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{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning " 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 18 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 "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 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 8 "Math 233" }{TEXT 456 12 " \+ Spring 2004" }}{PARA 18 "" 0 "" {TEXT 455 6 "Exam 3" }}{PARA 0 "" 0 " " {TEXT 259 2 " " }}}{SECT 1 {PARA 265 "" 0 "" {TEXT 533 1 " " } {TEXT -1 1 "1" }{TEXT 529 16 ". Suppose that " }{XPPEDIT 19 1 "u=<1/s qrt(5),2/sqrt(5)>" "/%\"uG-%-anglebracketG6$*&\"\"\"F(-%%sqrtG6#\"\"&! \"\"*&\"\"#F(-F*6#F,F-" }{TEXT 528 1 "," }{TEXT -1 2 " " }{XPPEDIT 19 1 "f(x,y) = sqrt(3*x+y)" "/-%\"fG6$%\"xG%\"yG-%%sqrtG6#,&*&\"\"$\" \"\"F&F.F.F'F." }{TEXT 530 6 ", and " }{XPPEDIT 19 1 "P= (1,2)" "/%\"P G6$\"\"\"\"\"#" }{TEXT 532 44 ". Calculate the directional derivative of " }{XPPEDIT 539 1 "f" "I\"fG6\"" }{TEXT 534 6 " at " }{XPPEDIT 537 1 "P" "I\"PG6\"" }{TEXT -1 1 " " }{TEXT 538 23 " in the direction of " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT 536 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 541 3 "a) " }{XPPEDIT 19 1 "s qrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 542 46 " b) 1/2 c) 3/2 d) " }{XPPEDIT 19 1 "sqrt(5)" "-%%sqrtG6#\"\"&" }{TEXT 543 11 " e) " }{XPPEDIT 19 1 "2*sqrt(2)/3" "*(\"\"#\"\"\"-%%sqr tG6#F#F$\"\"$!\"\"" }{TEXT 544 9 " \n\n f) " }{XPPEDIT 19 1 "2*sqrt( 5)/5" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 545 43 " g) \+ 1 h) 2 i) " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%s qrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 546 16 " j) " }{XPPEDIT 19 1 "sqrt(5)/2" "*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 527 14 "Solution: (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := (x,y) -> sqrt(3*x+y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG% &arrowGF)-%%sqrtG6#,&9$\"\"$9%\"\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "u := vector([1/sqrt(5),2/sqrt(5)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'VECTORG6#7$,$*$\"\"&#\"\"\"\"\"##F-F+,$F*#F .F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v := grad(f(x,y),vec tor([x,y])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7$, $*$,&%\"xG\"\"$%\"yG\"\"\"#!\"\"\"\"##F-F2,$F*#F/F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v := subs(\{x=1,y=2\} , [v[1],v[2]] );" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7$,$*$\"\"&#\"\"\"\"\"##\"\"$\" #5,$F'#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(u,v) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 16 "2. Suppose that " }{TEXT 548 2 " " }{XPPEDIT 299 1 " u=<4/5,3/5>" "/%\"uG-%-anglebracketG6$*&\"\"%\"\"\"\"\"&!\"\"*&\"\"$F) F*F+" }{TEXT 277 6 ", " }{XPPEDIT 297 1 "D[u](f)(P)=18" "/--&%\"DG 6#%\"uG6#%\"fG6#%\"PG\"#=" }{TEXT 298 4 ", " }{TEXT 553 3 "and" } {TEXT 554 2 " " }{XPPEDIT 279 1 "diff(f(P),y)= 10" "/-%%diffG6$-%\"fG 6#%\"PG%\"yG\"#5" }{TEXT 278 4 " . " }{TEXT 551 23 " What is the valu e of " }{TEXT 552 1 " " }{XPPEDIT 347 1 "diff(f(x,y),x)" "-%%diffG6$- %\"fG6$%\"xG%\"yGF(" }{TEXT 345 3 " " }{TEXT 555 2 "at" }{TEXT 556 1 " " }{XPPEDIT 348 1 "P" "I\"PG6\"" }{TEXT 346 1 " " }{TEXT 557 1 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 352 78 "a) \+ 1 b) 2 c) 3 d) 4 e ) 6" }{TEXT 349 1 " " }{TEXT 353 20 " " }}{PARA 0 "" 0 "" {TEXT 350 36 "f) 8 g) 9 " }{TEXT -1 1 " " }{TEXT 351 40 "h) 10 i) 12 j) " }{TEXT -1 1 " " }{TEXT 550 2 "15" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 531 14 "Solution: (j)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := Diff(f(x,y),x);\nB := Diff(f(x,y),y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%%Diff G6$-%\"fG6$%\"xG%\"yGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%%Di ffG6$-%\"fG6$%\"xG%\"yGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqn1 := A*(4/5)+B*(3/5)= 18; #Given information" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&-%%DiffG6$-%\"fG6$%\"xG%\"yGF-#\"\"%\"\"& -F(6$F*F.#\"\"$F1\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eq n2 := subs(B=10,eqn1); #Also given" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%eqn2G/,&-%%DiffG6$-%\"fG6$%\"xG%\"yGF-#\"\"%\"\"&\"\"'\"\"\"\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A := solve(eqn2, A );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 272 16 "3. Suppose that" } {TEXT 559 2 " " }{TEXT -1 2 " " }{XPPEDIT 19 1 "f(x,y)=x^3 + y^2" "/ -%\"fG6$%\"xG%\"yG,&*$F&\"\"$\"\"\"*$F'\"\"#F+" }{TEXT -1 1 " " } {TEXT 558 4 ". " }{TEXT -1 23 "For each unit vector " }{XPPEDIT 19 1 "v= " "/%\"vG-%-anglebracketG6$-%$cosG6#%& alphaG-%$sinG6#F*" }{TEXT -1 24 ", we obtain a function " }{XPPEDIT 19 1 "t -> f( <1,2>+t*v )" ":6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"fG6 #,&-%-anglebracketG6$\"\"\"\"\"#F1*&F$F1%\"vGF1F1F)F)" }{TEXT -1 77 " \+ of one variable. We obtain different instantaneous rates of change at " }{XPPEDIT 19 1 "t=0" "/%\"tG\"\"!" }{TEXT -1 28 " for differ ent values of " }{XPPEDIT 19 1 "alpha" "I&alphaG6\"" }{TEXT -1 24 " \+ . For what value of " }{XPPEDIT 19 1 "cos(alpha)" "-%$cosG6#%&alphaG " }{TEXT -1 62 " do we obtain the greatest instantaneous rate of chan ge at " }{XPPEDIT 19 1 "t=0" "/%\"tG\"\"!" }{TEXT -1 3 " ?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 280 4 "a) " } {XPPEDIT 281 1 "1/2" "*&\"\"\"F#\"\"#!\"\"" }{TEXT 282 14 " b ) " }{XPPEDIT 283 1 "1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT 284 16 " \+ c) " }{XPPEDIT 285 1 "2/3" "*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 286 14 " d) " }{XPPEDIT 287 1 "1/4" "*&\"\"\"F#\"\"%!\"\"" } {TEXT 288 15 " e) " }{XPPEDIT 289 1 "3/4" "*&\"\"$\"\"\"\" \"%!\"\"" }{TEXT 290 13 " \nf) " }{XPPEDIT 291 1 "1/5" "*&\"\" \"F#\"\"&!\"\"" }{TEXT 292 15 " g) " }{XPPEDIT 293 1 "2/5" "*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 294 16 " h) " }{XPPEDIT 295 1 "3/5" "*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 296 15 " i) " }{XPPEDIT 461 1 "4/5" "*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 462 16 " \+ j) " }{TEXT -1 1 " " }{XPPEDIT 460 1 "1" "\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 535 14 "Solution: (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := (x,y) -> x^3 + y^2:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "generalGradient := grad(f( x,y),[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0generalGradientG-%' VECTORG6#7$,$*$%\"xG\"\"#\"\"$,$%\"yGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "specificGradient := subs(\{x=1,y=2\},[generalGradient [1],generalGradient[2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1specif icGradientG7$\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 " cos(alpha) = specificGradient[1]/sqrt( (specificGradient[1])^2 + (spec ificGradient[2])^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&a lphaG#\"\"$\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 45 "4. The greatest directional derivative of " }{XPPEDIT 463 1 "f" "I\"fG6\"" }{TEXT 354 14 " at the point" }{TEXT 302 2 " " }{XPPEDIT 303 1 "P" "I\"PG6 \"" }{TEXT 301 1 " " }{TEXT 304 46 " is 2. The angle between the gra dient of " }{XPPEDIT 465 1 "f" "I\"fG6\"" }{TEXT 464 7 " at " } {XPPEDIT 561 1 "P" "I\"PG6\"" }{TEXT 560 30 " and a certain unit vec tor " }{XPPEDIT 563 1 "u" "I\"uG6\"" }{TEXT 562 1 " " }{TEXT 564 6 " \+ is " }{XPPEDIT 469 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 468 27 " with the unit vector " }{XPPEDIT 467 1 "u" "I\"uG6\"" }{TEXT 466 41 " . What is the directional derivative " }{XPPEDIT 471 1 "D[ u](f)(P)" "--&%\"DG6#%\"uG6#%\"fG6#%\"PG" }{TEXT 470 3 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 472 3 "a) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 473 46 " b) 1/2 \+ c) 3/2 d) " }{XPPEDIT 19 1 "sqrt(3)" "-%%sqrtG6#\"\"$" } {TEXT 474 11 " e) " }{XPPEDIT 19 1 "2*sqrt(2)/3" "*(\"\"#\"\"\" -%%sqrtG6#F#F$\"\"$!\"\"" }{TEXT 475 9 " \n\n f) " }{XPPEDIT 19 1 "2 *sqrt(3)/3" "*(\"\"#\"\"\"-%%sqrtG6#\"\"$F$F(!\"\"" }{TEXT 476 43 " \+ g) 1 h) 2 i) " }{XPPEDIT 19 1 "sqrt(2)/2" " *&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 477 16 " j) " } {XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 540 15 "Solution: (d) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg): " }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " D[u](f)(P) = 2*cos(Pi/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--&%\"DG6#%\"uG6#%\"fG6#%\"PG*$\"\" $#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 1 " " }{TEXT 307 18 "5. The function " }{XPPEDIT 306 1 "f(x,y)=2*x^3+6*x^2+6*x+2-24*x*y-24*y+16*y^3" "/-%\"fG6$%\"xG%\"yG,0 *&\"\"#\"\"\"*$F&\"\"$F+F+*&\"\"'F+*$F&F*F+F+*&F/F+F&F+F+F*F+*(\"#CF+F &F+F'F+!\"\"*&F3F+F'F+F4*&\"#;F+*$F'F-F+F+" }{TEXT 305 186 " has one critical point P = (a,b) for which a < 0. Choose the ordered list \+ [a, what] where a is the abscissa of the critical point P = (a,b) an d \"what\" describes the behavior of " }{XPPEDIT 356 1 "f" "I\"fG6\" " }{TEXT 355 10 " at P.\n " }}{PARA 0 "" 0 "" {TEXT 265 2 "a)" } {TEXT 256 25 " [-1,local minimum] " }{TEXT 308 2 "b)" }{TEXT 274 56 " [-2,local minimum] c) [-3,local minimum] " }{TEXT 309 7 " \nd) " }{TEXT 357 19 "[-1,local maximum] " }{TEXT 358 8 " \+ e) " }{TEXT 275 22 "[-2,local maximum] " }{TEXT 260 4 " " } {TEXT 361 3 " f)" }{TEXT 362 1 " " }{TEXT 359 18 "[-3,local maximum]" }{TEXT 360 1 " " }{TEXT 257 12 " \ng) " }{TEXT 363 17 "[-1,sadd le point]" }{TEXT 364 16 " h) " }{TEXT 365 17 "[-2,saddle \+ point]" }{TEXT 366 19 " i) " }{TEXT 369 17 "[-3,saddle \+ point]" }{TEXT 258 6 " \n " }{TEXT 370 3 "j) " }{TEXT 371 1 " " } {TEXT 367 18 "[-4,local minimum]" }{TEXT 368 2 " \n" }}{PARA 0 "" 0 " " {TEXT 687 15 "Solution: (g)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := (x,y) -> 2*x^3+6*x^2+6*x+2-24*x*y-24*y+16*y^3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "diff(2*x^3+6*x^2+6*x+2-24*x* y-24*y+16*y^3,x) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$%\"xG \"\"#\"\"'F&\"#7F(\"\"\"%\"yG!#C\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "diff(2*x^3+6*x^2+6*x+2-24*x*y-24*y+16*y^3,y) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG!#CF&\"\"\"*$%\"yG\"\"#\"#[\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y), x) = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$/%\"xG\"\"\"/%\"yGF&<$/F(\"\"!/F%!\"\"<$/F(,&#!\"$\" \"#F&-%'RootOfG6#,(*$%#_ZGF3F&F9\"\"%\"\"(F&#F-F3/F%F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The last solution involves complex numbers and is not relevant for our purpose s." }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"xG\"#7F'\"\"\"F(%\"yGF( \"#'*!$w&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=-1, y=0\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$w&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 1 " " }{TEXT 565 1 "6" }{TEXT -1 2 ". " }{TEXT 380 15 " The function " }{XPPEDIT 379 1 "f(x,y)=x^3+x*y^2-12*x" "/-%\"fG6$%\" xG%\"yG,(*$F&\"\"$\"\"\"*&F&F+*$F'\"\"#F+F+*&\"#7F+F&F+!\"\"" }{TEXT 378 206 " has one critical point P = (a,b) \n with a > 0. Choo se the ordered list [a,what] where a is the first entry of the critica l point P = (a,b) and \"what\" describes the behav ior of " }{XPPEDIT 382 1 "f" "I\"fG6\"" }{TEXT 381 8 " at P. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 387 2 "a)" } {TEXT 383 24 " [1,local minimum] " }{TEXT 390 2 "b)" }{TEXT 388 54 " [2,local minimum] c) [3,local minimum] " }{TEXT 391 7 " \nd) " }{TEXT 392 18 "[1,local maximum] " }{TEXT 393 8 " \+ e) " }{TEXT 389 21 "[2,local maximum] " }{TEXT 386 4 " " } {TEXT 396 3 " f)" }{TEXT 397 1 " " }{TEXT 394 17 "[3,local maximum]" } {TEXT 395 1 " " }{TEXT 384 12 " \ng) " }{TEXT 398 16 "[1,saddle point]" }{TEXT 399 16 " h) " }{TEXT 400 16 "[2,saddle poi nt]" }{TEXT 401 19 " i) " }{TEXT 404 16 "[3,saddle poin t]" }{TEXT 385 6 " \n " }{TEXT 405 3 "j) " }{TEXT 406 1 " " }{TEXT 402 17 "[2,local maximum]" }{TEXT 403 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 686 15 "Solution: (b)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := (x,y) -> x^3+x*y^2-12*x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "expand(diff(f(x,y),x)) = 0 ; expand(diff(f(x,y),y)) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$%\"xG\"\"#\"\"$*$%\"yGF'\"\"\"!#7F+\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%\"xG\"\"\"%\"yGF'\"\"#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y),x) \+ = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$/%\"xG\"\"!/%\"yG,$-%'RootOfG6#,&!\"$\"\"\"*$%#_ZG\"\"#F/F2<$/ F%F2/F(F&<$/F%!\"#F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "dif f(f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"#\"#7*$%\"yGF&!\"%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "subs(\{x=2,y=0\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs (\{x=2,y=0\},diff(f(x,y),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"# 7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 263 18 "7. The function " }{XPPEDIT 408 1 "f(x,y)=5-2*x^2-2*x-y^2+2*x*y+4*y" "/-%\"fG6$%\"x G%\"yG,.\"\"&\"\"\"*&\"\"#F**$F&F,F*!\"\"*&F,F*F&F*F.*$F'F,F.*(F,F*F&F *F'F*F**&\"\"%F*F'F*F*" }{TEXT 407 165 " has one critical point P = ( a,b). Choose the ordered list [a,what] where a is the abscissa of th e critical point P = (a,b) and \"what\" describes the behavior of " } {XPPEDIT 410 1 "f" "I\"fG6\"" }{TEXT 409 9 " at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 415 2 "a)" }{TEXT 411 24 " [0 ,local minimum] " }{TEXT 418 2 "b)" }{TEXT 416 54 " [1,local mini mum] c) [2,local minimum] " }{TEXT 419 7 " \nd) " } {TEXT 420 18 "[0,local maximum] " }{TEXT 421 8 " e) " }{TEXT 417 21 "[1,local maximum] " }{TEXT 414 4 " " }{TEXT 424 3 " f)" } {TEXT 425 1 " " }{TEXT 422 17 "[2,local maximum]" }{TEXT 423 1 " " } {TEXT 412 12 " \ng) " }{TEXT 426 16 "[0,saddle point]" }{TEXT 427 16 " h) " }{TEXT 428 16 "[1,saddle point]" }{TEXT 429 19 " i) " }{TEXT 432 16 "[2,saddle point]" }{TEXT 413 6 " \n " }{TEXT 433 3 "j) " }{TEXT 434 1 " " }{TEXT 430 18 "[-1,loc al maximum]" }{TEXT 431 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 685 15 "Solution: (e)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := (x,y) -> 5-2*x^2-2*x-y^2+2*x*y+4*y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(f(x,y));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,.\"\"&\"\"\"*$%\"xG\"\"#!\"#F'F)*$%\"yGF(!\"\"*&F'F% F+F%F(F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{di ff(f(x,y),x) = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/%\"xG\"\"\"/%\"yG\"\"$" }}}{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#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs(\{y = 3, x = 1\},diff(f(x,y),y$2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 33 "8. What is the maximum value of " }{TEXT 316 1 " " }{XPPEDIT 310 1 "f(x,y)=x^2+y" "/-%\"fG6$%\"xG%\"yG,&*$F&\"\"#\"\" \"F'F+" }{TEXT 311 2 " " }{TEXT 314 4 " if " }{TEXT 315 3 " " } {XPPEDIT 312 1 "x^2/2+y^2=1" "/,&*&%\"xG\"\"#F&!\"\"\"\"\"*$%\"yGF&F(F (" }{TEXT 313 3 " \n" }}{PARA 0 "" 0 "" {TEXT 317 158 "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 684 15 "Solution: (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := (x ,y) -> x^2 + y: phi := (x,y) -> x^2/2 + y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x,y),x) = lambda*diff(phi(x,y),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,$%\"xG\"\"#*&%'lambdaG \"\"\"F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff( f(x,y),y) = lambda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%eqn2G/\"\"\",$*&%'lambdaGF&%\"yGF&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x,y) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"##\"\"\"F)*$%\"yGF)F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,eqn3\}, \+ \{x,y,lambda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<%/%\"yG\"\"\"/%'l ambdaG#F&\"\"#/%\"xG\"\"!<%/F(#!\"\"F*/F%F1F+<%/F,,$-%'RootOfG6#,&*$%# _ZGF*F*!#:F&F)/F%#F&\"\"%/F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "allvalues(1/2*RootOf(2*_Z^2-15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$\"#I#\"\"\"\"\"##F'\"\"%,$F$#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f(0,1), \nf(0,-1), \nf(1/4*30^(1/2) ,1/4), \nf(-1/4*30^(1/2),1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\" \"!\"\"#\"#<\"\")F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 328 36 "9. When subject to the conditions " }{XPPEDIT 326 1 "x^2+y^2+z^2 =1" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'*$%\"zGF&F'F'" }{TEXT 327 1 " " } {TEXT 323 5 " and " }{TEXT 324 1 " " }{XPPEDIT 318 1 "x+ z=1" "/,&%\"x G\"\"\"%\"zGF%F%" }{TEXT 319 1 " " }{TEXT 325 12 "the function" } {TEXT 376 1 " " }{TEXT 321 1 " " }{XPPEDIT 322 0 "f(x,y,z)=x+2*y" "/-% \"fG6%%\"xG%\"yG%\"zG,&F&\"\"\"*&\"\"#F*F'F*F*" }{TEXT 320 47 " has a maximum at (a,b,c). What is f(a,b,c)?" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 372 31 "a) 1/2 b) 3/2 c) " }{XPPEDIT 19 1 "sqrt( 2)" "-%%sqrtG6#\"\"#" }{TEXT 490 14 " d) " }{XPPEDIT 19 1 "s qrt(3)" "-%%sqrtG6#\"\"$" }{TEXT 373 11 " e) " }{XPPEDIT 19 1 " 2*sqrt(2)" "*&\"\"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 374 9 " \n\n f) " } {XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"#\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 375 43 " g) 1 h) 2 i) " }{XPPEDIT 19 1 "sqrt( 2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 377 16 " j) \+ " }{XPPEDIT 19 1 "sqrt(3)/2" "*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 683 14 "Solution : (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "W arning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := (x,y,z) -> x+2*y: \nphi := (x,y,z) -> x^2+y^2+z ^2:\npsi := (x,y,z) -> x+z:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn1 := diff(f(x,y,z),x) = lambda*diff(phi(x,y,z),x)+mu*diff(psi(x ,y,z),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"\"\",&*&%'lamb daGF&%\"xGF&\"\"#%#muGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn2 := diff(f(x,y,z),y) = lambda*diff(phi(x,y,z),y)+mu*diff(psi(x,y, z),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"#,$*&%'lambdaG \"\"\"%\"yGF*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn3 := \+ diff(f(x,y,z),z) = lambda*diff(phi(x,y,z),z)+mu*diff(psi(x,y,z),z);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/\"\"!,&*&%'lambdaG\"\"\"%\"z GF*\"\"#%#muGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn4 := \+ phi(x,y,z) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,(*$%\"xG \"\"#\"\"\"*$%\"yGF)F**$%\"zGF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn5 := psi(x,y,z) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/,&%\"xG\"\"\"%\"zGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solve( \{eqn1,eqn2,eqn3,eqn4,eqn5\}, \{x,y,z,lambda,m u\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<'/%#muG!\"\"/%\"xG#\"\"#\"\" $/%\"zG#\"\"\"F+/%\"yGF)/%'lambdaG#F+F*<'/F%F*/F(F./F-F)/F1#!\"#F+/F3# !\"$F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f(2/3,2/3,1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 16 "10. The vector " }{XPPEDIT 329 1 "`<`*(-1),b,c*`>` " "6 %*&%\"GF%" }{TEXT 330 36 " is pe rpendicular to the surface " }{XPPEDIT 331 1 "x^2+2*y^3+3*z^2=6" "/, (*$%\"xG\"\"#\"\"\"*&F&F'*$%\"yG\"\"$F'F'*&F+F'*$%\"zGF&F'F'\"\"'" } {TEXT 332 38 " at the point (1,1,1). \n What is " }{XPPEDIT 567 1 "c" "I\"cG6\"" }{TEXT 566 2 " ?" }{TEXT 435 1 " " }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 495 72 "a) -5 b) - 3 c) -3 d) -2 e) " }{TEXT 492 3 "-1 \+ " }{TEXT 496 20 " " }}{PARA 0 "" 0 "" {TEXT 493 37 "f) 0 g) 1 " }{TEXT -1 1 " " }{TEXT 494 37 "h) 2 i) 3 j) " }{TEXT -1 1 " " }{TEXT 497 1 " 4" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 491 1 " " }}{PARA 0 "" 0 " " {TEXT 682 14 "Solution: (c)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, ne w definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new def inition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := ( x,y,z) -> x^2+2*y^3+3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6 %%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"\"*$9%\"\"$F1 *$9&F1F5F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(F(x,y, z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$%\"xG \"\"#,$*$%\"yGF)\"\"',$%\"zGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "v := subs(\{x=1,y=1,z=1\}, \" ); #Normal to surface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"#\"\"'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "solve(dotprod(v,[a,2,-1])=0,a); \n# Tangent vector is perpendicular to Normal vector" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 268 16 "11. Calculate " } {XPPEDIT 440 1 "int(int(`(`*y+3*x^2*`)`,x=0..1),y=-1..3)" "-%$intG6$-F #6$,&*&%\"(G\"\"\"%\"yGF*F**(\"\"$F**$%\"xG\"\"#F*%\")GF*F*/F/;\"\"!F* /F+;,$F*!\"\"F-" }{TEXT 441 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 438 73 "a) -2 b) -1 c) \+ 1 d) 2 e) " }{TEXT 439 21 "3 \+ " }}{PARA 0 "" 0 "" {TEXT 436 36 "f) 4 g) 5 \+ " }{TEXT -1 1 " " }{TEXT 437 40 "h) 6 i) 7 j ) 8" }}{PARA 0 "" 0 "" {TEXT 276 1 " " }}{PARA 0 "" 0 "" {TEXT 681 14 "Solution: (j)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 54 "integral := Int(Int(y+3*x^2, x = 0 .. 1),y = -1 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)integralG-%$IntG6$-F& 6$,&%\"yG\"\"\"*$%\"xG\"\"#\"\"$/F.;\"\"!F,/F+;!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value(integral);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 40 "12. Calculate the double integral of \+ " }{XPPEDIT 442 1 "5*x+10*y" ",&*&\"\"&\"\"\"%\"xGF%F%*&\"#5F%%\"yGF%F %" }{TEXT 443 83 " over the region in the first quadrant of the xy-pl ane that is bounded above by " }{XPPEDIT 333 1 "y=2*x-x^2" "/%\"yG,& *&\"\"#\"\"\"%\"xGF'F'*$F(F&!\"\"" }{TEXT 334 5 ". \n\n" }{TEXT 273 70 "a) 1 b) 2 c) 3 d) 4 e ) 5" }{TEXT 335 21 " " }}{PARA 0 "" 0 "" {TEXT 337 34 "f) 6 g) 7 " }{TEXT -1 3 " h)" }{TEXT 336 36 " 8 i) 9 j) 12" }{TEXT -1 2 " " }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 680 15 "Solution : (j)\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$$\"1LLL L3VfV!#<$\"1$\\gZH:)G&)F,7$$\"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$$\"1++]i'y]!HF4$\"1@(o97 4i'\\F47$$\"1LL$ezs$HLF4$\"1gJIqKF]bF47$$\"1++]7iI_PF4$\"1!RjPBKm4'F47 $$\"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'F 4$\"1qCpj![#**))F47$$\"1mmmm#pS1(F4$\"1zo!H2J!Q\"*F47$$\"1++]i`A3vF4$ \"1u6f:f5z$*F47$$\"1mmmm(y8!zF4$\"1[-!>*)y&f&*F47$$\"1++]i.tK$)F4$\"1i qe&>@?s*F47$$\"1++](3zMu)F4$\"1<]k>b6U)*F47$$\"1nmm\"H_?<*F4$\"1flAf-X J**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(zM#[t_*yF47$$\"1mm\"H!o-*\\\"F]s$\"12x q*\\A(4vF47$$\"1++DTO5T:F]s$\"1:*fU\\o?2(F47$$\"1nmmT9C#e\"F]s$\"1<#fg .\\*4mF47$$\"1++D1*3`i\"F]s$\"1*GWwr())*3'F47$$\"1LLL$*zym;F]s$\"1n]Y> x$Rb&F47$$\"1LL$3N1#4NEq\\F47$$\"1nm\"HYt7v\"F]s$\"1)*>F]s$\"1t-SU\")[2;F47 $$\"1++v.Uac>F]s$\"1cyAF'=B])F,7$$\"\"#F(F(-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(Fbz%(DEFAULTG" 2 462 208 208 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 0 318 32208 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "integral := Int(Int( 5 *x+10*y, y = 0 .. 2*x-x^2),x = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)integralG-%$IntG6$-F&6$,&%\"xG\"\"&%\"yG\"#5/F-;\"\"!,&F+\"\" #*$F+F3!\"\"/F+;F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "val ue(integral);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 270 133 "13. The triangular region in the first quadrant that is bound ed by y = 3 - x has mass density 1 + y.\n What is its ma ss?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 449 70 "a) 1 b) 2 c) 3 d) 4 e) \+ 5" }{TEXT 450 21 " \n" }{TEXT 452 34 "f) 6 \+ g) 7 " }{TEXT -1 3 " h)" }{TEXT 451 36 " 8 \+ i) 9 j) 12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 679 15 "Solution: (i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Mass := int(int(1+y,y = 0 .. 3-x),x=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MassG\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 271 1 " " }{TEXT 338 102 "14. W hat is the y-coordinate of the center of mass of the triangular region of the preceding problem?" }{TEXT 453 3 " \n\n" }{TEXT 339 155 "a) 5/ 4 b) 3/2 c) 4/3 d) 14/9 e) 5/3 \nf) 7/9 g) 8/9 h) 1 i) 10/9 \+ j) 11/9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 678 14 "Solution: (a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "centerOfMass[y] := Int(Int(y*(1+y), y = 0 .. 3-x),x=0..3)/Int(Int(1+y,y = 0 .. 3-x),x=0..3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%-centerOfMassG6#%\"yG*&-%$IntG6$-F*6$*&F'\"\" \",&F/F/F'F/F//F';\"\"!,&\"\"$F/%\"xG!\"\"/F6;F3F5F/-F*6$-F*6$F0F1F8F7 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value(centerOfMass[y]); " }}{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 "15. Calculate the integral of " }{XPPEDIT 19 1 "sqrt(x^2+y^2)/Pi" "*&-%%sqrtG6#,& *$%\"xG\"\"#\"\"\"*$%\"yGF)F*F*%#PiG!\"\"" }{TEXT -1 76 " over the re gion in the first quadrant that is bounded by y = x, y=0, and " } {XPPEDIT 19 1 "x^2+y^2=4" "/,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'\"\"%" } {TEXT -1 1 "." }}{PARA 3 "" 0 "" {TEXT 454 152 "a) 1/6 b) 1 /3 c) 1/2 d) 2/3 e) 5/6 \nf) 1/12 \+ g) 1/4 h) 5/12 i) 1/2 j) 7/12" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 676 15 "Solution : (d)\n" }{TEXT 523 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "integral := Int(Int(sqrt(x^2+y^2)/Pi*r,r=0..2),theta=0..Pi/4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)integralG-%$IntG6$-F&6$*(,&*$%\"xG \"\"#\"\"\"*$%\"yGF.F/#F/F.%#PiG!\"\"%\"rGF//F5;\"\"!F./%&thetaG;F8,$F 3#F/\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "integral := su bs((x^2+y^2)^(1/2)=r,integral);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%) integralG-%$IntG6$-F&6$*&%\"rG\"\"#%#PiG!\"\"/F+;\"\"!F,/%&thetaG;F1,$ F-#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "value(int egral);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 342 14 "16. Integrate " }{TEXT 445 1 " " }{XPPEDIT 446 1 "z/Pi" "*&%\"zG\" \"\"%#PiG!\"\"" }{TEXT 444 74 " over the solid region in the first oc tant that is bounded by the sphere " }{TEXT 447 1 " " }{XPPEDIT 448 1 "x^2+y^2+z^2=1" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'*$%\"zGF&F'F'" } {TEXT 343 2 " " }{TEXT 568 43 "and by the planes y = 0, z = 0, and x \+ = y. " }{TEXT 569 1 "\n" }{TEXT 341 159 "\na) 1 b) 1/2 \+ c) 1/3 d) 1/4 e) 1/5 \nf) 1/8 \+ g) 1/16 h) 1/24 i) 1/32 j) " } {TEXT 340 4 "1/48" }{TEXT 570 1 "\n" }}{PARA 0 "" 0 "" {TEXT 675 15 "S olution: (i)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 100 "spaceIntegral := Int(Int(Int((rho*cos(phi))/P i*rho^2*sin(phi),rho=0..1),phi=0..Pi/2),theta=0..Pi/4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%.spaceIntegralG-%$IntG6$-F&6$-F&6$**%$rhoG\"\" $-%$cosG6#%$phiG\"\"\"%#PiG!\"\"-%$sinGF1F3/F-;\"\"!F3/F2;F:,$F4#F3\" \"#/%&thetaG;F:,$F4#F3\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value( spaceIntegral );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\" \"#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 510 53 "17. Find the surface area \+ of that part of the sphere" }{TEXT 513 1 " " }{XPPEDIT 514 1 "x^2+y^2 +z^2=2" "/,(*$%\"xG\"\"#\"\"\"*$%\"yGF&F'*$%\"zGF&F'F&" }{TEXT 512 29 " that lies inside the cone " }{XPPEDIT 516 1 "z=sqrt(x^2+y^2)" "/% \"zG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+" }{TEXT 515 1 "." } {TEXT 511 3 " \n" }{TEXT 509 7 "\na) " }{XPPEDIT 581 1 "Pi" "I#PiG %*protectedG" }{TEXT 580 1 " " }{TEXT 582 23 " \nb) \+ " }{XPPEDIT 584 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 583 1 " " } {TEXT 585 26 " \nc) " }{XPPEDIT 587 1 "sqrt(2)*Pi " "*&-%%sqrtG6#\"\"#\"\"\"%#PiGF'" }{TEXT 586 1 " " }{TEXT 588 26 " \+ \nd) " }{XPPEDIT 590 1 "2*Pi(1+sqrt(2))" "*&\"\"#\" \"\"-%#PiG6#,&F$F$-%%sqrtG6#F#F$F$" }{TEXT 589 1 " " }{TEXT 591 28 " \+ \ne) " }{XPPEDIT 593 1 "2*Pi(2+sqrt(2))" "*&\"\" #\"\"\"-%#PiG6#,&F#F$-%%sqrtG6#F#F$F$" }{TEXT 592 1 " " }{TEXT 594 11 " \nf) " }{XPPEDIT 572 1 "2*Pi*(2-sqrt(2))" "*(\"\"#\"\"\"%#PiGF $,&F#F$-%%sqrtG6#F#!\"\"F$" }{TEXT 571 1 " " }{TEXT 573 12 " \ng) " }{XPPEDIT 575 1 "2*Pi*(4-sqrt(2))" "*(\"\"#\"\"\"%#PiGF$,&\"\"%F$ -%%sqrtG6#F#!\"\"F$" }{TEXT 574 1 " " }{TEXT 576 13 " \nh) " } {XPPEDIT 578 1 "2*Pi*(sqrt(2)-1)" "*(\"\"#\"\"\"%#PiGF$,&-%%sqrtG6#F#F $F$!\"\"F$" }{TEXT 577 1 " " }{TEXT 579 16 " \ni) " } {XPPEDIT 596 1 "Pi*(2*sqrt(2)-1)" "*&%#PiG\"\"\",&*&\"\"#F$-%%sqrtG6#F 'F$F$F$!\"\"F$" }{TEXT 595 1 " " }{TEXT 597 17 " \nj) " } {XPPEDIT 599 1 "2*Pi*(2*sqrt(2)-1)" "*(\"\"#\"\"\"%#PiGF$,&*&F#F$-%%sq rtG6#F#F$F$F$!\"\"F$" }{TEXT 598 1 " " }{TEXT 600 5 " " }{TEXT 508 1 "\n" }}{PARA 0 "" 0 "" {TEXT 674 15 "Solution: (f)\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := (x,y) -> sqrt (2-x^2-y^2); #surface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF)-%%sqrtG6#,(\"\"#\"\"\"*$9$F1!\"\"*$9 %F1F5F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "SA_factor := s qrt(1 + diff(f(x,y),x)^2 + diff(f(x,y),y)^2 ); #surface area factor" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*SA_factorG*$,(\"\"\"F'*&,(\"\"#F'* $%\"xGF*!\"\"*$%\"yGF*F-F-F,F*F'*&F)F-F/F*F'#F'F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 74 "SA_factor := subs( \{x=r*cos(t),y=r*sin(t)\} ,SA_factor); #polar coordinates" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *SA_factorG*$,(\"\"\"F'*(,(\"\"#F'*&%\"rGF*-%$cosG6#%\"tGF*!\"\"*&F,F* -%$sinGF/F*F1F1F,F*F-F*F'*(F)F1F,F*F3F*F'#F'F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "SA_factor := simplify(SA_factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*SA_factorG*$,$*$,&!\"#\"\"\"*$%\"rG\"\"#F*! \"\"F)#F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "SA := Int(In t(SA_factor*r,r=0..1),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#SAG-%$IntG6$-F&6$*&,$*$,&!\"#\"\"\"*$%\"rG\"\"#F/!\"\"F.#F/F2F1F//F1 ;\"\"!F//%\"tG;F7,$%#PiGF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "SA := value(SA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SAG,&%#PiG \"\"%*&F&\"\"\"\"\"##F)F*!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 518 47 "18. Find the surface area of th at part of the " }{TEXT 603 5 "plane" }{TEXT 607 1 " " }{XPPEDIT 604 1 "z =sqrt(737)+4*x+8*y" "/%\"zG,(-%%sqrtG6#\"$P(\"\"\"*&\"\"%F)%\"xGF )F)*&\"\")F)%\"yGF)F)" }{TEXT 602 111 " that lies above the region o f the xy-plane that is made up of each point with nonnegative polar c oordinate " }{XPPEDIT 631 1 "r" "I\"rG6\"" }{TEXT 630 1 " " }{TEXT 629 19 " that satisfies " }{XPPEDIT 606 1 "r<=1+cos(theta)" "1%\"rG ,&\"\"\"F%-%$cosG6#%&thetaGF%" }{TEXT 605 1 "." }{TEXT 601 2 " \n" } {TEXT 517 1 "\n" }{TEXT 618 4 "a) " }{XPPEDIT 619 1 "Pi" "I#PiG%*prot ectedG" }{TEXT 608 16 " b) " }{XPPEDIT 620 1 "3*Pi/2" "*( \"\"$\"\"\"%#PiGF$\"\"#!\"\"" }{TEXT 609 19 " c) " } {XPPEDIT 621 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 610 22 " \+ d) " }{XPPEDIT 622 1 "5*Pi/2" "*(\"\"&\"\"\"%#PiGF$\"\"#!\"\" " }{TEXT 611 17 " e) " }{XPPEDIT 623 1 "9*Pi/2" "*(\"\"* \"\"\"%#PiGF$\"\"#!\"\"" }{TEXT 612 13 " \nf) " }{XPPEDIT 624 1 "9*Pi" "*&\"\"*\"\"\"%#PiGF$" }{TEXT 613 14 " g) " } {XPPEDIT 625 1 "21*Pi/2" "*(\"#@\"\"\"%#PiGF$\"\"#!\"\"" }{TEXT 614 17 " h) " }{XPPEDIT 626 1 "12*Pi" "*&\"#7\"\"\"%#PiGF$" } {TEXT 615 17 " i) " }{XPPEDIT 627 1 "27*Pi/2" "*(\"#F\"\" \"%#PiGF$\"\"#!\"\"" }{TEXT 616 16 " j) " }{XPPEDIT 628 1 "15*Pi" "*&\"#:\"\"\"%#PiGF$" }{TEXT 617 7 " \n\n" }}{PARA 0 "" 0 "" {TEXT 672 15 "Solution: (i)\n" }}{PARA 3 "" 0 "" {TEXT 673 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := (x,y) -> sqrt(737)+4* x+8*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF),(-%%sqrtG6#\"$P(\"\"\"9$\"\"%9%\"\")F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "SA := Int(Int(sqrt(1+Diff(f(x,y),x) ^2+Diff(f(x,y),y)^2)*r,r=0..1+cos(theta)),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SAG-%$IntG6$-F&6$*&,(\"\"\"F,*$-%%DiffG6$,(* $\"$P(#F,\"\"#F,%\"xG\"\"%%\"yG\"\")F6F5F,*$-F/6$F1F8F5F,F4%\"rGF,/F=; \"\"!,&F,F,-%$cosG6#%&thetaGF,/FE;F@,$%#PiGF5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(value(SA));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"#F\"\"#" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 519 3 "19." }{TEXT 635 37 " Use polar co ordinates to convert " }{XPPEDIT 521 1 "int(int(f(x,y),x=y..sqrt(4- y^2)), y=1..sqrt(2) )" "-%$intG6$-F#6$-%\"fG6$%\"xG%\"yG/F*;F+-%%sqrtG 6#,&\"\"%\"\"\"*$F+\"\"#!\"\"/F+;F3-F/6#F5" }{TEXT 520 32 " \nto an in tegral of the form " }{XPPEDIT 633 1 "int(int( F(r,theta),r=g(theta )..h(theta)), theta=alpha..beta)" "-%$intG6$-F#6$-%\"FG6$%\"rG%&thetaG /F*;-%\"gG6#F+-%\"hG6#F+/F+;%&alphaG%%betaG" }{TEXT 632 14 " . What \+ is " }{XPPEDIT 636 1 "g(theta)" "-%\"gG6#%&thetaG" }{TEXT 634 8 "?\n \+ \na) " }{XPPEDIT 647 1 "sin(theta)" "-%$sinG6#%&thetaG" }{TEXT 637 15 " b) " }{XPPEDIT 648 1 "cos(theta)" "-%$cosG6#%&thetaG" }{TEXT 638 15 " c) " }{XPPEDIT 649 1 "tan(theta)" "-%$tanG6 #%&thetaG" }{TEXT 639 11 " d) " }{XPPEDIT 650 1 "cot(theta)" "- %$cotG6#%&thetaG" }{TEXT 640 12 " e) " }{XPPEDIT 651 1 "sec(th eta)" "-%$secG6#%&thetaG" }{TEXT 641 10 " \nf) " }{XPPEDIT 652 1 "csc(theta)" "-%$cscG6#%&thetaG" }{TEXT 642 15 " g) " } {XPPEDIT 653 1 "r*sin(theta)" "*&%\"rG\"\"\"-%$sinG6#%&thetaGF$" } {TEXT 643 12 " h) " }{XPPEDIT 654 1 "1" "\"\"\"" }{TEXT 644 24 " i) " }{XPPEDIT 655 1 "sqrt(2) " "-%%sqrtG6 #\"\"#" }{TEXT 645 25 " j) " }{XPPEDIT 656 1 "2" "\"\"#" }{TEXT 646 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 671 15 "Solution: (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "pl1 := plot([sqrt(4-y^2),y,y=1..sqrt(2)],scaling=con strained,view=[0..2,0..2]):\npl2 :=plot([y,y,y=1..sqrt(2)],scaling=con strained,view=[0..2,0..2]):\npl3 := plot(1,x=1..sqrt(3)):\ndisplay(pl1 ,pl2,pl3);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7S7$$\"1x)ov!30 K#>Ad-\"F*7$$\"16UVt2g6z#>iq\"F*$\"1Y22aLYV5F*7$$\"1/\"y*)ft6q\"F*$\"1.&z ILp;0\"F*7$$\"1&))e#G9*ep\"F*$\"1;U.\\h;g5F*7$$\"1]R07jO!p\"F*$\"1,%\\ #oN&*o5F*7$$\"1:xqr_z%o\"F*$\"1=(*41Grx5F*7$$\"1(G3)4y**y;F*$\"1ef()eF s'3\"F*7$$\"1\\i)))fMQn\"F*$\"1!\\&=I(eY4\"F*7$$\"1x'*4Vx&zm\"F*$\"1J. 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