{VERSION 2 3 "IBM INTEL NT" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2
D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "
" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 
0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 
0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 
0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 19 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 269 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 
0 0 0 0 0 0 0 0 }{CSTYLE "" 18 271 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 
0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 14 0 0 
0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 294 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "
" -1 295 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 14 
0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 297 "" 1 14 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 19 299 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
300 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 301 "" 1 14 0 0 
0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 1 14 0 0 0 0 0 1 0 0 0 0 
0 0 0 }{CSTYLE "" -1 303 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 304 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 18 308 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
309 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 310 "" 1 14 255 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE 
"" 19 313 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 1 
14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 315 "" 1 14 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 317 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
318 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 319 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 321 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 322 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 324 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 325 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 326 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
327 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 328 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 330 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 331 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 332 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 334 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 335 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
336 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 338 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 339 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 340 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 341 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 342 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 343 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 344 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
345 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 346 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 347 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 348 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 349 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 350 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 352 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 353 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
354 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 355 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 356 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 357 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 358 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 360 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 361 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 362 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
363 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 364 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 366 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 367 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 368 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 370 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 371 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
372 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 373 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 374 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 375 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 376 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 378 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 379 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 380 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
381 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 382 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 383 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 384 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 385 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 386 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 387 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 18 388 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 18 389 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 
390 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 391 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 392 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 18 393 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 18 394 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 395 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 396 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 397 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 398 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
399 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 400 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 401 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 402 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 403 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 404 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 405 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 406 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 407 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
408 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 409 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 410 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 411 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 412 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 413 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 414 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 415 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 416 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
417 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 418 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 419 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 420 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 18 421 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 422 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 423 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 424 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 18 425 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
426 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 427 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 428 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 18 429 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 430 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 431 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 432 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 18 433 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 434 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 
435 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 436 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 437 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 438 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 18 439 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 440 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 441 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 442 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 443 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 
444 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 445 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 446 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 447 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 448 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 449 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 450 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 451 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 452 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
453 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 454 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 455 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 456 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 457 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 458 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 459 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 460 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 461 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
462 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 463 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 464 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 465 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 466 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 467 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 468 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 469 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 470 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
471 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 472 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 473 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 474 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 475 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 476 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 477 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 478 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 479 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
480 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 481 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 482 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 483 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 484 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 485 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 486 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 487 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 488 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
489 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 490 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 491 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 492 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 493 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 494 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 495 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 496 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 497 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
498 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 499 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 500 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 501 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 502 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 503 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 504 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 505 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 506 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
507 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 508 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 509 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 510 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 511 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 512 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 513 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 514 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 515 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
516 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 517 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 518 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 519 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 520 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 521 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 522 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 523 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 524 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
525 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 526 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 527 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 528 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 529 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 530 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 531 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 532 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 533 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
534 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 535 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 536 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 537 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 538 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 539 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 540 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 541 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 542 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
543 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 544 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 545 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 546 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 547 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 548 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 549 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 550 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 551 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
552 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 553 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 554 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 555 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 556 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 557 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 558 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 559 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 560 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
561 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 562 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 563 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 564 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 565 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 566 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 567 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 568 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 569 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
570 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 571 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 572 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 573 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 574 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 575 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 576 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 577 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 578 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
579 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 580 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 581 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 582 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 583 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 584 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 585 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 586 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 587 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
588 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 589 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 590 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 591 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 592 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 593 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 594 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 595 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 596 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
597 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 598 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 599 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 600 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 601 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 602 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 603 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 604 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 605 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
606 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 607 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 608 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 609 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 610 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 611 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 612 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 613 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 614 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
615 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 616 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 617 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 618 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 619 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 620 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 621 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 622 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 19 623 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
624 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 625 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 626 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" 19 627 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 628 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 629 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 630 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" 19 631 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 632 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 
633 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 634 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 635 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 636 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" 19 637 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 638 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 639 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 }{CSTYLE "" -1 640 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE 
"" -1 641 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 642 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 643 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 }{CSTYLE "" -1 644 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE 
"" -1 645 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 646 "" 0 1 
0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 647 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 648 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }
{CSTYLE "" -1 649 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
650 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 651 "" 1 14 0 0 0 
0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 652 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 
0 }{CSTYLE "" -1 653 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
654 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 655 "" 0 1 0 0 0 
0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 656 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 
0 0 }{CSTYLE "" -1 657 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 658 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 659 "" 1 18 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 660 "" 1 18 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 661 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE 
"" -1 662 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 664 "" 1 
14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 665 "" 1 14 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 667 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 668 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
670 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 673 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 674 "" 1 14 0 0 0 0 0 0 0 0 0 0 
0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 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 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 "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 "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 "" 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 }{PSTYLE "" 0 261 
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 }{PSTYLE "" 0 262 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 }{PSTYLE "" 0 263 
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 265 47 "Notational Remarks:  In this exam, the s
ymbol  " }{XPPEDIT 268 1 "diff(y(x),x)" "-%%diffG6$-%\"yG6#%\"xGF(" }
{TEXT 267 9 " means   " }{XPPEDIT 269 0 "dy/dx;" "*&%#dyG\"\"\"%#dxG!
\"\"" }{TEXT 266 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT 296 3 "If " }{XPPEDIT 297 1 "M;" "I\"MG6\"" }{TEXT 298 19 " i
s a matrix, then " }{XPPEDIT 299 1 "M[i,j];" "&%\"MG6$%\"iG%\"jG" }
{TEXT 300 17 " is the entry of " }{XPPEDIT 301 1 "M;" "I\"MG6\"" }
{TEXT 302 21 " in the i'th row and " }{XPPEDIT 294 1 "j;" "I\"jG6\"" }
{TEXT 295 10 "'th column" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
321 17 "1. Suppose that  " }{XPPEDIT 322 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 323 7 "  and  " }
{XPPEDIT 324 1 "B = MATRIX([[3, 4,1], [4, 3,1],[1,1,1],[1,-2,0] ]);" "
/%\"BG-%'MATRIXG6#7&7%\"\"$\"\"%\"\"\"7%F*F)F+7%F+F+F+7%F+,$\"\"#!\"\"
\"\"!" }{TEXT 325 9 ".   If   " }{XPPEDIT 326 1 "C = AB;" "/%\"CG%#ABG
" }{TEXT 327 19 ",  then  what is   " }{XPPEDIT 328 1 "C[2,2];" "&%\"C
G6$\"\"#F%" }{TEXT 329 8 "?  \n\na) " }{XPPEDIT 330 1 "1;" "\"\"\"" }
{TEXT 331 18 "               b) " }{XPPEDIT 332 1 "2;" "\"\"#" }{TEXT 
333 16 "             c) " }{XPPEDIT 334 1 "3;" "\"\"$" }{TEXT 335 17 "
              d) " }{XPPEDIT 336 1 "4;" "\"\"%" }{TEXT 337 16 "       \+
     e)  " }{XPPEDIT 338 1 "5;" "\"\"&" }{TEXT 339 28 "               \+
        \nf)  " }{XPPEDIT 340 1 "6;" "\"\"'" }{TEXT 341 18 "          \+
     g) " }{XPPEDIT 342 1 "7;" "\"\"(" }{TEXT 343 17 "             h) \+
 " }{XPPEDIT 344 1 "8;" "\"\")" }{TEXT 345 16 "             i) " }
{XPPEDIT 346 1 "9;" "\"\"*" }{TEXT 347 17 "             j)  " }
{XPPEDIT 348 1 "10;" "\"#5" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 1 "
\n" }}{PARA 3 "" 0 "" {TEXT 641 6 "Answer" }{TEXT -1 8 ":  (e)  " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A := matrix([[1, 2, -3, 1], [2, -1,
 0, 0], [2, 2, 1, -1]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 
"B := matrix([[3, 4, 1], [4, 3, 1], [1, 1, 1], [1, -2, 0]]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C := evalm( A &* B);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7%7%\"\"*\"\"&\"\"!7
%\"\"#F+\"\"\"7%\"#9\"#<F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "C[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 3 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT -1 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 349 8 "2.  Let " }{XPPEDIT 350 1 "A;" "I\"AG6\"" }{TEXT 
351 5 " and " }{XPPEDIT 352 1 "B;" "I\"BG6\"" }{TEXT 353 49 "  be the \+
matrices of the preceding question. Let " }{XPPEDIT 354 1 "D = BA;" "/
%\"DG%#BAG" }{TEXT 355 10 ". What is " }{XPPEDIT 356 1 "D[4,4];" "&%\"
DG6$\"\"%F%" }{TEXT 357 7 "? \n\na) " }{XPPEDIT 358 1 "1" "\"\"\"" }
{TEXT 359 9 "      b) " }{XPPEDIT 360 1 "2" "\"\"#" }{TEXT 361 11 "   \+
    c)  " }{XPPEDIT 362 1 "3" "\"\"$" }{TEXT 363 10 "       d) " }
{XPPEDIT 364 1 "4" "\"\"%" }{TEXT 365 9 "      e) " }{XPPEDIT 366 1 "5
" "\"\"&" }{TEXT 367 11 "        f) " }{XPPEDIT 368 1 "6" "\"\"'" }
{TEXT 369 9 "     g)  " }{XPPEDIT 370 1 "7" "\"\"(" }{TEXT 371 20 "   \+
 \nh) The matrix  " }{XPPEDIT 372 1 "D;" "I\"DG%*protectedG" }{TEXT 
373 38 "  does not exist      \ni) The matrix  " }{XPPEDIT 374 1 "D;" 
"I\"DG%*protectedG" }{TEXT 375 28 "  does exist but the entry  " }
{XPPEDIT 376 1 "D[4,4]" "&%\"DG6$\"\"%F%" }{TEXT 377 70 "  does not ex
ist because there is no fourth row.       j) The matrix  " }{XPPEDIT 
378 1 "D;" "I\"DG%*protectedG" }{TEXT 379 28 "  does exist but the ent
ry  " }{XPPEDIT 380 1 "D[4,4]" "&%\"DG6$\"\"%F%" }{TEXT 381 51 "  does
 not exist because there is no fourth column." }}{PARA 3 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 639 6 "Answer" }{TEXT -1 7 ":  (
a) " }{TEXT 640 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eva
lm(B &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"#8\"
\"%!\")\"\"#7&\"#7\"\"(!#6\"\"$7&\"\"&F0!\"#\"\"!7&!\"$F)F6\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "\"[4,4];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 382 8 "3. If   " }{XPPEDIT 597 1 "F(s)" "-%\"FG6
#%\"sG" }{TEXT 598 42 "  is the Laplace transform of\n\n           " }
{XPPEDIT 599 1 "f (t)= PIECEWISE([0, t < 0 or 2-t < 0],[1, otherwise])
" "/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!52F&F+2,&\"\"#\"\"\"F&!\"\"F+7$
F1%*otherwiseG" }{TEXT 600 19 "  \n\nthen what is   " }{XPPEDIT 601 1 
"F(1/2)" "-%\"FG6#*&\"\"\"F&\"\"#!\"\"" }{TEXT 602 186 " ?\n\na)  2e  \+
           b)  1/2 + e       c)  2(1 + e)        d)  1 + 2/e        e)
  2 + 1/e \nf)  1 + 1/e       g)  2 + e          h) 1 + 2e            \+
i)  1 - 2/e         j)  2(1 - 1/e)\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 642 1 " " }}
{PARA 3 "" 0 "" {TEXT 643 6 "Answer" }{TEXT -1 7 ":  (j) " }{TEXT 644 
1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Int(1*exp(-s*t),t=0
..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*&%\"sG\"
\"\"%\"tGF,!\"\"/F-;\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"sG!\"\"-%$
expG6#,$F%!\"#\"\"\"F&*$F%F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "subs(s=1/2,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#
!\"\"!\"#\"\"#\"\"\"" }}}{PARA 3 "" 0 "" {TEXT 645 5 "    \n" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 383 33 "4. Ev
aluate the determinant of   " }{XPPEDIT 384 1 "MATRIX(   [[3, 2, 1], [
3, 3, -1], [-3, -2, 2]]   );" "-%'MATRIXG6#7%7%\"\"$\"\"#\"\"\"7%F'F',
$F)!\"\"7%,$F'F,,$F(F,F(" }{TEXT 385 7 ".\n\na)  " }{XPPEDIT 386 0 "1;
" "\"\"\"" }{TEXT 290 12 "        b)  " }{XPPEDIT 387 0 "2;" "\"\"#" }
{TEXT 291 12 "        c)  " }{XPPEDIT 388 0 "3;" "\"\"$" }{TEXT 292 
12 "        d)  " }{XPPEDIT 389 0 "4;" "\"\"%" }{TEXT 293 13 "        \+
e)   " }{XPPEDIT 390 0 "5;" "\"\"&" }{TEXT 287 10 "     \nf)  " }
{XPPEDIT 391 0 "6;" "\"\"'" }{TEXT 286 13 "         g)  " }{XPPEDIT 
392 0 "7;" "\"\"(" }{TEXT 285 12 "        h)  " }{XPPEDIT 393 0 "8;" "
\"\")" }{TEXT 288 11 "        i) " }{XPPEDIT 394 0 "9;" "\"\"*" }
{TEXT 289 14 "          j)  " }{XPPEDIT 395 0 "10;" "\"#5" }{TEXT -1 
1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 646 6 "
Answer" }{TEXT -1 7 ":  (i) " }{TEXT 647 1 "\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "det( matrix([[3, 2, 1], [3, 3, -1], [-3, -2, 2]]
) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{PARA 3 "" 0 "" 
{TEXT -1 19 "\nBy Row operations:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "A := matrix([[3, 2, 1], [3, 3, -1], [-3, -2, 2]]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"$\"\"#\"\"
\"7%F*F*!\"\"7%!\"$!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "A1 := addrow(A,1,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%
'MATRIXG6#7%7%\"\"$\"\"#\"\"\"7%F*F*!\"\"7%\"\"!F0F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "A2 := addrow(A1,1,2,-1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#A2G-%'MATRIXG6#7%7%\"\"$\"\"#\"\"\"7%\"\"!F,!\"
#7%F.F.F*" }}}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }{TEXT 648 52 "One can c
ompute det(A2) = (3)*(1)*(3) in one's head." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 396 7 "5. Let " }{XPPEDIT 
397 1 "A;" "I\"AG6\"" }{TEXT 398 3 ",  " }{XPPEDIT 399 1 "B;" "I\"BG6
\"" }{TEXT 400 6 ", and " }{XPPEDIT 401 1 "C;" "I\"CG6\"" }{TEXT 402 
82 "  be nonzero n x n   matrices. Which of these statements is always
 true: \n\nI)     " }{XPPEDIT 403 1 "AB = BA;" "/%#ABG%#BAG" }{TEXT 
404 12 "     \nII)   " }{XPPEDIT 405 1 "A*`(`*BC*`)` = `(`*A*B*`)`*C;
" "/**%\"AG\"\"\"%\"(GF%%#BCGF%%\")GF%*,F&F%F$F%%\"BGF%F(F%%\"CGF%" }
{TEXT 406 11 "    \nIII)  " }{XPPEDIT 407 1 "A" "I\"AG6\"" }{TEXT 408 
33 "  has a multiplicative inverse   " }{XPPEDIT 409 1 "A^(-1)" ")%\"A
G,$\"\"\"!\"\"" }{TEXT 410 179 ".\n\na)   I only        b)  II only   \+
  c)  III only       d) I and II only       e)  I and III only\nf)  II
 and III only           g) I, II, and III        h) None are always tr
ue \n" }}{PARA 3 "" 0 "" {TEXT 650 6 "Answer" }{TEXT -1 7 ":  (b) " }
{TEXT 651 1 "\n" }}{PARA 3 "" 0 "" {TEXT 649 1 "\n" }}{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 411 17 "6. Su
ppose that  " }{XPPEDIT 412 1 "A = MATRIX( [ [1,2,3], [1,3,-3],[0,1,-5
]]  );" "/%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%F)F+,$F+!\"\"7%\"\"!F
),$\"\"&F." }{TEXT 413 7 ".  If  " }{XPPEDIT 414 1 "B=A^(-1);" "/%\"BG
)%\"AG,$\"\"\"!\"\"" }{TEXT 415 16 " then what is   " }{XPPEDIT 416 1 
"B[2,3]" "&%\"BG6$\"\"#\"\"$" }{TEXT 417 132 "?\n\na)  0       b) 1   \+
      c) 2         d) 3          e) 4        \nf)  5        g) 6      \+
   h) 7         i)  8          j)  9\n\n\n\n\n" }{TEXT 652 6 "Answer" 
}{TEXT -1 7 ":  (g) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A :=
 matrix([[1, 2, 3], [1, 3, -3], [0, 1, -5]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%F*F,!\"$7%\"\"!
F*!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B := inverse(A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7%7%!#7\"#8!#:7%
\"\"&!\"&\"\"'7%\"\"\"!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "B[2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 3 "" 
0 "" {TEXT -1 4 "\nor\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Id := matrix([[1, 0, 0], [0, 1, 0],
 [0, 0, 1]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%'MATRIXG6#7%
7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "augA := augment(A,Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%a
ugAG-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F*F,!\"$F-F*F-7(F-F*!\"
&F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "augA1 := addrow(
augA,1,2,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augA1G-%'MATRIXG6#
7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F*!\"'!\"\"F*F-7(F-F*!\"&F-F-F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "augA2 := addrow(augA1,2,1,-2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augA2G-%'MATRIXG6#7%7(\"\"\"
\"\"!\"#:\"\"$!\"#F+7(F+F*!\"'!\"\"F*F+7(F+F*!\"&F+F+F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "augA3 := addrow(augA2,2,3,-1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augA3G-%'MATRIXG6#7%7(\"\"\"\"\"!\"
#:\"\"$!\"#F+7(F+F*!\"'!\"\"F*F+7(F+F+F*F*F1F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "augA4 := addrow(augA3,3,2,6); # We can actuall
y stop here." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augA4G-%'MATRIXG6#7
%7(\"\"\"\"\"!\"#:\"\"$!\"#F+7(F+F*F+\"\"&!\"&\"\"'7(F+F+F*F*!\"\"F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "augA5 := addrow(augA4,3,1
,-15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augA5G-%'MATRIXG6#7%7(\"
\"\"\"\"!F+!#7\"#8!#:7(F+F*F+\"\"&!\"&\"\"'7(F+F+F*F*!\"\"F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "\"[2,6]; " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 418 78 "7.  Which of the followi
ng numbers is a root of the indicial equation of\n \n   " }{XPPEDIT 
419 1 "4*x^2*diff(y(x),x,x)+3*x*diff(y(x),x)-15*Sum(x^n/2^(n+1),n=0..i
nfinity) *y(x)= 0" "/,(*(\"\"%\"\"\"*$%\"xG\"\"#F&-%%diffG6%-%\"yG6#F(
F(F(F&F&*(\"\"$F&F(F&-F+6$-F.6#F(F(F&F&*(\"#:F&-%$SumG6$*&)F(%\"nGF&)F
),&F=F&F&F&!\"\"/F=;\"\"!%)infinityGF&-F.6#F(F&F@FC" }{TEXT 420 10 "? \+
\n   \na) " }{XPPEDIT 421 0 "1/4" "*&\"\"\"F#\"\"%!\"\"" }{TEXT 422 
14 "          b)  " }{XPPEDIT 423 0 "1/2" "*&\"\"\"F#\"\"#!\"\"" }
{TEXT 424 14 "           c) " }{XPPEDIT 425 0 "3/4" "*&\"\"$\"\"\"\"\"
%!\"\"" }{TEXT 426 14 "           d) " }{XPPEDIT 427 0 "5/4" "*&\"\"&
\"\"\"\"\"%!\"\"" }{TEXT 428 18 "               e) " }{XPPEDIT 429 0 "
3/2" "*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 430 6 " \nf)  " }{XPPEDIT 431 0 
"7/4" "*&\"\"(\"\"\"\"\"%!\"\"" }{TEXT 432 16 "          g)    " }
{XPPEDIT 433 0 "9/4" "*&\"\"*\"\"\"\"\"%!\"\"" }{TEXT 434 14 "        \+
 h)   " }{XPPEDIT 435 0 "5/2" "*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT 436 16 
"         i)     " }{XPPEDIT 437 0 "11/4" "*&\"#6\"\"\"\"\"%!\"\"" }
{TEXT 438 14 "          j)  " }{XPPEDIT 439 0 "13/4" "*&\"#8\"\"\"\"\"
%!\"\"" }{TEXT 440 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 659 3 "(e)" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "4*x^2*dif
f(y(x),x,x)+3*x*diff(y(x),x)-15*Sum(x^n/(2^(n+1)),n = 0 .. infinity)*y
(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"#-%%diffG6$-F)
6$-%\"yG6#F&F&F&\"\"\"\"\"%*&F&F0F+F0\"\"$*&-%$SumG6$*&)F&%\"nGF0)F',&
F:F0F0F0!\"\"/F:;\"\"!%)infinityGF0F-F0!#:F@" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "\"/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"
xG\"\"#-%%diffG6$-F)6$-%\"yG6#F&F&F&\"\"\"F0*&F&F0F+F0#\"\"$\"\"%*&-%$
SumG6$*&)F&%\"nGF0)F',&F;F0F0F0!\"\"/F;;\"\"!%)infinityGF0F-F0#!#:F4FA
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"#-%%diffG6$-F)6$-%\"yG6#F&F&F&\"\"
\"F0*&F&F0F+F0#\"\"$\"\"%*&,&F&F0!\"#F0!\"\"F-F0#\"#:F4\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "indicial_equation := r*(r-1)
+3/4*r+(15/4/(0-2)) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2indicia
l_equationG/,(*&%\"rG\"\"\",&F(F)!\"\"F)F)F)F(#\"\"$\"\"%#!#:\"\")F)\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$%\"rG\"\"#\"\"\"F&#!\"\"\"\"%#!#:
\"\")F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"&\"\"%#\"\"$\"\"#" }}}{PARA 3 "
" 0 "" {TEXT -1 9 "\n\n\n\n\n\n\n\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 312 10 "8.  Let   \+
" }{XPPEDIT 313 1 "r" "I\"rG6\"" }{TEXT 314 66 "    denote the positiv
e root of the indicial equation of \n\n       " }{XPPEDIT 315 1 "x^2*d
iff(y(x),x,x)+x*diff(y(x),x)+ (x-1/4)*y(x)=0" "/,(*&%\"xG\"\"#-%%diffG
6%-%\"yG6#F%F%F%\"\"\"F-*&F%F--F(6$-F+6#F%F%F-F-*&,&F%F-*&F-F-\"\"%!\"
\"F7F--F+6#F%F-F-\"\"!" }{TEXT 316 35 ".\n\nFor the solution\n \n     \+
        " }{XPPEDIT 317 1 "y(x)=x^r+c[1]*x^(r+1)+`...`" "/-%\"yG6#%\"x
G,()F&%\"rG\"\"\"*&&%\"cG6#F*F*)F&,&F)F*F*F*F*F*%$...GF*" }{TEXT 318 
14 "   \n\nwhat is  " }{XPPEDIT 319 1 "c[1]" "&%\"cG6#\"\"\"" }{TEXT 
320 157 "?\n\n\na) -2/5         b) 2/5         c) -2/3           d)  2
/3        e) -1/2       \nf)  1/2         g) -1/3         h) 1/3      \+
      i) -3/5         j)  3/5\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 660 3 "(e)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "indicial_eqn := r*(r-1)+r-1/4 = 0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%-indicial_eqnG/,(*&%\"rG\"\"\",&F(F)!\"\"F)F)F)F(F)#F+\"\"%F)
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(indicial_eqn
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"\"##!\"\"F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:=1/2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "subs(y(x) = x^r+c[1]*x^(r+1)+c[2]*x^(r+2)+c[3]*x^(r+
3),x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x-1/4)*y(x)=0 );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"#-%%diffG6$-F)6$,**$F&#\"\"\"F'F0*&&
%\"cG6#F0F0F&#\"\"$F'F0*&&F36#F'F0F&#\"\"&F'F0*&&F36#F6F0F&#\"\"(F'F0F
&F&F0F0*&F&F0F+F0F0*&,&F&F0#!\"\"\"\"%F0F0F-F0F0\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,0*&&%\"cG6#\"\"\"F)%\"xG#\"\"$\"\"#F-*&&F'6#F-F)F*#\"
\"&F-\"\"'*&&F'6#F,F)F*#\"\"(F-\"#7*$F*F+F)*&F&F)F*F1F)*&F/F)F*F7F)*&F
5F)F*#\"\"*F-F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "col
lect(\",x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&&%\"cG6#\"\"$\"\"
\"%\"xG#\"\"*\"\"#F**&,&F&\"#7&F'6#F.F*F*F+#\"\"(F.F**&,&F2\"\"'&F'6#F
*F*F*F+#\"\"&F.F**&,&F9F.F*F*F*F+#F)F.F*\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "solve(2*c[1]+1=0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6##!\"\"\"\"#" }}}{PARA 3 "" 0 "" {TEXT 653 1 "\n" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT 441 37 "9. When the third order equation     " }{XPPEDIT 
442 1 "x*`'''`(t)+3*x*`''`(t)+2*x(t) = 0;" "/,(*&%\"xG\"\"\"-%$'''G6#%
\"tGF&F&*(\"\"$F&F%F&-%#''G6#F*F&F&*&\"\"#F&-F%6#F*F&F&\"\"!" }{TEXT 
443 79 "  \n\nis converted in the standard way to a first order system
\n   \n              " }{XPPEDIT 444 0 "MATRIX(   [[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#\"\"$6#F/F-%\"'GF-*&%\"AGF--F%6#7%7#-&F+6#F-6#F/7#-&F+6#F46#F/
7#-&F+6#F:6#F/F-" }{TEXT 445 9 "  \n\nwith " }{XPPEDIT 446 0 "x[1] = x
;" "/&%\"xG6#\"\"\"F$" }{TEXT 447 10 ", what is " }{XPPEDIT 448 1 "A[3
,2]" "&%\"AG6$\"\"$\"\"#" }{TEXT 449 3 "? \n" }{TEXT -1 1 "\n" }{TEXT 
257 5 "a)   " }{XPPEDIT 256 0 "0" "\"\"!" }{TEXT 272 15 "           b)
  " }{XPPEDIT 256 0 "1" "\"\"\"" }{TEXT 275 14 "           c) " }
{XPPEDIT 256 0 "2" "\"\"#" }{TEXT 306 15 "            d) " }{XPPEDIT 
256 0 "3" "\"\"$" }{TEXT 303 17 "             e)  " }{XPPEDIT 256 0 "4
" "\"\"%" }{TEXT 277 12 "       \nf)  " }{XPPEDIT 256 0 "-1" ",$\"\"\"
!\"\"" }{TEXT 273 14 "          g)  " }{XPPEDIT 256 0 "-2" ",$\"\"#!\"
\"" }{TEXT 305 12 "         h) " }{XPPEDIT 256 0 "-3" ",$\"\"$!\"\"" }
{TEXT 274 13 "         i)  " }{XPPEDIT 256 0 "-4" ",$\"\"%!\"\"" }
{TEXT 276 15 "           j)  " }{XPPEDIT 256 0 "-3/2" ",$*&\"\"$\"\"\"
\"\"#!\"\"F'" }{TEXT 304 3 "  \n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{PARA 260 "" 0 "" {TEXT 655 6 "Answer" }{TEXT -1 7 ":  (a) " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x[1] = x, x[2] = Diff(x[1],t), x[3]
 = Diff(x[2],t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"xG6#\"\"\"F%/
&F%6#\"\"#-%%DiffG6$F$%\"tG/&F%6#\"\"$-F-6$F)F/" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "Diff(x[3],t) = Diff(x,t$3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%%DiffG6$&%\"xG6#\"\"$%\"tG-F%6&F(F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lhs(\") = -3*Diff(x,t$2) -2*
x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&%\"xG6#\"\"$%\"tG,&-
F%6%F(F+F+!\"$F(!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "lhs
(\") = -3*x[3]-2*x[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&
%\"xG6#\"\"$%\"tG,&F'!\"$&F(6#\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "A := matrix([[0,1,0],[0,0,1],[-2,0,-3]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"!\"\"\"F*7%F*F*F+7
%!\"#F*!\"$" }}}{PARA 3 "" 0 "" {TEXT 654 3 "  \n" }}{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 450 10 "10.  \+
If   " }{XPPEDIT 451 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 452 42 "  is the \+
Laplace transform of\n\n           " }{XPPEDIT 453 1 "f (t)= PIECEWISE
([0,  t < 2],[sin(t-2), otherwise])" "/-%\"fG6#%\"tG-%*PIECEWISEG6$7$
\"\"!2F&\"\"#7$-%$sinG6#,&F&\"\"\"F-!\"\"%*otherwiseG" }{TEXT 454 19 "
  \n\nthen what is   " }{XPPEDIT 455 1 "F(1)" "-%\"FG6#\"\"\"" }{TEXT 
456 10 " ?\n\n\na)   " }{XPPEDIT 613 1 "1/e" "*&\"\"\"F#%\"eG!\"\"" }
{TEXT 614 17 "            b)   " }{XPPEDIT 615 1 "1/2/e" "*(\"\"\"F#\"
\"#!\"\"%\"eGF%" }{TEXT 616 14 "          c)  " }{XPPEDIT 617 1 "1/e^2
" "*&\"\"\"F#*$%\"eG\"\"#!\"\"" }{TEXT 618 16 "           d)   " }
{XPPEDIT 619 1 "1/2/e^2" "*(\"\"\"F#\"\"#!\"\"*$%\"eGF$F%" }{TEXT 620 
15 "           e)  " }{XPPEDIT 621 1 "2/e^2" "*&\"\"#\"\"\"*$%\"eGF#!
\"\"" }{TEXT 622 7 " \nf)   " }{XPPEDIT 623 1 "e" "I\"eG6\"" }{TEXT 
624 19 "              g)   " }{XPPEDIT 625 1 "2*e" "*&\"\"#\"\"\"%\"eG
F$" }{TEXT 626 15 "           h)  " }{XPPEDIT 627 1 "e^2" "*$%\"eG\"\"
#" }{TEXT 628 16 "            i)  " }{XPPEDIT 629 1 "2*e^2" "*&\"\"#\"
\"\"*$%\"eGF#F$" }{TEXT 630 19 "               j)  " }{XPPEDIT 631 1 "
e^2/2" "*&%\"eG\"\"#F$!\"\"" }{TEXT 632 4 "    " }}{PARA 3 "" 0 "" 
{TEXT 661 1 "\n" }}{PARA 3 "" 0 "" {TEXT 612 2 "\n\n" }{TEXT -1 3 "(d)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
664 1 "f (t)= PIECEWISE([0,  t < 2],[sin(t-2), otherwise])" "/-%\"fG6#
%\"tG-%*PIECEWISEG6$7$\"\"!2F&\"\"#7$-%$sinG6#,&F&\"\"\"F-!\"\"%*other
wiseG" }{MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "F := s -> exp(-2*s)*1/(s^2+1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"FG:6#%\"sG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$9$!\"#\"\"\",&F3
F3*$F1\"\"#F3!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(
1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"##\"\"\"\"\"#" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Alternat
ively:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "assume(s>0):\nint(sin(t-2)*exp(-s*t), t= 2 .. infinit
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$%#s|irG\"\"#\"\"\"F(F(!\"
\"-%$expG6#,$F&!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "su
bs(s=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"##\"\"\"
\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 457 9 "11. If   " }{XPPEDIT 458 1 "
F(s)" "-%\"FG6#%\"sG" }{TEXT 459 32 "  is the Laplace transform of   \+
" }{XPPEDIT 460 1 "f (t)= 16*cos(4*t)+13*sin(4*t)" "/-%\"fG6#%\"tG,&*&
\"#;\"\"\"-%$cosG6#*&\"\"%F*F&F*F*F**&\"#8F*-%$sinG6#*&F/F*F&F*F*F*" }
{TEXT 461 24 "  \n      then what is   " }{XPPEDIT 462 1 "F(3)" "-%\"F
G6#\"\"$" }{TEXT 463 164 " ?\n\na)  2             b) 3              c)
  4             d)  5             e)  3/5  \nf) 4/5            g)  18/
5         h)  52/25      i)  76/25        j)  116/25\n" }}{PARA 3 "" 
0 "" {TEXT 665 1 "\n" }}{PARA 0 "" 0 "" {TEXT 667 3 "(c)" }{TEXT -1 1 
"\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "laplace(16*cos(4*t)+1
3*sin(4*t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\",
&*$F%\"\"#F&\"#;F&!\"\"F**$F'F+\"#_" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "subs(s=3,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 1 " " }{TEXT 464 11 "12.  If    " }{XPPEDIT 465 1 "f(t)" "-%\"fG6#%
\"tG" }{TEXT 466 50 "  is the inverse Laplace transform of\n\n        \+
   " }{XPPEDIT 467 1 "F(s)=(3*s-1)/(s^2-s)" "/-%\"FG6#%\"sG*&,&*&\"\"$
\"\"\"F&F+F+F+!\"\"F+,&*$F&\"\"#F+F&F,F," }{TEXT 468 19 "  \n\nthen wh
at is   " }{XPPEDIT 469 1 "f(0)" "-%\"fG6#\"\"!" }{TEXT 470 7 " ?\n\na
) " }{XPPEDIT 258 1 "-1" ",$\"\"\"!\"\"" }{TEXT 259 16 "            b)
  " }{XPPEDIT 260 1 "0" "\"\"!" }{TEXT 261 16 "            c)  " }
{XPPEDIT 310 0 "2" "\"\"#" }{TEXT 309 13 "         d)  " }{XPPEDIT 
262 1 "3" "\"\"$" }{TEXT 263 15 "           e)  " }{XPPEDIT 256 1 "4" 
"\"\"%" }{TEXT 278 25 "                   \nf)   " }{XPPEDIT 308 0 "5
" "\"\"&" }{TEXT 307 17 "             g)  " }{XPPEDIT 271 0 "6" "\"\"'
" }{TEXT 270 16 "            h)  " }{XPPEDIT 256 1 "8" "\"\")" }{TEXT 
279 14 "          i)  " }{XPPEDIT 256 1 "10" "\"#5" }{TEXT 280 14 "   \+
       j)  " }{XPPEDIT 256 1 "12" "\"#7" }{TEXT 281 1 " " }}{PARA 3 "
" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 4 "(d)\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "F := s ->  (3*s-1)/(s^2-s);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"sG6\"6$%)operatorG%&arrowG
F(*&,&9$\"\"$!\"\"\"\"\"F1,&*$F.\"\"#F1F.F0F0F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "invlaplace(F(s),s,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&\"\"\"F$-%$expG6#%\"tG\"\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "subs(t=0,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&\"\"\"F$-%$expG6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 
3 "" 0 "" {TEXT 668 1 "\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 471 50 "13
. The Laplace transform of the Bessel function  " }{XPPEDIT 472 1 "J[0
](t)" "-&%\"JG6#\"\"!6#%\"tG" }{TEXT 473 6 "  is  " }{XPPEDIT 474 1 "1
/sqrt(1+s^2)" "*&\"\"\"F#-%%sqrtG6#,&F#F#*$%\"sG\"\"#F#!\"\"" }{TEXT 
475 8 ".\nLet   " }{XPPEDIT 476 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 477 
31 "  be the Laplace transform of  " }{XPPEDIT 478 1 "tJ[0](t)" "-&%#t
JG6#\"\"!6#%\"tG" }{TEXT 479 12 ". What  is  " }{XPPEDIT 480 1 "F(1" "
-%\"FG6#\"\"\"" }{TEXT 481 6 "?\n\na) " }{XPPEDIT 482 1 "sqrt(2)/4" "*
&-%%sqrtG6#\"\"#\"\"\"\"\"%!\"\"" }{TEXT 483 18 "             b)   " }
{XPPEDIT 484 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 
485 13 "          c) " }{XPPEDIT 486 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }
{TEXT 487 11 "        d) " }{XPPEDIT 488 1 "2*sqrt(2) " "*&\"\"#\"\"\"
-%%sqrtG6#F#F$" }{TEXT 489 10 "       e) " }{XPPEDIT 490 1 "3*sqrt(2)
" "*&\"\"$\"\"\"-%%sqrtG6#\"\"#F$" }{TEXT 491 14 "         \nf)  " }
{XPPEDIT 492 1 "-sqrt(2)/4" ",$*&-%%sqrtG6#\"\"#\"\"\"\"\"%!\"\"F*" }
{TEXT 493 14 "       g)     " }{XPPEDIT 494 1 "-sqrt(2)/2" ",$*&-%%sqr
tG6#\"\"#\"\"\"F'!\"\"F)" }{TEXT 311 8 "     h) " }{XPPEDIT 495 1 "-sq
rt(2)" ",$-%%sqrtG6#\"\"#!\"\"" }{TEXT 496 10 "      i)  " }{XPPEDIT 
497 1 "-2*sqrt(2) " ",$*&\"\"#\"\"\"-%%sqrtG6#F$F%!\"\"" }{TEXT 498 7 
"    j) " }{XPPEDIT 499 1 "-3*sqrt(2)" ",$*&\"\"$\"\"\"-%%sqrtG6#\"\"#
F%!\"\"" }{TEXT 500 4 "    " }}{PARA 3 "" 0 "" {TEXT 670 4 "\n\n\n\n" 
}{TEXT -1 3 "(a)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "-diff(1/sqrt
(1+s^2),s);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$%#s|irG\"\"#\"
\"\"F(F(#!\"$F'F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs
(s=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%#F'\"\"
%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "laplace(t*BesselJ(0,t),t,s);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$%#s|irG\"\"#\"\"\"F(F(#!\"$F
'F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=1,\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%#F'\"\"%" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 573 9 "14. Le
t  " }{XPPEDIT 563 1 "X(s)" "-%\"XG6#%\"sG" }{TEXT 564 30 "  be the La
place transform of " }{XPPEDIT 565 1 "x (t)" "-%\"xG6#%\"tG" }{TEXT 
570 5 " and " }{XPPEDIT 633 1 "Phi(s)" "-%$PhiG6#%\"sG" }{TEXT 634 32 
"  be the Laplace transform of   " }{XPPEDIT 635 1 "x*`''` (t)" "*&%\"
xG\"\"\"-%#''G6#%\"tGF$" }{TEXT 636 8 " .  If  " }{XPPEDIT 637 1 "X(2)
=3" "/-%\"XG6#\"\"#\"\"$" }{TEXT 638 3 ",  " }{XPPEDIT 566 1 "`x`*`(`*
0*`)`=5" "/**%\"xG\"\"\"%\"(GF%\"\"!F%%\")GF%\"\"&" }{TEXT 567 10 ",  \+
and    " }{XPPEDIT 569 1 "x*`'`(0)=-3" "/*&%\"xG\"\"\"-%\"'G6#\"\"!F%,
$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 568 13 "then what is " }{XPPEDIT 
571 1 "Phi(2)" "-%$PhiG6#\"\"#" }{TEXT 572 1 "?" }{TEXT -1 2 "\n\n" }
{TEXT 284 120 "a) - 4      b) - 3      c) -2      d) - 1        e) 0  \+
     \n\nf) 1         g) 2        h) 3        i)  4          j) 5 \n" 
}}{PARA 3 "" 0 "" {TEXT 673 2 "\n\n" }{TEXT -1 3 "(j)" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Phi(s) = \+
s^2*X(s)-s*x(0)-D(x)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$PhiG6#
%#s|irG,(*&F'\"\"#-%\"XGF&\"\"\"F-*&F'F--%\"xG6#\"\"!F-!\"\"--%\"DG6#F
0F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=2,\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$PhiG6#\"\"#,(-%\"XGF&\"\"%-%\"xG6#
\"\"!!\"#--%\"DG6#F-F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "subs(\{X(2)=3, x(0)=5, D(x)(0) = -3\}, \");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$PhiG6#\"\"#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 576 76 "15. Which of the following ex
pressions is the inverse Laplace transform of  " }{XPPEDIT 574 1 "(3*s
+1)/(s^2-2*s+5)" "*&,&*&\"\"$\"\"\"%\"sGF&F&F&F&F&,(*$F'\"\"#F&*&F*F&F
'F&!\"\"\"\"&F&F," }{TEXT 575 8 " ?\n\na)  " }{XPPEDIT 577 1 "2*exp(t)
*cos(3*t)+ 3*exp(t)*sin(3*t) " ",&*(\"\"#\"\"\"-%$expG6#%\"tGF%-%$cosG
6#*&\"\"$F%F)F%F%F%*(F.F%-F'6#F)F%-%$sinG6#*&F.F%F)F%F%F%" }{TEXT 578 
13 "         \nb) " }{XPPEDIT 579 1 "3*exp(t)*cos(3*t)+ exp(t)*sin(3*t
) " ",&*(\"\"$\"\"\"-%$expG6#%\"tGF%-%$cosG6#*&F$F%F)F%F%F%*&-F'6#F)F%
-%$sinG6#*&F$F%F)F%F%F%" }{TEXT 580 19 "               \nc) " }
{XPPEDIT 581 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 582 23 "                  \nd)  " }{XPPEDIT 583 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 584 17 "            \ne)  " }{XPPEDIT 585 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 
586 5 " \nf) " }{XPPEDIT 587 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$F%F*F%F%-%$sinG6#*&F/F%F*F%F%!\"\"" }{TEXT 588 9 "  \+
   \ng) " }{XPPEDIT 589 1 "2*exp(2*t)*cos(3*t) + 3*exp(2*t)*sin(2*t)  \+
 " ",&*(\"\"#\"\"\"-%$expG6#*&F$F%%\"tGF%F%-%$cosG6#*&\"\"$F%F*F%F%F%*
(F/F%-F'6#*&F$F%F*F%F%-%$sinG6#*&F$F%F*F%F%F%" }{TEXT 590 9 "     \nh)
 " }{XPPEDIT 591 1 "3*exp(2*t)*cos(3*t) + 2*exp(3*t)*sin(2*t)   " ",&*
(\"\"$\"\"\"-%$expG6#*&\"\"#F%%\"tGF%F%-%$cosG6#*&F$F%F+F%F%F%*(F*F%-F
'6#*&F$F%F+F%F%-%$sinG6#*&F*F%F+F%F%F%" }{TEXT 592 13 "        \ni)  \+
" }{XPPEDIT 593 1 "3*exp(3*t)*cos(t) + 2*exp(3*t)*sin(t)     " ",&*(\"
\"$\"\"\"-%$expG6#*&F$F%%\"tGF%F%-%$cosG6#F*F%F%*(\"\"#F%-F'6#*&F$F%F*
F%F%-%$sinG6#F*F%F%" }{TEXT 594 12 "       \nj)  " }{XPPEDIT 595 1 "2*
exp(3*t)*cos(2*t) + 3*exp(3*t)*sin(2*t)       " ",&*(\"\"#\"\"\"-%$exp
G6#*&\"\"$F%%\"tGF%F%-%$cosG6#*&F$F%F+F%F%F%*(F*F%-F'6#*&F*F%F+F%F%-%$
sinG6#*&F$F%F+F%F%F%" }{TEXT 596 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 3 "
(d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "A := (3*s+1)/(s^2-2*s+5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG*&,&%#s|irG\"\"$\"\"\"F)F),(*$F'\"\"#F)F'!\"#\"\"
&F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "B := map(z->stu
dent[completesquare](z,s),A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B
G*&,&%#s|irG\"\"$\"\"\"F)F),&*$,&F'F)!\"\"F)\"\"#F)\"\"%F)F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(identity(B = (a*(s-1) \+
+ b*2)/((s-1)^2+4),s),\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/
%\"bG\"\"#/%\"aG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a*
exp(t)*cos(2*t)+b*exp(t)*sin(2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&*(%\"aG\"\"\"-%$expG6#%\"tGF&-%$cosG6#,$F*\"\"#F&F&*(%\"bGF&F'F&-%$s
inGF-F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(\{b = 2, \+
a = 3\} , \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#%\"tG\"
\"\"-%$cosG6#,$F(\"\"#F)\"\"$*&F%F)-%$sinGF,F)F." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT 674 1 "\n" }{TEXT 
-1 2 " \n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 609 9 "16. Let  " }
{XPPEDIT 603 1 "H" "I\"HG6\"" }{TEXT 610 39 " denote the Heaviside fun
ction.   Let  " }{XPPEDIT 605 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 606 31 "
  be the Laplace transform of  " }{XPPEDIT 604 1 "f(t)=e^3*t^2*H(t-3" 
"/-%\"fG6#%\"tG*(%\"eG\"\"$F&\"\"#-%\"HG6#,&F&\"\"\"F)!\"\"F/" }{TEXT 
611 14 ".   What is   " }{XPPEDIT 607 1 "F(1)" "-%\"FG6#\"\"\"" }
{TEXT 608 133 " ?\n\n\na) 1           b) 3         c) 5          d) 7 \+
           e) 9\n\nf) 11         g) 13        h) 15         i)  17    \+
      j) 19 \n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" 
{TEXT -1 4 "(i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solve(i
dentity( t^2 = (t-3)^2+a*(t-3)+b, t ), \{a,b\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"aG\"\"'/%\"bG\"\"*" }}}{PARA 3 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "exp(3)*t^2*Heaviside(
t-3) = exp(3)*((t-3)^2+a*(t-3)+b)*Heaviside(t-3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(-%$expG6#\"\"$\"\"\"%\"tG\"\"#-%*HeavisideG6#,&F*F)!
\"$F)F)*(F%F),(*$F/F+F)*&%\"aGF)F/F)F)%\"bGF)F)F,F)" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The required Laplace tr
ansform is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 55 "exp(3)*exp(-3*s)*(GAMMA(3)/s^3 + a*GAMMA(2)/s^2 + b
/s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$expG6#\"\"$\"\"\"-F%6#,$%
\"sG!\"$F(,(*$F,F-\"\"#*&%\"aGF(F,!\"#F(*&%\"bGF(F,!\"\"F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(s=1, \");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*(-%$expG6#\"\"$\"\"\"-F%6#!\"$F(,(\"\"#F(%\"aGF
(%\"bGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(\{a = 6,
 b = 9\}, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#\"\"$\"
\"\"-F&6#!\"$F)\"#<" }}}{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 530 18 "17. Suppose that  " }{XPPEDIT 
526 1 "f(t+1)=f(t)" "/-%\"fG6#,&%\"tG\"\"\"F(F(-F$6#F'" }{TEXT 527 11 
"  for all  " }{XPPEDIT 533 1 "t" "I\"tG6\"" }{TEXT 534 7 "  and  " }
{XPPEDIT 528 1 "f (t)= exp(t) " "/-%\"fG6#%\"tG-%$expG6#F&" }{TEXT 
529 8 "  for   " }{XPPEDIT 539 1 "t" "I\"tG6\"" }{TEXT 540 8 "   in   \+
" }{XPPEDIT 537 1 "[0,1]" "7$\"\"!\"\"\"" }{TEXT 538 8 ".  If   " }
{XPPEDIT 531 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 532 32 "  is the Laplace \+
transform of   " }{XPPEDIT 536 1 "f(t)" "-%\"fG6#%\"tG" }{TEXT -1 1 " \+
" }{TEXT 535 15 "then what is   " }{XPPEDIT 541 1 "F(2)" "-%\"FG6#\"\"
#" }{TEXT 542 6 "?\n\na) " }{XPPEDIT 543 1 "exp(1)" "-%$expG6#\"\"\"" 
}{TEXT -1 1 " " }{TEXT 544 16 "            b)  " }{XPPEDIT 545 1 "exp(
1)/(1+exp(1))" "*&-%$expG6#\"\"\"F&,&F&F&-F$6#F&F&!\"\"" }{TEXT -1 1 "
 " }{TEXT 546 10 "      c)  " }{XPPEDIT 547 1 "1/(exp(1)-1)" "*&\"\"\"
F#,&-%$expG6#F#F#F#!\"\"F(" }{TEXT -1 1 " " }{TEXT 548 10 "      d)  \+
" }{XPPEDIT 549 1 "exp(1)/(exp(1)-1)" "*&-%$expG6#\"\"\"F&,&-F$6#F&F&F
&!\"\"F*" }{TEXT 550 12 "        e)  " }{XPPEDIT 551 1 "(exp(1)+1)/(ex
p(1)-1)" "*&,&-%$expG6#\"\"\"F'F'F'F',&-F%6#F'F'F'!\"\"F+" }{TEXT 552 
10 "       \nf)" }{XPPEDIT 553 1 "(exp(1)-1)/(exp(1)+1)" "*&,&-%$expG6
#\"\"\"F'F'!\"\"F',&-F%6#F'F'F'F'F(" }{TEXT 554 11 "        g) " }
{XPPEDIT 555 1 "2*exp(1)/(1+exp(1))" "*(\"\"#\"\"\"-%$expG6#F$F$,&F$F$
-F&6#F$F$!\"\"" }{TEXT -1 1 " " }{TEXT 556 11 "       h)  " }{XPPEDIT 
557 1 "2*exp(1)/(2+exp(1))" "*(\"\"#\"\"\"-%$expG6#F$F$,&F#F$-F&6#F$F$
!\"\"" }{TEXT -1 1 " " }{TEXT 558 11 "      i)   " }{XPPEDIT 559 1 "ex
p(1)/(2*exp(1)-1)" "*&-%$expG6#\"\"\"F&,&*&\"\"#F&-F$6#F&F&F&F&!\"\"F,
" }{TEXT 560 9 "     j)  " }{XPPEDIT 561 1 "2*exp(1)" "*&\"\"#\"\"\"-%
$expG6#F$F$" }{TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }
{TEXT 562 1 " " }{TEXT -1 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1/(1-exp(-s))*int(exp(t)*exp
(-s*t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG
6#,$%\"sG!\"\"F+F+,&*&,&F*F%F+F%F+-F'6#,&F%F%F*F+F%F+*$F.F+F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=2,\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG6#!\"#!\"\"F*,&-F'6#F*F*F%F%F
%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#!\"\"\"\"\"F(F)F),&F(F)-F&6#!\"
#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "testeq(\"=exp(1)/(
1+exp(1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 501 9 "18.  If  " }{XPPEDIT 502 1 "x*`'(`*t*`)` \+
+  x*`(`*t*`)`= delta(t-2" "/,&**%\"xG\"\"\"%#'(GF&%\"tGF&%\")GF&F&**F
%F&%\"(GF&F(F&F)F&F&-%&deltaG6#,&F(F&\"\"#!\"\"" }{TEXT 503 7 "  and  \+
" }{XPPEDIT 504 1 "x*`(`*0*`)`=0" "/**%\"xG\"\"\"%\"(GF%\"\"!F%%\")GF%
F'" }{TEXT 505 16 "  then what is  " }{XPPEDIT 506 1 "x*`(`*3*`)`" "**
%\"xG\"\"\"%\"(GF$\"\"$F$%\")GF$" }{TEXT 507 1 "?" }{TEXT -1 2 "\n\n" 
}{TEXT 282 133 "a) 1         b) 2           c) 3          d) e        \+
  e) 2e       \nf) 3e        g) 1/e        h) 2/e        i) 3/e       \+
 j) 3+e \n\n" }}{PARA 3 "" 0 "" {TEXT 662 3 "\n\n\n" }{TEXT -1 3 "(g)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "
map(z->laplace(z,t,s),diff(x(t),t)+x(t) = Dirac(t-2),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 27 "solve(\",laplace(x(t),t,s));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&-%\"xG6#\"\"!\"\"\"-%$expG6#,$%\"sG!\"#F)F),&F.F)F)
F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x(0)=0,\");
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$%\"sG!\"#\"\"\",&F(F*
F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(\",
s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%*HeavisideG6#,&%\"tG\"\"
\"!\"#F)F)-%$expG6#,&F(!\"\"\"\"#F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "subs(t=3,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%
*HeavisideG6#\"\"\"F'-%$expG6#!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6
#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 508 19 "19. Suppose that   " }{XPPEDIT 509 1 "x*`'(`*t*`)` = 2*x
(t)-y(t),`    `*y*`'(`*t*`)` =  x(t)+y(t)" "6$/**%\"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 510 18 "    \n      and    " }
{XPPEDIT 511 1 "x*`(`*0*`)`=3,`    `*y*`(`*0*`)`=0" "6$/**%\"xG\"\"\"%
\"(GF&\"\"!F&%\")GF&\"\"$/*,%%~~~~GF&%\"yGF&F'F&F(F&F)F&F(" }{TEXT 
512 8 " .  If  " }{XPPEDIT 513 1 "X(s)" "-%\"XG6#%\"sG" }{TEXT 514 31 
" is the Laplace transform of   " }{XPPEDIT 515 1 "x*`(`*t*`)`" "**%\"
xG\"\"\"%\"(GF$%\"tGF$%\")GF$" }{TEXT 516 21 ", then what is\n      " 
}{XPPEDIT 517 1 "X(3)" "-%\"XG6#\"\"$" }{TEXT 518 2 "? " }{TEXT -1 2 "
\n\n" }{TEXT 283 114 "a) 1        b) 2       c) 3        d) 4        e
) 5      \nf) 6        g) 7        h) 8        i)  9        j) 10 \n" 
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 656 6 "Answer
" }{TEXT 657 6 ": (b)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "e
qn1 := 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 39 "eqn1a
 := map(z->laplace(z,t,s) , eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&eqn1aG/,&*&%\"sG\"\"\"-%(laplaceG6%-%\"xG6#%\"tGF0F(F)F)-F.6#\"\"!!
\"\",&F*\"\"#-F+6%-%\"yGF/F0F(F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "eqn2a := map(z->laplace(z,t,s) , eqn2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&eqn2aG/,&*&%\"sG\"\"\"-%(laplaceG6%-%\"yG6#%
\"tGF0F(F)F)-F.6#\"\"!!\"\",&-F+6%-%\"xGF/F0F(F)F*F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "eqn1b := subs(x(0) = 3 , eqn1a);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn1bG/,&*&%\"sG\"\"\"-%(laplaceG6%
-%\"xG6#%\"tGF0F(F)F)!\"$F),&F*\"\"#-F+6%-%\"yGF/F0F(!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2b := subs(y(0) = 0 , eqn2a);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn2bG/*&%\"sG\"\"\"-%(laplaceG6%-
%\"yG6#%\"tGF/F'F(,&-F*6%-%\"xGF.F/F'F(F)F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 71 "eqn1c := subs(\{laplace(x(t),t,s)=X(s),laplace(y(t
),t,s)=Y(s)\} , eqn1b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn1cG/,
&*&%\"sG\"\"\"-%\"XG6#F(F)F)!\"$F),&F*\"\"#-%\"YGF,!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "eqn2c := subs(\{laplace(x(t),t,s)=X
(s),laplace(y(t),t,s)=Y(s)\} , eqn2b);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&eqn2cG/*&%\"sG\"\"\"-%\"YG6#F'F(,&-%\"XGF+F(F)F(" }}}{PARA 3 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solv
e(\{eqn1c,eqn2c\},\{X(s),Y(s)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<
$/-%\"YG6#%\"sG,$*$,(*$F(\"\"#\"\"\"F(!\"$\"\"$F.!\"\"F0/-%\"XGF',$*&,
&F(F.F1F.F.F+F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(s
=3, 3*(s-1)/(s^2-3*s+3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 519 9 "
20. Let  " }{XPPEDIT 520 1 "F(s)" "-%\"FG6#%\"sG" }{TEXT 521 34 " deno
te the Laplace transform of  " }{XPPEDIT 522 1 "f(t)=Int( (t-u)^2*exp(
2*u) ,u=0..t)" "/-%\"fG6#%\"tG-%$IntG6$*&,&F&\"\"\"%\"uG!\"\"\"\"#-%$e
xpG6#*&F/F,F-F,F,/F-;\"\"!F&" }{TEXT 523 11 ". What is  " }{XPPEDIT 
524 1 "F(3)" "-%\"FG6#\"\"$" }{TEXT 525 1 "?" }{TEXT -1 1 "\n" }{TEXT 
264 145 "\na) 1/3        b) 2/3          c) 1/9          d) 2/9       \+
   e) 1/27     \nf) 2/27       g) 3/2         h) 9/2           i) 27/2
          j) 9/4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" 
{TEXT 658 6 "Answer" }{TEXT -1 7 ":  (f) " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := t -> int((t-u)^2
*exp(2*u),u = 0 .. t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"
tG6\"6$%)operatorG%&arrowGF(-%$intG6$*&,&9$\"\"\"%\"uG!\"\"\"\"#-%$exp
G6#,$F3F5F2/F3;\"\"!F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "f(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#,$%\"tG\"\"##\"
\"\"\"\"%*$F(F)#!\"\"F)F(F.#F/F,F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "laplace(f(t) , t, s);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,**$,&%\"sG\"\"\"!\"#F'!\"\"#F'\"\"%*$F&!\"$F)*$F&F(#F)\"\"#*$F&
F)#F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=3,\");" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#F" }}}{PARA 259 "" 0 "" 
{TEXT -1 3 "\nor" }}{PARA 0 "" 0 "" {TEXT -1 30 "\nrecognizing the con
volution:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "laplace(t^3,t
,s)*laplace(exp(2*t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"s
G!\"%,&F%\"\"\"!\"#F(!\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "subs(s=3,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#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 "" }}}}
{MARK "15 1 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 }