{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 19 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 19 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 287 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 295 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 305 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 315 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 321 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 322 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 325 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 330 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 331 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 332 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 335 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 338 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 340 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 341 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 343 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 345 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "" 1 14 0 0 0 0 0 1 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 1 }{CSTYLE "" -1 348 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 350 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 351 "" 1 14 0 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 0 }{CSTYLE "" -1 353 "" 1 14 0 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 0 } {CSTYLE "" -1 355 "" 1 14 0 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 0 }{CSTYLE "" -1 357 "" 1 14 0 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 0 }{CSTYLE "" -1 359 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 360 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 361 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 363 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 365 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 366 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 367 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 1 14 0 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 0 } {CSTYLE "" 19 370 "" 1 14 0 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 0 }{CSTYLE "" 18 372 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 373 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 374 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 375 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 376 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 377 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 378 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 379 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 380 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 381 "" 1 14 0 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 0 }{CSTYLE "" 19 383 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 384 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 385 "" 1 14 0 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 0 }{CSTYLE "" 18 387 "" 1 14 0 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 0 }{CSTYLE "" 18 389 "" 1 14 0 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 0 }{CSTYLE "" 18 391 "" 1 14 0 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 0 }{CSTYLE "" 18 393 "" 1 14 0 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 0 } {CSTYLE "" -1 395 "" 1 14 0 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 0 }{CSTYLE "" 19 397 "" 1 14 0 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 0 }{CSTYLE "" 19 399 "" 1 14 0 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 0 }{CSTYLE "" 19 401 "" 1 14 0 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 0 }{CSTYLE "" -1 403 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 404 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 405 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 406 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 407 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 408 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 409 "" 1 14 0 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 0 }{CSTYLE "" -1 411 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 412 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 413 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 414 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 415 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 416 "" 1 14 0 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 0 }{CSTYLE "" 18 418 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 419 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 420 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 421 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 422 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 423 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 424 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 425 "" 1 14 0 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 0 }{CSTYLE "" 18 427 "" 1 14 0 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 0 }{CSTYLE "" 18 429 "" 1 14 0 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 0 }{CSTYLE "" -1 431 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 432 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 433 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 434 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 435 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 436 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 437 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 438 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 439 "" 1 14 0 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 0 }{CSTYLE "" 19 441 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 442 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 443 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 444 "" 1 14 0 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 0 }{CSTYLE "" 18 446 "" 1 14 0 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 0 }{CSTYLE "" 18 448 "" 1 14 0 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 0 } {CSTYLE "" 18 450 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 451 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 452 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 453 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 454 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 455 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 456 "" 1 14 0 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 0 }{CSTYLE "" 18 458 "" 1 14 0 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 0 } {CSTYLE "" 18 460 "" 1 14 0 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 0 }{CSTYLE "" 18 462 "" 1 14 0 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 0 }{CSTYLE "" 18 464 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 465 "" 1 14 0 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 0 }{CSTYLE "" 19 467 "" 1 14 0 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 0 }{CSTYLE "" 19 469 "" 1 14 0 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 0 }{CSTYLE "" 19 471 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 472 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 473 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 474 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 475 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 476 "" 1 14 0 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 0 }{CSTYLE "" -1 478 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 479 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 480 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 481 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 482 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 483 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 484 "" 1 14 0 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 0 }{CSTYLE "" -1 486 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 487 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 488 "" 1 14 0 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 0 } {CSTYLE "" -1 490 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 491 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 492 "" 1 14 0 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 0 }{CSTYLE "" -1 494 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 495 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 496 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 497 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 498 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 499 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 500 "" 1 14 0 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 0 }{CSTYLE "" -1 502 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 503 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 504 "" 1 14 0 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 0 }{CSTYLE "" 19 506 "" 1 14 0 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 0 }{CSTYLE "" 19 508 "" 1 14 0 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 0 } {CSTYLE "" -1 510 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 511 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 512 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 513 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 514 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 515 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 516 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 517 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 518 "" 1 14 0 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 0 } {CSTYLE "" 19 520 "" 1 14 0 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 0 }{CSTYLE "" 19 522 "" 1 14 0 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 0 }{CSTYLE "" 18 524 "" 1 14 0 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 0 }{CSTYLE "" -1 526 "" 1 14 0 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 0 }{CSTYLE "" -1 528 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 529 "" 1 14 0 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 0 }{CSTYLE "" -1 531 "" 1 14 0 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 0 }{CSTYLE "" -1 533 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 534 "" 1 14 0 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 0 }{CSTYLE "" -1 536 "" 1 14 0 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 0 }{CSTYLE "" -1 538 "" 1 14 0 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 0 } {CSTYLE "" -1 540 "" 1 14 0 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 0 }{CSTYLE "" -1 542 "" 1 14 0 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 0 }{CSTYLE "" -1 544 "" 1 14 0 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 0 }{CSTYLE "" -1 546 "" 1 14 0 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 0 }{CSTYLE "" -1 548 "" 1 14 0 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 0 } {CSTYLE "" -1 550 "" 1 14 0 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 0 }{CSTYLE "" -1 552 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 553 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 554 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 555 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 556 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 557 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 558 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 559 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 560 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 561 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 562 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 563 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 564 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 565 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 566 "" 1 14 0 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 0 }{CSTYLE "" 19 568 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 569 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 570 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 571 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 572 "" 1 14 0 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 0 }{CSTYLE "" -1 574 "" 1 14 0 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 0 }{CSTYLE "" 19 576 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 577 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 578 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 579 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 580 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 581 "" 1 14 0 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 0 }{CSTYLE "" -1 583 "" 1 14 0 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 0 } {CSTYLE "" 18 585 "" 1 14 0 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 0 }{CSTYLE "" 18 587 "" 1 14 0 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 0 }{CSTYLE "" 19 589 "" 1 14 0 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 0 }{CSTYLE "" -1 591 "" 1 14 0 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 0 }{CSTYLE "" -1 593 "" 1 14 0 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 0 } {CSTYLE "" 19 595 "" 1 14 0 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 0 }{CSTYLE "" 19 597 "" 1 14 0 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 0 }{CSTYLE "" 19 599 "" 1 14 0 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 0 }{CSTYLE "" -1 601 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 602 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 603 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 604 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 605 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 606 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 607 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 608 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 609 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 610 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 611 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 612 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 613 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 614 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 615 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 616 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 617 "" 1 18 0 0 0 0 0 1 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 0 }{CSTYLE "" -1 619 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 620 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 621 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 622 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 623 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 624 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 625 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 626 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 627 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 628 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 629 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 630 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 631 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 632 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 633 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 634 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 635 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 636 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 637 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 638 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 639 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 640 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 641 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 642 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 643 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 644 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 645 "" 1 18 0 0 0 0 0 1 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 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 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 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 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 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 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 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 28 " Math 217 Spring 2001 Exa m 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 272 47 "Notational Remarks: In this exam, the \+ symbol " }{XPPEDIT 275 1 "diff(y(x),x)" "6#-%%diffG6$-%\"yG6#%\"xGF) " }{TEXT 274 9 " means " }{XPPEDIT 276 0 "dy/dx;" "6#*&%#dyG\"\"\"%# dxG!\"\"" }{TEXT 273 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 340 3 "If " }{XPPEDIT 341 1 "M;" "6#%\"MG" }{TEXT 342 19 " is a matrix, then " }{XPPEDIT 343 1 "M[i,j];" "6#&%\"MG6$%\"iG%\"jG " }{TEXT 344 17 " is the entry of " }{XPPEDIT 345 1 "M;" "6#%\"MG" } {TEXT 346 21 " in the i'th row and " }{XPPEDIT 338 1 "j;" "6#%\"jG" } {TEXT 339 10 "'th column" }{TEXT -1 1 "." }{TEXT 319 0 "" }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 351 16 "1. Suppose that " } {XPPEDIT 352 1 "A = matrix([[1, 2], [2, -1], [2, 2]]);" "6#/%\"AG-%'ma trixG6#7%7$\"\"\"\"\"#7$F+,$F*!\"\"7$F+F+" }{TEXT 353 5 " and " } {XPPEDIT 354 1 "B = matrix([[3, 4, 3], [4, 3, 2]]);" "6#/%\"BG-%'matri xG6#7$7%\"\"$\"\"%F*7%F+F*\"\"#" }{TEXT 355 8 ". If " }{XPPEDIT 356 1 "C = AB;" "6#/%\"CG%#ABG" }{TEXT 357 18 ", then what is " } {XPPEDIT 358 1 "C[2,2];" "6#&%\"CG6$\"\"#F&" }{TEXT 359 1 "?" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 19 1 "1;" "6# \"\"\"" }{TEXT -1 18 " b) " }{XPPEDIT 19 1 "2;" "6#\"\"# " }{TEXT -1 16 " c) " }{XPPEDIT 19 1 "3;" "6#\"\"$" } {TEXT -1 17 " d) " }{XPPEDIT 19 1 "4;" "6#\"\"%" }{TEXT -1 16 " e) " }{XPPEDIT 19 1 "5;" "6#\"\"&" }{TEXT -1 25 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 4 "f) " } {XPPEDIT 19 1 "6;" "6#\"\"'" }{TEXT -1 18 " g) " } {XPPEDIT 19 1 "7;" "6#\"\"(" }{TEXT -1 17 " h) " } {XPPEDIT 19 1 "8;" "6#\"\")" }{TEXT -1 16 " i) " } {XPPEDIT 19 1 "9;" "6#\"\"*" }{TEXT -1 17 " j) " } {XPPEDIT 19 1 "10;" "6#\"#5" }{TEXT -1 5 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 602 9 "Solution:" }{TEXT 603 5 " (e)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning , the protected names norm and trace have been redefined and unprotect ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := matrix([[1, 2] , [2, -1], [2, 2]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B : = matrix([[3, 4, 3], [4, 3, 2]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C := evalm(A &* B);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"CG-%'matrixG6#7%7%\"#6\"#5\"\"(7%\"\"#\"\"&\"\"%7%\"#9F2F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "C[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 257 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 360 8 "2. Let " }{XPPEDIT 361 1 "A;" "6#%\"AG" }{TEXT 362 5 " and " }{XPPEDIT 363 1 "B;" "6#%\"BG" }{TEXT 364 49 " b e the matrices of the preceding question. Let " }{XPPEDIT 365 1 "D = B A;" "6#/%\"DG%#BAG" }{TEXT 366 10 ". What is " }{XPPEDIT 367 1 "D[3,2] ;" "6#&%\"DG6$\"\"$\"\"#" }{TEXT 368 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 19 1 "17;" "6# \"#<" }{TEXT -1 7 " b) " }{XPPEDIT 19 1 "8;" "6#\"\")" }{TEXT -1 11 " c) " }{XPPEDIT 19 1 "14;" "6#\"#9" }{TEXT -1 10 " d) " }{XPPEDIT 19 1 "9;" "6#\"\"*" }{TEXT -1 9 " e) " }{XPPEDIT 19 1 "11;" "6#\"#6" }{TEXT -1 11 " f) " }{XPPEDIT 19 1 "13;" "6#\" #8" }{TEXT -1 9 " g) " }{XPPEDIT 19 1 "19;" "6#\"#>" }{TEXT -1 4 " " }}{PARA 257 "" 0 "" {TEXT -1 15 "h) The matrix " }{XPPEDIT 19 1 "D;" "6#%\"DG" }{TEXT -1 22 " does not exist " }}{PARA 257 "" 0 "" {TEXT -1 15 "i) The matrix " }{XPPEDIT 19 1 "D;" "6#%\"DG" } {TEXT -1 28 " does exist but the entry " }{XPPEDIT 19 1 "D[3,2]" "6# &%\"DG6$\"\"$\"\"#" }{TEXT -1 54 " does not exist because there is no third row. " }}{PARA 257 "" 0 "" {TEXT -1 15 "j) The matrix " }{XPPEDIT 19 1 "D;" "6#%\"DG" }{TEXT -1 28 " does exist but the entry " }{XPPEDIT 19 1 "D[3,2]" "6#&%\"DG6$\"\"$\"\"#" }{TEXT -1 53 " doe s not exist because there is no second column. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 604 9 "Solution:" }{TEXT 605 5 " (i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(B &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' matrixG6#7$7$\"#<\"\")7$\"#9\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT 369 33 "3. Evaluate the determinant of \+ " }{XPPEDIT 370 1 "matrix([[3, 1], [-4, 2]]);" "6#-%'matrixG6#7$7$\" \"$\"\"\"7$,$\"\"%!\"\"\"\"#" }{TEXT 371 2 ".\n" }}{PARA 3 "" 0 "" {TEXT 257 4 "a) " }{XPPEDIT 372 0 "1;" "6#\"\"\"" }{TEXT 282 8 " b ) " }{XPPEDIT 373 0 "2;" "6#\"\"#" }{TEXT 283 8 " c) " }{XPPEDIT 374 0 "3;" "6#\"\"$" }{TEXT 284 9 " d) " }{XPPEDIT 375 0 "4;" "6# \"\"%" }{TEXT 285 9 " e) " }{XPPEDIT 376 0 "5;" "6#\"\"&" }{TEXT 279 9 " f) " }{XPPEDIT 377 0 "6;" "6#\"\"'" }{TEXT 278 8 " g) \+ " }{XPPEDIT 378 0 "7;" "6#\"\"(" }{TEXT 277 9 " h) " }{XPPEDIT 379 0 "8;" "6#\"\")" }{TEXT 280 9 " i) " }{XPPEDIT 380 0 "9;" "6# \"\"*" }{TEXT 281 10 " j) " }{XPPEDIT 381 0 "10" "6#\"#5" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 606 9 "Solution: " }{TEXT 607 5 " (j)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "det(matrix([[3, 1], [-4, 2]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 382 33 "4. Evaluate the determinant of \+ " }{XPPEDIT 383 1 "matrix([[3, 2, -2], [3, 3, -2], [-2, -2, 3]]);" "6 #-%'matrixG6#7%7%\"\"$\"\"#,$F)!\"\"7%F(F(,$F)F+7%,$F)F+,$F)F+F(" } {TEXT 384 3 ".\n\n" }{TEXT 326 4 "a) " }{XPPEDIT 385 0 "1;" "6#\"\"\" " }{TEXT 332 8 " b) " }{XPPEDIT 386 0 "2;" "6#\"\"#" }{TEXT 333 8 " c) " }{XPPEDIT 387 0 "3;" "6#\"\"$" }{TEXT 334 9 " d) " } {XPPEDIT 388 0 "4;" "6#\"\"%" }{TEXT 335 9 " e) " }{XPPEDIT 389 0 "5;" "6#\"\"&" }{TEXT 329 9 " f) " }{XPPEDIT 390 0 "6;" "6#\"\" '" }{TEXT 328 8 " g) " }{XPPEDIT 391 0 "7;" "6#\"\"(" }{TEXT 327 9 " h) " }{XPPEDIT 392 0 "8;" "6#\"\")" }{TEXT 330 9 " i) " }{XPPEDIT 393 0 "9;" "6#\"\"*" }{TEXT 331 10 " j) " }{XPPEDIT 394 0 "10;" "6#\"#5" }}{PARA 0 "" 0 "" {TEXT 395 0 "" }}{PARA 0 "" 0 " " {TEXT 608 9 "Solution:" }{TEXT 609 5 " (e)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "det(matrix([ [3, 2, -2], [3, 3, -2], [-2, -2, 3]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Doing this by hand it is easier to create some zeros first:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "M1 := matrix([[3, 2, -2], [3, 3, -2], [-2, -2, 3]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7%7%\"\"$\"\"#!\"#7%F*F*F,7%F, F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "M2 := addrow(M1,1,2 ,-1); #adds (-1)*row1 of M1 to row2 of M1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'matrixG6#7%7%\"\"$\"\"#!\"#7%\"\"!\"\"\"F.7%F, F,F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 " Thus, the determinant of M1 equals the determinant of M2 which is:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "1*(3*3-(-2)*(-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 396 7 "5. Let " }{XPPEDIT 397 1 "A;" "6# %\"AG" }{TEXT 398 3 ", " }{XPPEDIT 399 1 "B;" "6#%\"BG" }{TEXT 400 6 ", and " }{XPPEDIT 401 1 "C;" "6#%\"CG" }{TEXT 402 68 " be n x n \+ matrices. Which of these statements is always true: " }}{PARA 3 "" 0 "" {TEXT 403 3 "I) " }{XPPEDIT 404 1 "AB = BA;" "6#/%#ABG%#BAG" } {TEXT 405 9 " II) " }{XPPEDIT 406 1 "A*`(`*BC*`)` = `(`*A*B*`)`*C; " "6#/**%\"AG\"\"\"%\"(GF&%#BCGF&%\")GF&*,F'F&F%F&%\"BGF&F)F&%\"CGF&" }{TEXT 407 9 " III) " }{XPPEDIT 408 1 "A*(B+C) = A*B+A*C;" "6#/*&% \"AG\"\"\",&%\"BGF&%\"CGF&F&,&*&F%F&F(F&F&*&F%F&F)F&F&" }}{PARA 0 "" 0 "" {TEXT 409 0 "" }}{PARA 0 "" 0 "" {TEXT 410 0 "" }}{PARA 0 "" 0 " " {TEXT 336 95 "a) I only b) II only c) III only \+ d) I and II only e) I and III only" }}{PARA 0 "" 0 "" {TEXT 337 79 "f) II and III only g) I, II, and III h) None are always true " }}{PARA 0 "" 0 "" {TEXT 411 0 "" }}{PARA 0 "" 0 "" {TEXT 610 9 "Solution:" }{TEXT 611 5 " (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "It is not true that " } {XPPEDIT 256 1 "AB = BA;" "6#/%#ABG%#BAG" }{TEXT -1 10 " always:\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := matrix([[0, 0], [0, 1] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"!F*7$ F*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "B := matrix([[0 , 0], [1, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7 $7$\"\"!F*7$\"\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eva lm(A &* B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!F( 7$\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(B &* A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!F(7$F(\"\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 412 18 " 6. Suppose that " }{XPPEDIT 413 1 "A = matrix([[3, -2], [2, -1]]);" "6#/%\"AG-%'matrixG6#7$7$\"\"$ ,$\"\"#!\"\"7$F,,$\"\"\"F-" }{TEXT 414 42 ". What is the second row o f its inverse " }{XPPEDIT 415 1 "A^(-1);" "6#)%\"AG,$\"\"\"!\"\"" } {TEXT 416 1 "?" }}{PARA 3 "" 0 "" {TEXT 417 4 "a) " }{XPPEDIT 418 0 " [1, -1];" "6#7$\"\"\",$F$!\"\"" }{TEXT 419 12 " b) " } {XPPEDIT 420 0 "[-1, 1];" "6#7$,$\"\"\"!\"\"F%" }{TEXT 421 11 " \+ \nc) " }{XPPEDIT 422 0 "[2, -2];" "6#7$\"\"#,$F$!\"\"" }{TEXT 423 11 " d) " }{XPPEDIT 424 0 "[-2, 2];" "6#7$,$\"\"#!\"\"F%" }{TEXT 425 15 " " }}{PARA 3 "" 0 "" {TEXT 426 3 "e) " } {XPPEDIT 427 0 "[3, -2];" "6#7$\"\"$,$\"\"#!\"\"" }{TEXT 428 12 " \+ f) " }{XPPEDIT 429 0 "[-3, 2];" "6#7$,$\"\"$!\"\"\"\"#" }{TEXT 430 4 " " }}{PARA 3 "" 0 "" {TEXT 431 3 "g) " }{XPPEDIT 432 0 "[2, \+ 3];" "6#7$\"\"#\"\"$" }{TEXT 433 14 " h) " }{XPPEDIT 434 0 " [2, -3];" "6#7$\"\"#,$\"\"$!\"\"" }{TEXT 435 8 " \ni) " }{XPPEDIT 436 0 "[-2, 3];" "6#7$,$\"\"#!\"\"\"\"$" }{TEXT 437 12 " j) " }{XPPEDIT 438 0 "[3, 2];" "6#7$\"\"$\"\"#" }{TEXT 439 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 621 9 "Solution:" } {TEXT 622 5 " (i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A := matrix( [[3, -2], [2, -1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matri xG6#7$7$\"\"$!\"#7$\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$! \"\"\"\"#7$!\"#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 440 19 " 7. Suppose that " }{XPPEDIT 441 1 "A = matrix([[1, 2, 2], [1, 1, 0], [1, 1, 1]] );" "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F+7%F*F*\"\"!7%F*F*F*" }{TEXT 442 62 ". What is the last element of the second row of its inverse \+ " }{XPPEDIT 443 1 "A^(-1);" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT 444 1 "?" } }{PARA 3 "" 0 "" {TEXT 445 3 "a) " }{XPPEDIT 446 0 "-5;" "6#,$\"\"&!\" \"" }{TEXT 447 7 " b) " }{XPPEDIT 448 0 "-4;" "6#,$\"\"%!\"\"" } {TEXT 449 7 " c) " }{XPPEDIT 450 0 "-3;" "6#,$\"\"$!\"\"" }{TEXT 451 8 " d) " }{XPPEDIT 452 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT 453 9 " e) " }{XPPEDIT 454 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 455 6 " \nf) " }{XPPEDIT 456 0 "5;" "6#\"\"&" }{TEXT 457 12 " g) " } {XPPEDIT 458 0 "4;" "6#\"\"%" }{TEXT 459 10 " h) " }{XPPEDIT 460 0 "3;" "6#\"\"$" }{TEXT 461 12 " i) " }{XPPEDIT 462 0 "2; " "6#\"\"#" }{TEXT 463 10 " j) " }{XPPEDIT 464 0 "1;" "6#\"\"\" " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 623 9 "Solut ion:" }{TEXT 624 5 " (d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := \+ matrix([[1, 2, 2], [1, 1, 0], [1, 1, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F+7%F*F*\"\"!7%F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"\"\"\"!\"\"#7%\"\"\"F,!\" #7%F)F(F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 465 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 466 46 " 8. The general sol ution of a certain system " }{XPPEDIT 467 1 "diff(matrix([[x[1](t)], \+ [x[2](t)]]),t) = A*matrix([[x[1](t)], [x[2](t)]]);" "6#/-%%diffG6$-%'m atrixG6#7$7#-&%\"xG6#\"\"\"6#%\"tG7#-&F.6#\"\"#6#F2F2*&%\"AGF0-F(6#7$7 #-&F.6#F06#F27#-&F.6#F76#F2F0" }{TEXT 468 30 " of differential equatio ns is " }{XPPEDIT 469 1 "matrix([[x[1](t)], [x[2](t)]]) = a*matrix([[e xp(t)], [2*exp(t)]])+b*matrix([[exp(2*t)], [exp(2*t)]]);" "6#/-%'matri xG6#7$7#-&%\"xG6#\"\"\"6#%\"tG7#-&F+6#\"\"#6#F/,&*&%\"aGF--F%6#7$7#-%$ expG6#F/7#*&F4F--F>6#F/F-F-F-*&%\"bGF--F%6#7$7#-F>6#*&F4F-F/F-7#-F>6#* &F4F-F/F-F-F-" }{TEXT 470 7 ". If " }{XPPEDIT 471 1 "x[1](0) = 3;" " 6#/-&%\"xG6#\"\"\"6#\"\"!\"\"$" }{TEXT 472 6 " and " }{XPPEDIT 473 1 "x[2](0) = 5;" "6#/-&%\"xG6#\"\"#6#\"\"!\"\"&" }{TEXT 474 14 " then wh at is " }{XPPEDIT 475 1 "x[2](ln(2));" "6#-&%\"xG6#\"\"#6#-%#lnG6#F'" }{TEXT 476 1 "?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 258 65 "a) 3 b) 4 c) 5 d) 6 e) 8 " }}{PARA 3 "" 0 "" {TEXT 324 59 "f) 9 g) 10 h ) 12 i) 15 j) 16" }{TEXT 477 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 625 9 "Solution:" }{TEXT 626 5 " (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x[1] := t -> a*exp(t)+b*exp(2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"f*6#%\"tG6\"6$%)operatorG%&arrowGF+,&*&% \"aGF'-%$expG6#9$F'F'*&%\"bGF'-F36#,$F5\"\"#F'F'F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x[2] := t -> 2*a*exp(t)+b*exp(2*t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#f*6#%\"tG6\"6$%)oper atorG%&arrowGF+,&*&%\"aG\"\"\"-%$expG6#9$F2F'*&%\"bGF2-F46#,$F6F'F2F2F +F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "solve(\{x[1](0) = \+ 3, x[2](0) = 5\}, \{a,b\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\" bG\"\"\"/%\"aG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "subs (\{b = 1, a = 2,t=ln(2)\},x[2](t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&-%$expG6#-%#lnG6#\"\"#\"\"%-F%6#,$F'F*\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 478 35 "9. When the second order e quation " }{XPPEDIT 479 1 "x*`''`(t)+3*x*`'`(t)+2*x(t) = 0;" "6#/,(*& %\"xG\"\"\"-%#''G6#%\"tGF'F'*(\"\"$F'F&F'-%\"'G6#F+F'F'*&\"\"#F'-F&6#F +F'F'\"\"!" }{TEXT 480 39 " is converted to a first order system " } {XPPEDIT 481 0 "matrix([[x[1](t)], [x[2](t)]])*`'` = A*matrix([[x[1](t )], [x[2](t)]]);" "6#/*&-%'matrixG6#7$7#-&%\"xG6#\"\"\"6#%\"tG7#-&F,6# \"\"#6#F0F.%\"'GF.*&%\"AGF.-F&6#7$7#-&F,6#F.6#F07#-&F,6#F56#F0F." } {TEXT 482 7 " with " }{XPPEDIT 483 0 "x[1] = x;" "6#/&%\"xG6#\"\"\"F% " }{TEXT 484 12 ", what is A?" }}{PARA 3 "" 0 "" {TEXT 259 6 "\na) \+ " }{XPPEDIT 256 0 "matrix([[1, 0], [2, 3]]);" "6#-%'matrixG6#7$7$\"\" \"\"\"!7$\"\"#\"\"$" }{TEXT 289 11 " b) " }{XPPEDIT 256 0 "matr ix([[1, 0], [-2, -3]]);" "6#-%'matrixG6#7$7$\"\"\"\"\"!7$,$\"\"#!\"\", $\"\"$F-" }{TEXT 292 8 " c) " }{XPPEDIT 256 0 "matrix([[0, 1], [2, -3]]);" "6#-%'matrixG6#7$7$\"\"!\"\"\"7$\"\"#,$\"\"$!\"\"" }{TEXT 350 8 " d) " }{XPPEDIT 256 0 "matrix([[1, 1], [-2, -3]]);" "6#-%'m atrixG6#7$7$\"\"\"F(7$,$\"\"#!\"\",$\"\"$F," }{TEXT 347 8 " e) " } {XPPEDIT 256 0 "matrix([[3, 2], [-2, -3]]);" "6#-%'matrixG6#7$7$\"\"$ \"\"#7$,$F)!\"\",$F(F," }{TEXT 294 7 " " }}{PARA 3 "" 0 "" {TEXT 288 4 "f) " }{XPPEDIT 256 0 "matrix([[0, 1], [-2, -3]]);" "6#-% 'matrixG6#7$7$\"\"!\"\"\"7$,$\"\"#!\"\",$\"\"$F-" }{TEXT 290 11 " \+ g) " }{XPPEDIT 256 0 "matrix([[1, 1], [-3, -2]]);" "6#-%'matrixG6#7 $7$\"\"\"F(7$,$\"\"$!\"\",$\"\"#F," }{TEXT 349 8 " h) " }{XPPEDIT 256 0 "matrix([[0, 1], [2, 3]]);" "6#-%'matrixG6#7$7$\"\"!\"\"\"7$\"\" #\"\"$" }{TEXT 291 9 " i) " }{XPPEDIT 256 0 "matrix([[1, 1], [2, \+ 3]]);" "6#-%'matrixG6#7$7$\"\"\"F(7$\"\"#\"\"$" }{TEXT 293 10 " j ) " }{XPPEDIT 256 0 "matrix([[3, 2], [2, 3]]);" "6#-%'matrixG6#7$7$\" \"$\"\"#7$F)F(" }{TEXT 348 4 " " }}{PARA 0 "" 0 "" {TEXT 485 0 "" } }{PARA 0 "" 0 "" {TEXT 634 9 "Solution:" }{TEXT 635 5 " (c)" }}{PARA 0 "" 0 "" {TEXT 633 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 256 0 "x[1] = x" "6#/&%\"xG6#\"\"\"F%" }{TEXT -1 9 " and " }{XPPEDIT 256 0 "x[2] = x*`'`;" "6#/&%\"xG6#\"\"#*&F%\"\"\" %\"'GF)" }{TEXT -1 10 " . Then " }{XPPEDIT 256 1 "x*`''`(t) = -2*x(t )-3*x*`'`(t);" "6#/*&%\"xG\"\"\"-%#''G6#%\"tGF&,&*&\"\"#F&-F%6#F*F&!\" \"*(\"\"$F&F%F&-%\"'G6#F*F&F0" }{TEXT -1 10 " and so " }{XPPEDIT 256 0 "matrix([[x[1](t)], [x[2](t)]])*`'` = A*matrix([[x[1](t)], [x[2] (t)]])" "6#/*&-%'matrixG6#7$7#-&%\"xG6#\"\"\"6#%\"tG7#-&F,6#\"\"#6#F0F .%\"'GF.*&%\"AGF.-F&6#7$7#-&F,6#F.6#F07#-&F,6#F56#F0F." }{TEXT -1 8 " \+ with " }{XPPEDIT 19 1 "A = matrix([[0, 1], [-2, -3]]);" "6#/%\"AG-%' matrixG6#7$7$\"\"!\"\"\"7$,$\"\"#!\"\",$\"\"$F/" }{TEXT -1 1 "." }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 486 34 " 10. What are the eigenvalues of " }{XPPEDIT 487 0 "matrix([[2, -1], [-1, 2]]);" "6#-%'matrixG6#7$7$ \"\"#,$\"\"\"!\"\"7$,$F*F+F(" }{TEXT 488 2 "?\n" }{TEXT 260 193 "\na) \+ \{2,3\} b) \{2,-3\} c) \{-2,3\} d) \+ \{-2,-3\} e) \{-1,3\} \nf) \{1,3\} g) \{1,-3\} h) \{-1,-3\} i) \{1,-1\} j) \{2,2\} " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 631 9 "Solut ion:" }{TEXT 632 5 " (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eigen vals( matrix([[2, -1], [-1, 2]]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $\"\"$\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 489 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 490 67 "11. Find two distinct eigenvecto rs [-1,a] and [-1,b] of the matrix " }{XPPEDIT 491 0 "matrix([[3, 3], \+ [4, 2]]);" "6#-%'matrixG6#7$7$\"\"$F(7$\"\"%\"\"#" }{TEXT 492 26 ". \n What is the sum a + b ?" }}{PARA 3 "" 0 "" {TEXT 261 160 "a) 0 \+ b) 1/3 c) 2/3 d) 1 e) 4/3 \+ \nf) 5/3 g) 2 h) 7/3 i) 8/3 \+ j) 3" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 629 9 "Solution:" }{TEXT 630 5 " (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eigenvects(matrix([[3, 3], [4, 2]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%!\"\"\"\"\"<#-%'vectorG6#7$F%#!\"%\"\"$7%\"\"'F%<#-F( 6#7$F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a := 4/3; b := -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{PARA 3 "" 0 "" {TEXT 493 0 "" }{TEXT 325 0 "" }{TEXT 494 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 495 9 " 12. If " }{XPPEDIT 496 1 "u = m atrix([[3], [1]]);" "6#/%\"uG-%'matrixG6#7$7#\"\"$7#\"\"\"" }{TEXT 497 7 " and " }{XPPEDIT 498 1 "v = matrix([[1], [1]]);" "6#/%\"vG-%' matrixG6#7$7#\"\"\"7#F*" }{TEXT 499 23 " are eigenvectors of " } {XPPEDIT 500 1 "A = matrix([[2, -3], [1, -2]]);" "6#/%\"AG-%'matrixG6# 7$7$\"\"#,$\"\"$!\"\"7$\"\"\",$F*F-" }{TEXT 501 2 ", " }}{PARA 3 "" 0 "" {TEXT 502 48 "then what is the general solution of the system " } {XPPEDIT 503 1 "x*`'` = A*x;" "6#/*&%\"xG\"\"\"%\"'GF&*&%\"AGF&F%F&" } {TEXT 504 2 " " }}{PARA 3 "" 0 "" {TEXT 505 7 "(where " }{XPPEDIT 506 1 "x = matrix([[x[1]], [x[2]]]);" "6#/%\"xG-%'matrixG6#7$7#&F$6#\" \"\"7#&F$6#\"\"#" }{TEXT 507 2 ")?" }}{PARA 3 "" 0 "" {TEXT 262 3 "a) \+ " }{XPPEDIT 263 1 "c[1]*u+c[2]*v;" "6#,&*&&%\"cG6#\"\"\"F(%\"uGF(F(*&& F&6#\"\"#F(%\"vGF(F(" }{TEXT 264 36 " b ) " }{XPPEDIT 265 1 "c[1]*exp(-t)*u+c[2]*exp(2*t)*v;" "6#,&*(&%\"cG6# \"\"\"F(-%$expG6#,$%\"tG!\"\"F(%\"uGF(F(*(&F&6#\"\"#F(-F*6#*&F3F(F-F(F (%\"vGF(F(" }{TEXT 266 8 " " }}{PARA 3 "" 0 "" {TEXT 320 5 " c) " }{XPPEDIT 508 1 "c[1]*exp(2*t)*u+c[2]*exp(-t)*v;" "6#,&*(&%\"cG6# \"\"\"F(-%$expG6#*&\"\"#F(%\"tGF(F(%\"uGF(F(*(&F&6#F-F(-F*6#,$F.!\"\"F (%\"vGF(F(" }{TEXT 295 11 " d) " }{XPPEDIT 267 1 "c[1]*exp(-t)* u+c[2]*exp(t)*v;" "6#,&*(&%\"cG6#\"\"\"F(-%$expG6#,$%\"tG!\"\"F(%\"uGF (F(*(&F&6#\"\"#F(-F*6#F-F(%\"vGF(F(" }{TEXT 268 7 " " }}{PARA 3 "" 0 "" {TEXT 321 4 "e) " }{XPPEDIT 256 1 "c[1]*exp(t)*u+c[2]*exp(-t) *v;" "6#,&*(&%\"cG6#\"\"\"F(-%$expG6#%\"tGF(%\"uGF(F(*(&F&6#\"\"#F(-F* 6#,$F,!\"\"F(%\"vGF(F(" }{TEXT 296 22 " f) " } {XPPEDIT 509 1 "c[1]*exp(2*t)*u+c[2]*exp(t)*v;" "6#,&*(&%\"cG6#\"\"\"F (-%$expG6#*&\"\"#F(%\"tGF(F(%\"uGF(F(*(&F&6#F-F(-F*6#F.F(%\"vGF(F(" } {TEXT 297 12 " " }}{PARA 3 "" 0 "" {TEXT 322 4 "g) " } {XPPEDIT 287 0 "c[1]*exp(t)*u+c[2]*exp(2*t)*v;" "6#,&*(&%\"cG6#\"\"\"F (-%$expG6#%\"tGF(%\"uGF(F(*(&F&6#\"\"#F(-F*6#*&F1F(F,F(F(%\"vGF(F(" } {TEXT 286 22 " h) " }{XPPEDIT 256 1 "(exp(-t)+exp(t) )*(c[1]*u+c[2]*v);" "6#*&,&-%$expG6#,$%\"tG!\"\"\"\"\"-F&6#F)F+F+,&*&& %\"cG6#F+F+%\"uGF+F+*&&F16#\"\"#F+%\"vGF+F+F+" }{TEXT 298 10 " \+ " }}{PARA 3 "" 0 "" {TEXT 323 4 "i) " }{XPPEDIT 256 1 "(exp(t)+exp( 2*t))*(c[1]*u+c[2]*v);" "6#*&,&-%$expG6#%\"tG\"\"\"-F&6#*&\"\"#F)F(F)F )F),&*&&%\"cG6#F)F)%\"uGF)F)*&&F16#F-F)%\"vGF)F)F)" }{TEXT 299 14 " \+ j) " }{XPPEDIT 256 1 "(exp(-t)+exp(2*t))*(c[1]*u+c[2]*v);" "6# *&,&-%$expG6#,$%\"tG!\"\"\"\"\"-F&6#*&\"\"#F+F)F+F+F+,&*&&%\"cG6#F+F+% \"uGF+F+*&&F36#F/F+%\"vGF+F+F+" }{TEXT 300 2 " \n" }{TEXT 510 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 636 9 "Solution: " }{TEXT 637 5 " (e)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A := matr ix([[2, -3], [1, -2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'ma trixG6#7$7$\"\"#!\"$7$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "u := matrix([[3], [1]]); v := matrix([[1], [1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'matrixG6#7$7#\"\"$7#\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'matrixG6#7$7#\"\"\"F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(A &* u); evalm(A &* v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7#\"\"$7#\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7#!\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "So " }{TEXT 638 1 "u" }{TEXT -1 32 " corresponds to eigenvalue 1, " }{TEXT 639 1 "v " }{TEXT -1 18 " to eigenvalue -1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 511 57 "13. The general solution of a system of three equations " }{XPPEDIT 512 0 "matrix([[x[1](t)], [x[2](t)], [x[3](t)]])*`'` = A*matrix([[x[1] (t)], [x[2](t)], [x[3](t)]])" "6#/*&-%'matrixG6#7%7#-&%\"xG6#\"\"\"6#% \"tG7#-&F,6#\"\"#6#F07#-&F,6#\"\"$6#F0F.%\"'GF.*&%\"AGF.-F&6#7%7#-&F,6 #F.6#F07#-&F,6#F56#F07#-&F,6#F;6#F0F." }{TEXT 513 8 " is " } {XPPEDIT 514 1 "a*exp(t)*u+b*exp(2*t)*v+c*exp(3*t)*w;" "6#,(*(%\"aG\" \"\"-%$expG6#%\"tGF&%\"uGF&F&*(%\"bGF&-F(6#*&\"\"#F&F*F&F&%\"vGF&F&*(% \"cGF&-F(6#*&\"\"$F&F*F&F&%\"wGF&F&" }{TEXT 515 9 " \nwhere " } {XPPEDIT 516 1 "u = matrix([[1], [0], [1]]);" "6#/%\"uG-%'matrixG6#7%7 #\"\"\"7#\"\"!7#F*" }{TEXT 517 3 ", " }{XPPEDIT 518 1 "v = matrix([[0 ], [1], [1]]);" "6#/%\"vG-%'matrixG6#7%7#\"\"!7#\"\"\"7#F," }{TEXT 519 7 ", and " }{XPPEDIT 520 1 "w = matrix([[3], [3], [5]]);" "6#/%\" wG-%'matrixG6#7%7#\"\"$7#F*7#\"\"&" }{TEXT 521 11 ". What is " } {XPPEDIT 522 1 "c;" "6#%\"cG" }{TEXT 523 6 " if " }{XPPEDIT 524 0 "m atrix([[x[1](0)], [x[2](0)], [x[3](0)]]) = matrix([[2], [1], [2]]);" " 6#/-%'matrixG6#7%7#-&%\"xG6#\"\"\"6#\"\"!7#-&F+6#\"\"#6#F/7#-&F+6#\"\" $6#F/-F%6#7%7#F47#F-7#F4" }{TEXT 525 1 "?" }}{PARA 0 "" 0 "" {TEXT 526 0 "" }}{PARA 3 "" 0 "" {TEXT 317 104 "a) - 4 b) - 3 c) - 2 d) - 1 e) 0 f) 1 g) 2 h) 3 i) 4 \+ j) 5 " }}{PARA 0 "" 0 "" {TEXT 527 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 642 9 "Solu tion:" }{TEXT 643 5 " (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a := 'a': b := 'b': c := 'c':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x[1] := t -> a*exp(t)+3*c*exp(3*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"f*6#%\"tG6\"6$%)operatorG%&arrowGF+,&*&%\"aGF'-%$ expG6#9$F'F'*(\"\"$F'%\"cGF'-F36#,$F5F7F'F'F+F+F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "x[2] := t -> b*exp(2*t)+3*c*exp(3*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#f*6#%\"tG6\"6$%)operator G%&arrowGF+,&*&%\"bG\"\"\"-%$expG6#,$9$F'F2F2*(\"\"$F2%\"cGF2-F46#,$F7 F9F2F2F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x[3] := t - > a*exp(t)+b*exp(2*t)+5*c*exp(3*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"xG6#\"\"$f*6#%\"tG6\"6$%)operatorG%&arrowGF+,(*&%\"aG\"\"\"-%$ex pG6#9$F2F2*&%\"bGF2-F46#,$F6\"\"#F2F2*(\"\"&F2%\"cGF2-F46#,$F6F'F2F2F+ F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{x[1](0) = 2 , x[2](0) = 1, x[3](0) = 2\}, \{a,b,c\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"cG\"\"\"/%\"bG!\"#/%\"aG!\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 528 58 "14. The vector u = [2,b] is an eigenvector of the \+ matrix " }{XPPEDIT 529 0 "matrix([[-1, 4], [-1, 3]]);" "6#-%'matrixG6# 7$7$,$\"\"\"!\"\"\"\"%7$,$F)F*\"\"$" }{TEXT 530 120 ". \nLet v be the \+ vector [a,2]. If \{ u , v \} is a length 2 chain of generalized eige nvectors based on u, then what is a?" }}{PARA 3 "" 0 "" {TEXT 318 104 "a) - 4 b) - 3 c) -2 d) - 1 e) 0 f) 1 g ) 2 h) 3 i) 4 j) 5 " }}{PARA 0 "" 0 "" {TEXT 531 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 627 9 "Solutio n:" }{TEXT 628 5 " (h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A := matrix([[-1, 4], [-1, 3]]); Id := matrix([[1, 0], [0, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A G-%'matrixG6#7$7$!\"\"\"\"%7$F*\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#IdG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "eigenvects(matrix([[-1, 4], [-1, 3]])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"#<#-%'vectorG6#7$F%F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "lambda := 1; u := vector([2 , 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "linsolve(A-lambda*Id, u); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,&!\"\"\"\"\"*&\"\"#F)&% #_tG6#F)F)F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(_t[1 ] = 2 , -1+2*_t[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 532 24 "15. Consider the matrix " }{XPPEDIT 256 0 "A = matrix([[0, 0, 1], [0, 0, 2], [0, 0, 1]]);" "6#/%\"AG-%'matrixG6#7%7%\"\"!F*\"\"\"7%F*F*\"\" #7%F*F*F+" }{TEXT 533 55 ". What is the entry in the second row, third column of " }{XPPEDIT 534 1 "exp(t*A);" "6#-%$expG6#*&%\"tG\"\"\"%\"A GF(" }{TEXT 535 1 "?" }}{PARA 3 "" 0 "" {TEXT 536 16 "a) 0 b) \+ " }{XPPEDIT 537 1 "2*t;" "6#*&\"\"#\"\"\"%\"tGF%" }{TEXT 538 18 " \+ c) " }{XPPEDIT 539 1 "2*t^2;" "6#*&\"\"#\"\"\"*$%\"tGF$F%" } {TEXT 540 22 " d) " }{XPPEDIT 541 1 "t^2;" "6#*$%\"t G\"\"#" }{TEXT 542 15 " e) " }{XPPEDIT 543 1 "t^2/2;" "6#*& %\"tG\"\"#F%!\"\"" }{TEXT 544 5 " \nf) " }{XPPEDIT 545 1 "-t^2/2;" "6# ,$*&%\"tG\"\"#F&!\"\"F'" }{TEXT 546 8 " g) " }{XPPEDIT 547 1 "-t^2 /2+t;" "6#,&*&%\"tG\"\"#F&!\"\"F'F%\"\"\"" }{TEXT 548 8 " h) " } {XPPEDIT 549 1 "-t^2/2+2*t;" "6#,&*&%\"tG\"\"#F&!\"\"F'*&F&\"\"\"F%F)F )" }{TEXT 550 12 " i) " }{XPPEDIT 551 1 "2*exp(t)-2;" "6#,&*& \"\"#\"\"\"-%$expG6#%\"tGF&F&F%!\"\"" }{TEXT 552 8 " j) " } {XPPEDIT 257 1 "exp(2*t)-1;" "6#,&-%$expG6#*&\"\"#\"\"\"%\"tGF)F)F)!\" \"" }{TEXT 618 6 " \n\n" }}{PARA 0 "" 0 "" {TEXT 619 9 "Solution:" }{TEXT 620 5 " (i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := matrix([[0, 0, 1], [0, 0, 2], [0, 0, 1 ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"!F* \"\"\"7%F*F*\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eval m(A^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(\"\" \"7%F(F(\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A^ 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(\"\"\"7% F(F(\"\"#F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Note that this implies that " }{XPPEDIT 19 1 "A^n = A;" "6#/)% \"AG%\"nGF%" }{TEXT -1 10 " for all " }{XPPEDIT 19 1 "1 <= n;" "6#1\" \"\"%\"nG" }{TEXT -1 12 ". Since " }{XPPEDIT 19 1 "exp(t*A) = Sum( (t*A)^n/n!,n = 0 .. infinity);" "6#/-%$expG6#*&%\"tG\"\"\"%\"AGF)-%$Su mG6$*&)*&F(F)F*F)%\"nGF)-%*factorialG6#F1!\"\"/F1;\"\"!%)infinityG" } {TEXT -1 20 ", it follows that \n" }{XPPEDIT 19 1 "exp(t*A)[2,3] = Su m(2*t^n/n!,n = 1 .. infinity);" "6#/&-%$expG6#*&%\"tG\"\"\"%\"AGF*6$\" \"#\"\"$-%$SumG6$*(F-F*)F)%\"nGF*-%*factorialG6#F4!\"\"/F4;F*%)infinit yG" }{TEXT -1 6 ". \n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "value(Sum(2*t^n/n!,n = 1 .. infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#%\"tG\"\"\",&F)F)-F&6#,$F(!\"\"F.F)\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"tG\"\"#F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Or, in one line:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "exponential(A,t); exponent ial(A,t)[2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\" \"\"\"!,&-%$expG6#%\"tGF(F(!\"\"7%F)F(,&F+\"\"#F2F/7%F)F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#%\"tG\"\"#F(!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 553 32 "16. Consider the system system " } {XPPEDIT 554 0 "matrix([[x[1](t)], [x[2](t)], [x[3](t)]])*`'` = A*matr ix([[x[1](t)], [x[2](t)], [x[3](t)]]);" "6#/*&-%'matrixG6#7%7#-&%\"xG6 #\"\"\"6#%\"tG7#-&F,6#\"\"#6#F07#-&F,6#\"\"$6#F0F.%\"'GF.*&%\"AGF.-F&6 #7%7#-&F,6#F.6#F07#-&F,6#F56#F07#-&F,6#F;6#F0F." }{TEXT 555 177 ". Sup pose that u, v, and w are (column) vectors that satisfy Au = u, Av = 2 v, and Aw = 2w + v. Then which of the following matrices is a fundamen tal matrix for the given system?" }}{PARA 3 "" 0 "" {TEXT 269 4 "a) [ " }{XPPEDIT 556 0 "u,v,w;" "6%%\"uG%\"vG%\"wG" }{TEXT 304 50 "] \+ b) [" }{XPPEDIT 557 0 "exp(t)*u, exp(2*t)*v,exp(2*t)*w;" "6%*&-%$expG6#%\"tG\"\"\"%\"uGF(*&-F%6#*&\"\"# F(F'F(F(%\"vGF(*&-F%6#*&F.F(F'F(F(%\"wGF(" }{TEXT 303 9 "] " }} {PARA 3 "" 0 "" {TEXT 313 5 "c) [" }{XPPEDIT 558 0 "exp(t)*u,exp(2*t) *v,exp(2*t)*w+v;" "6%*&-%$expG6#%\"tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F( F(%\"vGF(,&*&-F%6#*&F.F(F'F(F(%\"wGF(F(F/F(" }{TEXT 305 17 "] \+ d) [" }{XPPEDIT 559 0 "exp(t)*u,exp(2*t)*v,exp(2*t)*(w+v);" "6%*&-% $expG6#%\"tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(*&-F%6#*&F.F(F'F (F(,&%\"wGF(F/F(F(" }{TEXT 306 11 "] " }}{PARA 3 "" 0 "" {TEXT 314 5 "e) [" }{XPPEDIT 560 0 "exp(t)*u,exp(2*t)*v,exp(2*t)*(2*w +v);" "6%*&-%$expG6#%\"tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(*&- F%6#*&F.F(F'F(F(,&*&F.F(%\"wGF(F(F/F(F(" }{TEXT 307 11 "] f) [" } {XPPEDIT 561 0 "exp(t)*u,exp(2*t)*v,2*w+exp(2*t)*v;" "6%*&-%$expG6#%\" tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(,&*&F.F(%\"wGF(F(*&-F%6#*& F.F(F'F(F(F/F(F(" }{TEXT 308 8 "] " }}{PARA 3 "" 0 "" {TEXT 315 6 "g) [" }{XPPEDIT 562 0 "exp(t)*u,exp(2*t)*v,w+exp(2*t)*v;" "6%*&-% $expG6#%\"tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(,&%\"wGF(*&-F%6# *&F.F(F'F(F(F/F(F(" }{TEXT 309 18 "] h) [" }{XPPEDIT 563 0 "exp(t)*u,exp(2*t)*v,t*w+exp(2*t)*v;" "6%*&-%$expG6#%\"tG\"\"\"%\"uG F(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(,&*&F'F(%\"wGF(F(*&-F%6#*&F.F(F'F(F(F/F (F(" }{TEXT 310 10 "] " }}{PARA 3 "" 0 "" {TEXT 316 5 "i) [" }{XPPEDIT 564 0 "exp(t)*u,exp(2*t)*v,t*v+exp(2*t)*w;" "6%*&-%$expG6#% \"tG\"\"\"%\"uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(,&*&F'F(F/F(F(*&-F%6#*&F .F(F'F(F(%\"wGF(F(" }{TEXT 311 18 "] j) [" }{XPPEDIT 565 0 "exp(t)*u,exp(2*t)*v,(t*v+w)*exp(2*t);" "6%*&-%$expG6#%\"tG\"\"\"%\" uGF(*&-F%6#*&\"\"#F(F'F(F(%\"vGF(*&,&*&F'F(F/F(F(%\"wGF(F(-F%6#*&F.F(F 'F(F(" }{TEXT 312 4 "] " }}{PARA 0 "" 0 "" {TEXT 566 0 "" }}{PARA 0 "" 0 "" {TEXT 640 9 "Solution:" }{TEXT 641 5 " (j)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 567 33 "17. What is the approximation of " }{XPPEDIT 568 1 "y(3);" "6#-%\"yG6#\"\"$" }{TEXT 569 117 " that results when Euler's M ethod with step size 1 is used to approximate the solution of the init ial value problem " }{XPPEDIT 570 1 "y*`'`(x) = x*y(x)-1;" "6#/*&%\"y G\"\"\"-%\"'G6#%\"xGF&,&*&F*F&-F%6#F*F&F&F&!\"\"" }{TEXT 571 4 ", " }{XPPEDIT 572 1 "y(1) = 2;" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT 573 1 "?" }}{PARA 3 "" 0 "" {TEXT 270 100 "a) 4 b) 5 c) 6 d) 7 \+ e) 8 f) 9 g) 10 h) 11 i) 12 j) 13 " }} {PARA 0 "" 0 "" {TEXT 574 0 "" }}{PARA 0 "" 0 "" {TEXT 616 9 "Solution :" }{TEXT 617 5 " (e)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x[0] := 1: y[0] := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := 1:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "x[1] := x[0]+h; x[2] := x[1]+h;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := (u,v) -> u*v - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\" uG%\"vG6\"6$%)operatorG%&arrowGF),&*&9$\"\"\"9%F0F0F0!\"\"F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k := f(x[0],y[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y[1] := y[0] + h*k;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"yG6#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k := f(x[1],y[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y[2] := y[1] + h*k;" }} {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 575 34 "18. What is the approximation of " }{XPPEDIT 576 1 "y(3);" "6#-%\"yG6#\"\"$" }{TEXT 577 128 " that results when the Imp roved Euler Method with step size 2 is used to approximate the solutio n of the initial value problem " }{XPPEDIT 578 1 "y*`'`(x) = x*y(x)-2 ;" "6#/*&%\"yG\"\"\"-%\"'G6#%\"xGF&,&*&F*F&-F%6#F*F&F&\"\"#!\"\"" } {TEXT 579 4 ", " }{XPPEDIT 580 1 "y(1) = 2;" "6#/-%\"yG6#\"\"\"\"\"# " }{TEXT 581 1 "?" }}{PARA 0 "" 0 "" {TEXT 582 0 "" }}{PARA 3 "" 0 "" {TEXT 301 101 "a) 4 b) 5 c) 6 d) 7 e) 8 f) 9 g) 10 h) 11 i) 12 j) 13 " }}{PARA 0 "" 0 "" {TEXT 583 0 "" }}{PARA 0 "" 0 "" {TEXT 614 9 "Solution:" }{TEXT 615 5 " (c)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x[0] := 1: y[0] := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " x[1] := x[0]+h; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := (u,v) -> u*v - \+ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"uG%\"vG6\"6$%)opera torG%&arrowGF),&*&9$\"\"\"9%F0F0\"\"#!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k1 := f(x[0],y[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k2 := f(x[0]+h,y[0]+h*k1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k 2G\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "y[1] := y[0] + ( h/2)*(k1 + k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"\"\"\" '" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 584 26 "19. Consider the system " }{XPPEDIT 585 0 "matrix([[x[1](t)], [x[2] (t)]])*`'` = matrix([[2, 1], [1, -1]])*matrix([[x[1](t)], [x[2](t)]]); " "6#/*&-%'matrixG6#7$7#-&%\"xG6#\"\"\"6#%\"tG7#-&F,6#\"\"#6#F0F.%\"'G F.*&-F&6#7$7$F5F.7$F.,$F.!\"\"F.-F&6#7$7#-&F,6#F.6#F07#-&F,6#F56#F0F. " }{TEXT 586 21 " with initial value " }{XPPEDIT 587 0 "matrix([[x[1] (0)], [x[2](0)]]) = matrix([[2], [1]]);" "6#/-%'matrixG6#7$7#-&%\"xG6# \"\"\"6#\"\"!7#-&F+6#\"\"#6#F/-F%6#7$7#F47#F-" }{TEXT 588 15 ". Appro ximate " }{XPPEDIT 589 1 "x[2](2);" "6#-&%\"xG6#\"\"#6#F'" }{TEXT 590 42 " by using Euler's Method with step size 1." }}{PARA 0 "" 0 "" {TEXT 591 0 "" }}{PARA 0 "" 0 "" {TEXT 302 98 "a) 1 b) 2 c) \+ 3 d) 4 e) 5 f) 6 g) 7 h) 8 i) 9 j ) 10 " }}{PARA 0 "" 0 "" {TEXT 592 0 "" }}{PARA 0 "" 0 "" {TEXT 593 0 "" }}{PARA 0 "" 0 "" {TEXT 644 9 "Solution:" }{TEXT 645 5 " (g)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A := matrix([[2, 1], [1, -1]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"#\"\"\"7$F+!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x[0] := vector([2,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!-%'vectorG6#7$\"\"#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[1] := x[0]+ h*A &* x[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\",&&F%6#\"\"!F '-%#&*G6$%\"AGF)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[2] \+ := x[1]+ h*A &* x[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\" \"#,(&F%6#\"\"!\"\"\"-%#&*G6$%\"AGF)F,-F.6$F0,&F)F,F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(x[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"#B\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 594 34 "20. What is the approximation of " }{XPPEDIT 595 1 "y(2);" "6#-%\"yG6#\"\"#" }{TEXT 596 125 " that re sults when the Runge-Kutta Method with step size 2 is used to approxim ate the solution of the initial value problem " }{XPPEDIT 597 1 "y*`' `(x) = x+y(x);" "6#/*&%\"yG\"\"\"-%\"'G6#%\"xGF&,&F*F&-F%6#F*F&" } {TEXT 598 3 ", " }{XPPEDIT 599 1 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\" " }{TEXT 600 1 "?" }}{PARA 3 "" 0 "" {TEXT 271 97 "\na) 10 b) 11 \+ c) 12 d) 13 e) 14 f) 15 g) 16 h) 17 i) 18 j) 19" }}{PARA 0 "" 0 "" {TEXT 601 0 "" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 612 9 "Solution:" }{TEXT 613 5 " (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x[0] := 0: \+ y[0] := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h := 2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x[1] := x[0]+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := (u,v) -> u + v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF),&9$\"\"\"9%F/F)F) F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k1 := f(x[0],y[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "k2 := f(x[0]+h/2,y[0]+(h/2)*k1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#k2G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k3 := f(x[0]+h/2,y[0]+(h/2)*k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k3G\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "k4 := f(x[0]+h,y[0]+h*k3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# k4G\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "y[1] := y[0] + ( h/6)*(k1 + 2*k2 + 2*k3 + k4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" yG6#\"\"\"\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 0 2" 112 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }