{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 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 12 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 12 0 0 0 0 0 1 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 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 305 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 306 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 307 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 308 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 309 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 310 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 315 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 316 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 317 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 318 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 320 "" 1 12 0 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 1 }{CSTYLE "" -1 322 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 325 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 332 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 333 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 334 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 335 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 336 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 338 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 339 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 340 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 341 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 343 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 345 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 346 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 348 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 349 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 350 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 351 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 352 "" 1 12 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 1 }{CSTYLE "" -1 354 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 355 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 356 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 357 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 360 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 363 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 366 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 370 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 371 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 372 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 373 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 375 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 376 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 380 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 381 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 382 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 383 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 384 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 385 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 386 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 387 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 388 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 390 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 393 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 394 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 395 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 396 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 397 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 398 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 399 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 400 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 401 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 402 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 403 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 404 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 405 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 406 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 407 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 408 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 409 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 410 "" 1 12 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 1 }{CSTYLE "" -1 412 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 413 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 414 "" 1 12 0 0 0 0 1 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 415 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 416 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 417 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 418 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 419 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 420 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 421 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 422 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 423 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 424 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 425 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 426 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 427 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 428 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 429 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 430 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 431 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 432 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 433 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 434 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 435 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 436 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 437 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 438 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 439 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 440 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 441 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 442 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 443 "" 0 14 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 444 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 445 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 446 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 447 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 448 "" 0 14 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 449 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 450 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 451 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 452 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 453 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 454 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 455 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 456 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 457 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 458 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 459 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 460 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 461 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 462 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 463 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 464 "" 0 14 0 0 0 0 1 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 465 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 466 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 467 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 468 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 469 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 470 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 471 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 472 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 473 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 474 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 475 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 476 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 477 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 478 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 479 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 480 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 481 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 482 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 483 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 484 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 485 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 486 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 487 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 488 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 489 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 490 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 491 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 492 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 493 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 494 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 495 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 496 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 497 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 498 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 499 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 500 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 501 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 502 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 503 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 504 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 505 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 506 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 507 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 508 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 509 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 510 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 511 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 512 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 513 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 514 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 515 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 516 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 517 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 518 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 519 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 520 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 521 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 522 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 523 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 524 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 525 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 526 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 527 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 528 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 529 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 530 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 531 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 532 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 533 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 534 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 535 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 536 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 537 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 538 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 539 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 540 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 541 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 542 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 543 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 544 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 545 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 546 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 547 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 548 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 549 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 550 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 551 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 552 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 553 "" 1 12 0 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 0 } {CSTYLE "" 18 555 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 556 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 557 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 558 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 559 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 560 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 561 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 562 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 563 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 564 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 565 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 566 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 567 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 568 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 569 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 570 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 571 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 572 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 573 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 574 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 575 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 576 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 577 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 578 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 579 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 580 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 581 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 582 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 583 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 584 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 585 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 586 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 587 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 588 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 589 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 590 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 591 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 592 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 593 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 594 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 595 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 596 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 597 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 598 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 599 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 600 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 601 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 602 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 603 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 604 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 605 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 606 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 607 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 608 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 609 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 610 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 611 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 612 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 613 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 614 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 615 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 616 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 617 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 618 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 619 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 620 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 621 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 622 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 623 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 624 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 625 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 626 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 627 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 628 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 629 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 630 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 631 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 632 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 633 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 634 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 635 "" 1 12 0 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 0 }{CSTYLE "" 18 637 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 638 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 639 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 640 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 641 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 642 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 643 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 644 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 645 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 646 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 647 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 648 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 649 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 650 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 651 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 652 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 653 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 654 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 655 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 656 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 657 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 658 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 659 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 660 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 661 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 662 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 663 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 664 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 665 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 666 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 667 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 668 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 669 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 670 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 671 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 672 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 673 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 674 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 675 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 676 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 677 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 678 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 679 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 680 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 681 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 682 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 683 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 684 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 685 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 686 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 687 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 688 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 689 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 690 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 691 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 692 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 693 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 694 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 695 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 696 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 697 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 698 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 699 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 700 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 701 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 702 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 703 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 704 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 705 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 706 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 707 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 708 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 709 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 710 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 711 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 712 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 713 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 714 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 715 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 716 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 717 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 718 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 719 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 720 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 721 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 722 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 723 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 724 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 725 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 726 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 727 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 728 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 729 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 730 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 731 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 732 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 733 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 734 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 735 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 736 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 737 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 738 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 739 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 740 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 741 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 742 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 743 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 744 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 745 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 746 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 747 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 748 "" 1 12 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 749 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 750 "" 1 12 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 751 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 752 "" 1 12 0 0 1 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 753 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 754 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 755 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 756 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 757 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 758 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 759 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 760 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 761 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 762 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 763 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 764 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 765 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 766 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 767 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 768 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 769 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 770 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 771 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 772 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 773 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 774 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 775 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 776 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 777 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 778 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 779 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 18 780 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 781 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 782 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 783 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 784 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 785 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 786 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 787 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 788 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 789 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 790 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 791 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 792 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 793 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 794 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 795 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 796 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 797 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 798 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 799 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 800 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 801 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 802 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 803 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 804 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 805 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 806 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 807 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 808 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 809 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 810 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 811 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 812 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 813 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 814 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 815 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 816 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 817 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 818 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 819 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 820 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 821 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 822 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 823 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 824 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 825 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 826 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 827 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 828 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 829 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 830 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 831 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 832 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 833 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 834 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 835 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 836 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 837 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 838 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 839 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 840 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 841 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 842 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 843 "" 0 1 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 844 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 845 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 846 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 847 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 848 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 849 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 850 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 851 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 852 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 853 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 854 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 855 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 856 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 857 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 861 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 871 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 872 "" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 873 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 874 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 875 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 "Maple Output" -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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 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" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 275 40 " Math 132 \n Fall 2007 Final Exam" }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 15 "1. Calculate " }{XPPEDIT 410 0 "Int(co s(x)*sin(x)^3,x = 0 .. Pi/2);" "6#-%$IntG6$*&-%$cosG6#%\"xG\"\"\"*$-%$ sinG6#F*\"\"$F+/F*;\"\"!*&%#PiGF+\"\"#!\"\"" }{TEXT 409 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 415 4 "a) " }{XPPEDIT 263 0 "1;" "6#\"\"\"" }{TEXT 416 14 " b) " }{XPPEDIT 264 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 417 12 " \+ c) " }{XPPEDIT 265 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 418 12 " d) " }{XPPEDIT 266 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 419 14 " e) " }{TEXT 414 1 " " }{XPPEDIT 267 0 "1/5;" "6#*& \"\"\"F$\"\"&!\"\"" }{TEXT 413 19 " " }}{PARA 0 "" 0 "" {TEXT 412 4 "f) " }{XPPEDIT 268 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$! \"\"" }{TEXT 420 13 " g) " }{XPPEDIT 269 0 "3/4;" "6#*&\"\"$ \"\"\"\"\"%!\"\"" }{TEXT 421 13 " h) " }{XPPEDIT 271 0 "3/2; " "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 422 12 " i) " }{XPPEDIT 272 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 423 15 " j) \+ " }{XPPEDIT 273 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT 424 3 " " }}{PARA 3 "" 0 "" {TEXT 411 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 363 12 "Solution: d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J := Int(cos(x)*si n(x)^3, x = 0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG 6$*&-%$cosG6#%\"xG\"\"\")-%$sinGF+\"\"$F-/F,;\"\"!,$*&\"\"#!\"\"%#PiGF -F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "K := student[changev ar](u = sin(x), J, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$Int G6$*$)%\"uG\"\"$\"\"\"/F*;\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\" \"%" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 261 3 "2. " }{TEXT 277 5 "Let " }{XPPEDIT 18 0 "F(x) = Int((5+t^4)/sqrt(1+t^3),t = x .. 2);" "6#/- %\"FG6#%\"xG-%$IntG6$*&,&\"\"&\"\"\"*$%\"tG\"\"%F.F.-%%sqrtG6#,&F.F.*$ F0\"\"$F.!\"\"/F0;F'\"\"#" }{TEXT 457 62 ". Calculate the derivativ e D( F )( 2 ) of F at 2. " }{TEXT 278 1 " " }}{PARA 3 "" 0 " " {TEXT 276 6 "a) " }{XPPEDIT 306 0 "4;" "6#\"\"%" }{TEXT 301 18 " \+ b) " }{XPPEDIT 307 0 "5;" "6#\"\"&" }{TEXT 302 17 " \+ c) " }{XPPEDIT 308 0 "6;" "6#\"\"'" }{TEXT 303 17 " \+ d) " }{XPPEDIT 309 0 "7;" "6#\"\"(" }{TEXT 304 17 " e) \+ " }{XPPEDIT 310 0 "8;" "6#\"\")" }{TEXT 305 14 " \nf) " } {XPPEDIT 315 0 "-4;" "6#,$\"\"%!\"\"" }{TEXT 311 16 " g) \+ " }{XPPEDIT 316 0 "-5;" "6#,$\"\"&!\"\"" }{TEXT 312 13 " h) \+ " }{XPPEDIT 317 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT 313 16 " i) " }{XPPEDIT 318 0 "-7;" "6#,$\"\"(!\"\"" }{TEXT 314 15 " \+ j) " }{XPPEDIT 320 0 "-8;" "6#,$\"\")!\"\"" }{TEXT 319 3 " " }{TEXT 321 6 " \n" }{TEXT 364 0 "" }}{PARA 0 "" 0 "" {TEXT 365 13 "Soluti on: i\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "F := (x) -> Int( (5+t^4)/sqrt(1+t^3),t = x .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$*&,&\"\"&\"\"\"*$)%\"t G\"\"%F2F2F2-%%sqrtG6#,&F2F2*$)F5\"\"$F2F2!\"\"/F5;9$\"\"#F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"&\"\"\"*$)%\"xG\"\"%F'F'F',&F'F'*$)F*\"\"$F'F '#!\"\"\"\"#F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"(\"\"\"\"\"$!\"\"\"\"*#F&\" \"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(D(F)(2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 14 "3. Calculate " }{TEXT 552 1 " " }{XPPEDIT 553 0 "Int(x/(x+1)/ (x+2),x = 0 .. 1);" "6#-%$IntG6$*(%\"xG\"\"\",&F'F(F(F(!\"\",&F'F(\"\" #F(F*/F';\"\"!F(" }{TEXT 551 2 ". " }{TEXT 323 2 "\n\n" }{TEXT 322 4 " a) " }{XPPEDIT 332 0 "ln(9/8);" "6#-%#lnG6#*&\"\"*\"\"\"\"\")!\"\"" } {TEXT 324 13 " b) " }{XPPEDIT 333 0 "ln(7/6);" "6#-%#lnG6#*& \"\"(\"\"\"\"\"'!\"\"" }{TEXT 325 13 " c) " }{XPPEDIT 334 0 " ln(5/4);" "6#-%#lnG6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT 326 11 " d) " }{XPPEDIT 335 0 "ln(4/3);" "6#-%#lnG6#*&\"\"%\"\"\"\"\"$!\"\"" } {TEXT 327 12 " e) " }{TEXT 287 1 " " }{XPPEDIT 336 0 "ln(3/2); " "6#-%#lnG6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 281 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 280 4 "f) " }{XPPEDIT 337 0 "ln(9/5);" "6 #-%#lnG6#*&\"\"*\"\"\"\"\"&!\"\"" }{TEXT 328 14 " g) " } {XPPEDIT 338 0 "ln(8/3);" "6#-%#lnG6#*&\"\")\"\"\"\"\"$!\"\"" }{TEXT 329 13 " h) " }{XPPEDIT 339 0 "ln(9/4);" "6#-%#lnG6#*&\"\"*\" \"\"\"\"%!\"\"" }{TEXT 330 12 " i) " }{XPPEDIT 340 0 "ln(16/3) ;" "6#-%#lnG6#*&\"#;\"\"\"\"\"$!\"\"" }{TEXT 331 12 " j) " } {XPPEDIT 341 0 "ln(16/9);" "6#-%#lnG6#*&\"#;\"\"\"\"\"*!\"\"" }{TEXT 342 3 " " }}{PARA 3 "" 0 "" {TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT 366 13 "Solution: a\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "J := Int(x/(x+1)/(x+2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*(%\"xG\"\"\",&F)F*F*F*!\"\",&F)F*\" \"#F*F,/F);\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R := \+ student[integrand](J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG*(%\"x G\"\"\",&F&F'F'F'!\"\",&F&F'\"\"#F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PFE := convert(R, parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PFEG,&*&\"\"\"F',&%\"xGF'F'F'!\"\"F**&\"\"#F',&F)F'F ,F'F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "antiderivative : = int(PFE, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,& -%#lnG6#,&%\"xG\"\"\"F+F+!\"\"*&\"\"#F+-F'6#,&F*F+F.F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "definiteIntegral := subs(x=1,antide rivative) - subs(x=0,antiderivative); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1definiteIntegralG,(*&\"\"$\"\"\"-%#lnG6#\"\"#F(!\"\"*&F,F(-F* 6#F'F(F(-F*6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Answer := combine(definiteIntegral, ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'AnswerG,$-%#lnG6##\"\")\"\"*!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 297 12 "4. Calculate" }{TEXT 554 1 " " }{XPPEDIT 555 0 "int((8*x ^2+2*x+6)/(1+x)/(1+x^2),x = 0 .. 1);" "6#-%$intG6$*(,(*&\"\")\"\"\"*$% \"xG\"\"#F*F**&F-F*F,F*F*\"\"'F*F*,&F*F*F,F*!\"\",&F*F**$F,F-F*F1/F,; \"\"!F*" }{TEXT 763 7 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 299 3 "a) " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$ \"\"%!\"\"" }{TEXT 382 1 " " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"# " }{TEXT 558 1 " " }{TEXT 383 14 " b) " }{XPPEDIT 18 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 556 1 " " }{XPPEDIT 18 0 "ln(2);" "6 #-%#lnG6#\"\"#" }{TEXT 557 17 " c) " }{XPPEDIT 19 1 "ln(2 );" "6#-%#lnG6#\"\"#" }{TEXT 384 16 " d) " }{XPPEDIT 19 1 "2*ln(2);" "6#*&\"\"#\"\"\"-%#lnG6#F$F%" }{TEXT 385 9 " e)" } {TEXT 300 1 " " }{XPPEDIT 19 1 "3*ln(2);" "6#*&\"\"$\"\"\"-%#lnG6#\"\" #F%" }{TEXT 386 1 " " }{TEXT 387 11 " " }}{PARA 0 "" 0 "" {TEXT 298 5 "f) " }{XPPEDIT 19 1 "4*ln(2);" "6#*&\"\"%\"\"\"-%#lnG6# \"\"#F%" }{TEXT 381 15 " g) " }{XPPEDIT 19 1 "5*ln(2);" "6# *&\"\"&\"\"\"-%#lnG6#\"\"#F%" }{TEXT 388 11 " " }{TEXT 389 5 " h) " }{XPPEDIT 19 1 "6*ln(2);" "6#*&\"\"'\"\"\"-%#lnG6#\"\"#F%" } {TEXT 390 15 " i) " }{XPPEDIT 19 1 "7*ln(2);" "6#*&\"\"(\" \"\"-%#lnG6#\"\"#F%" }{TEXT 391 12 " j) " }{XPPEDIT 19 1 "8*ln (2);" "6#*&\"\")\"\"\"-%#lnG6#\"\"#F%" }{TEXT 392 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 367 13 "Solu tion: i\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "J := Int((8*x^2+2*x+6)/(1+x)/(1+x^2),x = 0 .. 1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"JG-%$IntG6$*(,(*&\"\")\"\"\")%\"xG\"\"#F,F,*&F/F, F.F,F,\"\"'F,F,,&F.F,F,F,!\"\",&F,F,*$F-F,F,F3/F.;\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R := student[integrand](J);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG*(,(*&\"\")\"\"\")%\"xG\"\"#F)F) *&F,F)F+F)F)\"\"'F)F),&F+F)F)F)!\"\",&F)F)*$F*F)F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PFE := convert(R, parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PFEG,&*(\"\"#\"\"\"%\"xGF(,&F(F(*$)F)F'F( F(!\"\"F(*&\"\"'F(,&F)F(F(F(F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "antiderivative := int(PFE, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,&-%#lnG6#,&\"\"\"F**$)%\"xG\"\"#F*F* F**&\"\"'F*-F'6#,&F-F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "definiteIntegral := subs(x=1,antiderivative) - subs(x=0,antide rivative); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1definiteIntegralG,&* &\"\"(\"\"\"-%#lnG6#\"\"#F(F(*&F'F(-F*6#F(F(!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "Answer := combine(definiteIntegral, ln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,$*&\"\"(\"\"\"-%#lnG6#\"\"# F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 289 14 "5. Calculate " } {XPPEDIT 559 0 "Int(x^2*ln(x),x = 1 .. exp(1));" "6#-%$IntG6$*&%\"xG\" \"#-%#lnG6#F'\"\"\"/F';F,-%$expG6#F," }{TEXT 459 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 3 "a) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F $\"\"$!\"\"" }{TEXT 615 1 " " }{XPPEDIT 18 0 "exp(3);" "6#-%$expG6#\" \"$" }{TEXT 616 1 " " }{TEXT 617 21 " b) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 595 1 " " }{TEXT 597 1 "( " }{TEXT 598 1 " " }{XPPEDIT 18 0 "2*exp(3)-1;" "6#,&*&\"\"#\"\"\"-%$e xpG6#\"\"$F&F&F&!\"\"" }{TEXT 596 1 " " }{TEXT 599 1 ")" }{TEXT 600 8 " c) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 601 1 " " }{TEXT 603 1 "(" }{TEXT 604 1 " " }{XPPEDIT 18 0 "exp(3)-2;" "6# ,&-%$expG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 602 1 " " }{TEXT 605 1 ")" } {TEXT 606 2 " " }{TEXT 607 9 " d) " }{XPPEDIT 18 0 "2/3;" "6#*& \"\"#\"\"\"\"\"$!\"\"" }{TEXT 608 1 " " }{TEXT 610 1 "(" }{TEXT 611 1 " " }{XPPEDIT 18 0 "exp(3)-1;" "6#,&-%$expG6#\"\"$\"\"\"F(!\"\"" } {TEXT 609 1 " " }{TEXT 612 1 ")" }{TEXT 613 2 " " }{TEXT 614 8 " e ) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 583 1 " " } {TEXT 585 1 "(" }{TEXT 586 1 " " }{XPPEDIT 18 0 "2*exp(3)+1;" "6#,&*& \"\"#\"\"\"-%$expG6#\"\"$F&F&F&F&" }{TEXT 584 1 " " }{TEXT 587 1 ")" } {TEXT 588 5 " " }{TEXT 293 19 " \n" }{TEXT 290 4 "f) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 589 1 " \+ " }{TEXT 591 1 "(" }{TEXT 592 1 " " }{XPPEDIT 18 0 "exp(3)+2;" "6#,&-% $expG6#\"\"$\"\"\"\"\"#F(" }{TEXT 590 1 " " }{TEXT 593 1 ")" }{TEXT 594 9 " g) " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT 577 1 " " }{TEXT 579 1 "(" }{TEXT 580 1 " " }{XPPEDIT 18 0 "exp( 3)+1;" "6#,&-%$expG6#\"\"$\"\"\"F(F(" }{TEXT 578 1 " " }{TEXT 581 1 ") " }{TEXT 582 7 " " }{TEXT 295 1 " " }{TEXT 291 4 "h) " } {XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT 560 1 " " }{TEXT 562 1 "(" }{TEXT 563 1 " " }{XPPEDIT 18 0 "2*exp(3)+1;" "6#,&*&\"\"#\" \"\"-%$expG6#\"\"$F&F&F&F&" }{TEXT 561 1 " " }{TEXT 564 1 ")" }{TEXT 565 9 " i) " }{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" } {TEXT 566 1 " " }{TEXT 568 1 "(" }{TEXT 569 1 " " }{XPPEDIT 18 0 "exp( 3)+2;" "6#,&-%$expG6#\"\"$\"\"\"\"\"#F(" }{TEXT 567 1 " " }{TEXT 570 1 ")" }{TEXT 294 10 " j) " }{XPPEDIT 18 0 "2/9;" "6#*&\"\"#\"\" \"\"\"*!\"\"" }{TEXT 571 1 " " }{TEXT 573 1 "(" }{TEXT 574 1 " " } {XPPEDIT 18 0 "exp(3)+1;" "6#,&-%$expG6#\"\"$\"\"\"F(F(" }{TEXT 572 1 " " }{TEXT 575 1 ")" }{TEXT 576 4 " " }{TEXT 296 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 368 12 "Solution: h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J := Int(x^2*ln(x), x = 1 .. exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&)%\"xG\" \"#\"\"\"-%#lnG6#F*F,/F*;F,-%$expG6#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "K := student[intparts](J, ln(x)); #Integration by Par ts with u=ln(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG,&*&#\"\"\"\" \"$F(*$)-%$expG6#F(F)F(F(F(-%$IntG6$,$*&F)!\"\"%\"xG\"\"#F(/F5;F(F,F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&#\"\"#\"\"*\"\"\"*$)-%$expG6#F(\"\"$F(F(F(#F (F'F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 285 4 "6. " }{TEXT 393 28 "W hat is the derivative of " }{TEXT 547 1 " " }{XPPEDIT 548 1 "x^(1/x) ;" "6#)%\"xG*&\"\"\"F&F$!\"\"" }{TEXT 546 18 " with respect to " } {XPPEDIT 620 0 "x;" "6#%\"xG" }{TEXT 618 7 " at " }{XPPEDIT 621 0 " x = 1/2;" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT 619 3 " ? " }{TEXT 394 1 "\n" }{TEXT 284 1 " " }}{PARA 3 "" 0 "" {TEXT 283 4 "a) " } {XPPEDIT 257 0 "-ln(2);" "6#,$-%#lnG6#\"\"#!\"\"" }{TEXT 460 12 " \+ b) " }{XPPEDIT 624 0 "-1/2;" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT 622 1 " " }{XPPEDIT 625 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 623 13 " \+ c) " }{XPPEDIT 348 0 "1-ln(2);" "6#,&\"\"\"F$-%#lnG6#\"\"#!\" \"" }{TEXT 343 14 " d) " }{XPPEDIT 349 0 "1-1/2;" "6#,&\"\" \"F$*&F$F$\"\"#!\"\"F'" }{TEXT 344 1 " " }{XPPEDIT 627 0 "ln(2);" "6#- %#lnG6#\"\"#" }{TEXT 626 12 " e) " }{XPPEDIT 350 0 "ln(2);" "6 #-%#lnG6#\"\"#" }{TEXT 345 7 " \nf) " }{XPPEDIT 629 0 "1/2;" "6#*&\" \"\"F$\"\"#!\"\"" }{TEXT 628 1 " " }{XPPEDIT 256 0 "ln(2);" "6#-%#lnG6 #\"\"#" }{TEXT 397 12 " g) " }{XPPEDIT 257 0 "1+ln(2);" "6#,& \"\"\"F$-%#lnG6#\"\"#F$" }{TEXT 396 12 " h) " }{XPPEDIT 631 0 "1+1/2;" "6#,&\"\"\"F$*&F$F$\"\"#!\"\"F$" }{TEXT 630 1 " " }{XPPEDIT 351 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 346 11 " i) " } {XPPEDIT 633 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 632 1 " " } {XPPEDIT 352 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 347 19 " \+ j) " }{XPPEDIT 257 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 395 3 " " }{TEXT 282 1 "\n" }}{PARA 0 "" 0 "" {TEXT 378 11 "Solution: g" }} {PARA 0 "" 0 "" {TEXT 379 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn1 := f(x) = x^(1/x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq n1G/-%\"fG6#%\"xG)F)*&\"\"\"F,F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := map(z-> simplify(ln(z), symbolic), eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%#lnG6#-%\"fG6#%\"xG*&F,!\" \"-F'F+\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn3 := ma p(z -> diff(z,x), eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/* &-%%diffG6$-%\"fG6#%\"xGF-\"\"\"F*!\"\",&*&F-!\"#-%#lnGF,F.F/*&F.F.*$) F-\"\"#F.F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn4 := D( f)(x) = solve(eqn3, diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%eqn4G/--%\"DG6#%\"fG6#%\"xG,$*(-F*F+\"\"\",&-%#lnGF+F0F0!\"\"F0F,! \"#F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn5 := subs(x = 1 /2, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/--%\"DG6#%\"fG6 ##\"\"\"\"\"#,$*(\"\"%F--F*F+F-,&-%#lnGF+F-F-!\"\"F-F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn6 := subs(x = 1/2, eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/-%\"fG6##\"\"\"\"\"##F*\"\"% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(eqn6, eqn5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"fG6##\"\"\"\"\"#,&-%#lnG6 #F,F+F+F+" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 288 9 "7. If " } {XPPEDIT 637 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT 636 1 " " } {TEXT 638 10 " and " }{XPPEDIT 639 0 "dy/dx;" "6#*&%#dyG\"\"\"%#d xG!\"\"" }{TEXT 634 4 " = " }{XPPEDIT 640 0 "cos(x)*sqrt(1-y^2);" "6# *&-%$cosG6#%\"xG\"\"\"-%%sqrtG6#,&F(F(*$%\"yG\"\"#!\"\"F(" }{TEXT 635 16 ", then what is " }{TEXT 641 1 " " }{XPPEDIT 642 0 "y(x);" "6#-%\" yG6#%\"xG" }{TEXT 764 1 "?" }}{PARA 0 "" 0 "" {TEXT 462 3 "a) " } {XPPEDIT 266 0 "sin(Pi*cos(x));" "6#-%$sinG6#*&%#PiG\"\"\"-%$cosG6#%\" xGF(" }{TEXT 465 13 " b) " }{XPPEDIT 267 0 "sin(sin(x));" "6# -%$sinG6#-F$6#%\"xG" }{TEXT 466 7 " c) " }{XPPEDIT 256 0 "cos(Pi*co s(x)/2);" "6#-%$cosG6#*(%#PiG\"\"\"-F$6#%\"xGF(\"\"#!\"\"" }{TEXT 472 7 " d) " }{XPPEDIT 268 0 "cos(sin(x));" "6#-%$cosG6#-%$sinG6#%\"xG " }{TEXT 469 10 " -1 e) " }{XPPEDIT 269 0 "arcsin(x^2);" "6#-%'arcs inG6#*$%\"xG\"\"#" }{TEXT 464 2 " " }{TEXT 463 20 " \+ \n" }{TEXT 461 4 "f) " }{XPPEDIT 270 0 "arcsin(arcsin(x));" "6#-%'ar csinG6#-F$6#%\"xG" }{TEXT 468 5 " g) " }{XPPEDIT 766 0 "sin(tan(x)); " "6#-%$sinG6#-%$tanG6#%\"xG" }{TEXT 765 6 " h) " }{XPPEDIT 257 0 "t an(sin(x));" "6#-%$tanG6#-%$sinG6#%\"xG" }{TEXT 471 14 " i) \+ " }{XPPEDIT 273 0 "arcsin(tan(x));" "6#-%'arcsinG6#-%$tanG6#%\"xG" } {TEXT 467 7 " j) " }{XPPEDIT 274 0 "arcsin(arctan(x));" "6#-%'arcsi nG6#-%'arctanG6#%\"xG" }{TEXT 470 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 376 11 "Solution: b" }}{PARA 0 "" 0 "" {TEXT 377 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "By the Method of Separa tion of Variables:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "eqn1 := Int(1/sqrt(1-t^2),t = 0 .. y(x)) = I nt(cos(t),t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%$IntG 6$*&\"\"\"F**$,&F*F**$)%\"tG\"\"#F*!\"\"#F*F0F1/F/;\"\"!-%\"yG6#%\"xG- F'6$-%$cosG6#F//F/;F5F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " eqn2 := map(value, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/ -%'arcsinG6#-%\"yG6#%\"xG-%$sinGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Answer := y(x) = solve(eqn2, y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG/-%\"yG6#%\"xG-%$sinG6#-F+F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "For those who are in terested, here is how to get " }{TEXT 843 6 "MAPLE " }{TEXT -1 79 " \+ to solve this differential equation without the user supplying any gui dance: \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode := diff(y(x ),x)=cos(x)*sqrt(1-y(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG /-%%diffG6$-%\"yG6#%\"xGF,*&-%$cosGF+\"\"\",&F0F0*$)F)\"\"#F0!\"\"#F0F 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "initialCondition := y( 0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1initialConditionG/-%\"yG6# \"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "IVP := \{ode, in itialCondition\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IVPG<$/-%\"yG6 #\"\"!F*/-%%diffG6$-F(6#%\"xGF1*&-%$cosGF0\"\"\",&F5F5*$)F/\"\"#F5!\" \"#F5F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "dsolve(IVP, y(x) ); \n#Using Maple's differential equation solver" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$sinG6#-F)F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 544 2 "8." }{TEXT -1 1 " " }{TEXT 643 39 "Consider the follow ing three statements" }{TEXT 644 16 " about a series " }{TEXT 646 1 " \+ " }{XPPEDIT 258 1 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#% \"nG/F);\"\"\"%)infinityG" }{TEXT 645 55 " with positive terms:\nI: \+ The series converges because" }{TEXT 648 3 " " }{XPPEDIT 259 1 "limi t(a[n],n = infinity) = 0;" "6#/-%&limitG6$&%\"aG6#%\"nG/F*%)infinityG \"\"!" }{TEXT 647 35 ".\n\nII: The series converges because" }{TEXT 650 3 " " }{XPPEDIT 257 1 "limit(a[``[n+1]]/b[n],n = infinity) = 1.1 ;" "6#/-%&limitG6$*&&%\"aG6#&%!G6#,&%\"nG\"\"\"F0F0F0&%\"bG6#F/!\"\"/F /%)infinityG-%&FloatG6$\"#6F4" }{TEXT 649 9 " and " }{TEXT 651 1 " " }{XPPEDIT 256 1 "sum(b[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"bG6# %\"nG/F);\"\"\"%)infinityG" }{TEXT 652 50 " converges.\n \nIII: The s eries converges because " }{XPPEDIT 256 1 "limit(a[``[n+1]]/a[n],n = \+ infinity) = 1;" "6#/-%&limitG6$*&&%\"aG6#&%!G6#,&%\"nG\"\"\"F0F0F0&F)6 #F/!\"\"/F/%)infinityGF0" }{TEXT 659 1 " " }{TEXT 658 87 ".\n\nFor eac h statement, determine whether the reasoning is correct or incorrec t. \n " }}{PARA 261 "" 0 "" {TEXT -1 7 "a) I: " }{TEXT 660 7 "corre ct" }{TEXT -1 13 ", II: " }{TEXT 669 7 "correct" }{TEXT 653 13 ", III: " }{TEXT 661 7 "correct" }{TEXT -1 17 " \nb) I :" }{TEXT 662 8 " correct" }{TEXT -1 13 ", II: " }{TEXT 663 7 " correct" }{TEXT 654 12 ", III:" }{TEXT -1 1 " " }{TEXT 670 9 "in correct" }{TEXT -1 13 " \nc) I: " }{TEXT 664 7 "correct" }{TEXT -1 13 ", II: " }{TEXT 671 9 "incorrect" }{TEXT 655 10 ", III : " }{TEXT 665 7 "correct" }{TEXT -1 14 " \nd) I: " }{TEXT 666 7 "correct" }{TEXT -1 13 ", II: " }{TEXT 672 9 "incorrect" } {TEXT 656 10 ", III: " }{TEXT 673 9 "incorrect" }{TEXT -1 17 " \+ \ne) I: " }{TEXT 674 9 "incorrect" }{TEXT -1 10 ", II: " } {TEXT 668 7 "correct" }{TEXT 657 13 ", III: " }{TEXT 667 7 "corr ect" }{TEXT -1 24 " \nf) I:" }{TEXT 678 10 " incorrec t" }{TEXT -1 10 ", II: " }{TEXT 679 7 "correct" }{TEXT 675 12 ", \+ III:" }{TEXT -1 1 " " }{TEXT 683 9 "incorrect" }{TEXT -1 13 " \+ \ng) I: " }{TEXT 680 9 "incorrect" }{TEXT -1 10 ", II: " }{TEXT 684 9 "incorrect" }{TEXT 676 10 ", III: " }{TEXT 681 7 "correct" } {TEXT -1 14 " \nh) I: " }{TEXT 682 9 "incorrect" }{TEXT -1 10 ", II: " }{TEXT 685 9 "incorrect" }{TEXT 677 10 ", III: " }{TEXT 686 9 "incorrect" }{TEXT -1 53 " \ni) Wrong answer \nj) B onus wrong answer" }}{PARA 3 "" 0 "" {TEXT 545 13 "\nSolution: f " }} {PARA 0 "" 0 "" {TEXT 819 21 "I) Incorrect: When " }{XPPEDIT 18 0 "a [n] = 1/n;" "6#/&%\"aG6#%\"nG*&\"\"\"F)F'!\"\"" }{TEXT 844 33 " the t erms of the series satisfy" }{TEXT 820 3 " " }{XPPEDIT 256 1 "limit( a[n],n = infinity) = 0;" "6#/-%&limitG6$&%\"aG6#%\"nG/F*%)infinityG\" \"!" }{TEXT -1 26 " but the series diverges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "II) " }{TEXT 846 7 "Correc t" }{TEXT -1 150 ": The assertion follows from the Limit Comparison Te st. It is true that the Limit Comparison Test is stated with the sli ghtly different hypothesis\n " }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }{XPPEDIT 256 1 "limit(a[``[n]]/b[n],n = infinity) = L ;" "6#/-%&limitG6$*&&%\"aG6#&%!G6#%\"nG\"\"\"&%\"bG6#F.!\"\"/F.%)infin ityG%\"LG" }{TEXT -1 94 ". \nTo see why the Limit Compar ison Test applies nonetheless, define the series " }{TEXT 861 1 " " } {XPPEDIT 256 1 "sum(c[n],n = 0 .. infinity);" "6#-%$sumG6$&%\"cG6#%\"n G/F);\"\"!%)infinityG" }{TEXT -1 6 " by " }{XPPEDIT 18 0 "c[n] = a[n +1];" "6#/&%\"cG6#%\"nG&%\"aG6#,&F'\"\"\"F,F," }{TEXT -1 24 ". The su m of the first " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 10 " terms of " }{XPPEDIT 256 1 "sum(c[n],n = 0 .. infinity);" "6#-%$sumG6$&%\"cG6# %\"nG/F);\"\"!%)infinityG" }{TEXT -1 6 " is " }{XPPEDIT 256 1 "sum(c [n],n = 0 .. N-1);" "6#-%$sumG6$&%\"cG6#%\"nG/F);\"\"!,&%\"NG\"\"\"F/! \"\"" }{TEXT -1 18 " , which equals " }{XPPEDIT 256 1 "sum(a[n],n = \+ 1 .. N);" "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%\"NG" }{TEXT -1 23 " . \+ Since the series " }{XPPEDIT 256 1 "sum(a[n],n = 1 .. infinity);" "6 #-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 5 " and " } {XPPEDIT 256 1 "sum(c[n],n = 0 .. infinity)" "6#-%$sumG6$&%\"cG6#%\"nG /F);\"\"!%)infinityG" }{TEXT -1 92 " have the same partial sums, they both converge or both diverge. \nSince\n\n " } {XPPEDIT 256 1 "limit(c[``[n]]/b[n],n = infinity);" "6#-%&limitG6$*&&% \"cG6#&%!G6#%\"nG\"\"\"&%\"bG6#F-!\"\"/F-%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 257 1 "limit(a[``[n+1]]/b[n],n = infinity) = 1.1;" "6#/-% &limitG6$*&&%\"aG6#&%!G6#,&%\"nG\"\"\"F0F0F0&%\"bG6#F/!\"\"/F/%)infini tyG-%&FloatG6$\"#6F4" }{TEXT 871 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 874 4 "and " }{TEXT 872 1 " " }{XPPEDIT 256 1 "sum(b[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"bG6#%\"nG/F);\"\" \"%)infinityG" }{TEXT 873 30 " converges, we conclude from " }{TEXT -1 25 "the Limit Comparison Test" }{TEXT 875 8 " that " }{XPPEDIT 256 1 "sum(c[n],n = 0 .. infinity)" "6#-%$sumG6$&%\"cG6#%\"nG/F);\"\"! %)infinityG" }{TEXT -1 25 " converges. Therefore " }{XPPEDIT 256 1 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%)in finityG" }{TEXT -1 21 " also converges. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "III) " }{TEXT 845 9 "Incorrect" }{TEXT -1 73 ": The limit condition is the inconclusive case of the Ra tio Test. (When " }{XPPEDIT 18 0 "a[n] = 1/n;" "6#/&%\"aG6#%\"nG*&\" \"\"F)F'!\"\"" }{TEXT -1 9 " we have " }{XPPEDIT 18 0 "limit(a[n+1]/a[ n],n = infinity) = 1;" "6#/-%&limitG6$*&&%\"aG6#,&%\"nG\"\"\"F-F-F-&F) 6#F,!\"\"/F,%)infinityGF-" }{TEXT -1 26 " but the series diverges.)" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 43 "9. Consider the following three statements" }{TEXT 398 16 " about a series " }{TEXT 520 1 " " }{XPPEDIT 521 1 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$& %\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 519 55 " with positive term s:\nI: The series converges because" }{TEXT 523 3 " " }{XPPEDIT 524 1 "a[n] < 1/(10+sqrt(n));" "6#2&%\"aG6#%\"nG*&\"\"\"F),&\"#5F)-%%s qrtG6#F'F)!\"\"" }{TEXT 522 35 " .\n\nII: The series diverges because " }{TEXT 525 3 " " }{XPPEDIT 259 1 "1/(n^2) < a[n];" "6#2*&\"\"\"F%* $%\"nG\"\"#!\"\"&%\"aG6#F'" }{TEXT 526 38 " .\n\nIII: The series conv erges because" }{TEXT 543 3 " " }{XPPEDIT 259 1 "limit(a[n+1]/a[n],n = infinity) = 0;" "6#/-%&limitG6$*&&%\"aG6#,&%\"nG\"\"\"F-F-F-&F)6#F, !\"\"/F,%)infinityG\"\"!" }{TEXT 542 99 ".\n\nFor each statement, dete rmine whether the reasoning is correct ( C ) or incorrect ( F ). \n " }}{PARA 261 "" 0 "" {TEXT -1 7 "a) I: " }{TEXT 528 1 "C" }{TEXT -1 9 ", II: " }{TEXT 529 13 "C, III: C" }{TEXT -1 18 " \+ \nb) I: " }{TEXT 530 1 "C" }{TEXT -1 9 ", II: " }{TEXT 531 13 "C, III: F" }{TEXT -1 24 " \nc) I: C, II: F" }{TEXT 533 13 ", III: C" }{TEXT -1 24 " \nd) I: C, II: F" }{TEXT 534 13 " , III: F" }{TEXT -1 16 " \ne) I:" }{TEXT 536 2 " F" } {TEXT -1 10 ", II: " }{TEXT 535 13 "C, III: C" }{TEXT -1 24 " \+ \nf) I: " }{TEXT 537 1 "F" }{TEXT -1 11 ", II: " }{TEXT 532 13 "C, III: F" }{TEXT -1 15 " \ng) I: " }{TEXT 541 1 "F" }{TEXT -1 11 ", II: F" }{TEXT 540 13 ", III: C" } {TEXT -1 13 " \nh) I: " }{TEXT 538 1 "F" }{TEXT -1 10 ", II: \+ " }{TEXT 539 14 "F, III: F" }{TEXT -1 51 " \ni) Wrong answe r \nj) Bonus wrong answer" }}{PARA 0 "" 0 "" {TEXT 527 1 "\n" } {TEXT -1 0 "" }{TEXT 374 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 375 1 " " }{TEXT -1 4 "I) I" }{TEXT 847 29 "ncorrect ( F ): The series \+ " }{XPPEDIT 18 0 "sum(1/(10+sqrt(n)),n = 1 .. infinity);" "6#-%$sumG6$ *&\"\"\"F',&\"#5F'-%%sqrtG6#%\"nGF'!\"\"/F-;F'%)infinityG" }{TEXT 848 59 " is divergent (by comparison with the divergent p-series " } {XPPEDIT 18 0 "sum(1/sqrt(n),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\" F'-%%sqrtG6#%\"nG!\"\"/F+;F'%)infinityG" }{TEXT 849 25 ".) No informa tion about " }{XPPEDIT 18 0 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6 $&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 856 31 " can be deduc ed from " }{XPPEDIT 18 0 "a[n] < b[n];" "6#2&%\"aG6#%\"nG&%\"bG6# F'" }{TEXT 850 12 " when " }{XPPEDIT 18 0 "sum(b[n],n = 1 .. inf inity);" "6#-%$sumG6$&%\"bG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 851 16 " is divergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "II) I" }{TEXT 852 29 "ncorrect ( F ): The series " } {XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F' *$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT 853 42 " is convergent. \+ No information about " }{XPPEDIT 18 0 "sum(a[n],n = 1 .. infinity); " "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 857 29 " ca n be deduced from\n " }{XPPEDIT 18 0 "b[n] < a[n];" "6#2&%\"bG6#% \"nG&%\"aG6#F'" }{TEXT 854 9 " when " }{XPPEDIT 18 0 "sum(b[n],n = \+ 1 .. infinity);" "6#-%$sumG6$&%\"bG6#%\"nG/F);\"\"\"%)infinityG" } {TEXT 855 16 " is convergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "III) Cor rect (C): Since the limit, namely 0, is less than 1, this assertion fo llows from the Ratio Test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 407 5 "10. " } {TEXT 504 42 "Consider the three series \n I: " }{XPPEDIT 256 1 "Sum(n^5/(3^n),n = 0 .. infinity);" "6#-%$SumG6$*&%\"nG\"\"&)\" \"$F'!\"\"/F';\"\"!%)infinityG" }{TEXT 767 20 " , II: " } {XPPEDIT 256 1 "Sum(10^n/sqrt(n!),n = 0 .. infinity);" "6#-%$SumG6$*&) \"#5%\"nG\"\"\"-%%sqrtG6#-%*factorialG6#F)!\"\"/F);\"\"!%)infinityG" } {TEXT 505 29 " , and III: " }{XPPEDIT 256 1 "Sum(1/(n *ln(n)),n = 2 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F) F'!\"\"/F);\"\"#%)infinityG" }{TEXT 768 2 " \n" }{TEXT 506 149 "and th e statements\n\n( C ) The series converges \n( D ) The series diverg es\n\nFor each series, decide which of statements (C), (D) is co rrect. " }}{PARA 261 "" 0 "" {TEXT -1 19 "a) I: C, II: C" } {TEXT 507 12 ", III: C" }{TEXT -1 29 " \nb) I: C, II : " }{TEXT 508 11 "C, III:" }{TEXT -1 14 " D \nc) I: " }{TEXT 509 1 "C" }{TEXT -1 11 ", II: " }{TEXT 511 13 "D, III: C" } {TEXT -1 25 " \nd) I: C, II: " }{TEXT 512 12 "D, III: " }{TEXT -1 30 "D \ne) I: D, II: C" }{TEXT 513 12 ", II I: C" }{TEXT -1 23 " \nf) I: " }{TEXT 514 2 " D" } {TEXT -1 12 ", II: C" }{TEXT 510 11 ", III: " }{TEXT -1 15 "D \ng) I: " }{TEXT 518 1 "D" }{TEXT -1 11 ", II: " }{TEXT 517 13 "D, III: C" }{TEXT -1 13 " \nh) I: " }{TEXT 515 1 "D" }{TEXT -1 11 ", II: " }{TEXT 516 13 "D, III: D" }{TEXT -1 52 " \ni) Wrong answer \n j) Bonus wrong answer" }}{PARA 3 "" 0 "" {TEXT 503 1 "\n" }{TEXT 408 12 "Solution: b " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "I)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a := n -> n^ 5/3^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operat orG%&arrowGF(*&9$\"\"&)\"\"$F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "a(n+1), a(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*& ,&%\"nG\"\"\"F&F&\"\"&)\"\"$F$!\"\"*&F%F')F)F%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "'a(n+1)/a(n)' = a(n+1)/a(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"aG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\"**F(\"\" &)\"\"$F(F-F)!\"&)F1F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'a(n+1)/a(n)' = simplify(a(n+1)/a(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"aG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\",$*(\"\"$F-F(\"\"&F)! \"&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Limit('a(n+1)/a(n) ', n=infinity) = limit(1/3*(n+1)^5/n^5, n=infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%\"aG6#,&%\"nG\"\"\"F-F-F--F)6#F,! \"\"/F,%)infinityG#F-\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Since this limi t is less than 1, the Ratio Test gives convergence (C)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "II)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "b := n -> 10^n/sqrt(n!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&)\"#59$\"\" \"-%%sqrtG6#-%*factorialG6#F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b(n+1), b(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&) \"#5,&%\"nG\"\"\"F(F(F(-%*factorialG6#F&#!\"\"\"\"#*&)F%F'F(-F*6#F'F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "'b(n+1)/b(n)' = b(n+1)/ b(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"bG6#,&%\"nG\"\"\"F*F*F *-F&6#F)!\"\"**)\"#5F(F*-%*factorialGF'#F-\"\"#)F0F)F--F2F,#F*F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "'b(n+1)/b(n)' = 10*sqrt(n!/( n+1)!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"bG6#,&%\"nG\"\"\"F*F *F*-F&6#F)!\"\",$*&\"#5F**&-%*factorialGF,F*-F3F'F-#F*\"\"#F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "'b(n+1)/b(n)' = simplify(10* sqrt(n!/(n+1)!)) assuming (n,posint);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"bG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\",$*&\"#5F*F(#F-\"\"#F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Limit('b(n+1)/b(n)', n=inf inity) = limit(10/(n+1)^(1/2), n=infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&-%\"bG6#,&%\"nG\"\"\"F-F-F--F)6#F,!\"\"/F ,%)infinityG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Since this limit is less than 1, the Ratio Test gives con vergence (C)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "III)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> 1/x/ln(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*&9$F--%#lnG6 #F/F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Int(1/x /ln(x),x= 2 .. N) = int(1/x/ln(x),x= 2 .. N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*&%\"xGF(-%#lnG6#F*F(!\"\"/F*;\"\"# %\"NG,&-F,6#-F,6#F2F(-F,6#-F,6#F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Int(1/(x*ln(x)),x = 2 .. infinity) = Limit( ln(ln(N)) -ln(ln(2)), N = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&\"\"\"F(*&%\"xGF(-%#lnG6#F*F(!\"\"/F*;\"\"#%)infinityG-%&LimitG6$, &-F,6#-F,6#%\"NGF(-F,6#-F,6#F1F./F;F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Int(1/(x*ln(x)),x = 2 .. infinity) = limit( ln(ln(N)) -ln(ln(2)), N = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$*&\"\"\"F(*&%\"xGF(-%#lnG6#F*F(!\"\"/F*;\"\"#%)infinityGF2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The given series diverges (D) by the Integral Test." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 286 5 "11. " }{TEXT 489 39 "Consider the two se ries \n I: " }{XPPEDIT 259 1 "Sum((-1)^n*(n/(1+n))^n,n = 1 . . infinity);" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"nGF))*&F+F),&F)F)F+F)F*F+ F)/F+;F)%)infinityG" }{TEXT 490 17 " and II: " }{XPPEDIT 256 1 "Sum((-1)^n*n^(2/3)/(1+n^(4/3)),n = 0 .. infinity);" "6#-%$SumG6$*() ,$\"\"\"!\"\"%\"nGF))F+*&\"\"#F)\"\"$F*F),&F)F))F+*&\"\"%F)F/F*F)F*/F+ ;\"\"!%)infinityG" }{TEXT 491 2 " \n" }{TEXT 492 18 "and the statement s" }}{PARA 3 "" 0 "" {TEXT 769 188 "\n( AC ) The series converges abs olutely\n( CC ) The series converges conditionally\n( D ) The seri es diverges\n\nFor each series, decide which of statements (AC), (CC), (D) is correct. " }}{PARA 261 "" 0 "" {TEXT -1 7 "a) I: " }{TEXT 758 8 "AC, " }{TEXT -1 5 " II: " }{TEXT 759 2 "AC" }{TEXT -1 14 " \+ \nb) I: " }{TEXT 760 2 "AC" }{TEXT -1 11 ", II: " }{TEXT 502 2 "CC" }{TEXT -1 14 " \nc) I: " }{TEXT 761 2 "AC" }{TEXT -1 12 ", II: " }{TEXT 493 1 "D" }{TEXT -1 16 " \nd) I: " } {TEXT 495 2 "CC" }{TEXT -1 11 ", II: " }{TEXT 494 2 "AC" }{TEXT -1 14 " \ne) I: " }{TEXT 496 2 "CC" }{TEXT -1 11 ", II: " } {TEXT 497 2 "CC" }{TEXT -1 23 " \nf) I: " }{TEXT 498 2 "CC" }{TEXT -1 11 ", II: " }{TEXT 499 2 " D" }{TEXT -1 30 " \+ \ng) I: D, II: " }{TEXT 762 2 "AC" }{TEXT -1 29 " \nh) \+ I: D, II: " }{TEXT 500 3 " CC" }{TEXT -1 30 " \ni) I: \+ D, II: " }{TEXT 501 1 "D" }{TEXT -1 27 " \n j) Wrong \+ answer" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 373 12 "Solution: h " }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a := n -> (n/(1+n))^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf* 6#%\"nG6\"6$%)operatorG%&arrowGF()*&9$\"\"\",&F.F/F/F/!\"\"F.F(F(F(" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We stud ied the limit of this sequence in conjunction with continuous compound ing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(a(n), n = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "Since this number is not 0, it follows fr om the divergence test that the series diverges (D)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "II)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a := n -> n^ (2/3)/(1+n^(4/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6 \"6$%)operatorG%&arrowGF(*&9$#\"\"#\"\"$,&\"\"\"F2*$)F-#\"\"%F0F2F2!\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "b := n -> 1/n^( 2/3); #Captures the size of a(n)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$)9$#\"\"#\"\"$F-!\" \"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(a(n)/b(n), n = infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Since this number is not 0 and not " }{XPPEDIT 18 0 "infinity; " "6#%)infinityG" }{TEXT -1 62 ", we conclude from the Limit Compariso n Test that the series " }{XPPEDIT 18 0 "sum(n^(2/3)/(1+n^(4/3)),n = \+ 1 .. infinity);" "6#-%$sumG6$*&)%\"nG*&\"\"#\"\"\"\"\"$!\"\"F+,&F+F+)F (*&\"\"%F+F,F-F+F-/F(;F+%)infinityG" }{TEXT -1 40 " has the same be havior as the series " }{XPPEDIT 18 0 "sum(1/(n^(2/3)),n = 1 .. infini ty);" "6#-%$sumG6$*&\"\"\"F')%\"nG*&\"\"#F'\"\"$!\"\"F-/F);F'%)infinit yG" }{TEXT -1 53 ", which is divergent since it is a p-series with \+ " }{XPPEDIT 18 0 "p = 2/3;" "6#/%\"pG*&\"\"#\"\"\"\"\"$!\"\"" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 199 ". As a result, we conclude that the given series is not absolutely convergent. Sinc e the given series converges by the Alternating Series Test, we conclu de that it is conditionally convergent (CC)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 44 "12. Consider the two series \n I: " } {XPPEDIT 256 1 "Sum(1/(n^Pi),n = 0 .. infinity);" "6#-%$SumG6$*&\"\"\" F')%\"nG%#PiG!\"\"/F);\"\"!%)infinityG" }{TEXT 799 19 " and I I: " }{XPPEDIT 256 1 "Sum(n!/(10^(100*n)),n = 0 .. infinity);" "6#-%$ SumG6$*&-%*factorialG6#%\"nG\"\"\")\"#5*&\"$+\"F+F*F+!\"\"/F*;\"\"!%)i nfinityG" }{TEXT 473 2 " \n" }{TEXT 474 267 "and the statements\n\n( C ) The Ratio Test establishes convergence\n( D ) The Ratio Test esta blishes divergence\n( F ) The Ratio Test is not conclusive.\n\nApply the Ratio Test to series I and II and for each, decide which of sta tements (C), (D), (F) is correct. " }}{PARA 261 "" 0 "" {TEXT -1 7 "a) I: " }{TEXT 475 1 "C" }{TEXT -1 11 ", II: " }{TEXT 476 1 " C" }{TEXT -1 9 " " }}{PARA 261 "" 0 "" {TEXT -1 19 "b) I: C, \+ II: " }{TEXT 488 1 "D" }{TEXT -1 4 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "c) I: " }{TEXT 477 1 "C" }{TEXT -1 11 ", II: " } {TEXT 478 1 "F" }{TEXT -1 6 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "d ) I: " }{TEXT 480 1 "D" }{TEXT -1 11 ", II: " }{TEXT 479 1 "C" } {TEXT -1 8 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "e) I: " }{TEXT 481 1 "D" }{TEXT -1 11 ", II: " }{TEXT 482 1 "D" }{TEXT -1 22 " \+ \nf) I: " }{TEXT 483 2 " D" }{TEXT -1 19 ", II: F \+ " }}{PARA 261 "" 0 "" {TEXT -1 20 "g) I: F, II: " }{TEXT 484 1 "C" }{TEXT -1 6 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "h) I: \+ " }{TEXT 485 1 "F" }{TEXT -1 12 ", II: " }{TEXT 486 1 "D" } {TEXT -1 5 " " }}{PARA 261 "" 0 "" {TEXT -1 8 "i) I: " }{TEXT 487 1 "F" }{TEXT -1 19 ", II: F " }}{PARA 261 "" 0 "" {TEXT -1 16 "j) Wrong answer" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 399 12 "Solution: h " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a := n -> 1/n^Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#% \"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-)9$%#PiG!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Limit(a(n+1)/a(n), n = infin ity) = limit(a(n+1)/a(n), n = infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&),&%\"nG\"\"\"F+F+%#PiG!\"\")F*F,F+/F*%)i nfinityGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Because this is equal to 1, " }{TEXT 841 37 "the Ratio Test is \+ not conclusive (F)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "II" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a := n -> n! /(10^(100*n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$ %)operatorG%&arrowGF(*&-%*factorialG6#9$\"\"\")\"#5,$*&\"$+\"F1F0F1F1! \"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r := simplify (a(n+1)/a(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rG,&*&\"`q++++++ ++++++++++++++++++++++++++++++++++++++++++++\"!\"\"%\"nG\"\"\"F*#F*F'F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(r, n = infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Because this limit is gre ater than 1, " }{TEXT 842 42 "the Ratio Test establishes divergence (D )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 800 44 "13. Consider the two series \n \+ I: " }{XPPEDIT 256 1 "Sum(((1+n^3)/(10+100*n^2+n^3))^n,n = 0 .. infinity);" "6#-%$SumG6$)*&,&\"\"\"F)*$%\"nG\"\"$F)F),(\"#5F)*&\"$ +\"F)*$F+\"\"#F)F)*$F+F,F)!\"\"F+/F+;\"\"!%)infinityG" }{TEXT 817 25 " and II: " }{XPPEDIT 256 1 "Sum(((3+n)/3/n)^n,n = 1 .. infinity);" "6#-%$SumG6$)*(,&\"\"$\"\"\"%\"nGF*F*F)!\"\"F+F,F+/F+;F*% )infinityG" }{TEXT 801 2 " \n" }{TEXT 802 263 "and the statements\n\n( C ) The Root Test establishes convergence\n( D ) The Root Test esta blishes divergence\n( F ) The Root Test is not conclusive.\n\nApply \+ the Root Test to series I and II and for each, decide which of state ments (C), (D), (F) is correct. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 7 "a) I: " }{TEXT 803 1 "C" }{TEXT -1 11 ", II: " }{TEXT 804 1 "C" }{TEXT -1 9 " " }}{PARA 261 "" 0 "" {TEXT -1 19 "b) I: C, II: " }{TEXT 816 1 "D" }{TEXT -1 4 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "c) I: " }{TEXT 805 1 "C" } {TEXT -1 11 ", II: " }{TEXT 806 1 "F" }{TEXT -1 6 " " }} {PARA 261 "" 0 "" {TEXT -1 7 "d) I: " }{TEXT 808 1 "D" }{TEXT -1 11 " , II: " }{TEXT 807 1 "C" }{TEXT -1 8 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "e) I: " }{TEXT 809 1 "D" }{TEXT -1 11 ", II: " } {TEXT 810 1 "D" }{TEXT -1 22 " \nf) I: " }{TEXT 811 2 " \+ D" }{TEXT -1 19 ", II: F " }}{PARA 261 "" 0 "" {TEXT -1 20 "g) I: F, II: " }{TEXT 812 1 "C" }{TEXT -1 6 " " }}{PARA 261 "" 0 "" {TEXT -1 7 "h) I: " }{TEXT 813 1 "F" }{TEXT -1 12 ", \+ II: " }{TEXT 814 1 "D" }{TEXT -1 5 " " }}{PARA 261 "" 0 "" {TEXT -1 8 "i) I: " }{TEXT 815 1 "F" }{TEXT -1 19 ", II: F \+ " }}{PARA 261 "" 0 "" {TEXT -1 16 "j) Wrong answer" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 818 13 "Solution: g " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a := n -> ((1+n^3)/(10+100*n^2+n^3))^n;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF()*&,&\" \"\"F/*$)9$\"\"$F/F/F/,(\"#5F/*&\"$+\"F/)F2\"\"#F/F/F0F/!\"\"F2F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r := combine(a(n)^(1/n) , symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*&,&\"\"\"F'*$)% \"nG\"\"$F'F'F',(\"#5F'*&\"$+\"F')F*\"\"#F'F'F(F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Limit(a(n)^(1/n), n = infinity) = l imit(r, n = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$ ))*&,&\"\"\"F+*$)%\"nG\"\"$F+F+F+,(\"#5F+*&\"$+\"F+)F.\"\"#F+F+F,F+!\" \"F.*&F+F+F.F6/F.%)infinityGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Because t his is equal to 1, " }{TEXT 840 36 "the Root Test is not conclusive (F )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "II" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a := n -> ((3+n)/3/n)^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG f*6#%\"nG6\"6$%)operatorG%&arrowGF(),$*&#\"\"\"\"\"$F0*&,&F1F09$F0F0F4 !\"\"F0F0F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r := c ombine(a(n)^(1/n), symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"r G*&,&\"\"\"F'*&\"\"$!\"\"%\"nGF'F'F'F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Limit(a(n)^(1/n), n = infinity) = limit(r, n = infini ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)),$*(\"\"$!\"\",& F+\"\"\"%\"nGF.F.F/F,F.F/*&F.F.F/F,/F/%)infinityG#F.F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Because this is less than 1, " }{TEXT 839 42 "the Root Te st establishes convergence (C)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 260 10 " 14. Let " }{TEXT 745 1 " " } {TEXT 798 1 " " }{XPPEDIT 271 0 "f(x) = 1/72;" "6#/-%\"fG6#%\"xG*&\"\" \"F)\"#s!\"\"" }{XPPEDIT 272 0 "x^3*exp(2*x^2);" "6#*&%\"xG\"\"$-%$exp G6#*&\"\"#\"\"\"*$F$F*F+F+" }{TEXT 744 1 " " }{TEXT 747 5 " . " } {TEXT 736 9 "What is " }{TEXT 746 1 " " }{XPPEDIT 274 0 "f;" "6#%\"fG " }{XPPEDIT 275 0 "``^(``*7*``);" "6#)%!G*(F$\"\"\"\"\"(F&F$F&" } {TEXT 743 7 "( 0 ) ?" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 733 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 737 73 "a) 20 b) 40 \+ c) 60 d) 80 e) 100 " }{TEXT 740 3 " \+ " }{TEXT 739 18 " " }}{PARA 0 "" 0 "" {TEXT 738 64 "f ) 120 g) 140 h) 160 i) 180 j)" } {TEXT -1 1 " " }{TEXT 742 1 " " }{TEXT 741 4 "200 " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 734 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 735 1 " \+ " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> x^3*exp(2*x^2)/72;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"#sF/*&)9$\"\"$F/-%$expG6#,$*&\" \"#F/)F3F:F/F/F/F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Maclaurin := series(f(x), x = 0, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*MaclaurinG+-%\"xG#\"\"\"\"#s\"\"$#F(\"#O\"\"&F+\"\"( #F(\"#a\"\"*-%\"OG6#F(\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p := convert(Maclaurin, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,**&\"#s!\"\"%\"xG\"\"$\"\"\"*&\"#OF(F)\"\"&F+*&F-F(F)\"\"(F+ *&\"#aF(F)\"\"*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Answer = 7!*coeff(p, x^7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'AnswerG\"$S \"" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "By direc t calculation (not recommended!):\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@7)(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*& \"$S\"\"\"\"-%$expG6#,$*&\"\"#F&)%\"xGF,F&F&F&F&*(\"%!3$F&F-F&F'F&F&*( \"%g*)F&)F.\"\"%F&F'F&F&*&#\"&'HB\"\"$F&*&)F.\"\"'F&F'F&F&F&*&#\"%orF8 F&*&)F.\"\")F&F'F&F&F&*&#\"%[?\"\"*F&*&)F.\"#5F&F'F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@7)(f)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$S\"" }}}{PARA 3 "" 0 "" {TEXT 732 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 458 5 "15. " } {TEXT 693 12 "Calculate " }{XPPEDIT 259 1 "L = limit(120*sin(2*x^5)/ (x*cos(5*x^2)-x),x = 0);" "6#/%\"LG-%&limitG6$*(\"$?\"\"\"\"-%$sinG6#* &\"\"#F**$%\"xG\"\"&F*F*,&*&F1F*-%$cosG6#*&F2F**$F1F/F*F*F*F1!\"\"F:/F 1\"\"!" }{TEXT 692 2 " " }{TEXT 694 152 " by finding the Maclaurin s eries of the numerator and the Maclaurin series of the denominator. T hese two Maclaurin series begin with the same degree " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT 695 115 " monomial. (In other words, for the \+ Maclaurin series for both the numerator and denominator, the coefficie nts of " }{XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT 696 15 " are \+ 0 for " }{XPPEDIT 18 0 "n < p;" "6#2%\"nG%\"pG" }{TEXT 697 27 " and the coefficients of " }{XPPEDIT 18 0 "x^p;" "6#)%\"xG%\"pG" }{TEXT 698 51 " are nonzero.) What is the value of the product " } {XPPEDIT 18 0 "p*L;" "6#*&%\"pG\"\"\"%\"LGF%" }{TEXT 699 3 " ? " }} {PARA 260 "" 0 "" {TEXT 838 1 " " }}{PARA 0 "" 0 "" {TEXT 687 4 "a) \+ " }{TEXT 701 1 "-" }{TEXT 702 17 " 36 b) " }{TEXT 705 1 "-" }{TEXT 706 19 " 48 c) " }{TEXT 711 1 "-" }{TEXT 712 17 " 6 0 d) " }{TEXT 713 1 "-" }{TEXT 714 15 " 72 e) " } {TEXT 717 1 "-" }{TEXT 718 5 " 84 " }{TEXT 690 3 " " }{TEXT 689 18 " " }}{PARA 0 "" 0 "" {TEXT 688 5 "f) " }{TEXT 703 1 "-" }{TEXT 704 17 " 96 g) " }{TEXT 707 1 "-" }{TEXT 708 18 "108 h) " }{TEXT 709 1 "-" }{TEXT 710 17 "120 i) " }{TEXT 715 2 "- " }{TEXT 716 11 "132 j)" }{TEXT -1 2 " " } {TEXT 700 2 "- " }{TEXT 719 4 "144 " }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 691 13 "Solution: f " }} {PARA 0 "" 0 "" {TEXT 837 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ratio := x -> 120*sin(2*x^5)/(x*cos(5*x^2)-x); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,$*(\"$?\"\"\"\"-%$sinG6#,$*&\"\"#F/)9$\"\"&F/F/F/,&*&F7F/-% $cosG6#,$*&F8F/)F7F5F/F/F/F/F7!\"\"FAF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Limit(ratio(x), x = 0) = limit(ratio(x), x = 0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,$*(\"$?\"\"\"\"-%$sin G6#,$*&\"\"#F*)%\"xG\"\"&F*F*F*,&*&F2F*-%$cosG6#,$*&F3F*)F2F0F*F*F*F*F 2!\"\"F " 0 "" {MPLTEXT 1 0 71 "series(numer(ratio(x)), x = 0, 10);\nseries(denom(ratio(x)), x = 0 , 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG\"$S#\"\"&-%\"OG6#\" \"\"\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"xG#!#D\"\"#\"\"&#\"$D '\"#C\"\"*-%\"OG6#\"\"\"\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Answer = 5*(-96/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'Answer G!#'*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 5 "16. " }{TEXT 402 42 "Calcul ate the interval of convergence of " }{XPPEDIT 256 1 "Sum((-1)^n*(x+3 )^n/(sqrt(n+1)*` `*4^n),n = 0 .. infinity);" "6#-%$SumG6$*(),$\"\"\"! \"\"%\"nGF)),&%\"xGF)\"\"$F)F+F)*(-%%sqrtG6#,&F+F)F)F)F)%\"~GF))\"\"%F +F)F*/F+;\"\"!%)infinityG" }{TEXT 720 215 ". Let R be the radius o f convergence. You will need to calculate the sum of four integers and it might help to record them as you go. \n\nLet c be the base poi nt of the power series. ( c = ________ ) \nSet" }{TEXT 730 2 " \+ " }{XPPEDIT 731 0 "rho;" "6#%$rhoG" }{TEXT 729 51 " = R if R is a n integer and -1 otherwise. ( " }{XPPEDIT 795 0 "rho;" "6#%$rhoG" } {TEXT 794 20 " = ________ )\nSet " }{XPPEDIT 728 0 "sigma;" "6#%&sig maG" }{TEXT 721 86 " = 1 if the left endpoint belongs to the interval of convergence and 0 otherwise. ( " }{XPPEDIT 258 0 "sigma;" "6#%&si gmaG" }{TEXT 796 22 " = ________ ) \nSet " }{XPPEDIT 726 0 "tau;" " 6#%$tauG" }{TEXT 722 1 " " }{TEXT 727 88 "= 3 if the right endpoint \+ belongs to the interval of convergence and 0 otherwise. ( " } {XPPEDIT 258 0 "tau;" "6#%$tauG" }{TEXT 797 38 " = ________ )\n\nWhat is the value of " }{TEXT 724 1 " " }{XPPEDIT 725 0 "c+rho+sigma+tau; " "6#,*%\"cG\"\"\"%$rhoGF%%&sigmaGF%%$tauGF%" }{TEXT 723 3 "?\n " }} {PARA 0 "" 0 "" {TEXT 403 78 "a) -4 b) -3 c) - 2 d) 2 e) 3 " }{TEXT 406 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 404 36 "f) 4 g) 7 \+ " }{TEXT -1 1 " " }{TEXT 405 44 "h) 8 i) 10 \+ j) 11 " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 372 14 "Solution: f " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c := -3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG!\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a := n -> (-1)^n/sqrt(n +1)/4^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)oper atorG%&arrowGF(*()!\"\"9$\"\"\"-%%sqrtG6#,&F/F0F0F0F.)\"\"%F/F.F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn := R = Limit(abs(a( n)/a(n+1)), n = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/ %\"RG-%&LimitG6$-%$absG6#*.)!\"\"%\"nG\"\"\",&F0F1F1F1#F/\"\"#)\"\"%F0 F/)F/F2F/,&F0F1F4F1#F1F4)F6F2F1/F0%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "R := limit(sqrt(n+2)*4^(n+1)/sqrt(n+1)/4^n, n = \+ infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"RG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rho := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The left endpoint is " }{XPPEDIT 18 0 "c-R;" " 6#,&%\"cG\"\"\"%\"RG!\"\"" }{TEXT -1 26 ", or -7. Substituting " } {XPPEDIT 18 0 "x = -7;" "6#/%\"xG,$\"\"(!\"\"" }{TEXT -1 26 " result s in the series " }{XPPEDIT 18 0 "Sum((-1)^n*(-1)^n/sqrt(n+1),n = 0 . . infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF)),$F)F*F+F)-%%sqrtG6# ,&F+F)F)F)F*/F+;\"\"!%)infinityG" }{TEXT -1 6 ", or " }{XPPEDIT 18 0 "Sum(1/sqrt(n+1),n = 0 .. infinity);" "6#-%$SumG6$*&\"\"\"F'-%%sqrtG6# ,&%\"nGF'F'F'!\"\"/F,;\"\"!%)infinityG" }{TEXT -1 14 ", which is a " }}{PARA 0 "" 0 "" {TEXT -1 22 "divergent p-series (" }{XPPEDIT 18 0 "p = 1/2;" "6#/%\"pG*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "`` <= 1;" "6 #1%!G\"\"\"" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sigma := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 23 "The right endpoint is " }{XPPEDIT 18 0 "c+R;" "6#,&%\"cG\"\"\"%\"RGF%" }{TEXT -1 25 ", or 1. Substituting \+ " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 26 " results i n the series " }{XPPEDIT 18 0 "Sum((-1)^n/sqrt(n+1),n = 0 .. infinity );" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"nGF)-%%sqrtG6#,&%\"nGF)F)F)F*/F0;\" \"!%)infinityG" }{TEXT -1 15 ", which is a \n" }}{PARA 0 "" 0 "" {TEXT -1 39 "convergent alternating series because " }{XPPEDIT 18 0 " 1/sqrt(n+1);" "6#*&\"\"\"F$-%%sqrtG6#,&%\"nGF$F$F$!\"\"" }{TEXT -1 18 " decreases to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tau := 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t auG\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Finally," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c+rho+sigma+tau;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 267 4 "17. " }{TEXT 273 6 " Let " } {XPPEDIT 749 0 "T(x);" "6#-%\"TG6#%\"xG" }{TEXT 748 41 " be the degr ee 2 Taylor polynomial of " }{XPPEDIT 751 0 "ln(x);" "6#-%#lnG6#%\"xG " }{TEXT 750 31 " with base point 2. What is " }{XPPEDIT 753 0 "T(3 )-ln(2);" "6#,&-%\"TG6#\"\"$\"\"\"-%#lnG6#\"\"#!\"\"" }{TEXT 752 1 "? " }}{PARA 0 "" 0 "" {TEXT 355 6 "a) " }{XPPEDIT 361 0 "1/8;" "6#*& \"\"\"F$\"\")!\"\"" }{TEXT 360 15 " b) " }{XPPEDIT 257 0 "1 /4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 356 20 " c) " } {XPPEDIT 258 0 "3/8;" "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 357 17 " \+ d) " }{XPPEDIT 256 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 359 19 " e) " }{XPPEDIT 259 0 "5/8;" "6#*&\"\"&\"\"\"\" \")!\"\"" }{TEXT 358 14 " \nf) " }{XPPEDIT 262 0 "3/4;" "6#*& \"\"$\"\"\"\"\"%!\"\"" }{TEXT 456 16 " g) " }{XPPEDIT 257 0 "7/8;" "6#*&\"\"(\"\"\"\"\")!\"\"" }{TEXT 452 19 " h) \+ " }{XPPEDIT 258 0 "1;" "6#\"\"\"" }{TEXT 453 19 " i) \+ " }{XPPEDIT 256 0 "5/4;" "6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT 455 20 " \+ j) " }{XPPEDIT 259 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\" " }{TEXT 454 3 " \n" }{TEXT 362 0 "" }}{PARA 0 "" 0 "" {TEXT 380 11 " Solution: c" }}{PARA 0 "" 0 "" {TEXT 836 2 " " }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c := 2:\nf := ln:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "T := x -> sum((D@@n)(f)(c)/n !*(x-c)^n, n = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%$sumG6$*(---%#@@G6$%\"DG%\"nG6#%\"fG6# %\"cG\"\"\"-%*factorialG6#F6!\"\"),&9$F;F:F?F6F;/F6;\"\"!\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "T(x); #The degree 2 Taylo r polynomial of ln(x) with base point 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%#lnG6#\"\"#\"\"\"*&F'!\"\"%\"xGF(F(F(F**&\"\")F*,&F+F(F'F*F' F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T(3)-ln(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 269 4 "18. " }{TEXT 755 15 "To approxima te " }{XPPEDIT 757 0 "Int((arctan(x)-x)/(x^2),x = 0 .. 1/2);" "6#-%$In tG6$*&,&-%'arctanG6#%\"xG\"\"\"F+!\"\"F,*$F+\"\"#F-/F+;\"\"!*&F,F,F/F- " }{TEXT 756 2 ", " }{TEXT 770 1 " " }{TEXT 771 26 "the Maclaurin seri es of " }{XPPEDIT 259 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT 772 114 " (and, from that, the Maclaurin series of the integrand) is used. An alternating series for the (exact) value " }{XPPEDIT 774 0 "S;" "6#%\"SG" }{TEXT 773 58 " of the definite integral results. A n approximation to " }{XPPEDIT 258 0 "S;" "6#%\"SG" }{TEXT 775 166 " \+ is obtained by using the minimum number of terms that, by the Altern ating Series Test, guarantee an absolute error less than 0.001. What is the approximation? \n " }{TEXT 451 2 " " }{TEXT 354 5 " \na) " } {XPPEDIT 793 0 "-96/2401;" "6#,$*&\"#'*\"\"\"\"%,C!\"\"F(" }{TEXT 792 18 " b) " }{XPPEDIT 791 0 "-209/5376;" "6#,$*&\"$4#\"\" \"\"%w`!\"\"F(" }{TEXT 790 12 " c) " }{XPPEDIT 789 0 "-19/480; " "6#,$*&\"#>\"\"\"\"$![!\"\"F(" }{TEXT 788 14 " d) " } {XPPEDIT 256 0 "-13/336;" "6#,$*&\"#8\"\"\"\"$O$!\"\"F(" }{TEXT 787 12 " e) " }{XPPEDIT 786 0 "-2089/53760;" "6#,$*&\"%*3#\"\"\"\" &gP&!\"\"F(" }{TEXT 785 8 " \nf) " }{XPPEDIT 782 0 "-25069/645120; " "6#,$*&\"&p]#\"\"\"\"'?^k!\"\"F(" }{TEXT 781 13 " g) " } {XPPEDIT 784 0 "-131/3360;" "6#,$*&\"$J\"\"\"\"\"%gL!\"\"F(" }{TEXT 783 12 " h) " }{XPPEDIT 780 0 "-523/13440;" "6#,$*&\"$B&\"\"\" \"&SM\"!\"\"F(" }{TEXT 779 12 " i) " }{XPPEDIT 778 0 "-3/80;" "6#,$*&\"\"$\"\"\"\"#!)!\"\"F(" }{TEXT 777 15 " j) " } {XPPEDIT 776 0 "-37/960;" "6#,$*&\"#P\"\"\"\"$g*!\"\"F(" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 353 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 369 14 "Solution: j \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "series(arctan(t), t=0, 11); #The starting point. \n \+ Subtract t then divide by t^2." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"tG\"\"\"F%#!\"\"\"\"$F(#F%\"\"&F*#F'\"\"(F,#F%\"\" *F.-%\"OG6#F%\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Maclau rin := series((arctan(t)-t)/t^2, t=0, 15); #More terms than needed" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*MaclaurinG+1%\"tG#!\"\"\"\"$\"\"\"# F*\"\"&F)#F(\"\"(F,#F*\"\"*F.#F(\"#6F0#F*\"#8F2-%\"OG6#F*F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p := convert(Maclaurin, poly nom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,.*&\"\"$!\"\"%\"tG\"\" \"F(*&\"\"&F(F)F'F**&\"\"(F(F)F,F(*&\"\"*F(F)F.F**&\"#6F(F)F0F(*&\"#8F (F)F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "eqn := Int((arct an(t)-t)/(t^2),t = 0 .. x) = int(p, t=0 .. x) + `...`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$*&,&-%'arctanG6#%\"tG\"\"\"F.!\" \"F/F.!\"#/F.;\"\"!%\"xG,0*&\"\"'F0F5\"\"#F0*&\"#?F0F5\"\"%F/*&\"#UF0F 5F8F0*&\"#sF0F5\"\")F/*&\"$5\"F0F5\"#5F0*&\"$c\"F0F5\"#7F/%$...GF/" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We need to identify the first summand on the right side of this equation that evaluates to less than 0.001 when " }{XPPEDIT 18 0 "x = 1/2;" "6#/% \"xG*&\"\"\"F&\"\"#!\"\"" }{TEXT 834 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "The term " }{XPPEDIT 18 0 "x^2/6;" "6#*&%\"xG\"\"#\"\"'!\"\"" } {TEXT -1 36 " clearly does not do the job. Next," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(x=1/2, x^4/20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$?$" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 32 "is also not small enough. Next, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(x=1/2, x^6/42);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"%)o#" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 134 "shows that this is the first term that is less than t he acceptable error. We therefore add the terms up to but not includin g this one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Answer := subs(x=1 /2, -1/6*x^2+1/20*x^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG# !#P\"$g*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "As a verification, " }{TEXT 835 5 "MAPLE" }{TEXT -1 27 "'s (exact) integration is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "definiteIntegral := int((arctan(x)-x)/(x^2),x = 0 \+ .. 1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1definiteIntegralG,**&\" \"#\"\"\"-%'arctanG6##F(F'F(!\"\"*&#F(F'F(-%#lnG6#\"\"&F(F--F16#F'F(F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 22 "The absolute error is:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "abs(evalf( Answer-definiteIntegral));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(kKD$ !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 270 33 "19. What is the coefficient of " }{XPPEDIT 258 0 "x^5; " "6#*$%\"xG\"\"&" }{TEXT 449 31 " in the Maclaurin series of " } {XPPEDIT 18 0 "8*x/(4-x^2);" "6#*(\"\")\"\"\"%\"xGF%,&\"\"%F%*$F&\"\"# !\"\"F+" }{TEXT 450 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 438 6 "a) " }{XPPEDIT 18 0 "1/16;" "6#*&\"\"\"F$ \"#;!\"\"" }{TEXT 439 16 " b) " }{XPPEDIT 18 0 "(-1)/16;" "6#*&,$\"\"\"!\"\"F%\"#;F&" }{TEXT 440 19 " c) " } {XPPEDIT 550 1 "1/8;" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT 441 16 " \+ d) " }{XPPEDIT 18 0 "-1/8;" "6#,$*&\"\"\"F%\"\")!\"\"F'" }{TEXT 442 19 " e) " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%! \"\"" }{TEXT 443 3 " \n" }{TEXT 444 4 "f) " }{XPPEDIT 18 0 "-1/4;" " 6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT 448 18 " g) " } {XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 447 17 " \+ h) " }{XPPEDIT 18 0 "-1/2;" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT 446 21 " i) " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT 445 19 " j) " }{XPPEDIT 18 0 "-2;" "6#,$\"\"#!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 14 "Solution : c \n" }{TEXT -1 1 "\n" }{TEXT 830 8 "Method 1" }{TEXT -1 41 ": The \+ geometric series way (as intended):" }}{PARA 0 "" 0 "" {TEXT -1 18 "We write " }{XPPEDIT 18 0 "8*x/(4-x^2) = 2*x;" "6#/*(\"\")\"\" \"%\"xGF&,&\"\"%F&*$F'\"\"#!\"\"F,*&F+F&F'F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/(1-u);" "6#*&\"\"\"F$,&F$F$%\"uG!\"\"F'" }{TEXT -1 17 " where " }{XPPEDIT 18 0 "u = x^2/4;" "6#/%\"uG*&%\"xG\" \"#\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 821 1 "." }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }{XPPEDIT 18 0 "8*x/(4-x^2);" "6#*(\"\")\"\"\"%\"xGF%,&\"\"%F%*$F& \"\"#!\"\"F+" }{TEXT -1 9 " = " }{XPPEDIT 18 0 "2*x;" "6#*&\"\"# \"\"\"%\"xGF%" }{TEXT -1 1 " " }{TEXT 832 1 "(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1+u+u^2+u^3+`...`;" "6#,,\"\"\"F$%\"uGF$*$F%\"\"#F$*$F% \"\"$F$%$...GF$" }{TEXT -1 3 " " }{TEXT 833 1 ")" }{TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " \+ " }{XPPEDIT 18 0 "8*x/(4-x^2) = 2*x*(1+x^2/4+x^4/16+ `...`);" "6#/*(\"\")\"\"\"%\"xGF&,&\"\"%F&*$F'\"\"#!\"\"F,*(F+F&F'F&,* F&F&*&F'F+F)F,F&*&F'F)\"#;F,F&%$...GF&F&" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "2*x+x^3/2+x^5/8+`...`;" "6#,**&\"\"#\"\"\"%\"xGF&F&*&F' \"\"$F%!\"\"F&*&F'\"\"&\"\")F*F&%$...GF&" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "The coefficient o f " }{XPPEDIT 18 0 "x^5;" "6#*$%\"xG\"\"&" }{TEXT -1 7 " is " } {XPPEDIT 18 0 "1/8;" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT 822 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 831 8 "Method 2" }{TEXT -1 55 ": The brute force compu tational way (not recommended!):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> 8*x/(4-x^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$*(\"\")\"\"\"9$F/,&\"\"%F/*$)F0\"\"#F/!\"\"F6F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@5)(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"&!3Y\"\"\",&\"\"%F&*$)%\"xG\"\"#F&!\"\"!\"&F+F(F& *(\"&!G " 0 "" {MPLTEXT 1 0 25 "Answer = (D@@5)(f)(0)/5!;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'AnswerG#\"\"\"\"\")" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 823 5 "Maple" }{TEXT -1 6 " way:\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Maclaurin := series(8*x/(4-x ^2),x=0,8);\np := convert(Maclaurin, polynom);\nAnswer := coeff(p, x^5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*MaclaurinG+-%\"xG\"\"#\"\"\"# F(F'\"\"$#F(\"\")\"\"&#F(\"#K\"\"(-%\"OG6#F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,**&\"\"#\"\"\"%\"xGF(F(*&F'!\"\"F)\"\"$F(*&\"\") F+F)\"\"&F(*&\"#KF+F)\"\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Ans werG#\"\"\"\"\")" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 30 "20. What i s the coefficient of" }{TEXT 754 2 " " }{XPPEDIT 426 0 "x^4;" "6#*$% \"xG\"\"%" }{TEXT 425 32 " in the Maclaurin series of " }{XPPEDIT 428 0 "1/((1+x^2)^(1/3));" "6#*&\"\"\"F$),&F$F$*$%\"xG\"\"#F$*&F$F$\" \"$!\"\"F," }{TEXT 427 3 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 400 0 "" }}{PARA 260 "" 0 "" {TEXT 271 3 "a) " }{XPPEDIT 18 0 "-1/9;" "6#,$*&\"\"\"F%\"\"*!\"\"F'" }{TEXT 429 13 " b) " }{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT 430 18 " c) \+ " }{XPPEDIT 549 1 "-1/6;" "6#,$*&\"\"\"F%\"\"'!\"\"F'" }{TEXT 431 16 " d) " }{XPPEDIT 18 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" } {TEXT 432 16 " e) " }{XPPEDIT 18 0 "-2/9;" "6#,$*&\"\"#\" \"\"\"\"*!\"\"F(" }{TEXT 433 2 " " }}{PARA 260 "" 0 "" {TEXT 401 4 "f ) " }{XPPEDIT 18 0 "2/9;" "6#*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT 437 16 " g) " }{XPPEDIT 18 0 "-1/3;" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT 436 18 " h) " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$ \"\"$!\"\"" }{TEXT 435 15 " i) " }{XPPEDIT 18 0 "-2/3;" "6# ,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT 434 19 " j) " } {XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 371 14 "Solution: f \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 828 8 "Method 1" }{TEXT -1 31 ": The N ewton way (as intended):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "In " }{TEXT 827 5 "MAPLE" }{TEXT -1 15 ", the num ber \"" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 10 " choose \+ " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 46 "\", namely\n \n \+ " }{XPPEDIT 18 0 "alpha*(alpha-1)*`...`*(alph a-n+1)/n!;" "6#*,%&alphaG\"\"\",&F$F%F%!\"\"F%%$...GF%,(F$F%%\"nGF'F%F %F%-%*factorialG6#F*F'" }{TEXT -1 21 ", \n\nis written as " }{TEXT 825 18 "binomial(alpha, n)" }{TEXT 826 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "NewtonFormula := (1+u)^alpha = Sum(binomial(alpha,n)* u^n, n = 0 .. infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.Newton FormulaG/),&\"\"\"F(%\"uGF(%&alphaG-%$SumG6$*&-%)binomialG6$F*%\"nGF() F)F2F(/F2;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "subs(\{u=x^2, alpha = -1/3\}, NewtonFormula); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&\"\"\"F%*$),&F%F%*$)%\"xG\"\"#F%F%#F%\"\"$F%!\"\"- %$SumG6$*&-%)binomialG6$#F/F.%\"nGF%)F)F8F%/F8;\"\"!%)infinityG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The coeff icient of " }{XPPEDIT 18 0 "x^4;" "6#*$%\"xG\"\"%" }{TEXT -1 9 " ( o r " }{XPPEDIT 18 0 "(x^2)^n;" "6#)*$%\"xG\"\"#%\"nG" }{TEXT -1 6 " \+ for " }{XPPEDIT 18 0 "n = 2;" "6#/%\"nG\"\"#" }{TEXT -1 10 " ) is: \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "binomial(-1/3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\" \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 829 8 "Method \+ 2" }{TEXT -1 55 ": The brute force computational way (not recommended! ):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> (1+x^2)^(-1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$), &F-F-*$)9$\"\"#F-F-#F-\"\"$F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D@@4)(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(** \"%![%\"\"\"\"#\")!\"\",&F&F&*$)%\"xG\"\"#F&F&#!#8\"\"$F,\"\"%F&**\"$[ %F&\"\"*F(F)#!#5F0F,F-F(*(\"#;F&F0F(F)#!\"(F0F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "(D@@4)(f)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#;\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(D@@4)(f)( 0)/4!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 824 5 "Maple" }{TEXT -1 5 " way:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Macl aurin := series((1+x^2)^(-1/3), x = 0, 5);\np := convert(Maclaurin, po lynom);\nAnswer := coeff(p, x^4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%*MaclaurinG++%\"xG\"\"\"\"\"!#!\"\"\"\"$\"\"##F,\"\"*\"\"%-%\"OG6#F' \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(\"\"\"F&*&\"\"$!\"\"% \"xG\"\"#F)*(F+F&\"\"*F)F*\"\"%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'AnswerG#\"\"#\"\"*" }}}}}{MARK "8 5 2" 127 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }