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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 "Tim es" 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 "" 0 260 1 {CSTYLE "" -1 -1 "" 1 18 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 " Math 217 Fall 2001 Exam \+ 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 48 "Notational Remarks: In this exam, the s ymbol " }{XPPEDIT 262 1 "diff(y(x),x)" "-%%diffG6$-%\"yG6#%\"xGF(" } {TEXT 261 11 " means " }{XPPEDIT 263 0 "dy/dx;" "*&%#dyG\"\"\"%#dx G!\"\"" }{TEXT 260 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 282 3 "If " }{XPPEDIT 283 1 "M;" "I\"MG6\"" }{TEXT 284 19 " is a matrix, then " }{XPPEDIT 285 1 "M[i,j];" "&%\"MG6$%\"iG%\"jG " }{TEXT 286 17 " is the entry of " }{XPPEDIT 287 1 "M;" "I\"MG6\"" } {TEXT 288 21 " in the i'th row and " }{XPPEDIT 280 1 "j;" "I\"jG6\"" } {TEXT 281 10 "'th column" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 363 77 "1. Suppose that f(t) = 2 for ln(2) < t < ln(3) and 0 oth erwise. If " }{XPPEDIT 361 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 362 301 " is the Laplace transform of f(t) then what is F(1) ?\n\na) 3 /ln(3) - 2/ln(2) \nb) 2 \nc) 1 \nd) ln(3) - \+ ln(2) \ne) 3ln(3) - 2ln(2) \nf) 1/3 \+ \ng) 2/3 \nh) 2/ln(2) - 2/ln(3) \ni) 1/ln(2) - 1/l n(3) \nj) ln(3)/3 - ln(2)/2\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 364 7 "Answer:" }{TEXT 365 5 " (f)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "F := s -> int(2*exp(-s*t), t = ln(2) .. ln(3) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"sG6\"6$%)operatorG%&arrowG F(-%$intG6$,$-%$expG6#,$*&9$\"\"\"%\"tGF6!\"\"\"\"#/F7;-%#lnG6#F9-F=6# \"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 367 9 "2. If " }{XPPEDIT 368 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 369 32 " is the Laplace transform of " }{XPPEDIT 370 1 "f (t)= 30*cos(3*t)+10*sin(3*t)" "/- %\"fG6#%\"tG,&*&\"#I\"\"\"-%$cosG6#*&\"\"$F*F&F*F*F**&\"#5F*-%$sinG6#* &F/F*F&F*F*F*" }{TEXT 371 24 " \n then what is " }{XPPEDIT 372 1 "F(4)" "-%\"FG6#\"\"%" }{TEXT 373 152 " ?\n\na) 2 b ) 3 c) 4 d) 5 e) 6 \nf) 8 \+ g) 10 h) 12 i) 16 j) 20" }}{PARA 3 "" 0 "" {TEXT 366 1 "\n" }}{PARA 0 "" 0 "" {TEXT 374 7 "Answer:" } {TEXT 375 5 " (e)" }{TEXT -1 2 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "with(inttrans):\nf := t -> 30*cos(3*t)+10*sin(3*t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "laplace(f(t),t,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"sG\"\"\",&*$F%\"\"#F&\"\"*F&!\" \"\"#I*$F'F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs( s = 4 , \" );" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 378 9 "3. If " }{XPPEDIT 379 1 "f(t)" "-%\"fG 6#%\"tG" }{TEXT 380 41 " is the inverse Laplace transform of " } {XPPEDIT 376 1 "(s+12)/(s*(s-3))" "*&,&%\"sG\"\"\"\"#7F%F%*&F$F%,&F$F% \"\"$!\"\"F%F*" }{TEXT 377 16 " then what is " }{XPPEDIT 382 1 "f(ln (2))" "-%\"fG6#-%#lnG6#\"\"#" }{TEXT 381 81 "? \n\na) 4\nb) 6\nc ) 8\nd) 12\ne) 16\nf) 18\ng) 24\nh) 30\ni) 36\nj) 42" } }{PARA 3 "" 0 "" {TEXT 383 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 384 7 "Answer:" }{TEXT 385 5 " (i)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "invlaplace( (s+12)/(s*(s-3)) , s, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$%\"tG\"\"$\" \"&!\"%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(t=ln( 2),\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$-%#lnG6#\"\"# \"\"$\"\"&!\"%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sim plify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 388 75 "4. Which of the following expressions is the inverse Laplace transform of " } {XPPEDIT 386 1 "(s+7)/(s^2+2*s+5)" "*&,&%\"sG\"\"\"\"\"(F%F%,(*$F$\"\" #F%*&F)F%F$F%F%\"\"&F%!\"\"" }{TEXT 387 11 " ?\n\na) " }{XPPEDIT 389 1 "2*exp(t)*cos(3*t)+ 3*exp(t)*sin(3*t) " ",&*(\"\"#\"\"\"-%$expG6 #%\"tGF%-%$cosG6#*&\"\"$F%F)F%F%F%*(F.F%-F'6#F)F%-%$sinG6#*&F.F%F)F%F% F%" }{TEXT 390 17 " \nb) " }{XPPEDIT 391 1 "exp(-t)*cos(2* t)+3*exp(-t)*sin(2*t)" ",&*&-%$expG6#,$%\"tG!\"\"\"\"\"-%$cosG6#*&\"\" #F*F(F*F*F**(\"\"$F*-F%6#,$F(F)F*-%$sinG6#*&F/F*F(F*F*F*" }{TEXT 392 23 " \nc) " }{XPPEDIT 393 1 "2*exp(t)*cos(2*t)+3*exp (t)*sin(2*t) " ",&*(\"\"#\"\"\"-%$expG6#%\"tGF%-%$cosG6#*&F$F%F)F%F%F% *(\"\"$F%-F'6#F)F%-%$sinG6#*&F$F%F)F%F%F%" }{TEXT 394 26 " \+ \nd) " }{XPPEDIT 395 1 "3*exp(t)*cos(2*t)+2*exp(t)*sin(2*t) \+ " ",&*(\"\"$\"\"\"-%$expG6#%\"tGF%-%$cosG6#*&\"\"#F%F)F%F%F%*(F.F%-F'6 #F)F%-%$sinG6#*&F.F%F)F%F%F%" }{TEXT 396 20 " \ne) " } {XPPEDIT 397 1 "2*exp(2*t)*cos(t) -3*exp(2*t)*sin(t) " ",&*( \"\"#\"\"\"-%$expG6#*&F$F%%\"tGF%F%-%$cosG6#F*F%F%*(\"\"$F%-F'6#*&F$F% F*F%F%-%$sinG6#F*F%!\"\"" }{TEXT 398 9 " \nf) " }{XPPEDIT 399 1 "2 *exp(2*t)*cos(3*t) -3*exp(2*t)*sin(3*t) " ",&*(\"\"#\"\"\"-%$expG6#* &F$F%%\"tGF%F%-%$cosG6#*&\"\"$F%F*F%F%F%*(F/F%-F'6#*&F$F%F*F%F%-%$sinG 6#*&F/F%F*F%F%!\"\"" }{TEXT 400 13 " \ng) " }{XPPEDIT 401 1 "2 *exp(-t)*cos(3*t) + 3*exp(-t)*sin(2*t) " ",&*(\"\"#\"\"\"-%$expG6#,$ %\"tG!\"\"F%-%$cosG6#*&\"\"$F%F*F%F%F%*(F0F%-F'6#,$F*F+F%-%$sinG6#*&F$ F%F*F%F%F%" }{TEXT 402 13 " \nh) " }{XPPEDIT 403 1 "3*exp(2*t) *cos(3*t) + 2*exp(3*t)*sin(2*t) " ",&*(\"\"$\"\"\"-%$expG6#*&\"\"#F% %\"tGF%F%-%$cosG6#*&F$F%F+F%F%F%*(F*F%-F'6#*&F$F%F+F%F%-%$sinG6#*&F*F% F+F%F%F%" }{TEXT 404 17 " \ni) " }{XPPEDIT 405 1 "3*exp(3* t)*cos(t) + 2*exp(3*t)*sin(t) " ",&*(\"\"$\"\"\"-%$expG6#*&F$F%%\" tGF%F%-%$cosG6#F*F%F%*(\"\"#F%-F'6#*&F$F%F*F%F%-%$sinG6#F*F%F%" } {TEXT 406 16 " \nj) " }{XPPEDIT 407 1 "2*exp(-t)*cos(2*t) + 3*exp(-t)*sin(2*t) " ",&*(\"\"#\"\"\"-%$expG6#,$%\"tG!\"\"F%-%$ cosG6#*&F$F%F*F%F%F%*(\"\"$F%-F'6#,$F*F+F%-%$sinG6#*&F$F%F*F%F%F%" } {TEXT 408 2 "\n " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 410 7 "Answer:" }{TEXT 411 5 " (b)" }{TEXT -1 2 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "invlaplace((s+7)/(s^2+2*s+5),s,t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$%\"tG!\"\"\"\"\"-%$cosG6#,$F)\"\" #F+F+*&F%F+-%$sinGF.F+\"\"$" }}}{PARA 3 "" 0 "" {TEXT 409 2 " \n" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 412 29 "5. The Laplace transform of " } {XPPEDIT 571 1 "f(t) = exp(t)/sqrt(Pi*t)" "/-%\"fG6#%\"tG*&-%$expG6#F& \"\"\"-%%sqrtG6#*&%#PiGF+F&F+!\"\"" }{TEXT 570 7 " is " }{XPPEDIT 435 1 "F(s)= 1/sqrt(s-1)" "/-%\"FG6#%\"sG*&\"\"\"F(-%%sqrtG6#,&F&F(F(! \"\"F-" }{TEXT 432 65 " . \nIf the Laplace transform of g(t) is " }{XPPEDIT 437 1 "G(s) = 1/2/(s-1)^(3/2)" "/-%\"GG 6#%\"sG*(\"\"\"F(\"\"#!\"\"),&F&F(F(F**&\"\"$F(F)F*F*" }{TEXT 436 18 " then what is " }{XPPEDIT 413 1 "g(4)" "-%\"gG6#\"\"%" }{TEXT 414 11 " ?\n\na) " }{XPPEDIT 415 1 "sqrt(Pi)*exp(4)" "*&-%%sqrtG6#%#Pi G\"\"\"-%$expG6#\"\"%F'" }{TEXT 416 17 " \n\nb) " } {XPPEDIT 574 1 "2*sqrt(Pi)*exp(4)" "*(\"\"#\"\"\"-%%sqrtG6#%#PiGF$-%$e xpG6#\"\"%F$" }{TEXT 417 24 " \n \nc) " }{XPPEDIT 418 1 "4*sqrt(Pi)*exp(4)" "*(\"\"%\"\"\"-%%sqrtG6#%#PiGF$-%$expG6#F#F$ " }{TEXT 419 27 " \n \nd) " }{XPPEDIT 420 1 "4*exp (4)/Pi" "*(\"\"%\"\"\"-%$expG6#F#F$%#PiG!\"\"" }{TEXT 421 21 " \n \+ \ne) " }{XPPEDIT 422 1 "2*exp(4)/Pi" "*(\"\"#\"\"\"-%$expG6# \"\"%F$%#PiG!\"\"" }{TEXT 423 10 " \n\nf) " }{XPPEDIT 424 1 "exp(4 )/Pi" "*&-%$expG6#\"\"%\"\"\"%#PiG!\"\"" }{TEXT 425 14 " \n\ng) \+ " }{XPPEDIT 426 1 "exp(4)/sqrt(Pi)" "*&-%$expG6#\"\"%\"\"\"-%%sqrtG6 #%#PiG!\"\"" }{TEXT 427 14 " \n\nh) " }{XPPEDIT 428 1 "2*exp(4 )/sqrt(Pi)" "*(\"\"#\"\"\"-%$expG6#\"\"%F$-%%sqrtG6#%#PiG!\"\"" } {TEXT 429 18 " \n \ni) " }{XPPEDIT 430 1 "4*exp(4)/sqrt(Pi) " "*(\"\"%\"\"\"-%$expG6#F#F$-%%sqrtG6#%#PiG!\"\"" }{TEXT 431 17 " \+ \n \nj) " }{XPPEDIT 433 1 "4*exp(4)/Pi^(3/2)" "*(\"\"%\"\"\"-%$ expG6#F#F$)%#PiG*&\"\"$F$\"\"#!\"\"F-" }{TEXT 434 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 572 7 "Answer:" }{TEXT 573 5 " (h)" }{TEXT -1 2 "\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "laplace(exp(t)/sqrt(Pi*t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&%\"sG\"\"\"!\"\"F&#F'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "laplace(t*exp(t)/sqrt(Pi*t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&%\"sG\"\"\"!\"\"F'#!\"$\"\"##F 'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "invlaplace(1/2/((s-1 )^(3/2)),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"tG#\"\"\"\"\"#% #PiG#!\"\"F'-%$expG6#F$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=4,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(\"\"%#\"\"\"\" \"#%#PiG#!\"\"F'-%$expG6#F$F&" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 480 17 "6. Suppose that " }{XPPEDIT 470 1 "X(s)" "-%\"XG6#% \"sG" }{TEXT 471 30 " is the Laplace transform of " }{XPPEDIT 472 1 " x (t)" "-%\"xG6#%\"tG" }{TEXT 477 11 " and that " }{XPPEDIT 484 1 "Xi (s)" "-%#XiG6#%\"sG" }{TEXT 481 32 " is the Laplace transform of " }{XPPEDIT 483 1 "x*`'`(t)" "*&%\"xG\"\"\"-%\"'G6#%\"tGF$" }{TEXT 482 10 ". If " }{XPPEDIT 473 1 "Xi(2)=14" "/-%#XiG6#\"\"#\"#9" } {TEXT 474 4 ", " }{XPPEDIT 485 1 "x(0)=-2" "/-%\"xG6#\"\"!,$\"\"#!\" \"" }{TEXT 486 10 ", and " }{XPPEDIT 476 1 "x*`'`(0)=-4" "/*&%\"xG \"\"\"-%\"'G6#\"\"!F%,$\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 475 15 "then what is " }{XPPEDIT 478 1 "X(2)" "-%\"XG6#\"\"#" }{TEXT 479 1 "?" } {TEXT -1 2 "\n\n" }{TEXT 469 141 "a) - 2 b) - 1 c) 0 \+ d) 1 e) 2 \n\nf) 3 g) 4 \+ h) 5 i) 6 j) 7 \n\n\n" }}{PARA 0 "" 0 "" {TEXT 487 7 "Answer:" }{TEXT 488 5 " (i)" }{TEXT -1 2 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Xi(s) = laplace(diff(x(t),t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#XiG6#%\"sG,&*&F'\"\"\"-%(laplaceG6%-%\"x G6#%\"tGF1F'F*F*-F/6#\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Xi(s) = s*X(s)-x(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#Xi G6#%\"sG,&*&F'\"\"\"-%\"XGF&F*F*-%\"xG6#\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "subs(\{Xi(s)=14, x(0)= -2, s=2\} , \" );" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"#9,&-%\"XG6#\"\"#F)F)\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X(2) = solve(\",X(2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"XG6#\"\"#\"\"'" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 438 1 "7" }{TEXT 467 1 "." }{TEXT 468 6 " If " }{XPPEDIT 439 1 "F(s)" "-%\"FG6#%\"sG" } {TEXT 440 42 " is the Laplace transform of\n\n " }{XPPEDIT 441 1 "f (t)= PIECEWISE([0, t < 2],[cos(t-2), otherwise])" "/-%\"fG6 #%\"tG-%*PIECEWISEG6$7$\"\"!2F&\"\"#7$-%$cosG6#,&F&\"\"\"F-!\"\"%*othe rwiseG" }{TEXT 442 19 " \n\nthen what is " }{XPPEDIT 443 1 "F(2)" " -%\"FG6#\"\"#" }{TEXT 444 10 " ?\n\n\na) " }{XPPEDIT 445 1 "exp(-4) " "-%$expG6#,$\"\"%!\"\"" }{TEXT 446 10 " b) " }{XPPEDIT 447 1 " exp(-4)/5" "*&-%$expG6#,$\"\"%!\"\"\"\"\"\"\"&F(" }{TEXT 448 10 " \+ c) " }{XPPEDIT 449 1 "2*exp(-4)/5" "*(\"\"#\"\"\"-%$expG6#,$\"\"%!\" \"F$\"\"&F*" }{TEXT 450 11 " d) " }{XPPEDIT 451 1 "3*exp(-4)/5 " "*(\"\"$\"\"\"-%$expG6#,$\"\"%!\"\"F$\"\"&F*" }{TEXT 452 11 " \+ e) " }{XPPEDIT 453 1 "4*exp(-4)/5" "*(\"\"%\"\"\"-%$expG6#,$F#!\"\"F$ \"\"&F)" }{TEXT 454 7 " \nf) " }{XPPEDIT 455 1 "exp(4)" "-%$expG6#\" \"%" }{TEXT 456 16 " g) " }{XPPEDIT 457 1 "exp(4)/5" "*&-% $expG6#\"\"%\"\"\"\"\"&!\"\"" }{TEXT 458 16 " h) " } {XPPEDIT 459 1 "2*exp(4)/5" "*(\"\"#\"\"\"-%$expG6#\"\"%F$\"\"&!\"\"" }{TEXT 460 16 " i) " }{XPPEDIT 461 1 "3*exp(4)/5" "*(\"\"$ \"\"\"-%$expG6#\"\"%F$\"\"&!\"\"" }{TEXT 462 19 " j) " }{XPPEDIT 463 1 "4*exp(4)/5" "*(\"\"%\"\"\"-%$expG6#F#F$\"\"&!\"\"" } {TEXT 464 5 " \n\n" }}{PARA 0 "" 0 "" {TEXT 465 7 "Answer:" }{TEXT 466 5 " (c)" }{TEXT -1 2 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "la place(Heaviside(t-2)*cos(t-2),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*(-%$expG6#,$%\"sG!\"#\"\"\"F(F*,&*$F(\"\"#F*F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=2,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"%#\"\"#\"\"&" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 489 7 "8. If " }{XPPEDIT 490 1 "x*`''(`*t*`)` + 2*x*`'(`*t*`)` + 3*x*`(`*t*`)`= 75" "/,(**%\" xG\"\"\"%$''(GF&%\"tGF&%\")GF&F&*,\"\"#F&F%F&%#'(GF&F(F&F)F&F&*,\"\"$F &F%F&%\"(GF&F(F&F)F&F&\"#v" }{TEXT 491 8 " , if " }{XPPEDIT 496 1 "x (0) = 5" "/-%\"xG6#\"\"!\"\"&" }{TEXT 494 9 ", and if " }{XPPEDIT 497 1 "D(x)(0)=4" "/--%\"DG6#%\"xG6#\"\"!\"\"%" }{TEXT 495 18 " , then wh at is " }{XPPEDIT 492 1 "X(3)" "-%\"XG6#\"\"$" }{TEXT 493 9 " (where " }{XPPEDIT 501 1 "X(s)" "-%\"XG6#%\"sG" }{TEXT 500 163 " is the Lap lace transform of x(t) ) ?\n\na) 2 b) 3 c) 4 \+ d) 5 e) 6\nf) 8 g) 9 h) 10 i) 12 \+ j) 15\n\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 498 7 "Answer:" }{TEXT 499 5 " (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := diff(x(t),t$2) + 2*diff(x(t),t) +3*x(t) = 75;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ eqnG/,(-%%diffG6$-F(6$-%\"xG6#%\"tGF/F/\"\"\"F*\"\"#F,\"\"$\"#v" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "laplace(eqn,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%\"sG\"\"\",&*&F&F'-%(laplaceG6%-%\"xG6 #%\"tGF0F&F'F'-F.6#\"\"!!\"\"F'F'--%\"DG6#F.F2F4F)\"\"#F1!\"#F*\"\"$,$ *$F&F4\"#v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "laplace(x(t), t,s) = solve(\",laplace(x(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%(laplaceG6%-%\"xG6#%\"tGF*%\"sG*(,**&F+\"\"#-F(6#\"\"!\"\"\"F3*&-- %\"DG6#F(F1F3F+F3F3*&F+F3F0F3F/\"#vF3F3F+!\"\",(*$F+F/F3F+F/\"\"$F3F; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs( \{s = 3, x(0) = 5 , D(x)(0) = 4\} , \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplace G6%-%\"xG6#%\"tGF*\"\"$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 502 8 "9. Let \+ " }{XPPEDIT 503 1 "H" "I\"HG6\"" }{TEXT 509 39 " denote the Heaviside \+ function. Let " }{XPPEDIT 505 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 506 31 " be the Laplace transform of " }{XPPEDIT 504 1 "f(t)=4*exp(4)*t^ 2*H(t-2)" "/-%\"fG6#%\"tG**\"\"%\"\"\"-%$expG6#F(F)F&\"\"#-%\"HG6#,&F& F)F-!\"\"F)" }{TEXT 510 14 ". What is " }{XPPEDIT 507 1 "F(2)" "-% \"FG6#\"\"#" }{TEXT 508 3 " ?\n" }}{PARA 3 "" 0 "" {TEXT 511 133 "\n\n a) 1 b) 3 c) 5 d) 7 e) 9\n\nf) 1 1 g) 13 h) 15 i) 17 j) 19 \n\n\n\n" } {TEXT 512 7 "Answer:" }{TEXT -1 5 " (g)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "laplace(4*exp(4)*t^2*Heaviside(t-2),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#\"\"%\"\"\",(*&-F&6#,$%\"sG!\"# F)F/!\"\"F(*&F,F)F/F0F(*&F,F)F/!\"$\"\"#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=2,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&-%$expG6#\"\"%\"\"\"-F&6#!\"%F)\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8 " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 513 18 "10. Suppose that " } {XPPEDIT 514 1 "f(t+2)=f(t)" "/-%\"fG6#,&%\"tG\"\"\"\"\"#F(-F$6#F'" } {TEXT 515 11 " for all " }{XPPEDIT 520 1 "t" "I\"tG6\"" }{TEXT 521 7 " and " }{XPPEDIT 516 1 "f (t)= exp(t) " "/-%\"fG6#%\"tG-%$expG6#F &" }{TEXT 517 8 " for " }{XPPEDIT 526 1 "t" "I\"tG6\"" }{TEXT 527 8 " in " }{XPPEDIT 524 1 "[0,1]" "7$\"\"!\"\"\"" }{TEXT 525 7 " a nd " }{XPPEDIT 549 1 "f (t)= 0 " "/-%\"fG6#%\"tG\"\"!" }{TEXT 550 8 " for " }{XPPEDIT 553 1 "t" "I\"tG6\"" }{TEXT 554 8 " in " } {XPPEDIT 551 1 "[1,2]" "7$\"\"\"\"\"#" }{TEXT 552 9 " . If " } {XPPEDIT 518 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 519 32 " is the Laplace \+ transform of " }{XPPEDIT 523 1 "f(t)" "-%\"fG6#%\"tG" }{TEXT -1 1 " \+ " }{TEXT 522 15 "then what is " }{XPPEDIT 528 1 "F(1)" "-%\"FG6#\"\" \"" }{TEXT 529 6 "?\n\na) " }{XPPEDIT 530 1 "(1-exp(-1))/(1-exp(-4))" "*&,&\"\"\"F$-%$expG6#,$F$!\"\"F)F$,&F$F$-F&6#,$\"\"%F)F)F)" }{TEXT -1 1 " " }{TEXT 531 16 " b) " }{XPPEDIT 532 1 "exp(1)/(1+e xp(1))" "*&-%$expG6#\"\"\"F&,&F&F&-F$6#F&F&!\"\"" }{TEXT -1 1 " " } {TEXT 533 10 " c) " }{XPPEDIT 534 1 "(1-exp(-2))/(1+exp(-2))" "* &,&\"\"\"F$-%$expG6#,$\"\"#!\"\"F*F$,&F$F$-F&6#,$F)F*F$F*" }{TEXT -1 1 " " }{TEXT 535 12 " d) " }{XPPEDIT 536 1 "exp(3)/(exp(4)-1) " "*&-%$expG6#\"\"$\"\"\",&-F$6#\"\"%F'F'!\"\"F," }{TEXT 537 15 " \+ e) " }{XPPEDIT 538 1 "(exp(1)+1)/(exp(1)-1)" "*&,&-%$expG6#\"\" \"F'F'F'F',&-F%6#F'F'F'!\"\"F+" }{TEXT 539 10 " \nf)" }{XPPEDIT 540 1 "(exp(1)-1)/(exp(1)+1)" "*&,&-%$expG6#\"\"\"F'F'!\"\"F',&-F%6#F' F'F'F'F(" }{TEXT 541 13 " g) " }{XPPEDIT 542 1 "2*exp(1)/(1+e xp(1))" "*(\"\"#\"\"\"-%$expG6#F$F$,&F$F$-F&6#F$F$!\"\"" }{TEXT -1 1 " " }{TEXT 543 13 " h) " }{XPPEDIT 544 1 "2*exp(1)/(2+exp(1)) " "*(\"\"#\"\"\"-%$expG6#F$F$,&F#F$-F&6#F$F$!\"\"" }{TEXT -1 1 " " } {TEXT 545 13 " i) " }{XPPEDIT 546 1 "exp(1)/(2*exp(1)-1)" "*& -%$expG6#\"\"\"F&,&*&\"\"#F&-F$6#F&F&F&F&!\"\"F," }{TEXT 547 11 " \+ j) " }{XPPEDIT 548 1 "(1-exp(-2))/(1-exp(-3))" "*&,&\"\"\"F$-%$expG 6#,$\"\"#!\"\"F*F$,&F$F$-F&6#,$\"\"$F*F*F*" }{TEXT -1 2 " \n" }}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }{TEXT 555 7 "Answer:" }{TEXT -1 6 " (a)\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "1/(1-exp(-p*s))*(Int(exp (t)*exp(-s*t),t=0..1)+Int(0*exp(s*t),t=1..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG6#,$*&%\"pGF%%\"sGF%!\"\"F-F-,&-%$In tG6$*&-F'6#%\"tGF%-F'6#,$*&F,F%F5F%F-F%/F5;\"\"!F%F%-F06$F " 0 "" {MPLTEXT 1 0 15 "lp := value(\");" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lpG*&,&\"\"\"F'-%$expG6#,$*&%\"pGF '%\"sGF'!\"\"F/F/,&*&,&F/F'F.F'F/-F)6#,&F'F'F.F/F'F/*$F2F/F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "lp := subs(\{p=2\},\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lpG*&,&\"\"\"F'-%$expG6#,$%\"sG!\"# !\"\"F.,&*&,&F.F'F,F'F.-F)6#,&F'F'F,F.F'F.*$F1F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(\{s=2\},lp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG6#!\"%!\"\"F*,&F%F%-F'6#F*F*F%" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 556 30 "11. The Laplace transform of " }{XPPEDIT 566 1 "J[0](t)" "-&%\"JG6#\"\"!6#%\"tG" }{TEXT 565 5 " is \+ " }{XPPEDIT 567 1 "1/ sqrt( s^2+1)" "*&\"\"\"F#-%%sqrtG6#,&*$%\"sG\"\" #F#F#F#!\"\"" }{TEXT 564 8 ". Let " }{XPPEDIT 558 1 "F(s)" "-%\"FG6# %\"sG" }{TEXT 559 34 " denote the Laplace transform of " }{XPPEDIT 560 1 "f(t)= sqrt(5)*Int( (t-u)*J[0](u),u=0..t)" "/-%\"fG6#%\"tG*&-%%s qrtG6#\"\"&\"\"\"-%$IntG6$*&,&F&F,%\"uG!\"\"F,-&%\"JG6#\"\"!6#F2F,/F2; F8F&F," }{TEXT 561 13 ". What is " }{XPPEDIT 562 1 "F(2)" "-%\"FG6# \"\"#" }{TEXT 563 1 "?" }{TEXT -1 1 "\n" }{TEXT 557 162 "\na) 1/3 \+ b) 2/3 c) 1/2 d) 2/5 e) 1/4 \+ \nf) 5 g) 5/2 h) 10 i) 3/2 \+ j) 9/4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 568 7 "Answer:" }{TEXT 569 5 " (e)" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "laplace(sqrt(5)*int((t-u)*BesselJ(0,u),u=0..t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(\"\"&#\"\"\"\"\"#,&*$%\"sGF'F&F&F&#!\"\"F' F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=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 589 9 "12. If " }{XPPEDIT 591 1 "x*`'(`*t*`)` + x*`(`*t*`)`= delta(t-5)" "/,&**%\"xG\"\"\"%#'(G F&%\"tGF&%\")GF&F&**F%F&%\"(GF&F(F&F)F&F&-%&deltaG6#,&F(F&\"\"&!\"\"" }{TEXT 592 7 " and " }{XPPEDIT 593 1 "x*`(`*0*`)`=0" "/**%\"xG\"\"\" %\"(GF%\"\"!F%%\")GF%F'" }{TEXT 594 16 " then what is " }{XPPEDIT 595 1 "x*`(`*6*`)`" "**%\"xG\"\"\"%\"(GF$\"\"'F$%\")GF$" }{TEXT 596 1 "?" }}{PARA 3 "" 0 "" {TEXT 590 132 "a) 1 b) 3 c) 6 \+ d) e e) 3e \nf) 6e g) 1/e h) 3/e i) 6/e j) 6+e \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT 597 7 "Answer:" }{TEXT -1 5 " (g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "laplace(diff (x(t),t)+ x(t) = Dirac(t-5), t, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,(*&%\"sG\"\"\"-%(laplaceG6%-%\"xG6#%\"tGF.F&F'F'-F,6#\"\"!!\"\"F(F' -%$expG6#,$F&!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "laplac e(x(t),t,s) = solve( \" , laplace(x(t),t,s) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(laplaceG6%-%\"xG6#%\"tGF*%\"sG*&,&-F(6#\"\"!\"\"\"- %$expG6#,$F+!\"&F1F1,&F+F1F1F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(x(0)=0, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%(laplaceG6%-%\"xG6#%\"tGF*%\"sG*&-%$expG6#,$F+!\"&\"\"\",&F+F2F2F2 !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "invlaplace(\", s, \+ t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&-%*HeavisideG6# ,&F'\"\"\"!\"&F-F--%$expG6#,&F'!\"\"\"\"&F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs( t = 6 , \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#\"\"'*&-%*HeavisideG6#\"\"\"F,-%$expG6#!\"\"F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#\"\"'-%$expG6#!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 575 19 "13. Suppose that " }{XPPEDIT 577 1 "x* `'(`*t*`)` = 2*x(t)-y(t),` `*y*`'(`*t*`)` = x(t)+y(t)" "6$/**%\"xG \"\"\"%#'(GF&%\"tGF&%\")GF&,&*&\"\"#F&-F%6#F(F&F&-%\"yG6#F(!\"\"/*,%%~ ~~~GF&F0F&F'F&F(F&F)F&,&-F%6#F(F&-F06#F(F&" }{TEXT 578 18 " \n \+ and " }{XPPEDIT 579 1 "x*`(`*0*`)`=3,` `*y*`(`*0*`)`=0" "6$/**% \"xG\"\"\"%\"(GF&\"\"!F&%\")GF&\"\"$/*,%%~~~~GF&%\"yGF&F'F&F(F&F)F&F( " }{TEXT 580 8 " . If " }{XPPEDIT 581 1 "Y(s)" "-%\"YG6#%\"sG" } {TEXT 582 31 " is the Laplace transform of " }{XPPEDIT 583 1 "y*`(`* t*`)`" "**%\"yG\"\"\"%\"(GF$%\"tGF$%\")GF$" }{TEXT 584 21 ", then what is\n " }{XPPEDIT 585 1 "Y(3)" "-%\"YG6#\"\"$" }{TEXT 586 2 "? " }{TEXT -1 2 "\n\n" }{TEXT 576 113 "a) 1 b) 2 c) 3 \+ d) 4 e) 5 \nf) 6 g) 7 h) 8 i) 9 \+ j) 10 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 587 7 "An swer:" }{TEXT 588 5 " (a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn1 := diff (x(t),t) = 2*x(t)-y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-% %diffG6$-%\"xG6#%\"tGF,,&F)\"\"#-%\"yGF+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn2 := diff(y(t),t) = x(t)+y(t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn2G/-%%diffG6$-%\"yG6#%\"tGF,,&-%\"xGF+\"\" \"F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn3 := laplace(e qn1,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*&%\"sG\"\"\"- %(laplaceG6%-%\"xG6#%\"tGF0F(F)F)-F.6#\"\"!!\"\",&F*\"\"#-F+6%-%\"yGF/ F0F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn4 := laplace(e qn2,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,&*&%\"sG\"\"\"- %(laplaceG6%-%\"yG6#%\"tGF0F(F)F)-F.6#\"\"!!\"\",&-F+6%-%\"xGF/F0F(F)F *F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn3 := subs(x(0) = \+ 3, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*&%\"sG\"\"\"- %(laplaceG6%-%\"xG6#%\"tGF0F(F)F)!\"$F),&F*\"\"#-F+6%-%\"yGF/F0F(!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn4 := subs(y(0) = 0, \+ eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/*&%\"sG\"\"\"-%(lap laceG6%-%\"yG6#%\"tGF/F'F(,&-F*6%-%\"xGF.F/F'F(F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "eqn3 := subs( \{laplace(x(t),t,s) = X(s) \+ , laplace(y(t),t,s) = Y(s)\} , eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*&%\"sG\"\"\"-%\"XG6#F(F)F)!\"$F),&F*\"\"#-%\"YGF,!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "eqn4 := subs( \{laplace (x(t),t,s) = X(s) , laplace(y(t),t,s) = Y(s)\} , eqn4);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn4G/*&%\"sG\"\"\"-%\"YG6#F'F(,&-%\"XGF+F(F) F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve( \{eqn3,eqn4\}, \{X(s),Y(s)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"YG6#%\"sG,$ *$,(*$F(\"\"#\"\"\"F(!\"$\"\"$F.!\"\"F0/-%\"XGF',$*&,&F(F.F1F.F.F+F1F0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs( s = 3, Y(s) = 3/( s^2-3*s+3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"YG6#\"\"$\"\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 293 19 "14. Suppose that " }{XPPEDIT 294 1 "A = MATRIX([[1, 2, -3,1], [2, -1,0,0], [2, 2,1,-1]]);" "/%\"AG- %'MATRIXG6#7%7&\"\"\"\"\"#,$\"\"$!\"\"F)7&F*,$F)F-\"\"!F07&F*F*F),$F)F -" }{TEXT 295 9 " and " }{XPPEDIT 296 1 "B = MATRIX([[3, 4,1], [4, 3,1],[6,1,1],[1,-2,0] ]);" "/%\"BG-%'MATRIXG6#7&7%\"\"$\"\"%\"\"\"7%F *F)F+7%\"\"'F+F+7%F+,$\"\"#!\"\"\"\"!" }{TEXT 297 9 ". If " } {XPPEDIT 298 1 "C = AB;" "/%\"CG%#ABG" }{TEXT 299 19 ", then what is " }{XPPEDIT 300 1 "Sum(C[j,j] , j=1..n);" "-%$SumG6$&%\"CG6$%\"jGF( /F(;\"\"\"%\"nG" }{TEXT 301 12 " where " }{XPPEDIT 355 1 "n" "I\" nG6\"" }{TEXT 356 30 " is the number of rows of " }{XPPEDIT 353 1 "C " "I\"CG6\"" }{TEXT 354 8 ". \n\n\na) " }{XPPEDIT 302 1 "1;" "\"\" \"" }{TEXT 303 18 " b) " }{XPPEDIT 304 1 "2;" "\"\"#" } {TEXT 305 16 " c) " }{XPPEDIT 306 1 "3;" "\"\"$" }{TEXT 307 17 " d) " }{XPPEDIT 308 1 "4;" "\"\"%" }{TEXT 309 16 " e) " }{XPPEDIT 310 1 "5;" "\"\"&" }{TEXT 311 28 " \+ \nf) " }{XPPEDIT 312 1 "6;" "\"\"'" }{TEXT 313 18 " \+ g) " }{XPPEDIT 314 1 "7;" "\"\"(" }{TEXT 315 17 " \+ h) " }{XPPEDIT 316 1 "8;" "\"\")" }{TEXT 317 16 " i) " }{XPPEDIT 318 1 "9;" "\"\"*" }{TEXT 319 17 " j) " } {XPPEDIT 320 1 "10;" "\"#5" }{TEXT -1 3 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 359 7 "Answer:" } {TEXT -1 7 " (d) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for \+ norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "A := matrix([[1, 2, -3 , 1], [2, -1, 0, 0], [2, 2, 1, -1]]):\nB := matrix([[3, 4, 1], [4, 3, \+ 1], [6, 1, 1], [1, -2, 0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG -%'MATRIXG6#7&7%\"\"$\"\"%\"\"\"7%F+F*F,7%\"\"'F,F,7%F,!\"#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C := evalm(A &* B);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7%7%!\"'\"\"&\"\"!7% \"\"#F+\"\"\"7%\"#>\"# " 0 "" {MPLTEXT 1 0 27 "sum(C[i,i],i=1..rowdim(C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 321 35 "15. Evaluate the determ inant of " }{XPPEDIT 322 1 "MATRIX( [[3, 2, 1], [2, 3, -1], [4, 6 ,-1]] );" "-%'MATRIXG6#7%7%\"\"$\"\"#\"\"\"7%F(F',$F)!\"\"7%\"\"%\" \"',$F)F," }{TEXT 323 7 ".\n\na) " }{XPPEDIT 324 0 "1;" "\"\"\"" } {TEXT 276 12 " b) " }{XPPEDIT 325 0 "2;" "\"\"#" }{TEXT 277 12 " c) " }{XPPEDIT 326 0 "3;" "\"\"$" }{TEXT 278 12 " \+ d) " }{XPPEDIT 327 0 "4;" "\"\"%" }{TEXT 279 13 " e) " } {XPPEDIT 328 0 "5;" "\"\"&" }{TEXT 273 10 " \nf) " }{XPPEDIT 329 0 "6;" "\"\"'" }{TEXT 272 13 " g) " }{XPPEDIT 330 0 "7;" "\" \"(" }{TEXT 271 12 " h) " }{XPPEDIT 331 0 "8;" "\"\")" }{TEXT 274 11 " i) " }{XPPEDIT 332 0 "9;" "\"\"*" }{TEXT 275 14 " \+ j) " }{XPPEDIT 333 0 "10;" "\"#5" }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 3 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 360 7 "Answer:" }{TEXT -1 6 " (e) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C := matrix([[3, 2, 1], [2, 3, -1], [4, 6, -1]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7%7%\"\"$\"\"#\" \"\"7%F+F*!\"\"7%\"\"%\"\"'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 334 18 "16. Suppose that " }{XPPEDIT 335 1 "A = MATRIX( [ [1,0,-3], [ 1,-2,0],[1,1,-4]] );" "/%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"!,$\"\"$!\"\"7 %F),$\"\"#F-F*7%F)F),$\"\"%F-" }{TEXT 336 8 ". If " }{XPPEDIT 337 1 "B=A^(-1);" "/%\"BG)%\"AG,$\"\"\"!\"\"" }{TEXT 338 16 " then what is " }{XPPEDIT 339 1 "B[1,3]" "&%\"BG6$\"\"\"\"\"$" }{TEXT 340 128 "? \n\na) 0 b) 1 c) 2 d) 3 e) 4 \n f) 5 g) 6 h) 7 i) 8 j) 9\n" }} {PARA 3 "" 0 "" {TEXT 357 2 "\n\n" }{TEXT 358 7 "Answer:" }{TEXT -1 5 " (g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A := matrix([[1, 0, -3], [1, -2, 0], [1, 1, -4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"!!\"$7%F*!\" #F+7%F*F*!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B := inver se(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7%7%!\")\" \"$\"\"'7%!\"%\"\"\"F+7%!\"$F/\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalm( A &* B ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'MATRIXG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 341 38 "17. When the th ird order equation " }{XPPEDIT 342 1 "x*`'''`(t)-5*x*`''`(t)-x(t) \+ = 0;" "/,(*&%\"xG\"\"\"-%$'''G6#%\"tGF&F&*(\"\"&F&F%F&-%#''G6#F*F&!\" \"-F%6#F*F0\"\"!" }{TEXT 343 79 " \n\nis converted in the standard wa y to a first order system\n \n " }{XPPEDIT 344 0 "MATRI X( [[x[1](t)], [x[2](t)] , [x[3](t)] ] )*`'` = A*MATRIX([[x[ 1](t)], [x[2](t)] ,[x[3](t)] ]);" "/*&-%'MATRIXG6#7%7#-&%\"xG6#\"\" \"6#%\"tG7#-&F+6#\"\"#6#F/7#-&F+6#\"\"$6#F/F-%\"'GF-*&%\"AGF--F%6#7%7# -&F+6#F-6#F/7#-&F+6#F46#F/7#-&F+6#F:6#F/F-" }{TEXT 345 9 " \n\nwith \+ " }{XPPEDIT 346 0 "x[1] = x;" "/&%\"xG6#\"\"\"F$" }{TEXT 347 10 ", wha t is " }{XPPEDIT 348 1 "A[3,3]" "&%\"AG6$\"\"$F%" }{TEXT 349 3 "? \n" }{TEXT -1 1 "\n" }{TEXT 257 5 "a) " }{XPPEDIT 256 0 "-5" ",$\"\"&!\" \"" }{TEXT 264 15 " b) " }{XPPEDIT 256 0 "-4" ",$\"\"%!\"\" " }{TEXT 267 15 " c) " }{XPPEDIT 256 0 "-3" ",$\"\"$!\"\"" }{TEXT 292 12 " d) " }{XPPEDIT 256 0 "-1" ",$\"\"\"!\"\"" } {TEXT 289 14 " e) " }{XPPEDIT 256 0 "0" "\"\"!" }{TEXT 269 16 " \nf) " }{XPPEDIT 256 0 "1" "\"\"\"" }{TEXT 265 17 " \+ g) " }{XPPEDIT 256 0 "2" "\"\"#" }{TEXT 291 17 " \+ h) " }{XPPEDIT 256 0 "3" "\"\"$" }{TEXT 266 16 " i) " } {XPPEDIT 256 0 "4" "\"\"%" }{TEXT 268 15 " j) " }{XPPEDIT 256 0 "5" "\"\"&" }{TEXT 290 6 " \n \n" }}{PARA 0 "" 0 "" {TEXT 648 7 "Answer:" }{TEXT 649 4 " (j)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x[1](t) := x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"xG6#\"\"\"6#%\"tG-F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x[2](t) := diff(x(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"xG6#\"\"#6#%\"tG-%%diffG6$-F&F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x[3](t) := diff(x(t),t$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"xG6#\"\"$6#%\"tG-%%diffG6$-F,6 $-F&F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "diff(x[3](t), t) = 5*x[3](t) + x[1](t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG 6$-F%6$-F%6$-%\"xG6#%\"tGF.F.F.,&F'\"\"&F+\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 352 17 "18. The matrix " }{XPPEDIT 19 1 "A;" "I\"AG6\"" }{TEXT 613 16 " of the system\n" } {TEXT 598 9 " " }{XPPEDIT 599 1 "diff(x(t),t) = 2*x + y;" "/-% %diffG6$-%\"xG6#%\"tGF),&*&\"\"#\"\"\"F'F-F-%\"yGF-" }{TEXT -1 8 " \n \+ " }{TEXT 614 2 " " }{XPPEDIT 615 1 "diff(y(t),t) = -x+4*y;" "/-% %diffG6$-%\"yG6#%\"tGF),&%\"xG!\"\"*&\"\"%\"\"\"F'F/F/" }{TEXT -1 7 " \+ \n" }{TEXT 600 52 "has which of the following numbers as an eigen value " }{XPPEDIT 617 1 "lambda" "I'lambdaG6\"" }{TEXT 616 7 "?\n\n\na ) " }{XPPEDIT 601 0 "-4;" ",$\"\"%!\"\"" }{TEXT 602 33 " b) -3 c) " }{XPPEDIT 603 0 "-2;" ",$\"\"#!\"\"" }{TEXT 604 32 " d) -1 e) " }{XPPEDIT 605 0 "0" "\"\"!" } {TEXT 606 35 " \nf) 1 g) " }{XPPEDIT 607 0 "2 ;" "\"\"#" }{TEXT 608 19 " h) " }{XPPEDIT 609 0 "3;" " \"\"$" }{TEXT 610 16 " i) " }{XPPEDIT 611 0 "4;" "\"\"%" } {TEXT 612 18 " j) 5" }}{PARA 3 "" 0 "" {TEXT 618 3 " \n\n " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(lin alg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M := matrix( [[2,1 ],[-1,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7$7$ \"\"#\"\"\"7$!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " eigenvals(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$F#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 350 18 "19. The matrix " }{XPPEDIT 19 1 "A;" "I\"AG 6\"" }{TEXT 640 16 " of the system\n" }{TEXT 637 9 " " } {XPPEDIT 638 1 "diff(x(t),t) = 3*x + y;" "/-%%diffG6$-%\"xG6#%\"tGF),& *&\"\"$\"\"\"F'F-F-%\"yGF-" }{TEXT -1 8 " \n " }{TEXT 641 2 " " }{XPPEDIT 642 1 "diff(y(t),t) = -x+y;" "/-%%diffG6$-%\"yG6#%\"tGF),&% \"xG!\"\"F'\"\"\"" }{TEXT -1 7 " \n" }{TEXT 639 17 "has eigenvec tor " }{XPPEDIT 643 1 "MATRIX([[-1],[1]])" "-%'MATRIXG6#7$7#,$\"\"\"! \"\"7#F(" }{TEXT 644 1 " " }{TEXT -1 2 ". " }{TEXT 646 63 "Correspondi ng to this eigenvector is a generalized eigenvector " }{XPPEDIT 645 1 "MATRIX([[1],[b]])" "-%'MATRIXG6#7$7#\"\"\"7#%\"bG" }{TEXT 647 13 ". W hat is b?" }{TEXT -1 3 "\n \n" }{TEXT 270 127 "a) -4 b) -3 \+ c) -2 d) -1 e) 0 \nf) 1 g) 2 h) \+ 3 i) 4 j) 5 \n\n\n\n\n\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M := matrix( [[3,1],[-1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7$7$\"\"$\"\"\"7$!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "linsolve(matrix( [[1,1],[-1,-1]]) , vector([-1,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$,&! \"\"\"\"\"&%#_tG6#F)F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(_t[1]=-2,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$ \"\"\"!\"#" }}}{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 351 44 "20. Suppose that A is a square matrix with " }{XPPEDIT 630 1 "(Au=lambda*u" "/%#AuG*&%'lambdaG\"\"\"%\"uGF&" }{TEXT 625 4 ", " } {XPPEDIT 631 1 "(Av=lambda*v + u" "/%#AvG,&*&%'lambdaG\"\"\"%\"vGF'F'% \"uGF'" }{TEXT 626 8 ", and " }{XPPEDIT 632 1 "(Aw=lambda*w + v" "/% #AwG,&*&%'lambdaG\"\"\"%\"wGF'F'%\"vGF'" }{TEXT 627 54 ".\nWhich of th e following are solutions of the system " }{XPPEDIT 629 1 "x*`'`(t)=A *`x` *`(`*t*`)`" "/*&%\"xG\"\"\"-%\"'G6#%\"tGF%*,%\"AGF%F$F%%\"(GF%F)F %%\")GF%" }{TEXT 628 7 "? \nI) " }{XPPEDIT 619 1 "exp(lambda*t)*(t*u+ v)" "*&-%$expG6#*&%'lambdaG\"\"\"%\"tGF(F(,&*&F)F(%\"uGF(F(%\"vGF(F(" }{TEXT 620 9 " \n\nII) " }{XPPEDIT 633 1 "exp(lambda*t)*(t*v+u)" "*& -%$expG6#*&%'lambdaG\"\"\"%\"tGF(F(,&*&F)F(%\"vGF(F(%\"uGF(F(" }{TEXT 634 8 " \nIII) " }{XPPEDIT 621 1 "exp(lambda*t)*(t*v+w)" "*&-%$expG6# *&%'lambdaG\"\"\"%\"tGF(F(,&*&F)F(%\"vGF(F(%\"wGF(F(" }{TEXT 622 7 " \+ \nIV) " }{XPPEDIT 623 1 "exp(lambda*t)*(t^2*u/2+w)" "*&-%$expG6#*&%'la mbdaG\"\"\"%\"tGF(F(,&*(F)\"\"#%\"uGF(F,!\"\"F(%\"wGF(F(" }{TEXT 624 9 " \n\nV) " }{XPPEDIT 635 1 "exp(lambda*t)*(t^2*u/2+tv+w)" "*&-%$e xpG6#*&%'lambdaG\"\"\"%\"tGF(F(,(*(F)\"\"#%\"uGF(F,!\"\"F(%#tvGF(%\"wG F(F(" }{TEXT 636 188 " \n\na) I, II b) I,III c) I,IV d) I,V e) II and III \nf) II and \+ IV g) II and V h) III and IV i) III and V j) IV and \+ V " }{TEXT -1 1 "\n" }{TEXT 258 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}}}{MARK "10 1 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }