\"\",&\"\"#F-F,\"\"'F-/F,;\"\"!F-/%\"xGF2F--F'6$-F'6$F.F1
F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value( answer );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 268 21 " 14.   The graph of  " }{XPPEDIT 456 1 "
y=sin(x)" "/%\"yG-%$sinG6#%\"xG" }{TEXT 453 2 ", " }{XPPEDIT 457 1 "x
" "I\"xG6\"" }{TEXT 455 2 "  " }{TEXT 460 2 "in" }{TEXT 461 1 " " }
{XPPEDIT 458 1 "[0,Pi/3]" "7$\"\"!*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 454 
1 " " }{TEXT 459 129 " is rotated about the x-axis. Which of the follo
wing integrals represents the surface area of the resulting surface of
 revolution" }{TEXT 452 3 "?\n\n" }{TEXT 280 5 "a)   " }{XPPEDIT 463 
1 "2*Pi*Int(sin(x)*sqrt(1+cos(x)^2),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-
%$IntG6$*&-%$sinG6#%\"xGF$-%%sqrtG6#,&F$F$*$-%$cosG6#F-F#F$F$/F-;\"\"!
*&F%F$\"\"$!\"\"F$" }{TEXT 462 15 "           b)  " }{XPPEDIT 465 1 "2
*Pi*Int(sin(x)*sqrt(1+sin(x)^2),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-%$In
tG6$*&-%$sinG6#%\"xGF$-%%sqrtG6#,&F$F$*$-F+6#F-F#F$F$/F-;\"\"!*&F%F$\"
\"$!\"\"F$" }{TEXT 464 9 "    \nc)  " }{XPPEDIT 467 1 "2*Pi*Int(cos(x)
*sqrt(1+sin(x)^2),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$cosG6
#%\"xGF$-%%sqrtG6#,&F$F$*$-%$sinG6#F-F#F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$
" }{TEXT 466 16 "            d)  " }{XPPEDIT 469 1 "2*Pi*Int(cos(x)*sq
rt(1+cos(x)^2),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$cosG6#%
\"xGF$-%%sqrtG6#,&F$F$*$-F+6#F-F#F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }
{TEXT 468 16 "           \ne)  " }{XPPEDIT 471 1 "2*Pi*Int(sin(x)*sqrt
(1+cos(x)),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$sinG6#%\"xGF
$-%%sqrtG6#,&F$F$-%$cosG6#F-F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 
470 21 "                f)   " }{XPPEDIT 472 1 "2*Pi*Int(sin(x)*sqrt(1
+sin(x)),x=0..Pi/3)" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$sinG6#%\"xGF$-
%%sqrtG6#,&F$F$-F+6#F-F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 473 13 "
       \ng)   " }{XPPEDIT 475 1 "2*Pi*Int(cos(x)*sqrt(1+sin(x)),x=0..P
i/3)" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$cosG6#%\"xGF$-%%sqrtG6#,&F$F$
-%$sinG6#F-F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 474 21 "           \+
    h)    " }{XPPEDIT 476 1 "2*Pi*Int(cos(x)*sqrt(1+cos(x)),x=0..Pi/3)
" "*(\"\"#\"\"\"%#PiGF$-%$IntG6$*&-%$cosG6#%\"xGF$-%%sqrtG6#,&F$F$-F+6
#F-F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 477 17 "           \ni)   \+
" }{XPPEDIT 479 1 "2*Pi*Int(sin(x)^2*sqrt(1+cos(x)),x=0..Pi/3)" "*(\"
\"#\"\"\"%#PiGF$-%$IntG6$*&-%$sinG6#%\"xGF#-%%sqrtG6#,&F$F$-%$cosG6#F-
F$F$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 478 18 "              j)  " }
{XPPEDIT 481 1 "2*Pi*Int(cos(x)^2*sqrt(1+sin(x)),x=0..Pi/3)" "*(\"\"#
\"\"\"%#PiGF$-%$IntG6$*&-%$cosG6#%\"xGF#-%%sqrtG6#,&F$F$-%$sinG6#F-F$F
$/F-;\"\"!*&F%F$\"\"$!\"\"F$" }{TEXT 480 6 "      " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 9 "Solution:" }{TEXT 308 5 "
 (a)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> sin(x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 48 "2*Pi*Int(f(x)*sqrt(1+diff(f(x),x)^2),x=0..Pi/
3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\"-%$IntG6$*&-%$si
nG6#%\"xGF&,&F&F&*$-%$cosGF-\"\"#F&#F&F3/F.;\"\"!,$F%#F&\"\"$F&F3" }}}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 
35 "15.  Calculate  the surface area   " }{XPPEDIT 19 1 "S" "I\"SG6\"
" }{TEXT -1 34 "  of that part of the paraboloid  " }{XPPEDIT 19 1 "z=
x^2+y^2" "/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(" }{TEXT -1 42 "  that \+
lies under the \nhorizontal plane   " }{XPPEDIT 19 1 "z=12" "/%\"zG\"#
7" }{TEXT -1 12 ".  What is  " }{XPPEDIT 19 1 "S/Pi" "*&%\"SG\"\"\"%#P
iG!\"\"" }{TEXT -1 2 " ?" }}{PARA 3 "" 0 "" {TEXT 291 145 "a)  33     \+
      b)  36          c)  39           d)  42        e) 45  \nf)   48 \+
          g)  51          h)  54           i)   57         j)  60" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 321 9 "Solutio
n:" }{TEXT 322 4 "   i" }{TEXT 485 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "unevaluated_answer :=
 Int(Int(sqrt(1+4*r^2)*r,r=0..sqrt(12)),theta=0..2*Pi);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%3unevaluated_answerG-%$IntG6$-F&6$*&,&\"\"\"F,
*$%\"rG\"\"#\"\"%#F,F/F.F,/F.;\"\"!,$*$\"\"$F1F//%&thetaG;F4,$%#PiGF/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "answer := value(unevalu
ated_answer );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,$%#PiG\"#
d" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf( \" );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+8yq!z\"!\"(" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 131 "int(int(sqrt(1+diff(x^2+y^2,x)^2 + diff(x^2+y^2,y)^2
),y=-sqrt(12-x^2)..sqrt(12-x^2)),x=-sqrt(12)..sqrt(12));  # Verificati
on step 1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%$intG6$,,*$,&\"#7\"\"\"
*$%\"xG\"\"#!\"\"#F*F-\"\"(-%#lnG6#,&F'F-F0F*#F*\"\"%*&F1F*F,F-F*-F26#
,&F'!\"#F0F*#F.F6*&F8F*F,F-F./F,;,$*$\"\"$F/F;,$FAF-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "evalf( \" ); # Verification step 2: shoul
d be 179.07..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8yq!z\"!\"(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 282 4 "
16. " }{TEXT 325 65 "What is the surface area of the surface that is p
arameterized by " }{XPPEDIT 487 1 "x=u*cos(v),y=u*sin(v),z=u^2" "6%/%
\"xG*&%\"uG\"\"\"-%$cosG6#%\"vGF'/%\"yG*&F&F'-%$sinG6#F+F'/%\"zG*$F&\"
\"#" }{TEXT 486 6 " for  " }{XPPEDIT 491 1 "u" "I\"uG6\"" }{TEXT 490 
19 "   in  [0,1]  and  " }{XPPEDIT 489 1 "v" "I\"vG6\"" }{TEXT 488 13 
" in  [1,13]. " }{TEXT 326 1 "\n" }{TEXT 281 5 "\na)  " }{XPPEDIT 496 
1 "2*sqrt(2)-1" ",&*&\"\"#\"\"\"-%%sqrtG6#F$F%F%F%!\"\"" }{TEXT 492 7 
"       " }{TEXT 514 2 "b)" }{TEXT 515 2 "  " }{XPPEDIT 497 1 "2*(sqrt
(2)-1)" "*&\"\"#\"\"\",&-%%sqrtG6#F#F$F$!\"\"F$" }{TEXT 493 4 "    " }
{TEXT 516 2 "c)" }{TEXT 517 3 "   " }{XPPEDIT 498 1 "3*sqrt(3)-1" ",&*
&\"\"$\"\"\"-%%sqrtG6#F$F%F%F%!\"\"" }{TEXT 494 5 "     " }{TEXT 518 
2 "d)" }{TEXT 519 3 "   " }{XPPEDIT 499 1 "3*(sqrt(3)-1)" "*&\"\"$\"\"
\",&-%%sqrtG6#F#F$F$!\"\"F$" }{TEXT 495 1 " " }{TEXT 500 5 "   e)" }
{TEXT 507 1 " " }{XPPEDIT 508 1 "5*sqrt(5)-1" ",&*&\"\"&\"\"\"-%%sqrtG
6#F$F%F%F%!\"\"" }{TEXT 501 3 "  \n" }{TEXT 528 2 "f)" }{TEXT 529 3 " \+
  " }{XPPEDIT 509 1 "5*(sqrt(5)-1)" "*&\"\"&\"\"\",&-%%sqrtG6#F#F$F$!
\"\"F$" }{TEXT 502 3 "   " }{TEXT 526 2 "g)" }{TEXT 527 3 "   " }
{XPPEDIT 510 1 "6*sqrt(6)-1" ",&*&\"\"'\"\"\"-%%sqrtG6#F$F%F%F%!\"\"" 
}{TEXT 503 6 "      " }{TEXT 524 2 "h)" }{TEXT 525 2 "  " }{XPPEDIT 
511 1 "6*(sqrt(6)-1)" "*&\"\"'\"\"\",&-%%sqrtG6#F#F$F$!\"\"F$" }{TEXT 
504 3 "   " }{TEXT 522 2 "i)" }{TEXT 523 1 " " }{XPPEDIT 512 1 "7*sqrt
(7)-1" ",&*&\"\"(\"\"\"-%%sqrtG6#F$F%F%F%!\"\"" }{TEXT 505 10 "       \+
   " }{TEXT 520 2 "j)" }{TEXT 521 1 " " }{XPPEDIT 513 1 "7*(sqrt(7)-1)
" "*&\"\"(\"\"\",&-%%sqrtG6#F#F$F$!\"\"F$" }{TEXT 506 2 " \n" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r \+
:= (u,v) -> [u*cos(v),u*sin(v),u^2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"rG:6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)7%*&9$\"\"\"-%$cosG6#9%
F0*&F/F0-%$sinGF3F0*$F/\"\"#F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "v1 := map(z->diff(z,u), r(u,v));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v1G7%-%$cosG6#%\"vG-%$sinGF(,$%\"uG\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v2 := map(z->diff(z,v), r(u,v));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%,$*&%\"uG\"\"\"-%$sinG6#%\"vG
F)!\"\"*&F(F)-%$cosGF,F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "cp := linalg[crossprod](v1,v2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#cpG-%'VECTORG6#7%,$*&%\"uG\"\"#-%$cosG6#%\"vG\"\"\"!\"#,$*&F+F,-
%$sinGF/F1F2,&*&F-F,F+F1F1*&F5F,F+F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "integrand := sqrt(simplify(cp[1]^2+cp[2]^2+cp[3]^2));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*integrandG*$,&*$%\"uG\"\"%F)*$F
(\"\"#\"\"\"#F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(in
t(integrand,u=0..1),v=1..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$
\"\"&#\"\"\"\"\"#F%!\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
262 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 310 14 "17.
 Integrate " }{TEXT 313 1 " " }{XPPEDIT 314 1 "24*z" "*&\"#C\"\"\"%\"z
GF$" }{TEXT 312 79 "  over the solid region in the first octant that i
s bounded above by the plane " }{TEXT 315 1 " " }{XPPEDIT 316 1 "z=1+y
" "/%\"zG,&\"\"\"F%%\"yGF%" }{TEXT 311 2 ", " }{TEXT 368 87 "\nbelow b
y the xy-plane, and on the sides by the xz-plane, the yz-plane, and th
e plane  " }{XPPEDIT 369 1 "x+y=1" "/,&%\"xG\"\"\"%\"yGF%F%" }{TEXT 
367 1 "." }{TEXT 370 2 " \n" }{TEXT 309 153 "\na)  10           b)  11
           c)  12           d)  13           e) 14 \nf)   15          \+
g)   16           h)  17           i)  18            j) 19  " }}{PARA 
3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "in
t(int(int(24*z,z=0..1+y), x=0..1-y ),y=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 318 55 "18.  Let  S  be the solid region that th
at lies under  " }{XPPEDIT 358 1 "z=8-(x^2+y^2)" "/%\"zG,&\"\")\"\"\",
&*$%\"xG\"\"#F&*$%\"yGF*F&!\"\"" }{TEXT 356 13 "  and over   " }
{XPPEDIT 359 1 "z=x^2+y^2" "/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(" }
{TEXT 357 3 ".\n " }{TEXT 360 38 "The volume of  S is what multiple of
  " }{XPPEDIT 366 1 "Pi" "I#PiG%*protectedG" }{TEXT 365 142 "?\n\na)  \+
1           b)  2            c)  3           d)  4           e) 5 \nf)
   8          g)   12         h)  15          i)  16          j)  " }
{TEXT 317 3 "20\n" }}{PARA 3 "" 0 "" {TEXT -1 10 "Answer   i" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "In
t(Int(Int(r,z=r^2..8-r^2),r=0..2),theta=0..2*Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$%\"rG/%\"zG;*$F*\"\"#,&\"\")\"\"\"F.
!\"\"/F*;\"\"!F//%&thetaG;F6,$%#PiGF/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "value( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#P
iG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "int(int(8-2*(x^2+
y^2),y=-sqrt(4-x^2)..sqrt(4-x^2)),x=-2..2); #Verify" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 320 54 
"19. Let  S  be the solid region that that lies under  " }{XPPEDIT 
354 1 "z=8-(x^2+y^2)" "/%\"zG,&\"\")\"\"\",&*$%\"xG\"\"#F&*$%\"yGF*F&!
\"\"" }{TEXT 352 13 "  and over   " }{XPPEDIT 355 1 "z=x^2+y^2" "/%\"z
G,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(" }{TEXT 353 1 " " }{TEXT 361 89 "(as \+
\nin the preceding problem). Suppose the mass density of S at the poin
t (x,y,z)  is   " }{XPPEDIT 363 1 "z/Pi" "*&%\"zG\"\"\"%#PiG!\"\"" }
{TEXT 362 27 ".\nCalculate the mass of  S." }}{PARA 3 "" 0 "" {TEXT 
364 143 "\na)  1/2          b)  1           c)  2            d)  4    \+
       e) 8 \nf)   16          g)   32         h)  64          i)  128
        j) 256" }{TEXT 319 1 "\n" }}{PARA 0 "" 0 "" {TEXT 540 11 "Answ
er:   h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( 8 - r^2 = r^2, r
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#!\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 66 "answer := Int(Int(Int(z/Pi*r,z=r^2..8-r^2),r=0
..2),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG-%$I
ntG6$-F&6$-F&6$*(%\"zG\"\"\"%#PiG!\"\"%\"rGF./F-;*$F1\"\"#,&\"\")F.F4F
0/F1;\"\"!F5/%&thetaG;F:,$F/F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "value( answer );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}
{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 324 
104 "20. The mass density at the point P = (x,y,z)  of the solid ball \+
of radius 2 centered at the  origin is " }}{PARA 263 "" 0 "" {TEXT 
350 2 "  " }{XPPEDIT 19 1 "3*(1+sqrt(x^2+y^2+z^2))/Pi" "*(\"\"$\"\"\",
&F$F$-%%sqrtG6#,(*$%\"xG\"\"#F$*$%\"yGF,F$*$%\"zGF,F$F$F$%#PiG!\"\"" }
{TEXT 351 175 ".  What is the mass of the ball?\n \n  \na)  10        \+
 b)  20          c)  30          d)  40        e) 50 \nf)   60        \+
 g)   70         h)  80          i)  90         j) 100" }{TEXT 323 1 "
\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "Answer h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "answer := Int(Int(Int(3*(1
+rho)/Pi*rho^2*sin(phi),rho=0..2),phi=0..Pi),theta=0..2*Pi);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'answerG-%$IntG6$-F&6$-F&6$,$**,&\"\"\"F/%
$rhoGF/F/%#PiG!\"\"F0\"\"#-%$sinG6#%$phiGF/\"\"$/F0;\"\"!F3/F7;F;F1/%&
thetaG;F;,$F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value( a
nswer );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 2 0" 10 }
{VIEWOPTS 1 1 0 1 1 1803 }