{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 259 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 0 262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Da rk Red Emphasis" -1 264 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" 264 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 257 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 257 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 257 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 257 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 264 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 301 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 307 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 310 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 320 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 321 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 325 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 258 335 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 338 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 340 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 345 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 347 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 348 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 350 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 352 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 353 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 355 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 356 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 357 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 358 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 365 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 366 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 370 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 371 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 372 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 373 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 375 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 376 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 380 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 381 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 382 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 383 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 384 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 385 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 386 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 388 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 390 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 393 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 394 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 395 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 396 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 397 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 398 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 399 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 400 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 401 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 402 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 403 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 404 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 405 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 406 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 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 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 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 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Implicit differentiation" }} {PARA 0 "" 0 "" {TEXT -1 14 "by Peter Stone" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 21.10.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "load calc ulus procedures " }}{PARA 0 "" 0 "" {TEXT -1 35 "RMIT file path to rea d Maple m-file" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "read \"J: \\\\Class_Notes/Peter Stone/MapleMath/procdrs/calculus.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Another file path " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "read \"E:\\\\MapleMath/procdrs/calculus.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "The method of implicit diffe rentiation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 86 "Consider the circle of unit radius with its centre at the origin given by the equation" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We can think of this equation as implicitly def ining " }{TEXT 278 1 "y" }{TEXT -1 18 " as a function of " }{TEXT 279 1 "x" }{TEXT -1 160 ", that is, if we restrict our attention to a suit able section of the curve, the equation (i) determines a functional re lationship from the independent variable " }{TEXT 280 1 "x" }{TEXT -1 27 " to the dependent variable " }{TEXT 277 1 "y" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 66 "In this example we can obtain explicit fo rmulas for two functions:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)=sqrt(1-x^2)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"\"F, *$F'\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g(x)=-sqrt(1-x^2)" " 6#/-%\"gG6#%\"xG,$-%%sqrtG6#,&\"\"\"F-*$F'\"\"#!\"\"F0" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 88 "whose graphs are the upper and lower \+ semi-circular arcs of the unit circle respectively." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "plot([sqr t(1-x^2),-sqrt(1-x^2)],x=-1..1,thickness=2,\n color=[red,magenta],scal ing=constrained,\n tickmarks=[4,4],legend=[\"y=f(x)\",\"y=g(x)\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6%7_ o7$$!\"\"\"\"!$F*F*7$$!3-n;HdNvs**!#=$\"3aIk@d'**oP(!#>7$$!3/MLe9r]X** F/$\"3^xD,?$RD/\"F/7$$!3/,](=ng#=**F/$\"3W*\\#GuA(fF\"F/7$$!3%pmm\"HU, \"*)*F/$\"3=cuB4yNs9F/7$$!3()***\\PM@l$)*F/$\"3gMpy'*3z+=F/7$$!3!RLL$e %G?y*F/$\"3iA8`Ih^w?F/7$$!3u****\\(oUIn*F/$\"3nSQYJr=ODF/7$$!3ommm;p0k &*F/$\"3Y%G&e[NT?HF/7$$!3E++vV5Su$*F/$\"3'eGkq$fY\"[$F/7$$!3wKL$3(eMbF/7$$!3\"QLL3i.9 !zF/$\"3^7**H-%f#HhF/7$$!3\"ommT!R=0vF/$\"3^qi^Nc\\3mF/7$$!3u****\\P8# \\4(F/$\"3ifl'faLr/(F/7$$!3+nm;/siqmF/$\"37dnjs%HaF/ $\"3e&R<6BowR)F/7$$!3Q+++]$*4)*\\F/$\"3]Y1e+6Nh')F/7$$!39+++]_&\\c%F/$ \"3S:_+\"3ds*))F/7$$!31+++]1aZTF/$\"3i'GE3]N$*4*F/7$$!3umm;/#)[oPF/$\" 33y-$=zZFE*F/7$$!3hLLL$=exJ$F/$\"3:IF>)f#eL%*F/7$$!3*RLLLtIf$HF/$\"3kU 7=H\\If&*F/7$$!3]++]PYx\"\\#F/$\"3)G16\">%yXo*F/7$$!3EMLLL7i)4#F/$\"3i 8>'>!*4tx*F/7$$!3c****\\P'psm\"F/$\"35Fh>K5.g)*F/7$$!3')****\\74_c7F/$ \"3JpBi\"oV2#**F/7$$!3)3LLL3x%z#)F2$\"3%4o%3*=mc'**F/7$$!3KMLL3s$QM%F2 $\"3#oX(\\%3h0***F/7$$!3]^omm;zr)*!#@$\"2sF&QF^******!#<7$$\"3%pJL$ezw 5VF2$\"3)*Q2K>Vq!***F/7$$\"3s*)***\\PQ#\\\")F2$\"3s^4RW'Rn'**F/7$$\"3G KLLe\"*[H7F/$\"3+'Q!p2+8C**F/7$$\"3I*******pvxl\"F/$\"3]s:1(pJ;')*F/7$ $\"3#z****\\_qn2#F/$\"3;GKN_W(>y*F/7$$\"3U)***\\i&p@[#F/$\"3!QaP4mWqo* F/7$$\"3B)****\\2'HKHF/$\"3+\\tky/Ug&*F/7$$\"3ElmmmZvOLF/$\"3%e0ez0!)o U*F/7$$\"3i******\\2goPF/$\"3wC!*\\-?qi#*F/7$$\"3UKL$eR<*fTF/$\"3CILE@ Ro$4*F/7$$\"3m******\\)Hxe%F/$\"3-\"f%eX^`&)))F/7$$\"3ckm;H!o-*\\F/$\" 35DIn2b'em)F/7$$\"3y)***\\7k.6aF/$\"3a'=*zS\"f&4%)F/7$$\"3#emmmT9C#eF/ $\"3H&p3sIf,8)F/7$$\"33****\\i!*3`iF/$\"3;c[JlKx.yF/7$$\"3%QLLL$*zym'F /$\"3o4V`oXZ_uF/7$$\"3wKLL3N1#4(F/$\"3!Q)Q?)e4+0(F/7$$\"3Nmm;HYt7vF/$ \"3+_Th$[5**f'F/7$$\"3Y*******p(G**yF/$\"3_pyiCh)>8'F/7$$\"3]mmmT6KU$) F/$\"3I@,y!QMT^&F/7$$\"3fKLLLbdQ()F/$\"3e&)*oZS?='[F/7$$\"3amm\"zW?)\\ *)F/$\"3ud/FE;-hWF/7$$\"3[++]i`1h\"*F/$\"3$*pj$>D\\$4SF/7$$\"3Y++++PDj $*F/$\"3v!)RQ\\fL6NF/7$$\"3W++]P?Wl&*F/$\"3:\"G)fhN(e\"HF/7$$\"3K+]7G: 3u'*F/$\"3fG;Bq;AKDF/7$$\"3A++v=5s#y*F/$\"37vO>U4Dt?F/7$$\"3;+D1k2/P)* F/$\"3mLW(eL^zz\"F/7$$\"35+]P40O\"*)*F/$\"31H.f%oH+Z\"F/7$$\"3k]7.#Q?& =**F/$\"3G#>z^*=&RF\"F/7$$\"31+voa-oX**F/$\"3(pV\\8H')3/\"F/7$$\"3[\\P MF,%G(**F/$\"3?&=hJ*p=ltF27$$\"\"\"F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\" )F+F+-%'LEGENDG6#Q'y=f(x)6\"-F$6%7_oF'7$F-$!3aIk@d'**oP(F27$F4$!3^xD,? $RD/\"F/7$F9$!3W*\\#GuA(fF\"F/7$F>$!3=cuB4yNs9F/7$FC$!3gMpy'*3z+=F/7$F H$!3iA8`Ih^w?F/7$FM$!3nSQYJr=ODF/7$FR$!3Y%G&e[NT?HF/7$FW$!3'eGkq$fY\"[ $F/7$Ffn$!3u)R\"3d;![&RF/7$F[o$!36'pC*=pM(eMbF/7$Fjo$!3^7**H-%f#HhF/7$F_p$!3^qi^Nc\\3mF/7$Fdp$!3ifl'fa Lr/(F/7$Fip$!37dnj)f#eL%*F/7$Ffs$!3kU7=H\\If& *F/7$F[t$!3)G16\">%yXo*F/7$F`t$!3i8>'>!*4tx*F/7$Fet$!35Fh>K5.g)*F/7$Fj t$!3JpBi\"oV2#**F/7$F_u$!3%4o%3*=mc'**F/7$Fdu$!3#oX(\\%3h0***F/7$Fiu$! 2sF&QF^******F^v7$F`v$!3)*Q2K>Vq!***F/7$Fev$!3s^4RW'Rn'**F/7$Fjv$!3+'Q !p2+8C**F/7$F_w$!3]s:1(pJ;')*F/7$Fdw$!3;GKN_W(>y*F/7$Fiw$!3!QaP4mWqo*F /7$F^x$!3+\\tky/Ug&*F/7$Fcx$!3%e0ez0!)oU*F/7$Fhx$!3wC!*\\-?qi#*F/7$F]y $!3CILE@Ro$4*F/7$Fby$!3-\"f%eX^`&)))F/7$Fgy$!35DIn2b'em)F/7$F\\z$!3a'= *zS\"f&4%)F/7$Faz$!3H&p3sIf,8)F/7$Ffz$!3;c[JlKx.yF/7$F[[l$!3o4V`oXZ_uF /7$F`[l$!3!Q)Q?)e4+0(F/7$Fe[l$!3+_Th$[5**f'F/7$Fj[l$!3_pyiCh)>8'F/7$F_ \\l$!3I@,y!QMT^&F/7$Fd\\l$!3e&)*oZS?='[F/7$Fi\\l$!3ud/FE;-hWF/7$F^]l$! 3$*pj$>D\\$4SF/7$Fc]l$!3v!)RQ\\fL6NF/7$Fh]l$!3:\"G)fhN(e\"HF/7$F]^l$!3 fG;Bq;AKDF/7$Fb^l$!37vO>U4Dt?F/7$Fg^l$!3mLW(eL^zz\"F/7$F\\_l$!31H.f%oH +Z\"F/7$Fa_l$!3G#>z^*=&RF\"F/7$Ff_l$!3(pV\\8H')3/\"F/7$F[`l$!3?&=hJ*p= ltF2F_`l-Fc`l6&Fe`lFf`lF+Ff`l-Fj`l6#Q'y=g(x)F]al-%(SCALINGG6#%,CONSTRA INEDG-%+AXESLABELSG6$Q\"xF]alQ!6\"-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$ \"\"%Fj^m-%%VIEWG6$;F(F``l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 1 "y=f(x)" "y=g(x)" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 30 " are differentiable functions." }}{PARA 0 "" 0 "" {TEXT -1 3 "If \+ " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 14 " is either of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 88 ", or, indeed, any differentiable function whose graph is an arc of the unit circle, the n" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+h(x)^2=1" " 6#/,&*$%\"xG\"\"#\"\"\"*$-%\"hG6#F&F'F(F(" }{TEXT -1 15 " ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 260 34 "method of implicit differentiation" }{TEXT -1 100 " i nvolves differentiating both sides of the given equation with respect \+ to the independent variable " }{TEXT 281 1 "x" }{TEXT -1 76 " to const ruct an equation involving the derivative of the implicit function." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "In our i ntroductory example, differentiating (ii) and applying the chain rule \+ gives" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x+2*h(x)*` h '`(x) = 0;" "6#/,&*&\"\"#\"\"\"%\"xGF'F'*(F&F'-%\"hG6#F(F'-%$h~'G6#F (F'F'\"\"!" }{TEXT -1 15 " ------- (iii)" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 161 "The usual way of doing this i s to omit any mention of any underlying function or functions, and use the Leibniz notation in connection with the original equation" }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$% \"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 13 " ------- (i) " }}{PARA 0 " " 0 "" {TEXT -1 8 "to write" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*x+2*y" "6#,&*&\"\"#\"\"\"%\"xGF&F&*&F%F&%\"yGF&F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\" \"!" }{TEXT -1 15 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 9 "The \+ term " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 41 " is first d ifferentiated with respect to " }{TEXT 282 1 "y" }{TEXT -1 55 ", and t hen the derivative of this term with respect to " }{TEXT 283 1 "x" } {TEXT -1 31 " is obtained by multiplying by " }{XPPEDIT 18 0 "dy/dx" " 6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 29 " according to the chain rule. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Thus, if we temporarily write " } {XPPEDIT 18 0 "u = y^2;" "6#/%\"uG*$%\"yG\"\"#" }{TEXT -1 8 ", then \+ " }{XPPEDIT 18 0 "du/dx=du/dy" "6#/*&%#duG\"\"\"%#dxG!\"\"*&F%F&%#dyGF (" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*y" "6#/*&%#dyG\"\"\"%#dxG !\"\"*&\"\"#F&%\"yGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Equ ation (iv) can now be solved for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\" \"\"%#dxG!\"\"" }{TEXT -1 8 " to give" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=-x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF &%\"yGF(F(" }{TEXT -1 14 " ------- (v) " }}{PARA 257 "" 0 "" {TEXT 259 9 "_________" }{TEXT -1 1 " " }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "This is formula for the \+ gradient of a tangent line to the unit circle at a point on the circle with coordinates" }{XPPEDIT 18 0 " ``(x,y)" "6#-%!G6$%\"xG%\"yG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, the point" }{XPPEDIT 18 0 " ``(3/5,4/5)" "6#-%!G6$*&\"\"$\"\"\"\"\"&!\"\"*&\"\"%F (F)F*" }{TEXT -1 80 " lies on the unit circle, and the gradient of the tangent line at this point is " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "eval(dy/dx,x = 3/5);" "6#-%%evalG6$*&%#dyG\"\"\"%#dxG! \"\"/%\"xG*&\"\"$F(\"\"&F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``[y=4/5] =-``(3/5)/``(4/5)" "6#/&%!G6#/%\"yG*&\"\"%\"\"\"\"\"&!\"\",$*&-F%6#* &\"\"$F+F,F-F+-F%6#*&F*F+F,F-F-F-" }{XPPEDIT 18 0 "``=-3/4" "6#/%!G,$* &\"\"$\"\"\"\"\"%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 296 271 271 {PLOTDATA 2 "6,-%'CURVESG6&7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$ \"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#* F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$ \"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$ \"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF /$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F /$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB [i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3J x$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[Rg F/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\" 3u3HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V ?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4 *>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$ F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7 $$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@ UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30 yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3Ezy OHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRq X8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0 I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$ *4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F /$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)! #F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-%*LINESTYLE G6#F)-F$6&7$7$F+F+7$$\"3w**************fF/$\"3U+++++++!)F/-Fjz6&F\\[lF +F+F][l-Fa[lFf[l-Fe[lFb[l-F$6&7$7$$\"3++++++++:Fdo$\"3+++++++]7F/7$$!3 5+++++++?F/$\"3!**************R\"Fdo-Fjz6&F\\[lF+F][lF+F`[lFd[l-%%TEXT G6$7$$\"\"*!\"\"Fg]lQ-P(~3/5,4/5~)6\"-Fd]l6$7$$\"$Y\"!\"#$Fi]lFi]lQ\"x F[^l-Fd]l6$7$$!#:Fa^l$\"#7Fi]lQ\"yF[^l-Fd]l6$7$$\"\")Fi]l$\"#9Fi]lQ;gr adient~of~tangent~=~-3/4F[^l-%*AXESTICKSG6$\"\"$Fg_l-%+AXESLABELSG6$Q! F[^lF[`l-%%VIEWG6$%(DEFAULTGF_`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT 257 42 "Some comments on the gradient formula (v)." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Rel ation to radius line" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 33 "Notice that, for a general point " } {XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$%\"xG%\"yG" }{TEXT -1 75 " on the un it circle, the gradient of the radius line joining the origin to " } {TEXT 284 1 "P" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "y/x" "6#*&%\"yG\"\" \"%\"xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Now recall the standard formula relating the g radients of perpendicular lines, namely that two lines are perpendicul ar exactly when the product of their gradients is" }{XPPEDIT 18 0 " `` -1" "6#,&%!G\"\"\"F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 85 "Applying this result here shows that the tangent line is perpendic ular to the radius." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 322 "Alternatively, if you want to start from the geometric al fact, proven by traditional Euclidean geometry techniques for examp le, that the tangent line to a circle at any point is perpendicular to the radius line at that point, then you can obtain the formula (v) fo r the gradient of the tangent line without using calculus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Vertical tangent lines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 142 "The formula (v) for the gradient of a ta ngent line to the unit circle applies at all points on the unit circle except at the two points where " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\" !" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the equation " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 21 " . . . . . . . . (i) " }}{PARA 0 "" 0 "" {TEXT -1 13 "al so defines " }{TEXT 286 1 "x" }{TEXT -1 18 " as a function of " } {TEXT 287 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "We can \+ differentiate equation (i) with respect to " }{TEXT 285 1 "y" }{TEXT -1 8 " to give" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2* x" "6#*&\"\"#\"\"\"%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dy+2*y \+ = 0;" "6#/,&*&%#dxG\"\"\"%#dyG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dy = -y/x" "6#/*&%#dxG\"\"\"%#dyG!\" \",$*&%\"yGF&%\"xGF(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "If we plotted the graph of equation (i) with the " }{TEXT 290 1 "y" } {TEXT -1 25 " axis horizontal and the " }{TEXT 289 1 "x" }{TEXT -1 83 " axis vertical, this last formula shows that the tangent lines at the points where " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 90 " ha ve gradient 0 in this new configuration. Thus, in the original configu ration, with the " }{TEXT 295 1 "x" }{TEXT -1 25 " axis horizontal and the " }{TEXT 288 1 "y" }{TEXT -1 54 " axis vertical, the tangent line s at the points where " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 14 " are vertical." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 294 276 276 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4 hRPij!**!#=$\"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\" 3-N5')yke[#*F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3S b](G._U!zF/$\"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3? H$>jubk6'F/$\"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\" 37C3nW'R,#QF/$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$ \"3OB^hAo8X8F/$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#< 7$$!33)\\T#fB[i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7 $$!3%*yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$ $!3[igNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&= e_b?/U!zF/$\"3u3HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT> +.2**F/$\"31V?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=! Q%3**F/$!3q#4*>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/ $!3q4&=j`Bsw$F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!3 7W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2ye GL'zF/7$$!3g@UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%* **4E*F/7$$!30yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C ;**F/7$$!3EzyOHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;* *F/7$$\"3EKRqX8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F /7$$\"3O7w4w0I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7 $$\"33/wY'Q+$*4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$ \"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F( $\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\" \"#-F$6%7$7$F($!3%**************>\"Fdo7$F($\"3%**************>\"Fdo-Fj z6&F\\[lF+F][lF+F`[l-F$6%7$7$$!\"\"F*Fh[l7$Fc\\lF[\\lF]\\lF`[l-%%TEXTG 6$7$$\"#7Fd\\l$Fd\\lFd\\lQ\"x6\"-Fg\\l6$7$F\\]lFj\\lQ\"yF^]l-%*AXESTIC KSG6$\"\"$Ff]l-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q!F^]lF^^l-%% VIEWG6$;$!#6Fd\\lFj\\lFb^l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }} {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Co mparison with derivatives of explicit functions" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Differentiat ing " }{XPPEDIT 18 0 "y = sqrt(1-x^2);" "6#/%\"yG-%%sqrtG6#,&\"\"\"F) *$%\"xG\"\"#!\"\"" }{TEXT -1 17 " with respect to " }{TEXT 291 1 "x" } {TEXT -1 25 " by the chain rule gives:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = 1/(2*sqrt(1-x^2));" "6#/*&%#dyG\"\"\"%# dxG!\"\"*&F&F&*&\"\"#F&-%%sqrtG6#,&F&F&*$%\"xGF+F(F&F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff([1-x^2],x);" "6#-%%DiffG6$7#,&\"\"\"F(*$%\"xG\" \"#!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/(2*sqrt(1-x^2));" "6#/%!G*&\"\"\"F&*&\"\"#F&-%%s qrtG6#,&F&F&*$%\"xGF(!\"\"F&F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`.`*`` (-2*x)" "6#*&%\".G\"\"\"-%!G6#,$*&\"\"#F%%\"xGF%!\"\"F%" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x/sqrt(1- x^2);" "6#/%!G,$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F/F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Substituting " }{TEXT 292 1 "y" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sqrt(1-x^2)" "6#-%%sqrtG 6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 74 " gives the formula (v) ob tained by the method of implicit differentiation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Similarly, differentiatin g " }{XPPEDIT 18 0 "y = -sqrt(1-x^2);" "6#/%\"yG,$-%%sqrtG6#,&\"\"\"F **$%\"xG\"\"#!\"\"F." }{TEXT -1 17 " with respect to " }{TEXT 293 1 "x " }{TEXT -1 26 " by the chain rule gives: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = -1/(2*sqrt(1-x^2));" "6#/*&%#dyG\"\" \"%#dxG!\"\",$*&F&F&*&\"\"#F&-%%sqrtG6#,&F&F&*$%\"xGF,F(F&F(F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([1-x^2],x);" "6#-%%DiffG6$7#,&\"\" \"F(*$%\"xG\"\"#!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(2*sqrt(1-x^2));" "6#/%!G,$*&\"\"\"F'*& \"\"#F'-%%sqrtG6#,&F'F'*$%\"xGF)!\"\"F'F0F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`.`*``(-2*x)" "6#*&%\".G\"\"\"-%!G6#,$*&\"\"#F%%\"xGF%! \"\"F%" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/sqrt(1-x^2);" "6#/%!G*&%\"xG\"\"\"-%%sqrtG6#,&F'F'*$F&\" \"#!\"\"F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Substituti ng " }{TEXT 294 1 "y" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-sqrt(1-x^2 );" "6#,$-%%sqrtG6#,&\"\"\"F(*$%\"xG\"\"#!\"\"F," }{TEXT -1 80 " again gives the formula (v) obtained by the method of implicit differentiat ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Implicit d ifferentiation in action .. examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 321 8 "Question" }{TEXT 316 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "y^2-2*x*y = 1;" "6#/,&*$%\"y G\"\"#\"\"\"*(F'F(%\"xGF(F&F(!\"\"F(" }{TEXT -1 25 ", find an expressi on for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 318 1 "x" }{TEXT -1 5 " and " }{TEXT 319 1 " y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the gradien t of the tangent line to the graph of the equation given in (a) at the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Sol ution" }{TEXT 317 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Differenti ating both sides of the given equation with respect to " }{TEXT 320 1 "x" }{TEXT -1 52 " (making use of the product and chain rules) gives: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\" \"\"%\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-2*y-2*x" "6#,(*&%#d yG\"\"\"%#dxG!\"\"F&*&\"\"#F&%\"yGF&F(*&F*F&%\"xGF&F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Here the term " }{XPPEDIT 18 0 "-2*x*y" "6#,$*(\"\"#\" \"\"%\"xGF&%\"yGF&!\"\"" }{TEXT -1 42 " has been differentiated with \+ respect to " }{TEXT 329 1 "x" }{TEXT -1 16 " as the product " } {XPPEDIT 18 0 "(-2*x)*`.`*y" "6#*(,$*&\"\"#\"\"\"%\"xGF'!\"\"F'%\".GF' %\"yGF'" }{TEXT -1 10 ". to give:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-2)*`.`*y+(-2*x)*`.`" "6#,&*(,$\"\"#!\"\"\"\"\"%\".GF (%\"yGF(F(*&,$*&F&F(%\"xGF(F'F(F)F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -2*y-2*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\"\"#F&%\"yGF&F(*&F +F&%\"xGF&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"% #dxG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "The same re sult can be obtained by differentiating the product " }{XPPEDIT 18 0 " x*`.`*y" "6#*(%\"xG\"\"\"%\".GF%%\"yGF%" }{TEXT -1 10 " to give: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1*`.`*y+x*`.`" "6#,&* (\"\"\"F%%\".GF%%\"yGF%F%*&%\"xGF%F&F%F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=y+x" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&%\"xGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and then multiplying this result by" }{XPPEDIT 18 0 " ``-2" "6#,&%!G\"\"\"\"\"#!\"\"" }{TEXT -1 10 " t o give: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-2*y-2*x " "6#,&*&\"\"#\"\"\"%\"yGF&!\"\"*&F%F&%\"xGF&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 31 "Collecting the terms involving " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 114 " on t he left hand side of the equation (i), and the remaining terms on the \+ right hand side of the equation gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\"\"\"%\"yGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx-2*x" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"#F&%\"xG F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=2*y" "6#/*&%#dyG\"\"\"%#dx G!\"\"*&\"\"#F&%\"yGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "Dividing both sides of the equation by 2 gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 322 1 "y" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx-x" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&%\"xGF(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=y" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out " }{XPPEDIT 18 0 "dy /dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on the left side gives : " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(y-x) = y;" "6#/ -%!G6#,&%\"yG\"\"\"%\"xG!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Dividing both sides by " }{XPPEDIT 18 0 "y-x;" "6#,&%\"y G\"\"\"%\"xG!\"\"" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y/(y-x);" "6#/*&%#dyG\"\"\"%#dxG!\"\" *&%\"yGF&,&F*F&%\"xGF(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(b) The point" }{XPPEDIT 18 0 "``( 0,1);" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 72 " lies on the curve, and the gradient of the tangent at this point is 1. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 364 364 364 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$ !\"$\"\"!$\"3B&z$o,mxA;!#=7$$!3ULLL8#)\\gG!#<$\"3;nO\\hbd(p\"F-7$$!3[m mma%=\"RFF1$\"3+?S%[*[KoF-7$$!3Ymmme\"\\%GBF1$\"3OyW?@M`c?F-7$$!3cLLL\\)e; ?#F1$\"3;I%*ROggk@F-7$$!3G+++G[Pq?F1$\"3UT[8Fs`)G#F-7$$!3RLLL02gM>F1$ \"3+qV<.xoJCF-7$$!3-+++7?E*z\"F1$\"3[Pg<)H*=#f#F-7$$!3]mmmI&\\+m\"F1$ \"3Y9j4S%*HzFF-7$$!3FLLL,6VP:F1$\"3+yX`wi1mHF-7$$!3%)*****>z\"R*R\"F1$ \"3R`:J&y$y0KF-7$$!39+++!o&yg7F1$\"31ZfbfoJ%[$F-7$$!3-+++3I@F6F1$\"3y! e4Ta5kz$F-7$$!3OLLLDi\"f+\"F1$\"3\"*p/=$4p[7%F-7$$!3Womm'=Eoh)F-$\"3S! oRu)4a$e%F-7$$!3?lmmY$y\\R(F-$\"3y`u4XCHU]F-7$$!3q+++S)yO(fF-$\"3%3QA4 p/Zn&F-7$$!3eqmmYze:ZF-$\"3cSdENf[SjF-7$$!3^******RGENLF-$\"3q(f0k,ti? (F-7$$!3d******>p'3-#F-$\"3GUN;m\\G\"=)F-7$$!3#*\\mmmmK%\\'!#>$\"3c_g/ %GL;P*F-7$$\"3AILLL4s*4'Fer$\"3G,[!>\"e&G1\"F17$$\"3'pmmmE5%o>F-$\"3lD g26,.;7F17$$\"3)\\mmmuX%zLF-$\"3A&\\*e&R/NR\"F17$$\"3P******ziv2YF-$\" 3+lvBDq#=c\"F17$$\"3%ommmIlV$fF-$\"3MWJy_OEcF17$$\"3q)*****zcmX')F-$\"3iU9J^w['=#F17$$\"3j(******fUH% **F-$\"3())=Hk-zWS#F17$$\"3z*****RuM$Q6F1$\"3=mJaxk_`EF17$$\"3aKLLD:wn 7F1$\"3]r\\ZG,X#)GF17$$\"3T+++SA&fS\"F1$\"3b7n1*Hk78$F17$$\"3PmmmmNRMDrIHh$F17$$\"3mKLLpd)oz\" F1$\"3mj2HE+H`QF17$$\"3K+++_;`J>F1$\"3$*G`**>_d1TF17$$\"3CLLL8EXT4E$F1$\"3yb'4r?oY F17$FN$!39&[8(yLgpVF17$FS$!3c.T)4=qB6%F17$FX$!3c.w\"Q&HudQF17$Fgn$!3mk HM0!H!)f$F17$F\\o$!3MC,KI)o9P$F17$Fao$!3$[:JD'>O>JF17$Ffo$!3'\\fbf/.+( GF17$F[p$!3*y&4Tqq1MEF17$F`p$!3r8Z)*f$>VU#F17$Fep$!3s,t2O$><=#F17$Fjp$ !3UyI%Q\"\\A$)>F17$F_q$!3AQA4Pi?irm!>)f$>$F-7$F` v$!3s%\\_[*z@wHF-7$Fev$!3m^V#F-7$Fdw$!3W=!G-NFdH#F-7$Fiw$!3Iv;(eE1_;#F-7$F^x$!3s%og*\\JE_?F -7$Fcx$!3=IM_^%3z%>F-7$Fhx$!3qZ9UpT1a=F-7$F]y$!3S#o0FwJ_x\"F-7$Fby$!3i (4ZLQBDp\"F-7$Fgy$!3_tEDoZlC;F-7$F\\z$!3yq^+>g!zb\"F-7$Faz$!35jl4JI&)) \\\"F-7$Ffz$!3`+(QLH!4S9F-Fjz-F$6%7S7$$!3++++++++DF1$!3++++++++:F17$$! 3#****\\i!G\">S#F1$!3#****\\i!G\">S\"F17$$!34+vVjwc;BF1$!34+vVjwc;8F17 $$!3')**\\PC\")e?AF1$!3')**\\PC\")e?7F17$$!3A+]iqB(R7#F1$!3A+]iqB(R7\" F17$$!30+vo9e\"y-#F1$!30+vo9e\"y-\"F17$$!3++vVyjmQ>F1$!3/+]P%yjmQ*F-7$ $!31+v$4Idj%=F1$!3c+]P4Idj%)F-7$$!3++v$47\"*3v\"F1$!3!***\\P47\"*3vF-7 $$!33+v=-6tb;F1$!3#3+v=-6tb'F-7$$!3))***\\iKZyb\"F1$!3)))***\\iKZybF-7 $$!3/+]7V7jr9F1$!3W++DJCJ;ZF-7$$!3')***\\PNsXP\"F1$!3k)***\\PNsXPF-7$$ !33++DJ\\6x7F1$!3%3++DJ\\6x#F-7$$!3')***\\ik'>$=\"F1$!3t)***\\ik'>$=F- 7$$!31+v$f%)4z4\"F1$!3I2+v$f%)4z*Fer7$$!3!)****\\7f&\\'**F-$\"3&4-++v3 W]$!#?7$$!3!)******\\T%e5*F-$\"3(>+++]e:%*)Fer7$$!3H+]PMH\\1\")F-$\"3q **\\ilq]$*=F-7$$!3I,++vx*=A(F-$\"3q)****\\A-\"yFF-7$$!3#z*\\P%oc8D'F-$ \"33-]i:Lk[PF-7$$!3v(*\\7`?$[n *F-$\"3%)*\\7G>$[n>F17$$\"3/++vVK/g5F1$\"3/++vVK/g?F17$$\"3!)*\\i!R]%p :\"F1$\"3!)*\\i!R]%p:#F17$$\"3]+++&)HF]7F1$\"3]+++&)HF]AF17$$\"3/+]P*G 9dM\"F1$\"3/+]P*G9dM#F17$$\"3E+Dc\"Hl.W\"F1$\"3E+Dc\"Hl.W#F17$$\"3x*** *\\K(Rt_\"F1$\"3x****\\K(Rt_#F17$$\"3p**\\(oDAqi\"F1$\"3p**\\(oDAqi#F1 7$$\"3W+++&\\zhr\"F1$\"3W+++&\\zhr#F17$$\"3m*\\ilqR7\"=F1$\"3m*\\ilqR7 \"GF17$$\"3))*\\P%eWA->F1$\"3))*\\P%eWA-HF17$$\"\"#F*$\"\"$F*-F[[l6&F] [lFa[lF^[lFa[l-%*THICKNESSG6#F^dm-%%TEXTG6$7$$\"#@!\"\"$\"#:F\\emQ8gra dient~of~tangent~=~16\"-Fgdm6$7$$\"\"%F\\em$\"\"\"F*Q&(0,1)F`em-Fgdm6$ 7$$\"#MF\\em$!\"#F\\emQ\"xF`em-Fgdm6$7$F^fmF\\fmQ\"yF`em-%+AXESLABELSG 6%%!GFhfm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F\\fmF`gm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can solve the equation " }{XPPEDIT 18 0 "y^2-2*x*y = 1" "6# /,&*$%\"yG\"\"#\"\"\"*(F'F(%\"xGF(F&F(!\"\"F(" }{TEXT -1 5 " for " } {TEXT 328 1 "y" }{TEXT -1 35 " to obtain an explicit formula for " } {TEXT 323 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 324 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 65 "Completing the square on the \+ left side of the equation by adding " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG \"\"#" }{TEXT -1 18 " (and also adding " }{XPPEDIT 18 0 "x^2" "6#*$%\" xG\"\"#" }{TEXT -1 27 " to the right side) gives: " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2-2*x*y+x^2=1+x^2" "6#/,(*$%\"yG\" \"#\"\"\"*(F'F(%\"xGF(F&F(!\"\"*$F*F'F(,&F(F(*$F*F'F(" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-x)^2=1+x^2" "6#/*$,&%\"yG\"\"\"%\"xG!\"\" \"\"#,&F'F'*$F(F*F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-x=`` " "6#/,& %\"yG\"\"\"%\"xG!\"\"%!G" }{TEXT 325 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+x^2)" "6#-%%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 1 " " } {TEXT 326 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+x^2);" "6#-%%sq rtG6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Taking " }{XPPEDIT 18 0 "y=x+sqrt(1+x^2)" "6#/%\"yG,&%\"x G\"\"\"-%%sqrtG6#,&F'F'*$F&\"\"#F'F'" }{TEXT -1 10 ", we have " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1+1/(2*sqrt(1+x ^2))" "6#/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*&F&F&*&\"\"#F&-%%sqrtG6#,&F&F& *$%\"xGF,F&F&F(F&" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Diff([1+x^2],x)" " 6#-%%DiffG6$7#,&\"\"\"F(*$%\"xG\"\"#F(F*" }{TEXT -1 1 " " }{TEXT 327 0 "" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1+1/(2*sqrt(1+x^2))" "6#/%!G,&\"\"\"F&*&F&F&*&\"\"#F&-%%sqrtG6#,&F &F&*$%\"xGF)F&F&!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`.`*2*x;" "6# *(%\".G\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+x/sqrt(1+x^2)" "6#/%!G,&\"\"\"F& *&%\"xGF&-%%sqrtG6#,&F&F&*$F(\"\"#F&!\"\"F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(sqrt(1+x^2)+x)/sqrt(1+x^2)" "6#/%!G*&,&-%%sqrtG6#,&\"\"\"F+* $%\"xG\"\"#F+F+F-F+F+-F(6#,&F+F+*$F-F.F+!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "On the ot her hand, when " }{XPPEDIT 18 0 "y= (x+sqrt(1+x^2))" "6#/%\"yG,&%\"xG \"\"\"-%%sqrtG6#,&F'F'*$F&\"\"#F'F'" }{TEXT -1 26 ", the last expressi on for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " is equal to " }{XPPEDIT 18 0 "y/(y-x)" "6#*&%\"yG\"\"\",&F$F%%\"x G!\"\"F(" }{TEXT -1 46 " as was obtained by implicit differentiation. \+ " }}{PARA 0 "" 0 "" {TEXT -1 41 "One can check in a similar way that w hen " }{XPPEDIT 18 0 "y=x-sqrt(1+x^2)" "6#/%\"yG,&%\"xG\"\"\"-%%sqrtG6 #,&F'F'*$F&\"\"#F'!\"\"" }{TEXT -1 32 ", direct differentiation gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (sqrt(1+ x^2)-x)/sqrt(1+x^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&-%%sqrtG6#,&F&F&* $%\"xG\"\"#F&F&F0F(F&-F,6#,&F&F&*$F0F1F&F(" }{XPPEDIT 18 0 "``=(x-sqrt (1+x^2))/(-sqrt(1+x^2))" "6#/%!G*&,&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\" \"#F(!\"\"F(,$-F*6#,&F(F(*$F'F.F(F/F/" }{XPPEDIT 18 0 "``=y/(y-x)" "6# /%!G*&%\"yG\"\"\",&F&F'%\"xG!\"\"F*" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 393 8 "Question" }{TEXT 386 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "x^3*y+x*y^3 = 2;" "6#/,&*&%\"xG\"\"$%\"yG\"\"\"F)*&F&F)*$F(F'F)F)\"\"#" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\" %#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 388 1 "x" }{TEXT -1 5 " and " }{TEXT 389 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the gradient of the tangent line to the graph of the equa tion given in (a) at the point" }{XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\" \"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 " " 0 "" {TEXT 264 8 "Solution" }{TEXT 387 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Differentiating both sides of the given equation with respect to " }{TEXT 390 1 "x" }{TEXT -1 52 " (making use of the produ ct and chain rules) gives: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "3*x^2*y+x^3" "6#,&*(\"\"$\"\"\"*$%\"xG\"\"#F&%\"yGF&F&* $F(F%F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y^3+3*x*y^2" "6#,(*&%#d yG\"\"\"%#dxG!\"\"F&*$%\"yG\"\"$F&*(F+F&%\"xGF&F*\"\"#F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Collecting the terms i nvolving " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 110 " on the left hand side of the equation, and the remaining term s on the right hand side of the equation gives: " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx + 3*x*y^2" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*(\"\"$ F&%\"xGF&%\"yG\"\"#F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -3*x^2* y-y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*(\"\"$F&*$%\"xG\"\"#F&%\"yGF&F(*$ F/F+F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out \+ " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on the left side gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(x^3+3*x*y^2) = -3*x^2*y-y^3;" "6#/-%!G6#,&*$%\"xG\"\"$\"\"\"*(F* F+F)F+%\"yG\"\"#F+,&*(F*F+*$F)F.F+F-F+!\"\"*$F-F*F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Dividing both sides by " }{XPPEDIT 18 0 "x^3+3*x*y^2" "6#,&*$%\"xG\"\"$\"\"\"*(F&F'F%F'%\"yG\"\"#F'" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx=-(3*x^2*y+y^3)/(x^3+3*x*y^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*(\" \"$F&*$%\"xG\"\"#F&%\"yGF&F&*$F1F-F&F&,&*$F/F-F&*(F-F&F/F&F1F0F&F(F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = -y*(3*x^2+y^2)/(x*(x^2+3*y^2));" "6#/%!G,$*(%\"yG\"\"\",&*&\"\"$F(*$ %\"xG\"\"#F(F(*$F'F.F(F(*&F-F(,&*$F-F.F(*&F+F(*$F'F.F(F(F(!\"\"F5" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(b) The point" }{XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F& " }{TEXT -1 68 " lies on the curve, and the gradient of the tangent at this point is" }{XPPEDIT 18 0 " ``-1" "6#,&%!G\"\"\"F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 358 310 310 {PLOTDATA 2 "6+-%'CURVESG6%7hn7$$!\"$\"\" !$!3:;XmB'**GS(!#>7$$!3pQmf)49Y$H!#<$!3l0^191\"z!zF-7$$!3u)3-a]wRW)o*)F-7$$!3M;?!GBO$\\FF1$!3P C_^$pZ?h*F-7$$!31%H:QJP_o#F1$!3^+&[cbM9.\"!#=7$$!3$f$=o5%4ei#F1$!3a*zA /-XF5\"FH7$$!3qHc=f_FkDF1$!3S)\\\"HZ&=O=\"FH7$$!3dRFo^lj+DF1$!3!RlwTo- dF\"FH7$$!3Y->KM>?PCF1$!3;yT&QyP#F1$!3q+SQ^)zF\\\" FH7$$!38t%pD8zWJ#F1$!3;Mkp'*)3ag\"FH7$$!3YUf64(y(\\AF1$!3+>)p7*\\$eu\" FH7$$!3mAu]>E\"[=#F1$!3O$e+,*eG.>FH7$$!3/WfCad?A@F1$!3CRZc:wts?FH7$$!3 MSOuAFNl?F1$!3?K?!37NOC#FH7$$!3mjWMp*[x*>F1$!3[4-'3fr1Z#FH7$$!3)*GkhA( z/%>F1$!3)ySCtuFdo#FH7$$!3][r5u>'Q(=F1$!3+Zk*)QgMlHFH7$$!3BaLtyR*[\"=F 1$!3!yiL%f5:UKFH7$$!3O,Q)=u'>]FsHf\"4Vi\"F1$!3?=%\\Nb0SN%FH7$$!3aG5m\"ez_c\"F1 $!33X!*QW#*erZFH7$$!3kX9c2#3;]\"F1$!3m=)\\My%3h_FH7$$!3EocQW:ZN9F1$!3c :[um:%)4eFH7$$!35-0n%Q**yP\"F1$!3+p\\fJ#)=dJ\"F1$!3q )47!)z83*oFH7$$!3/JM6dB[^7F1$!3ntY>6_$[](FH7$$!3k^+Ey%Q')=\"F1$!3SX1)f V,*>\")FH7$$!3Hht4%zLy7\"F1$!3Q'[bXO%*Hs)FH7$$!3QY&\\#f0Kg5F1$!3o$>0') egoR*FH7$$!3+!fj'\\\"pl***FH$!3mZO.&3V.+\"F17$$!3cuW)HEb)[$*FH$!3]:oZk M7l5F17$$!3uL4W7-$>w)FH$!3CjpBoY#R7\"F17$$!3A:d(\\5m-7)FH$!3mN>c@:g)= \"F17$$!34mT!f=5l^(FH$!3YlW\"z%zF]7F17$$!35wQ6'G6a)oFH$!3_7I^aUH;8F17$ $!3g6B>dnRoiFH$!3WYEW$R;LQ\"F17$$!3tM=MF(RCi&FH$!335uR_#QvX\"F17$$!3(* 4%)[`\"F17$$!3+n%\\@3&3kVFH$!3!>](*p%G#Gi\"F17$$!33; >qj?8LPFH$!3O&z+-.UKs\"F17$$!3QhT*zw^L:$FH$!3q,1Mk([J$=F17$$!3gr-%47e) )[#FH$!37mzD`9n#*>F17$$!31'**G:3FX*=FH$!3i7J5&*=C)=#F17$$!3Q\\n0dmox:F H$!3K!R:N$f7GBF17$$!3o-XeKi%3E\"FH$!3qcXfZn[5DF17$$!3)HZtV=&*ed*F-$!3% >*)eR,.Gv#F17$$!3<>>!H/GLa'F-$!3CH$)fNzdn3OppCF-$!3\">%G!f%fPE VF17$$!3%)z/$=sm\\l\"F-$!3$es%o7=+W\\F17$$!3))fySyKgZ7F-$!3'>JUC$)*HKa F17$$!3&*)R_)\\$)R-%)!#?$!3w44JztQ(>'F17$$!33*HRF8\"eljF[]l$!3KnfU]3D) z'F17$$!3B*>Ec\"RwGVF[]l$!3L-&ywNc2t(F17$$!3!)*48&)pY>H#F[]l$!3C9bwX#y fb*F17$$!3')*****R\"[H^D!#@$!3C4Rm^b]')>!#;7hn7$$\"3G+++3Z&[(QF`^l$\"3 Ja-!\\9!>GJXtdj%['F17$$\"3!))Gl:B?-l *F[]l$\"3J@0Z)*)[y\"fF17$$\"3M$z%>h(e\"G9F-$\"3g#)zE'3\")H>&F17$$\"3!y 0L#*\\&H\"*=F-$\"3Ar7ut#R)GZF17$$\"3s'e4`(*ov\"GF-$\"3;cnU%yE/9%F17$$ \"3I:hQ^C%Qu$F-$\"3U.5$yd#3mPF17$$\"39t\"RNS*Q'f&F-$\"3Sz;9$*)fNH$F17$ $\"3GIApbj$*[uF-$\"3#>78sy'z$*HF17$$\"3'G&*z](=Fn5FH$\"3%pcF1p9Zl#F17$ $\"3c#o!f9,l*Q\"FH$\"3_d91L,zHCF17$$\"3IP!39;)>_F17$$ \"3)GL)[6@3rNFH$\"3YF52u*H>v\"F17$$\"3()R*H^')yXC%FH$\"3bRohFRmS;F17$$ \"3![PrIuX>%\\FH$\"3)Gx(=G`_U:F17$$\"3sO?$ecjJm&FH$\"3WYpGq4q_9F17$$\" 3oK$))QVo?Q'FH$\"3'R]C6*3tq8F17$$\"34$*\\Kb/b@rFH$\"3?gXa!f_8H\"F17$$ \"3gf[JAm)Gx(FH$\"3]BZDhS&RA\"F17$$\"3trQzxo81&)FH$\"3'4'e<2hj\\6F17$$ \"3Z96FHzRU#*FH$\"3g;9f&owd2\"F17$$\"3U:ZAy=#>&**FH$\"3J\"oWC\"y![+\"F 17$$\"3Um$[(fPif5F1$\"33KNpyb#QS*FH7$$\"3%oJB28Ri8\"F1$\"3B\\8Q.nKR')F H7$$\"3\\bQf*RU6?\"F1$\"3OD&yNf*o'*zFH7$$\"3)p#QoI.kw7F1$\"3A/!Q*Q!=AE (FH7$$\"3gk0L#zoMM\"F1$\"3Kb@?*)\\!>j'FH7$$\"3\">U)\\1-z;9F1$\"3k,c8v& f;(fFH7$$\"3on-zJ&4m[\"F1$\"3=3oF>D2#Q&FH7$$\"3e6T?t'e%f:F1$\"3@-vV**o n9[FH7$$\"3w#)*p)3XX$FH7$$\"3(z0`l+:(Q=F1$\"3eG%R)) QRo7$FH7$$\"3%))z#>zI=4>F1$\"3Cl5)4BRH\"GFH7$$\"31?b`$\\$)>)>F1$\"3*Gs >qg&oFDFH7$$\"3)*o^1'\\/K0#F1$\"3%eiS#)GSCG#FH7$$\"3VB$G\"yW6A@F1$\"3E oV)zY)*H2#FH7$$\"3oU\"zVEF')>#F1$\"3gED>kiJo=FH7$$\"3#y9peSxtE#F1$\"3d **z;n!3hq\"FH7$$\"3h1OLcGySBF1$\"3aoIDU6`_:FH7$$\"3yN&\\Y5*H2CF1$\"3Na )>.!ygG9FH7$$\"3!)eBIL*=+[#F1$\"3`BR\\CMb28FH7$$\"3C)*e([iU%[DF1$\"3:# R\\Pb'o07FH7$$\"3M43'z4l*>EF1$\"3BjMY$41,6\"FH7$$\"3ah\">&)Q\"*)*o#F1$ \"3Y:gH/$F1$\"3%\\)yG^B^%4(F-7$$\"3IwpS5nA=JF1$ \"3'**>h,G^Mf'F-7$$\"31M$\\)zAe&=$F1$\"3mcdP9EV%='F-7$$\"3E\\TPltRdKF1 $\"3w'=SY()*p%y&F-7$$\"3`M)>ccLhK$F1$\"3J-p^YCpLaF-7$$\"3!************ **R$F1$\"39Z8\\*e,u3&F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fham-F$6%7 $7$$!3++++++++]FH$\"3++++++++DF17$F_bmF]bm-Fbam6&FdamFhamFeamFham-%*TH ICKNESSG6#\"\"#-F$6&7#7$$\"\"\"F*F\\cm-Fbam6&FdamF*F*F*-%&STYLEG6#%&PO INTG-%'SYMBOLG6#%'CIRCLEG-%%TEXTG6$7$$\"#@!\"\"$\"#>F^dmQ9gradient~of~ tangent~=~-16\"-Ficm6$7$$\"#9F^dm$\"$D\"!\"#Q&(1,1)Fbdm-Ficm6$7$$\"#MF ^dm$FjdmF^dmQ\"xFbdm-Ficm6$7$FaemF_emQ\"yFbdm-%+AXESLABELSG6%%!GFjem-% %FONTG6#%(DEFAULTG-%%VIEWG6$;F(F_emFbfm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 7 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 260 4 "Note" }}{PARA 0 "" 0 "" {TEXT -1 71 "As you can see from the pr evious graph, it turns out that the equation:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3*y+x*y^3 = 2;" "6#/,&*&%\"xG\"\"$% \"yG\"\"\"F)*&F&F)*$F(F'F)F)\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "defines a one-to-one function " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 6 " from " }{TEXT 391 1 "x" }{TEXT -1 4 " to \+ " }{TEXT 392 1 "y" }{TEXT -1 57 " with the domain the set of all real \+ numbers excluding 0." }}{PARA 0 "" 0 "" {TEXT -1 42 "An explicit formu la for this function for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 15 " can be found. " }}{PARA 0 "" 0 "" {TEXT -1 49 "(The comp uter algebra system Maple can do this.) " }}{PARA 0 "" 0 "" {TEXT -1 9 "In fact: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) \+ = (x^2*(27+3*sqrt(3*x^8+81)))^(1/3)/(3*x)-x^3/((x^2*(27+3*sqrt(3*x^8+8 1)))^(1/3));" "6#/-%\"gG6#%\"xG,&*&)*&F'\"\"#,&\"#F\"\"\"*&\"\"$F/-%%s qrtG6#,&*&F1F/*$F'\"\")F/F/\"#\")F/F/F/F/*&F/F/F1!\"\"F/*&F1F/F'F/F;F/ *&F'F1)*&F'F,,&F.F/*&F1F/-F36#,&*&F1F/*$F'F8F/F/F9F/F/F/F/*&F/F/F1F;F; F;" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 75 "We can obtain an explicit, but complicated, formula fo r the derivative of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 15 " has the for m: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = phi(x)^ (1/3)/(3*x)-x^3/(phi(x)^(1/3));" "6#/-%\"gG6#%\"xG,&*&)-%$phiG6#F'*&\" \"\"F/\"\"$!\"\"F/*&F0F/F'F/F1F/*&F'F0)-F,6#F'*&F/F/F0F1F1F1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x) = x^2*(27+3*sqrt(3*x^8+81));" "6 #/-%$phiG6#%\"xG*&F'\"\"#,&\"#F\"\"\"*&\"\"$F,-%%sqrtG6#,&*&F.F,*$F'\" \")F,F,\"#\")F,F,F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = 1/3;" "6#/-%\"gG6#%\"xG*&\"\"\"F)\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x)^(1/3)*x^(-1)-x^3*phi(x)^(-1 /3);" "6#,&*&)-%$phiG6#%\"xG*&\"\"\"F+\"\"$!\"\"F+)F),$F+F-F+F+*&F)F,) -F'6#F),$*&F+F+F,F-F-F+F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g '`(x) = 1/3;" "6#/-%$g~'G6#%\"xG*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(``(1/3)*phi(x)^(-2/3)*phi*`'`(x)*`.`*x^(-1 )+phi(x)^(1/3)*`.`*(-1)*x^(-2))-(3*x^2*phi(x)^(-1/3)+x^3*`.`*``(-1/3)* phi(x)^(-4/3)*phi*`'`(x));" "6#,&-%!G6#,&*.-F%6#*&\"\"\"F,\"\"$!\"\"F, )-%$phiG6#%\"xG,$*&\"\"#F,F-F.F.F,F1F,-%\"'G6#F3F,%\".GF,)F3,$F,F.F,F, **)-F16#F3*&F,F,F-F.F,F:F,,$F,F.F,)F3,$F6F.F,F,F,,&*(F-F,*$F3F6F,)-F16 #F3,$*&F,F,F-F.F.F,F,*.F3F-F:F,-F%6#,$*&F,F,F-F.F.F,)-F16#F3,$*&\"\"%F ,F-F.F.F,F1F,-F86#F3F,F,F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = phi*`' `(x)/(9*x*phi(x)^(2/3))-phi(x)^(1/3)/(3*x^2)-3*x^2/(phi(x)^(1/3))+x^3* phi*`'`(x)/(3*phi(x)^(4/3));" "6#/%!G,**(%$phiG\"\"\"-%\"'G6#%\"xGF(*( \"\"*F(F,F()-F'6#F,*&\"\"#F(\"\"$!\"\"F(F5F(*&)-F'6#F,*&F(F(F4F5F(*&F4 F(*$F,F3F(F5F5*(F4F(*$F,F3F()-F'6#F,*&F(F(F4F5F5F5**F,F4F'F(-F*6#F,F(* &F4F()-F'6#F,*&\"\"%F(F4F5F(F5F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x) = x^2*(27+3*sqrt(3*x^8+81));" "6#/-%$phiG6#%\"xG*&F'\"\"#,&\"#F\"\"\"*&\"\"$F,-%%sqrtG6#,&*&F.F,*$F' \"\")F,F,\"#\")F,F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "implies that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " phi*`'`(x) = 2*x*`.`*(27+3*sqrt(3*x^8+81))+x^2*`.`*``(3/(2*sqrt(3*x^8+ 81)))*`.`*24*x^7;" "6#/*&%$phiG\"\"\"-%\"'G6#%\"xGF&,&**\"\"#F&F*F&%\" .GF&,&\"#FF&*&\"\"$F&-%%sqrtG6#,&*&F2F&*$F*\"\")F&F&\"#\")F&F&F&F&F&*. F*F-F.F&-%!G6#*&F2F&*&F-F&-F46#,&*&F2F&*$F*F9F&F&F:F&F&!\"\"F&F.F&\"#C F&F*\"\"(F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*x*(27+3*sqrt(3*x^8+8 1))+36*x^9/sqrt(3*x^8+81);" "6#/%!G,&*(\"\"#\"\"\"%\"xGF(,&\"#FF(*&\" \"$F(-%%sqrtG6#,&*&F-F(*$F)\"\")F(F(\"#\")F(F(F(F(F(*(\"#OF(*$F)\"\"*F (-F/6#,&*&F-F(*$F)F4F(F(F5F(!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = (54*x*sqrt(3*x^8+81)+54*x^9+486*x)/sqrt(3*x^8+81);" "6#/%!G*&,(*(\"# a\"\"\"%\"xGF)-%%sqrtG6#,&*&\"\"$F)*$F*\"\")F)F)\"#\")F)F)F)*&F(F)*$F* \"\"*F)F)*&\"$'[F)F*F)F)F)-F,6#,&*&F0F)*$F*F2F)F)F3F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 54*x*(sqrt(3*x^8+81)+x^8+9)/sqrt(3*x^8+81);" " 6#/%!G**\"#a\"\"\"%\"xGF',(-%%sqrtG6#,&*&\"\"$F'*$F(\"\")F'F'\"#\")F'F '*$F(F1F'\"\"*F'F'-F+6#,&*&F/F'*$F(F1F'F'F2F'!\"\"" }{TEXT -1 2 ". " } }{PARA 260 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g '`(x) = phi*`'`(x)/(9*x*phi(x)^(2/3))-phi(x)^(1 /3)/(3*x^2)-3*x^2/(phi(x)^(1/3))+x^3*phi*`'`(x)/(3*phi(x)^(4/3));" "6# /-%$g~'G6#%\"xG,**(%$phiG\"\"\"-%\"'G6#F'F+*(\"\"*F+F'F+)-F*6#F'*&\"\" #F+\"\"$!\"\"F+F7F+*&)-F*6#F'*&F+F+F6F7F+*&F6F+*$F'F5F+F7F7*(F6F+*$F'F 5F+)-F*6#F'*&F+F+F6F7F7F7**F'F6F*F+-F-6#F'F+*&F6F+)-F*6#F'*&\"\"%F+F6F 7F+F7F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 6*(sqrt(3*x^8+81)+x^8+9)/(s qrt(3*x^8+81)*(x^2*(27+3*sqrt(3*x^8+81)))^(2/3))-(x^2*(27+3*sqrt(3*x^8 +81)))^(1/3)/(3*x^2)-3*x^2/((x^2*(27+3*sqrt(3*x^8+81)))^(1/3))+18*x^4* (sqrt(3*x^8+81)+x^8+9)/((x^2*(27+3*sqrt(3*x^8+81)))^(4/3)*sqrt(3*x^8+8 1));" "6#/%!G,**(\"\"'\"\"\",(-%%sqrtG6#,&*&\"\"$F(*$%\"xG\"\")F(F(\"# \")F(F(*$F1F2F(\"\"*F(F(*&-F+6#,&*&F/F(*$F1F2F(F(F3F(F()*&F1\"\"#,&\"# FF(*&F/F(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(F(F(*&F>F(F/!\"\"F(FHF(*&)*&F1F> ,&F@F(*&F/F(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(F(F(*&F(F(F/FHF(*&F/F(*$F1F>F (FHFH*(F/F(*$F1F>F()*&F1F>,&F@F(*&F/F(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(F(F (*&F(F(F/FHFHFH**\"#=F(*$F1\"\"%F(,(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(*$F1F 2F(F5F(F(*&)*&F1F>,&F@F(*&F/F(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(F(F(*&F_oF( F/FHF(-F+6#,&*&F/F(*$F1F2F(F(F3F(F(FHF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "g := x -> (x^2*(27+3*sqrt(3*x^8+81)))^(1/3)/ (3*x)-x^3/((x^2*(27+3*sqrt(3*x^8+81)))^(1/3)):\n'g(x)'=g(x);\nh := x - > 6*(sqrt(3*x^8+81)+x^8+9)/(sqrt(3*x^8+81)*(x^2*(27+3*sqrt(3*x^8+81))) ^(2/3))-(x^2*(27+3*sqrt(3*x^8+81)))^(1/3)/(3*x^2)-3*x^2/((x^2*(27+3*sq rt(3*x^8+81)))^(1/3))+18*x^4*(sqrt(3*x^8+81)+x^8+9)/((x^2*(27+3*sqrt(3 *x^8+81)))^(4/3)*sqrt(3*x^8+81)): 'h(x)'=h(x);\nplot([g(x),h(x)],x=-3. .3,-5..5,discont=true,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*(\"\"$!\"\"*&)F'\"\"#\"\"\",&\"#FF/*&F*F/,&*&F* F/)F'\"\")F/F/\"#\")F/#F/F.F/F/#F/F*F'F+F/*&F'F*F,#F+F*F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,***\"\"'\"\"\",(*$,&*&\"\"$F+)F' \"\")F+F+\"#\")F+#F+\"\"#F+*$F1F+F+\"\"*F+F+F.#!\"\"F5*&)F'F5F+,&\"#FF +*&F0F+F.F4F+F+#!\"#F0F+*(F0F9F:#F+F0F'F@F9*(F0F+F'F5F:#F9F0F9*,\"#=F+ F'\"\"%F,F+F:#!\"%F0F.F8F+" }}{PARA 13 "" 1 "" {GLPLOT2D 466 466 466 {PLOTDATA 2 "6&-%'CURVESG6%7gn7$$!\"$\"\"!$!3:;XmB'**GS(!#>7$$!3*GyIw` 3Y$H!#<$!3_N4\"G[bz!zF-7$$!3Ew&pex6x(GF1$!3')\\!*[DxJ&Q)F-7$$!3;\\Di;a s8GF1$!3Obt`qa**o*)F-7$$!3?q8D9\\J\\FF1$!3\"H_A$o0F7'*F-7$$!3yyXvV0@&o #F1$!373Hg$Gl9.\"!#=7$$!3sWMP'exdi#F1$!3:F>(*G\\y-6FH7$$!3[Bl\\,#QUc#F 1$!375!GVdpO=\"FH7$$!3p6Qi\"3%f+DF1$!3Y!)zx8FH7$$!3m.iv=#)*=P#F1$!33+%4=lzG\\\"FH7$$!3ie67I3U9BF1$!355* G.pGbg\"FH7$$!3Sq0+/\\r\\AF1$!3/mE:&>\")fu\"FH7$$!3S808*GVZ=#F1$!3s,7! e]jM!>FH7$$!3%ytb#*4J@7#F1$!3ow^Ok4&H2#FH7$$!3Q`W\\KKFl?F1$!3>Lfr?j)QC #FH7$$!3ksY]HPm(*>F1$!3q$)\\WOb(4Z#FH7$$!3y?#>@h*QS>F1$!3'3c$GbQ3'o#FH 7$$!3ynu(y>mP(=F1$!3i_!)RCDxlHFH7$$!3t8/P(=$z9=F1$!3*Qn1nJ\\EC$FH7$$!3 =>[7[/4]C;F1$!3_*>,N=)zaVFH7$$!3%[o%*4cd^c\"F1$!3)3!>Dy*)[sZFH7$$!3lQqur 2[,:F1$!3cL!e(pQ5i_FH7$$!30BVv$[Q`V\"F1$!3e*G#ejS)4\"eFH7$$!3=x%>wUhxP \"F1$!3h__2/NV=jFH7$$!3pY)='Hmd:8F1$!3c!Re,Th@*oFH7$$!3)HL(\\[OL^7F1$! 3-?x:VkF1vFH7$$!3IJI([U%[)=\"F1$!3\"=;P9d@97)FH7$$!30^'p$pXnF6F1$!3=)Q 3t**zXs)FH7$$!31*oHEfb,1\"F1$!3ylC\"R`4&)R*FH7$$!3Ko-,!*y'[***FH$!3\"y +**4@80+\"F17$$!3q-eI;*)4Z$*FH$!3k8;c'>*Hl5F17$$!3Ua(H([R7g()FH$!3N\"[ WQ)f5C6F17$$!3\"p=j(o_S=\")FH$!3Es)[v5!z)=\"F17$$!3!p!3A,!)f9vFH$!3irZ fv^Z]7F17$$!3G5J[FaW$)oFH$!3?Y'\\^I.lJ\"F17$$!3uWsYA%yjE'FH$!3%4e[>(4a $Q\"F17$$!37p#4]Xm.i&FH$!3'fC(>q^yd9F17$$!3MRO+],=)*\\FH$!3T&4tUEh^`\" F17$$!3#>>w()y/>O%FH$!3uN^()oS9B;F17$$!3;.#)y3\")*3t$FH$!3;Rw'oCGOs\"F 17$$!3]iyp.&o5:$FH$!3$>WxF)fiL=F17$$!3Okp_U$=l[#FH$!3!*G*=G6AL*>F17$$! 3%)Hd@cn8#*=FH$!3o2Dw48=*=#F17$$!3G&>LP,-%e7FH$!3$*zo^x>77DF17$$!3/7w* 33&>^&*F-$!3o&='e!e\"=bFF17$$!3TrKYC+P=lF-$!3,t*4?Sn,8$F17$$!3[`uMowx) )[F-$!3WKAh/A7F-$!3AGCgzn opaF17$$!3C)3z0ynz9)!#?$!3o?0$GtM7E'F17$$!3#eJMaLx46'Ff\\l$!3%4a2`/#Q \"*oF17$$!3GW&*G!*o)R2%Ff\\l$!3'H^PODj'))yF17$$!3HsZ9Xk*p.#Ff\\l$!3lFR %Rv)3R**F17$$!3'***************f!#E$!3&3N1Nu7O$p!#:7gn7$$\"3'********* ******fF[^l$\"3&3N1Nu7O$pF^^l7$$\"3NL!Q!*>*[V?Ff\\l$\"3n=!y]8^&G**F17$ $\"3Qmg2)Rsp3%Ff\\l$\"3'pn*>>&*H!)yF17$$\"3U*49rfb/8'Ff\\l$\"3'Q/>mjvS )oF17$$\"3YK@:'zQR<)Ff\\l$\"3_\")oq.kfaiF17$$\"3')>GU>04E7F-$\"3oK6B=w )QY&F17$$\"3YE/BfryM;F-$\"3oAF)Q%>Ek\\F17$$\"3/Sc%)Q/=_CF-$\"3enS!**)= lOVF17$$\"3C`3Y=PdpKF-$\"3w$f2EOh+%RF17$$\"3O!G\"px-O/\\F-$\"3HKxfzn!= W$F17$$\"3z1<#p$o9RlF-$\"3@!3Xf([%o7$F17$$\"3kI?)\\#\\)RQ*F-$\"3=/')R[ 4]rFF17$$\"3XNUI,B)GA\"FH$\"3fI$H)4$oj`#F17$$\"353Xx$*eui=FH$\"3$>&R0$ fe3?#F17$$\"3^)H'[<4&o]#FH$\"3#[1S87$p()>F17$$\"3O5UXAY*y9$FH$\"3A*poe ,!HM=F17$$\"39^bE'>CAu$FH$\"3\\l8u.Fn@Qa)pN:F17$$\"3i\"3$\\n$f%GcFH$\"3# QX5#)p@oX\"F17$$\"3/lzVsy,\"G'FH$\"3M%=6>&H\">Q\"F17$$\"3_;%)ye8F17$$\"3#))H%**>5&G](FH$\"3SonX-ro^7F17$$\"3Il[porc_\")FH$ \"3ij,xf\"H`=\"F17$$\"35AEWn!*oy()FH$\"3'>#QnIDCA6F17$$\"3[ka0NxEZ$*FH $\"3Al()G0BGl5F17$$\"3UF`\\wiL-5F1$\"3C#o&>Nsjw**FH7$$\"3Ez2)QR5'f5F1$ \"3^c)[2=fRS*FH7$$\"3FKD73QBE6F1$\"3E_2#)o)G*Q()FH7$$\"3K'eH'=o?&=\"F1 $\"37Wdr/_x`\")FH7$$\"3)3=vyb4*\\7F1$\"3(QEu@@#3?vFH7$$\"3Nc>]m=_68F1$ \"3J<]K+ANIpFH7$$\"3K%Q)*p%y!eP\"F1$\"3V^?w4<3OjFH7$$\"3@:`+XC%[V\"F1$ \"33+%)**)[W_\"eFH7$$\"3ShHDM#>&)\\\"F1$\"37-bi++$eG&FH7$$\"3+xcCA:mk: F1$\"3_yD7>\"[hx%FH7$$\"3)G_!Qy&QAi\"F1$\"3])\\w&)\\`!oVFH7$$\"3O`6QwL U%o\"F1$\"3Q'))e`^*4lRFH7$$\"33nE]djm[sHFH7$$\"3*4J qLTW)R>F1$\"3%Q8MvlR#)o#FH7$$\"3At*)*p@80+#F1$\"3)oH4d0S3Y#FH7$$\"3y>% pV6!Hl?F1$\"3.AOuoH$QC#FH7$$\"3hCq76w)R7#F1$\"3+QGaLPln?FH7$$\"3O\"oB \"z%f\")=#F1$\"3m;a9tpt%*=FH7$$\"3O>z(e?S&[AF1$\"3A;([Dvv'[=+DF1$\"3cLS(R#QRw7FH7$$\" 3'3QAr_4Qc#F1$\"3Y2p)Q0gU=\"FH7$$\"3uz67&>5pi#F1$\"3r_#Q:yl85\"FH7$$\" 3!Q@Ic:$*[o#F1$\"32ZPD*oH=.\"FH7$$\"3i.tur\"[8v#F1$\"3e*[aFnW5f*F-7$$ \"32F%y.L'y5GF1$\"3)[M3Nb$3(**)F-7$$\"3_!oEY!)fT(GF1$\"3Q7Sa_QP;%)F-7$ $\"3Mn`v0j\"[$HF1$\"3e,p*>Lzi!zF-7$$\"\"$F*$\"3:;XmB'**GS(F--%'COLOURG 6&%$RGBG$\"*++++\"!\")$F*F*F^am-F$6%7gn7$F($!3mfE1'y+4R(F-7$F/$!3-!pAx 5Z&o!)F-7$F5$!3b=(QZ\"*3>s)F-7$F:$!3BU*GU=Ip`*F-7$F?$!3/)4K.;ma/\"FH7$ FD$!3)*G1`2/(y9\"FH7$FJ$!3mkfFKs0a7FH7$FO$!3So^5n8+x8FH7$FT$!3+t0p9c,? :FH7$FY$!3'Qo#RI`&4o\"FH7$Fhn$!3kP0hzg\\o=FH7$F]o$!3]2Kx[Hza?FH7$Fbo$! 3S@Y!HeP:H#FH7$Fgo$!3'R>/N,G?c#FH7$F\\p$!35=/FAX)z&GFH7$Fap$!3uo'o;Ee. ;$FH7$Ffp$!3y!Ra2bTgc$FH7$F[q$!3-Y-R[HA_RFH7$F`q$!3Cj0^ptJ`WFH7$Feq$!3 sG'*4.(*)Q%\\FH7$Fjq$!3c`A6%R^*HbFH7$F_r$!30_%o/i`o7'FH7$Fdr$!3c$*4jIM >vnFH7$Fir$!3e#*>Tkp*FH7$Fgt$!3_nc:A )[x')*FH7$F\\u$!3gxAQi_`e**FH7$Fau$!3%*=yI0![c***FH7$Ffu$!3kW.F++++5F1 7$F[v$!3H;ty+pb+5F17$F`v$!3paC:(=CQ+\"F17$Fev$!3/y&R)3xY85F17$Fjv$!3$4 @M!\\IYJ5F17$F_w$!3%H0,e/pM1\"F17$Fdw$!3<#\\aNq!z76F17$Fiw$!3'H9i/&*46 >\"F17$F^x$!3WKuOv6M.8F17$Fcx$!3]<+^H;lt9F17$Fhx$!3dOx*z'\\SH6GY v)p'F17$Faz$!3S&4P'4&)GY'*F17$Ffz$!3kDq9N5a-;!#;7$F[[l$!3yW!*4UI\\]BFe jm7$F`[l$!37W(=$>=rMSFejm7$Fe[l$!3hXkH&fF0#fFejm7$Fj[l$!3WgRa\")pa;5F^ ^l7$F_\\l$!3gq.33gy\"\\\"F^^l7$Fd\\l$!3G(GxqZ#[hDF^^l7$Fj\\l$!3;$)>QIe -fPF^^l7$F_]l$!3+pF[eN]akF^^l7$Fd]l$!3^.1YL*Gki\"!#97$Fi]l$!3+mdszq+_Q !\"(7gn7$Fa^lFc\\n7$Ff^l$!3B9yO3Ca>;Fa\\n7$F[_l$!3A_[;'GurU'F^^l7$F`_l $!3qjVG>'4Ju$F^^l7$Fe_l$!3U(*\\#e'oj]DF^^l7$Fj_l$!3Udxb5(pa[\"F^^l7$F_ `l$!3gaRa&)HC75F^^l7$Fd`l$!3]eR^1HY&*eFejm7$Fi`l$!3kQv\"\\pLw,%Fejm7$F ^al$!3huADk+bSBFejm7$Fcal$!3C3,9x$pdf\"Fejm7$Fhal$!3!z%zH*ov[()*F17$F] bl$!3C]DELzXcpF17$Fbbl$!3:x4[O2V8SF17$Fgbl$!3S$HN1&)GXv#F17$F\\cl$!3OT A9W%3M4#F17$Facl$!3h*\\mIyoPs\"F17$Ffcl$!3QShDy.0v9F17$F[dl$!3C,W%>yQU I\"F17$F`dl$!3mP)oaj<**=\"F17$Fedl$!3/)R)GOiO66F17$Fjdl$!3)=[^@Ga_1\"F 17$F_el$!3xxVQu?#>.\"F17$Fdel$!325:b.-u75F17$Fiel$!3/e)*=gWl.5F17$F^fl $!3)4&yqokb+5F17$Fcfl$!2QGj\\u*******F17$Fhfl$!3?[FW>_w&***FH7$F]gl$!3 9bX$*3!=*f**FH7$Fbgl$!3eZA;DmWu)*FH7$Fggl$!3'=x:&)empp*FH7$F\\hl$!3EDw Ux\">$H%*FH7$Fahl$!3>,[X'HH0/*FH7$Ffhl$!3idz$zN_4f)FH7$F[il$!37;@x\"pl `-)FH7$F`il$!3g!*4U0d@ztFH7$Feil$!3+i#Rp5%3&z'FH7$Fjil$!31Ie;z-(ya$FH7$Fh[m$!3+*=)*>Sj-;$FH7$F]\\m$!3Q*yD-> q'[GFH7$Fb\\m$!3IDA+Ph\"pa#FH7$Fg\\m$!3!zi%pFN7'H#FH7$F\\]m$!3)Ho%\\&p +V1#FH7$Fa]m$!3y(yZv4@S'=FH7$Ff]m$!3Q%=&**4exy;FH7$F[^m$!3tiQ$Q3**4_\" FH7$F`^m$!3ca%)p#G3zP\"FH7$Fe^m$!3%G*y!*3N#>D\"FH7$Fj^m$!3!*==#oV2%[6F H7$F__m$!3@_?`ilSU5FH7$Fd_m$!33'Q&*Rm$ew&*F-7$Fi_m$!3%[\\PJ?M[w)F-7$F^ `m$!3o<5^UYFm!)F-7$Fc`mFcam-Fh`m6&Fj`mF^amF^amF[am-%+AXESLABELSG6%Q\"x 6\"Q!Fjgn-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fc`m;$!\"&F*$\"\"&F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 24 ": Bec ause the equation " }{XPPEDIT 18 0 "x^3*y+x*y^3 = 2" "6#/,&*&%\"xG\" \"$%\"yG\"\"\"F)*&F&F)*$F(F'F)F)\"\"#" }{TEXT -1 20 " is symmetrical \+ in " }{TEXT 394 1 "x" }{TEXT -1 5 " and " }{TEXT 395 1 "y" }{TEXT -1 15 ", the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 28 " is a self-inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "evalf(g(g(0.25)));\nevalf(g (g(0.75)));\nevalf(g(g(3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*++ +]#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/+++v!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+-+++I!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "solve" } {TEXT -1 37 " can find a formula for the function " }{XPPEDIT 18 0 "g( x)" "6#-%\"gG6#%\"xG" }{TEXT -1 36 " defined implicitly by the equatio n " }{XPPEDIT 18 0 "x^3*y+x*y^3 = 2" "6#/,&*&%\"xG\"\"$%\"yG\"\"\"F)*& F&F)*$F(F'F)F)\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "( We must remove the non-real solutions.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "[solve(x^3*y+x*y^3=2,y) ]:\nop(remove(hastype,%,nonreal)):\ng := unapply(%,x):\n'g(x)'=g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*(\"\"$!\"\"F'F+*&,& \"#F\"\"\"*(F*F/F*#F/\"\"#,&*$)F'\"\")F/F/F.F/F1F/F/)F'F2F/#F/F*F/*&F' F*F,#F+F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "phi := 'phi':\ng := x -> phi(x)^(1/3)/(3*x)-x^3/p hi(x)^(1/3):\n'g(x)'=g(x);\nDiff(g(x),x)=diff(g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*&#\"\"\"\"\"$F+*&-%$phiGF&F*F'!\" \"F+F+*&F'F,F.#F0F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,& *&#\"\"\"\"\"$F**&-%$phiG6#%\"xGF)F0!\"\"F*F**&F0F+F-#F1F+F1F0,**&#F* \"\"*F**(F-#!\"#F+F0F1-%%diffG6$F-F0F*F*F**&#F*F+F**&F-F)F0F:F*F1*(F+F *F0\"\"#F-F3F1*&F)F**(F0F+F-#!\"%F+F;F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "phi := x -> x^2*(27+3*sqrt(3*x^8+81)):\n'phi(x)'=phi (x);\ng := x -> phi(x)^(1/3)/(3*x)-x^3/phi(x)^(1/3):\n'g(x)'=g(x);\nDi ff('g(x)',x)=map(normal,diff(g(x),x));\nh := unapply(rhs(%),x):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG*&)F'\"\"#\"\"\",&\"#FF +*&\"\"$F+,&*&F/F+)F'\"\")F+F+\"#\")F+#F+F*F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*(\"\"$!\"\"*&)F'\"\"#\"\"\",&\"#FF/*&F *F/,&*&F*F/)F'\"\")F/F/\"#\")F/#F/F.F/F/#F/F*F'F+F/*&F'F*F,#F+F*F+" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"gG6#%\"xGF*,**,\"\"#\" \"\"\"\"$#F.F/,(*$,&*&F/F.)F*\"\")F.F.\"#\")F.#F.F-F.*$F5F.F.\"\"*F.F. *&)F*F-F.,&F:F.F2F.F.#!\"#F/F3#!\"\"F-F.**F/FAF/F0F;F0F*F?FA*(F*F-F/#F -F/F;#FAF/FA*.F-F.F*\"\"%F/FDF1F.F;#!\"%F/F3F@F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The derivative where " } {XPPEDIT 18 0 "x =1" "6#/%\"xG\"\"\"" }{TEXT -1 14 " should be -1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(h(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 40 "We can obtain an alternative expression \+ " }{XPPEDIT 18 0 "k(x)" "6#-%\"kG6#%\"xG" }{TEXT -1 20 " for the deriv ative " }{XPPEDIT 18 0 "`g '`(x);" "6#-%$g~'G6#%\"xG" }{TEXT -1 17 " b y substituting " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" } {TEXT -1 19 " in the expression " }{XPPEDIT 18 0 "-y*(3*x^2+y^2)/(x*(x ^2+3*y^2))" "6#,$*(%\"yG\"\"\",&*&\"\"$F&*$%\"xG\"\"#F&F&*$F%F,F&F&*&F +F&,&*$F+F,F&*&F)F&*$F%F,F&F&F&!\"\"F3" }{TEXT -1 53 " obtained by imp licit differentiation. The graphs of " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG 6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k(x)" "6#-%\"kG6#%\"xG" } {TEXT -1 21 " appear to coincide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "subs(y=g(x),-y*(3*x^2+y^2)/ (x*(x^2+3*y^2))):\nk := unapply(simplify(%),x):\n'k(x)'=k(x);\nplot([h (x),k(x)],x=0..3,-5..0,color=[blue,cyan],thickness=[1,3]);\n\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"kG6#%\"xG,$*0\"\"*!\"\"\"\"$#\"\" \"F,,&*$)*&)F'\"\"#F.,&F*F.*$,&*&F,F.)F'\"\")F.F.\"#\")F.#F.F4F.F.#F4F ,F.F+*&)F'\"\"%F.)F,F-F.F.F.F'!\"#,**(\"#@F.F3F.F1F.F.*(F*F.)F,F=F.)F2 F-F.F.*(FGF.FHF.F7F;!#97$$\"3s*****\\Smp3%F*$!3=&RbC&o=Fk!#:7$$\"3z******4'\\ /8'F*$!3O(=!R$\\9Ju$F37$$\"3Y******4G$R<)F*$!3QieSc$R1b#F37$$\"3****** *>gT<-\"!#>$!3=#y75?\\U*=F37$$\"3'******>#**3E7FA$!3s&G7cnqa[\"F37$$\" 3#******>CQ/V\"FA$!3OZ_`&f&\\47F37$$\"3*)*****>c'yM;FA$!3qiUeyMC75F37$ $\"3.+++#)[8R=FA$!31E*o%*z&Q^')!#;7$$\"3\"******H?$[V?FA$!3$e[6%4NiS2@&>$FX7$$\"3#)*****fSmp3%FA$!3?80o`I5%)HFX 7$$\"3%******fs98H%FA$!3cQDv%Gpiz#FX7$$\"3;+++ZIm&\\%FA$!3gV'3#Q1@GEFX 7$$\"3p*****zO6+q%FA$!39/20oD3xCFX7$$\"3$)*****zofV!\\FA$!3'pDG\"R/bSB FX7$$\"38+++5!3(3^FA$!31y4>)\\\"o;AFX7$$\"3F+++Ij08`FA$!30eRr\"*H'Q5#F X7$$\"3s******\\YSFX7$$\"3++++!H,h#fFA$!3x.tdD)e\">=FX7$$\"38+++5'\\/8'FA$!3g'*G,d))))Q [xs')FA$!3YqAFLZN'4\"FX7$$\"3=+++XV)RQ*FA$!3M?&45!#=$!3%*p`94g\\j*)Fdw7$$\"3%*******Q.k!3 \"Fhw$!3'p$**)=bn4>)Fdw7$$\"31+++#Hh<:\"Fhw$!3,&3#o-k-HvFdw7$$\"31+++W A)GA\"Fhw$!3EK&>KOek&pFdw7$$\"3'******>9[GQ\"Fhw$!3HKsH;<4GF$Fdw7$$\"3=+++i3&o]#Fhw$!3])enh#*GXv#Fdw7$$\"3')******oX*y9$Fhw$!3' f\\bb[3M4#Fdw7$$\"3\")*****R9CAu$Fhw$!3+b5(*3)oPs\"Fdw7$$\"3#)*****R*z hdVFhw$!3k\\=$\\R]]Z\"Fdw7$$\"33+++%>fS*\\Fhw$!3lmtz#zQUI\"Fdw7$$\"3&) ******=$f%GcFhw$!3%y<&fUw\"**=\"Fdw7$$\"3Q+++Dy,\"G'Fhw$!3%R0#)3Cm86\" Fdw7$$\"3/+++7. \"Fdw7$$\"3!)*****\\7nD:)Fhw$!3SCXY/-u75Fdw7$$\"3[+++D!*oy()Fhw$!3P))> dgWl.5Fdw7$$\"3!******RpnsM*Fhw$!3<.H\")okb+5Fdw7$$\"3(******>FOB+\"Fd w$!2Qvk\\u*******Fdw7$$\"3-+++!R5'f5Fdw$!3'o^p-Ald***Fhw7$$\"3&******R !QBE6Fdw$!3e%yYG,=*f**Fhw7$$\"3!******\\\"o?&=\"Fdw$!3+SWXKmWu)*Fhw7$$ \"3/+++a&4*\\7Fdw$!3q&y)=-m'pp*Fhw7$$\"32+++j=_68Fdw$!3P<)eb>>$H%*Fhw7 $$\"35+++Wy!eP\"Fdw$!36[89<$H0/*Fhw7$$\"36+++UC%[V\"Fdw$!3mY.%GQ_4f)Fh w7$$\"3'******4B>&)\\\"Fdw$!3&*ekNAdOD!)Fhw7$$\"3%*******=:mk:Fdw$!37F t-Qd@ztFhw7$$\"3*******fdQAi\"Fdw$!3MU)y68%3&z'Fhw7$$\"33+++uLU%o\"Fdw $!3'zHsGIqr;'Fhw7$$\"3!******\\Nm'[hFdw$!3kc`@JY4cRFhw7$$\"3z*****\\@80+#Fdw$!3-V$p`%>(ya$Fhw7$ $\"3-+++7,Hl?Fdw$!3K+=J:MEgJFhw7$$\"3$)******3w)R7#Fdw$!3OXx(3?q'[GFhw 7$$\"3?+++y%f\")=#Fdw$!3a^g&>9;pa#Fhw7$$\"3A+++/-a[AFdw$!3'**[M]`BhH#F hw7$$\"30+++ial6BFdw$!3yx*4)*p+V1#Fhw7$$\"3#)*****>;iLP#Fdw$!3=!\\**G5 @S'=Fhw7$$\"3$)******eL'zV#Fdw$!3G0#HS\"exy;Fhw7$$\"3>+++!*>=+DFdw$!3? p/A'3**4_\"Fhw7$$\"3++++E&4Qc#Fdw$!3#*\\R2&G3zP\"Fhw7$$\"39+++%>5pi#Fd w$!3Oyy,6N#>D\"Fhw7$$\"39+++bJ*[o#Fdw$!3Ms%))yV2%[6Fhw7$$\"31+++r\"[8v #Fdw$!3')\\XljlSU5Fhw7$$\"3++++Ijy5GFdw$!3;eT6pOew&*FA7$$\"3/+++/)fT(G Fdw$!3/&)Qt5U$[w)FA7$$\"35+++1j\"[$HFdw$!31-e$)RYFm!)FA7$$\"\"$\"\"!$! 31/F1'y+4R(FA-%'COLOURG6&%$RGBG$FbhlFbhlFihl$\"*++++\"!\")-%*THICKNESS G6#\"\"\"-F$6%7hp7$F($!3X'fgbu[&>;F-7$F/$!3O)QbC&o=FkF37$F5$!36\"=!R$ \\9Ju$F3F97$F?$!3K\"y75?\\U*=F37$FE$!3Y$G7cnqa[\"F37$FJ$!3AZ_`&f&\\47F 37$FO$!3*HE%eyMC75F37$FT$!3Q?*o%*z&Q^')FX7$FZ$!3*H[6%4Ni)\\\"o;AFX7$Fds$!3qdRr\"*H'Q5#FX7$Fis$!3G([+NN R2+#FX7$F^t$!3'znTG_jh!>FX7$Fct$!3k,tdD)e\">=FX7$Fht$!3#[*G,d))))Q )Fdw7$Fax$!3y'3#o-k-HvFdw7$Ffx$!3gH&>KOek&pFdw7$F[y$!3&eB(H;<4GF$Fdw7$Fdz$!3s!fnh#*GXv#Fdw7$Fiz$!3=%\\bb[3M4#Fdw7$F^[l$!3Lc5 (*3)oPs\"Fdw7$Fc[l$!3K]=$\\R]]Z\"FdwFg[l7$F]\\l$!31y^fUw\"**=\"Fdw7$Fb \\l$!3;a?)3Cm86\"Fdw7$Fg\\l$!3?xB?&Ga_1\"Fdw7$F\\]l$!3/c9:w?#>.\"Fdw7$ Fa]l$!3iCXY/-u75Fdw7$Ff]l$!3/*)>dgWl.5FdwFj]l7$F`^l$!2qvk\\u*******Fdw Fd^l7$Fj^l$!3o&yYG,=*f**Fhw7$F__l$!3))QWXKmWu)*FhwFc_l7$Fi_l$!3E;)eb>> $H%*Fhw7$F^`l$!3L]89<$H0/*FhwFb`l7$Fh`l$!3RjkNAdOD!)Fhw7$F]al$!3+Et-Qd @ztFhw7$Fbal$!3-R)y68%3&z'Fhw7$Fgal$!3G,B(GIqr;'Fhw7$F\\bl$!3sGk!)y=OV bFhw7$Fabl$!3/$=J>hMp`%>(ya$Fhw7$Fecl$!3W,=J:MEgJFhwFicl7$F_dl$!3k_g&>9;pa#Fh wFcdl7$Fidl$!3)\\)*4)*p+V1#Fhw7$F^el$!3e\"\\**G5@S'=Fhw7$Fcel$!3o1#HS \"exy;Fhw7$Fhel$!3qr/A'3**4_\"Fhw7$F]fl$!3'=&R2&G3zP\"Fhw7$Fbfl$!3e!)y ,6N#>D\"Fhw7$Fgfl$!3+u%))yV2%[6Fhw7$F\\gl$!3s\\XljlSU5Fhw7$Fagl$!3%G>9 \"pOew&*FA7$Ffgl$!3SzRt5U$[w)FA7$F[hl$!3qVe$)RYFm!)FA7$F`hl$!3=eF1'y+4 R(FA-Ffhl6&FhhlFihlFjhlFjhl-F^il6#Fahl-%+AXESLABELSG6$Q\"x6\"Q!F_im-%% VIEWG6$;FihlF`hl;$!\"&FbhlFihl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(h(x)-k(x),x=0.1..3,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 387 241 241 {PLOTDATA 2 "6&-%'CURVESG6#7edn7$$\"3/+++++ ++5!#=$!3G],k.T\"e1\"!#J7$$\"3sTN'4n`(>5F*$\"3Y]-SRoNwM 2&R5F*$\"3%4]+)yOr_NF37$$\"3/D1*G,h#f5F*$!3Y]-SRoNw\"F*$\"3R^2?=02H`F37$$\"3Oe*)f!Q!H<7F*$\"3;E1])4#*3W%F37$$ \"3.+Dc^S/P7F*FQ7$$\"3#=/EDs(zc7F*$!3jw3!z$*[s@'F37$$\"3O$e*[$R^lF\"F* Fhn7$$\"3'[7`W10jH\"F*Fhn7$$\"3ommTN(egJ\"F*F<7$$\"3[3-Q1C\"eL\"F*FY7$ $\"3+]PMxgcb8F*FY7$$\"3a\"H2$[(>`P\"F*F<7$$\"3ML3F>M2&R\"F*FQ7$$\"39vV B!4F[T\"F*FG7$$\"3n;z>h2eM9F*F<7$$\"3=e9;KWLa9F*FG7$$\"3)***\\7.\")3u9 F*FQ7$$\"3!=a)3u<%Q\\\"F*$\"3jw3!z$*[s@'F37$$\"3K$3_]W&f8:F*$\"3qv.5f_ `kEF37$$\"3&[i:g6\\L`\"F*F[r7$$\"3mm\"zpy-Jb\"F*$!3;E1])4#*3W%F37$$\"3 Y3F%zXcGd\"F*F`o7$$\"3)*\\i!*G,h#f\"F*FG7$$\"3]\"zp)*zjBh\"F*F77$$\"3J LL$3Z<@j\"F*F`o7$$\"3=1kJM]I\\;F*FG7$$\"3Mz%*z(f#\\m;F*FG7$$\"3A_DGh,o $o\"F*FQ7$$\"34DcwCx'3q\"F*F<7$$\"3%3xJ<&GCN%y\")))!#L7$$\"3NiSm0J*R!=F*F17$$\"3]Nr9p1 =@=F*F17$$\"3Q3-jK#o$Q=F*F[r7$$\"3E\"G8hzbb&=F*Fhn7$$\"3SajffLus=F*$!3 38.D\\gW?AF37$$\"3GF%zI#4$**)=F*$\"338.D\\gW?AF37$$\"3;+Dc'[=r!>F*Fhu7 $$\"3/tb/]gIC>F*$!3K_7+(>%y\")))Fht7$$\"3#fkGNh$\\T>F*Fev7$$\"3!)=<,x6 oe>F*$FinFht7$$\"3m\"z%\\S(oe(>F*F]w7$$\"3_ky(RIcI*>F*FG7$$\"3UP4YnQC5 ?F*$FdrFht7$$\"3d5S%4VJu-#F*Fft7$$\"3X$3FW**=Y/#F*Fhu7$$\"3Kc,\"zb1=1# F*FG7$$\"3YHKR@T**y?F*$\"3&y=]&HwEK8F37$$\"3M-j([o\"='4#F*Fgw7$$\"3Bv$ f$[#pL6#F*F]w7$$\"36[C%=\"obI@F*$\"3JQ/&*oWi3JF37$$\"3)4_D`PWx9#F*Fft7 $$\"3'Qf3)Q>$\\;#F*Fgw7$$\"3um;H-&>@=#F*F]w7$$\"3QJ&>]r[9?#F*Fev7$$\"3 -'RZx#zx?AF*Fhu7$$\"3Qg_ZSr5SAF*FG7$$\"3-DJ?`jVfAF*FQ7$$\"3o*)4$fcl(yA F*F<7$$\"3/a)e'yZ4)H#F*Fdx7$$\"3n=nQ\"*RUG)[?)\\`m#F*Fev7$$\"3KYh@L!zYo#F* Fft7$$\"3o5S%fC3Sq#F*$!3&y=]&HwEK8F37$$\"30v=neuLBFF*Fft7$$\"3SR(*Rrmm UFF*F]w7$$\"3K/w7%)e*>w#F*F17$$\"3Cpa&o4D8y#F*Fdx7$$\"3hLLe4Vl+GF*F]w7 $$\"3g3-))[;6?GF*F17$$\"3f$3x\"))*o&RGF*Fg^l7$$\"3eeRZFj-fGF*F17$$\"3c L3xmO[yGF*Fgw7$$\"3a3x115%z*GF*Fgw7$$\"3`$ek`M)R6$F*FG7$$\"3Q$3FpT')3:$ F*Fft7$$\"3OL3_&4,)*=$F*Fdx7$$\"3K$e9Tx:(GKF*Fft7$$\"3IL$3FXIwE$F*F]w7 $$\"3G3_+#z(3(G$F*$!3CR4vZ\"Q8m'Fht7$$\"3F$3-88XlI$F*FG7$$\"3Ee*)fqC+E LF*$FiuFht7$$\"3CLe*)4)faM$F*$\"3ac^iCIA56F37$$\"3C3F>\\r\"\\O$F*Fft7$ $\"3A$e*[)[uVQ$F*F_dl7$$\"3wekyF=$QS$F*Fft7$$\"3vLL3n\"*GBMF*FG7$$\"3p pH/HSlUMF*F_dl7$$\"310E+\"*)=?Y$F*Fft7$$\"3WSA'Hv$Q\"[$F*Fev7$$\"3$e(= #\\h[2]$F*Fgcl7$$\"3fY6%)Q$y%RNF*FG7$$\"3O/wiS'e5QF*Fcfl7$$\"3LxcBo7&*HQF*FG7 $$\"3s7`>IhJ\\QF*Fcdl7$$\"35[\\:#*4ooQF*Fft7$$\"3[$e9T&e/))QF*F_dl7$$ \"3+DJ&>I0b'RF*$\"3CR4vZ\"Q8m'Fht7$$\"3am;z\\Z'H/%F*$\"3;>_ZMAJa:F37$$ \"3y*\\PMdzZ6%F*Fcdl7$$\"3/LL3(R%f'=%F*Fcfl7$$\"3w\"z%*H5[X?%F*FG7$$\" 3$*\\i!*3=]AUF*F17$$\"353x\"[^b/C%F*Fcfl7$$\"3%o;H2A4%eUF*Fgw7$$\"3+D1 kEHOwUF*F]w7$$\"3=$3_Dj;VH%F*Fdx7$$\"3\">aj%Q.F7VF*Fev7$$\"33+]PWSAIVF *F]w7$$\"3LL3-o)Q?S%F*Fcfl7$$\"39nmm\"p`QZ%F*Fgw7$$\"3Q+DJ:&oca%F*FG7$ $\"3jL$e*QL[N?mZF*Fgcl7$$\"3u$3F>a$z%y%F*F iil7$$\"3qm\"zWc$Q.[F*Fcfl7$$\"3B]7.(et>#[F*F]w7$$\"3wLLe4OcS[F*Fgw7$$ \"3E+voaOux[F*F_dl7$$\"3xm;z*pB\\\"\\F*Fft7$$\"3yK$3-)Qkj]F*FG7$$\"3\" ***\\igSO7_F*Fgw7$$\"3il;a)))*eJ_F*Fcfl7$$\"3WK$ekr:3D&F*F]w7$$\"3G** \\PW:/q_F*Fiil7$$\"35m;HstE*G&F*Fcfl7$$\"3v**\\7G!>xK&F*Fcfl7$$\"3UL$e RoqhO&F*Fcfl7$$\"3A+](=^'R&Q&F*F]w7$$\"31n;zRBi/aF*FG7$$\"3*QL3x;[QU&F *F]w7$$\"3s+]i&*R2VaF*F]w7$$\"3an;aB)*HiaF*F]w7$$\"3OM$e9lD:[&F*Fiil7$ $\"3?,]Pz9v+bF*FG7$$\"3\"pm\"H2t(*>bF*Fcfl7$$\"3jK$3_8.#RbF*FG7$$\"3Y* *\\7j*G%ebF*Fiil7$$\"3Gm;/\"zawd&F*Fcfl7$$\"35L$e*=1)of&F*Fcfl7$$\"3%* **\\(oW1hh&F*F_dl7$$\"3wm;zuALNcF*F]w7$$\"3eL$3F5eXl&F*Fcdl7$$\"3U+]iI RytcF*F]w7$$\"3Cn;ae(4Ip&F*F_dl7$$\"32M$ekeNAr&F*Fgw7$$\"3*3+vVTh9t&F* Fgw7$$\"3sn;HUso]dF*Fcfl7$$\"3cM$3-28*pdF*Fcfl7$$\"3Q,]7)*)Q\"*y&F*F_d l7$$\"3?o;/EZO3eF*FG7$$\"3#RLeRb!fFeF*Fcfl7$$\"3an\"HdBZU!fF*FG7$$\"31 ++]lF*Fcfl7$$\"3Y;a8Zca)f 'F*Fcfl7$$\"3')=<,_$e#=mF*FG7$$\"3D@!))o0rzj'F*Fcfl7$$\"3kBVwhPodmF*Fg cl7$$\"3#\\iSmY'RxmF*Fgcl7$$\"3?Ep^r\"4rp'F*Fiil7$$\"3eGKRw=#or'F*F]w7 $$\"3(4`p7eMlt'F*Fcdl7$$\"3OLe9'GZiv'F*Fgcl7$$\"3E]i:D*[R\"pF*Fgw7$$\" 3;nm;k0lrqF*Fcfl7$$\"3#Hc^`_8!*3(F*F]w7$$\"3qek`'[wj5(F*Fcfl7$$\"3[a8s Z%RP7(F*Fgw7$$\"3D]i!*3C5TrF*Fgw7$$\"3-Y64q`YerF*FG7$$\"3!=/w7LGe<(F*F cfl7$$\"3eP4Y#H\">$>(F*FG7$$\"3OLek`Ub5sF*F_dl7$$\"3Y;aQ)41+G(F*F]w7$$ \"3c**\\7VzX\\tF*Fdx7$$\"3M&*)4V!4#oO(F*F]w7$$\"37\"z%\\lQ=%Q(F*F_dl7$ $\"3*oozm#oa,uF*Fgw7$$\"3m#ekyy4*=uF*FG7$$\"3AuVB5dj`uF*FG7$$\"3wlTgK; O)[(F*FG7$$\"3ah!*y$fCd](F*Fgcl7$$\"3JdR(\\b(3BvF*Fgcl7$$\"34`)eh^]/a( F*Fcfl7$$\"3')[PMxM\"yb(F*Fcfl7$$\"3SSNr*RRDf(F*FG7$$\"33LL3A`EFwF*Fgw 7$$\"3#3](=xzj$y(F*Fcfl7$$\"3[n;HK1,SzF*Fiil7$$\"37MeR(G$Q'4)F*FG7$$\" 3y++]Ufv_#)F*F_dl7$$\"3S++vy1x4%)F*FG7$$\"3-+++:aym&)F*Fft7$$\"3?+D1*4 Rgg)F*Fiil7$$\"3Q+]7$y#HX')F*FG7$$\"3#*\\i:D'>\\m)F*Fcfl7$$\"3W*\\(=nk a%o)F*FG7$$\"35](=#4LvI-) )F*FG7$$\"3`\\iShV$>#))F*Fiil7$$\"32*\\PM?h:%))F*FG7$$\"3g[(oa/)=h))F* Fiil7$$\"3E****\\()[\"3)))F*Fcfl7$$\"3W******zw7K!*F*Fcfl7$$\"3i****\\ s/W$=*F*Fcfl7$$\"3A+]iq'o7A*F*Fcfl7$$\"3q***\\(oo4f#*F*Fcfl7$$\"3?**\\ (o1DpH*F*Fgw7$$\"3!)*****\\E`ZL*F*Fev7$$\"3*)***\\7m4/T*F*F_dl7$$\"3)* ****\\dg1'[*F*FG7$$\"3wm\"H23z3w*F*FG7$$\"3OLeR5#pN+\"!#5FfbnFcfl7$$\"3qX'y_lx!G5FfbnFG7$$\"3 .]i!p8Zi.\"FfbnF_dl7$$\"3G^JJ2&*GQ5FfbnF_dl7$$\"3J_+sx=LS5FfbnFjbn7$$ \"3M`p7[UPU5Ffbn$!3q#yDJ7:6b&Fht7$$\"3PaQ`=mTW5FfbnFjbn7$$\"3Sb2%*))*e k/\"FfbnFG7$$\"3UcwMf8][5FfbnFgw7$$\"3YdXvHPa]5FfbnFjbn7$$\"3[e9;+he_5 FfbnFgw7$$\"3_f$o0ZGY0\"Ffbn$FddlFht7$$\"3ag_(4%3nc5FfbnF_en7$$\"3ch@Q 6Kre5FfbnFiil7$$\"3gi!*y\"eb21\"FfbnFcfn7$$\"3mkGgA.%[1\"FfbnFcfn7$$\" 3qmmTj]#*o5FfbnFcfn7$$\"3Oz%\\!>_lq5FfbnFcfl7$$\"3y\"H#ou`Qs5FfbnFiil7 $$\"3?/^JIb6u5Ffbn$\"3q#yDJ7:6b&Fht7$$\"3'o\"z%foXe2\"FfbnFjbn7$$\"3GH 2eTedx5FfbnFG7$$\"3qTN@(*fIz5FfbnF_en7$$\"3Maj%G:O53\"FfbnFcfl7$$\"3ym \"z%3jw#3\"FfbnFcfn7$$\"3?z>6kk\\%3\"FfbnF]w7$$\"3%=zW(>mA'3\"Ffbn$\"3 ipa(Q2p1L$Fht7$$\"3E/wPvn&z3\"FfbnF\\hn7$$\"3q;/,Jpo*3\"FfbnF_en7$$\"3 MHKk'3<94\"FfbnFcfn7$$\"3wTgFUs9$4\"FfbnF_dl7$$\"3=a)3zRx[4\"FfbnFjbn7 $$\"3#omTNb2m4\"Ffbn$!3ipa(Q2p1L$Fht7$$\"3EzW<4xL)4\"FfbnFgcl7$$\"3o\" H2['y1+6FfbnFcfl7$$\"3L/,W?!)z,6FfbnFcfl7$$\"3w;H2w\"GN5\"FfbnFgjn7$$ \"3=HdqJ$e_5\"FfbnF\\hn7$$\"3$=aQt[))p5\"FfbnFcfn7$$\"3Da8(Hk=(36FfbnF jbn7$$\"3omTg)z[/6\"FfbnFev7$$\"3KzpBa*y@6\"FfbnFgjn7$$\"3v\"zp)4\"4R6 \"FfbnFcin7$$\"36FfbnFiil7$$\"3nT5SK(H37\"FfbnFjbn7$$\"3KaQ.)))fD7\"FfbnFcfn7$$\" 3ummmV+HC6FfbnFG7$$\"3Z6f(fi-j7\"FfbnFjbn7$$\"3Tc^G3_JG6FfbnFgw7$$\"39 ,Wf!zF.8\"FfbnFcfn7$$\"3(ek.HPSB8\"FfbnFcfn7$$\"3\"3*G@bHNM6FfbnFcin7$ $\"3aN@_PbOO6FfbnFcfl7$$\"3F!QJ)>\"y$Q6FfbnFgjn7$$\"3@D19-2RS6FfbnFG7$ $\"3M/wPJ5W[6FfbnFjbn7$$\"3Y$e91O\"\\c6FfbnFgjn7$$\"3ei:&)*oTX;\"FfbnF G7$$\"3qT&)3>?fs6FfbnFcfn7$$\"3m'y(R,Ygu6FfbnF_en7$$\"3QJqq$=\"FfbnFcfl7$$\"3[gxGIcD#>\"FfbnFgw7$$\"3u! R]m5PS>\"FfbnFG7$$\"3+@I,$e=e>\"FfbnFjbn7$$\"3/^cPf+g(>\"FfbnFgw7$$\"3 K\"GQd`\"Q*>\"FfbnFcin7$$\"3f6457I;,7FfbnFG7$$\"3jTNY)[WH?\"FfbnFG7$$ \"3YiS\"RRq+@\"FfbnFcfn7$$\"3H$ek$*H'><7FfbnFcin7$$\"37/^\"[?AVA\"Ffbn Fjbn7$$\"3&\\il-6[9B\"FfbnFgjn7$$\"3Ab#GmeHKB\"FfbnFcfn7$$\"3[&)3*H16] B\"FfbnFgjn7$$\"3_:NNRDzO7FfbnF_dl7$$\"3yXhr:SdQ7FfbnFcfn7$$\"3419Wop8 U7FfbnFG7$$\"3imm;@**pX7FfbnF_dl7$$\"3['G&3$[awC\"FfbnF]w7$$\"3O1R+X!4 '\\7FfbnFjbn7$$\"3+ED#pgj:D\"FfbnFgw7$$\"3'e9T)o\"=ND\"FfbnFgjn7$$\"3Q &QyEHFuD\"FfbnFG7$$\"37Dc^;kLh7FfbnFcfl7$$\"3wWUVy4Hj7FfbnFjbn7$$\"3kk GNSbCl7FfbnFjbn7$$\"3]%[rA5+sE\"FfbnFjbn7$$\"39/,>kY:p7FfbnFG7$$\"3-C( 3hA46F\"FfbnFgjn7$$\"3)QMF!)yjIF\"FfbnFgjn7$$\"3_jf%*\\$=]F\"FfbnFcfn7 $$\"3S$ek=\"H(pF\"FfbnFcfl7$$\"3E.Kytu#*y7FfbnFG7$$\"3!H#=qN?)3G\"Ffbn Fjbn7$$\"3yU/i(fOGG\"FfbnFG7$$\"3ki!R&f6z%G\"FfbnF]w7$$\"3I#od9sXnG\"F fbnFcfn7$$\"3;-jP$G+()G\"FfbnFG7$$\"3-A\\HX[l!H\"FfbnF]w7$$\"3mTN@2%4E H\"FfbnFcfn7$$\"3ah@8pRc%H\"FfbnFgw7$$\"3S\"y]5`=lH\"FfbnFcin7$$\"31,% pH4t%)H\"FfbnFG7$$\"3#4-))[lF/I\"FfbnFcfn7$$\"3Wg_synL/8FfbnFjbn7$$\"3 <+Dc-fC38FfbnF_dl7$$\"3[cEs2r558FfbnFgw7$$\"3z7G)GJo>J\"FfbnF]w7$$\"35 pH/=&HQJ\"FfbnFgw7$$\"3>DJ?B2p:8FfbnFcfn7$$\"3G\"Gj$G>b<8FfbnFG7$$\"3f PM_LJT>8FfbnF_dl7$$\"3!Rf$oQVF@8FfbnFjbn7$$\"3@]P%QaNJK\"FfbnFjbn7$$\" 3_1R+\\n*\\K\"FfbnFG7$$\"3$G1kT&z&oK\"FfbnF\\hn7$$\"39>UKf\">(G8FfbnFc fn7$$\"3AvV[k.eI8FfbnFcin7$$\"3JJXkp:WK8FfbnFjbn7$$\"3i(o/[x-VL\"FfbnF cin7$$\"3$R%['*zR;O8FfbnFgjn7$$\"3C+]7&=D!Q8FfbnFgjn7$$\"3&GJXafZ\\8 FfbnFgjn7$$\"3'R4Y7i`5N\"FfbnFgw7$$\"3E]iSE[\"HN\"FfbnFcfn7$$\"3e1kcJg xa8Ffbn$!3y&4wB%>q/9FfbnFcfn7$$\"3Q/^J9Rk19FfbnFG7$$\"3l%[ Ym)ee39FfbnFG7$$\"3#\\'y(*ey_59FfbnFgw7$$\"3>X#48$)pCT\"FfbnF]w7$$\"3C D1k.=T99FfbnFcin7$$\"3H0?(fx`jT\"FfbnFjbn7$$\"3c&Q.$[dH=9FfbnF_en7$$\" 3$ewM1sP-U\"FfbnFgcl7$$\"35Yh'Hpz@U\"FfbnF_dl7$$\"3PEvHl;7C9FfbnFjbn7$ $\"3k1*GwjjgU\"FfbnFcfn7$$\"3\"pGg*4c+G9FfbnFcfn7$$\"3'pm\"H#eZ*H9Ffbn FG7$$\"3EE+6>4tJ9FfbnFgcl7$$\"3x&QGfD9NV\"FfbnFcfl7$$\"31Xnu#f(HN9Ffbn FG7$$\"3O/^cH43P9FfbnFjbn7$$\"3(QY$QmU')Q9FfbnF_dl7$$\"3aQoF9UW\"FfbnF\\hn7$$\"3Qz>6CwM^9Ffb nFcfn7$$\"3w;aQr4[e9FfbnFgcl7$$\"3eN@-Xw/i9FfbnFjbn7$$\"3Sa)e'=Vhl9Ffb nFcfl7$$\"3o8sZbwRn9FfbnFd^p7$$\"3)Hd&H#*4=p9FfbnFcfn7$$\"3]KR6HV'4Z\" FfbnF_dl7$$\"3y\"HKfmZFZ\"FfbnFcfn7$$\"3g5!p&RVJw9FfbnFgjn7$$\"3=Hd?85 ))z9FfbnFd^p7$$\"3+[C%ooZM[\"FfbnFcfn7$$\"3!o;z/O9q[\"FfbnFcfl7$$\"3I. #e,$z$*)[\"FfbnFG7$$\"3#)Rs$)*\\h3\\\"FfbnFgcl7$$\"3Kwi^p]y#\\\"FfbnFc in7$$\"3g7`>R'3Z\\\"FfbnF_dl7$$\"3))[V()3Aj'\\\"FfbnFG7$$\"3S&Q`&ydb) \\\"FfbnFcin7$$\"3\">UK#[$z/]\"FfbnF_dl7$$\"3Te9\"z\"HS-:FfbnF_dl7$$\" 3A/wi'>(45:FfbnFG7$$\"3.]PMv9z<:FfbnFcin7$$\"3/B=q9'Q;_\"FfbnFG7$$\"3% e*)fSv&[D:FfbnFgjn7$$\"3jozT$*GLH:FfbnF_dl7$$\"3lTgxK+=L:FfbnFG7$$\"3: y]X-O5N:FfbnFG7$$\"3m9T8sr-P:FfbnFgjn7$$\"3<^J\"=u]*Q:FfbnFgjn7$$\"3o( =#\\6V(3a\"FfbnFgjn7$$\"3pg-&3X@Za\"FfbnF_dl7$$\"3[L$3-fo&[:FfbnFjbn7$ $\"33Va\\Qmc]:FfbnFcin7$$\"3W_Dy'okDb\"FfbnFjbn7$$\"3!=mp]tiXb\"FfbnFg w7$$\"3=rnN$yglb\"FfbnF_dl7$$\"3a!)QkJ)e&e:FfbnFcfl7$$\"3!**)4$*zobg:F fbnFcin7$$\"3G*4=#G\\bi:FfbnF_dl7$$\"3k3_]wHbk:FfbnFjbn7$$\"35YOlp^as: FfbnFcfn7$$\"3c$3-GOP0e\"FfbnFcin7$$\"3%H>*36a`#e\"FfbnFG7$$\"3I-jPfM` %e\"FfbnFgjn7$$\"3m6Mm2:`'e\"FfbnFgcl7$$\"3/@0&fbH&)e\"FfbnFcin7$$\"3S IwB/w_!f\"FfbnF_en7$$\"3wRZ__c_#f\"FfbnF_dl7$$\"37\\=\"3qBXf\"FfbnFcfn 7$$\"3]e*)4\\<_'f\"FfbnF_dl7$$\"3BxJnXy^+;FfbnF_dl7$$\"3'fRZA%R^/;Ffbn F_dl7$$\"3L0X`!*>^1;FfbnFcin7$$\"3q9;#)Q+^3;FfbnFgjn7$$\"31C(3r330h\"F fbnFcfn7$$\"3VLeRNh]7;FfbnFG7$$\"3S&Q.BWXUh\"FfbnFcfn7$$\"3gP4@\\Z)fh \"FfbnF_dl7$$\"3!)*[=h0Cxh\"FfbnFcfl7$$\"3xTg-jLY>;FfbnFcfn7$$\"3u$fL* pE?@;FfbnFcfn7$$\"3%f9To(>%Hi\"FfbnFjbn7$$\"39)p[PG\"oC;FfbnFcfn7$$\"3 6]il!f?ki\"FfbnFcfn7$$\"34-Qc(*)f\"G;FfbnFd^p7$$\"3Ga8Z/#**)H;FfbnF\\h n7$$\"3[1*y8^Q;j\"FfbnFcfl7$$\"3XekG=yPL;FfbnF]w7$$\"3U5S>Dr6N;FfbnF_d l7$$\"3ii:5Kk&oj\"FfbnF_en7$$\"3#[64!RdfQ;FfbnF]w7$$\"3zmm\"f/N.k\"Ffb nFG7$$\"3w=U#GNu?k\"FfbnFcin7$$\"3'4xJ(fO\"Qk\"FfbnF_dl7$$\"3;B$Rm'HbX ;FfbnFcfl7$$\"39voatAHZ;Ffbn$\"3y&4wBO()3x];FfbnFG7$$ \"3[$3x6]\\Ul\"FfbnFgcl7$$\"3#=H2)Gn?h;FfbnF_dl7$$\"39+vVcR;o;FfbnFcfl 7$$\"3a3xJ\\>>$o\"FfbnFG7$$\"3%p\"z>U*>#)p\"FfbnF_en7$$\"3**ozTSp(>q\" FfbnFcfn7$$\"3.@!Q'QRt0w?/E:iQgt\"FfbnF]w7$$\"3/'*)4$o*>*REkc=Ffbn$!3#[tPp`M`m\"Fht7$$\"3o(ozOez%e=FfbnFgjn7$$\" 3mf$o![lJg=Ffbn$\"3O\"*GchvbvFFht7$$\"3UJqX7N:i=FfbnFjbn7$$\"3=.d%oZ!* R'=FfbnFcfn7$$\"3;vVBTu#e'=FfbnF_dl7$$\"3ki!*y)HvJ(=FfbnFcfl7$$\"35]PM cJ_!)=FfbnFcin7$$\"3%Q4@^3(>%)=Ffbn$\"3s#yDJ7:6b&!#M7$$\"3eP%)*Q,ry)=F fbnFdiq7$$\"3K\"yvE%\\a\"*=FfbnF`[r7$$\"31DJXr)=_*=FfbnF]w7$$\"3a7y+Hn c->Ffbn$!3s#yDJ7:6b&Fb[r7$$\"3-+Dc'e9*4>FfbnFgjn7$$\"3**\\(=(e([D%>Ffb nFdiq7$$\"3'***\\(3$H=v>FfbnF`[r7$$\"3a9;KMc,x>FfbnFihq7$$\"35H#oxL[)y >FfbnF_\\r7$$\"3oV[@T5o!)>Ffbn$!3!z/)='e!y&)QFht7$$\"3Ce9mWP^#)>FfbnFc in7$$\"3Q(oa::zh)>FfbnF\\jq7$$\"3u;zWeX%)*)>Ffbn$\"3#[tPp`M`m\"Fht7$$ \"3aJX*=Ex;*>FfbnFihq7$$\"35Y6Ml*4N*>FfbnFihq7$$\"3ogxyoEM&*>FfbnF_\\r 7$$\"3CvVBs`<(*>Ffbn$!3W/K\"3h.g*\\Fht7$$\"3#)*)4ov!3!**>FfbnF\\jq7$$ \"3R/w7z2%3+#FfbnFihq7$$\"3&*=Ud#[tE+#FfbnF`[r7$$\"3_L3-'=1X+#FfbnFjbn 7$$\"33]Pf8y;>?FfbnF^^r7$$\"3kmm;T%HQ.#Ffbn$\"3(4OQajEi5'Fht7$$\"3:$e9 '\\mlT?FfbnFcin7$$\"35+D1eQ[\\?FfbnFcfl7$$\"3-zW)4jm3#FfbnF^^r7$$\"35$3-.!*>')3#FfbnFcin7$$\"3 YiST-nd!4#FfbnFG7$$\"3!=/EX]LD4#FfbnFiil7$$\"3;@!QmI!\\%4#FfbnF_\\r7$$ \"31++v3rW'4#FfbnF^^r7$$\"3aK9Ii-A)4#FfbnF\\jq7$$\"3ekG&eT$***4#FfbnFc ]r7$$\"32(H/%plw,@FfbnFdiq7$$\"3bHd&HsRN5#FfbnFcfn7$$\"3fhr]wGJ0@FfbnF iil7$$\"32%fe+.'32@FfbnFG7$$\"3bE+h$=f)3@FfbnF]w7$$\"3fe9;PBj5@FfbnFci n7$$\"3j!*Gr!\\0C6#FfbnF\\jq7$$\"36BVEW'yT6#Ffbn$\"3^'=\\5)\\>@FfbnF cfl7$$\"3;_+Ze7F@@FfbnFdiq7$$\"3k%[@?TWI7#FfbnF_\\r7$$\"3o;Hdlv\"[7#Ff bnF_\\r7$$\"3sX'y(z,\">8#FfbnFdiq7$$\"3wuV)Rz-!R@FfbnF_dl7$$\"3GRs3,\" \\D9#FfbnF]w7$$\"3!Q5!>3a4Y@FfbnFcbr7$$\"3yoHH:>#FfbnFcbr7$$\"3;ekyEEr*>#Ffbn$\"3#\\QK&Gp6\\!)Fht7$$\"3bVB:x'*e.A FfbnF]w7$$\"3#*G#=vsmu?#Ffbn$!3c$\\ppOfNF&Fht7$$\"3$=<,FD0%4AFfbnFgjn7 $$\"3u9T)yxV8@#FfbnF]w7$$\"3?dq1.BG8AFfbnFiil7$$\"36++DG3A:AFfbnFcfn7$ $\"3@\\VPH[/Fht7$$\"3)p /B;$Gp?AFfbnFiil7$$\"34'RZF$o^AAFfbnF]w7$$\"3vW<(Q$3MCAFfbn$\"3?-mS0=+ )\\#Fht7$$\"3'Q4'*\\$[;EAFfbn$\"3&R-%4$H!*G%>Fht7$$\"3'HW?h$)))zA#Ffbn F`[r7$$\"3i\"zWs$G\")HAFfbnFcin7$$\"3tS\"p$QojJAFfbnFcfl7$$\"3%)*[$\\R 3YLAFfbn$!3O\"*GchvbvFFb[r7$$\"3]QyhS[GNAFfbnF_dl7$$\"3h(=U<%)3rB#Ffbn F\\hn7$$\"3P&)3*R%ovSAFfbn$!3[!=>xJ8J0$Fht7$$\"3f$eRi%[SWAFfbnFbir7$$ \"35vVBbo**eAFfbnFa[s7$$\"3im\"HU'))etAFfbn$\"3-PVMUjLjTFht7$$\"3?DJqx ?SxAFfbn$\"31u')o%osmK)Fb[r7$$\"3K$3x6H:7G#FfbnF_\\r7$$\"3iiS\"z*=7$G# FfbnFcfn7$$\"3YT5l/&G]G#Ffbn$\"3%=2#Gz3nGeFht7$$\"3u?!)Q6^$pG#Ffbn$\"3 I:playW=ZFht7$$\"3/+]7=<%))G#FfbnF^^r7$$\"3Kz>'[K[2H#FfbnF\\hn7$$\"3ge *)fJ\\l#H#FfbnFG7$$\"3YPfLQ:c%H#FfbnF\\hn7$$\"3u;H2X\"okH#Ffbn$F]jqFb[ r7$$\"3-'*)4=vu$)H#FfbnFcfn7$$\"3)[(oae8G+BFfbnF``s7$$\"33WBFfbnF_^s7$$\"3;YOlKSD@BFfbnFcfl 7$$\"3XD1RR1;BBFfbnFj]s7$$\"3H/w7Ys1DBFfbnFcfn7$$\"3e$ekG&Q(pK#FfbnFj] s7$$\"3(Gc,'f/))GBFfbnFf[s7$$\"3rT&Qj1(yIBFfbnFcbr7$$\"3+@b2tOpKBFfbnF ^^r7$$\"3H+D\")z-gMBFfbnF__s7$$\"3'4EvHRG$QBFfbn$\"3[!=>xJ8J0$Fht7$$\" 3=@!Qh]c?M#FfbnFa[s7$$\"3_,%>Fc?RM#FfbnF\\jq7$$\"3U\"y+$>YyXBFfbnF\\jq 7$$\"3uh@)en[wM#FfbnF`as7$$\"33UNYKF^\\BFfbnF]w7$$\"3)>#\\/*yw8N#FfbnF a[s7$$\"3K-jiX3C`BFfbnF_\\r7$$\"3k#o2A!\\5bBFfbnFcfn7$$\"3ai!*ye*opN#F fbnFd\\s7$$\"3)GWq`,L)eBFfbnFG7$$\"3?B=&>2(pgBFfbnF``r7$$\"35.K`G6ciBF fbn$!3oX9y!yyxQ\"Fht7$$\"3W$e9^=DWO#FfbnFa[s7$$\"3yjfpT#*GmBFfbnF\\jq7 $$\"3mVtF)H`\"oBFfbnFa[s7$$\"3+C(e[N<+P#FfbnF\\jq7$$\"3M/,W69)=P#FfbnF diq7$$\"3A%[@!oautBFfbn$!31u')o%osmK)Fb[r7$$\"3ckGgC&4cP#FfbnF_]s7$$\" 3!\\C%=\"etuP#FfbnFgw7$$\"3yCcwPwLzBFfbnF`as7$$\"370qM%p,7Q#Ffbn$Fe\\s Fht7$$\"3Y&QG4vlIQ#Ffbn$!35]Yf\"R#y$Q'Fht7$$\"3Ml(4v!)H\\Q#FfbnF^^r7$$ \"3oX64kQz'Q#FfbnFfgs7$$\"3-EDn?zl)Q#FfbnFcin7$$\"3!f!RDx>_!R#FfbnFcin 7$$\"3C'GNQ.'Q#R#FfbnF``s7$$\"3emmT!4]UR#FfbnFcin7$$\"39JXRzI:)R#FfbnF ``s7$$\"3r&Rs$og0-CFfbnFbfs7$$\"3Gg-Nd!ffS#FfbnF_^s7$$\"3%[7Gj/i)4CFfb nF\\ds7$$\"3(R&QGC!owT#FfbnF\\hn7$$\"34$eRA+uaU#FfbnFher7$$\"3mZu@\"*p PHCFfbnFiil7$$\"3A7`>!)*zKV#FfbnFa[s7$$\"3sWUou9BNCFfbnF_dl7$$\"3ywJ

*)[CFfbnF_]s7$$\"3[NY3D\\z_CFfbnF``s7$$ \"3/+D19zpcCFfbnFa[s7$$\"3sjf>NudeCFfbnFbfs7$$\"3'pUHj&pXgCFfbnFcfn7$$ \"3k!*GYxkLiCFfbnF``r7$$\"3Kajf)*f@kCFfbnFj]s7$$\"3c<)H(>b4mCFfbnFcfl7 $$\"3C\"Gj3/vzY#Ffbn$\"3c$\\ppOfNF&Fht7$$\"3#\\u'*>ca)pCFfbnF^^r7$$\"3 ;3-8$3Mj\\ %3WGUg)Fht7$$\"3M*)4$*47,$[#Ffbn$\"3we<.I[A3OFht7$$\"3q;z>_-x'[#FfbnF \\jq7$$\"3)4xJnL)G%\\#FfbnFcfn7$$\"3DDcE@k!=]#FfbnF_dl7$$\"3h_D`jac0DF fbnF`[r7$$\"3_z%*z0XK4DFfbnF\\hn7$$\"3W1k1[N38DFfbnF]w7$$\"3!QLL.fUo^# FfbnFd\\s7$$\"3t_v:sYw=DFfbnF``s7$$\"3Ar<)Rv'o?DFfbn$\"3QGs!R!R*)QpFht 7$$\"3r*)f!e$)3E_#FfbnF``r7$$\"3l3-j<4`CDFfbnFher7$$\"3eFWX**HXEDFfbnF `[r7$$\"33Y'y73v$GDFfbnFgw7$$\"3dkG5jrHIDFfbnF]w7$$\"3]$3F\\C>A`#FfbnF ``r7$$\"3W-8vE89MDFfbn$!3-PVMUjLjTFht7$$\"3$4_v&3M1ODFfbnFcfn7$$\"3UR( *R!\\&)z`#FfbnFiil7$$\"3NeRAsv!*RDFfbn$\"3k1)>iT0S\\(Fht7$$\"3A'Rset^P a#FfbnFj^s7$$\"3mL3_**efZDFfbnF]w7$$\"33r#pJ1S9b#FfbnFa_t7$$\"3%*3x\"o A%GbDFfbnFa[s7$$\"3)y#>k3j?dDFfbnF__s7$$\"3QYhY!RG\"fDFfbnF_dl7$$\"3'[ O!Hs/0hDFfbnF^^t7$$\"3!Qe9TbsHc#FfbnFjhr7$$\"3A@IwpOHJg#FfbnFd\\s7$$\"3F)>OH`N]g#FfbnFcbr7$$\"3_uC&*)pTpg#FfbnFf[ s7$$\"3K](o\\'y%)3EFfbnF_^s7$$\"3O-8+(>gEh#FfbnFfgs7$$\"3'[&Q.HDZ;EFfb n$!3I:playW=ZFht7$$\"35J,0&py$=EFfbnFcfl7$$\"3!pSm5'[G?EFfbnFcfn7$$\"3 q#o#3F5>AEFfbnFc]r7$$\"3&*e*)4$>(4CEFfbnFdiq7$$\"3*4^J^_4zi#FfbnFa[s7$ $\"3/jS;d=sJEFfbnFiil7$$\"3HR.=B!GOj#FfbnF_\\r7$$\"34:m>*=Mbj#Ffbn$\"3 A^zH%H$yKwFht7$$\"3*3*G@b.WPEFfbnFcfn7$$\"38n\"H7_Y$REFfbnFcfl7$$\"3K% fL%e'\\Gk#Ffbn$\"3yYZ[$oznj#Fht7$$\"3]@!Qcz_jk#Ffbn$\"3/Dtz&>\"*=>$Fht 7$$\"3E[C%G$f&)\\EFfbnFcfn7$$\"3Wvo/q!fLl#Ffbn$!3]o@yrw2(Fht7$$\"3/#H#onTQ\"o#FfbnFh^u7$$\"3A>n)[I()[o#FfbnFcin7$ $\"3'f9\"4U/R)o#Ffbn$!3C!ff)embE:Fht7$$\"39tbHzN*=p#FfbnFcfn7$$\"3M++] ;nR&p#FfbnFcin7$$\"3]_Dy6US(p#FfbnF\\\\u7$$\"3o/^129q#FfbnFjbn7$$\"3g3-j(pEMq#Ffbn$\"3kd%Gt#RAfBFht7$$\"3]7`>)oTuq #FfbnF\\jq7$$\"3&oTg(ymX6FFfbnFh^u7$$\"35D1*)fm[>FFfbnF\\bu7$$\"3NL3-T m^FFFfbnF_\\r7$$\"3`&Q.j8C&HFFfbnFg[u7$$\"3rPfeJ;`JFFfbnF_\\r7$$\"3W*[ oo7RNt#FfbnFG7$$\"3hT5:AmaNFFfbnFg[u7$$\"3y$fLu6avt#FfbnFbit7$$\"3^Xhr 7;cRFFfbnF\\ds7$$\"3p(p)*z5p:u#FfbnFcin7$$\"3')\\7G.mdVFFfbn$!3#f.T'Ra L9HFht7$$\"3/-Qc)4%eXFFfbnFjbn7$$\"3Aaj%Qf\"fZFFfbn$\"3f\"[A/A9@I%Fht7 $$\"3Q1*G\"*3*f\\FFfbn$\"3Qze,:C6/=Fht7$$\"3ce9T%e1;v#FfbnF_eu7$$\"3u5 SpzSh`FFfbnFcfn7$$\"3!Hcw\\d@cv#FfbnFcin7$$\"3k9\"f-2Hwv#FfbnF``r7$$\" 3#omTbcO'fFFfbn$\"35,Lq-4+\\7Fht7$$\"3w$eRsy+Sx#FfbnFj]s7$$\"3G+v$*3]O )y#Ffbn$\"3fIQJ4d*oV*Fht7$$\"3z;ajI#HF!GFfbnF^at7$$\"3ILLL_M4GFfbn$\"3RGs!R!R*)QpFb[r7$$\"34c,TmA#4#GF fbnF_\\r7$$\"3qn&[MnOG#GFfbn$!3t>,Zl9X9(*Fb[r7$$\"3()yp[!3^Z#GFfbn$\"3 t>,Zl9X9(*Fb[r7$$\"3\\!RDv[lm#GFfbn$!3qX9y!yyxQ\"Fb[r7$$\"35-Qc%*)z&GG FfbnFjbn7$$\"3F8Ag,V\\IGFfbn$\"3i=owi0Xl%)Fht7$$\"3)[iS'3(3C$GFfbnFa[s 7$$\"3*3F%zOj1SGFfbnFcbr7$$\"3Y;z%\\'RsZGFfbn$\"35]Yf\"R#y$Q'Fht7$$\"3 qRZ-zFb^GFfbnFjdu7$$\"3[i:5$f\"QbGFfbnFfjr7$$\"3E&QyrS5#fGFfbnF\\ds7$$ \"3[3_D@#RI'GFfbn$\"3EFR?,I*)*o&Fht7$$\"3m>OHGO&\\'GFfbnF`fu7$$\"3GJ?L N!oo'GFfbn$\"37,Lq-4+\\7F37$$\"3)GWqBW#yoGFfbnFjiu7$$\"31a)3%\\opqGFfb nF]hu7$$\"3mlsWc7hsGFfbnFfjr7$$\"3Gxc[jc_uGFfbnFh^u7$$\"3Y)3C02Sk(GFfb nF_^s7$$\"31+DcxWNyGFfbn$\"3?9O&>&pWpMFht7$$\"3w\\7.KJ,$*GFfbnFa[s7$$ \"3\"*****\\'yrw!HFfbnF_eu7$$\"3o$f3$=T]4HFfbnFdhs7$$\"3X(=<,XO8\"HFfb nF\\jq7$$\"3A\"yD>yoJ\"HFfbnFfjr7$$\"3)\\PMP6,]\"HFfbn$!3f\"[A/A9@I%Fh t7$$\"3voHaXM$o\"HFfbnF`fu7$$\"3_i:Nxdm=HFfbnFd\\v7$$\"3Gc,;4\")\\?HFf bn$\"3a0l^8X+XiFht7$$\"30](o4WIB#HFfbn$\"3qX9y!yyxQ\"Fb[r7$$\"3#QMxFxi T#HFfbn$!3?9O&>&pWpMFht7$$\"3fPfe/^*f#HFfbnF\\\\u7$$\"3OJXROu#y#HFfbn$ \"30u')o%osmK)Fht7$$\"37DJ?o(f'HHFfbnF\\jq7$$\"3*)=<,+@\\JHFfbnFe_v7$$ \"3m7.#=VCL$HFfbnF_]s7$$\"3V1*GOwc^$HFfbnF]_u7$$\"3?+vV&4*)p$HFfbnFg[u 7$$\"3lcE(pFF4%HFfbnFj]s7$$\"3n7y]ea'[%HFfbn$\"3NSUX]!R.\"zFht7$$\"3oo H/SO!)[HFfbnFc^v7$$\"39D\"y:#=u_HFfbnFd\\v7$$\"3Q.dM74raHFfbnF\\\\u7$$ \"3g\"G8J+!ocHFfbn$\"3]H0h1[*y=)Fht7$$\"3%)f3)Q4\\'eHFfbnFiil7$$\"3iP% [Y==1'HFfbnFd\\v7$$\"3U:gTvseiHFfbnFd\\v7$$\"3k$f$=mjbkHFfbnF^at7$$\"3 )=<^pXDl'HFfbn$\"3E%3'e&3M3/\"F37$$\"35](=xa%\\oHFfbnF``r7$$\"3MGj[QOY qHFfbn$\"3]o@mUYeX-%Fht7$$\"3ei!*y54PwHFfbnF^at7$$\"3g=UK#44.)HFfbnFa[s7$$\"31v$f QFZU)HFfbnFh^v7$$\"3_JXRba=))HFfbnF``r7$$\"3_(oHpjB@*HFfbnFiil7$$\"3Il spFF4%*HFfbnFcfn7$$\"3aV[Y==1'*HFfbn$\"3C!ff)embE:Fht7$$\"3w@CB44.)*HF fbnFi\\u7$$\"\"$FH$\"39Qw/XsL7aFht-%'COLOURG6&%$RGBGFGFG$\"*++++\"!\") -%+AXESLABELSG6$Q\"x6\"Q!Fgfv-%%VIEWG6$;$\"\"\"!\"\"Fhev%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The follo wing loop uses random numbers between -3 and 3 to test whether the exp ressions " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "k(x)" "6#-%\"kG6#%\"xG" }{TEXT -1 48 " produce value s which are essentially the same. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "randomize():\nfor i to 100 \+ do \n xx:= Float(rand(),-12)*6-3;\n a := evalf(evalf(h(xx),12));\n b : = evalf(evalf(h2(xx),12));\n tst := testfloat(a,b,1);\n if tst<>true then print(a,b,tst) end if;\nend do: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+CElk!*!#6$!+EElk!*F%7%%&falseG$\"\"#\"\"!%&ulps~G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "g:=x->(x^2*(27+3*sqrt(3*x^8+81)))^ (1/3)/(3*x)-x^3/((x^2*(27+3*sqrt(3*x^8+81)))^(1/3)):\n'g(x)'=g(x);\nde rivplot(g(x),x=-3..3,-5..5,discont=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*(\"\"$!\"\"*&)F'\"\"#\"\"\",&\"#FF/*&F *F/,&*&F*F/)F'\"\")F/F/\"#\")F/#F/F.F/F/#F/F*F'F+F/*&F'F*F,#F+F*F+" }} {PARA 13 "" 1 "" {GLPLOT2D 353 401 401 {PLOTDATA 2 "6#-%(ANIMATEG6J7+- %'CURVESG6%7gn7$$!\"$\"\"!$!3:;XmB'**GS(!#>7$$!3*GyIw`3Y$H!#<$!3_N4\"G [bz!zF17$$!3Ew&pex6x(GF5$!3')\\!*[DxJ&Q)F17$$!3;\\Di;as8GF5$!3Obt`qa** o*)F17$$!3?q8D9\\J\\FF5$!3\"H_A$o0F7'*F17$$!3yyXvV0@&o#F5$!373Hg$Gl9. \"!#=7$$!3sWMP'exdi#F5$!3:F>(*G\\y-6FL7$$!3[Bl\\,#QUc#F5$!375!GVdpO=\" FL7$$!3p6Qi\"3%f+DF5$!3Y!)zx8FL7$ $!3m.iv=#)*=P#F5$!33+%4=lzG\\\"FL7$$!3ie67I3U9BF5$!355*G.pGbg\"FL7$$!3 Sq0+/\\r\\AF5$!3/mE:&>\")fu\"FL7$$!3S808*GVZ=#F5$!3s,7!e]jM!>FL7$$!3%y tb#*4J@7#F5$!3ow^Ok4&H2#FL7$$!3Q`W\\KKFl?F5$!3>Lfr?j)QC#FL7$$!3ksY]HPm (*>F5$!3q$)\\WOb(4Z#FL7$$!3y?#>@h*QS>F5$!3'3c$GbQ3'o#FL7$$!3ynu(y>mP(= F5$!3i_!)RCDxlHFL7$$!3t8/P(=$z9=F5$!3*Qn1nJ\\EC$FL7$$!3=>[7[/4]C;F5$!3_* >,N=)zaVFL7$$!3%[o%*4cd^c\"F5$!3)3!>Dy*)[sZFL7$$!3lQqur2[,:F5$!3cL!e(p Q5i_FL7$$!30BVv$[Q`V\"F5$!3e*G#ejS)4\"eFL7$$!3=x%>wUhxP\"F5$!3h__2/NV= jFL7$$!3pY)='Hmd:8F5$!3c!Re,Th@*oFL7$$!3)HL(\\[OL^7F5$!3-?x:VkF1vFL7$$ !3IJI([U%[)=\"F5$!3\"=;P9d@97)FL7$$!30^'p$pXnF6F5$!3=)Q3t**zXs)FL7$$!3 1*oHEfb,1\"F5$!3ylC\"R`4&)R*FL7$$!3Ko-,!*y'[***FL$!3\"y+**4@80+\"F57$$ !3q-eI;*)4Z$*FL$!3k8;c'>*Hl5F57$$!3Ua(H([R7g()FL$!3N\"[WQ)f5C6F57$$!3 \"p=j(o_S=\")FL$!3Es)[v5!z)=\"F57$$!3!p!3A,!)f9vFL$!3irZfv^Z]7F57$$!3G 5J[FaW$)oFL$!3?Y'\\^I.lJ\"F57$$!3uWsYA%yjE'FL$!3%4e[>(4a$Q\"F57$$!37p# 4]Xm.i&FL$!3'fC(>q^yd9F57$$!3MRO+],=)*\\FL$!3T&4tUEh^`\"F57$$!3#>>w()y />O%FL$!3uN^()oS9B;F57$$!3;.#)y3\")*3t$FL$!3;Rw'oCGOs\"F57$$!3]iyp.&o5 :$FL$!3$>WxF)fiL=F57$$!3Okp_U$=l[#FL$!3!*G*=G6AL*>F57$$!3%)Hd@cn8#*=FL $!3o2Dw48=*=#F57$$!3G&>LP,-%e7FL$!3$*zo^x>77DF57$$!3/7w*33&>^&*F1$!3o& ='e!e\"=bFF57$$!3TrKYC+P=lF1$!3,t*4?Sn,8$F57$$!3[`uMowx))[F1$!3WKAh/A7F1$!3AGCgznopaF57$$!3C)3z 0ynz9)!#?$!3o?0$GtM7E'F57$$!3#eJMaLx46'Fj\\l$!3%4a2`/#Q\"*oF57$$!3GW&* G!*o)R2%Fj\\l$!3'H^PODj'))yF57$$!3HsZ9Xk*p.#Fj\\l$!3lFR%Rv)3R**F57$$!3 '***************f!#E$!3&3N1Nu7O$p!#:7gn7$$\"3'***************fF_^l$\"3 &3N1Nu7O$pFb^l7$$\"3NL!Q!*>*[V?Fj\\l$\"3n=!y]8^&G**F57$$\"3Qmg2)Rsp3%F j\\l$\"3'pn*>>&*H!)yF57$$\"3U*49rfb/8'Fj\\l$\"3'Q/>mjvS)oF57$$\"3YK@:' zQR<)Fj\\l$\"3_\")oq.kfaiF57$$\"3')>GU>04E7F1$\"3oK6B=w)QY&F57$$\"3YE/ BfryM;F1$\"3oAF)Q%>Ek\\F57$$\"3/Sc%)Q/=_CF1$\"3enS!**)=lOVF57$$\"3C`3Y =PdpKF1$\"3w$f2EOh+%RF57$$\"3O!G\"px-O/\\F1$\"3HKxfzn!=W$F57$$\"3z1<#p $o9RlF1$\"3@!3Xf([%o7$F57$$\"3kI?)\\#\\)RQ*F1$\"3=/')R[4]rFF57$$\"3XNU I,B)GA\"FL$\"3fI$H)4$oj`#F57$$\"353Xx$*eui=FL$\"3$>&R0$fe3?#F57$$\"3^) H'[<4&o]#FL$\"3#[1S87$p()>F57$$\"3O5UXAY*y9$FL$\"3A*poe,!HM=F57$$\"39^ bE'>CAu$FL$\"3\\l8u.Fn@Qa)pN:F57$$\"3i\"3$\\n$f%GcFL$\"3#QX5#)p@oX\"F57$ $\"3/lzVsy,\"G'FL$\"3M%=6>&H\">Q\"F57$$\"3_;%)ye8F57 $$\"3#))H%**>5&G](FL$\"3SonX-ro^7F57$$\"3Il[porc_\")FL$\"3ij,xf\"H`=\" F57$$\"35AEWn!*oy()FL$\"3'>#QnIDCA6F57$$\"3[ka0NxEZ$*FL$\"3Al()G0BGl5F 57$$\"3UF`\\wiL-5F5$\"3C#o&>Nsjw**FL7$$\"3Ez2)QR5'f5F5$\"3^c)[2=fRS*FL 7$$\"3FKD73QBE6F5$\"3E_2#)o)G*Q()FL7$$\"3K'eH'=o?&=\"F5$\"37Wdr/_x`\") FL7$$\"3)3=vyb4*\\7F5$\"3(QEu@@#3?vFL7$$\"3Nc>]m=_68F5$\"3J<]K+ANIpFL7 $$\"3K%Q)*p%y!eP\"F5$\"3V^?w4<3OjFL7$$\"3@:`+XC%[V\"F5$\"33+%)**)[W_\" eFL7$$\"3ShHDM#>&)\\\"F5$\"37-bi++$eG&FL7$$\"3+xcCA:mk:F5$\"3_yD7>\"[h x%FL7$$\"3)G_!Qy&QAi\"F5$\"3])\\w&)\\`!oVFL7$$\"3O`6QwLU%o\"F5$\"3Q')) e`^*4lRFL7$$\"33nE]djm[sHFL7$$\"3*4JqLTW)R>F5$\"3%Q 8MvlR#)o#FL7$$\"3At*)*p@80+#F5$\"3)oH4d0S3Y#FL7$$\"3y>%pV6!Hl?F5$\"3.A OuoH$QC#FL7$$\"3hCq76w)R7#F5$\"3+QGaLPln?FL7$$\"3O\"oB\"z%f\")=#F5$\"3 m;a9tpt%*=FL7$$\"3O>z(e?S&[AF5$\"3A;([Dvv'[=+DF5$\"3cLS(R#QRw7FL7$$\"3'3QAr_4Qc#F5$\" 3Y2p)Q0gU=\"FL7$$\"3uz67&>5pi#F5$\"3r_#Q:yl85\"FL7$$\"3!Q@Ic:$*[o#F5$ \"32ZPD*oH=.\"FL7$$\"3i.tur\"[8v#F5$\"3e*[aFnW5f*F17$$\"32F%y.L'y5GF5$ \"3)[M3Nb$3(**)F17$$\"3_!oEY!)fT(GF5$\"3Q7Sa_QP;%)F17$$\"3Mn`v0j\"[$HF 5$\"3e,p*>Lzi!zF17$$\"\"$F.$\"3:;XmB'**GS(F1-%'COLOURG6&%$RGBG$\"*++++ \"!\")$F.F.Fbam-F(6$7#7$F,$!3GEF1'y+4R(F1-F\\am6&F^amFbamFbamF_am-%'PO INTSG6%7$F,$!)(**GS(!\"*-%'SYMBOLG6#%'CIRCLEG-F\\am6&F^amF.F.F.-F\\bm6 %F^bm-Fcbm6#%(DIAMONDGFfbm-F\\bm6%F^bm-Fcbm6#%&CROSSGFfbm-F\\bm6%7$F,$ !)3!4R(FabmFbbmFiam-F(6%7$7$$!+94T*f$Fabm$!+(48F(H!#67$$!+'3*e+CFabm$! +I'3L=\"!#5-%*THICKNESSG6#\"\"#Ffbm-%+AXESLABELSG6%%\"xG%!G-%%FONTG6#% (DEFAULTG-%%VIEWG6$;F,Fg`m;$!\"&F.$\"\"&F.7+F'-F(6$7%Ffam7$F3$!3UMFs2r ao!)F17$F9$!3b=(QZ\"*3>s)F1Fiam-F\\bm6%7$$!+YQ:YGFabm$!)Cnm')FabmFbbmF fbm-F\\bm6%FhfmFjbmFfbm-F\\bm6%FhfmF_cmFfbm-F\\bm6%7$Fifm$!)hh7\"*Fabm FbbmFiam-F(6%7$7$$!+A!f_W$Fabm$!+HpD2KF_dm7$$!+q'[qC#Fabm$!+(y3ET\"Fed mFfdmFfbmFjdmFcem7+F'-F(6$7'FfamF`fmFcfm7$F>$!3BU*GU=Ip`*F17$FC$!3/)4K .;ma/\"FLFiam-F\\bm6%7$$!+#p2Bp#Fabm$!*4hL-\"FabmFbbmFfbm-F\\bm6%F_imF jbmFfbm-F\\bm6%F_imF_cmFfbm-F\\bm6%7$F`im$!*7FabmFbbmFfbm-F\\ bm6%Fi[nFjbmFfbm-F\\bm6%Fi[nF_cmFfbm-F\\bm6%7$Fj[n$!*)p&GV\"FabmFbbmFi am-F(6%7$7$$!+IkDOJFabm$!+$[NKj$F_dm7$$!+[mmS>Fabm$!+7FVw?FedmFfdmFfbm FjdmFcem7+F'-F(6$7,FfamF`fmFcfmFghmFjhmF^[nFa[nFd[n7$FX$!3+t0p9c,?:FL7 $Fgn$!3'Qo#RI`&4o\"FLFiam-F\\bm6%7$$!+&Q:YQ#Fabm$!*$3c#FedmFfd mFfbmFjdmFcem7+F'-F(6$7/FfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^n7$F\\o $!3?L0hzg\\o=FL7$Fao$!318Kx[Hza?FL7$Ffo$!3]AY!HeP:H#FLFiam-F\\bm6%7$$! +J#p2B#Fabm$!*i/,z\"FabmFbbmFfbm-F\\bm6%Fj`nFjbmFfbm-F\\bm6%Fj`nF_cmFf bm-F\\bm6%7$F[an$!+Fk#oO#FedmFbbmFiam-F(6%7$7$$!+5#4[#GFabm$!**4:TQFed m7$$!+_#Hnj\"Fabm$!+TT4'>$FedmFfdmFfbmFjdmFcem7+F'-F(6$71FfamF`fmFcfmF ghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`n7$F[p$!3u\">/N,G?c#FL7$F`p$!35= /FAX)z&GFLFiam-F\\bm6%7$$!+xI#p2#Fabm$!*4Zu?#FabmFbbmFfbm-F\\bm6%FacnF jbmFfbm-F\\bm6%FacnF_cmFfbm-F\\bm6%7$Fbcn$!+ZDj&4$FedmFbbmFiam-F(6%7$7 $$!+7Q$om#Fabm$!*>-I\"QFedm7$$!+UB,([\"Fabm$!+hRfLSFedmFfdmFfbmFjdmFce m7+F'-F(6$74FfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF \\cn7$Fep$!3%)p'o;Ee.;$FL7$Fjp$!3y!Ra2bTgc$FL7$F_q$!3-Y-R[HA_RFLFiam-F \\bm6%7$$!+Bp2B>Fabm$!+/@ebFFedmFbbmFfbm-F\\bm6%F[fnFjbmFfbm-F\\bm6%F[ fnF_cmFfbm-F\\bm6%7$F\\fn$!+I/+xSFedmFbbmFiam-F(6%7$7$$!+:F*e]#Fabm$!* q=Wz$Fedm7$$!+J6ES8Fabm$!+QBsJ^FedmFfdmFfbmFjdmFcem7+F'-F(6$76FfamF`fm FcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfen7$Fdq $!3Mk0^ptJ`WFL7$Fiq$!3sG'*4.(*)Q%\\FLFiam-F\\bm6%7$$!+q2Bp%Fedm7$$!+:(fz>\"Fabm$! +,**RMlFedmFfdmFfbmFjdmFcem7+F'-F(6$79FfamF`fmFcfmFghmFjhmF^[nFa[nFd[n Fh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF]hn7$F^r$!3naA6%R^*HbFL7 $Fcr$!3%4Xo/i`o7'FL7$Fhr$!3c$*4jIM>vnFLFiam-F\\bm6%7$$!+;YQ:;Fabm$!+_f '[T%FedmFbbmFfbm-F\\bm6%F\\[oFjbmFfbm-F\\bm6%F\\[oF_cmFfbm-F\\bm6%7$F] [o$!+2n)['oFedmFbbmFiam-F(6%7$7$$!+tj;q@Fabm$!*T?N1'Fedm7$$!+fGgg5Fabm $!+j)zLA)FedmFfdmFfbmFjdmFcem7+F'-F(6$7;FfamF`fmFcfmFghmFjhmF^[nFa[nFd [nFh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjn7$F]s$! 3![*>TkFabm$!+z:B/6Fedm7$$!+:If_#*Fedm$ !+/.O25FabmFfdmFfbmFjdmFcem7+F'-F(6$7>FfamF`fmFcfmFghmFjhmF^[nFa[nFd[n Fh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]o7 $Fgs$!3)[,pYaRoe)FL7$F\\t$!3mAH0@B+F!*FL7$Fat$!3IM,m]453%*FLFiam-F\\bm 6%7$$!+3Bp28Fabm$!+M+]mpFedmFbbmFfbm-F\\bm6%F]`oFjbmFfbm-F\\bm6%F]`oF_ cmFfbm-F\\bm6%7$F^`o$!+Df%*[%*FedmFbbmFiam-F(6%7$7$$!+S^kH=Fabm$!+XffM ?Fedm7$$!+cZRdyFedm$!+7/%)*=\"FabmFfdmFfbmFjdmFcem7+F'-F(6$7@FfamF`fmF cfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF] hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_o7$Fft$!3SWn'4'e\">p*FL7$F[u$!3_nc:A) [x')*FLFiam-F\\bm6%7$$!+ah%Q:\"Fabm$!+b(GVY)FedmFbbmFfbm-F\\bm6%FdboFj bmFfbm-F\\bm6%FdboF_cmFfbm-F\\bm6%7$Febo$!+HVmF**FedmFbbmFiam-F(6%7$7$ $!+/jKp;Fabm$!+b[\"oM$Fedm7$$!+P+m$Q'Fedm$!+mU=e8FabmFfdmFfbmFjdmFcem7 +F'-F(6$7BFfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF\\c nF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_oF\\boF_bo7$F`u$!3 QvAQi_`e**FL7$Feu$!3;@yI0![c***FLFiam-F\\bm6%7$$!+,+++5Fabm$!+')****** **FedmFbbmFfbm-F\\bm6%F[eoFjbmFfbm-F\\bm6%F[eoF_cmFfbm-F\\bm6%7$F\\eo$ !+%*********FedmFbbmFiam-F(6%7$7$$!+cd\\9:Fabm$!+NC/b[Fedm7$$!+cC/b[Fe dm$!+ad\\9:FabmFfdmFfbmFjdmFcem7+F'-F(6$7EFfamF`fmFcfmFghmFjhmF^[nFa[n Fd[nFh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF ^]oFb_oFe_oFh_oF\\boF_boFcdoFfdo7$Fju$!3kW.F++++5F57$F_v$!32;ty+pb+5F5 7$Fdv$!3paC:(=CQ+\"F5Fiam-F\\bm6%7$$!*ZQ:Y)Fabm$!+@s7a6FabmFbbmFfbm-F \\bm6%FegoFjbmFfbm-F\\bm6%FegoF_cmFfbm-F\\bm6%7$Ffgo$!+%HEt+\"FabmFbbm Fiam-F(6%7$7$$!+%4^'f8Fabm$!+F%z'ojFedm7$$!+0gcELFedm$!+*\\'Qr;FabmFfd mFfbmFjdmFcem7+F'-F(6$7GFfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb `nFe`nFibnF\\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_oF\\ boF_boFcdoFfdoFjfoF]goF`go7$Fiv$!3/y&R)3xY85F57$F^w$!3g6U.\\IYJ5F5Fiam -F\\bm6%7$$!*$p2BpFabm$!+*e$H78FabmFbbmFfbm-F\\bm6%F\\joFjbmFfbm-F\\bm 6%F\\joF_cmFfbm-F\\bm6%7$F]jo$!+WC(41\"FabmFbbmFiam-F(6%7$7$$!+#=a%)> \"Fabm$!+66'Gv(Fedm7$$!+O?hh=Fedm$!+n5I\\=FabmFfdmFfbmFjdmFcem7+F'-F(6 $7JFfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFc enFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_oF\\boF_boFcdoFfdoFjfoF]g oF`goFdioFgio7$Fcw$!3%H0,e/pM1\"F57$Fhw$!3R#\\aNq!z76F57$F]x$!3'H9i/&* 46>\"F5Fiam-F\\bm6%7$$!*R:YQ&Fabm$!+aqH'[\"FabmFbbmFfbm-F\\bm6%Ff\\pFj bmFfbm-F\\bm6%Ff\\pF_cmFfbm-F\\bm6%7$Fg\\p$!+x0iG7FabmFbbmFiam-F(6%7$7 $$!+k)=9-\"Fabm$!+ruDH*)Fedm7$$!*V@/b&Fedm$!+h$o'z?FabmFfdmFfbmFjdmFce m7+F'-F(6$7LFfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF \\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_oF\\boF_boFcdoF fdoFjfoF]goF`goFdioFgioF[\\pF^\\pFa\\p7$Fbx$!3WKuOv6M.8F57$Fgx$!3]<+^H ;lt9F5Fiam-F\\bm6%7$$!*&Q:YQFabm$!+)))=Sq\"FabmFbbmFfbm-F\\bm6%F]_pFjb mFfbm-F\\bm6%F]_pF_cmFfbm-F\\bm6%7$F^_p$!+bz4u;FabmFbbmFiam-F(6%7$7$$! +w;\"Fabm7$$\"*t*picFedm$ !+ojDBHFabmFfdmFfbmFjdmFcem7+F'-F(6$7RFfamF`fmFcfmFghmFjhmF^[nFa[nFd[n Fh]nF[^nF_`nFb`nFe`nFibnF\\cnF`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oF b_oFe_oFh_oF\\boF_boFcdoFfdoFjfoF]goF`goFdioFgioF[\\pF^\\pFa\\pFe^pFh^ pF\\apF_apFbap7$F[z$!3CI/MfnELRF57$F`z$!3o8>6GYv)p'F57$Fez$!3%*)4P'4&) GY'*F5Fiam-F\\bm6%7$$!)yI#p(Fabm$!+q-$='HFabmFbbmFfbm-F\\bm6%FadpFjbmF fbm-F\\bm6%FadpF_cmFfbm-F\\bm6%7$Fbdp$!+Ayw&G\"FaamFbbmFiam-F(6%7$7$$! +mL_S:Fedm$!+\"3F,(>Fabm7$$\")l!=1#F_dm$!+fM``RFabmFfdmFfbmFjdmFcem7,F '-F(6$7gnFfamF`fmFcfmFghmFjhmF^[nFa[nFd[nFh]nF[^nF_`nFb`nFe`nFibnF\\cn F`enFcenFfenFjgnF]hnFajnFdjnFgjnF[]oF^]oFb_oFe_oFh_oF\\boF_boFcdoFfdoF jfoF]goF`goFdioFgioF[\\pF^\\pFa\\pFe^pFh^pF\\apF_apFbapFfcpFicpF\\dp7$ Fjz$!3$\\-Z^.TDg\"!#;7$F_[l$!31W!*4UI\\]BFcfp7$Fd[l$!37W(=$>=rMSFcfp7$ Fi[l$!3.ZkH&fF0#fFcfp7$F^\\l$!3HhRa\")pa;5Fb^l7$Fc\\l$!3gq.33gy\"\\\"F b^l7$Fh\\l$!3G(GxqZ#[hDFb^l7$F^]l$!3I%)>QIe-fPFb^l7$Fc]l$!3smF[eN]akFb ^l7$Fh]l$!3^.1YL*Gki\"!#97$F]^l$!3+mdszq+_Q!\"(Fiam-F(6$7.7$Fe^lFahp7$ Fj^l$!3p9yO3Ca>;F_hp7$F__l$!3\\a[;'GurU'Fb^l7$Fd_l$!3%[O%G>'4Ju$Fb^l7$ Fi_l$!3%o*\\#e'oj]DFb^l7$F^`l$!3Udxb5(pa[\"Fb^l7$Fc`l$!3YbRa&)HC75Fb^l 7$Fh`l$!3]eR^1HY&*eFcfp7$F]al$!31Sv\"\\pLw,%Fcfp7$Fbal$!3!RF_U1]0M#Fcf p7$Fgal$!3g3,9x$pdf\"Fcfp7$$\"3q++++wI#p(F1$!3'******f'yw&G\"FcfpFiam- F\\bm6%7$$\")wI#p(Fabm$\"+&HI='HFabmFbbmFfbm-F\\bm6%F][qFjbmFfbm-F\\bm 6%F][qF_cmFfbm-F\\bm6%7$F^[q$!+myw&G\"FaamFbbmFiam-F(6%7$7$$!)1!=1#F_d m$\"+&[LN&RFabm7$$\"+@L_S:Fedm$\"+0r7q>FabmFfdmFfbmFjdmFcem7,F'F]fp-F( 6$71FghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjp7$F\\bl$!3MWzH*ov[()*F 57$Fabl$!3ebDELzXcpF57$Ffbl$!3:x4[O2V8SF57$$\"3#********H#p2BFL$!3%*** ***pj$QaIF5Fiam-F\\bm6%7$$\"*I#p2BFabm$\"+X!Qa/#FabmFbbmFfbm-F\\bm6%F \\^qFjbmFfbm-F\\bm6%F\\^qF_cmFfbm-F\\bm6%7$F]^q$!+POQaIFabmFbbmFiam-F( 6%7$7$$!*n*picFedm$\"+xjDBHFabm7$$\"+nXl\"=&Fedm$\"+8(>w;\"FabmFfdmFfb mFjdmFcem7,F'F]fp-F(6$74FghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF \\]qF_]qFb]q7$F[cl$!3S$HN1&)GXv#F57$F`cl$!3!=CUTW3M4#F57$Fecl$!30+l1$y oPs\"F57$$\"3/+++SQ:YQFL$!33+++dz4u;F5Fiam-F\\bm6%7$$\"*%Q:YQFabm$\"+* ))=Sq\"FabmFbbmFfbm-F\\bm6%F[aqFjbmFfbm-F\\bm6%F[aqF_cmFfbm-F\\bm6%7$F \\aq$!+dz4u;FabmFbbmFiam-F(6%7$7$$!*F*QqQFedm$\"+Dop7CFabm7$$\"+2mMz!) Fedm$\"+F&4M&**FedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$76FghpFhhpF[ipF^ipFaip FdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`q7$Fjcl$!3%*RhDy.0v9F5 7$F_dl$!38-W%>yQUI\"F57$$\"3!*******p`h%Q&FL$!32+++!e?'G7F5Fiam-F\\bm6 %7$$\"*P:YQ&Fabm$\"+dqH'[\"FabmFbbmFfbm-F\\bm6%FgcqFjbmFfbm-F\\bm6%Fgc qF_cmFfbm-F\\bm6%7$Fhcq$!+!e?'G7FabmFbbmFiam-F(6%7$7$$\"*J@/b&Fedm$\"+ k$o'z?Fabm7$$\"+h)=9-\"Fabm$\"+'\\d#H*)FedmFfdmFfbmFjdmFcem7,F'F]fp-F( 6$79FghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qF a`qFjbqF]cq7$Fddl$!3mP)oaj<**=\"F57$Fidl$!3[)R)GOiO66F57$F^el$!3)=[^@G a_1\"F57$$\"3.+++5p2BpFL$!31+++YC(41\"F5Fiam-F\\bm6%7$$\"*\"p2BpFabm$ \"+\"f$H78FabmFbbmFfbm-F\\bm6%FffqFjbmFfbm-F\\bm6%FffqF_cmFfbm-F\\bm6% 7$Fgfq$!+YC(41\"FabmFbbmFiam-F(6%7$7$$\"+??hh=Fedm$\"+q5I\\=Fabm7$$\"+ !=a%)>\"Fabm$\"+D6'Gv(FedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7;FghpFhhpF[ipF ^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFi eqF\\fq7$Fcel$!3xxVQu?#>.\"F57$Fhel$!3%)4:b.-u75F57$$\"3;+++]%Q:Y)FL$! 3,+++'HEt+\"F5Fiam-F\\bm6%7$$\"*XQ:Y)Fabm$\"+Bs7a6FabmFbbmFfbm-F\\bm6% FbiqFjbmFfbm-F\\bm6%FbiqF_cmFfbm-F\\bm6%7$Fciq$!+'HEt+\"FabmFbbmFiam-F (6%7$7$$\"+!*fcELFedm$\"+-lQr;Fabm7$$\"+\"4^'f8Fabm$\"+U%z'ojFedmFfdmF fbmFjdmFcem7,F'F]fp-F(6$7=FghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjp F\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhq7$F]fl$!3Fe)*=gW l.5F57$Fbfl$!3U^yqokb+5F57$$\"2-+++*********F5$!33+++,+++5F5Fiam-F\\bm 6%7$$\"**********Fabm$\"+,+++5FabmFbbmFfbm-F\\bm6%F^\\rFjbmFfbm-F\\bm6 %F^\\rF_cmFfbm-F\\bm6%7$F_\\rF\\eoFbbmFiam-F(6%7$7$$\"+OC/b[Fedm$\"+dd \\9:Fabm7$$\"+ad\\9:Fabm$\"+^C/b[FedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7@Fg hpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjb qF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[r7$Fgfl$!2;Gj\\u*******F57$F\\gl$!3w` FW>_w&***FL7$Fagl$!3[eX$*3!=*f**FL7$$\"3%******H:YQ:\"F5$!3H+++LVmF**F LFiam-F\\bm6%7$$\"+`h%Q:\"Fabm$\"+j(GVY)FedmFbbmFfbm-F\\bm6%F[_rFjbmFf bm-F\\bm6%F[_rF_cmFfbm-F\\bm6%7$F\\_r$!+LVmF**FedmFbbmFiam-F(6%7$7$$\" +F+m$Q'Fedm$\"+nU=e8Fabm7$$\"+.jKp;Fabm$\"+g[\"oM$FedmFfdmFfbmFjdmFcem 7,F'F]fp-F(6$7BFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb ]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^r7$Ffgl$ !3-_A;DmWu)*FL7$F[hl$!3kpd^)empp*FL7$$\"33+++1Bp28F5$!3))*****z#f%*[%* FLFiam-F\\bm6%7$$\"+1Bp28Fabm$\"+h+]mpFedmFbbmFfbm-F\\bm6%FgarFjbmFfbm -F\\bm6%FgarF_cmFfbm-F\\bm6%7$Fhar$!+Gf%*[%*FedmFbbmFiam-F(6%7$7$$\"+O ZRdyFedm$\"+:/%)*=\"Fabm7$$\"+Q^kH=Fabm$\"+rffM?FedmFfdmFfbmFjdmFcem7, F'F]fp-F(6$7EFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]q F[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]ar7 $F`hl$!3EDwUx\">$H%*FL7$Fehl$!3I-[X'HH0/*FL7$Fjhl$!3idz$zN_4f)FL7$$\"3 5+++g%Q:Y\"F5$!3%)*****He&fi$)FLFiam-F\\bm6%7$$\"+g%Q:Y\"Fabm$\"+Dt\"* )e&FedmFbbmFfbm-F\\bm6%FfdrFjbmFfbm-F\\bm6%FfdrF_cmFfbm-F\\bm6%7$Fgdr$ !+$e&fi$)FedmFbbmFiam-F(6%7$7$$\"++If_#*Fedm$\"+1.O25Fabm7$$\"+?w\"y*> Fabm$\"+!fJU5\"FedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7GFghpFhhpF[ipF^ipFaip FdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fq FehqFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFicrF\\dr7$F_il$!3B<@x\"pl`-)F L7$Fdil$!39'*4U0d@ztFL7$$\"36+++9YQ:;F5$!3_+++\"Fabm$\"+B**RMlFedm7$$\"+B=]SBFabm$\" *H1\")>%FedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7LFghpFhhpF[ipF^ipFaipFdipFgi pFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhh qFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFicrF\\drFefrFhfrFairFdirFgir7$Fhjl$! 3'[HwLf#p2B>F5$!33+++V/+xSF LFiam-F\\bm6%7$$\"+Ap2B>Fabm$\"+8@ebFFedmFbbmFfbm-F\\bm6%F]]sFjbmFfbm- F\\bm6%F]]sF_cmFfbm-F\\bm6%7$F^]s$!+V/+xSFedmFbbmFiam-F(6%7$7$$\"+I6ES 8Fabm$\"+aBsJ^Fedm7$$\"+9F*e]#Fabm$\"*s=Wz$FedmFfdmFfbmFjdmFcem7,F'F]f p-F(6$7OFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF ^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFi crF\\drFefrFhfrFairFdirFgirF`\\sFc\\s7$Fb[m$!3wFZP9Y4cRFL7$Fg[m$!3)H'p lK>(ya$FL7$F\\\\m$!3A\">)*>Sj-;$FL7$$\"3%)*****\\2Bp2#F5$!3!)******pDj &4$FLFiam-F\\bm6%7$$\"+vI#p2#Fabm$\"*5Zu?#FabmFbbmFfbm-F\\bm6%F\\`sFjb mFfbm-F\\bm6%F\\`sF_cmFfbm-F\\bm6%7$F]`s$!+qDj&4$FedmFbbmFiam-F(6%7$7$ $\"+SB,([\"Fabm$\"+%)RfLSFedm7$$\"+5Q$om#Fabm$\"*;-I\"QFedmFfdmFfbmFjd mFcem7,F'F]fp-F(6$7QFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF _]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj `rF]arFfcrFicrF\\drFefrFhfrFairFdirFgirF`\\sFc\\sF\\_sF__sFb_s7$Fa\\m$ !3s#zD->q'[GFL7$Ff\\m$!3UEA+Ph\"pa#FL7$$\"3&)******G#p2B#F5$!3++++Pk#o O#FLFiam-F\\bm6%7$$\"+H#p2B#Fabm$\"*i/,z\"FabmFbbmFfbm-F\\bm6%FhbsFjbm Ffbm-F\\bm6%FhbsF_cmFfbm-F\\bm6%7$Fibs$!+Pk#oO#FedmFbbmFiam-F(6%7$7$$ \"+]#Hnj\"Fabm$\"+YT4'>$Fedm7$$\"+3#4[#GFabm$\"*%4:TQFedmFfdmFfbmFjdmF cem7,F'F]fp-F(6$7TFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_] qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj`r F]arFfcrFicrF\\drFefrFhfrFairFdirFgirF`\\sFc\\sF\\_sF__sFb_sF[bsF^bs7$ F[]m$!3CJYpFN7'H#FL7$F`]m$!3)Ho%\\&p+V1#FL7$Fe]m$!3c&yZv4@S'=FL7$$\"3' )*****HQ:YQ#F5$!3$*******\\![+$=FLFiam-F\\bm6%7$$\"+$Q:YQ#Fabm$\"*$3c#Fedm7$$\"+j2.\")HF abm$\"*oV*yPFedmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7VFghpFhhpF[ipF^ipFaipFdi pFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFeh qFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFicrF\\drFefrFhfrFairFdirFgirF`\\ sFc\\sF\\_sF__sFb_sF[bsF^bsFgdsFjdsF]es7$Fj]m$!3]&=&**4exy;FL7$F_^m$!3 &\\'Q$Q3**4_\"FL7$$\"3))*****p`h%QDF5$!3*)******zp&GV\"FLFiam-F\\bm6%7 $$\"+P:YQDFabm$\"*8y)>7FabmFbbmFfbm-F\\bm6%FchsFjbmFfbm-F\\bm6%FchsF_c mFfbm-F\\bm6%7$FdhsFe\\nFbbmFiam-F(6%7$7$$\"+YmmS>Fabm$\"+7FVw?Fedm7$$ \"+GkDOJFabm$\"+$[NKj$F_dmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7YFghpFhhpF[ipF ^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFi eqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFicrF\\drFefrFhfrFairFd irFgirF`\\sFc\\sF\\_sF__sFb_sF[bsF^bsFgdsFjdsF]esFfgsFigs7$Fd^m$!3M_%) p#G3zP\"FL7$Fi^m$!3%G*y!*3N#>D\"FL7$F^_m$!3MB=#oV2%[6FL7$$\"3))*****4p 2Bp#F5$!3'********>Uf8\"FLFiam-F\\bm6%7$$\"+\"p2Bp#Fabm$\"*4hL-\"FabmF bbmFfbm-F\\bm6%F`[tFjbmFfbm-F\\bm6%F`[tF_cmFfbm-F\\bm6%7$Fa[t$!*?Uf8\" FabmFbbmFiam-F(6%7$7$$\"+[kp$4#Fabm$\"+i)[Lq\"Fedm7$$\"+M*=4H$Fabm$\"+ yJtLMF_dmFfdmFfbmFjdmFcem7,F'F]fp-F(6$7enFghpFhhpF[ipF^ipFaipFdipFgipF jipF]jpF`jpFcjpF\\]qF_]qFb]qF[`qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqF a[rFd[rF[^rF^^rFa^rFj`rF]arFfcrFicrF\\drFefrFhfrFairFdirFgirF`\\sFc\\s F\\_sF__sFb_sF[bsF^bsFgdsFjdsF]esFfgsFigsF`jsFcjsFfjs7$Fc_m$!3**\\?`il SU5FL7$Fh_m$!3[Ia*Rm$ew&*F17$$\"3#)*****R%Q:YGF5$!3i*******H;E6*F1Fiam -F\\bm6%7$$\"+WQ:YGFabm$\")Bnm')FabmFbbmFfbm-F\\bm6%F\\^tFjbmFfbm-F\\b m6%F\\^tF_cmFfbm-F\\bm6%7$F]^t$!)jh7\"*FabmFbbmFiam-F(6%7$7$$\"+o'[qC# Fabm$\"+*y3ET\"Fedm7$$\"+?!f_W$Fabm$\"+4nD2KF_dmFfdmFfbmFjdmFcem7,F'F] fp-F(6$7gnFghpFhhpF[ipF^ipFaipFdipFgipFjipF]jpF`jpFcjpF\\]qF_]qFb]qF[` qF^`qFa`qFjbqF]cqFfeqFieqF\\fqFehqFhhqFa[rFd[rF[^rF^^rFa^rFj`rF]arFfcr FicrF\\drFefrFhfrFairFdirFgirF`\\sFc\\sF\\_sF__sFb_sF[bsF^bsFgdsFjdsF] esFfgsFigsF`jsFcjsFfjsF_]tFb]t7$F]`m$!3ksu8.U$[w)F17$Fb`m$!3o<5^UYFm!) F17$$\"3%)*****z*******HF5$!3#)*******z+4R(F1Fiam-F\\bm6%7$$\"+)****** *HFabm$\")(**GS(FabmFbbmFfbm-F\\bm6%Fh`tFjbmFfbm-F\\bm6%Fh`tF_cmFfbm-F \\bm6%7$Fi`tFecmFbbmFiam-F(6%7$7$$\"+%3*e+CFabm$\"+I'3L=\"Fedm7$$\"+74 T*f$Fabm$\"+(48F(HF_dmFfdmFfbmFjdmFcem" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 307 8 "Question" }{TEXT 302 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "x^3+y^3 = 2;" "6#/,&*$% \"xG\"\"$\"\"\"*$%\"yGF'F(\"\"#" }{TEXT -1 25 ", find an expression fo r " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " \+ in terms of " }{TEXT 304 1 "x" }{TEXT -1 5 " and " }{TEXT 305 1 "y" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the gradient of the tangent line to the graph of the equation given in (a) at the poi nt" }{XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\"\"\"F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solutio n" }{TEXT 303 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Differentiatin g both sides of the given equation with respect to " }{TEXT 306 1 "x" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2+3*y^2" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&F%F&*$%\"yGF)F& F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\" \"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "Dividing both sides of this equation by 3 gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "Isolating the ter m involving " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 117 " on the left hand side of the equation, and transposing \+ the other term to the right hand side of the equation gives: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"# " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=-x^2" "6#/*&%#dyG\"\"\"%#dxG! \"\",$*$%\"xG\"\"#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "H ence " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x^ 2/(y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xG\"\"#*$%\"yGF,F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(b) The point" }{XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\"\"\"F &" }{TEXT -1 68 " lies on the curve, and the gradient of the tangent a t this point is" }{XPPEDIT 18 0 " ``-1" "6#,&%!G\"\"\"F%!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 350 317 317 {PLOTDATA 2 "6+-%'CURVESG6$7gn7$$!\"$\"\"!$\"33ZeoDoJsI!#<7$$!3!****** \\2<#pGF-$\"3u7#4z/9![HF-7$$!3#)***\\7bBav#F-$\"3()o$RWrs0%GF-7$$!36++ ]K3XFEF-$\"3%G`U%[Nn?FF-7$$!3%)****\\F)H')\\#F-$\"3k&G-zb\\6g#F-7$$!3# ****\\i3@/P#F-$\"3y9?4(Q!e$[#F-7$$!3;++Dr^b^AF-$\"37K-D<+0wBF-7$$!3$** **\\7Sw%G@F-$\"37;dG3z[mAF-7$$!3*****\\7;)=,?F-$\"3]TV%ztea:#F-7$$!3/+ +DO\"3V(=F-$\"3WczQ5#ov/#F-7$$!3#******\\V'zVF-7$$!3* *****\\d;%)G;F-$\"3;k_hw`,\\=F-7$$!3!******\\!)H%*\\\"F-$\"31sV#G)4G^< F-7$$!3/+++vl[p8F-$\"37H#*\\BTHf;F-7$$!3\"******\\>iUC\"F-$\"3=(HaBF)f x:F-7$$!3-++DhkaI6F-$\"3$>^pNA)H5:F-7$$!3s******\\XF`**!#=$\"3L#*fd\\- ,S9F-7$$!3u*******>#z2))Ffp$\"3/.Ib?qf*Q\"F-7$$!3S++]7RKvuFfp$\"3\"o62 =x`@M\"F-7$$!3s,+++P'eH'Ffp$\"3-(o.zE%G58F-7$$!3q)***\\7*3=+&Ffp$\"3wx Pyk\"ocG\"F-7$$!3[)***\\PFcpPFfp$\"35@YxQ)p5F\"F-7$$!3;)****\\7VQ[#Ffp $\"3A9LD228j7F-7$$!32)***\\i6:.8Ffp$\"3?>Gm#e&Qg7F-7$$!3Wb+++v`hH!#?$ \"3/L#\\`0@*f7F-7$$\"3]****\\(QIKH\"Ffp$\"3Wpon9nYf7F-7$$\"38****\\7:x WCFfp$\"31@?h#=XoD\"F-7$$\"3E,++vuY)o$Ffp$\"3ioC:dVH\\7F-7$$\"3!z***** *4FL(\\Ffp$\"3$*e+(y3UNB\"F-7$$\"3A)****\\d6.B'Ffp$\"34r#4\\5Tp?\"F-7$ $\"3s****\\(o3lW(Ffp$\"3u'o#e6AXm6F-7$$\"35*****\\A))oz)Ffp$\"3IzL^*p` n4\"F-7$$\"3e******Hk-,5F-$\"3;B;Y&e9(*)**Ffp7$$\"3%)****\\FL!e1\"F-$ \"3/p`DAAmT#*Ffp7$$\"36+++D-eI6F-$\"3f.!)*3Wpt@)Ffp7$$\"3!***\\(=sx#*= \"F-$\"3))RQ\"z(H'\\#oFfp7$$\"3u***\\(=_(zC\"F-$\"3'>aX^UMQ$QFfp7$$\"3 )[ilAq'*fD\"F-$\"3-'*y+:`)4l#Ffp7$$\"3!)\\7y&==SE\"F-$!3o)*H?)f[]p#Ffp 7$$\"3uuoHp'R?F\"F-$!3q!fW!4T$o(QFfp7$$\"3))*\\7G:h+G\"F-$!3vh\\d]f$=g %Ffp7$$\"3'*\\P%)>T5'H\"F-$!3uS?Q:p\"zh&Ffp7$$\"3/+](o3Z@J\"F-$!3OgZ(* 3ykvjFfp7$$\"3>+v$4-LUM\"F-$!37k%zD+h=a(Ffp7$$\"3M+++b*=jP\"F-$!3Ud4)z R`uY)Ffp7$$\"3(***\\(=o*pO9F-$!3w[s=:Hl$))*Ffp7$$\"3g***\\(3/3(\\\"F-$ !3i0c1ZBm16F-7$$\"33++vB4JB;F-$!3CkB%*3jr:8F-7$$\"3u*****\\KCnu\"F-$!3 k7X1EU?$\\\"F-7$$\"3s***\\(=n#f(=F-$!3c,OXrLHj;F-7$$\"3P+++!)RO+?F-$!3 4X>Y'\\hv\"=F-7$$\"30++]_!>w7#F-$!3#Q_)eEUyo>F-7$$\"3O++v)Q?QD#F-$!3\" QY4(yg49@F-7$$\"3G+++5jypBF-$!3?'*G\">SwXC#F-7$$\"3<++]Ujp-DF-$!390HJh $[FF-$!3I_^^n21dEF -7$$\"37++D6EjpGF-$!3W>De)[xiy#F-7$$\"\"$F*$!3.mG@Qx,CHF--%'COLOURG6&% $RGBG$\"*++++\"!\")$F*F*Fc^l-F$6%7S7$$!\"\"F*Fh]l7$$!3PLLLLQ6G\"*Ffp$ \"3cLLL$Q6G\"HF-7$$!3immmT.\\p$)Ffp$\"3bmm;M!\\p$GF-7$$!3LLLL$))Qj^(Ff p$\"37LLL))Qj^FF-7$$!3ULLL$=Kvl'Ffp$\"3ALLL=KvlEF-7$$!3hnmmTs!G!eFfp$ \"3wmm;C2G!e#F-7$$!3iLLL3yO5]Ffp$\"39LL$3yO5]#F-7$$!3i+++vE%)*=%Ffp$\" 3&*****\\nU)*=CF-7$$!3)RLL$3WDTLFfp$\"3iLL$3WDTL#F-7$$!3'4++]d(Q&\\#Ff p$\"3))****\\d(Q&\\AF-7$$!3:mmmm&4`i\"Ffp$\"3gmmmc4`i@F-7$$!3GKLLLQW*e )!#>$\"3KLLLQW*e3#F-7$$\"3HI#*******H,Q!#@$\"33+++q)>'**>F-7$$\"3Q(*** ****\\*3q)F_bl$\"3.+++]5*H\">F-7$$\"3!********p=\\q\"Ffp$\"3,+++I\"3&H =F-7$$\"3_mmm\"fBIY#Ffp$\"3OLL$3k(p`9F-7$$\"3'3++]sgam'Ffp$\"3\" *****\\FRXL8F-7$$\"3G+++v\"ep[(Ffp$\"3)*****\\#=/8D\"F-7$$\"3#QLLLe/TM )Ffp$\"3immmT&*el6F-7$$\"39LLLeDBJ\"*Ffp$\"3omm;Wn(o3\"F-7$$\"3Immm;kD !)**Ffp$\"3PLLLeV(>+\"F-7$$\"3Mmm;f`@'3\"F-$\"3hOLL3k%y8*Ffp7$$\"3y*** *\\nZ)H;\"F-$\"31-++DB:q$)Ffp7$$\"3YmmmJy*eC\"F-$\"3XNLL$o@5a(Ffp7$$\" 3')******R^bJ8F-$\"3S,+++'[Wo'Ffp7$$\"3f*****\\5a`T\"F-$\"3;/++]*ek%eF fp7$$\"3o****\\7RV'\\\"F-$\"39.++v3mN]Ffp7$$\"3k*****\\@fke\"F-$\"3b.+ +]ySNTFfp7$$\"3/LLL`4Nn;F-$\"3[pmmm/\\ELFfp7$$\"3#*******\\,s`$=F-$\"3;NLL3_;!o\"Ffp7$$\"3$*******pfa<>F-$ \"3q1+++ISX#)F_bl7$$\"3#HLLeg`!)*>F-$\"3)p3nm;%RY>F_s7$$\"3w****\\#G2A 3#F-$!3lv****\\#G2A)F_bl7$$\"3;LLL$)G[k@F-$!3jJLLL)G[k\"Ffp7$$\"3#)*** *\\7yh]AF-$!3:)****\\7yh]#Ffp7$$\"3xmmm')fdLBF-$!3onmmm)fdL$Ffp7$$\"3b mmm,FT=CF-$!3almm;q7%=%Ffp7$$\"3FLL$e#pa-DF-$!3pKLLe#pa-&Ffp7$$\"3!*** ****Rv&)zDF-$!3%*)******Rv&)z&Ffp7$$\"3ILLLGUYoEF-$!3+LLL$GUYo'Ffp7$$ \"3_mmm1^rZFF-$!3=lmmm5:xuFfp7$$\"34++]sI@KGF-$!3&4++]sI@K)Ffp7$$\"34+ +]2%)38HF-$!3)3++]2%)38*Ffp7$Fh]lFh^l-F]^l6&F_^lFc^lF`^lFc^l-%*THICKNE SSG6#\"\"#-F$6%7S7$F(Fh]l7$F/$\"3!******\\2<#pGF-7$F4$\"3#)***\\7bBav# F-7$F9$\"36++]K3XFEF-7$F>$\"3%)****\\F)H')\\#F-7$FC$\"3#****\\i3@/P#F- 7$FH$\"3;++Dr^b^AF-7$FM$\"3$****\\7Sw%G@F-7$FR$\"3*****\\7;)=,?F-7$FW$ \"3/++DO\"3V(=F-7$Ffn$\"3#******\\V'zViUC\" F-7$F_p$\"3-++DhkaI6F-7$Fdp$\"3s******\\XF`**Ffp7$Fjp$\"3u*******>#z2) )Ffp7$F_q$\"3S++]7RKvuFfp7$Fdq$\"3s,+++P'eH'Ffp7$Fiq$\"3q)***\\7*3=+&F fp7$F^r$\"3[)***\\PFcpPFfp7$Fcr$\"3;)****\\7VQ[#Ffp7$Fhr$\"32)***\\i6: .8Ffp7$F]s$\"3Wb+++v`hHF_s7$Fcs$!3]****\\(QIKH\"Ffp7$Fhs$!38****\\7:xW CFfp7$F]t$!3E,++vuY)o$Ffp7$Fbt$!3!z******4FL(\\Ffp7$Fgt$!3A)****\\d6.B 'Ffp7$F\\u$!3s****\\(o3lW(Ffp7$Fau$!35*****\\A))oz)Ffp7$Ffu$!3e******H k-,5F-7$F`v$!36+++D-eI6F-7$Fjv$!3u***\\(=_(zC\"F-7$Fby$!3M+++b*=jP\"F- 7$F\\z$!3g***\\(3/3(\\\"F-7$Faz$!33++vB4JB;F-7$Ffz$!3u*****\\KCnu\"F-7 $F[[l$!3s***\\(=n#f(=F-7$F`[l$!3P+++!)RO+?F-7$Fe[l$!30++]_!>w7#F-7$Fj[ l$!3O++v)Q?QD#F-7$F_\\l$!3G+++5jypBF-7$Fd\\l$!3<++]Ujp-DF-7$Fi\\l$!3++ ++gEd@EF-7$F^]l$!39++v3'>$[FF-7$Fc]l$!37++D6EjpGF-7$Fh]lF(-F]^l6&F_^lF *F*F*-%*LINESTYLEGF\\^m-%%TEXTG6$7$$\"#>Fi^l$\"#=Fi^lQ9gradient~of~tan gent~=~-16\"-Fegm6$7$$\"#8Fi^l$\"#7Fi^lQ&(1,1)F]hm-Fegm6$7$$\"#MFi^l$! \"#Fi^lQ\"xF]hm-Fegm6$7$F[imFihmQ\"yF]hm-%+AXESLABELSG6%%!GFeim-%%FONT G6#%(DEFAULTG-%%VIEWG6$;F(FihmF]jm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equati on " }{XPPEDIT 18 0 "x^3+y^3=2" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(\" \"#" }{TEXT -1 19 " can be solved for " }{TEXT 385 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 384 1 "x" }{TEXT -1 10 " to give: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (2-x^3)^(1/3);" "6#/%\" yG),&\"\"#\"\"\"*$%\"xG\"\"$!\"\"*&F(F(F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx=1/3" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&\"\"$F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "(2-x^3)^(-2/3)*(-3*x^2);" "6#*&),&\"\"# \"\"\"*$%\"xG\"\"$!\"\",$*&F&F'F*F+F+F',$*&F*F'*$F)F&F'F+F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -x^2/(2-x)^(2/3)" "6#/%!G,$*&%\"xG\"\"#), &F(\"\"\"F'!\"\"*&F(F+\"\"$F,F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-x^2/y^2" "6#/%!G,$*&%\"xG\"\"#*$% \"yGF(!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 335 8 "Question" }{TEXT 330 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "x^2-x*y+y^2 = 9;" "6#/,(*$%\"xG\"\"#\" \"\"*&F&F(%\"yGF(!\"\"*$F*F'F(\"\"*" }{TEXT -1 25 ", find an expressio n for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 332 1 "x" }{TEXT -1 5 " and " }{TEXT 333 1 " y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 121 "(b) Find the equati ons of the tangent lines to the graph of the equation given in (a) at \+ the points where it crosses the " }{TEXT 336 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "So lution" }{TEXT 331 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Different iating both sides of the given equation with respect to " }{TEXT 334 1 "x" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*x-y-x;" "6#,(*&\"\"#\"\"\"%\"xGF&F&%\"yG!\"\"F'F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y;" "6#,&*&%#dyG\"\"\"%#dxG!\" \"F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/* &%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Collecting the terms involving " }{XPPEDIT 18 0 "dy/dx" "6#*&%# dyG\"\"\"%#dxG!\"\"" }{TEXT -1 110 " on the left hand side of the equa tion, and the remaining terms on the right hand side of the equation g ives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y;" "6#*& \"\"#\"\"\"%\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-x;" "6#,&*&% #dyG\"\"\"%#dxG!\"\"F&%\"xGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y-2*x;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&*&\"\"#F&%\"xGF&F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on th e left side gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " ``(2*y-x) = y-2*x;" "6#/-%!G6#,&*&\"\"#\"\"\"%\"yGF*F*%\"xG!\"\",&F+F* *&F)F*F,F*F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Dividing both sides by " }{XPPEDIT 18 0 "5*y-x;" "6#,&*&\"\"&\"\"\"%\"yGF&F&% \"xG!\"\"" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = (y-2*x)/(2*y-x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*& ,&%\"yGF&*&\"\"#F&%\"xGF&F(F&,&*&F-F&F+F&F&F.F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) Subst ituting " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 17 " in the e quation " }{XPPEDIT 18 0 "x^2-x*y+y^2 = 9" "6#/,(*$%\"xG\"\"#\"\"\"*&F &F(%\"yGF(!\"\"*$F*F'F(\"\"*" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "y^ 2 = 9;" "6#/*$%\"yG\"\"#\"\"*" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "y =``" "6#/%\"yG%!G" }{TEXT 337 1 "+" }{TEXT -1 4 " 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "At both of the po ints" }{XPPEDIT 18 0 "``(0,3);" "6#-%!G6$\"\"!\"\"$" }{TEXT -1 4 " and " }{XPPEDIT 18 0 "``(0,-3);" "6#-%!G6$\"\"!,$\"\"$!\"\"" }{TEXT -1 38 ", the gradient of the tangent line is " }{XPPEDIT 18 0 "1/2;" "6#*&\" \"\"F$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The t angent lines at the points" }{XPPEDIT 18 0 " ``(0,3)" "6#-%!G6$\"\"!\" \"$" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(0,-3)" "6#-%!G6$\"\"!,$\" \"$!\"\"" }{TEXT -1 21 " have the equations: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x/2+3" "6#/%\"yG,&*&%\"xG\"\"\"\"\"#! \"\"F(\"\"$F(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y=x/2-3" "6#/%\"y G,&*&%\"xG\"\"\"\"\"#!\"\"F(\"\"$F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The curve is an ellipse which has the lines " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y=-x" "6#/%\"yG,$%\"xG!\"\"" }{TEXT -1 43 " as its major and mi nor axes respectively." }}{PARA 0 "" 0 "" {TEXT -1 11 "The points " } {XPPEDIT 18 0 "``(3,3)" "6#-%!G6$\"\"$F&" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(-sqrt(3),sqrt(3))" "6#-%!G6$,$-%%sqrtG6#\"\"$!\"\"-F(6#F*" } {TEXT -1 1 "," }{XPPEDIT 18 0 "``(-3,-3)" "6#-%!G6$,$\"\"$!\"\",$F'F( " }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(sqrt(3),-sqrt(3))" "6#-%!G6$-%%sq rtG6#\"\"$,$-F'6#F)!\"\"" }{TEXT -1 21 " lie on the ellipse. " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 278 280 280 {PLOTDATA 2 "6 *-%'CURVESG6$7\\x7$$!3=+++ln>jM!#<$!3QU.%3jLIm\"F*7$$!31+++;c=ZMF*$!3L U(o`\")yuU\"F*7$$!3&******pYu6V$F*$!36W>*ekNHI\"F*7$$!3++++V?;&R$F*$!3 J/$eL)40-6F*7$$!3?+++14\"*eLF*$!3kz([5DUxX*!#=7$$!3)******R1KGK$F*$!3x #f^U1TU8)FA7$$!3%)******fBQ*G$F*$!3Q1u@!o7'RqFA7$$!3++++GruaKF*$!36LH* 3f#e-gFA7$$!3()*****>KF*=KF*$!35a2!)G)\\\"4]FA7$$!31+++!RAK=$F*$!3Ik^x [Uf#3%FA7$$!3%*******ya\\YJF*$!3wbIZ(GtQ=$FA7$$!3@+++'RYT6$F*$!3-,O0!3 M3V#FA7$$!3y*******)*Gx2$F*$!34<3'4jX-i\"FA7$$!39+++^?;TIF*$!37TQqnQA4 %)!#>7$$!3%*******HI#f+$F*$!3s4YNt^8)=\"Fio7$$!3!******>iAR(HF*$\"3+E \"[2j'=\\^Fio7$$!37+++V5(e$HF*$\"3k'G'f\"R&)QC\"FA7$$!3%******HOOO!HF* $\"3[T4&)ydNU=FA7$$!3-+++o(Rh'GF*$\"3vc%Hb1K%=DFA7$$!3(******z#*[H$GF* $\"3U=N\\]'\\-5$FA7$$!35+++;L`'z#F*$\"3<#**3_\"o*>s$FA7$$!3%******>/d= w#F*$\"3e!eORvD!*H%FA7$$!3!)*****p)fnDFF*$\"3Q\"*fAgq_')[FA7$$!3;+++&[ ]Cp#F*$\"3%)zR#R+XPT&FA7$$!3;+++#H7ml#F*$\"35nn^C>,qfFA7$$!3\"******>[ 'Q>EF*$\"3Bt%>Y1d\\`'FA7$$!3/+++/8)pe#F*$\"3-=&pzI\\m,(FA7$$!31+++DG)> b#F*$\"3AT[s_S(o_(FA7$$!3%******4(f#e^#F*$\"3CS6KDweV!)FA7$$!3'******R a`/[#F*$\"3omp5MzMR&)FA7$$!3-+++,*GiW#F*$\"37'*\\606E5!*FA7$$!3.+++r#G #3CF*$\"3I'Hq=`wM_*FA7$$!33+++:I3uBF*$\"3Y*pI@\"yJw**FA7$$!3-+++lciPBF *$\"3'4-czg\\^/\"F*7$$!3?+++)))*e/BF*$\"3mzYP)y#\\(3\"F*7$$!35+++eIZoA F*$\"3!z)zz2b.L6F*7$$!3/+++N**[MAF*$\"3[LE]!>(>v6F*7$$!3!)******zx'*)> #F*$\"3e,QXNod=7F*7$$!35+++\"QQU;#F*$\"3)zl&o.qKg7F*7$$!34+++=*zy7#F*$ \"3CP#RU3dLI\"F*7$$!3?+++(RiG4#F*$\"3$*=j(f'R;W8F*7$$!37+++D>0d?F*$\"3 y:hUg$o_Q\"F*7$$!3#)*****4)z`@?F*$\"3*e0R5QCaU\"F*7$$!33+++bV!*))>F*$ \"3'RTw(QS!=Y\"F*7$$!35+++PD]^>F*$\"3BB(yIG.H]\"F*7$$!3%******\\!*\\!= >F*$\"3g!)p9cq8R:F*7$$!31+++ACQ#)=F*$\"3Ms*)pOjBx:F*7$$!3/+++*3W#[=F*$ \"3i*)fhf%)>8;F*7$$!3-+++eyL5=F*$\"3Y_.5Mfc_;F*7$$!3!******z%Q^x'[ \"H$*FA$\"3:,xJhPqACF*7$$!3[+++SOIy*)FA$\"3k.Gwz3d[CF*7$$!3b******>$H, j)FA$\"3QDLV.M!RZ#F*7$$!3y******>5pd#)FA$\"33D4,g?j+DF*7$$!3M+++]..HzF A$\"3Zee_#Q/R_#F*7$$!3;+++qT#)\\vFA$\"3O'R;W=#R]DF*7$$!3]+++I)>w?(FA$ \"31;BRML'Rd#F*7$$!3-+++?l-poFA$\"3QE6b>y(pf#F*7$$!3Z+++S*H\\]'FA$\"37 YPh#='Q@EF*7$$!3Q+++50ORhFA$\"31(=zwRTbk#F*7$$!3h******fku1eFA$\"3othJ )[9sm#F*7$$!3S+++5^_`aFA$\"3c%4)HIV\"**o#F*7$$!3d*******f:)3^FA$\"3)Rl B(p[v6FF*7$$!3\"*******\\o9NZFA$\"3.>EN\"[$3NFF*7$$!3=+++!\\>=T%FA$\"3 MF2Yo$z\\v#F*7$$!3#*******piYSSFA$\"3WF?5&4+vx#F*7$$!3%*******4ZP\"p$F A$\"37J@=E)\\$)z#F*7$$!3;++++]lXLFA$\"3T%y'[4Fp=GF*7$$!3m******p-m**HF A$\"3GtPrq$[(QGF*7$$!39+++SvwYEFA$\"3+6J^#3#*)eGF*7$$!37+++grqsAFA$\"3 s#)o)>=,*zGF*7$$!36+++?/tH>FA$\"3A-pyP]&))*GF*7$$!3%)******R$o6f\"FA$ \"3beJ3T^F;7o)Fio $\"3k^=6F*HF*7$$\"3)********4gDh\"Fio$\"3Ydz%fHI!3IF*7$$\"3s****** *H..P&Fio$\"3KVz'G*4\\EIF*7$$\"3m++++bqc*)Fio$\"38Q4ox0yVIF*7$$\"3(*** ****HN*e@\"FA$\"3='*>i5h%*eIF*7$$\"3++++]x,w:FA$\"3_A'z#*[%pvIF*7$$\"3 ()******R\"H&Q>FA$\"3M#=^YSDA4$F*7$$\"3;+++]vJ*H#FA$\"34Xh/1+N3JF*7$$ \"3%)*******e9Qj#FA$\"3!\\$3MPp+BJF*7$$\"3E+++?p;!)HFA$\"3j=1f%)f)y8$F *7$$\"35+++q\\OQLFA$\"3s\"3<=!\\&H:$F*7$$\"3w*******G9ap$FA$\"3?sy%Hw^ w;$F*7$$\"31++++MoiSFA$\"3aj%Gv)4V#=$F*7$$\"3y******>U<'Q%FA$\"3#fU[&* Qj^>$F*7$$\"3=++++$[.v%FA$\"3%QALQGw\"4KF*7$$\"37+++!oT)y&FA$\"3)o[$p]>C ZKF*7$$\"3w******zx#*ohFA$\"33I`&eb$\\gKF*7$$\"3?+++]XF\"\\'FA$\"3!*\\ M[^AUrKF*7$$\"3%)******>0CmoFA$\"3oB^Ft**y$G$F*7$$\"3')********)[\")>( FA$\"3_$>vm7EWH$F*7$$\"3E+++S]IivFA$\"3TvI;iqv0LF*7$$\"30+++qx14zFA$\" 3#[`n,D:iJ$F*7$$\"3C+++?$y3F)FA$\"3>#)3'p'*znK$F*7$$\"3S+++gL8.')FA$\" 3kqwxVp;OLF*7$$\"3u******z_^h*)FA$\"3y>h_%f^fM$F*7$$\"3,+++!QtPL*FA$\" 3kXLpIytbLF*7$$\"33+++]^#yl*FA$\"3=^'oN3TRO$F*7$$\"3&******\\*4y+5F*$ \"3#oc?K!yYsLF*7$$\"33+++\\y$p.\"F*$\"3EE+#o<54Q$F*7$$\"3#******pF5B2 \"F*$\"3/%4\\-J0))Q$F*7$$\"34+++?\\`16F*$\"3%RtJy(z4'R$F*7$$\"3******* *[b`W6F*$\"3)*pB-%)=z.MF*7$$\"3++++.3oy6F*$\"3$4sJH5R.T$F*7$$\"3\"**** **R:Q^@\"F*$\"3H#[yZdUpT$F*7$$\"3!******H$R<[7F*$\"3*H83M)pdAMF*7$$\"3 )*******f2H%G\"F*$\"3_abA!>yLY$F*7$$\"3)******4t>Xq\"F*$\"3cR%)R5l&RY$F*7$$\"3++++if UUF*$\"3+o%*fazOdMF*7$$\"3%***** *\\tV@&>F*$\"3?L2M0xMaMF*7$$\"3%******f\"zR()>F*$\"3K:=F9&p3X$F*7$$\"3 ++++n!HE-#F*$\"3[)3U[fGoW$F*7$$\"37+++51td?F*$\"3d:?RKdAUMF*7$$\"39+++ -AX\"4#F*$\"3#y(*HypYsV$F*7$$\"3%)*****R81z7#F*$\"3!yB&eOCBJMF*7$$\"3A +++y$\\E;#F*$\"3)eUHqIr[U$F*7$$\"3-+++/]@*>#F*$\"3NYv%*=B\\&)e:sfAR$F*7$$\"3%)*****z8#*)QBF*$\"3/()3]+^S#Q$F *7$$\"33+++%3-aP#F*$\"39Hpp$p)HrLF*7$$\"35+++y!H!4CF*$\"37**3+'Q0.O$F* 7$$\"3()*****prnMW#F*$\"3?4[BRmD[LF*7$$\"3')*****\\1#\\\"[#F*$\"3ru2ON Y)RL$F*7$$\"3$**********>f^#F*$\"3,>C28d9?LF*7$$\"3!******\\,H6b#F*$\" 3'RC8/d[]I$F*7$$\"3++++rc$pe#F*$\"3i>jH%zn')G$F*7$$\"3()*****p&*[)>EF* $\"3))pmHP0lsKF*7$$\"3%)*****pXL\\l#F*$\"3A@icD,^aKF*7$$\"3?+++()3v*o# F*$\"3)R!)*RT&f`B$F*7$$\"3#)*****zr%*ps#F*$\"3oHrb.6`8KF*7$$\"3;+++$yg )fFF*$\"39Y]uFF/$>$F*7$$\"3-+++-9y(z#F*$\"3AL9JNv(y;$F*7$$\"3'******p$ =+KGF*$\"3L*=$f%HlO9$F*7$$\"3!******z;he'GF*$\"3qIKZc!*>=JF*7$$\"3;+++ F3F-HF*$\"37*R5CQ0!*3$F*7$$\"3*)*****zwF)QHF*$\"3qG!*o))>kdIF*7$$\"3#) *****\\<*3sHF*$\"3OPg/xF:FIF*7$$\"34+++48T2IF*$\"3;biM_L`#*HF*7$$\"3&* ******fA)=/$F*$\"3'[Yt0x&GcHF*7$$\"31+++O\"\\#zIF*$\"3>^7MaH\"R\"HF*7$ $\"36+++p=e6JF*$\"39`*[tSHV(GF*7$$\"3)******H><([JF*$\"3c[JVEC.DGF*7$$ \"37+++Zji$=$F*$\"3o!o@ii\\Vx#F*7$$\"3%*******>$)>=KF*$\"3&zT@F9M#>FF* 7$$\"39+++%z(z_KF*$\"3+B3*4CN\"eEF*7$$\"37+++mq3)G$F*$\"3Yp\"4&QN;)e#F *7$$\"3!******H5$\\DLF*$\"3Q@l>:&yG]#F*7$$\"3!)*****zx!zfLF*$\"3$>L=n# )[0T#F*7$$\"3;+++&)pk$R$F*$\"3?hhfS`z)H#F*7$$\"3********\\Gu6MF*$\"3J \\ql2W\\DAF*7$$\"3!******frQ)HMF*$\"3J$==W&p\"e8#F*7$$\"31+++^p')QMF*$ \"3cs*o)QF)33#F*7$$\"3A+++'=&*yW$F*$\"3&[ckSV*z8?F*7$$\"3/+++.$4CX$F*$ \"3-YX>T>[s>F*7$$\"3%******4UBpX$F*$\"3hl01-6\\@>F*7$$\"3-+++z/=fMF*$ \"3/\"\\+OqU%*)=F*7$$\"3?+++QvVhMF*$\"31u6<;AN[=F*7$$\"3/+++ngciMF*$\" 3oGV!)e!)f?=F*7$$\"3%******pf%pjMF*$\"3u6//(HMyx\"F*7$%*undefinedGF`\\ o-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fi\\oFh\\o-F$6$7\\x7$F($!3zd'fT 8j,!=F*7$F.$!3'zDJ1!oq>?F*7$F3$!3&e036#)Q#G@F*7$F8$!3q&pT'f56$H#F*7$F= $!3BA^*3oOJT#F*7$FC$!3qS[ddzS4DF*7$FH$!3)*e#y>4@ae#F*7$FM$!3o12\"*o))[ aEF*7$FR$!3YC*>\"RB,=FF*7$FW$!3k$[A^'H'\\x#F*7$Ffn$!38%p_-:3\"GGF*7$F[ o$!35SY*z)H1rGF*7$F`o$!3H=R!pU/d\"HF*7$Feo$!3_hHK7)pq&HF*7$F[p$!3%QXm# y;/%*HF*7$F`p$!3<\"[2$)[9a-$F*7$Fep$!3yG'f@ef-1$F*7$Fjp$!33%4&)3%>(y3$ F*7$F_q$!3\\XHbuH)z6$F*7$Fdq$!3#=N\\I*Q(H9$F*7$Fiq$!35**3_(*HtoJF*7$F^ r$!3?eOR<'f<>$F*7$Fcr$!3%*)fAIpGV@$F*7$Fhr$!3;)R#R&)\\#QB$F*7$F]s$!3kw ;X%[8OD$F*7$Fbs$!3YZ>Y)=#)GF$F*7$Fgs$!3i^pzMik)G$F*7$F\\t$!3T%[s-BqYI$ F*7$Fat$!3'R6KNt%=?LF*7$Fft$!3S'p5uL)QMLF*7$F[u$!3%)*\\6:,bsM$F*7$F`u$ !3))Hq=CfdgLF*7$Feu$!3-qI@'z9j(HOEqV$F*7$Fbx$!37;hU&G? BW$F*7$Fgx$!3qb!R?OipW$F*7$F\\y$!3!QTwPR32X$F*7$Fay$!3aB(y+#eSaMF*7$Ff y$!3w!)p9hp=dMF*7$F[z$!3Ss*)pe(='fMF*7$F`z$!3))*)fh[DWhMF*7$Fez$!3E_.5 #z.HY$F*7$Fjz$!3?`45+.qjMF*7$F_[l$!3)Q:(eI')3kMF*7$Fd[l$!3)yW\\amuRY$F *7$Fi[l$!3Kg/T'\\\\LY$F*7$F^\\l$!3AB_=UxRiMF*7$Fc\\l$!3OhN)>Ed3Y$F*7$F h\\l$!3'y-.uy:*eMF*7$F]]l$!3q-0A*HQlX$F*7$Fb]l$!3%3#\\+k@u`MF*7$Fg]l$! 3QNSOP1m]MF*7$F\\^l$!3EWk;M0!pW$F*7$Fa^l$!3%)))RM]%3HW$F*7$Ff^l$!3\"yi O!4HGQMF*7$F[_l$!3bt:]?fsLMF*7$F`_l$!3aET\")*['GGMF*7$Fe_l$!3]f-PI[oAM F*7$Fj_l$!3_xVzM*>nT$F*7$F_`l$!3[O&Q5m#45MF*7$Fd`l$!3xHx8dTj.MF*7$Fi`l $!3cfoA$QrmR$F*7$F^al$!33!['3x'y&)Q$F*7$Fcal$!3'f#4b2())3Q$F*7$Fhal$!3 khm+r9nsLF*7$F]bl$!3-)z*)pU_RO$F*7$Fbbl$!3y+xJB'=cN$F*7$Fgbl$!3Y.GwV7S YLF*7$F\\cl$!3bDLVNj\"pL$F*7$Facl$!3%[#4,i6SELF*7$Ffcl$!3iee_$ F*7$F]gl$!3/F?5Ana\"=$F*7$Fbgl$!3GJ@=(H([nJF*7$Fggl$!3[%y'[4#eK:$F*7$F \\hl$!3DtPr(R9(QJF*7$Fahl$!3!468l$)oN7$F*7$Ffhl$!3J#)o)z*=<2JF*7$F[il$ !3W-pyz!G=4$F*7$F`il$!3KeJ3v>RwIF*7$Feil$!3=3p$[8t&fIF*7$Fjil$!3%=&=6* )QYUIF*7$F_jl$!3.@)Q5SHo-$F*7$Fdjl$!3vO%)yW#pz+$F*7$Fijl$!3udz%\\p/>*H F*7$F^[m$!3kVz')fzysHF*7$Fc[m$!3GQ4oAN@aHF*7$Fh[m$!35'*>idnNPHF*7$F]\\ m$!3pA'zUr#4=HF*7$Fb\\m$!3_#=^1\\s$)*GF*7$Fg\\m$!37Xh/^#=%yGF*7$F\\]m$ !3CN3MyaifGF*7$Fa]m$!3g=1f#Hp)RGF*7$Ff]m$!3l\"3<[S=\">GF*7$F[^m$!3Msy% RL5\")z#F*7$F`^m$!3)RYGvkihx#F*7$Fe^m$!3%eU[v'facFF*7$Fj^m$!3gBK$QXTTt #F*7$F__m$!3khT]'p,8r#F*7$Fd_m$!3yn;?LW'*)o#F*7$Fi_m$!3%o[$p`2SoEF*7$F ^`m$!3aI`&yx+Ok#F*7$Fc`m$!3a\\M['z%HAEF*7$Fh`m$!3oB^F@f;(f#F*7$F]am$!3 ?$>vmB6Yd#F*7$Fbam$!31vI;el_\\DF*7$Fgam$!3#[`nJZ3`_#F*7$F\\bm$!3/#)3' \\8#p*\\#F*7$Fabm$!3sqwx2O&eZ#F*7$Ffbm$!3#*>h_m+!)\\CF*7$F[cm$!3(fM$p# \\gBU#F*7$F`cm$!3G^'o&o&e\")R#F*7$Fecm$!3Wm0A3oorBF*7$Fjcm$!3=E+#yKsRM #F*7$F_dm$!3M%4\\K.&\\;BF*7$Fddm$!3IM<$y0j&*G#F*7$Fidm$!3WqB-NjDfAF*7$ F^em$!3;@<$**He;B#F*7$Fcem$!3P#[y2U/=?#F*7$Fhem$!3_L\"3/0.W<#F*7$F]fm$ !3KaX3#F*7$F\\gm$!3Ik' *['*4>a?F*7$Fagm$!3lU`0%\\>?-#F*7$Ffgm$!3SevR,?i!*>F*7$F[hm$!35g;@L$)3 e>F*7$F`hm$!3\\hF7wDRD>F*7$Fehm$!3sqgUcN'\\*=F*7$Fjhm$!39b\\X\">F'f=F* 7$F_im$!3;]X#*esfF=F*7$Fdim$!3O>bA\"z'*Hz\"F*7$Fiim$!3MR%)RznVf n3eGY1:;F*7$Fb[n$!31Y')owR%fd\"F*7$Fg[n$!32o%*f$4@C9F*7$F[]n$!3W:?RA^\\ %Q\"F*7$F`]n$!3ox*He\\%zX8F*7$Fe]n$!3'zB&e-jK.8F*7$Fj]n$!3mD%H!H>Ai7F* 7$F_^n$!3KYv%\\Jx#=7F*7$Fd^n$!3O3tK)3\"F*7$Fc_n$!3*p)3]iH^V5F*7$Fh_n$!3e!Hpp4m*e**FA7$F]`n$!3' z)*3+3jF^*FA7$Fb`n$!3M$4[BA*)y/*FA7$Fg`n$!3QYxg.d#\\_)FA7$F\\an$!3d)=C 28dA/)FA7$Faan$!3pSC8ab>RvFA7$Ffan$!3*R>jHB@t,(FA7$F[bn$!39+n'H!e,GlFA 7$F`bn$!3#Q@icomd*fFA7$Febn$!3]N!)*Ra'3caFA7$Fjbn$!3!3Irb&QOl[FA7$F_cn $!36g/XZ%>=L%FA7$Fdcn$!3%)HV6L8'4q$FA7$Ficn$!3k$*=$fdMm6$FA7$F^dn$!3)z IKZ))yL_#FA7$Fcdn$!3i*)R5abMn=FA7$Fhdn$!3#eG!*o?U\")=\"FA7$F]en$!3*Ht. Y?gj]&Fio7$Fben$\"3s\"\\u`m&z([\"Fio7$Fgen$\"33'[`E%*['f&)Fio7$F\\fn$ \"3g)[(e;=O`;FA7$Fafn$\"3Un/^;Y_sBFA7$Fffn$\"3!>^ocmZoB$FA7$F[gn$\"3Y% >$y2sw#4%FA7$F`gn$\"3yC\\*FA7$Fihn$\"3'*Q QSW;&[4\"F*7$F^in$\"3o]HMU%[i=\"F*7$Fcin$\"3e;=eh<-%H\"F*7$Fhin$\"3[F5 87U)zN\"F*7$F]jn$\"3PNa$>v&4M9F*7$Fbjn$\"3/aa!=OF*z9F*7$Fgjn$\"3LM%R*= BVN:F*7$F\\[o$\"3)*3&*Rvxtp:F*7$Fa[o$\"39E)G=K&38;F*7$Ff[o$\"3Nrc>3!o> k\"F*7$F[\\o$\"3=)ef**Hgeo\"F*F_\\oFa\\o-F$6%7S7$$!3++++++++NF*$\"3+++ ++++]7F*7$$!3]LL$3#*>uM$F*$\"3DLLeR+HE8F*7$$!3Ym;z43m9KF*$\"3wmT5&fpER \"F*7$$!3;LLe/$f`1$F*$\"3UL$3xM?tY\"F*7$$!3PLL3K\"o]\"HF*$\"3JL$eR$fYU :F*7$$!3]m;Hn7\\lFF*$\"3vmTNmVD<;F*7$$!3WL$ekO9oi#F*$\"3GL3x;Gf'o\"F*7 $$!3q**\\7oCA$[#F*$\"3;+v$fw)QeZL#F*$\"3TL3FR-kK=F*7$$! 3A+]iDGp'=#F*$\"3*)*\\(=(e`m!>F*7$$!3Cmm;u\"HW.#F*$\"3)om;HT&y#)>F*7$$ !3?LL3n_J+>F*$\"3SL$ekOU)\\?F*7$$!3q****\\sZL\\CF *7$$!3MmmmcddF5F*$\"31nmm@@@'[#F*7$$!3(*)**\\7B67s)FA$\"31+vVQ%RRc#F*7 $$!3?nmm;VByR%FA$\"3G+DJS)3,y#F*7$$!3-jm;zp\"y*GFA$\"3%omT5:4^&GF*7$$!3o jm\"H-V._\"FA$\"3/nT&)[G)R#HF*7$$!3-)RLL3F^X$!#?$\"3_L$ekVs#)*HF*7$$\" 3fnmT&yo(3:FA$\"3;L3FR%Qa2$F*7$$\"3'>+]7VLA&GFA$\"3K+Dcr;hUJF*7$$\"3jp m;a?@.VFA$\"3[L$3Fgg^@$F*7$$\"3w******\\\\@-eFA$\"3)*****\\Z26!H$F*7$$ \"3%Q++v$oposFA$\"3=+](=%[VjLF*7$$\"3g0+voMf(o)FA$\"3G+vVt'zVV$F*7$$\" 3#)***\\ii.j-\"F*$\"3\"***\\78=:8NF*7$$\"3%GLL$oT'y;\"F*$\"3Umm;%3KRe$ F*7$$\"3'3++DE5!>8F*$\"3W++DJ^]fOF*7$$\"3Mm;a)3rfX\"F*$\"3HF*$\"3#pm\"z*>1*f WF*7$$\"3'pmmmV,&eIF*$\"3[LLL=2DHXF*7$$\"3<+](o(GP1KF*$\"3`+vVQk=.YF*7 $$\"3g+]78Z!zM$F*$\"3I+DccB&Rn%F*7$$\"3++++++++NF*$\"3+++++++]ZF*-Fb\\ o6&Fd\\oFh\\oFe\\oFh\\o-%*THICKNESSG6#\"\"#-F$6%7S7$F^dq$!3+++++++]ZF* 7$Fcdq$!3ummTg*4Pn%F*7$Fhdq$!3XLe*[SItg%F*7$F]eq$!3\"om\"H_'zE`%F*7$Fb eq$!39n;/mS`dWF*7$Fgeq$!3YLekLcu#Q%F*7$F\\fq$!3sm\"HK=2MJ%F*7$Fafq$!3& )*\\iSB6;C%F*7$Fffq$!39m\"H2wft;%F*7$F[gq$!35+D\"GTYL4%F*7$F`gq$!37LL3 (e9s,%F*7$Fegq$!3gm;aLw:]RF*7$Fjgq$!3i***\\iQnY(QF*7$F_hq$!3=++vor'))z $F*7$Fdhq$!3q***\\Ph>es$F*7$Fihq$!3im\"HdV&[fOF*7$F^iq$!3YLL3#o21e$F*7 $Fciq$!3&HLL$yyy8NF*7$Fhiq$!3&**\\i:cggV$F*7$F]jq$!3eLL$eresO$F*7$Fbjq $!3e*\\il=sKF*7$F\\[r$!3:L$e*[3*[9$F*7$Fa[r$! 3'H$e9^r,wIF*7$Ff[r$!3[m;ajvs,IF*7$F\\\\r$!3%o;H2chX#HF*7$Fa\\r$!3o*\\ P%G$)QdGF*7$Ff\\r$!3_m;H(RR[y#F*7$F[]r$!3,++]_#*))4FF*7$F`]r$!3\")**\\ 7e^cOEF*7$Fe]r$!3s*\\ilK?cc#F*7$Fj]r$!35+](o=[o[#F*7$F_^r$!3eLL$e\"z1; CF*7$Fd^r$!3d***\\(o[\\SBF*7$Fi^r$!3#o;HdX9?F#F*7$F^_r$!3]***\\iFZr>#F *7$Fc_r$!3VLe*[4.n7#F*7$Fh_r$!3))*\\7yioI0#F*7$F]`r$!3ELL3xu2\")>F*7$F b`r$!3;+D19%4d!>F*7$Fg`r$!33mmm657L=F*7$F\\ar$!3]m;/')))))e " 0 "" {MPLTEXT 1 0 37 "x^2-x*y+y^2 = 9;\nimplicitdiff(%,y, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&F'F)% \"yGF)!\"\"*$)F+F(F)F)\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*& \"\"#\"\"\"%\"xGF'F'%\"yG!\"\"F',&F(F'*&F&F'F)F'F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 259 "" 0 "" {TEXT 258 8 "Question" }{TEXT 373 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 51 "Find the equation of \+ the tangent line to the curve " }{XPPEDIT 18 0 "x^2*y^3-x^3*y^2=12" "6 #/,&*&%\"xG\"\"#%\"yG\"\"$\"\"\"*&F&F)F(F'!\"\"\"#7" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\"\"!\"\"\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solution" }{TEXT 374 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 19 "Note that the point" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\"\"! \"\"\"\"#" }{TEXT -1 61 " does indeed lie on the given curve, since, i f we substitute " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 42 " in the left side of the equation, we get " }{XPPEDIT 18 0 "(-1)^2*`.` *2^3-(-1)^3*`.`*2^2 = 8+4;" "6#/,&*(,$\"\"\"!\"\"\"\"#%\".GF'F)\"\"$F' *(,$F'F(F+F*F'F)F)F(,&\"\")F'\"\"%F'" }{TEXT -1 70 " = 12, which is th e value appearing on the right side of the equation." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Differentiating both si des of the equation: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y^3-x^3*y^2 = 12" "6#/,&*&%\"xG\"\"#%\"yG\"\"$\"\"\"*&F&F)F( F'!\"\"\"#7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "with respe ct to " }{TEXT 375 1 "x" }{TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x*`.`*y^3+x^2*`.`*3*y^2" "6#,&**\"\"# \"\"\"%\"xGF&%\".GF&%\"yG\"\"$F&**F'F%F(F&F*F&F)F%F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-3*x^2*`.`*y^2-x^3*`.`*2*y" "6#,(*&%#dyG\"\"\"%#d xG!\"\"F&**\"\"$F&*$%\"xG\"\"#F&%\".GF&%\"yGF-F(**F,F*F.F&F-F&F/F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Collecting the te rms involving " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 109 " on the left hand side of the equation, and the remainin g terms on the right hand side of the equation gives:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2*y^2" "6#*(\"\"$\"\"\"*$%\"xG \"\"#F%%\"yGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx -2*x^3*y" "6#,&* &%#dyG\"\"\"%#dxG!\"\"F&*(\"\"#F&*$%\"xG\"\"$F&%\"yGF&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 3*x^2*y^2-2*x*y^2;" "6#/*&%#dyG\"\"\"%#dxG !\"\",&*(\"\"$F&*$%\"xG\"\"#F&%\"yGF.F&*(F.F&F-F&F/F.F(" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {XPPEDIT 18 0 "``(3*x^2*y^2-2*x^3*y) = 3*x^2*y^2-2*x*y^3;" "6#/-%!G6#, &*(\"\"$\"\"\"*$%\"xG\"\"#F*%\"yGF-F**(F-F**$F,F)F*F.F*!\"\",&*(F)F**$ F,F-F*F.F-F**(F-F*F,F*F.F)F1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx = (3*x^2*y^2-2*x*y^3)/(3*x^2*y^2-2*x^3*y);" "6#/*&%#dyG\"\"\"%#dx G!\"\"*&,&*(\"\"$F&*$%\"xG\"\"#F&%\"yGF/F&*(F/F&F.F&F0F,F(F&,&*(F,F&*$ F.F/F&F0F/F&*(F/F&*$F.F,F&F0F&F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x *y^2*(3*x-2*y))/(x^2*y*(3*y-2*x))" "6#/%!G**%\"xG\"\"\"*$%\"yG\"\"#F', &*&\"\"$F'F&F'F'*&F*F'F)F'!\"\"F'*(F&F*F)F',&*&F-F'F)F'F'*&F*F'F&F'F/F 'F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(y*(3*x-2*y))/(x*(3*y-2*x))" "6#/ %!G*(%\"yG\"\"\",&*&\"\"$F'%\"xGF'F'*&\"\"#F'F&F'!\"\"F'*&F+F',&*&F*F' F&F'F'*&F-F'F+F'F.F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The gradient of the tangent line at \+ the point" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\"\"!\"\"\"\"#" } {TEXT -1 40 " is obtained by substituting the values " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2 " "6#/%\"yG\"\"#" }{TEXT -1 55 " in the expression obtained for the de rivative giving: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 2*(3*`.`*(-1)-2*`.`*2)/((-1)*(3*`.`*2-2*`.`*(-1)));" "6#*(\"\"#\"\"\", &*(\"\"$F%%\".GF%,$F%!\"\"F%F%*(F$F%F)F%F$F%F+F%*&,$F%F+F%,&*(F(F%F)F% F$F%F%*(F$F%F)F%,$F%F+F%F+F%F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(-14)/(-8)" "6#/%!G*&,$\"#9!\"\"\"\" \",$\"\")F(F(" }{XPPEDIT 18 0 "``=7/4" "6#/%!G*&\"\"(\"\"\"\"\"%!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The equation of the tangent line can then be obtained by using the point-slope form for the equation of a straight line, namel y" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y-y[1]=m*(x-x[1 ])" "6#/,&%\"yG\"\"\"&F%6#F&!\"\"*&%\"mGF&,&%\"xGF&&F-6#F&F)F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 0 "x[ 1]=-1,y[1]=2" "6$/&%\"xG6#\"\"\",$F'!\"\"/&%\"yG6#F'\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " m=7/4" "6#/%\"mG*&\"\"(\"\"\"\"\"%!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-2=7/4" "6#/,&%\"yG\"\"\" \"\"#!\"\"*&\"\"(F&\"\"%F(" }{XPPEDIT 18 0 "``(x+1)" "6#-%!G6#,&%\"xG \"\"\"F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=7*x/4+15/4" "6#/% \"yG,&*(\"\"(\"\"\"%\"xGF(\"\"%!\"\"F(*&\"#:F(F*F+F(" }{TEXT -1 2 ". \+ " }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 369 340 340 {PLOTDATA 2 "6*-%'CURVESG6_^l7$7$$!3T9dG9dG9x!#<$!3xUtZK4[L;!#=7$$!3+#*)oi(3!)=x F*$!3;(\\C71`Ej\"F-7$F.7$$!\")\"\"!$!39xyysI**[z!#>7$7$F5$\"3\"RZ9JpQ- S\"F-7$$!3`o#*He7FGxF*$\"3:S%Q\"GmcG9F-7$7$F($\"3asL!z9VjW\"F-F?7$7$F( $!3]UtZK4[L;F-7$$!3+%Qi5(H>9xF*$!3Dkc+n.gL;F-7$7$$!3#)G9dG9dGuF*$!3kst >Wp<)o\"F-FL7$7$$!3I:dG9dG9xF*FF7$$!3_Km'p%p6ZuF*$\"3mMCK4N9v9F-7$7$FS $\"334'Q*3qj*\\\"F-Fen7$FR7$$!3b_I`#*pJAuF*$!3y,A!zlPlp\"F-7$7$$!3NUr& G9dG9(F*$!3)oG**er&G_sUr/C6x\"F-7$7$$!3wcG9dG9doF*$!3s2-O))H\"z#=F-F[q7$Fdp7 $$!3cvu09y/()oF*$\"3_%e8*f?<\"f\"F-7$7$$!3kdG9dG9doF*$\"3`onm&)f,M;F-F gq7$7$F^r$!3*z?g$))H\"z#=F-7$$!3$yr$GZ>([$oF*$!3EeGk(\\M,'=F-7$7$$!31s &G9dG9d'F*$!3,4-QG`v<>F-Ffr7$F]r7$$!31wdg`4X3mF*$\"3)Rs&4v9(Qm\"F-7$7$ F]s$\"3DVzJ.(f#>'e%y'QlF*$!3KSejq?F-7$7$$!3[' G9dG9dG'F*$!3o_<+#)GID?F-F\\t7$7$$!3=r&G9dG9d'F*$\"3`VzJ.(f#>S!>K%G?=F-F]u7$7$$!3Fe%=\"RmV3 gF*$!\"'F77$$!3)=xSEF6Y+'F*$!3coSP9**G&*fF*7$7$$!3)3+++++++'F*$!3WE,Yc (p(*)fF*F\\v7$Fbt7$$!3<2`h?mDSiF*$!3'Gz\\/kqp4#F-7$7$Fcv$!3'41P*=d7b@F -Fhv7$7$Fct$\"3z=S!>K%G?=F-7$$!33&)*=O%fEbgF*$\"3%R=3TU;-&=F-7$7$Fcv$ \"3yVY,kt#4%>F-Few7$Fbv7$$!3_d9CGr$3'eF*$!3u\\Bg)>3.&eF*7$7$$!3IRX+#z&*)*QfF*$!3e#QB1$4$eD#F-7$7$ Fby$!3-vfgRYG8BF-Fgy7$7$$!3y,++++++gF*$\"3]VY,kt#4%>F-7$$!3)QIhvzJ8y&F *$\"3IN\\JO@aq>F-7$7$Fby$\"3-Cz*[?/i3#F-Ffz7$7$$!3T9dG9dG9dF*Fdy7$$!3# \\4`YpWKd&F*$!3KA%3y&)*QgbF*7$7$$!32*yfKGT+V&F*$!3cWAhIlK;aF*Fc[l7$7$$ !3'**yfKGT+V&F*$!3XXAhIlK;aF*7$$!3AsA(*zsQHaF*$!3i%Q\"o'>$\\:aF*7$7$$! 3rH9dG9dGaF*$!3BzShs/\\9aF*Fd\\l7$7$Fby$!3IvfgRYG8BF-7$$!3d$***e)\\WSj &F*$!3aCASk(fAX#F-7$7$F[]l$!3#GLsYzIz]#F-Fc]l7$7$Fa[l$\"3YBz*[?/i3#F-7 $$!3_XB2JOx4bF*$\"3#3#zo#yO^6#F-7$7$F[]l$\"3ELeLwEviAF-F`^l7$7$$!3#)G9 dG9dGaF*$!3MyShs/\\9aF*7$$!3)yF<7u\"*fG&F*$!39^sGiZ7q_F*7$7$$!3,Xr&G9d G9&F*$!3#Rsa7&F*F__l7$Fi]l7$$!3&ofPg+zUK&F*$!3ix+#GmDzp#F-7$7$$!38 Wr&G9dG9&F*$!3cj?c&o(>]FF-F[`l7$7$$!3gI9dG9dGaF*$\"3aLeLwEviAF-7$$!3lC `Z#3F7C&F*$\"3y49naH\\!H#F-7$7$Fb`l$\"3mo*pzS![zCF-F\\al7$7$Fb`lFh_l7$ $!3up:,\\fJU^F*$!3)*=\\x'eU]7&F*7$7$$!3/\"3>5JW;9&F*$!3=o$=fz*[C^F*Fga l7$F]bl7$$!3GNX\">V1#**\\F*$!3])H(o,?Qz\\F*7$7$$!3aeG9dG9d[F*$!3`'fHS- '3N[F*Fcbl7$Fa`l7$$!3ldh;%G)43]F*$!3l&GDka=\"4IF-7$7$Fjbl$!3+X'GCLcb0$ F-F_cl7$Fbal7$$!3!=(QqURaw\\F*$\"3bP.B)36`]#F-7$7$$!3kdG9dG9d[F*$\"3'f 5Y()=D&[FF-Ficl7$Fibl7$$!3-yr*f'pzb[F*$!3G@Z*GzFS$[F*7$7$$!37\")y%Rk;T &[F*$!3!4\\C71`E$[F*Fedl7$F[el7$$!3b)4#f?s28ZF*$!3'[gkQ3pzo%F*7$7$$!3' HdG9dG9d%F*$!3a47\\kiGVXF*Fael7$Fecl7$$!3a5d!RWJKo%F*$!3?RR+DR.4MF-7$7 $Fhel$!3=/ObNH#fW$F-F]fl7$7$FjblFbdl7$$!3K/mxnJ)or%F*$\"3Om\"p74H9x#F- 7$7$Fhel$\"3]>q*ec4n3$F-Fhfl7$Fgel7$$!3[$e;avb+d%F*$!3$QdG+c=Aa%F*7$7$ $!3EGx\"3MI$oXF*$!3i81`Ej\"3a%F*Fbgl7$7$$!3;Hx\"3MI$oXF*F[hl7$$!3F(*ym %f()yU%F*$!3Cb3nYkf&R%F*7$7$$!3O(G9dG9dG%F*$!3_z;^PYVF*$!3y&=<.dX9$RF-7$7$Fhhl$!3ogg'p:CI&RF-F]il7$F^gl7$$!33/# 3)>&[QY%F*$\"3K.]C[#G_5$F-7$7$Fhhl$\"3B6Pdiaz<=5%F*7$7$$!37GnY;gI/SF*$!3 ieG9dG9dRF*F`[m7$Ff[m7$$!3+3\"f]zLD+%F*$!3mU`ZmZbaRF*7$7$$!3y,++++++SF *$!3uO$p-ZB6&RF*F\\\\m7$7$Fc\\m$!3w!**)[jYl$f%F-7$$!3_Ux>V#[b-%F*$!39s K;3/-^XF-7$7$F\\]m$!3qsK;3/-^XF-Fcil7$F]jl7$$!3c$zWDc,(>UF*$\"3>,=U<)3 )HNF-7$7$Fc\\m$\"3+)HQCFqg2%F-Fe]m7$Fb\\m7$$!3)p`M\\([LiQF*$!3lM,[97#f !QF*7$7$$!3Bi?[NicFPF*$!3!4)*[C71`m$F*F_^m7$Fe^m7$$!3\"pQBJnOAs$F*$!3_ J>#fy%=dOF*7$7$$!3>;dG9dG9PF*$!3k>`U*)G6YOF*F[_m7$Fh\\m7$$!3Oq,w\">$\\ $*RF*$!30yHh1G[Z%F-F]am7$7$$!3u:dG9dG9 PF*Fd_m7$$!3wC&4.44Qe$F*$!3yFzQR>u1NF*7$7$$!3;H-O_^BdMF*$!3u-^v(QpMP$F *Fjam7$7$$!3gH-O_^BdMF*$!3=.^v(QpMP$F*7$$!3!Q4fVJ%3YMF*$!3c=M%eA\"ebLF *7$7$$!3gI9dG9dGMF*$!3)>,.V^#RILF*F[cm7$F]`m7$$!3ie]:c[\\@OF*$!3'*fNyW F\"))\\&F-7$7$Fbcm$!3%*3t%G!\\j$*eF-Fgcm7$7$Fb_m$\"3!>nj.Vl>Z%F-7$$!39 Az(>nQ$yMF*$\"3#\\O>gZ=Cr%F-7$7$Fbcm$\"34t9_3;#Q!\\F-Fddm7$Facm7$$!35u 6*f\"yW5LF*$!3\\f5\"f[(G-KF*7$7$$!3E5L*4&G0(>$F*$!3YD71`Ej\"3$F*F^em7$ 7$$!3q5L*4&G0(>$F*Fgem7$$!3)QvYX!4ywJF*$!3MD6i`>)p/$F*7$7$$!3,Xr&G9dG9 $F*$!3CTW!)e2I'*HF*F]fm7$F]dm7$$!3)[\"*f0-:0?$F*$!3![hm&=PX!)oF-7$7$Fd fm$!3T`)Riu:N,(F-Fifm7$7$$!3;I9dG9dGMF*$\"3`s9_3;#Q!\\F-7$$!3h#)[Y\"GZ AB$F*$\"3_!4K%[!Qr6&F-7$7$Fdfm$\"31x*=FS;>\\&F-Fhgm7$Fcfm7$$!3[4b\"HnM d/$F*$!3/ZM4%e***))GF*7$7$$!3sO\"*4%HJO&HF*$!3IZtO=fz*y#F*Fbhm7$7$Fihm $!3wZtO=fz*y#F*7$$!3**\\4Zy'R$>HF*$!31)z5!3jEEFF*7$7$$!3UfG9dG9dGF*$!3 0I0v7$R!HEF*Faim7$7$Fhim$!3YK]4Y]vd&)F-7$$!3]:g#RFWr/$F*$!3A[?5bxQpuF- 7$F`jmF_gm7$F^hm7$$!3'=wR5=\\&**HF*$\"3aErX70mecF-7$7$Fhim$\"3K#Rv/)>( \\J'F-Fgjm7$7$Fhim$!3gH0v7$R!HEF*7$$!3Y)[c?/Oiz#F*$!3[1uzW6L$Q#F*7$7$$!3/@ek.mOwDF*$!3v#fz*[C71AF*Fc\\n7$7$Fj\\n$!3 J#fz*[C71AF*7$$!3/j?vjUmvDF*$!3g*f?=*fz,AF*7$7$$!3%QdG9dG9d#F*$!3A2+xI L/*=#F*Fb]n7$7$Fi]n$!3J/>5^5HD6F*7$$!3T7()=74GJEF*$!3K#3/-^v(Q5F*7$Fa^ nF]jm7$F][n7$$!3aK=@7Bs'y#F*$\"3tA1^8M:.kF-7$7$$!3A5*\\D#HF*7$7$$!3)e`:O;LUW# F*$!3\"[r&G9dG9>F*F^`n7$7$$!3WNbhjJBWCF*$!3f9dG9dG9>F*7$$!3[D\")H'y5UV #F*$!3mBA$e92Ew\"F*7$7$$!3iKx$*py!pR#F*$!3KP=fz*[Ci\"F*F_an7$Fean7$$!3 '=R-&)ysyV#F*$!3Eu#eu$*HqY\"F*7$7$$!33\"R.=t.YW#F*$!3#)fz*[C71L\"F*F[b n7$FabnF^^n7$Fj_n7$$!3mSS([O:'eBF*$\"3e*H45V\"3nyF-7$7$$!3E)G9dG9dG#F* $\"3s[R\"4jRNQ)F-Fhbn7$7$F_cn$\"3#)\\R\"4jRNQ)F-7$$!3K\"pm]$=(p:#F*$\" 3^@(e?:/es)F-7$7$$!3m-++++++?F*$\"3%eV\"y\"y=-***F-Fgcn7$7$F^dn$\"3uM9 y\"y=-***F-7$$!3!R4p$Rf9'*>F*$\"39K'3])\\9+5F*7$7$$!3kjn,,^$>*>F*$\"3R gIlK;3/5F*Ffdn7$F\\en7$$!3S/6Y1.K%z\"F*$\"3gMO!=DJe3\"F*7$7$$!33^:P'Q9j\"F*$\"3o)R1d*fH67F*7$7$$ !3MIOtZ!)Rm:F*$\"3)y$pMn$=fH\"F*F^fn7$Fdfn7$$!3]l)[#e\"f)y9F*$\"3iX=F*7$7$$! 31Ia#=+#eh5F*$\"35$pMn$=fz=F*Fain7$7$Fhin$\"3)GpMn$=fz=F*7$$!3tG8l/nsA '*F-$\"3395xJW(p)>F*7$7$$!3)*pmk&>Qf+*F-$\"3Oq&G9dG9<#F*F`jn7$Ffjn7$$! 3@'eI&)yA(4))F-$\"3E()RgB'od>#F*7$7$$!3#4gG9dG9d)F-$\"3K+glr?ctAF*F\\[ o7$Fb[o7$$!3;%RZ49wY\"yF-$\"35'HB2XofQ#F*7$7$$!3'4)>gbPfgwF-$\"35[C71` EjCF*Fh[o7$7$$!33#)>gbPfgwF-Fa\\o7$$!3-$o=bD\"zCpF-$\"3wS7p+)4pe#F*7$7 $$!3I\"4s],veh'F-$\"3#eK;3/-^v#F*Fg\\o7$F]]o7$$!3K@!\\SWryH'F-$\"3;Y@k a6r9GF*7$7$$!3vEz&zi/C!fF-$\"3a.-^v(Qp/$F*Fc]o7$Fi]o7$$!3q*epy(\\?SeF- $\"3%=)>')G0!)fIF*7$7$$!31XdG9dG9dF-$\"3h)eM\\sC\"RJF*F_^o7$Fe^o7$$!3m D7rV\"=$p^F-$\"3/)R&\\l46$G$F*7$7$$!3w!H.^3:38&F-$\"3\"33/-^v(QLF*F[_o 7$Fa_o7$$!3x%zB@V,%QXF-$\"3Yf&[Y1/0^$F*7$7$$!3_kE$=lq2Y%F-$\"3)*ez*[C7 1j$F*Fg_o7$7$$!33lE$=lq2Y%F-$\"3Ufz*[C71j$F*7$$!3'f8['4/goSF-$\"3'y8=1 )RNaPF*7$7$$!3-0v\\(GYm'RF-$\"3pO=fz*[C#RF*Fh`o7$7$$!3e0v\\(GYm'RF-$\" 3DO=fz*[C#RF*7$$!3ND>a,j+6PF-$\"3B&f79.l'4SF*7$7$$!3!)y'QA=0Tf$F-$\"3T 9dG9dG9UF*Fiao7$F_bo7$$!30XT3U5eLMF-$\"3?d=YdZ;tUF*7$7$$!3PPyPd&*z2LF- $\"3p\"fz*[C71XF*Febo7$F[co7$$!3T=:XfBs9KF-$\"3q:6+TmkUXF*7$7$$!3ajq*e ]]S3$F-$\"3'*oMn$=fzz%F*Faco7$Fgco7$$!3SiM%4#G\\RIF-$\"3Wcf(*H\\e;[F*7 $7$$!3T*4;Hf&e1HF-$\"3CYtO=fz*3&F*F]do7$Fcdo7$$!3?ox)z(>L(*GF-$\"3W-l* *\\4!R4&F*7$7$$!33))G9dG9dGF-$\"3M,lN5)31=&F*Fido7$F_eo7$$!3)pIT([(H)e DF-$\"3i=f%)*4i6N&F*7$7$Ffeo$\"3_B71`Ej\"Q&F*Feeo7$F[fo7$$!3WoLiE:g*=# F-$\"3c#*[**3[G0cF*7$7$F`fo$\"3!**4bxQpMn&F*F_fo7$Fefo7$$!3s6'pARM\")) =F-$\"3Y27DH)Gj'eF*7$7$Fjfo$\"33y*[C71`'fF*Fifo7$F_go7$$!3PlfT\\!p&R;F -$\"3q>$R'*QwF8'F*7$7$Fdgo$\"3MbG9dG9diF*Fcgo7$Figo7$$!3\"G$3yU8zK9F-$ \"3qPSeeA\\.kF*7$7$F^ho$\"3iKn$=fz*[lF*F]ho7$7$F^ho$\"3]Ln$=fz*[lF*7$$ !3UFyK-cPf7F-$\"3m*\\uS\"ehxmF*7$7$F[io$\"3*)41`Ej\"3%oF*Fjho7$F`io7$$ !3,z5:Xa&G6\"F-$\"3;C`Ghl[apF*7$7$Feio$\"3<([C71`E8(F*Fdio7$Fjio7$$!3) RQe)f97#))*F:$\"34_(yK(=fLsF*7$7$F_jo$\"3Wk$=fz*[CuF*F^jo7$Fdjo7$$!3h8 R[G=5:))F:$\"3)z,&fk(HX^(F*7$7$Fijo$\"3sTAhIlK;xF*Fhjo7$7$$!3A7R[G=5:) )F:$\"3hUAhIlK;xF*7$$!31XsD*))Rj*yF:$\"3oB#QB+#)pz(F*7$7$Fh[p$\"3))>hI lK;3!)F*Fg[p7$F]\\p7$$!33nnx!e`45(F:$\"3k^S-MWp!3)F*7$7$Fb\\p$\"3;(*** *********H)F*Fa\\p7$7$$\"3!f#G9dG9dGF-$\"3Gp2O@o@u`F*7$$\"3w/D]+IwXGF- F\\fo7$F`]p7$Fa]p$\"37UB8C]z#Q&F*7$7$$\"3#4Jo:*>$=U#F-$\"3y+^v(QpMn&F* Fd]p7$Fh]p7$Fi]p$\"3A6-J$GLzr&F*7$7$$\"3AJ#49?,\"y?F-F`goF^^p7$7$$\"3] J#49?,\"y?F-F`go7$Fc^p$\"3+4*)*Hmz[/'F*7$7$$\"3+-(o\"4cY'z\"F-FjgoFi^p 7$F]_p7$F^_p$\"3CbtvhM[ljF*7$7$$\"3o%ok0'>^j:F-FdhoFa_p7$Fe_p7$Ff_p$\" 3#)*\\,xu96o'F*7$7$$\"3wU\\uy#f\"p8F-FaioFi_p7$F]`p7$F^`p$\"3GA)41A.G* pF*7$7$$\"3kj&)[2;t07F-F[joFa`p7$Fe`p7$Ff`p$\"3)*H?eCHL,tF*7$7$$\"33cj !*=VJn5F-FejoFi`p7$F]ap7$F^ap$\"31h'4m*zI2wF*7$7$$\"3%=rj'[.+$\\*F:F_[ pFaap7$Feap7$Ffap$\"3n^!fu/*>6zF*7$7$$\"3-kEt_cj![)F:F^\\pFiap7$F]bp7$ F^bp$\"3OCUSkjP8#)F*7$7$$\"36FdXJ/E2wF:Fh\\pFabp7$7$$\"3)Go&G9dG9dF-$ \"3(e+:!=+`?NF*7$$\"3a'fS$*\\wdO&F-F``o7$F^cp7$$\"3[st.F&HlK&F-$\"3;A$ Q(p(=-n$F*7$7$$\"3krdRuH<)e%F-FaaoFbcp7$7$$\"33rdRuH<)e%F-Fgao7$$\"3++ EqWK8uWF-$\"3OBy7'p@\"\\SF*7$7$$\"3us-Zaz[CSF-FbboF_dp7$Fedp7$$\"3gFk1 sn.pQF-$\"3OCQQ8Zw-WF*7$7$$\"3I'4#R-K/1OF-F^coFidp7$F_ep7$$\"3AOR(=3y$ GMF-$\"3r\"z=de6'RZF*7$7$$\"3#)=%41H8!*G$F-FjcoFcep7$Fiep7$$\"3p$G(3'> m/5$F-$\"3A>!=t6SWvR@CF*7$$\"32/*\\+1(pH8F*Fa\\o7$Ffip7$Fchp$\"3?F+v\\SXWD F*7$Faip7$$\"3#=V1[m6WY\"F*$\"3+j_#[1dmU#F*7$7$$\"336dG9dG9)>F*$\"3kj$GH\" op\"[#F*7$7$$\"3W'*************>F*$\"3e\\H9uc:#[#F*Fg[q7$F]\\q7$$\"3sK r&=kv/=#F*$\"3IOlxj!f2d#F*7$7$$\"3/#G9dG9dG#F*$\"3(fd!=IQX1EF*Fc\\q7$F i\\q7$$\"3))>.*y5Y!oBF*$\"3KWzIlf+rEF*7$7$$\"3[X/qHJ&o[#F*F`]oF_]q7$Fe ]q7$$\"3=@2!Q6WFa#F*$\"3ESLRl6S%y#F*7$7$$\"3in&G9dG9d#F*$\"3'Ga'RlSu&z #F*Fi]q7$F_^q7$$\"30['4r=H(4FF*$\"3:&Q2_duc!HF*7$7$$\"3?`G9dG9dGF*$\"3 bJ2jF*RL,$F*Fe^q7$F[_q7$$\"3_UUQ'p;G(GF*$\"3Zq]Go!H4.$F*7$7$$\"39rb'*R u_')GF*F\\^oFa_q7$7$Fh_q$\"35.-^v(Qp/$F*7$$\"3TIV\"eK\\*HIF*$\"3)zNvd/ mA;$F*7$7$$\"3!)Qr&G9dG9$F*$\"3%p2mDL&yYKF*F^`q7$Fd`q7$$\"3)3WTwx8#4V$F*7$7$$\"3QC9dG9dGMF*$\"3Q<\"\\l M1)*\\$F*Fgaq7$F]bq7$$\"3eFFrU+^!\\$F*$\"3OU$RZPYtc$F*7$7$$\"3m,Y3(fQr a$F*F``oFcbq7$Fibq7$$\"3'*QZj>f;%F*FbboFbeq7$Fheq7$$ \"3jRr?v!*)>B%F*$\"3mAPIz@;pUF*7$7$$\"39\"G9dG9dG%F*$\"3]q$Ht.L>J%F*F \\fq7$Fbfq7$$\"3]9(e\"e96zVF*$\"3/1Ub3Rs5WF*7$7$$\"3St0ZS)*>oWF*F^coFh fq7$F^gq7$$\"3av``C])\\_%F*$\"3[zZt97c`XF*7$7$$\"3um&G9dG9d%F*$\"3Sxq' zU24f%F*Fbgq7$Fhgq7$$\"3MO@tZ;TrYF*$\"3izw(GhLep%F*7$7$$\"3IH*e4^sxw%F *$\"33oMn$=fzz%F*F^hq7$7$$\"3=I*e4^sxw%F*Fjco7$$\"3ce.,4+#o\"[F*$\"3tt t$G4Y\"R[F*7$7$$\"3K_G9dG9d[F*$\"3EDj.,U&=([F*F]iq7$Fciq7$$\"3ss.q$)3w i\\F*$\"3y#f$Q[Y\">)\\F*7$7$$\"31c8?zdZl]F*FfdoFiiq7$F_jq7$$\"3n&G&fCZ &y5&F*$\"3L&4\\%)Q[b7&F*7$7$$\"3\"z8dG9dG9&F*$\"33G$f&>'eT:&F*Fcjq7$7$ $\"3-Pr&G9dG9&F*$\"3?F$f&>'eT:&F*7$$\"3))p>/n![MD&F*$\"3%\\9eL#>no_F*7 $7$$\"35j7_l%e=O&F*$\"3SC71`Ej\"Q&F*Fd[r7$Fj[r7$$\"3?d=QrbK)R&F*$\"3m& *>wNm_7aF*7$7$$\"3]B9dG9dGaF*$\"3Mi'*=BmTPaF*F`\\r7$Ff\\r7$$\"3;B0`WZm VbF*$\"3y?SX'y4fb&F*7$7$$\"3'4x%Gs))GdcF*F[^pF\\]r7$Fb]r7$$\"3g%*zk*H \"R)o&F*$\"3%4[1Fo=**p&F*7$7$$\"334dG9dG9dF*$\"3679R[kN@dF*Ff]r7$7$F]^ r$\"3A69R[kN@dF*7$$\"3xkh)e&3aLeF*$\"3$H^GN^&\\VeF*7$7$$\"3/+a\"=w>?&f F*$\"3=x*[C71`'fF*Fe^r7$F[_r7$$\"3A>/(*[3;yfF*$\"3[HjGbKh()fF*7$7$$\"3 m%*************fF*$\"3?&*o6O:z0gF*Fa_r-%'COLOURG6&%$RGBG$\"*++++\"F6$F 7F7Fb`r-F$6%7S7$F5$!3+++++++D5!#;7$$!3+nmmT)R[p(F*$!3;om\"Hspfr*F*7$$! 3#HL$e>;KHuF*$!3SK3FMGJ^#*F*7$$!3Lmm;4'=28(F*$!3vm;/mvvG()F*7$$!3vmm;k i8IoF*$!3Pm;Hi%QF?)F*7$$!3+LLeMD)4`'F*$!3SL3_N%>#zwF*7$$!3)om;HtGOD'F* $!3-nTg#G]Q>(F*7$$!3S***\\i$\\WmfF*$!3#*)\\P%Q'y7p'F*7$$!3Mmm\"H/R%pcF *$!3amT5D$=:<'F*7$$!3W++D^cQt`F*$!3!**\\(o*)[U`cF*7$$!3[KLL[$e)o]F*$!3 '4L$e4@]?^F*7$$!3Tmm;M0j+[F*$!3mm;zMM5^YF*7$$!3Q*****\\ap')\\%F*$!39** *\\PqrE7%F*7$$!3')*****\\noa>%F*$!3a***\\7=q?f$F*7$$!3q*****\\XyK!RF*$ !3q***\\iHP23$F*7$$!3]mm\"HuTzj$F*$!3PmT5]!)R;EF*7$$!3(HLL$G2VALF*$!3[ KLeuPDk?F*7$$!3oKLL8::bIF*$!3SKLL[^^'f\"F*7$$!3!)***\\iCUUu#F*$!3k*\\P 4$RU_5F*7$$!3WLLLj[.pCF*$!3*HLL$3,63dF-7$$!3C***\\iu)3n@F*$!3h\"*)\\Pf I0C%F:7$$!3!*)**\\(QYcz=F*$\"39>](=#)=wg%F-7$$!3gKL$eRj&z:F*$\"3Szm\"H dSw&)*F-7$$!3uKLe/'oSI\"F*$\"3]n\"z>%*zyY\"F*7$$!3!ommTD5p+\"F*$\"37L$ 3_02z)>F*7$$!3#[mm\"HCY#)pF-$\"3)Q$e*[2p!GDF*7$$!35'***\\PJ`&H%F-$\"3Y +v$4q\"G)*HF*7$$!3ugmm\"*ed$R\"F-$\"3PM$e*=U71NF*7$$\"3_********)HWg\" F-$\"3#*****\\K_xISF*7$$\"3o2++vORPXF-$\"3z,]7$*Q/WXF*7$$\"3?6+]Pp=vtF -$\"3'>]iSrd1/&F*7$$\"3k****\\_sg_5F*$\"3Q**\\(=pi?f&F*7$$\"3olmmO$GdL \"F*$\"3%\\mm\"*eCv3'F*7$$\"3t,++D0-Q;F*$\"3-.+v=f`;mF*7$$\"3qKL3x@%> \">F*$\"3?Ke*)4))*e4(F*7$$\"3*>++]*3T6AF*$\"3#R+]i1p*>wF*7$$\"3ImmT?w= $\\#F*$\"3#p;HdLyI6)F*7$$\"3[++v)[Dxy#F*$\"3H,DJ0'>&G')F*7$$\"3'pmm;4! pvIF*$\"3inmTgwXK\"*F*7$$\"3M***\\PMirP$F*$\"3U)\\i:5M+m*F*7$$\"3oNLL` f^nOF*$\"3vLL$=H:o,\"Fi`r7$$\"31ML$eXWW'RF*$\"3cL3xzxxo5Fi`r7$$\"3momT SU\"*eUF*$\"3$p\"H2#**4.7\"Fi`r7$$\"3&=+++R,&HXF*$\"3Q++DVFmn6Fi`r7$$ \"3lnm;*zC'R[F*$\"3vmT&)RV$>A\"Fi`r7$$\"3#RLLL(G+<^F*$\"3WLL$G]v/F\"Fi `r7$$\"3L++v`du7aF*$\"3-]i!p]IAK\"Fi`r7$$\"3=,+DE%4ep&F*$\"3A]Pf\\mwr8 Fi`r7$$\"\"'F7$\"3+++++++D9Fi`r-F]`r6&F_`rFb`rF``rFb`r-%*THICKNESSG6# \"\"#-F$6%7S7$F5F57$F[arF[ar7$F`arF`ar7$FearFear7$FjarFjar7$F_brF_br7$ FdbrFdbr7$FibrFibr7$F^crF^cr7$FccrFccr7$FhcrFhcr7$F]drF]dr7$FbdrFbdr7$ FgdrFgdr7$F\\erF\\er7$FaerFaer7$FferFfer7$F[frF[fr7$F`frF`fr7$FefrFefr 7$FjfrFjfr7$F_grF_gr7$FdgrFdgr7$FigrFigr7$F^hrF^hr7$FchrFchr7$FhhrFhhr 7$F]irF]ir7$FbirFbir7$FgirFgir7$F\\jrF\\jr7$FajrFajr7$FfjrFfjr7$F[[sF[ [s7$F`[sF`[s7$Fe[sFe[s7$Fj[sFj[s7$F_\\sF_\\s7$Fd\\sFd\\s7$Fi\\sFi\\s7$ F^]sF^]s7$Fc]sFc]s7$Fh]sFh]s7$F]^sF]^s7$Fb^sFb^s7$Fg^sFg^s7$F\\_sF\\_s 7$Fa_sFa_s7$Ff_sFf_s-%&COLORG6&F_`r$\"\"$!\"\"FgcsFgcs-%*LINESTYLEGF^` s-F$6%7S7$F5$\"\")F77$F[ar$\"3+nmmT)R[p(F*7$F`ar$\"3#HL$e>;KHuF*7$Fear $\"3Lmm;4'=28(F*7$Fjar$\"3vmm;ki8IoF*7$F_br$\"3+LLeMD)4`'F*7$Fdbr$\"3) om;HtGOD'F*7$Fibr$\"3S***\\i$\\WmfF*7$F^cr$\"3Mmm\"H/R%pcF*7$Fccr$\"3W ++D^cQt`F*7$Fhcr$\"3[KLL[$e)o]F*7$F]dr$\"3Tmm;M0j+[F*7$Fbdr$\"3Q***** \\ap')\\%F*7$Fgdr$\"3')*****\\noa>%F*7$F\\er$\"3q*****\\XyK!RF*7$Faer$ \"3]mm\"HuTzj$F*7$Ffer$\"3(HLL$G2VALF*7$F[fr$\"3oKLL8::bIF*7$F`fr$\"3! )***\\iCUUu#F*7$Fefr$\"3WLLLj[.pCF*7$Fjfr$\"3C***\\iu)3n@F*7$F_gr$\"3! *)**\\(QYcz=F*7$Fdgr$\"3gKL$eRj&z:F*7$Figr$\"3uKLe/'oSI\"F*7$F^hr$\"3! ommTD5p+\"F*7$Fchr$\"3#[mm\"HCY#)pF-7$Fhhr$\"35'***\\PJ`&H%F-7$F]ir$\" 3ugmm\"*ed$R\"F-7$Fbir$!3_********)HWg\"F-7$Fgir$!3o2++vORPXF-7$F\\jr$ !3?6+]Pp=vtF-7$Fajr$!3k****\\_sg_5F*7$Ffjr$!3olmmO$GdL\"F*7$F[[s$!3t,+ +D0-Q;F*7$F`[s$!3qKL3x@%>\">F*7$Fe[s$!3*>++]*3T6AF*7$Fj[s$!3ImmT?w=$\\ #F*7$F_\\s$!3[++v)[Dxy#F*7$Fd\\s$!3'pmm;4!pvIF*7$Fi\\s$!3M***\\PMirP$F *7$F^]s$!3oNLL`f^nOF*7$Fc]s$!31ML$eXWW'RF*7$Fh]s$!3momTSU\"*eUF*7$F]^s $!3&=+++R,&HXF*7$Fb^s$!3lnm;*zC'R[F*7$Fg^s$!3#RLLL(G+<^F*7$F\\_s$!3L++ v`du7aF*7$Fa_s$!3=,+DE%4ep&F*7$Ff_sFjuFdcsFjcs-%%TEXTG6$7$$\"#lFics$! \"$FicsQ\"x6\"-Fa]t6$7$Ff]t$\"\"*F7Q\"yFi]t-%+AXESLABELSG6%%!GFc^t-%%F ONTG6#%(DEFAULTG-%%VIEWG6$;F5Fd]t;FjuF]^t" 1 2 0 1 10 0 2 9 1 4 2 1.000000 47.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 260 5 "Notes" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 37 "By writing the \+ equation in the form " }{XPPEDIT 18 0 "y-x=12/(x^2*y^2)" "6#/,&%\"yG \"\"\"%\"xG!\"\"*&\"#7F&*&F'\"\"#F%F,F(" }{TEXT -1 14 ", we see that \+ " }{XPPEDIT 18 0 "x " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 372 8 "Question" }{TEXT 367 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given \+ " }{XPPEDIT 18 0 "y*exp(y) = x;" "6#/*&%\"yG\"\"\"-%$expG6#F%F&%\"xG" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 369 1 "x" } {TEXT -1 5 " and " }{TEXT 370 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the gradient of the tangent line to the graph of the equation given in (a) at the point" }{XPPEDIT 18 0 "``(exp(1),1); " "6#-%!G6$-%$expG6#\"\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solution" }{TEXT 368 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Differentiating both sides of \+ the given equation with respect to " }{TEXT 371 1 "x" }{TEXT -1 8 " gi ves: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*& %#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(y) + y * ex p(y)" "6#,&-%$expG6#%\"yG\"\"\"*&F'F(-F%6#F'F(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 1" "6#/*&%#dyG\"\"\"%#dxG!\"\"F&" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out " }{XPPEDIT 18 0 "dy/d x" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on the left side gives: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG \"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(exp(y)+y*exp(y)) \+ = 1;" "6#/-%!G6#,&-%$expG6#%\"yG\"\"\"*&F+F,-F)6#F+F,F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Dividing both sides by " } {XPPEDIT 18 0 "exp(y)+exp(y)*y;" "6#,&-%$expG6#%\"yG\"\"\"*&-F%6#F'F(F 'F(F(" }{TEXT -1 9 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 1/(exp(y)+y*exp(y));" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&F&F&,&-%$expG6#%\"yGF&*&F.F&-F,6#F.F&F&F(" }{TEXT -1 2 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(exp(y)*(1+y))" "6#/%!G*&\"\"\"F&*&-%$expG6#%\"yGF&,&F&F&F+F&F&!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) A t the point " }{XPPEDIT 18 0 "``(exp(1),1);" "6#-%!G6$-%$expG6#\"\"\"F )" }{TEXT -1 38 " the gradient of the tangent line is " }{XPPEDIT 18 0 "1/(exp(1)*(1+1))=1/(2*exp(1))" "6#/*&\"\"\"F%*&-%$expG6#F%F%,&F%F%F %F%F%!\"\"*&F%F%*&\"\"#F%-F(6#F%F%F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 " The tangent line has equation: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1=1/(2*exp(1))" "6#/,&%\"yG\"\" \"F&!\"\"*&F&F&*&\"\"#F&-%$expG6#F&F&F'" }{XPPEDIT 18 0 "``(x-exp(1)) " "6#-%!G6#,&%\"xG\"\"\"-%$expG6#F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-1=x/(2*exp(1))-1/2" "6#/,&%\"yG\"\"\"F&!\"\",&*&%\"xG F&*&\"\"#F&-%$expG6#F&F&F'F&*&F&F&F,F'F'" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x/(2*exp(1))+1/2" "6#/%\"yG,&*&%\"xG\"\"\"*&\"\"#F(-%$expG6#F( F(!\"\"F(*&F(F(F*F.F(" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 447 247 247 {PLOTDATA 2 "6)-%'CURVESG6$7Z7$$!+1[,nO!# 5$!+$fT.A*F*7$$!+z'REj$F*$!+`Q(R\\)F*7$$!+_XE)f$F*$!+'RE4/)F*7$$!+)H9& HNF*$!+!R@4R(F*7$$!+WSwgMF*$!+0tG**oF*7$$!+ONEBLF*$!+%=]>9'F*7$$!+GIw& =$F*$!+2rn[bF*7$$!+7?w5HF*$!+&[t,i%F*7$$!+'*4wNEF*$!+G-\\))QF*7$$!+*=E s,#F*$!+y-&>i#F*7$$!+#Q\"p)R\"F*$!+b$H&\\;F*7$$!*n;U`\"F*$!++=Je:!#67$ $\"+)\\Hf3\"F*$\"+lV[T)*F^o7$$\"+wm'\\B#F*$\"+9tJc=F*7$$\"+>\"GZU$F*$ \"+yE;KEF*7$$\"+06=bYF*$\"+[.-NLF*7$$\"*,)o\")e!\"*$\"+*zk)eRF*7$$\"*7 ,L9(Ffp$\"+bBlPXF*7$$\"*kIXD)Ffp$\"+^cT/]F*7$$\"*)=^0&*Ffp$\"+7+!)*[&F *7$$\"+xH;w5Ffp$\"+`b.TfF*7$$\"+6K@(>\"Ffp$\"+t#4mM'F*7$$\"+?%QrI\"Ffp $\"++ND$p'F*7$$\"+E,&yV\"Ffp$\"+My!>3(F*7$$\"+(3!e[:Ffp$\"+OAQ$R(F*7$$ \"+b`Qx;Ffp$\"+b\"out(F*7$$\"+U)*R\"z\"Ffp$\"+W(=t-)F*7$$\"+/=\\;>Ffp$ \"+r0*4L)F*7$$\"+N*3c.#Ffp$\"+QB[2')F*7$$\"+k^*)f@Ffp$\"+ok'R)))F*7$$ \"+?(GSF#Ffp$\"+j%fz7*F*7$$\"+!=PrR#Ffp$\"+[xN\"Q*F*7$$\"+qA,DDFfp$\"+ &>]Yj*F*7$$\"+7zKOEFfp$\"+YA_Z)*F*7$$\"+b=bcFFfp$\"+([-q+\"Ffp7$$\"+_ \\v!)GFfp$\"+0eBH5Ffp7$$\"+^ME-IFfp$\"+5_H]5Ffp7$$\"+s\"H)>JFfp$\"+B=1 q5Ffp7$$\"+heO]KFfp$\"+'4^84\"Ffp7$$\"+\"))ewO$Ffp$\"+\">D*46Ffp7$$\"+ ;U*G\\$Ffp$\"+x7@H6Ffp7$$\"+WgP1OFfp$\"+'QFi9\"Ffp7$$\"+b;WIPFfp$\"+Rt Nk6Ffp7$$\"+Fx\"Ffp7$$\"+! =+&)3%Ffp$\"+-7997Ffp7$$\"+FeR8UFfp$\"+w@pI7Ffp7$$\"+z^oLVFfp$\"+6iEY7 Ffp7$$\"+;%)pcWFfp$\"+o%Q=E\"Ffp7$$\"+TIpyXFfp$\"+GR%pF\"Ffp7$$\"+KMz! p%Ffp$\"+o0a!H\"Ffp7$$\"+IJF>[Ffp$\"+(4/eI\"Ffp7$$\"+.p=M\\Ffp$\"+;'y \">8Ffp7$$\"+a*3n0&Ffp$\"+t@;L8Ffp7$$\"+*=yR<&Ffp$\"+X&*GY8Ffp7$$\"#`! \"\"$\"+/!G,O\"Ffp-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe]lFd]l-F$6%7 S7$$!3++++++++]!#=$\"3;%RrqR,.3%F\\^l7$$!3]LLLe]wNPF\\^l$\"39mi**=W%GJ %F\\^l7$$!3kmmT&*4wNEF\\^l$\"3I;B(*e)y^^%F\\^l7$$!3WLL$3Q\"p)R\"F\\^l$ \"3,gQA)4DFu%F\\^l7$$!35ELLem@M:!#>$\"3+@[lh'z<(\\F\\^l7$$\"3]LLe*\\Hf 3\"F\\^l$\"3]]\"H(obu*>&F\\^l7$$\"3fmm\"znm\\B#F\\^l$\"3$*)Qai9*46aF\\ ^l7$$\"3:***\\77GZU$F\\^l$\"3^'y/P`V*HcF\\^l7$$\"3=nm\"z5\"=bYF\\^l$\" 3c>hE7FFceF\\^l7$$\"3g***\\i,)o\")eF\\^l$\"3a49G]g(=3'F\\^l7$$\"3vMLLG 6IVrF\\^l$\"3!4I0O\"o$RJ'F\\^l7$$\"3gmm;W1`a#)F\\^l$\"3Y*Ga-1O$=lF\\^l 7$$\"3*>++])=^0&*F\\^l$\"3y^6:+7W[nF\\^l7$$\"3!*****\\xH;w5!#<$\"3-x/' Q<\"\\zpF\\^l7$$\"3-++]6K@(>\"F_bl$\"3')Q%[gj]@?(F\\^l7$$\"3wm;z?%QrI \"F_bl$\"3e*zs)enM/uF\\^l7$$\"3YLL$o7]yV\"F_bl$\"3Kr'Qd]xZk(F\\^l7$$\" 3ILLL(3!e[:F_bl$\"31]qo&)QX[yF\\^l7$$\"3?+]7b`Qx;F_bl$\"3)>ti`$zP&3)F \\^l7$$\"3/LLLU)*R\"z\"F_bl$\"3!=)4cke4&H)F\\^l7$$\"3<+]70=\\;>F_bl$\" 3:p^Rr'*=D&)F\\^l7$$\"3)***\\PN*3c.#F_bl$\"3#)[n%zQ$HW()F\\^l7$$\"3wLL ek^*)f@F_bl$\"3M'3z180H(*)F\\^l7$$\"3UL$e4sGSF#F_bl$\"373)p`2UG=*F\\^l 7$$\"3ymmT!=PrR#F_bl$\"3F_[rJuG4%*F\\^l7$$\"3om;zqA,DDF_bl$\"3sl$\\c^+ Xk*F\\^l7$$\"37+](G\"zKOEF_bl$\"3%*\\cm'>a#\\)*F\\^l7$$\"3'om;f&=bcFF_ bl$\"3xit:y$Rq+\"F_bl7$$\"3%)*****H&\\v!)GF_bl$\"3H_28h_))H5F_bl7$$\"3 !)***\\AXjA+$F_bl$\"3a(eJ0]NA0\"F_bl7$$\"3')**\\7t\"H)>JF_bl$\"3#Rzzj] gQ2\"F_bl7$$\"3w***\\<'eO]KF_bl$\"3wh;T)Qry4\"F_bl7$$\"35LLL#))ewO$F_b l$\"3L:W%QBY%>6F_bl7$$\"3'*****\\[U6F_bl7$$\"3Sm ;zWgP1OF_bl$\"3G8F_bl7$$\"33M$eC/$pyXF_bl$\"3hrv()=N?U8F_bl7$$\"3]+++LM z!p%F_bl$\"3#fN\"RLK#GO\"F_bl7$$\"3WLL3JJF>[F_bl$\"3%>udJvbkQ\"F_bl7$$ \"3Vmmm/p=M\\F_bl$\"3#46i0'Hf29F_bl7$$\"3'***\\7b*3n0&F_bl$\"3Z1!*GK'H ,V\"F_bl7$$\"3A+](3>yR<&F_bl$\"3oihu-,q^9F_bl7$$\"3#)*************H&F_ bl$\"3;AV5>0)[Z\"F_bl-F^]l6&F`]lFd]lFa]lFd]l-%*THICKNESSG6#\"\"#-%%TEX TG6$7$$\"#dFj\\l$Fj\\lFj\\lQ\"x6\"-Fh]m6$7$$!#:!\"#$\"#9Fj\\lQ\"yF_^m- %*AXESTICKSG6$\"\"&\"\"$-%+AXESLABELSG6%%!GFa_m-%%FONTG6#%(DEFAULTG-%% VIEWG6$;$!\"&Fj\\lF[^m;$Fj\\lFe]l$\"$X\"Fe^m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y*exp(y) =x;\nimplicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"yG \"\"\"-%$expG6#F%F&%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$ *&-%$expG6#%\"yGF$,&F$F$F)F$F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 381 8 "Question" }{TEXT 376 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "x+sin(x*y) = y;" "6#/,&%\"xG\"\"\"-%$sinG6#*&F%F&%\"yGF&F&F+" }{TEXT -1 26 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\" \"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 378 1 "x" }{TEXT -1 5 " and " }{TEXT 379 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the equation of the tangent line to the graph of the e quation given in (a) at the point" }{XPPEDIT 18 0 "``(0, 0);" "6#-%!G6 $\"\"!F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solution" }{TEXT 377 2 ": " }}{PARA 0 "" 0 " " {TEXT -1 69 "(a) Differentiating both sides of the given equation wi th respect to " }{TEXT 380 1 "x" }{TEXT -1 8 " gives: " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+cos(x*y)" "6#,&\"\"\"F$-%$cosG 6#*&%\"xGF$%\"yGF$F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x*y],x)=dy /dx" "6#/-%%DiffG6$7#*&%\"xG\"\"\"%\"yGF*F)*&%#dyGF*%#dxG!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+cos(x*y)" "6#,&\"\"\"F$-%$cosG 6#*&%\"xGF$%\"yGF$F$" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "y+x" "6#,&%\"y G\"\"\"%\"xGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 2 " )" }{XPPEDIT 18 0 "`` =dy/dx" "6#/%!G*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+cos(x*y)*y + co s(x*y)*x" "6#,(\"\"\"F$*&-%$cosG6#*&%\"xGF$%\"yGF$F$F+F$F$*&-F'6#*&F*F $F+F$F$F*F$F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = dy/dx" "6#/*&%# dyG\"\"\"%#dxG!\"\"*&F%F&F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Collecting the terms involving " }{XPPEDIT 18 0 "dy/dx" " 6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 109 " on the left hand side of th e equation, and the remaining terms on the right hand side of the equa tion gives:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cos(x *y)*x" "6#*&-%$cosG6#*&%\"xG\"\"\"%\"yGF)F)F(F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx-dy/dx = -1-cos(x*y)*y;" "6#/,&*&%#dyG\"\"\"%#dxG! \"\"F'*&F&F'F(F)F),&F'F)*&-%$cosG6#*&%\"xGF'%\"yGF'F'F2F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on the left side \+ gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6# *&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos(x*y)*x -1) = -1-cos(x*y)*y;" "6#/-%!G6#,&*&-%$cosG6#*&%\"xG\"\"\"%\"yGF.F.F-F .F.F.!\"\",&F.F0*&-F*6#*&F-F.F/F.F.F/F.F0" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 9 "so that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = (-1-cos(x*y)*y)/(cos(x*y)*x-1);" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&,&F&F(*&-%$cosG6#*&%\"xGF&%\"yGF&F&F1F&F(F&,&*&-F-6#*& F0F&F1F&F&F0F&F&F&F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+cos(x*y)*y) /(1-cos(x*y)*x)" "6#/%!G*&,&\"\"\"F'*&-%$cosG6#*&%\"xGF'%\"yGF'F'F.F'F 'F',&F'F'*&-F*6#*&F-F'F.F'F'F-F'!\"\"F4" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) At the point \+ " }{XPPEDIT 18 0 "``(0,0);" "6#-%!G6$\"\"!F&" }{TEXT -1 40 " the gradi ent of the tangent line is 1. " }}{PARA 0 "" 0 "" {TEXT -1 36 " Th e tangent line has equation " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" } {TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 336 321 321 {PLOTDATA 2 "6(-%'CURVESG6gy7$7$$!3u+()[=dcTR!#<$!\"%\"\"!7$F+$!3Q H^U@3R,RF*7$7$F+$!3E=I43n!yK$F*7$$!3iTP9A\"yP*RF*$!3l/x'yu_\"37M]LF*$!3s%GHfY%pdOF*7$7$$!3O.-^v(QpM$F*$!3'G&3 6jUQfOF*Fin7$7$$!3!Q?5bxQpM$F*$!3]c$=DXPs]#F*7$$!3%e&=@'>))>O$F*$!3OK; 3/-^vCF*7$7$$!3Sb=@'>))>O$F*F]p7$$!3%>NQo]*GpLF*$!3yx'G.K@BX#F*7$7$F`o $!3/K-gvL^.CF*Fcp7$7$$!3=axQpMn$=$F*$!3[8BZe\\%)\\PF*7$$!3nxT#HWAMM$F* $!3I=fz*[C7m$F*7$7$Fcq$!3'y\"fz*[C7m$F*F_o7$7$F^q$!3ymrxs(*=_GF*7$$!3w ^nYrO'\\?$F*$!319dG9dG9GF*7$7$$!3@_nYrO'\\?$F*Fbr7$$!3\"o^Hf6&ePKF*$!3 AN')*4Q_$eFF*7$7$$!3WcFGr]o!H$F*$!3@tO=fz*[k#F*Fhr7$F^s7$F`oFho7$Fip7$ $!3\"oDqCrPIK$F*$!3?ON)=!)>4L#F*7$7$$!3u`xQpMn$=$F*$!3*QUtM'*H%4BF*Ffs 7$7$$!3.0`Ej\"3/-$F*$!3CXrpg=vRQF*7$$!3IYDH-ElNIF*$!3:fz*[C71$QF*7$Fgt F]q7$7$Fct$!3;()=1U\\'><$F*7$$!3ut['*444HIF*$!3w&z*[C71`JF*7$Fau7$$!3' 3A]`eZ%QIF*$!3U&pwSKXV8$F*7$7$$!3Tz&Qo!He8JF*$!3#\\v(QpMn$)HF*Fgu7$7$$ !3')z&Qo!He8JF*F`vF\\r7$7$F^q$!3LCMZj*H%4BF*7$$!3%Q'3R()yC\"=$F*$!3bl8 I!**Q'3BF*7$7$Fct$!3i14)=@#ROBF*Fjv7$Fbt7$$!3wz\\j#GW$F*7$7$$!3cSLsag>aHF*$ !3gO=fz*[CK$F*Fgx7$F]y7$Fct$!3g()=1U\\'><$F*7$F`w7$$!3)Hj\"o#pTs)HF*$! 3@^,@iE`SBF*7$7$F[x$!3'f$yb^mKzBF*Fgy7$Fjw7$$!3Ms\"4\\'RbXGF*$!3.1/f?d jUQF*7$7$$!3m.zeY@OYGF*FjtFaz7$7$$!3?.zeY@OYGF*Fjt7$$!3dJWHlN#\\$GF*$! 3ie')30qF%o$F*7$7$$!3%3h%>\\w#=%GF*FeqF^[l7$Fd[l7$F[x$!347uRng1/OF*7$F ]z7$$!3uT+)z\\Aqv#F*$!3K65h@t**4CF*7$7$$!3q1/-^v(Qp#F*$!38'42L8g@V#F*F \\\\l7$7$$!3bdz*[C71`#F*$!3AZs@6))z'\\#F*7$$!39v2sE&GEe#F*F]p7$F]]lFb \\l7$Fh\\l7$$!3a?3!eDyS\\#F*$!3O7*px@9M^#F*7$7$$!3Q3bxQpMnBF*$!39()>r6 hCnDF*Fb]l7$Fh]l7$$!3#)Q+38TZ0AF*$!3a\"oS2E`Mk#F*7$7$$!3BfIlK;3/AF*$!3 lzv2*HbRk#F*F^^l7$7$$!3251`Ej\"3/#F*$!3Ze*=z&3'>r#F*7$$!3f$>sTdC;?#F*F as7$7$F`_l$!3ltO=fz*[k#F*Fd^l7$Fj^l7$$!3GyWt]iKO>F*$!3#)f@#GT1Lv#F*7$7 $$!37h\"3/-^v(=F*$!35qV,FtffFF*Fg_l7$7$F^`l$!3oZG5kGa79F*7$$!3:tl'e*zV 7>F*$!3UYtO=fz*G\"F*7$Ff`l7$$!3pLm$>8\"yE>F*$!3Lcllx'>(Q7F*7$7$$!3yM1) GIp0%>F*$!3d0`Ej\"3/7\"F*F\\al7$Fbal7$$!3'>PpzzrD%>F*$!3=\\ba1\"\\H0\" F*7$7$$!3#zL'o-d=<>F*$!33YEj\"3/-^*!#=Fhal7$7$F_bl$!3>ZEj\"3/-^*Fcbl7$ $!3)ziB3oWT!>F*$!3Fz@#[&[HM#*Fcbl7$7$F^`l$!3OAH+$\\dl%*)FcblFhbl7$7$$! 3!4;3/-^v(=F*F``l7$$!3Qy[z-?\"3x\"F*$!3Cau&**oRcv#F*7$7$$!3>7dG9dG9F*7$Ffdl7$$!3g1#o8O4gs\"F*$!3u@u#\\l$=b>F*7$7$$!3#*)oM))z Vcw\"F*$!3uoMn$=fzz\"F*F\\el7$7$$!39*oM))zVcw\"F*$!3'*oMn$=fzz\"F*7$$! 3ms)y*G*R+z\"F*$!3<:$Hf=k$>'=F*$!3a+'R#HtRv9F*7$7$$!3'*44]@)Hb'=F*$!3E(QpMn$=f9 F*Fifl7$7$$!3=54]@)Hb'=F*FbglFc`l7$F^cl7$$!3w%o5oS'QG=F*$!3Hbgo'R4kK)F cbl7$7$$!3aC*oTS'>h$e!>$4]lvFcblFehl7$F[dl7$$!3eUX,%\\r5p\"F* $!3k6]r7F)*oEF*7$7$$!3QEMs7Ai(o\"F*FasF_il7$7$$!3gEMs7Ai(o\"F*Fd_l7$$! 3C>)[>,F:m\"F*$!3hehvdtCIDF*7$7$$!3!G6$3c.\\i;F*F]pF\\jl7$7$$!3e7J3c. \\i;F*F]p7$$!3ai?*f\"Gwh;F*$!33`#ff&\\hgBF*7$7$$!3-7]C;$pvm\"F*$!3^\"f z*[C71BF*Fijl7$7$$!3!=,XiJpvm\"F*Fb[m7$$!3U9brOG5%o\"F*$!3-JmNH%\\!o@F *7$7$$!3gu9^s&p!*o\"F*$!3m]v(QpMn8#F*Fh[m7$7$$!3#[Z6Ddp!*o\"F*Fa\\m7$F \\dlFddl7$F[il7$$!3))eR\")fO))=;F*$!3Vsit%HXA6(Fcbl7$7$$!3/jK;3/-^:F*$ !3-!f:4s.%\\oFcblFi\\m7$7$$!3EjK;3/-^:F*Fb]m7$$!3vnq=06A+:F*$!35g)pzu \"\\\\mFcbl7$7$$!3K93/-^v(Q\"F*$!3Q`B`e&H%fiFcblFh]m7$F^^m7$$!3MM%ysG[ xP\"F*$!3$H_^FopiA'Fcbl7$7$$!39%QPx%o4X8F*$!3@I=fz*[C7'FcblFd^m7$Fj^m7 $$!3s#pd-p..D\"F*$!3?48#e#)HY&eFcbl7$7$$!3Ql$=fz*[C7F*$!3w!*)*[1]UsdFc blF`_m7$7$$!3:l$=fz*[C7F*Fi_m7$$!3lZ!*fV6k@6F*$!37TVQ'>Ec\\&Fcbl7$7$$! 3W;fz*[C71\"F*$!3/;:T(3y0I&FcblF_`m7$Fe`m7$$!3m()\\%p[U0%**Fcbl$!39H') **zEYD^Fcbl7$7$$!3)\\nMn$=fz*)Fcbl$!3Nh_U)*ox0[FcblF[am7$Faam7$$!3yf,` ^*)G)o)Fcbl$!3v16(><)zIZFcbl7$7$$!3!z-![7Fv@yFcbl$!3s@9dG9dGWFcblFgam7 $F]bm7$$!3A,]4![DzX(Fcbl$!3HuYcAFU8VFcbl7$7$$!3g&=5bxQpM(Fcbl$!3c!)RM% Q'zlUFcblFcbm7$7$Fjbm$!3+!)RM%Q'zlUFcbl7$$!3:'**4`RoGB'Fcbl$!374(3q*=a !*QFcbl7$7$$!3@'p&G9dG9dFcbl$!3Kw+H$>guk$FcblFbcm7$7$Ficm$!3)o2!H$>guk $Fcbl7$$!3%)\\5J@!=I1&Fcbl$!3,5A,4TQ5MFcbl7$7$$!3#o?hIlK;3%Fcbl$!3P%) \\aGFcbl7$7$$!3A\"ztw*pMhPFcbl$!3B85bxQpMFFcblF`em7$Ffem7$$!3u[![ztmb) GFcbl$!3120Y8Zt\"G#Fcbl7$7$$!3;FcblF\\fm7$7$$ !3)orO=fz*[CFcbl$!39n-71JWn>Fcbl7$$!3!*yDgez]x=Fcbl$!3RB5,Y\"=Pj\"Fcbl 7$7$$!33))3M!zB<;\"Fcbl$!3,01`Ej\"3/\"FcblF]gm7$7$$!3A))3M!zB<;\"FcblF fgm7$$!3I`)*y'R7zd*!#>$!3(pVQ(3pYS*)F_hm7$7$$!3'[FAhIlK;)F_hm$!3g:.)*[C71`'F_hmFihm7$F`im7$$\"3K-_[x%*>trF_hm$\"3Sb$4@a0yb (F_hm7$7$$\"3#=iAhIlK;)F_hm$\"3kMNMa(o)))))F_hmFfim7$F\\jm7$$\"3u$zp6G N)z9Fcbl$\"3Qsn!oW[&e;Fcbl7$7$$\"3;!G54))**4!>Fcbl$\"3U6-^v(QpM#FcblFb jm7$Fhjm7$$\"3Won1=rqR@Fcbl$\"3Mpl/n*3ym#Fcbl7$7$$\"3%=vO=fz*[CFcbl$\" 3*)z7b@g\">C$FcblF^[n7$7$Fe[n$\"3Mz7b@g\">C$Fcbl7$$\"3KAV&=*y=[FFcbl$ \"3W!Ri_r(QIPFcbl7$7$$\"3A')R!)Qw:!)GFcbl$\"3i>1`Ej\"3/%FcblF]\\n7$7$F d\\n$\"3=?1`Ej\"3/%Fcbl7$$\"3fpT65/!=J$Fcbl$\"3!o?)eys^R[Fcbl7$7$$\"35 *)=^mERbOFcbl$\"36G5bxQpMdFcblF\\]n7$Fb]n7$$\"3Grv#)414)z$Fcbl$\"3-hfS Z(o)GgFcbl7$7$$\"3oS71`Ej\"3%Fcbl$\"3$[(HP9!=K$oFcblFh]n7$7$$\"3CT71`E j\"3%FcblFa^n7$$\"39)=pgAusC%Fcbl$\"3c\"pvlH=nD(Fcbl7$7$$\"3c0aq+/$[H% Fcbl$\"3gO9dG9dGuFcblFg^n7$7$$\"351aq+/$[H%Fcbl$\"3rP9dG9dGuFcbl7$$\"3 ?zD$GQAKo%Fcbl$\"3!=3a\\j*H)\\)Fcbl7$7$$\"3kc+eya[)G/e N&Fcbl$\"3CD71`Ej\"3\"F*Fd`n7$Fj`n7$$\"3iD8&*>MZbumm[\"eFcbl$ \"3%Q$>FM^eS7F*7$7$$\"3$3v+/r!3WeFcbl$\"35mK;3/-^7F*F\\bn7$7$Fcbn$\"3K mK;3/-^7F*7$$\"3.WqijUp:iFcbl$\"3]X6FEqQo8F*7$7$$\"3M,Xx?%>@Q'Fcbl$\"3 %pIlK;3/U\"F*F[cn7$Facn7$$\"3*\\&)QJpCKj'Fcbl$\"3AO8J**fX%\\\"F*7$7$$ \"3'QakntFv)pFcbl$\"3!yMn$=fz*e\"F*Fgcn7$7$$\"3'\\akntFv)pFcbl$\"3-[tO =fz*e\"F*7$$\"3I!RaDUfv3(Fcbl$\"3,:lBElq;;F*7$7$$\"3!*=-^v(QpM(Fcbl$\" 3s%QlI!fuv;F*Fhdn7$7$$\"3-?-^v(QpM(FcblFaen7$$\"3@L'*Hb+$=j(Fcbl$\"30P 7>.iiHK)Fcbl$\"3P.l8RFTF=F*7$7$$\"3S4ZtO=fz*)Fcbl$\"3MZ_\"*)p3\"))=F*Fcfn7 $Fifn7$$\"3r(\\FoLL\\@*Fcbl$\"3i-o$zuaT!>F*7$7$$\"3')*pT+ghr+\"F*$\"3F H9dG9dG>F*F_gn7$7$Ffgn$\"3rH9dG9dG>F*7$$\"3s^^W8y>[5F*$\"3g^`G0m3U>F*7 $7$$\"3+?fz*[C71\"F*$\"30(HN'H$=B%>F*F^hn7$7$$\"3$*o$=fz*[C7F*$\"3#[hi Y@]\"G>F*7$$\"3ut(eeI;!>7F*$\"3\\H9dG9dG>F*7$F_in7$Fehn$\"3F(HN'H$=B%> F*7$Fjhn7$$\"3n#o?nvN\\A\"F*$\"3yG!RUu3\"G>F*7$7$$\"3k<3/-^v(Q\"F*$\"3 E)Q7/U/e)=F*Fiin7$7$$\"3'y\"3/-^v(Q\"F*Fbjn7$$\"32?Pu;?\\S9F*$\"3Sorcz o&Q(=F*7$7$$\"3!oEj\"3/-^:F*$\"3G*34TmKp$=F*Fhjn7$7$$\"3-nK;3/-^:F*$\" 3^*34TmKp$=F*7$$\"3GTkomegr;F*$\"3lNON:SY.=F*7$7$$\"3u:dG9dG9^v(QpMnBF*$\"3;ILBpJvk;F*Fi ^o7$7$F`_o$\"3Admg4\\F\"*GF*7$$\"3k[$3<,I9N#F*$\"3OuO=fz*[%HF*7$Fh_o7$ $\"3zk(f&\\%[[M#F*$\"3]6]uT,CoHF*7$7$$\"3]Q'yO9'*=K#F*$\"3@:dG9dG9JF*F ^`o7$Fd`o7$$\"3]jx\"4D!)3J#F*$\"3aH\"QH!*pG<$F*7$7$$\"3p!z\"z*e\"=HBF* $\"3]cxQpMn$G$F*Fj`o7$F`ao7$$\"3F91_)RY2L#F*$\"3l7A!*Hlk@LF*7$7$F`_o$ \"33S[8vPVYLF*Ffao7$F__o7$$\"3Mpk*)o)GSY#F*$\"3]&*)o4dor\"F*Faeo7$7$Fheo$\"35)R)Hyl^!4#F*7$$\"3Q:)e@y?;p#F*$\"37qMn$=fz4#F *7$F`fo7$$\"3uFWzu)=/p#F*$\"3AFzDgva,@F*7$7$$\"32'RbhkwFh#F*$\"3)4^v(Q pMnAF*Fffo7$7$$\"3i&RbhkwFh#F*F_go7$$\"3k.(*=:i/WDF*$\"3#*o)*p\\pzACF* 7$7$$\"3'Hxp(=%Qs`#F*$\"3\"=bxQpMnV#F*Fego7$F[hoF\\co7$7$Fheo$\"3!=u'[ &='poKF*7$$\"3[;Y#3\"QvcEF*$\"31cxQpMn$G$F*7$7$$\"3#phC3\"QvcEF*FhhoF] eo7$Fgeo7$$\"3N[,.^5YCFF*$\"3c&G4AH`us\"F*7$7$$\"34P4p2\\&Ru#F*F]]oF_i o7$Feio7$$\"3YGq+ZtTnFF*$\"3'GJN-*QF_=F*7$7$$\"3CGC!ev0*\\FF*FbinFiio7 $F_joF]fo7$Fbho7$$\"3IIxk/d-=FF*$\"3M6%***f(>'eKF*7$7$$\"3UfG9dG9dGF*$ \"3=0,DS;S!=$F*Fdjo7$7$$\"3f3`Ej\"3/-$F*$\"3k'>`c-.[4$F*7$$\"3mk!\\'*= >?)HF*$\"3m:dG9dG9JF*7$7$Ff[pFg`oFjjo7$F`[p7$$\"3'*fg+E'HV0$F*$\"3O&=# p'>#4zIF*7$7$$\"3udxQpMn$=$F*$\"3c\\ru\"Hgc,$F*F]\\p7$7$$\"3!p?5bxQpM$ F*$\"3-Nh/>x2THF*7$$\"3c+`_qG>QLF*$\"3#Rn$=fz*[%HF*7$7$F_]pF[`oFc\\p7$ 7$Fj\\p$\"39$zuBof'\\RF*7$$\"3X\"Q^MQn=M$F*$\"3Y>fz*[C7'RF*7$Fi]p7$$\" 3F]9+E\\yPLF*$\"3!\\Dh'3;sqRF*7$7$$\"3)*)>$\\MaE)H$F*$\"3')fz*[C718%F* F_^p7$Fe^p7$$\"3)*eV(e_#GvKF*$\"3b_^*Qgb\\?%F*7$7$$\"3'R)=!*)=<-'QHF*7$7$$\"32cE j\"3/-^$F*$\"3?bO)R[$*e)GF*Fg_p7$7$F^`p$\"3X.\"[M]Eug$F*7$$\"3[vQ3())y @]$F*$\"3wP=fz*[Ci$F*7$Ff`p7$$\"3/'eNzRoN\\$F*$\"34[vU,&3(ROF*7$7$$\"3 \\aw<#[M+U$F*$\"3gyQpMn$=z$F*F\\ap7$7$Fcap$\"3/zQpMn$=z$F*7$Fj\\p$\"3g $zuBof'\\RF*7$7$F^`p$\"3S_(fhw@:B%F*7$$\"3cmCM[@WdLF*Fd_p7$F]`p7$$\"35 5')[QF`'f$F*$\"3IAcpx>LbGF*7$7$$\"3A0^v(QpMn$F*$\"3IA;@4d3]GF*Ffbp7$7$ F]cp$\"3jT52)>(>?LF*7$$\"3AHs%**otef$F*$\"3!pz*[C71`MF*7$FecpFc`p7$7$F ^`p$\"3^^(fhw@:B%F*7$$\"3'pAPpkh9k$F*$\"3g3l*fF?Q;%F*7$7$F]cp$\"3d%eg# oF7UTF*F_dp7$F\\cp7$$\"3'\\jx;bvVy$F*$\"3j\\S69G$)HGF*7$7$$\"3Rav(QpMn $QF*$\"3S)o@j-*3()GF*Fidp7$7$F`ep$\"377%Q)yXO'*HF*7$$\"3wYVb%)4o)y$F*F g`o7$Fhep7$$\"32K]S/SuPPF*$\"37dMha&))p@$F*7$7$$\"3A;l]wZ'ep$F*FhhoF\\ fp7$7$Fcfp$\"3ibxQpMn$G$F*Fbcp7$7$F`ep$\"3'e7em\"*y%QSF*7$$\"31(4N@X!) 4p$F*Fh^p7$F]gp7$$\"3m0^v(QpMn$F*Ffdp7$F_ep7$$\"3=m>lj:irQF*$\"3:AFzJG q3HF*7$7$$\"3YN\"*y.nmhQF*F[`oFegp7$7$F\\hpFa]pFeep7$7$$\"3c.++++++SF* $\"3ot5=(y\\5%RF*7$$\"3sXN:hfSlRF*$\"3!*>fz*[C7'RF*7$FfhpFjfp-%'COLOUR G6&%$RGBG$\"*++++\"!\")$F-F-Fcip-F$6%7S7$F+F+7$$!3ommmmFiDQF*Fiip7$$!3 5LLLo!)*Qn$F*F\\jp7$$!3nmmmwxE.NF*F_jp7$$!3YmmmOk]JLF*Fbjp7$$!3_LLL[9c gJF*Fejp7$$!3smmmhN2-IF*Fhjp7$$!3!******\\`oz$GF*F[[q7$$!3!omm;)3DoEF* F^[q7$$!3?+++:v2*\\#F*Fa[q7$$!3BLLL8>1DBF*Fd[q7$$!3kmmmw))yr@F*Fg[q7$$ !3;+++S(R#**>F*Fj[q7$$!30++++@)f#=F*F]\\q7$$!3-+++gi,f;F*F`\\q7$$!3qmm m\"G&R2:F*Fc\\q7$$!3XLLLtK5F8F*Ff\\q7$$!3eLLL$HsV<\"F*Fi\\q7$$!3+-++]& )4n**FcblF\\]q7$$!37PLLL\\[%R)FcblF_]q7$$!3G)*****\\&y!pmFcblFb]q7$$!3 Y******\\O3E]FcblFe]q7$$!3NKLLL3z6LFcblFh]q7$$!3sLLL$)[`P(******z-6j'FcblFj^q7$$\"3q\"******4 #32$)FcblF]_q7$$\"3r$*****\\#y'G**FcblF`_q7$$\"3G******H%=H<\"F*Fc_q7$ $\"35mmm1>qM8F*Ff_q7$$\"3%)*******HSu]\"F*Fi_q7$$\"3'HLL$ep'Rm\"F*F\\` q7$$\"3')******R>4N=F*F_`q7$$\"3#emm;@2h*>F*Fb`q7$$\"3]*****\\c9W;#F*F e`q7$$\"3Lmmmmd'*GBF*Fh`q7$$\"3j*****\\iN7]#F*F[aq7$$\"3aLLLt>:nEF*F^a q7$$\"35LLL.a#o$GF*Faaq7$$\"3ammm^Q40IF*Fdaq7$$\"3y******z]rfJF*Fgaq7$ $\"3gmmmc%GpL$F*Fjaq7$$\"3/LLL8-V&\\$F*F]bq7$$\"3=+++XhUkOF*F`bq7$$\"3 =+++:o " 0 "" {MPLTEXT 1 0 34 "x+sin(x*y)=y;\nim plicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"-% $sinG6#*&F%F&%\"yGF&F&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&-% $cosG6#*&%\"xG\"\"\"%\"yGF,F,F-F,F,F,F,F,,&*&F'F,F+F,F,F,!\"\"F0F0" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 301 8 "Question" }{TEXT 296 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "y*exp(x*y) = x+1;" "6#/*&%\"yG\"\"\"- %$expG6#*&%\"xGF&F%F&F&,&F+F&F&F&" }{TEXT -1 26 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 298 1 "x" }{TEXT -1 5 " and " }{TEXT 299 1 " y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Show that the po int" }{XPPEDIT 18 0 "``(0, 0);" "6#-%!G6$\"\"!F&" }{TEXT -1 65 " is a \+ stationary point on the graph of the equation given in (a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solution" } {TEXT 297 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 69 "(a) Differentiating bo th sides of the given equation with respect to " }{TEXT 300 1 "x" } {TEXT -1 8 " gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " exp(x*y) +y" "6#,&-%$expG6#*&%\"xG\"\"\"%\"yGF)F)F*F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff([exp(x*y)],x)=1" "6#/-%%DiffG6$7#-%$expG6#*&%\" xG\"\"\"%\"yGF-F,F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "th at is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6# *&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x*y) +y * exp(x*y)" "6#,&-%$expG6#*&%\"xG\"\"\"%\"yGF)F)*&F*F)-F%6#*&F(F)F*F)F)F )" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x*y],x)=1" "6#/-%%DiffG6$7#*& %\"xG\"\"\"%\"yGF*F)F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x*y) +y *exp (x*y)" "6#,&-%$expG6#*&%\"xG\"\"\"%\"yGF)F)*&F*F)-F%6#*&F(F)F*F)F)F)" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "y+x" "6#,&%\"yG\"\"\"%\"xGF%" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 2 " )" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(x*y)+y^2*exp(x*y)+y*exp(x*y)*x;" "6#,(-%$expG6#*&% \"xG\"\"\"%\"yGF)F)*&F*\"\"#-F%6#*&F(F)F*F)F)F)*(F*F)-F%6#*&F(F)F*F)F) F(F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1" "6#/*&%#dyG\"\"\"%# dxG!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Collecting the terms involving " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 110 " on the left hand side of the equation, and the remaining terms on the right hand side \+ of the equation gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x*y) + x*y*exp(x*y)" "6#,&-%$expG6#*&%\"xG\"\"\"%\"yGF)F)*(F(F) F*F)-F%6#*&F(F)F*F)F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx =1-y^2* exp(x*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*&%\"yG\"\"#-%$expG6#*&%\"x GF&F+F&F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Factoring out " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " on the left side gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 " ``( exp(x*y) + x*y*exp(x*y)) = 1-y^2*exp(x*y)" "6#/-%!G6#,&-%$expG6#*&%\"x G\"\"\"%\"yGF-F-*(F,F-F.F--F)6#*&F,F-F.F-F-F-,&F-F-*&F.\"\"#-F)6#*&F,F -F.F-F-!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "so that: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(1-y^2*exp( x*y))/(exp(x*y)+x*y*exp(x*y))" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*&% \"yG\"\"#-%$expG6#*&%\"xGF&F,F&F&F(F&,&-F/6#*&F2F&F,F&F&*(F2F&F,F&-F/6 #*&F2F&F,F&F&F&F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1-y^2*exp(x*y))/( exp(x*y)*(1+x*y)" "6#/%!G*&,&\"\"\"F'*&%\"yG\"\"#-%$expG6#*&%\"xGF'F)F 'F'!\"\"F'*&-F,6#*&F/F'F)F'F',&F'F'*&F/F'F)F'F'F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) At \+ the point " }{XPPEDIT 18 0 "``(0,1);" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 40 " the gradient of the tangent line is 0. " }}{PARA 0 "" 0 "" {TEXT -1 11 " Hence" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" } {TEXT -1 40 " is a stationary point on the graph of " }{XPPEDIT 18 0 "y*exp(x*y) = x+1" "6#/*&%\"yG\"\"\"-%$expG6#*&%\"xGF&F%F&F&,&F+F&F&F& " }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 472 201 201 {PLOTDATA 2 "6(-%'CURVESG6ix7$7$$!\"'\"\"!$!39To-s<\"e9%!#=7$$ !3#*)R9^37X(f!#<$!3Wyy@Z>vaTF-7$7$$!33D:\\p,h'z&F1$!3)Gs7jA*y%>%F-F.7$ F57$$!3l0\\jm$Rgs&F1$!3B([V=P>'=UF-7$7$$!39]I)*Q.A$f&F1$!3Q8A'3IokC%F- F;7$FA7$$!3m2Z(yNXkZ&F1$!35IM.\"HvSG%F-7$7$$!3AvXZ30$)*Q&F1$!3I>KC_Dl+ VF-FG7$FM7$$!39Ng!)*=EjA&F1$!3av.\"f=v-N%F-7$7$$!3I+h'znSk=&F1$!31I$=E PYqN%F-FS7$7$$!3GEwXZ30$)\\F1$!3%y2&yN&yPT%F-7$$!3XXR`YL#)4]F1$!3i\\p, h'znS%F-7$F^oFY7$7$$!3SDwXZ30$)\\F1F\\o7$$!3iVXy=&Rj(\\F1$!35\\rjwtG;W F-7$7$$!3Y]\"\\p,h'zZF1$!35KV(=,QLY%F-Fho7$F^p7$$!39buxdKeEZF1$!3%3&o< `J(>[%F-7$7$$!3av1W'=rid%F1$!3CocrK3#R^%F-Fdp7$Fjp7$$!3_\"p:7(3ByWF1$! 3[wce2,nXXF-7$7$$!3i+A$fN\")GP%F1$!3s/k`9+qkXF-F`q7$Ffq7$$!3v*>,7()\\D B%F1$!3`+#>xf#e0YF-7$7$$!3qDPUD:\\pTF1$!3lU8Zyxd9YF-F\\r7$7$Fcr$!3@V8Z yxd9YF-7$$!3CO\"yg*e'4*RF1$!3!e`c#p4pfYF-7$7$$!3A^_\"\\p,h'RF1$!3w5@)o .<@m%F-F[s7$7$$!3uwnSk=riPF1$!3jtDGd$4Sq%F-7$$!3JY>cG&3w!QF1$!3])GPUD: \\p%F-7$F\\tFas7$Fgs7$$!3ObHbwv\\aPF1$!3b--.i@b1ZF-7$7$$!3#=I)*Q.A$fNF 1$!3Y6?\\>L)ft%F-Fct7$Fit7$$!3uvd-z%>R_$F1$!3#[.SO()o]u%F-7$7$$!3AG)*Q .A$fN$F1$!3%fys$eR%*fZF-F_u7$7$$!3yF)*Q.A$fN$F1$!3Q&ys$eR%*fZF-7$$!3'G Bc%fMy-LF1$!33:K93$4-x%F-7$7$$!3'GN\")GPUD:$F1$!3Wa,8yeHsZF-F`v7$7$$!3 I`8)GPUD:$F1$!3)\\:I\"yeHsZF-7$$!3-5(\\[@(*H4$F1$!35;@y%*3FzZF-7$7$$!3 #)yGPUD:\\HF1$!3g&)4L(e3%oZF-Faw7$Fgw7$$!3eDNkrUH'*GF1$!3o/sV/yzpZF-7$ 7$$!3O/W'=ridu#F1$!3StIP,7LUZF-F]x7$Fcx7$$!3xW\"pB8HTr#F1$!3osb&=\\H(R ZF-7$7$$!3uKK')=ntpDF1F_tFix7$F_y7$$!3ytAhC&yra#F1$!37MTifs5)o%F-7$7$$ !3()HfN\")GPUDF1$!3mHn*[smao%F-Fcy7$Fiy7$$!3KTLnS&*z#R#F1$!3G\\R)pAv'= YF-7$7$$!3Rbu%30$)*QBF1$!3a*)pVa5)*zXF-F_z7$Fez7$$!3kiqyV+$QD#F1$!3GRe o(38u_%F-7$7$$!3Z!)*Q.A$fN@F1$!3BD_8-**p=WF-F[[l7$7$$!3\"4)*Q.A$fN@F1F d[l7$$!3S4$*oS*)p6KtB@F1FaoFj[l7$F`\\l7$ $!3\"e^>fAfg,#F1$!3]'=CK*Q)zG%F-7$7$$!3V10$)*Q.A$>F1$!3Qd\\pu%f`;%F-Fd \\l7$Fj\\l7$$!3@zbC-5P:>F1$!3,\"f3-&**[UTF-7$7$$!3#pW`$>=$=!>F1$!3u5mz nSk=TF-F`]l7$Ff]l7$$!3\"Q1)HIDhA=F1$!3%>BZA>id)RF-7$7$$!3])y-x?S'[ 5,9QF-7$7$$!3'>.A$fN\")GP0.W&y$F-F[_l7$Fa_l7$$!3yL9q7;>k;F1$! 3)pWo!R1#Rj$F-7$7$$!39\\OJu#p=j\"F1$!3(H$fN\")GPUNF-Fg_l7$F]`l7$$!3eYB BkzR&f\"F1$!3=[E^9FCVMF-7$7$$!3gfDX)RCm`\"F1$!35%fN\")GPUD$F-Fc`l7$7$$ !3$)fDX)RCm`\"F1F\\al7$$!32R3SvD7L:F1$!3!=h.))eILC$F-7$7$$!3[dN\")GPUD :F1$!3L`x/$exa@$F-Fbal7$Fhal7$$!396K^enzu9F1$!3BzSqoK\"F1$!3!GMhA4zHQ#F-7$7$$!3+$30$)*Q.A 8F1$!3ig*y@PApN#F-Fhdl7$F^el7$$!3cl$*[1T)RG\"F1$!3qrJ*py)fb@F-7$7$$!3] $ouTJ(yu7F1$!3dQUD:\\p,@F-Fdel7$Fjel7$$!3>p=Q)=dSC\"F1$!3n_%exKES#>F-7 $7$$!3lT.Kc`dE7F1$!3p**Q.A$fN\"=F-F`fl7$Fffl7$$!3LVgH(*yv17F1$!3k+(4pR J()o\"F-7$7$$!3CAL\"4D@F=\"F1$!3\"3c8)GPUD:F-F\\gl7$Fbgl7$$!3m3%p)GZtr 6F1$!3KF/'*e>@]9F-7$7$$!3QSx!))z?D9\"F1$!3z@KfN\")GP7F-Fhgl7$F^hl7$$!3 +))f-&Q&pQ6F1$!3s+;&G5#))37F-7$7$$!3_3mznSk=6F1$!31dLyHBS_5F-Fdhl7$Fjh l7$$!3)G3cTcwt5\"F1$!3W1'p>R[6l*!#>7$7$$!3f*ep]A;a5\"F1$!3oF)GPUD:\\*F eilF`il7$Fgil7$$!35A'y%*o*ox5F1$!3[4pp(Feil7$7$$!3;f4%*3>&42\"F1$ !3[Pa_\"\\p,h'FeilF]jl7$Fcjl7$$!3)*\\./=*G&\\5F1$!3p4d`Z[%zq%Feil7$7$$ !3u4F\"f.h(Q5F1$!3GZ?KfN\")GPFeilFijl7$F_[m7$$!3jh*\\QR7)GPUD: \\FeilF^]m7$Fd]m7$$!3%*G]-&)*f&*[*F-$\"35$o\"\\aep#R&Feil7$7$$!3kFmC!4 AZF*F-$\"397:\\p,h'z(FeilFj]m7$F`^m7$$!3G(*p;`*)4i#*F-$\"3'G>'*[K:=&zF eil7$7$$!3?Q8)GPUD:*F-$\"3%e\"[,)z(pS#*FeilFf^m7$7$$!3KR8)GPUD:*F-$\"3 A<[,)z(pS#*Feil7$$!3&o`t(Hh%R/*F-$\"3)[)eRU@T_5F-7$7$$!3r)[`(*3X..*F-$ \"3C!\\p,h'zn5F-Fg_m7$7$F^`m$\"35!\\p,h'zn5F-7$$!3)*GwUW%=[$))F-$\"3u` b_88#4J\"F-7$7$$!39(f1K4ckz)F-$\"37H)*Q.A$fN\"F-Ff`m7$F\\am7$$!3,zPg&Q fCj)F-$\"3]8DTt$*Qq:F-7$7$$!3?9pj,v(=d)F-$\"3+o,h'znSk\"F-Fbam7$Fham7$ $!3b9E3QK7O%)F-$\"38at1z1rI=F-7$7$$!3OA\"RHKhbN)F-$\"3)o]I)*Q.A$>F-F^b m7$7$$!3YB\"RHKhbN)F-Fgbm7$$!3]fUZ/!Q^C)F-$\"3/V)f3.!z\"4#F-7$7$$!3O-2 j(**yl9)F-$\"3wX30$)*Q.A#F-F]cm7$7$$!3E,2j(**yl9)F-Ffcm7$$!3'GYrB\\&*) e!)F-$\"3K_*)>o7a`BF-7$7$$!3_Y1c/R5WzF-$\"3k%=ridu%3DF-F\\dm7$Fbdm7$$! 3iV2%o#H%o(yF-$\"3FRCI\"3')eh#F-7$7$$!3m0>@WoRZxF-$\"3_B:\\p,h'z#F-Fhd m7$F^em7$$!35.#)4BeZ)p(F-$\"3%\\e#\\,IvyGF-7$7$$!3=(z:\"HRzbvF-$\"3Ui= ridu%3$F-Fdem7$Fjem7$$!305w>1>LBvF-$\"3St'=)Rl2UJF-7$7$$!3G)=ALV'potF- $\"3G,A$fN\")GP$F-F`fm7$Fffm7$$!3;baW0f)4N(F-$\"3JB\"*f\\kz0MF-7$7$$!3 m[biYAc&=(F-$\"3sSD:\\p,hOF-F\\gm7$7$Fcgm$\"3=SD:\\p,hOF-7$$!3O%>MO3X5 =(F-$\"37nG1$4d)pOF-7$7$$!3U$f'znSk=rF-$\"3UxfjGUOpPF-F[hm7$Fahm7$$!3W %F-7$7$$!3EX3c%o3a#oF-$\"3$z@$fN\")GPU F-Fcim7$Fiim7$$!3=8b2(3;Ap'F-$\"3#RS6#)48]Y%F-7$7$$!3Ez9w2PsZmF-$\"3\" ob8)GPUDXF-F_jm7$7$Ffjm$\"3PdN\")GPUDXF-7$$!3&[VYbW?:`'F-$\"3Sc%[bZ$QI ZF-7$7$$!3))f[i#*o/rkF-$\"3p&*Q.A$fN\"[F-F^[n7$Fd[n7$$!3c$4<&y@*3P'F-$ \"3Q0jg(Rjd*\\F-7$7$$!3Cia?0F![H'F-$\"3dMUD:\\p,^F-Fj[n7$F`\\n7$$!3VU. 4Sl%)4iF-$\"3/@&\\?>%3h_F-7$7$$!3$H9vhh9%=hF-$\"3WtXZ30$)*Q&F-Ff\\n7$F \\]n7$$!3m0)3.Y))y/'F-$\"3Nvu'Rsvi_&F-7$7$$!3r+yXv&)HTfF-$\"3W8\\p,h'z n&F-Fb]n7$7$Fi]n$\"3M7\\p,h'zn&F-7$$!3-*\\P(4r]%)eF-$\"3azo&eil7z&F-7$ 7$$!3m3mMQi&Gw&F-$\"3A^_\"\\p,h'fF-Fa^n7$Fg^n7$$!3n[<+S+<>dF-$\"3'*>8` n%yf0'F-7$7$$!3'*[GdtxY#e&F-$\"35!fN\")GPUD'F-F]_n7$Fc_n7$$!3K,@>^nJ^b F-$\"3&*ep$[%[L?jF-7$7$$!3CS2XkS[*R&F-$\"3)*GfN\")GPUlF-Fi_n7$7$$!3MT2 XkS[*R&F-Fb`n7$$!3KN#R>^^.Q&F-$\"3Wvk\\i.D%e'F-7$7$$!3hP[3'3=K@&F-$\"3 (yEwXZ30$oF-Fh`n7$F^an7$$!3hi))QC_j0_F-$\"3#z5XtfS2%F-$\"3GiznSk=r#)F-F `en7$7$$!3w%>5XtfS2%F-Fien7$$!3KQJvwZ8yRF-$\"3J<3AW@b-%)F-7$7$$!3[Hm&3 cx6z$F-$\"3/+$)*Q.A$f&)F-F_fn7$Fefn7$$!3D!)yps7p?PF-$\"3#*fmOul@a')F-7 $7$$!3%o:Fh7%[![$F-$\"3/S'=ridu%))F-F[gn7$Fagn7$$!3v6.Cy)*zQMF-$\"3Ly@ P`vT-*)F-7$7$$!3:u'4\\^'=NJF-$\"3#*y*Q.A$fN\"*F-Fggn7$7$$!3qu'4\\^'=NJ F-F`hn7$$!3m`*[O_:o7$F-$\"3)zSDIObj9*F-7$7$$!3k,ridu%30$F-$\"3UxyeL*Q< ?*F-Ffhn7$F\\in7$$!3k(>aJoGtt#F-$\"3uAB%Ql8$z$*F-7$7$$!3#R\"4)=Xx&*f#F -$\"3!yJfN\")GPU*F-Fbin7$Fhin7$$!3_t6f2C[^AF-$\"3+M7pC5i)f*F-7$7$$!3T' ef&G8aQ=F-$\"3oc'znSk=r*F-F^jn7$Fdjn7$$!3NpCAzkHO;F-$\"37S$f!y_g*z*F-7 $7$$!3)eNUD:\\p,\"F-$\"31@sur<&ol!fq*F-7$$\"3=abuM;i#)HF-$\"3zd'znS k=r*F-7$7$F[]oFgjn7$$\"3;!RUD:\\p,\"F-Fb\\o7$7$$\"3%>)=ridu%3&F-$\"351 Z'R;OxO*F-7$$\"3wzW*\\NI:l%F-$\"3p;$fN\")GPU*F-7$7$F[^oF[jnFe\\o7$7$$ \"3tEmznSk=rF-$\"30(4qkYo7,*F-7$$\"3e90U!QtoF'F-$\"3\"y(*Q.A$fN\"*F-7$ Fg^oFe]o7$7$$\"3ut8)GPUD:*F-$\"3))p>)eu#*Rm)F-7$$\"3k&fBDm4S&zF-Fdgn7$ Fc_o7$$\"3%yi'znSk=rF-$\"3;)4qkYo7,*F-7$7$$\"337mznSk=6F1$\"3#=sT\"3N\"F1$\"3]CwXZ30$)zF-7$7$FaboF]enF^ao7$7$$\"3^N?KfN\")G F1$\"3!HqC'3`kOsF-7$$\"3epU2Q/-&y\"F1Fecn7$7$F[do $\"3_Wp,h'znS(F-Fhbo7$7$$\"3Z%)*Q.A$fN@F1$\"3N7fZ)[_*4qF-7$$\"35?.b>7! >.#F1Fibn7$FgdoFeco7$7$$\"3%*eu%30$)*QBF1$\"3U5Ro+A%4!oF-7$$\"3y;%oaGY vI#F1Faan7$FaeoFbdo7$F\\eo7$$\"3W+PKC\"3tR#F1$\"3_#f^rG\")yu'F-7$7$$\" 3ULfN\")GPUDF1$\"3oY$*[yFK.mF-Ffeo7$7$$\"3Y2W'=ridu#F1$\"3!4g^hho&=kF- 7$$\"3A1hiM/-1EF1Fb`n7$7$$\"3m1hiM/-1EF1Fb`nF\\fo7$7$$\"3Q#)GPUD:\\HF1 $\"3#4:Vnu:/D'F-7$$\"3#*R2P!>WU%HF1$\"3A\"fN\")GPUD'F-7$7$FegoFf_n7$$ \"3!zSk=ridu#F1Fefo7$F_go7$$\"3#34%)GnCX&HF1$\"3my8\"*pniYiF-7$7$$\"3' oN\")GPUD:$F1$\"31HY/8@@&3'F-F_ho7$7$$\"3MJ)*Q.A$fN$F1$\"3Yu+Ep(*4ZLPF1F_^n7$FfjoF[jo7$Fajo7$$\"3'Q/gSI;nyk&F-7$7$$\"3Ma_\"\\p,h'RF1$\"3,q(3,)*yw_&F-F[[p7$7$$\"3ya_ \"\\p,h'RF1Fd[p7$$\"3I*)*G\")GI79%F1$\"3F(y38En)HaF-7$7$$\"3CHPUD:\\pT F1$\"3dcIP8@w5aF-Fj[p7$7$$\"3;/A$fN\")GP%F1$\"3[6iS%f)f#H&F-7$$\"3LqeY X&Rg?%F1F_]n7$F[]pF`\\p7$Ff\\p7$$\"3%>[EDiShZ%F1$\"3n8gYZlaV_F-7$7$$\" 35z1W'=rid%F1$\"39&pfR7tP=&F-F`]p7$7$$\"3!\\:\\p,h'zZF1$\"3Era+F_%Q3&F -7$$\"36@8f*46Xu%F1Fc\\n7$Fa^p7$Fg]p$\"3-%pfR7tP=&F-7$7$$\"3-a\"\\p,h' zZF1F_^p7$$\"3aR_xRX:(z%F1$\"3)e%ReCC\"p2&F-7$7$$\"3%*GwXZ30$)\\F1$\"3 ]=eTT'[+)\\F-F\\_p7$Fb_p7$$\"3!R=M)o-t.^F1$\"3')\\uj,CtI\\F-7$7$$\"3'Q 5mznSk=&F1$\"3k+F2$>+o)[F-Fh_p7$7$$\"3yyXZ30$)*Q&F1$\"3gAd,n[&4![F-7$$ \"3y1%*em+qh`F1$\"39&*Q.A$fN\"[F-7$7$Fj`pFg[n7$F_`p$\"3?,F2$>+o)[F-7$F d`p7$$\"3Wx9EIjZ,aF1$\"3cbC]u51(z%F-7$7$$\"3q`I)*Q.A$f&F1$\"3%z6y?!p;4 ZF-Fdap7$Fjap7$$\"3Yc#)G0Ib!p&F1$\"3mC:5y8nvYF-7$7$$\"3iG:\\p,h'z&F1$ \"3*3ioZ=1ii%F-F`bp7$Ffbp7$$\"3_q]phG_sfF1$\"3)eaXZn\\Vc%F-7$7$$\"3c.+ +++++gF1$\"3IfwNN(o8b%F-F\\cp-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F^dp -%%TEXTG6$7$$\"#j!\"\"$!\"#FedpQ\"x6\"-F`dp6$7$Ffdp$\"#8FedpQ\"yFidp-% +AXESLABELSG6%%!GFcep-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$\"\"&\"\"%-%%V IEWG6$;F(Fcdp;$!\"(FedpF]ep" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}{TEXT -1 1 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y*exp(x*y)=x+1;\nimplicitdiff(%,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"yG\"\"\"-%$expG6#*&%\"xGF&F%F&F&,&F+F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&*&)%\"yG\"\"#\"\"\"-%$expG6#*& %\"xGF*F(F*F*F*F*!\"\"F*F+F0,&F.F*F*F*F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "Some standard expressions which occur when differentiatin g implicitly" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 259 "" 0 "" {TEXT 258 8 "Degree 2" }{TEXT 265 2 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[y^2]=2*y" "6#/7#*$%\"yG\"\"#*&F'\"\"\" F&F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\" \"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x*y] = y + x" "6#/7#*&%\"xG\"\"\"%\"y GF',&F(F'F&F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\" %#dxG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 258 8 "Degree 3" }{TEXT 266 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[y^3] = 3*y^2" "6#/7#*$%\"yG\"\"$*&F'\"\"\"*$F&\"\"#F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2*y] \+ = 2*x*y+x^2;" "6#/7#*&%\"xG\"\"#%\"yG\"\"\",&*(F'F)F&F)F(F)F)*$F&F'F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x*y^2] \+ = y^2 + 2*x*y" "6#/7#*&%\"xG\"\"\"*$%\"yG\"\"#F',&*$F)F*F'*(F*F'F&F'F) F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\" \"" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT 258 8 " Degree 4" }{TEXT 267 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[y^4] = 4*y^3;" "6#/7#*$%\"yG\"\"%*&F'\"\"\"*$F&\"\"$F) " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^3*y] \+ = 3*x^2+ x^3 " "6#/7#*&%\"xG\"\"$%\"yG\"\"\",&*&F'F)*$F&\"\"#F)F)*$F&F 'F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\" " }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d /dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x*y^ 3] = y^3 + 3*x*y^2" "6#/7#*&%\"xG\"\"\"*$%\"yG\"\"$F',&*$F)F*F'*(F*F'F &F'F)\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"% #dxG!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[x^2*y^2] = 2*x*y^2 + 2*x^2*y" "6#/7#*&%\"xG\"\"#%\"yGF ',&*(F'\"\"\"F&F+F(F'F+*(F'F+*$F&F'F+F(F+F+" }{TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 " Implicit differentiation using " }{TEXT 0 4 "diff" }{TEXT -1 8 " and/o r " }{TEXT 0 12 "implicitdiff" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 259 "" 0 "" {TEXT 258 8 "Question" }{TEXT 270 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG \"\"\"%#dxG!\"\"" }{TEXT -1 5 " for " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3*y+x*y^3=2" "6#/,&*&%\"xG\"\"$%\"yG\"\"\"F)*&F& F)*$F(F'F)F)\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 271 8 "Sol ution" }{TEXT -1 16 " using Maple's " }{TEXT 0 4 "diff" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 6 "Ste p 1" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "Substitute y(x) f or y in the given equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eq1 := x^3*y+x*y^3=2;\neq2 := subs( y=y(x),x^3*y+x*y^3=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&*& )%\"xG\"\"$\"\"\"%\"yGF+F+*&F)F+)F,F*F+F+\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&*&)%\"xG\"\"$\"\"\"-%\"yG6#F)F+F+*&F)F+)F,F*F +F+\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 6 "Step 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Differen tiate with respect to x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eq3 := diff(eq2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,**&)%\"xG\"\"#\"\"\"-%\"yG6#F)F+\"\"$*&)F )F/F+-%%diffG6$F,F)F+F+*$)F,F/F+F+**F/F+F)F+)F,F*F+F2F+F+\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 6 "Step 3" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solve for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 25 ", which appears here as " }{XPPEDIT 18 0 "diff(y(x),x)" "6#-%%diffG6$-%\"yG6#%\"xGF) " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "gradient := solve(eq3,diff(y(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)gradientG,$*&*&-%\"yG6#%\"xG\"\"\",&*$)F+ \"\"#F,\"\"$*$)F(F0F,F,F,F,*&F+F,,&F.F,*&F1F,F3F,F,F,!\"\"F7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 6 "Step 4" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "Substitute y back for y( x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(y(x)=y,gradient);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"yG\"\"\",&*$)%\"xG\"\"#F'\"\"$*$)F&F,F'F'F'F'*&F+F',&F) F'*&F-F'F/F'F'F'!\"\"F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 97 "We can put these commands together to provide a te mplate for performing implicit differentiation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "x^3*y+x*y^3= 2;\nsubs(y=y(x),%);\ndiff(%,x);\nsolve(%,diff(y(x),x));\nsubs(y(x)=y,% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"xG\"\"$\"\"\"%\"yGF)F)* &F'F))F*F(F)F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"xG\"\" $\"\"\"-%\"yG6#F'F)F)*&F'F))F*F(F)F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&)%\"xG\"\"#\"\"\"-%\"yG6#F'F)\"\"$*&)F'F-F)-%%diff G6$F*F'F)F)*$)F*F-F)F)**F-F)F'F))F*F(F)F0F)F)\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&*&-%\"yG6#%\"xG\"\"\",&*$)F)\"\"#F*\"\"$*$)F&F.F* F*F*F**&F)F*,&F,F**&F/F*F1F*F*F*!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"yG\"\"\",&*$)%\"xG\"\"#F'\"\"$*$)F&F,F'F'F'F'*&F+F',&F) F'*&F-F'F/F'F'F'!\"\"F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 191 "You can use the following modification of this te mplate, if you want to see steps which look more like what you would w rite down if you were performing the implicit differentiation \"by han d\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "x^3*y+x*y^3=2;\nsubs(\{y(x)=y,diff(y(x),x)=`dy/dx`\} ,diff(subs(y=y(x),%),x));\n`dy/dx`=solve(%,`dy/dx`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"xG\"\"$\"\"\"%\"yGF)F)*&F'F))F*F(F)F)\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&)%\"xG\"\"#\"\"\"%\"yGF)\"\"$ *&)F'F+F)%&dy/dxGF)F)*$)F*F+F)F)**F+F)F'F))F*F(F)F.F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&dy/dxG,$*&*&%\"yG\"\"\",&*$)%\"xG\"\"#F) \"\"$*$)F(F.F)F)F)F)*&F-F),&F+F)*&F/F)F1F)F)F)!\"\"F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 16 " using Maple's " }{TEXT 0 12 "implicitdiff" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "implicitdif f(x^3*y+x*y^3=2,y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"yG\"\" \",&*$)%\"xG\"\"#F&\"\"$*$)F%F+F&F&F&F*!\"\",&F(F&*&F,F&F.F&F&F/F/" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "Tangent lines to gene ral curves defined by an equation involving two variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 259 "" 0 "" {TEXT 258 8 "Question" }{TEXT 268 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 51 "Find the e quation of the tangent line to the curve " }{XPPEDIT 18 0 "x^2*y^3-x^3 *y^2=12" "6#/,&*&%\"xG\"\"#%\"yG\"\"$\"\"\"*&F&F)F(F'!\"\"\"#7" } {TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\" \"!\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 264 8 "Solution" }{TEXT 269 3 ": " }}{PARA 0 "" 0 " " {TEXT -1 62 "This problem has been solved \"by hand\" in an earlier \+ section. " }}{PARA 0 "" 0 "" {TEXT -1 112 "We can use the template fro m the previous section to perform the implicit differentiation, or use the procedure " }{TEXT 0 12 "implicitdiff" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "x^2* y^3-x^3*y^2=12;\nsubs(y=y(x),%);\ndiff(%,x);\nsolve(%,diff(y(x),x));\n deriv := subs(y(x)=y,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"x G\"\"#\"\"\")%\"yG\"\"$F)F)*&)F'F,F))F+F(F)!\"\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"xG\"\"#\"\"\")-%\"yG6#F'\"\"$F)F)*&)F'F.F)) F+F(F)!\"\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&%\"xG\"\"\")-% \"yG6#F&\"\"$F'\"\"#**F,F')F&F-F')F)F-F'-%%diffG6$F)F&F'F'*(F,F'F/F'F0 F'!\"\"**F-F')F&F,F'F)F'F1F'F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&*&-%\"yG6#%\"xG\"\"\",&F&!\"#*&\"\"$F*F)F*F*F*F**&F)F*,&F&!\"$*& \"\"#F*F)F*F*F*!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG,$* &*&%\"yG\"\"\",&F(!\"#*&\"\"$F)%\"xGF)F)F)F)*&F.F),&F(!\"$*&\"\"#F)F.F )F)F)!\"\"F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Or, more simply, we could use " }{TEXT 0 12 "implicitdiff " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "deriv := implicitdiff(x^2*y^3-x^3*y^2=12,y,x); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The \+ gradient of the tangent line at the point" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\"\"!\"\"\"\"#" }{TEXT -1 40 " is obtained by substitutin g the values " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 47 " in th e expression obtained for the derivative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m := subs(\{x=-1,y=2 \},deriv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG#\"\"(\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The equation of the tangent li ne can then be obtained by using the point-slope form for the equation of a straight line, namely" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y-y[1]=m*(x-x[1])" "6#/,&%\"yG\"\"\"&F%6#F&!\"\"*&%\"mG F&,&%\"xGF&&F-6#F&F)F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " with " }{XPPEDIT 18 0 "x[1]=-1,y[1]=2" "6$/&%\"xG6#\"\"\",$F'!\"\"/&% \"yG6#F'\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " m=7/4" "6#/%\"mG*& \"\"(\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 " This gives" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y-2 = \+ 7/4;" "6#/,&%\"yG\"\"\"\"\"#!\"\"*&\"\"(F&\"\"%F(" }{TEXT -1 2 " (" } {TEXT 263 1 "x" }{TEXT -1 6 " + 4) " }}{PARA 0 "" 0 "" {TEXT -1 2 "or " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 7/4*x+15/4; " "6#/%\"yG,&*(\"\"(\"\"\"\"\"%!\"\"%\"xGF(F(*&\"#:F(F)F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "curve := plots[implicitplot](x^2*y^3-x^3*y^2=12,x=-3 ..3,\n y=-3..8.5,grid=[50,50],color=red):\ntgt := plot((7*x+15)/4 ,x=-3..3,color=green,thickness=2):\nplots[display]([curve,tgt]);" }} {PARA 13 "" 1 "" {GLPLOT2D 327 338 338 {PLOTDATA 2 "6&-%'CURVESG6gy7$7 $$!\"$\"\"!$!3Z2Pd8yjDG!#<7$$!3#=vU30'))zHF-$!3F0PLey&Q!GF-7$7$$!3/QNL ]()yVHF-$!3mz*[C71`w#F-F.7$F47$$!3x&z:ZN(e2HF-$!36%o#p\"[Oxq#F-7$7$$!3 7j\"3/-^v(GF-$!3)>y^_B&fjEF-F:7$7$F($!3*o&>Z!*o#fq(!#=7$$!3H-nz>:)y!HF -$!3#y\\%z[k@'H)FI7$7$FA$!3\"HaI?*4(oU)FIFJ7$7$F($\"3[(yg)R\"Q[&fFI7$$ !3r1bA3!*eBHF-$\"35V@:*Rwk3'FI7$7$FA$\"3!4Ty>%pm#G'FIFW7$7$$!3oi\"3/-^ v(GF-FC7$$!3ZgTNr0IOGF-$!3;(z%e0cn4EF-7$7$$!3ULrntR*Gx#F-$!3Lfz*[C71`# F-F^o7$Fdo7$$!3#4I8$QAIkFF-$!3'zvGI`yH^#F-7$7$$!3EEj\"3/-^v#F-$!3)))zI Vp*R'\\#F-Fjo7$7$Fap$!34u529hb:$*FI7$$!38!H(>%Gn5\"GF-$!3cn\"3/-^v())F I7$7$Fjp$!3oo\"3/-^v())FIFP7$Fgn7$$!3*zdh052q\"GF-$\"3Q$QqxF&f!R'FI7$7 $Fap$\"3k8]4bVKlmFIFcq7$7$$!3qEj\"3/-^v#F-$!3M*zIVp*R'\\#F-7$$!3SJ92mS t)p#F-$!3@()zg&)p&RS#F-7$7$$!3R*[C71`Ej#F-$!3jW!o+-Z]I#F-Fbr7$7$Fap$!3 )H2rS6cbJ*FI7$$!35L6HWDv/FF-$!3yK59PIeU)*FI7$7$Fir$!3-k,o@[BY5F-Fas7$7 $Fap$\"3w9]4bVKlmFI7$$!3m/7G\"H`Hr#F-$\"3(*onSv$pJu'FI7$7$Fir$\"3(zAXn M]\\6(FIF^t7$7$Fir$!32X!o+-Z]I#F-7$$!3H#Grv]K)HEF-$!3$ee:&GWK,BF-7$7$$ !3'R24Sn7bi#F-$!3)*QpMn$=fH#F-F[u7$Fau7$$!3+:s1o0yrDF-$!3MhtMon*y<#F-7 $7$$!3__Ej\"3/-^#F-$!3A>)fFwp@1#F-Fgu7$7$F^v$!3>d?0ok**>7F-7$$!3[la%yX kcc#F-$!3KP=fz*[C7\"F-7$7$$!3#fYXyXkcc#F-FivFgs7$Fdt7$$!3H66P;$p>h#F-$ \"33#=S^'>earFI7$7$$!37Fg]rS0IDF-$\"3[hK;3/-^vFIF`w7$Ffw7$$!3uCUD^'fF^ #F-$\"3](RV#f@++wFI7$7$F^v$\"3mx3BYu$[h(FIF\\x7$F]v7$$!3'*)QE*yH%*4DF- $!3A!f\\m$\\sh?F-7$7$$!32,'>X%eh4DF-$!3m=fz*[C71#F-Ffx7$7$$!3^,'>X%eh4 DF-F_y7$$!3WcfF5$e_Y#F-$!3+**=,umn7>F-7$7$$!35'HkV8Q;V#F-$!3K)*[C71`E= F-Fey7$F[z7$$!3[&RAnEM%GCF-$!3Mq=F8Ac[1'=Ld$H:F-7$7$$!3Cm8A?&)3WCF-$!3kd G9dG9d8F-F][l7$Fc[l7$F^v$!3Td?0ok**>7F-7$Fbx7$$!3]_FKzk#pS#F-$\"3#*eq1 1=Z=zFI7$7$$!3k:3/-^v(Q#F-$\"3$>2C9J,\\.)FIF]\\l7$7$Fd\\l$\"3/tSU68!\\ .)FI7$$!3`A84<:#RI#F-$\"3M[[Z0)[6H)FI7$7$$!3yy*[C71`E#F-$\"3%GK15;dq`) FIF\\]l7$Fb]l7$$!3CIxaqcT/AF-$\"3)Qa**))=#*3t)FI7$7$$!3\">9dG9dG9#F-$ \"3t9L5ey+V\"*FIFh]l7$7$F_^l$\"3i8L5ey+V\"*FI7$$!3Acn9-iG4@F-$\"3Eq5R` h^a#*FI7$7$$!3.0`Ej\"3/-#F-$\"3mHSG0;%=))*FIFg^l7$F]_l7$$!3%4/>qC'p>?F -$\"3WTnGATJ%))*FI7$7$$!3***pR8doy,#F-$\"3%fYtO=fz*)*FIFc_l7$Fi_l7$$!3 &*pl[)Hl3#>F-$\"3*yX#40$*pL5F-7$7$$!3QoMn$=fz*=F-$\"3[]n$f]2M0\"F-F_`l 7$7$$!3goMn$=fz*=F-Fh`l7$$!3Vfxe\"F-F\\fl7$7$Fc fl$\"3)pRBI*3&>e\"F-7$$!35S*>CZ8gC\"F-$\"3&)z&RbK%y<;F-7$7$$!3ujPir\\! )*>\"F-$\"3#yS?5bxQp\"F-F[gl7$7$Fbgl$\"3e2/-^v(Qp\"F-7$$!3cn\"zR@ug<\" F-$\"3s&yXx&zU=24:5\"F-$\"3Gya48`?5=F-7$7$$!3JydL3B.U5F-$\"3%zUr&G9dG>F-Ffhl7$F\\ il7$$!3ih5;=pHT5F-$\"3%)*yHDc#\\H>F-7$7$$!3&=hIlK;3/\"F-$\"3F-Fbil7$7$Fiil$\"3RS*)*HkI0$>F-7$$!3Mr8m>%Rwp*FI$\"3gQ0/p:3F?F-7$7$$! 3)>v(QpMn$=*FI$\"3IO$)*4dW9:#F-Fajl7$7$$!3'3v(QpMn$=*FIFjjl7$$!3w`Ansh '*[\"*FI$\"3-y>DuIhc@F-7$7$$!3Gvuqg\\([7*FI$\"3E[C71`Ej@F-F`[m7$7$Fg[m $\"3#yWAhIlK;#F-7$$!3Kb\\`PNv*\\)FI$\"3s\\$o+`uoE#F-7$7$$!3qv?[d]%)f!) FI$\"3goMn$=fzR#F-F_\\m7$Fe\\m7$$!3z=W%o0s#3!)FI$\"3*p,([!zntS#F-7$7$$ !3G$QpMn$=fzFI$\"3Kfw_rI1BCF-F[]m7$Fa]m7$$!396;6AYIGuFI$\"3d#eZIC,4`#F -7$7$$!3s+^Ne%4p:(FI$\"3&*)[C71`Ej#F-Fg]m7$F]^m7$$!3eO8+`EAhpFI$\"3+M \">l2rgn#F-7$7$$!3c95bxQpMnFI$\"3e&ohX@Jow#F-Fc^m7$7$$!3n:5bxQpMnFIF\\ _m7$$!3E&4J$e/13lFI$\"3Sd$en')3R#GF-7$7$$!3/XW$[\\'3>kFI$\"3G4bxQpMnGF -Fb_m7$Fh_m7$$!30OJ$>=9X/'FI$\"3QOkmujvpHF-7$7$$!3O<8bY36!z&FI$\"3iHlK ;3/-JF-F^`m7$Fd`m7$$!3oBTJ%f^)zcFI$\"3+z/f*ecX8$F-7$7$$!33[Ej\"3/-^&FI $\"3h0bB)ea\"GKF-Fj`m7$F`am7$$!3K=VIwb_2`FI$\"3g'Gc.)z)yH$F-7$7$$!3w8R lW$fJD&FI$\"3&*\\v(QpMnL$F-Ffam7$F\\bm7$$!3+C3xm'fF#\\FI$\"3Ca+ee]$)eM F-7$7$$!3ciJ\"z6()Qx%FI$\"3Gq&G9dG9d$F-Fbbm7$7$$!37jJ\"z6()Qx%FIF[cm7$ $!3jNXBHSS9YFI$\"38>m$*QuUMOF-7$7$$!3'[M_eLn#*R%FI$\"3i!fz*[C71QF-Facm 7$7$$!3IWB&eLn#*R%FIFjcm7$$!3OJyT'37PO%FI$\"3>rxWF?2@QF-7$7$$!3[!G9dG9 dG%FI$\"3kF+M)Qxl)QF-F`dm7$Ffdm7$$!3?I(3ul;f1%FI$\"3-Z?xZ%)o)*RF-7$7$$ !3m[m]?n7JSFI$\"3'4hIlK;3/%F-F\\em7$Fbem7$$!3'3,qRXhvx$FI$\"3izRmWU6yT F-7$7$$!3tI!*z\"=-5q$FI$\"3IJ;3/-^vUF-Fhem7$F^fm7$$!3c7z613hSNFI$\"3i \"ow8e#RnVF-7$7$$!3j.2%=()3NV$FI$\"3j^Ej\"3/-^%F-Fdfm7$Fjfm7$$!3MXf(z \"z&QM$FI$\"3#3\\+()[vVc%F-7$7$$!3Q8Y2(>@W@$FI$\"3(>n$=fz*[u%F-F`gm7$7 $$!3$Rhuq>@W@$FIFigm7$$!3)f%f6)fG*yJFI$\"3yW$yK\"zXnZF-7$7$$!3K7fz*[C7 1$FI$\"3D:L+^'G)Q\\F-F_hm7$7$$!3)G\"fz*[C71$FIFhhm7$$!3H8z'*Q]t?IFI$\" 3CD$f'e8$=(\\F-7$7$$!32GkE3t!y,$FI$\"3I#pMn$=fz\\F-F^im7$Fdim7$$!3qf!Q b#\\t,GFI$\"3eB'pW@]X;&F-7$7$$!3;F!f\\5TPy#FI$\"3k7dG9dG9_F-Fjim7$F`jm 7$$!3_D6*Gf+qh#FI$\"3y:LEym$QO&F-7$7$$!3aF**y4/T(e#FI$\"3)HtO=fz*[aF-F fjm7$7$$!3)p#**y4/T(e#FIF_[n7$$!3_aV$y![-gCFI$\"3'47X.`V%obF-7$7$$!3N( zo)*e'\\@CFI$\"3I`xQpMn$o&F-Fe[n7$F[\\n7$$!3Q#*z;,PtDBFI$\"3UA%=\"z#)R xdF-7$7$$!3m`Sc08K!G#FI$\"3kt(QpMn$=fF-Fa\\n7$Fg\\n7$$!3;7j$['385AFI$ \"3mODPY\\$**)fF-7$7$$!319yt(*HVf@FI$\"3)Rz*[C71`hF-F]]n7$Fc]n7$$!3)ed ()e3R+6#FI$\"3q:FI$\"3aN=fz*[Ci'F-Fh^n7$7$$!3/!\\#)e8M^'>FI$\"3lM=fz*[Ci'F-7$$!35<`?0 eoY>FI$\"3fBjfUH_VmF-7$7$$!3%e4ecVXm)=FI$\"3)e&G9dG9doF-Fi_n7$7$F``n$ \"3)\\&G9dG9doF-7$$!3CI#oJi(oz=FI$\"3gpN#>]v`'oF-7$7$$!3dXv(QpMn$=FI$ \"3%4oeLm:FI$\"3+;fz*[C7c(F-F\\cn7$Fbcn7$$!3%3& *zFh]TZ\"FI$\"31kl8&3Bks(F-7$7$$!3)yv*)>bd-Z\"FI$\"3KOpMn$=fz(F-Fhcn7$ 7$F_dn$\"3WNpMn$=fz(F-7$$!3!GP[;#\\m*Q\"FI$\"3:lt)=(Q#\\%zF-7$7$$!3)z/ 7'=#)*\\Q\"FI$\"3bdz*[C71.)F-Fgdn7$F]en7$$!3kc]/P'yVJ\"FI$\"3%4vA#yy=l \")F-7$7$$!3(*>v6\"ft!48FI$\"3!z(*[C71`E)F-Fcen7$7$$!3q>v6\"ft!48FIF\\ fn7$$!33E]2P32Z7FI$\"3wZG6%4\")pQ)F-7$7$$!3eUY+%)*R7C\"FI$\"3A)******* *****\\)F-Fbfn7$7$$\"3Rcv(QpMn$=FI$\"31gBd3ruXrF-7$$\"3[!e,Lkfpt\"FIFi bn7$Fcgn7$$\"3()RXMw%yct\"FI$\"3k!R01s**eM(F-7$7$$\"3V[CbT*\\/i\"FIFec nFggn7$7$$\"3:[CbT*\\/i\"FIFecn7$$\"3=V3WvXw<;FI$\"3w&G'\\iP>.wF-7$7$$ \"3dtSQ()3n<:FIFadnFdhn7$Fjhn7$$\"3`c;Nei$Q^\"FI$\"33og:Ms!y&yF-7$7$$ \"3s$RfAIemU\"FI$\"3ybz*[C71.)F-F^in7$7$$\"3+%RfAIemU\"FIF`en7$$\"3AsK V_F)=U\"FI$\"31zZnLb75\")F-7$7$$\"3Sd*pjuvdM\"FIF\\fnF]jn7$Fcjn7$$\"3h *)p!3LI-M\"FI$\"3!*pTqe%p/O)F-7$7$$\"3]DBT_Clt7FIF[gnFgjn7$7$$\"3)R#fz *[C71$FI$\"3sYDdDq?U^F-7$$\"3!35$49Tz#)HFIFcjm7$Ff[o7$$\"3!z+IO#zjvHFI $\"3yzT81)*oI_F-7$7$$\"33t(f\"=hB[FFIF_[nFj[o7$F`\\o7$$\"3V$HHWuS6s#FI $\"30q')p4B;9bF-7$7$$\"31gwkgAx^DFIF^\\nFd\\o7$Fj\\o7$$\"3'f7Qi)y\")4D FI$\"3%Rmp]tf$*y&F-7$7$$\"3w@HsD3)fQ#FIFj\\nF^]o7$Fd]o7$$\"3F'3El-dHL# FI$\"3k\"\\lS%>&z0'F-7$7$$\"3qP[Im$4^C#FIFf]nFh]o7$7$$\"3;P[Im$4^C#FI$ \"35$z*[C71`hF-7$$\"3?L)H!=%pQ=#FI$\"39&Hyl%4A@jF-7$7$$\"3fg$))**f_Y7# FIFf^nFg^o7$7$$\"3(3O))**f_Y7#FIFb^n7$$\"3#)QR7;AOd?FI$\"3%[%z#y.i,e'F -7$7$$\"3'p7YH-[5-#FIFg_nFd_o7$7$$\"3SEh%H-[5-#FIFg_n7$$\"3MZpuPCO\\>F I$\"3z@7!H(ebNoF-7$7$$\"3+p+7bNXJ>FIFf`nFa`o7$7$Fh`o$\"35aG9dG9doF-7$$ \"3)GM0E@Xm&=FI$\"3=;x$>b?!)3(F-7$7$$\"3!\\x'ReJe`=FIF]bnF^ao7$FdaoF^g n7$7$$\"3e\"H9dG9dG%FI$\"3!zmJHNpA<%F-7$$\"3>&))=)*o\"G-TFIFafm7$F^bo7 $$\"37>4hx8%o0%FI$\"3y5\"o9^x$>VF-7$7$$\"3B+#Q*>ssNPFIF]gmFbbo7$Fhbo7$ $\"3%)p_qM(z%4OFI$\"3.\"Q4Rq:)RYF-7$7$$\"3h-![u;K:W$FIFigmF\\co7$Fbco7 $$\"3Y4GEn1+hKFI$\"3z0-s-9IT\\F-7$7$$\"3(4Fme9GF?$FIFgimFfco7$F\\doFa[ o7$7$$\"3=fEj\"3/-^&FI$\"3OC-taPa)f$F-7$$\"3//oG*H^y(\\FIFjcm7$Ffdo7$$ \"33!R1-eSdp%FI$\"3UD'>M&zAiRF-7$7$$\"38&H6#**4f1XFIFeemFjdo7$F`eoFiao 7$7$$\"3xE5bxQpMnFI$\"3bBai\"*)*Qw(QpMn$=*FI$\"3-4D3(*pglFF-7$$\"34&G/lNKL])FIF[`m7$7$$\"3)RG/lNKL]) FIF[`mF_go7$F^ho7$$\"3Q?S\\H4/P5F-$\"3,ez()Q#*))REF-7$7$$\"3'HhIlK;3/ \"F-$\"3d[FOMHFNEF-F[io7$7$$\"3g\\C71`Ej6F-$\"3N9$p$\\X]KDF-7$$\"3vpGb (oTS/\"F-F`^m7$F\\joFaio7$Fgio7$$\"3#Q&3i$3L7B\"F-$\"3w!)eo([\"Q-DF-7$ 7$$\"3E'G9dG9dG\"F-$\"39(R64f/pY#F-Fajo7$Fgjo7$$\"3=V59-J2(Q\"F-$\"3IQ lS'*>QQCF-7$7$$\"3!H71`Ej\"39F-$\"3vf+?Tq>GCF-F][p7$Fc[p7$$\"3m,Gvr,5B :F-$\"3Wa$oQKdBT#F-7$7$$\"3bfz*[C71`\"F-$\"3Y1(RcS'))4CF-Fi[p7$7$F`\\p $\"3\"pqRcS'))4CF-7$$\"35,T8!G=sk\"F-$\"3o<_Ng\"e\"4CF-7$7$$\"3?'z*[C7 1`;F-$\"3TF-$\"3y`\"3hsOcZ#F-7$7$$\"3P1`Ej\"3/-#F-$\"3K]P&fH8][#F-F\\_p7$Fb_p 7$$\"3kDTorZ'>3#F-$\"3g&3b%)>#Hx 2^GF-7$7$$\"3GkeU2\"F-7$$!36++]K3XFEF-$!3;/+vo&*Q![)FI7$$ !3%)****\\F)H')\\#F-$!3='**\\7)>-EiFI7$$!3#****\\i3@/P#F-$!3G&*\\P4!pB )RFI7$$!3;++Dr^b^AF-$!3e,](o\\:A!>FI7$$!3$****\\7Sw%G@F-$\"3f,+D\"yHm^ #!#>7$$!3*****\\7;)=,?F-$\"3-)*\\7yr?zCFI7$$!3/++DO\"3V(=F-$\"31,]i:wg *p%FI7$$!3#******\\V'zV++vQiN)pFI7$$!3******\\d;%)G;F-$\"3%>+ ]P*4F&**)FI7$$!3!******\\!)H%*\\\"F-$\"3C++DTy*f7\"F-7$$!3/+++vl[p8F-$ \"3))***\\P\\)R`8F-7$$!3\"******\\>iUC\"F-$\"3+++ve6as:F-7$$!3-++DhkaI 6F-$\"3!**\\7GpV:x\"F-7$$!3s******\\XF`**FI$\"30++v`p<3?F-7$$!3u****** *>#z2))FI$\"3F+++ljj3AF-7$$!3S++]7RKvuFI$\"3:+DJ:$==W#F-7$$!3s,+++P'eH 'FI$\"3q****\\_QA[EF-7$$!3q)***\\7*3=+&FI$\"3M+DJSMouGF-7$$!3[)***\\PF cpPFI$\"3O+v$4_E.4$F-7$$!3;)****\\7VQ[#FI$\"3b+]7`uK:LF-7$$!32)***\\i6 :.8FI$\"3B+DcY&[>_$F-7$$!3Wb+++v`hH!#?$\"3o**\\P4t\"[u$F-7$$\"3]****\\ (QIKH\"FI$\"3#**\\7yJ:j(RF-7$$\"38****\\7:xWCFI$\"31+vo9]$y<%F-7$$\"3E ,++vuY)o$FI$\"3++]73=[&R%F-7$$\"3!z******4FL(\\FI$\"3(*****\\UAL?YF-7$ $\"3A)****\\d6.B'FI$\"3C**\\iDXIS[F-7$$\"3s****\\(o3lW(FI$\"3'**\\7.-R J0&F-7$$\"35*****\\A))oz)FI$\"3u**\\PRaX*G&F-7$$\"3e******Hk-,5F-$\"3E ****\\_iz,bF-7$$\"36+++D-eI6F-$\"3T++v$R:&GdF-7$$\"3u***\\(=_(zC\"F-$ \"3_*\\7GjcR$fF-7$$\"3M+++b*=jP\"F-$\"3;++Dr\"e&ehF-7$$\"3g***\\(3/3( \\\"F-$\"3%))\\7`r!*)pjF-7$$\"33++vB4JB;F-$\"3!**\\il6%z!f'F-7$$\"3u** ***\\KCnu\"F-$\"3x***\\(ovw1oF-7$$\"3s***\\(=n#f(=F-$\"32*\\7yvrG.(F-7 $$\"3P+++!)RO+?F-$\"3)3++]'pj]sF-7$$\"30++]_!>w7#F-$\"3m**\\(=MLLZ(F-7 $$\"3O++v)Q?QD#F-$\"3g+DJ!o&=%p(F-7$$\"3G+++5jypBF-$\"3/++]Ug7(*yF-7$$ \"3<++]Ujp-DF-$\"3')**\\P*f=(H\")F-7$$\"3++++gEd@EF-$\"3W+++b@vP$)F-7$ $\"39++v3'>$[FF-$\"3!)*\\7`Jf&f&)F-7$$\"37++D6EjpGF-$\"37*\\(opq&=x)F- 7$$\"\"$F*$\"\"*F*-Fehp6&FghpF[ipFhhpF[ip-%*THICKNESSG6#\"\"#-%+AXESLA BELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F`hqFbiq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "We can get a better picture by setting up a procedure for the function whose graph is the upper part of the curve. Then we can use " }{TEXT 0 4 "p lot" }{TEXT -1 12 " instead of " }{TEXT 0 12 "implicitplot" }{TEXT -1 17 ". The range for " }{TEXT 339 1 "x" }{TEXT -1 39 " is reduced so a s to exclude values of " }{TEXT 338 1 "x" }{TEXT -1 45 " which would g ive points on the lower branch." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "f := proc(x) fsolve(x^2*y^3 -x^3*y^2=12,y=0..infinity) end:\ncurve := plot('f(x)',x=-2.4..3,y=-0.4 ..8.5,color=red):\ntgt := plot((7*x+15)/4,x=-2.4..3,color=green):\nplo ts[display]([curve,tgt],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7jp7$$!#C!\"\"$\"+\"zJ51)!#57$$ !+o`H#G#!\"*$\"+$[2pc)F-7$$!+'>\"))z@F1$\"+W6+[!*F-7$$!+\\dqk?F1$\"+qZ ^S'*F-7$$!+Xow[>F1$\"+w*\\*H5F17$$!+y*yL$=F1$\"+uQp-6F17$$!+a'*REG^j8F17$$!+Bt(o Q\"F1$\"+P_F17$$!*=z`K*F1$\"+F;cC@F17$$!*wf$)>)F1$\"+AD%pO#F17 $$!*:=\\<(F1$\"+GqMOEF17$$!*5Zz&fF1$\"+)*f_YIF17$$!*)H,F\\F1$\"+z3%H^$ F17$$!*_\"zFPF1$\"+oh?*H%F17$$!+DW.(>$F-$\"+K\"GBz%F17$$!*Lxim#F1$\"+< K7RaF17$$!+vE&R3#F-$\"+q>eWkF17$$!*-G;]\"F1$\"+sP*R0)F17$$!+IEPC7F-$\" +pbUW#*F17$$!++CYF_s$\"+U`!ox\"Fbs7$$!+Q QjsUF_s$\"+62$=(=Fbs7$$!)Y1ERF1$\"+\\Vi!)>Fbs7$$!+gfXkNF_s$\"+yuh7@Fbs 7$$!+>t%G?$F_s$\"+]s)*oAFbs7$$!+z'Q7%GF_s$\"+2h%yX#Fbs7$$!+Q+jzCF_s$\" +2/b\"p#Fbs7$$!+(R@!=@F_s$\"+o()***)HFbs7$$!+cFTcL5F_s$\"+rn&e#[Fbs7$$!+V$oer'!#7$\"+#) RZJkFbs7$$!+P>y*4$Ffx$\"+SP)o2\"!\"(7$$\"*qWI;&Ffx$\"+hNKdNF^y7$$\"+v3 RKTFfx$\"+jqj!*))Fbs7$$\"+$Gx%[xFfx$\"+%[Iq%eFbs7$$\"+pjXO6F_s$\"+UyhH XFbs7$$\"+5]1)\\\"F_s$\"+oT)yw$Fbs7$$\"+]Onf=F_s$\"+^#fAE$Fbs7$$\"+\"H #G@AF_s$\"+.e-)*GFbs7$$\"+J4*Ge#F_s$\"+PC*4i#Fbs7$$\"+s&*\\WHF_s$\"+:$ [>S#Fbs7$$\"+8#3hI$F_s$\"+%yROA#Fbs7$$\"+`ornOF_s$\"+9];v?Fbs7$$\"+%\\ D$HSF_s$\"+3)o#\\>Fbs7$$\"+NT$4R%F_s$\"+l'>4%=Fbs7$$\"+vFa_ZF_s$\"+B\" 4lu\"Fbs7$$\"+d+wvaF_s$\"+D;^*e\"Fbs7$$\"+Qt(*)>'F_s$\"+j:tj9Fbs7$$\"+ >Y>ApF_s$\"+4aIg8Fbs7$$\")>TXwF1$\"+1y\\t7Fbs7$$\"+].pt*)F_s$\"+S,@X6F bs7$$\"+!)o>I5F-$\"+>bBX5Fbs7$$\"+DZ-j6F-$\"+/SpZ'*F17$$\"+qD&eH\"F-$ \"+;!*z$)*)F17$$\"+g#3:c\"F-$\"+DZhZzF17$$\"*&R;F=F1$\"+h^_rrF17$$\"+b ]D+CF-$\"+gO$)3gF17$$\"*;YL(HF1$\"+kAES_F17$$\"+bnioNF-$\"+eOisYF17$$ \"*N2R;%F1$\"+\\(\\$\\UF17$$\"*O%H+_F1$\"+V4UAPF17$$\"*t?'>jF1$\"+<#>W L$F17$$\"*R%*fZ(F1$\"+>eC_IF17$$\"*U!G2')F1$\"+Lx._GF17$$\"*#y&=q*F1$ \"+p#*H3FF17$$\"+S*><4\"F1$\"+@fA\"f#F17$$\"+(yB4?\"F1$\"+-;n:DF17$$\" +-A_<8F1$\"+L#G*fCF17$$\"+(pxJU\"F1$\"+km4GCF17$$\"+gqoQ:F1$\"+gJy5CF1 7$$\"+oBPZ;F1$\"+4\\>4CF17$$\"+J)z4w\"F1$\"+JO7@CF17$$\"+#*=0s=F1$\"+Z a)[W#F17$$\"+ZSL))>F1$\"+\"*>K\"[#F17$$\"+#eF.5#F1$\"+'p([EDF17$$\"+Zr &[@#F1$\"+#QI=e#F17$$\"+]$Q%GBF1$\"+a$[[k#F17$$\"+zw!GV#F1$\"+\"*=24FF 17$$\"+3nU_DF1$\"+=>>*y#F17$$\"+%R:%fEF1$\"+sF)f'GF17$$\"+[w[tFF1$\"+K 2X_HF17$$\"+]$pE)GF1$\"+)3G!RIF17$$\"\"$\"\"!$\"+X3hNJF1-%'COLOURG6&%$ RGBG$\"*++++\"Fbs$F\\ilF\\ilFeil-F$6$7S7$$!3!**************R#!#<$!3w,+ +++++X!#=7$$!3s****\\n`H#G#F\\jl$!3B&**\\7$*o,W#F_jl7$$!3u**\\7'>\"))z @F\\jl$!37z\\(=K4#zk!#>7$$!3%)***\\#\\dqk?F\\jl$\"3q+]7)Q\\wO\"F_jl7$$ !3s***\\Z%ow[>F\\jl$\"3+2](o@!e'R$F_jl7$$!3z**\\ix*yL$=F\\jl$\"3[-Dc\" *y'eT&F_jl7$$!3'***\\7a'*REF\\jl7$$!3D*****\\)F_jl$\"3`+](G/(G:BF\\jl7$$!3?)**\\7:=\\<(F _jl$\"3?]7`B$*Q%\\#F\\jl7$$!3)z****\\4Zz&fF_jl$\"3O+]Pe#ftq#F\\jl7$$!3 V)*****zH,F\\F_jl$\"3K++]GFx()GF\\jl7$$!3!o**\\7_\"zFPF_jl$\"3m]7y$[Ow 4$F\\jl7$$!3y******HtFmEF_jl$\"3/++Dn9S$G$F\\jl7$$!3h'**\\7-G;]\"F_jl$ \"3[]7G'4:s[$F\\jl7$$!3#>)**\\PY1ERFjjl$\"3U]P%)oQH\"o$F\\jl7$$\"3;_++ v=TXwFjjl$\"3-,D\"yq%z$)QF\\jl7$$\"3'R+]P&R;F=F_jl$\"3#4D1>p`(pSF\\jl7 $$\"3G,+]ihMtHF_jl$\"3!**\\P%ybLqUF\\jl7$$\"3C-+v[t!R;%F_jl$\"31]7.'y$ oyWF\\jl7$$\"3)=+]7O%H+_F_jl$\"3M](=K^^+m%F\\jl7$$\"3\"Q++vs?'>jF_jl$ \"3)3]7tiLf&[F\\jl7$$\"3K+++!R%*fZ(F_jl$\"3]++D=!*He]F\\jl7$$\"3R)*** \\<4\"F\\jl$\"3]*\\Pa*)40m&F\\jl7$$\"3!)*****pyB4?\"F \\jl$\"3U***\\si;;&eF\\jl7$$\"3=++]-A_<8F\\jl$\"35+]PaQmbgF\\jl7$$\"3% )**\\(opxJU\"F\\jl$\"3G]7`p4cSiF\\jl7$$\"3S++]fqoQ:F\\jl$\"39+]7aBqUkF \\jl7$$\"33+](yOstk\"F\\jl$\"3-]7yV;!Hj'F\\jl7$$\"3E+]PJ)z4w\"F\\jl$\" 3a]i!\\q9<$oF\\jl7$$\"3e****\\#*=0s=F\\jl$\"3=**\\(=\"34EqF\\jl7$$\"3Q +](o/M$))>F\\jl$\"3w]7.#e%eHsF\\jl7$$\"3p+++#eF.5#F\\jl$\"3)3++&oKdDuF \\jl7$$\"3\\++DZr&[@#F\\jl$\"3v+vo2++EwF\\jl7$$\"3w+]()\\$Q%GBF\\jl$\" 3W^7G7rwCyF\\jl7$$\"3<+++zw!GV#F\\jl$\"3H,+DQMT2!)F\\jl7$$\"31++D3nU_D F\\jl$\"3A+vVRnu;#)F\\jl7$$\"3E+++%R:%fEF\\jl$\"3Y,+]Rp(RS)F\\jl7$$\"3 e+](yk([tFF\\jl$\"3y^7y$Q.Og)F\\jl7$$\"3Y+]7]$pE)GF\\jl$\"3/_(=FOrYz)F \\jl7$Fjhl$\"\"*F\\il-F`il6&FbilFeilFcilFeil-%*THICKNESSG6#\"\"#-%+AXE SLABELSG6%Q\"x6\"Q\"yFiim-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fjhl;$!\"%F* $\"#&)F*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Exi stence of derivatives of implicit functions" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 295 "In the examples \+ considered in this worksheet we have often used Maple to investigate a ctual explicit functions defined by an equation involving two variable s. The question arises as to the circumstances under which the method \+ of implicit differentiation gives a valid formula for the derivative. " }}{PARA 0 "" 0 "" {TEXT -1 204 "The method itself does not establish the existence of the derivative. However, if the derivative can be sh own to exist for specified values of the variables, then the formula g iven by the method is valid." }}{PARA 0 "" 0 "" {TEXT -1 18 "Be carefu l though!" }}{PARA 0 "" 0 "" {TEXT -1 30 "Suppose you are asked to fin d " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 8 ", \+ given " }{XPPEDIT 18 0 "x^2+y^2+2=0" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF' F(F'F(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "There are \+ no real values of " }{TEXT 312 1 "x" }{TEXT -1 5 " and " }{TEXT 313 1 "y" }{TEXT -1 45 " which satisfy this equation, so it would be " } {TEXT 260 11 "meaningless" }{TEXT -1 47 " to differentiate the equatio n with respect to " }{TEXT 308 1 "x" }{TEXT -1 5 " and " }{TEXT 309 1 "y" }{TEXT -1 22 " and give the formula " }{XPPEDIT 18 0 "dy/dx = -x/y ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yGF(F(" }{TEXT -1 17 ", wh en, in fact, " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 34 " does not exist for any values of " }{TEXT 310 1 "x" } {TEXT -1 5 " and " }{TEXT 311 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 8 " Given " }{XPPEDIT 18 0 " x^4+y^4=16" "6#/,&*$%\"xG\"\"%\"\"\"*$% \"yGF'F(\"#;" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " } {TEXT 340 1 "x" }{TEXT -1 5 " and " }{TEXT 341 1 "y" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "A ns " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=-x^3/y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xG\"\"$*$%\"yGF,F(F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "___________________________________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "x^2+3*x*y+y^2=31" "6#/,(*$%\"xG\"\"#\"\"\"*(\"\"$F(F&F(%\"yGF(F( *$F+F'F(\"#J" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 12 " interms of " } {TEXT 342 1 "x" }{TEXT -1 5 " and " }{TEXT 343 1 "y" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 101 "(b) Find the gradient of tangent lines \+ to the curve given by the equation in (a) at the points where " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "dy/dx=-(2*x+3*y)/(3*x+2*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"#F&%\"xGF&F&*&\"\"$F&%\"yGF&F&F &,&*&F0F&F.F&F&*&F-F&F1F&F&F(F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "(b) When " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=3" "6#/%\"yG\"\"$" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "-9" "6#,$\"\"*!\"\"" }{TEXT -1 32 ". The gradient of th e tangent at" }{XPPEDIT 18 0 "``(2,3)" "6#-%!G6$\"\"#\"\"$" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "-13/12;" "6#,$*&\"#8\"\"\"\"#7!\"\"F(" } {TEXT -1 36 ", and the gradient of the tangent at" }{XPPEDIT 18 0 "``( 2,-9)" "6#-%!G6$\"\"#,$\"\"*!\"\"" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "-23/12" "6#,$*&\"#B\"\"\"\"#7!\"\"F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q 3" }}{PARA 0 "" 0 "" {TEXT -1 7 " Given " }{XPPEDIT 18 0 "2*y^2+x*y-x^ 2=3" "6#/,(*&\"\"#\"\"\"*$%\"yGF&F'F'*&%\"xGF'F)F'F'*$F+F&!\"\"\"\"$" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 346 1 "x" } {TEXT -1 5 " and " }{TEXT 347 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(2*x-y)/(4*y+x)" "6#/*&%#d yG\"\"\"%#dxG!\"\"*&,&*&\"\"#F&%\"xGF&F&%\"yGF(F&,&*&\"\"%F&F.F&F&F-F& F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "_________________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 8 " Given \+ " }{XPPEDIT 18 0 "x^3+x-x^2*y-2*y^2 = 0;" "6#/,**$%\"xG\"\"$\"\"\"F&F( *&F&\"\"#%\"yGF(!\"\"*&F*F(*$F+F*F(F,\"\"!" }{TEXT -1 25 ", find an ex pression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 13 " in terms of " }{TEXT 348 1 "x" }{TEXT -1 5 " and " } {TEXT 349 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(3*x^2+1-2*x*y)/(x^2+4*y)" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&,(*&\"\"$F&*$%\"xG\"\"#F&F&F&F&*(F/F&F.F&%\"yGF&F(F&,&*$ F.F/F&*&\"\"%F&F1F&F&F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 " " {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "sqrt(x)+sqrt(y)=4" "6#/,&-% %sqrtG6#%\"xG\"\"\"-F&6#%\"yGF)\"\"%" }{TEXT -1 25 ", find an expressi on for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 344 1 "x" }{TEXT -1 5 " and " }{TEXT 345 1 " y" }{TEXT -1 49 " by using the method of implicit differentiation." }} {PARA 0 "" 0 "" {TEXT -1 54 "(b) Solve the equation given in (a) for y in terms of " }{TEXT 351 1 "x" }{TEXT -1 30 " and obtain an expressio n for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 34 " which involves only the variable " }{TEXT 350 1 "x" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 50 "(c) Demonstrate that the expression s obtained for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 32 " in (a) and (b) are equivalent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "dy/dx = -sqrt(y/x);" "6#/*&% #dyG\"\"\"%#dxG!\"\",$-%%sqrtG6#*&%\"yGF&%\"xGF(F(" }{TEXT -1 8 " (b ) " }{XPPEDIT 18 0 "y=x-8*sqrt(x)+16" "6#/%\"yG,(%\"xG\"\"\"*&\"\")F' -%%sqrtG6#F&F'!\"\"\"#;F'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "d y/dx=1-4/sqrt(x)" "6#/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*&\"\"%F&-%%sqrtG6# %\"xGF(F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "__________ __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________ _____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 8 " Given " }{XPPEDIT 18 0 "y^5+5*x^2+y^2-x^4 = 4" "6#/,**$%\"yG\"\"&\" \"\"*&F'F(*$%\"xG\"\"#F(F(*$F&F,F(*$F+\"\"%!\"\"F/" }{TEXT -1 25 ", fi nd an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\" \"" }{TEXT -1 13 " in terms of " }{TEXT 354 1 "x" }{TEXT -1 5 " and " }{TEXT 355 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=2*x*(2*x^2-5)/y/(5*y^3+2)" "6#/*&%#dyG\" \"\"%#dxG!\"\"*,\"\"#F&%\"xGF&,&*&F*F&*$F+F*F&F&\"\"&F(F&%\"yGF(,&*&F/ F&*$F0\"\"$F&F&F*F&F(" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 " " {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "x^2+y^2=10*x" "6#/,&*$%\"xG \"\"#\"\"\"*$%\"yGF'F(*&\"#5F(F&F(" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 356 1 "x" }{TEXT -1 5 " and " }{TEXT 357 1 " y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the equati on of the tangent line to the graph of the equation given in (a) at th e point" }{XPPEDIT 18 0 "``(2, 4);" "6#-%!G6$\"\"#\"\"%" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "dy/dx= (5-x)/y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&\"\"&F&%\"xGF(F&%\"yGF(" } {TEXT -1 9 ", (b) " }{XPPEDIT 18 0 "y=3*x/4+5/2" "6#/%\"yG,&*(\"\"$ \"\"\"%\"xGF(\"\"%!\"\"F(*&\"\"&F(\"\"#F+F(" }{TEXT -1 2 ". " }}} {PARA 0 "" 0 "" {TEXT -1 44 "_________________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "y^2=x^3*(2-x)" "6#/*$%\"yG\"\"#*&%\"xG\"\"$,&F&\"\"\"F(!\"\"F+" }{TEXT -1 24 " find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%# dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 358 1 "x" } {TEXT -1 5 " and " }{TEXT 359 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 96 "(b) Find the equation of the tangent line to the graph \+ of the equation given in (a) at the point" }{XPPEDIT 18 0 " ``(1,1)" " 6#-%!G6$\"\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "dy/dx=x^2*(3-2*x)/y" "6#/*&%#dyG\"\"\"%#dxG! \"\"*(%\"xG\"\"#,&\"\"$F&*&F+F&F*F&F(F&%\"yGF(" }{TEXT -1 9 ", (b) \+ " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q 9" }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "x*cos*y +y*cos*x=1" "6#/,&*(%\"xG\"\"\"%$cosGF'%\"yGF'F'*(F)F'F(F'F&F'F'F'" } {TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%# dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 352 1 "x" } {TEXT -1 5 " and " }{TEXT 353 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 90 "(b) Find the gradient of the tangent line to the graph \+ of the equation in (a) at the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6 $\"\"!\"\"\"" }{TEXT -1 31 ", correct to 4 decimal places. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "dy/dx = (cos*y-y*sin *x)/(x*sin*y-cos*x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&*&%$cosGF&%\"yGF& F&*(F-F&%$sinGF&%\"xGF&F(F&,&*(F0F&F/F&F-F&F&*&F,F&F0F&F(F(" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "-cos(1)" "6#,$-%$cosG6#\"\"\"!\"\"" } {TEXT -1 1 " " }{TEXT 360 1 "~" }{XPPEDIT 18 0 "``-0" "6#,&%!G\"\"\"\" \"!!\"\"" }{TEXT -1 7 ".5403. " }}}{PARA 0 "" 0 "" {TEXT -1 44 "______ ______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________ ________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q10" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "exp(x*y)=1+sin*y" "6#/-%$expG6#*& %\"xG\"\"\"%\"yGF),&F)F)*&%$sinGF)F*F)F)" }{TEXT -1 25 ", find an expr ession for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 13 " in terms of " }{TEXT 363 1 "x" }{TEXT -1 5 " and " } {TEXT 364 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=exp(x*y)*y/(cos*y-exp(x*y)*x)" "6#/*&%#d yG\"\"\"%#dxG!\"\"*(-%$expG6#*&%\"xGF&%\"yGF&F&F/F&,&*&%$cosGF&F/F&F&* &-F+6#*&F.F&F/F&F&F.F&F(F(" }{TEXT -1 4 ". " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "__ __________________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q11" }} {PARA 0 "" 0 "" {TEXT -1 11 "(a) Given " }{XPPEDIT 18 0 "y^2*tan*x = \+ exp(y-1);" "6#/*(%\"yG\"\"#%$tanG\"\"\"%\"xGF(-%$expG6#,&F%F(F(!\"\"" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 361 1 "x" } {TEXT -1 5 " and " }{TEXT 362 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 96 "(b) Find the equation of the tangent line to the graph \+ of the equation given in (a) at the point" }{XPPEDIT 18 0 "``(Pi/4,1); " "6#-%!G6$*&%#PiG\"\"\"\"\"%!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "dy/dx = y^2*sec^2*x/(exp (y-1)-2*y*tan*x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"**%\"yG\"\"#%$secGF+%\"x GF&,&-%$expG6#,&F*F&F&F(F&**F+F&F*F&%$tanGF&F-F&F(F(" }{TEXT -1 7 " ( b) " }{XPPEDIT 18 0 "y=-2*x+Pi/2+1" "6#/%\"yG,(*&\"\"#\"\"\"%\"xGF(! \"\"*&%#PiGF(F'F*F(F(F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 196 229 229 {PLOTDATA 2 "6)-%'CURVESG6h[l7$7$$\"3YfQ1 I:EW7!#>$!\"#\"\"!7$$\"3I\")*41tuNN\"F*$!3UV__Q?F=>!#<7$7$$\"3QIBCo1lk ;F*$!3?dG9dG9d=F3F.7$7$$\"3/IBCo1lk;F*F87$$\"39***=Kd-nx\"F*$!3#Rt6]*Q _%z\"F37$7$$\"3->[of^m`AF*$!3T9dG9dG97$7$$\"3s>[of^m`AF*FG7$$\"3D 65djyOVBF*$!3_2z%y3esn\"F37$7$$\"3e5s.\\k\"R4$F*$!3ir&G9dG9d\"F3FM7$FS 7$$\"3`AX#>NsH5$F*$!31L'>Qm0(o:F37$7$$\"3w[C71`EjJF*$!3a^n+h[vh:F3FY7$ Fin7$$\"3W&)o)[HJM)QF*$!3!zYqQ0&4h9F37$7$$\"3OO&)**)e>iJ%F*$!3#)G9dG9d G9F3F_o7$Feo7$$\"3-Q()G_IT*)[F*$!3)*p&=_MpOO\"F37$7$$\"3.-ndW&yU9'F*$! 3/'G9dG9dG\"F3F[p7$Fap7$$\"3ueHzX#Gc?'F*$!3+jA%)yTD!G\"F37$7$$\"3\\(*[ C71`EjF*$!3)GvBv!)RUF\"F3Fgp7$F]q7$$\"3v&*pTyI%)QvF*$!3/DO5Qng(>\"F37$ 7$$\"3s?`T?(\\q&*)F*$!3CVr&G9dG9\"F3Fcq7$Fiq7$$\"3-%4SLff$4#*F*$!3&)*e llF#>I6F37$7$$\"3%pMn$=fz*[*F*$!39')\\5Q@k@6F3F_r7$7$$\"3aXtO=fz*[*F*F hr7$$\"3'*Gcpm?r\"4\"!#=$!31IcNF)fW1\"F37$7$$\"3]z*[C71`E\"Fas$!3ot)*y![Cg*Fas 7$7$$\"3IC71`Ej\"e\"Fas$!3#Qty+K+IX*FasFgt7$7$F^u$!3#\\ty+K+IX*Fas7$$ \"3\"fcf$fHK5jp)) FasFfu7$7$$\"3QpMn$=fz*=FasF_v7$$\"3w@)R$zG=V>Fas$!3M&Rtjhkcx)Fas7$7$$ \"3%[M/81,L4#Fas$!3]w&G9dG9d)FasFev7$F[w7$$\"3]^,wTc;#=#Fas$!34\\wPz,P E%)Fas7$7$$\"3\"Rr&G9dG9AFas$!3A=u2&*3e$R)FasFaw7$7$$\"3>9dG9dG9AFasFj w7$$\"3v`f(>)*)[ACFas$!37PSok*HJ3)Fas7$7$$\"3+fz*[C71`#Fas$!3y()=r;-V \")zFasF`x7$7$Fgx$!3!*))=r;-V\")zFas7$$\"3!3&Q5(z8Hn#Fas$!39i&4TJ4by(F as7$7$$\"3!Q?5bxQp%GFas$!3NC0!pi+Lj(FasF_y7$7$Ffy$!3CB0!pi+Lj(Fas7$$\" 3Cs=i'Q5:$HFas$!3/#R.6X$zCvFas7$7$$\"3g[C71`EjJFas$!3u+i)*[gzMtFasF^z7 $7$$\"3/[C71`EjJFasFgz7$$\"3!*HO&*fH\"o>$Fas$!3#=H)GwEO%H(Fas7$7$$\"3+ \\fwY$)e#R$Fas$!3YZr&G9dG9(FasF][l7$7$Fd[l$!3d[r&G9dG9(Fas7$$\"3E<8:@X ukMFas$!3Wl***)\\Z!e2(Fas7$7$$\"3S$pMn$=fzMFas$!3M)*)4\"yNenqFasF\\\\l 7$Fb\\l7$$\"3pm;riw8IPFas$!3-19*G*HyXoFas7$7$$\"3?QpMn$=fz$Fas$!3RtDM \\,(4\"oFasFh\\l7$F^]l7$$\"3.V%QJ!RO+SFas$!3r\"H*)R8)ePmFas7$7$$\"3-$= fz*[C7TFas$!31'=V(y*32e'FasFd]l7$Fj]l7$$\"35_5e3!=ZF%Fas$!3)yTEhCN![kF as7$7$$\"3QG9dG9dGWFas$!3;^zHMHasjFasF`^l7$7$$\"3$yUr&G9dGWFasFi^l7$$ \"3!H3KJQEEb%Fas$!3k^D<'*Q`uiFas7$7$$\"3jsO=fz*[u%Fas$!3u,f^cc2$='FasF __l7$Fe_l7$$\"3\"[fo](zhL[Fas$!3+6gw`n&\\6'Fas7$7$$\"35>fz*[C71&Fas$!3 'zuK\\xj&4gFasF[`l7$7$$\"3+=fz*[C71&Fas$!3&ouK\\xj&4gFas7$$\"3x!=(>BEI <^Fas$!3U-Sh+=anfFas7$7$$\"3Oj\"3/-^vP&Fas$!3E5Ta,lw\\eFasF\\al7$Fbal7 $$\"3#yD]]saLS&Fas$!30%pfRd=3$eFas7$7$$\"3c_XB!RLem&Fas$!3j?dG9dG9dFas Fhal7$F^bl7$$\"3KDM#z'*H3p&Fas$!3%H@$yNC_+dFas7$7$$\"3q3/-^v(Qp&Fas$!3 9plRZji*p&FasFdbl7$Fjbl7$$\"3g*)HZR5GtfFas$!3]s\"H;\\Ova&Fas7$7$$\"32a Ej\"3/-,'Fas$!387>xJtzObFasF`cl7$7$Fgcl$\"35>&H)\\p@C=F37$$\"3e+RSAJV$ *fFas$\"3)e&G9dG9d=F37$F_dl7$$\"3SrtH[nN%*fFas$\"3')*yetr*Hk=F37$7$$\" 3KfS2%[k!ofFas$\"3)))*************>F3Fedl7$7$F\\el$\"3m)*************> F37$$\"3-MgJ'4La(fFas$\"3**foiDIq:?F37$7$$\"3=dslK,7\"*fFas$\"3YTr&G9d G9#F3Fdel7$Fjel7$$\"3Un3SYfB'*fFas$\"3+>='HLl\"\\@F37$7$Fgcl$\"31&*)*e \"*)z0=#F3F`fl7$7$$\"3=bEj\"3/-,'Fas$!3C8>xJtzObFas7$$\"3W59AOFkdiFas$ !3%QW0uiFas$\"3Gq&G9dG9d\"F37$F^hl7$$\"3>)*G_ujUS hFas$\"3=H)Glkval\"F37$7$$\"3Omgi.$GJ2'Fas$\"338dG9dG90'Fas$ \"3=))4VTF%oE#F37$7$$\"3_H!o^el\\0'Fas$\"3E%G9dG9dG#F3Fgil7$7$$\"3iI!o ^el\\0'FasF`jl7$$\"30X7ptqNHhFas$\"3[0)**)*RgZP#F37$7$$\"3!*o6lUKdbhFa s$\"30F9dG9dGCF3Ffjl7$F\\[m7$$\"3TZ)H/I2wkg\\ lE&Fas7$7$$\"3zWr&G9dGk'Fas$!3(=W'oqk0Q_FasFh\\m7$7$F_]m$\"3O`68:S1Q8F 37$$\"3#4DJPtirW'Fas$\"3\\F9dG9dG9F37$Fg]mF[hl7$Fd\\m7$$\"3cukEB'R$>kF as$\"3GVFPn>PsEF37$7$$\"3w-\"R=OdnX'Fas$\"3k7dG9dG9FF3F^^m7$Fd^m7$$\"3 ^@iGc))\\NlFas$\"3yb$p<5qFw#F37$7$$\"3oVr&G9dGk'Fas$\"3-@TC)3p'[GF3Fj^ m7$F^]m7$$\"3=Z%Rq+F8$oFas$!3W?P06&oo8&Fas7$7$$\"3;!RpMn$=fpFas$!3uM0T (fk,5&FasFf_m7$7$F]`m$\"3Y&pQw#p7A7F37$$\"3IuQyL4%ex'Fas$\"3q%G9dG9dG \"F37$Fe`m7$F_]m$\"3e`68:S1Q8F37$7$F_]mFc_m7$$\"3klFk&*y&)\\mFas$\"3/v D&oA\")R&GF37$7$$\"3q'GG(=d$Hl'Fas$\"3UbG9dG9dGF3F`am7$7$$\"3#yGG(=d$H l'FasFiam7$$\"3im!*)oTYgy'Fas$\"3]YOCwQLNHF37$7$$\"3)zr_AZ&esoFas$\"3A )*************HF3F_bm7$Febm7$$\"3>y(o\\.yZ\"pFas$\"3$\\&GKfT0?IF37$7$F ]`m$\"3UwESvq'e/$F3F[cm7$F\\`m7$$\"3\"=WSGGF.7(Fas$!3y^xN='fM,&Fas7$7$ $\"3]N;3/-^vsFas$!37%z?s**=)o\\FasFecm7$7$F\\dm$\"30![48v&QJ6F37$$\"34 bh!=bcNB(Fas$\"3\">9dG9dG9\"F37$FddmFb`m7$Facm7$$\"3'>A88%3@bqFas$\"3s O')zM+\\*4$F37$7$$\"3!y>6A+Tu6(Fas$\"3-Tr&G9dG9$F3F[em7$Faem7$$\"3#4gK \"=)Qi>(Fas$\"33=$3O7d'yJF37$7$F\\dm$\"3&)\\>bDD3AKF3Fgem7$F[dm7$$\"37 eNV_he5uFas$!3G(>EwpMd*[Fas7$7$$\"3'3)QpMn$=f(Fas$!3b!o'Gh!>L%[FasFafm 7$Fadm7$$\"3I\\rMV\"e`K(Fas$\"3(*>cGq^M?6F37$7$Fhfm$\"3k>u&R=>L0\"F3F] gm7$F]fm7$$\"3X(R;=(eJStFas$\"3y+J'*4sWcKF37$7$$\"3IEn)fd+`Q(Fas$\"3#Q G9dG9dG$F3Fggm7$F]hm7$$\"3u<'[)Hkh!\\(Fas$\"3Ta3X\"oE9L$F37$7$Fhfm$\"3 m>Z(\\CJPQ$F3Fchm7$Fgfm7$$\"3$4\"pv--*>q(Fas$!3+^w$3c\"=$y%Fas7$7$$\"3 7DhIlK;3zFas$!3z%y$Hfw/BZFasF]im7$7$$\"3BEhIlK;3zFas$\"3WYjq%ynZ))*Fas 7$$\"3?\\(3)p5d_yFas$\"27***************F37$F^jm7$$\"3)>)QpMn$=f(FasFd gm7$7$Fejm$\"35?Z(\\CJPQ$F37$$\"3S];;3=wQwFas$\"3&Hc3*f%ztS$F37$7$$\"3 S'zTvu_Nn(Fas$\"3gE9dG9dGMF3F[[n7$Fa[n7$$\"3UOQD!3/ez(Fas$\"3#o(pL'G9$ zMF37$7$Fjim$\"3Md$Qfr5U`$F3Fg[n7$7$Fjim$!3M&y$Hfw/BZFas7$$\"37bC!ztSW *zFas$!3A>k`%[a`n%Fas7$7$$\"3fr$=fz*[C#)Fas$!3]44B(=eug%FasFd\\n7$7$Fj im$\"3KXjq%ynZ))*Fas7$$\"3URv\"==+![zFas$\"3]eryW84?)*Fas7$7$F[]n$\"3G >nfenIz#*FasFc]n7$7$Fjim$\"3*oNQfr5U`$F37$$\"39vzfF<6[zFas$\"3#3a$RItQ `NF37$7$$\"39Y&)y2lhzzFas$\"3Sp&G9dG9d$F3F`^n7$Ff^n7$$\"3W4kF([:)4\")F as$\"3^x(z)op@BOF37$7$$\"3[q$=fz*[C#)Fas$\"3;5\"fDcsjn$F3F\\_n7$7$$\"3 qs$=fz*[C#)FasF]]n7$$\"3iK!p\"3@&yG)Fas$!3w+0ijm'=d%Fas7$7$$\"3&phIlK; 3a)Fas$!3%R7V&yQ1'\\%FasF[`n7$Fi]n7$$\"3O8)QP>;lY)Fas$\"3sb-Bev(p!*)Fa s7$7$Fb`n$\"3G0mY,6Wo()FasFg`n7$7$F[]n$\"3g5\"fDcsjn$F37$$\"3gs2k_M8m# )Fas$\"37qBd,*yap$F37$7$$\"3%\\RHB,!*3I)Fas$\"3>7dG9dG9PF3Fdan7$7$F[bn $\"3v6dG9dG9PF37$$\"3P,BXT^\"3V)Fas$\"374s$)oO'Rw$F37$7$Fb`n$\"3/7*Hq* fa7QF3Fcbn7$Fa`n7$$\"3?9$4'G/:#e)Fas$!3y9U8CWQsWFas7$7$$\"3?hG9dG9d))F as$!3IV[eA*G%)Q%FasF]cn7$7$$\"3UjG9dG9d))Fas$\"37FtAR37#G)Fas7$$\"3'eZ bm&Qbu')Fas$\"3=j&G9dG9d)Fas7$7$$\"3uualcQbu')FasFadn7$$\"3%ehIlK;3a)F asF^an7$7$$\"31=1`Ej\"3a)FasFjbn7$$\"3Uk#y]P63f)Fas$\"3zO87[WcMQF37$7$ $\"3+Lf38frM')Fas$\"3)\\&G9dG9dQF3F^en7$Fden7$$\"3!)\\z2?t7d()Fas$\"3 \")GW?k6J-RF37$7$$\"3KiG9dG9d))Fas$\"3zz$Qo>2Y%RF3Fjen7$Fccn7$$\"3[7-s [EFx))Fas$!3!p1q=-BmP%Fas7$7$$\"3M'RX&QO\\o\"*Fas$!39#H9dG9dG%FasFffn7 $F\\gn7$$\"3yeL$>LWI<*Fas$!3UetxyYz$G%Fas7$7$$\"3o2^v(QpM<*Fas$!3=#H%* 4#*=PG%FasFbgn7$Ficn7$$\"3zl'Q%om4W*)Fas$\"3M&4q(RRty\")Fas7$7$Fign$\" 3\"fIv?v&yEyFasF^hn7$F`fn7$$\"3)>atM))Q.#*)Fas$\"3Qlk2f)f9(RF37$7$$\"3 ?Qkv*)3Oy*)Fas$\"3y(*************RF3Fhhn7$F^in7$$\"37ER&e\"3B(3*Fas$\" 3QPj#=XY*QSF37$7$$\"3y3^v(QpM<*Fas$\"3m0N(y>iS2%F3Fdin7$7$Fign$!3k\"H% *4#*=PG%Fas7$$\"3oY?!oKm:Y*Fas$!3yCun2eAeTFas7$7$$\"3-`tO=fz*[*Fas$!3+ TT;)fQI:%FasFcjn7$7$$\"3c1^v(QpM<*Fas$\"3![Iv?v&yEyFas7$$\"35_WHAY<&R* Fas$\"3KX=7D\"z,d(Fas7$7$Fjjn$\"3nmDg9kXGuFasFd[o7$7$FignF]jn7$$\"3#\\ )z[[!*4`#*Fas$\"3sz1fg`*o5%F37$7$$\"3+Y%om=3!H$*Fas$\"3dSr&G9dG9%F3F_ \\o7$Fe\\o7$$\"3Z'eOjWW(>%*Fas$\"33'>;:H$\\uTF37$7$Fjjn$\"3Da)\\^mF@?% F3F[]o7$7$Fjjn$!3YST;)fQI:%Fas7$$\"3UX@#e%ol](*Fas$!3T:K+lWANSFas7$7$$ \"3G(fz*[C71)*Fas$!3d!HT'=qnCSFasFh]o7$7$$\"3R)fz*[C71)*Fas$\"3W$yN![_ +fqFas7$$\"3[TjBHP:T(*Fas$\"3CNr&G9dG9(Fas7$Fi^o7$$\"3#>Nn$=fz*[*FasF[ \\o7$7$F`_oFb]o7$$\"3)z;zqqQwe*Fas$\"3/KW%)>s_TUF37$7$$\"3ic0xzGw$o*Fa s$\"3#QG9dG9dG%F3Fd_o7$Fj_o7$$\"3wGy>4oS`(*Fas$\"3[M:%)[8_4VF37$7$Fe^o $\"3)42^hc.)HVF3F``o7$F^^o7$$\"3Tf+&\\]ES+\"F3$!3[NT/`5c9RFas7$7$$\"3E %=fz*[C75F3$!3Oq=)4XA$)*QFasFj`o7$Fd^o7$$\"3Dn*3Mrh$H)*Fas$\"3MR$zgi0z .(Fas7$7$Faao$\"3)fInxjiVm'FasFfao7$Ff`o7$$\"35$e_$*>RE#**Fas$\"3;Z5mg 4&fP%F37$7$$\"3x7oCMq'R+\"F3$\"30F9dG9dGWF3F`bo7$7$$\"3*H\"oCMq'R+\"F3 Fibo7$$\"3[`%\\#\\_q35F3$\"3qMI.\"F3$!3E%Q+6BPgz$Fas7$7$$\"3!)3/-^v(Q/\"F3$!3W1UTn 2otPFasFico7$F\\bo7$$\"3[Z/'H+([D5F3$\"34*)HT%fE[a'Fas7$7$F`do$\"3Nr\\ N\\9l6jFasFedo7$Feco7$$\"3V$e:&o&)oD5F3$\"36.(G0A:2^%F37$7$$\"3GztM-\\ PR5F3$\"3Gq&G9dG9d%F3F_eo7$Feeo7$$\"3=QDx,'e>/\"F3$\"3k4$QQz%4!e%F37$7 $F`do$\"3K&p>Ncyse%F3F[fo7$7$F`do$!3)e?9uw!otPFas7$$\"3M\\2^:m3i5F3$!3 -=P40x[zOFas7$7$$\"3ML;3/-^v5F3$!3A*o8Nas/l$FasFhfo7$F[eo7$$\"3y3BT>U* o1\"F3$\"3(Q(>N*p%R.hFas7$7$F_go$\"3S'e@xIAR*fFasFdgo7$Fafo7$$\"3]b$*o0h;ZF37$7$F_go$\"3nCUI8g`=ZF3Fjho7$F^go7$ $\"3]?_Z)Rz64\"F3$!3)ygsU*QykNFas7$7$$\"3(y&G9dG926F3$!3nXeyK/VGNFasFd io7$7$F[jo$\"3rv'3)R'3Fq&Fas7$$\"3(*o_.eH416F3$\"3J2dG9dG9dFas7$7$$\"3 vo_.eH416F3FfjoFjgo7$F`io7$$\"3Y%*)p'[!f=4\"F3$\"3gNRXl$4Ly%F37$7$F[jo $\"3Wq)4*>gD^[F3F][p7$Fjio7$$\"3%HD`Z:6.7\"F3$!3%41!eo,%=X$Fas7$7$$\"3 T#3/-^v(Q6F3$!3yi9ihnH2MFasFg[p7$7$F[jo$\"3#on3)R'3Fq&Fas7$$\"3n.^J`lW 26F3$\"3eb5G./d+dFas7$7$F^\\p$\"3aP:el)Q3N&FasFf\\p7$Fc[p7$$\"3#[Mwv*y x26F3$\"3#*GXgK\\Fa[F37$7$$\"3`au!fS4$36F3$\"3wcG9dG9d[F3F`]p7$Ff]p7$$ \"3C?\"QT5'QC6F3$\"3v)HM;FF@#\\F37$7$F^\\p$\"3g=65?=C()\\F3F\\^p7$F]\\ p7$$\"3?0j\"*e6[\\6F3$!3e@rsmPiSLFas7$7$$\"3%pIlK;3/<\"F3$!3/;A7(H@oG$ FasFf^p7$F\\]p7$$\"3'H-9A%3N[6F3$\"3[$o\\aO^=G&Fas7$7$$\"3;2`Ej\"3/<\" F3$\"3&Rr[&z&=0.&FasFb_p7$Fb^p7$$\"3k%37P\\I,9\"F3$\"3m!GT'>2)Q*\\F37$ 7$$\"3HNzuaFBT6F3$\"\"&F-F^`p7$7$F]_p$!3g;A7(H@oG$Fas7$$\"3m2+;(>!py6F 3$!3K%*GK%>q6B$Fas7$7$$\"3[JlK;3/-7F3$!3/q[nX\"fn;$FasF]ap7$7$F]_pF[`p 7$$\"3wUs_)\\P&)=\"F3$\"3#>-m6aTb*[Fas7$7$Fdap$\"3M(o#)*p[7PZFasFjap7$ 7$Fdap$!3gq[nX\"fn;$Fas7$$\"39Sri&4Tz?\"F3$!3gd>i$32O7$Fas7$7$$\"3-cxQ pMnL7F3$!3GFk8W(oo/$FasFgbp7$F`bp7$$\"3vu,\\J&y\"G7F3$\"3e*\\69xsQ`%Fa s7$7$F^cp$\"3')yg#ph!)oY%FasFccp7$F]cp7$$\"3A^_u[&RsB\"F3$!3Xk+K-;>=IF as7$7$$\"3c!)*[C71`E\"F3$!3)H:p#3*4p#HFasF]dp7$7$$\"3y!)*[C71`E\"F3$\" 33)>Y5J+R=%Fas7$$\"3#\\O<)Hhrb7F3$\"3EyUr&G9dG%Fas7$F^ep7$$\"3CcxQpMnL 7F3$\"3Szg#ph!)oY%Fas7$Fcdp7$$\"3/j4'oW&fm7F3$!3F&RK7Tq`\"HFas7$7$$\"3 %Q(R0f_%GG\"F3$!3kjG9dG9dGFasFjep7$F`fp7$$\"3#p#[(3mr_H\"F3$!3=IN44Q&= y#Fas7$7$$\"3'[?5bxQpH\"F3$!3#RCU][f&zFFasFffp7$7$FddpF\\ep7$$\"3gQzTt 3*yE\"F3$\"3k+Oz'f$)*oTFas7$7$$\"330-^v(QpH\"F3$\"3:)e*G0;vVQFasFcgp7$ F\\gp7$$\"3$\\f]%zu)GK\"F3$!3Qy!=2Q[/g#Fas7$7$$\"3jH9dG9dG8F3$!3=2Gi>( [;f#FasF_hp7$Figp7$$\"3&R@I*QRH38F3$\"3'*3H%QX+Hx$Fas7$7$Ffhp$\"3D1Yv! e[,`$FasF[ip7$7$Ffhp$!3i1Gi>([;f#Fas7$$\"3[!H\"\\U[W]8F3$!3HbA&\\-.kT# Fas7$7$$\"3;aEj\"3/-O\"F3$!3MZ9xX6:*R#FasFhip7$Faip7$$\"3)o=`Q`5$[8F3$ \"3)z%oP@3F%R$Fas7$7$F_jp$\"3p&z:l*3aRKFasFdjp7$7$F_jp$!3iZ9xX6:*R#Fas 7$$\"3Yx0v(=^zP\"F3$!39:'[.k`+B#Fas7$7$$\"3oyQpMn$=R\"F3$!3Zq#QdV'\\,A FasFa[q7$Fjjp7$$\"3G\"of24*>)Q\"F3$\"3n9`%4bB9-$Fas7$7$Fh[q$\"39>wA(fm *oHFasF]\\q7$7$Fh[q$!3?q#QdV'\\,AFas7$$\"3*e!*\\TREaS\"F3$!3F)3]Y)zHU? Fas7$7$$\"3B.^v(QpMU\"F3$!3A9&R\"=?1)*>FasFj\\q7$7$Fa]q$\"3vyQ2pA?3EFa s7$$\"3B:0A\"**)3/9F3$\"3K]G9dG9dGFas7$Fi]qFc\\q7$7$Fa]q$!3%R^R\"=?1)* >Fas7$$\"3XECnUi\"HV\"F3$!3?kz7'e._&=Fas7$7$$\"3wFj\"3/-^X\"F3$!3u_))> tW;)y\"FasFc^q7$Ff]q7$$\"3zKB2\"[g'H9F3$\"3%\\DHW4Xvd#Fas7$7$Fj^q$\"3k S2k$G]$)>#FasF__q7$Fi^q7$$\"3#Q`v0LE0Y\"F3$!3QR2=`$RNn\"Fas7$7$$\"3I_v (QpMn[\"F3$!3kO2qg%\\5d\"FasFi_q7$Fe_q7$$\"3Oe@/^&[@Z\"F3$\"3Cqs%)f/I( 3#Fas7$7$F``q$\"3k_#y^m:?#=FasFe`q7$F_`q7$$\"3#H#*['=U_)[\"F3$!3N`2Jw' )Q4:Fas7$7$$\"3^mUQMZF.:F3$!3;N9dG9dG9FasF_aq7$Feaq7$$\"31T8p-6J8:F3$! 3O06<9\\A+7Fas7$7$$\"3%oxQpMn$=:F3F^bqF[bq7$7$$\"3a_v(QpMn[\"F3$\"3#HD y^m:?#=Fas7$$\"3ppe\\ds=;:F3$\"3\\Q<$33Fq_\"Fas7$7$Fbbq$\"3sWIvD1eu9Fa sFjbq7$Fabq7$$\"3/w_&Gej'H:F3$!3u!)Qx(*e_,^F*7$7$$\"3Q,+++++]:F3$!3#G) Qx(*e_,^F*Fdcq7$7$F[dq$\"3g2zH>$yLo&F*7$$\"3+/5t4?E?:F3$\"3$=Ur&G9dG9F as7$Fcdq7$Fbbq$\"3;WIvD1eu9Fas-%'COLOURG6&%$RGBG$\"*++++\"!\")$F-F-Fce q-F$6%7S7$Fceq$\"3c'*[zEjzqDF37$$\"3YLL$e*)e&yLF*$\"3yH#y)[^A.DF37$$\" 3-n;/E\\A=jF*$\"3NjSFG=VWCF37$$\"3iLL3Fo=C'*F*$\"3gHKD!f7$yBF37$$\"3]L $e*yi?&H\"Fas$\"3vHK+r]v6BF37$$\"3vmT&Q>7ki\"Fas$\"3/jS-))Q^XAF37$$\"3 UL3FuC[L>Fas$\"3')H2%>$)*4%=#F37$$\"3-+vVjfV^AFas$\"3]'R2T8407#F37$$\" 3cL3x;RE!e#Fas$\"3'*H2WVNua?F37$$\"3$**\\(o9t.3HFas$\"3o'RdQ'))=*)>F37 $$\"3Enm\"Ha#>XKFas$\"36j:@=yv@>F37$$\"3)RLe90f@a$Fas$\"3')HK];XOi=F37 $$\"3e++v.IZwQFas$\"3W'*[/E<]&z\"F37$$\"31++D\"of@@%Fas$\"3V'*[a!Rk$G< F37$$\"3$4+]i*flNXFas$\"3O'*[aF^mj;F37$$\"3bL3xT;UH[Fas$\"3&)H2W)*>\" \\g\"F37$$\"3/nm\"zSP(y^Fas$\"3:j:@X)[]`\"F37$$\"3)omm;o`YZ&Fas$\"3=j: Y!fleZ\"F37$$\"3U+v$flu)=eFas$\"3Z'R2cR@qS\"F37$$\"3_mm;W&oN7'Fas$\"3E j:'zh#3Y8F37$$\"3M,v$4.myX'Fas$\"3I'R217B#z7F37$$\"3!3]7GH'>wnFas$\"3S 'RK#oqb:7F37$$\"3mn;/E0M3rFas$\"3-jle@#G\"\\6F37$$\"3UnTN;EN8uFas$\"33 jS_.e7)3\"F37$$\"3&QLe9O\\Bu(Fas$\"3zHK]akKA5F37$$\"3=M3x;X3%3)Fas$\"3 C(H2WB%zR&*Fas7$$\"3O+D1ufc\"Q)Fas$\"3%['R#)>8$[%*)Fas7$$\"3iM$3x4aGq) Fas$\"3L'HKD2bAI)Fas7$$\"3X++]nhxM!*Fas$\"3nk*[H$4TQwFas7$$\"3s+](o:( \\f$*Fas$\"37k*)>a*o*))pFas7$$\"3M,v$4T\"ot'*Fas$\"3!H'R2Y/ggjFas7$$\" 3-+D\"e%HD-5F3$\"3Al*)p^V!Hm&Fas7$$\"3ummT%\\)fL5F3$\"3(3j:'zL*f.&Fas7 $$\"31+]7el1n5F3$\"3Ik*[a5KmO%Fas7$$\"3E$3x\")f$R(4\"F3$\"3K+tS/84gPFa s7$$\"3;+]Pj!\\08\"F3$\"3Ki*[/+#)p4$Fas7$$\"3w;/Esduh6F3$\"3]I1uAy/tCF as7$$\"39](o>KbV>\"F3$\"3OiRdGo&3#=Fas7$$\"3im;HPfsSvnV\"Fas 7$$\"3C++v'z%>(Q\"F3$!3:R50nE$f.#Fas7$$\"3&p;z%)))H:U\"F3$!3PtVj,XjAFF as7$$\"3SLL$QgRAX\"F3$!3F-xr3)GoL$Fas7$$\"33]PflD)\\[\"F3$!3/Og#R/)o\" *RFas7$$\"3B]i!zv@j^\"F3$!35Rg " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q12" }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) Gi ven " }{XPPEDIT 18 0 "x*y^2-x^3+x*exp(x*y)+6 = 0;" "6#/,**&%\"xG\"\" \"*$%\"yG\"\"#F'F'*$F&\"\"$!\"\"*&F&F'-%$expG6#*&F&F'F)F'F'F'\"\"'F'\" \"!" }{TEXT -1 25 ", find an expression for " }{XPPEDIT 18 0 "dy/dx" " 6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 365 1 " x" }{TEXT -1 5 " and " }{TEXT 366 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the equation of the tangent line to the gra ph of the equation given in (a) at the point" }{XPPEDIT 18 0 "``(2, 0) ;" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "dy/dx = (3*x^2-y^2-exp(x*y)-x*exp(x *y)*y)/(x*(2*y+x*exp(x*y)));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,**&\"\"$F& *$%\"xG\"\"#F&F&*$%\"yGF/F(-%$expG6#*&F.F&F1F&F(*(F.F&-F36#*&F.F&F1F&F &F1F&F(F&*&F.F&,&*&F/F&F1F&F&*&F.F&-F36#*&F.F&F1F&F&F&F&F(" }{TEXT -1 9 ", (b) " }{XPPEDIT 18 0 "y=11*x/4-11/2" "6#/%\"yG,&*(\"#6\"\"\"% \"xGF(\"\"%!\"\"F(*&F'F(\"\"#F+F+" }{TEXT -1 2 ". " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 228 218 218 {PLOTDATA 2 "6(-%'CURVESG6by7$ 7$$!\"&\"\"!$!3CJz(**z(Hmi!#=7$$!334=WmG,l[!#<$!37s%3>:7>X'F-7$7$$!3)* QpMn$=fz%F1$!3!>(>/3#))yS'F-F.7$7$F6$\"3h2!)HL6jB\\F17$$!3ihb/MsGs[F1$ \"3y$*************\\F17$F57$$!3]3!\\&Q#yfk%F1$!3vleI/@W,mF-7$7$$!3)z(Q pMn$=f%F1$!3EYehZG_flF-FD7$7$FK$\"3]kvWS^-JZF17$$!3y\"Gt:'*Hnl%F1$\"3y KpMn$=fz%F17$FSF;7$FJ7$$!3k(\\))o!eKEWF1$!3Sm.Q%4]rv'F-7$7$$!3)p\"3/-^ v(Q%F1$!3J8$*=1/E@nF-FZ7$7$F[o$\"3=#Hn)*RQ+a%F17$$!3Rf\\wOMbRWF1$\"3wr QpMn$=f%F17$FcoFP7$Fjn7$$!3(*z)pD3dh?%F1$!3#>$f.65-=pF-7$7$$!3'fv(QpMn $=%F1$!3Y6SopT6$*oF-Fjo7$7$Fap$\"3eSH,T0*4N%F17$$!37$[&)*H!Q/A%F1$\"3w 53/-^v(Q%F17$7$$!3,%[&)*H!Q/A%F1F\\qF`o7$F`p7$$!3u&)*HqM$e&)RF1$!3]nU! *R?%H3(F-7$7$$!3'\\pMn$=fzRF1$!3gvK!\\@0^2(F-Fcq7$7$Fjq$\"3[,#oj8%GkTF 17$$!35S\">&o6)*)*RF1$\"3u\\xQpMn$=%F17$7$$!3aS\">&o6)*)*RF1FerFfp7$7$ $!3'Rj\"3/-^vPF1$!3%p2z:Cb6?(F-7$$!3Eu*oN*yt'*QF1$!3*H_,=$y$F1$\"3#f*f%[yhs)RF17$7$F]s$\" 3sV&=\\#**Q!)RF1F[t7$F\\s7$$!3+`D!fF/vw$F1$!3MzzkCk\"HA(F-7$7$$!3'HdG9 dG9d$F1$!3gU!4;C:HI(F-Fet7$7$F\\u$\"3%[(y!*>6!))z$F17$$!3+t.?%f7Xx$F1$ \"3w)oMn$=fzRF17$7$$!3cs.?%f7Xx$F1Fgu7$$!3_L;3/-^vPF1Fbt7$7$F\\u$!3\\T !4;C:HI(F-7$$!3C/*yyRU&\\NF1$!3=hQNy)=^v(QpMnLF1$!3H3D;- I\\>uF-Fdv7$7$F[w$\"3A$R3RM47i$F17$$!3VR%yAzE>a$F1$\"3uF;3/-^vPF17$Fcw 7$F\\u$\"3Quy!*>6!))z$F17$Fjv7$$!3lEw)z*ftHLF1$!3/Fgt]l'*=vF-7$7$$!3%4 XAhIlK;$F1$!3+*f^Q(RS`vF-F]x7$7$Fdx$\"3#3oc\"yfi[MF17$$!3YB1`ENM.LF1$ \"3um&G9dG9d$F17$F\\yF`w7$Fcx7$$!3j\\]#GeTy5$F1$!3Q*=JeP%4(p(F-7$7$$!3 [*QpMn$=fHF1$!3H[R4m^`3xF-Fcy7$7$$!3#**QpMn$=fHF1$\"3#p()z]A.CG$F17$$! 3eTB&G;zQpV$)GF1$!3)y ([j*[C.!zF-7$7$$!3#*Gj\"3/-^v#F1$!3s,$eAg\">\"*yF-F\\[l7$7$Fc[l$\"3#30 sB`!QCJF17$$!3#zb%3n$[.!GF1$\"3sWC71`EjJF17$7$$!3PeX3n$[.!GF1F^\\lF_z7 $7$Fc[l$!3i+$eAg\">\"*yF-7$$!3COvM%*zvbEF1$!37-^a2wHO\")F-7$7$$!3#zEj \"3/-^DF1$!3gQKd\"4)*=6)F-Fh\\l7$Fh[l7$$!3@Y8]go:4EF1$\"3,iu!e7?t,$F17 $7$F_]l$\"3a+&=Sd8f(HF1Fd]l7$7$F_]l$!3\\PKd\"4)*=6)F-7$$!32yBk\"[2MU#F 1$!3zug13k)*=%)F-7$7$$!3!p?5bxQpM#F1$!3Efp;$y$\\)Q)F-Fa^l7$7$Fh^l$\"3x y-Ck;$y$GF17$$!3pfr!y+zF_#F1$\"3s$QpMn$=fHF17$F`_lFj]l7$7$Fh^l$!3Qgp;$ y$\\)Q)F-7$$!3g\"pGu-BQ=#F1$!3?GBnBY,u()F-7$7$$!3!f9dG9dG9#F1$!3]Q'oS% pX^()F-Fj_l7$7$Fa`l$\"3h/r,>$yqr#F17$$!3?<%['eo/+AF1$\"3qAj\"3/-^v#F17 $Fi`l7$Fh^l$\"3@z-Ck;$y$GF17$7$$!3n%3/-^v(Q>F1$!30^)f_,X:B*F-7$$!3#\\. -w+_)f>F1$!3G&y(QpMn$=*F-7$7$$!3;N?g2?&)f>F1F[bl7$Fa`l$!3iR'oS%pX^()F- 7$Ff`l7$$!3YdMg!G8>1#F1$\"3EMEcy\"eTn#F17$7$Fdal$\"3Zc;*[t-]h#F1Febl7$ Fcal7$$!39*G5)3\"G%H>F1$!3eSdK$[ZrF*F-7$7$$!3BB5bxQpM%)fc_I:1\"F1F[el7$7$F\\dl$\"3`ge(=08/ a#F17$$!3s<8OMuZ>M\"[(p\\'\\#F1Fjel 7$7$$!3*4!\\C71`E8F1$!3En!o6>3m=\"F17$$!3H-0^co+B9F1$!3)*R=fz*[C7\"F17 $7$$!32-0^co+B9F1$!3wR=fz*[C7\"F1Fael7$F`fl7$$!3mH7&pEj.Z\"F1$\"39Hl@I 9x!\\#F17$7$Fefl$\"3q)zqaC2w\\#F1Fegl7$7$Fbgl$!3aULdYbVv8F17$$!3KB5-_* Hg;\"F1Fefl7$7$$!3aB5-_*Hg;\"F1Fefl7$Fefl$!3/n!o6>3m=\"F17$7$Fefl$\"3E )zqaC2w\\#F17$$!3ew@g>sjz7F1$\"3HP0_:q7/DF17$7$$!3e\"fuccUj9\"F1FcdlF` il7$Ffil7$$!3G7bqF'pv7\"F1$\"3WMpFc59cDF17$7$Fbgl$\"3#>72>(*[(eDF1Fjil 7$7$F[bl$!3i6:BAM.g;F17$$!3%f5GS5`h%**F-$!3Wiz*[C71`\"F17$7$FhjlFbel7$ Fbgl$!3KULdYbVv8F17$F`jl7$$!3K0e%*))Rvf**F-$\"3rj!>,YG'GEF17$7$F[bl$\" 3'[[$H3t$*)p#F1Fb[m7$7$Fds$!3Jge,B#[05#F17$$!3&fuUyvpTr(F-Fdal7$F_\\m7 $$!3w0V&e69%3%)F-$!3SrV!H\")>A\"=F17$7$$!3#fP\\Wx4bs)F-F\\dlFc\\m7$7$$ !3.x$\\Wx4bs)F-F\\dlFdjl7$Fh[m7$$!3]A2&p]K&*)))F-$\"3w;EdWzoDFF17$7$$! 3'yH.`#=8*o)F-F\\alFa]m7$Fg]m7$$!3^=QD\\P;NzF-$\"3P()fX,FLMGF17$7$$!3/ 7gT;M2yrF-Fc_lF[^m7$Fa^m7$$!3+uP;26)>;(F-$\"3#R/+*pg4hHF17$7$Fds$\"3em $[Dt'ekHF1Fe^m7$7$$!3qglK;3/-^F-$!3-l5Z$R`&*z#F17$$!3#*\\bZ_7@'=&F-Fc[ l7$Fd_m7$$!3Z&QC>fh!H_F-$!3MYlDj**RUFF17$7$$!3kN*Qj8yxi&F-F_]lFh_m7$F^ `m7$$!3#>!pJKHeCfF-$!3!QBkl>m(oCF17$7$$!3Uo&o)>y&\\?'F-Fh^lFb`m7$Fh`m7 $$!3$RX$ef/G/pF-$!3elF3MF-FerFbjm7$Fhjm7$$!3Usu*)eT@PLF-$ \"3#G!zHOCF6UF17$7$$!3%=!\\![(>(34$F-F\\qF\\[n7$Fb[n7$$!3O\")RiQS/&3$F -$\"3W9O#p0P,R%F17$7$F^dm$\"3%Hl]_<4!3WF1Ff[n7$7$$!35rbaI9vz?F-F(7$$!3 >(p*e>[A6AF-$!3;kvOM$=4)[F17$7$$!3S8`N7N[TAF-F6Fc\\n7$Fi\\n7$$!31ffoN# f#*R#F-$!3%o([5gK.eYF17$7$$!3wgX*[6BQU#F-FKF]]n7$Fc]n7$$!3+GQfKc$Hh#F- $!3pG5w()ReKWF17$7$$!3e*zI/=*HIEF-F[oFg]n7$F]^n7$$!3qga?G2&p&GF-$!3_/o aX35/UF17$7$$!3i[:nf9AlGF-FapFa^n7$Fg^nF]dm7$F\\\\n7$$!3g)HTJ-DE$GF-$ \"372%G!)yw*oXF17$7$$!3WsM()>WsAGF-FfoF\\_n7$Fb_n7$$!3+u(fa$>H2EF-$\"3 i:L\">6D0v%F17$7$$!3+(fSG&\\V)e#F-FVFf_n7$7$$!3a(fSG&\\V)e#F-$\"3mLpMn $=fz%F17$$!3uIA6i'H!3CF-$\"3:A;B<0oM\\F17$7$$!3]3Q#F-FAFe`n7$7$$ \"3oyS?5bxQ>F1$!36'Hy\"RG\\4cF-7$$\"3_Z\"3O%**[K>F1F`_m7$Fdan7$$\"3or' GV%o`K>F1$!3p!\\sv:a'R]F-7$7$$\"3Zu=,Zt?N>F1F^dmFhan7$F^bn7$$\"3\"pZe1 /ec$>F1$!3;I*RVz\\+.$F-7$7$F`an$!3=4CxRDJiGF-Fbbn7$7$$\"3pRr&G9dG9#F1$ !3'Hd2+(\\8:8F17$$\"332lmEOlf?F1F\\gl7$7$$\"3k1lmEOlf?F1Fbgl7$$\"3&p'3 [&)\\oB?F1$!3-nb@AoF.5F17$7$$\"3o*[5!fp%f*>F1F[blFhcn7$F^dn7$$\"3))3NB ZMle>F1$!39x9:8ljTtF-7$7$$\"3UlmSq(GU&>F1FdsFbdn7$FhdnF_an7$Fhbn7$$\"3 cg\">yDJ>'>F1$!3ubhTRc'>D\"F-7$7$$\"3%)>k49@1m>F1$!37O`Ej\"3/-\"F-F]en 7$Fcen7$$\"3rgn8VH%)3?F1$\"3!Gd%QR$Qt>$!#>7$7$$\"33DitJuIQ?F1$\"3^&=-k.V$GF-7$7$$\"3=,(Qd`_[=#F1$\"3Y))ez*[C71$F-Fjhn7$F`in 7$$\"3q[R,nrvpAF1$\"3)*3%eZdSI$QF-7$7$Fagn$\"3Gc4b1D\"\\@%F-Ffin7$7$Fc dl$!3u^Qj:VfM?F17$$\"3k\\(G1\\z8[#F1Fdal7$Fcjn7$$\"3yBe@#e-vP#F1$!3aYm D%od_w\"F17$7$$\"3/mVL(o#eeBF1F\\dlFgjn7$7$F^[oFfclF`gn7$F\\jn7$$\"39, w%y.*[%R#F1$\"3p&\\_H>Qli%F-7$7$$\"3)p1*H*\\^T\\#F1$\"3v+lK;3/-^F-Fc[o 7$7$$\"3am!*H*\\^T\\#F1F\\\\o7$$\"3&*))oUw*oa`#F1$\"3IH-pL^bd_F-7$7$Fc dl$\"3)G'=3#QL]E&F-Fb\\o7$7$F\\al$!3<3^'>2O`K#F17$$\"3!y,!zu&F-7$7$F\\al$\"3'*o7 pbf!4w&F-Fd]o7$7$Fc_l$!39gZ$35>/$F1$\"3$)*))G*p>f:jF-7$7$F^\\l$\"3m2]u1U]-jF-Fe`o 7$7$Fgz$!3')plUTo\"**3$F17$$\"3k^/@>*y8D$F1F`z7$Fbao7$F^\\l$!3C_]a'F-7$7$$\" 3=n&G9dG9d$F1$\"37OV)\\LZVZ'F-Fbco7$7$Ffw$!3M*GpPe%[eNF17$$\"35q^Xi)fP g$F1F[w7$FadoFgbo7$7$F_y$\"3+NV)\\LZVZ'F-7$$\"3PO$Q0@g\"GOF1$\"3))=%f< vSbd'F-7$7$Ffim$\"330)R8\\n!ykF-Fido7$7$Fgu$!3k!*3\"Q%)y^y$F17$$\"3W0H $**>B*pRF1F]s7$Ffeo7$$\"3!*H3Y'HmS)QF1$!35vx!Qm%)*zOF17$7$$\"3k[P@a')) fy$F1F\\uFjeo7$7$$\"33\\P@a'))fy$F1$!3St&G9dG9d$F1F^do7$7$FfwF`eo7$$\" 3@!oi'zL:LQF1$\"3_!fYqQDkc'F-7$7$$\"3I)oMn$=fzRF1$\"3#[eN(o'ymW'F-F[go 7$7$Fer$!3hxA$f`!*z+%F17$$\"3)R#4\">xu_:%F1Fjq7$FjgoFceo7$7$FguFdgo7$$ \"3k]wNHq:TSF1$\"3(=\\Fm@0s_'F-7$7$$\"3k]xQpMn$=%F1$\"3Ev]\"[2L1R'F-F` ho7$7$F\\q$!3r))pB<-AGUF17$$\"3&*G&)oa$3KM%F1Fap7$F_ioFggo7$7$Fer$\"39 u]\"[2L1R'F-7$$\"37z0?`RR^UF1$\"3Y>)GZI_cY'F-7$7$F\\q$\"3d4'ewj0zJ'F-F gio7$7$Ffo$!3Y!H:Ie@jW%F17$$\"3Kt+$QDqK`%F1F[o7$FdjoF\\io7$F]jo7$$\"3) e'R:0wAjWF1$\"3%=cD<6K\")Q'F-7$7$$\"3msQpMn$=f%F1$\"3))*p[r@nXB'F-Fijo 7$7$$\"3)=$pMn$=fz%F1$!3SW?TsGliYF17$$\"3BD'\\'HA5DZF1FK7$Fj[pFajo7$7$ Ffo$\"3x)p[r@nXB'F-7$$\"3%Qr[GiQhn%F1$\"31!p38EQ)*H'F-7$7$FV$\"3Mw/bPw @XhF-Fb\\p7$7$FA$!3%e1(******\\x[F17$$\"318LNn$=%=\\F1F67$F_]p7$FVFh[p 7$Fh\\p7$$\"3)\\1C;b1(*)[F1$\"3,\"z&3+`(\\?'F-7$7$FA$\"3m^]t`gJ`gF-Fe] p-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fe^p-F$6%7S7$$!\"'F*$!#AF*7$$!3z ******\\TVQdF1$!3')***\\7Rp!G@!#;7$$!3l****\\-r%3^&F1$!3')*\\(=`H[l?Fc _p7$$!3A+++l;!\\D&F1$!35+](y&z4&*>Fc_p7$$!3o*****\\lfs*\\F1$!3)***\\70 kCC>Fc_p7$$!3%)****\\s@%3u%F1$!3!)*\\PufJP&=Fc_p7$$!3J++]U.6.XF1$!3;+v =W`N)y\"Fc_p7$$!3')****\\-G&pD%F1$!3;+vo??m?F1$!3!****\\-5Iu4\"Fc_p7$$!3'*******R%e:w\"F1$!3%** ****4dGW.\"Fc_p7$$!33++]#yk]\\\"F1$!3x**\\(=:G9h*F17$$!3M+++SFam%**F-$!3Q,++&4qYw#F17$ $\"3k*****\\JigC\"F1$!3)4+]PjGL2#F17$$\"3%*****\\P=(F-7$$\"3Y****\\P/&f\\#F1$\" 3])*\\7.P'QO\"F17$$\"3q+++5zj_FF1$\"3)=++DDa(p?F17$$\"3=****\\<3;%*HF1 $\"3(o*\\7[A%Rt#F17$$\"3;++]Z=iYKF1$\"3****\\i!35#GMF17$$\"3[******\\' [M\\$F1$\"3/****\\(y$)p5%F17$$\"3W****\\PM&=v$F1$\"3e(*\\7`pf<[F17$$\" 3v+++gzs+SF1$\"3]-++!*=+-bF17$$\"35+++0\"Q_D%F1$\"3S***\\()y/>?'F17$$ \"3q++]x2k2XF1$\"3!>+D\"Q@,'*oF17$$\"3d+++?EdRZF1$\"3o+++0Z#Q`(F17$$\" 3M+++&o#R0]F1$\"31++v$))H[E)F17$$\"3++++?`9V_F1$\"3)3+++j\\'=*)F17$$\" 3G++]<#Rm\\&F1$\"3!***\\7[yv:'*F17$$\"3F++]A_ERdF1$\"3.+v=OzHG5Fc_p7$$ \"\"'F*$\"#6F*-F_^p6&Fa^pFe^pFb^pFe^p-%*THICKNESSG6#\"\"#-%%TEXTG6$7$$ \"#`!\"\"$!\"$F\\_qQ\"x6\"-Fg^q6$7$F]_qFj^qQ\"yF`_q-%+AXESLABELSG6%%!G Fh_q-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fj^qF``q" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "In quest ions 13 to 15 find an expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 12 " interms of " }{TEXT 382 1 "x" } {TEXT -1 5 " and " }{TEXT 383 1 "y" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q13" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "x*y-x+2*y=1" "6#/,(*&%\"xG\"\"\"%\"yGF'F'F&!\"\"*&\"\"# F'F(F'F'F'" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "_________ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________ _____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q14" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3*y+x*y^5=2" "6#/,&*&%\"xG\"\"$%\"yG\"\"\"F)*&F &F)*$F(\"\"&F)F)\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q15" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^2*y^3=2*x-y" "6#/*&%\"xG\"\"#%\"y G\"\"$,&*&F&\"\"\"F%F+F+F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "__ __________________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q16" }} {PARA 0 "" 0 "" {TEXT -1 51 "Find the equation of the tangent line to \+ the curve " }{XPPEDIT 18 0 "x^2*y-y^3=8" "6#/,&*&%\"xG\"\"#%\"yG\"\"\" F)*$F(\"\"$!\"\"\"\")" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ` `(-3,1)" "6#-%!G6$,$\"\"$!\"\"\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 71 "Plot the graphs of the curve and the tangent line in th e same picture. " }}{PARA 0 "" 0 "" {TEXT -1 44 "_____________________ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________ _____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q17" }}{PARA 0 "" 0 "" {TEXT -1 82 "The following curves are plotted in the worksheet on graphs of imp licit functions." }}{PARA 0 "" 0 "" {TEXT -1 36 "In each case find an \+ expression for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 13 " in terms of " }{TEXT 314 1 "x" }{TEXT -1 5 " and " } {TEXT 315 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "y^3-2*x*y-1 = 0" "6#/,(*$%\"yG\"\"$\"\"\"*(\"\"#F(%\"xG F(F&F(!\"\"F(F,\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "x^4-y^3 = 2" "6#/,&*$%\"xG\"\"%\"\"\"*$%\"yG\"\"$!\" \"\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*y^2 = 1" "6#/*&%\"xG\"\"#%\"yGF&\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/, &*$%\"xG\"\"$\"\"\"*$%\"yGF'F(*(F'F(F&F(F*F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "Code for ta ngent line picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 310 "p1 := plot([[cos(t),sin(t),t=0..2*Pi],[[0,0 ],[0.6,0.8]],\n[[1.5,.125],[-.2,1.4]]],color=[red,blue,green],thicknes s=[2,1,2],\n linestyle=[1,2,1]):\nt1 := plots[textplot]([[0.9,0.9,`P( 3/5,4/5 )`],[1.46,-0.1,`x`],\n [-0.15,1.2,`y`],[0.8,1.4,`gradient of tangent = -3/4`]]):\nplots[display]([p1,t1],tickmarks=[3,3]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 39 "Code for vertical tange nt lines picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 267 "p1 := plot([[cos(t),sin(t),t=0..2*Pi],[[1,-1. 2],[1,1.2]],\n [[-1,-1.2],[-1,1.2]]],color=[red,green,green],thicknes s=2):\nt1 := plots[textplot]([[1.2,-0.1,`x`],[-0.1,1.2,`y`]]):\nplots[ display]([p1,t1],tickmarks=[3,3],\n view=[-1.1..1.2,-1.1..1.2],scali ng=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 47 "Co de for tangent line pictures for the examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "p1 := plot( [x+sqrt(x^2+1),x-sqrt(x^2+1)],x=-3..3.4,-3..3,color=red):\np2 := plot( x+1,x=-2.5..2,-3..3,color=green,thickness=2):\nt1 := plots[textplot]([ [2.1,1.5,`gradient of tangent = 1`],\n [.4,1,`(0,1)`],[3.4,-.2,`x`], [-.2,3.4,`y`]]):\nplots[display]([p1,p2,t1],labels=[``,``],\n vie w=[-3..3.4,-3..3.4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 472 "g := x -> 1/3*surd((27+3*sqrt(3)*s qrt(x^8+27))*x^2,3)/x-x^3/\n surd((27+3*sqrt(3)*sqrt(x^8+27))*x^2,3 ):\np1 := plot(g(x),x=-3..3.4,y=-3..3,color=red,discont=true):\np2 := \+ plot([[-.5,2.5],[2.5,-.5]],color=green,thickness=2):\np3 := plot([[1,1 ]],color=black,style=point,symbol=circle):\nt1 := plots[textplot]([[2. 1,1.9,`gradient of tangent = -1`],\n [1.4,1.25,`(1,1)`],[3.4,-.2,`x` ],[-.2,3.4,`y`]]):\nplots[display]([p1,p2,p3,t1],labels=[``,``],\n \+ view=[-3..3.4,-3..3.4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "p1 := plot(surd(2-x^3,3),x=-3..3,- 3..3,color=red):\np2 := plot(2-x,x=-1..3,color=green,thickness=2):\np3 := plot(-x,x=-3..3,color=black,linestyle=2):\nt1 := plots[textplot]([ [1.9,1.8,`gradient of tangent = -1`],\n [1.3,1.2,`(1,1)`],[3.4,-.2,` x`],[-.2,3.4,`y`]]):\nplots[display]([p1,p2,p3,t1],labels=[``,``],\n \+ view=[-3..3.4,-3..3.4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x^2-x*y+y^2 = 9;\nsolve(%,y);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "p1 := plot([(x+sqrt(36-3*x^2))/2,(x-sqrt(36-3*x^2))/2],\n x=-3. 5..3.5,-3.5..3.5,numpoints=200,color=red):\np2 := plot([x/2+3,x/2-3], \n x=-3.5..3.5,color=green,thickness=2):\nt1 := plots[textplot]([ [3.5,-.15,`x`],[-.15,3.7,`y`]]):\nplots[display]([p1,p2,t1],labels=[`` ,``],\n view=[-3.5..3.5,-3.7..3.7]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "p1 := plots[implicit plot](x^2*y^3-x^3*y^2=12,x=-8..6,\n y=-6..8.3,grid=[50,50],color= red):\np2 := plot((7*x+15)/4,x=-8..6,color=green,thickness=2):\np3 := \+ plot([x,-x],x=-8..6,color=COLOR(RGB,.3,.3,.3),linestyle=2):\nt1 := plo ts[textplot]([[6.5,-.3,`x`],[-.3,9,`y`]]):\nplots[display]([p1,p2,p3,t 1],labels=[``,``],view=[-8..6.5,-6..9]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "p1 := plot(LambertW(x),x=-.5..5.3,y=-.5..1.3,color=r ed):\np2 := plot(x/(2*exp(1))+1/2,x=-.5..5.3,color=green,thickness=2): \nt1 := plots[textplot]([[5.7,-.1,`x`],[-.15,1.4,`y`]]):\nplots[displa y]([p1,p2,t1],labels=[``,``],\n tickmarks=[5,3],view=[-.5..5.7,-1.. 1.45]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "p1 := plots[implicitplot](x+sin(x*y)=y,x=-4..4,\n \+ y=-4..4.3,grid=[50,50],color=red):\np2 := plot(x,x=-4..4,color=gree n,thickness=2):\nt1 := plots[textplot]([[4.5,-.2,`x`],[-.2,4.7,`y`]]): \nplots[display]([p1,p2,t1],labels=[``,``],view=[-4..4.5,-4..4.7]);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "p1 := plots[implicitplot](y*exp(x*y)=x+1,\n x=-6..6,y=-0.7 ..1,grid=[60,60],color=red):\nt1 := plots[textplot]([[6.3,-.2,`x`],[-. 2,1.3,`y`]]):\nplots[display]([p1,t1],labels=[``,``],\n tickmarks=[5 ,4],view=[-6..6.3,-.7..1.3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 49 "Code for tangent line pictures for the questions " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "p1 := plots[implicitplot](x*y^2-x^3+x*exp(x*y)+6=0,x=0..5,\n \+ y=-5..5,grid=[50,50],color=red,resolution=200):\np2 := plot(11/4*x-11 /2,x=-6..6,color=green,thickness=2):\nt1 := plots[textplot]([[6.5,-.3, `x`],[-.3,9,`y`]]):\nplots[display]([p1,p2,t1],labels=[``,``],view=[-5 ..5.5,-5..5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 290 "p1 := plots[implicitplot](x*y^2-x^3+x*exp(x*y )+6=0,x=-5..5,\n y=-5..5,grid=[50,50],color=red,resolution=200): \np2 := plot(11/4*x-11/2,x=-6..6,color=green,thickness=2):\nt1 := plot s[textplot]([[5.3,-.3,`x`],[-.3,5.3,`y`]]):\nplots[display]([p1,p2,t1] ,labels=[``,``],view=[-5..5.3,-5..5.3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "p1 := plots[implicit plot](y^2*tan(x)=exp(y-1),x=0..1.55,\n y=-2..5,grid=[50,50],color =red):\np2 := plot(-2*x+Pi/2+1,x=0..1.55,color=green,thickness=2):\nt1 := plots[textplot]([[1.59,-.2,`x`],[-.08,4.9,`y`]]):\nplots[display]( [p1,p2,t1],labels=[``,``],\n view=[-.08..1.59,-2..4.9],tickmarks=[2 ,7]);" }}{PARA 13 "" 1 "" {GLPLOT2D 269 369 369 {PLOTDATA 2 "6)-%'CURV ESG6h[l7$7$$\"3YfQ1I:EW7!#>$!\"#\"\"!7$$\"3I\")*41tuNN\"F*$!3UV__Q?F=> !#<7$7$$\"3QIBCo1lk;F*$!3?dG9dG9d=F3F.7$7$$\"3/IBCo1lk;F*F87$$\"39***= Kd-nx\"F*$!3#Rt6]*Q_%z\"F37$7$$\"3->[of^m`AF*$!3T9dG9dG97$7$$\"3s >[of^m`AF*FG7$$\"3D65djyOVBF*$!3_2z%y3esn\"F37$7$$\"3e5s.\\k\"R4$F*$!3 ir&G9dG9d\"F3FM7$FS7$$\"3`AX#>NsH5$F*$!31L'>Qm0(o:F37$7$$\"3w[C71`EjJF *$!3a^n+h[vh:F3FY7$Fin7$$\"3W&)o)[HJM)QF*$!3!zYqQ0&4h9F37$7$$\"3OO&)** )e>iJ%F*$!3#)G9dG9dG9F3F_o7$Feo7$$\"3-Q()G_IT*)[F*$!3)*p&=_MpOO\"F37$7 $$\"3.-ndW&yU9'F*$!3/'G9dG9dG\"F3F[p7$Fap7$$\"3ueHzX#Gc?'F*$!3+jA%)yTD !G\"F37$7$$\"3\\(*[C71`EjF*$!3)GvBv!)RUF\"F3Fgp7$F]q7$$\"3v&*pTyI%)QvF *$!3/DO5Qng(>\"F37$7$$\"3s?`T?(\\q&*)F*$!3CVr&G9dG9\"F3Fcq7$Fiq7$$\"3- %4SLff$4#*F*$!3&)*ellF#>I6F37$7$$\"3%pMn$=fz*[*F*$!39')\\5Q@k@6F3F_r7$ 7$$\"3aXtO=fz*[*F*Fhr7$$\"3'*Gcpm?r\"4\"!#=$!31IcNF)fW1\"F37$7$$\"3]z* [C71`E\"Fas$!3ot)*y![Cg*Fas7$7$$\"3IC71`Ej\"e\"Fas$!3#Qty+K+IX*FasFgt7$7$F^u$!3# \\ty+K+IX*Fas7$$\"3\"fcf$fHK5jp))FasFfu7$7$$\"3QpMn$=fz*=FasF_v7$$\"3w@)R$zG=V>Fas$!3M& Rtjhkcx)Fas7$7$$\"3%[M/81,L4#Fas$!3]w&G9dG9d)FasFev7$F[w7$$\"3]^,wTc;# =#Fas$!34\\wPz,PE%)Fas7$7$$\"3\"Rr&G9dG9AFas$!3A=u2&*3e$R)FasFaw7$7$$ \"3>9dG9dG9AFasFjw7$$\"3v`f(>)*)[ACFas$!37PSok*HJ3)Fas7$7$$\"3+fz*[C71 `#Fas$!3y()=r;-V\")zFasF`x7$7$Fgx$!3!*))=r;-V\")zFas7$$\"3!3&Q5(z8Hn#F as$!39i&4TJ4by(Fas7$7$$\"3!Q?5bxQp%GFas$!3NC0!pi+Lj(FasF_y7$7$Ffy$!3CB 0!pi+Lj(Fas7$$\"3Cs=i'Q5:$HFas$!3/#R.6X$zCvFas7$7$$\"3g[C71`EjJFas$!3u +i)*[gzMtFasF^z7$7$$\"3/[C71`EjJFasFgz7$$\"3!*HO&*fH\"o>$Fas$!3#=H)GwE O%H(Fas7$7$$\"3+\\fwY$)e#R$Fas$!3YZr&G9dG9(FasF][l7$7$Fd[l$!3d[r&G9dG9 (Fas7$$\"3E<8:@XukMFas$!3Wl***)\\Z!e2(Fas7$7$$\"3S$pMn$=fzMFas$!3M)*)4 \"yNenqFasF\\\\l7$Fb\\l7$$\"3pm;riw8IPFas$!3-19*G*HyXoFas7$7$$\"3?QpMn $=fz$Fas$!3RtDM\\,(4\"oFasFh\\l7$F^]l7$$\"3.V%QJ!RO+SFas$!3r\"H*)R8)eP mFas7$7$$\"3-$=fz*[C7TFas$!31'=V(y*32e'FasFd]l7$Fj]l7$$\"35_5e3!=ZF%Fa s$!3)yTEhCN![kFas7$7$$\"3QG9dG9dGWFas$!3;^zHMHasjFasF`^l7$7$$\"3$yUr&G 9dGWFasFi^l7$$\"3!H3KJQEEb%Fas$!3k^D<'*Q`uiFas7$7$$\"3jsO=fz*[u%Fas$!3 u,f^cc2$='FasF__l7$Fe_l7$$\"3\"[fo](zhL[Fas$!3+6gw`n&\\6'Fas7$7$$\"35> fz*[C71&Fas$!3'zuK\\xj&4gFasF[`l7$7$$\"3+=fz*[C71&Fas$!3&ouK\\xj&4gFas 7$$\"3x!=(>BEI<^Fas$!3U-Sh+=anfFas7$7$$\"3Oj\"3/-^vP&Fas$!3E5Ta,lw\\eF asF\\al7$Fbal7$$\"3#yD]]saLS&Fas$!30%pfRd=3$eFas7$7$$\"3c_XB!RLem&Fas$ !3j?dG9dG9dFasFhal7$F^bl7$$\"3KDM#z'*H3p&Fas$!3%H@$yNC_+dFas7$7$$\"3q3 /-^v(Qp&Fas$!39plRZji*p&FasFdbl7$Fjbl7$$\"3g*)HZR5GtfFas$!3]s\"H;\\Ova &Fas7$7$$\"32aEj\"3/-,'Fas$!387>xJtzObFasF`cl7$7$Fgcl$\"35>&H)\\p@C=F3 7$$\"3e+RSAJV$*fFas$\"3)e&G9dG9d=F37$F_dl7$$\"3SrtH[nN%*fFas$\"3')*yet r*Hk=F37$7$$\"3KfS2%[k!ofFas$\"3)))*************>F3Fedl7$7$F\\el$\"3m) *************>F37$$\"3-MgJ'4La(fFas$\"3**foiDIq:?F37$7$$\"3=dslK,7\"*f Fas$\"3YTr&G9dG9#F3Fdel7$Fjel7$$\"3Un3SYfB'*fFas$\"3+>='HLl\"\\@F37$7$ Fgcl$\"31&*)*e\"*)z0=#F3F`fl7$7$$\"3=bEj\"3/-,'Fas$!3C8>xJtzObFas7$$\" 3W59AOFkdiFas$!3%QW0uiFas$\"3Gq&G9dG9d\"F37$F^hl7 $$\"3>)*G_ujUShFas$\"3=H)Glkval\"F37$7$$\"3Omgi.$GJ2'Fas$\"338dG9dG90'Fas$\"3=))4VTF%oE#F37$7$$\"3_H!o^el\\0'Fas$\"3E%G9dG9dG#F3Fgi l7$7$$\"3iI!o^el\\0'FasF`jl7$$\"30X7ptqNHhFas$\"3[0)**)*RgZP#F37$7$$\" 3!*o6lUKdbhFas$\"30F9dG9dGCF3Ffjl7$F\\[m7$$\"3TZ)H/I2wkg\\lE&Fas7$7$$\"3zWr&G9dGk'Fas$!3(=W'oqk0Q_FasFh\\m7$7$F_]m$ \"3O`68:S1Q8F37$$\"3#4DJPtirW'Fas$\"3\\F9dG9dG9F37$Fg]mF[hl7$Fd\\m7$$ \"3cukEB'R$>kFas$\"3GVFPn>PsEF37$7$$\"3w-\"R=OdnX'Fas$\"3k7dG9dG9FF3F^ ^m7$Fd^m7$$\"3^@iGc))\\NlFas$\"3yb$p<5qFw#F37$7$$\"3oVr&G9dGk'Fas$\"3- @TC)3p'[GF3Fj^m7$F^]m7$$\"3=Z%Rq+F8$oFas$!3W?P06&oo8&Fas7$7$$\"3;!RpMn $=fpFas$!3uM0T(fk,5&FasFf_m7$7$F]`m$\"3Y&pQw#p7A7F37$$\"3IuQyL4%ex'Fas $\"3q%G9dG9dG\"F37$Fe`m7$F_]m$\"3e`68:S1Q8F37$7$F_]mFc_m7$$\"3klFk&*y& )\\mFas$\"3/vD&oA\")R&GF37$7$$\"3q'GG(=d$Hl'Fas$\"3UbG9dG9dGF3F`am7$7$ $\"3#yGG(=d$Hl'FasFiam7$$\"3im!*)oTYgy'Fas$\"3]YOCwQLNHF37$7$$\"3)zr_A Z&esoFas$\"3A)*************HF3F_bm7$Febm7$$\"3>y(o\\.yZ\"pFas$\"3$\\&G KfT0?IF37$7$F]`m$\"3UwESvq'e/$F3F[cm7$F\\`m7$$\"3\"=WSGGF.7(Fas$!3y^xN ='fM,&Fas7$7$$\"3]N;3/-^vsFas$!37%z?s**=)o\\FasFecm7$7$F\\dm$\"30![48v &QJ6F37$$\"34bh!=bcNB(Fas$\"3\">9dG9dG9\"F37$FddmFb`m7$Facm7$$\"3'>A88 %3@bqFas$\"3sO')zM+\\*4$F37$7$$\"3!y>6A+Tu6(Fas$\"3-Tr&G9dG9$F3F[em7$F aem7$$\"3#4gK\"=)Qi>(Fas$\"33=$3O7d'yJF37$7$F\\dm$\"3&)\\>bDD3AKF3Fgem 7$F[dm7$$\"37eNV_he5uFas$!3G(>EwpMd*[Fas7$7$$\"3'3)QpMn$=f(Fas$!3b!o'G h!>L%[FasFafm7$Fadm7$$\"3I\\rMV\"e`K(Fas$\"3(*>cGq^M?6F37$7$Fhfm$\"3k> u&R=>L0\"F3F]gm7$F]fm7$$\"3X(R;=(eJStFas$\"3y+J'*4sWcKF37$7$$\"3IEn)fd +`Q(Fas$\"3#QG9dG9dG$F3Fggm7$F]hm7$$\"3u<'[)Hkh!\\(Fas$\"3Ta3X\"oE9L$F 37$7$Fhfm$\"3m>Z(\\CJPQ$F3Fchm7$Fgfm7$$\"3$4\"pv--*>q(Fas$!3+^w$3c\"=$ y%Fas7$7$$\"37DhIlK;3zFas$!3z%y$Hfw/BZFasF]im7$7$$\"3BEhIlK;3zFas$\"3W Yjq%ynZ))*Fas7$$\"3?\\(3)p5d_yFas$\"27***************F37$F^jm7$$\"3)>) QpMn$=f(FasFdgm7$7$Fejm$\"35?Z(\\CJPQ$F37$$\"3S];;3=wQwFas$\"3&Hc3*f%z tS$F37$7$$\"3S'zTvu_Nn(Fas$\"3gE9dG9dGMF3F[[n7$Fa[n7$$\"3UOQD!3/ez(Fas $\"3#o(pL'G9$zMF37$7$Fjim$\"3Md$Qfr5U`$F3Fg[n7$7$Fjim$!3M&y$Hfw/BZFas7 $$\"37bC!ztSW*zFas$!3A>k`%[a`n%Fas7$7$$\"3fr$=fz*[C#)Fas$!3]44B(=eug%F asFd\\n7$7$Fjim$\"3KXjq%ynZ))*Fas7$$\"3URv\"==+![zFas$\"3]eryW84?)*Fas 7$7$F[]n$\"3G>nfenIz#*FasFc]n7$7$Fjim$\"3*oNQfr5U`$F37$$\"39vzfF<6[zFa s$\"3#3a$RItQ`NF37$7$$\"39Y&)y2lhzzFas$\"3Sp&G9dG9d$F3F`^n7$Ff^n7$$\"3 W4kF([:)4\")Fas$\"3^x(z)op@BOF37$7$$\"3[q$=fz*[C#)Fas$\"3;5\"fDcsjn$F3 F\\_n7$7$$\"3qs$=fz*[C#)FasF]]n7$$\"3iK!p\"3@&yG)Fas$!3w+0ijm'=d%Fas7$ 7$$\"3&phIlK;3a)Fas$!3%R7V&yQ1'\\%FasF[`n7$Fi]n7$$\"3O8)QP>;lY)Fas$\"3 sb-Bev(p!*)Fas7$7$Fb`n$\"3G0mY,6Wo()FasFg`n7$7$F[]n$\"3g5\"fDcsjn$F37$ $\"3gs2k_M8m#)Fas$\"37qBd,*yap$F37$7$$\"3%\\RHB,!*3I)Fas$\"3>7dG9dG9PF 3Fdan7$7$F[bn$\"3v6dG9dG9PF37$$\"3P,BXT^\"3V)Fas$\"374s$)oO'Rw$F37$7$F b`n$\"3/7*Hq*fa7QF3Fcbn7$Fa`n7$$\"3?9$4'G/:#e)Fas$!3y9U8CWQsWFas7$7$$ \"3?hG9dG9d))Fas$!3IV[eA*G%)Q%FasF]cn7$7$$\"3UjG9dG9d))Fas$\"37FtAR37# G)Fas7$$\"3'eZbm&Qbu')Fas$\"3=j&G9dG9d)Fas7$7$$\"3uualcQbu')FasFadn7$$ \"3%ehIlK;3a)FasF^an7$7$$\"31=1`Ej\"3a)FasFjbn7$$\"3Uk#y]P63f)Fas$\"3z O87[WcMQF37$7$$\"3+Lf38frM')Fas$\"3)\\&G9dG9dQF3F^en7$Fden7$$\"3!)\\z2 ?t7d()Fas$\"3\")GW?k6J-RF37$7$$\"3KiG9dG9d))Fas$\"3zz$Qo>2Y%RF3Fjen7$F ccn7$$\"3[7-s[EFx))Fas$!3!p1q=-BmP%Fas7$7$$\"3M'RX&QO\\o\"*Fas$!39#H9d G9dG%FasFffn7$F\\gn7$$\"3yeL$>LWI<*Fas$!3UetxyYz$G%Fas7$7$$\"3o2^v(QpM <*Fas$!3=#H%*4#*=PG%FasFbgn7$Ficn7$$\"3zl'Q%om4W*)Fas$\"3M&4q(RRty\")F as7$7$Fign$\"3\"fIv?v&yEyFasF^hn7$F`fn7$$\"3)>atM))Q.#*)Fas$\"3Qlk2f)f 9(RF37$7$$\"3?Qkv*)3Oy*)Fas$\"3y(*************RF3Fhhn7$F^in7$$\"37ER&e \"3B(3*Fas$\"3QPj#=XY*QSF37$7$$\"3y3^v(QpM<*Fas$\"3m0N(y>iS2%F3Fdin7$7 $Fign$!3k\"H%*4#*=PG%Fas7$$\"3oY?!oKm:Y*Fas$!3yCun2eAeTFas7$7$$\"3-`tO =fz*[*Fas$!3+TT;)fQI:%FasFcjn7$7$$\"3c1^v(QpM<*Fas$\"3![Iv?v&yEyFas7$$ \"35_WHAY<&R*Fas$\"3KX=7D\"z,d(Fas7$7$Fjjn$\"3nmDg9kXGuFasFd[o7$7$Fign F]jn7$$\"3#\\)z[[!*4`#*Fas$\"3sz1fg`*o5%F37$7$$\"3+Y%om=3!H$*Fas$\"3dS r&G9dG9%F3F_\\o7$Fe\\o7$$\"3Z'eOjWW(>%*Fas$\"33'>;:H$\\uTF37$7$Fjjn$\" 3Da)\\^mF@?%F3F[]o7$7$Fjjn$!3YST;)fQI:%Fas7$$\"3UX@#e%ol](*Fas$!3T:K+l WANSFas7$7$$\"3G(fz*[C71)*Fas$!3d!HT'=qnCSFasFh]o7$7$$\"3R)fz*[C71)*Fa s$\"3W$yN![_+fqFas7$$\"3[TjBHP:T(*Fas$\"3CNr&G9dG9(Fas7$Fi^o7$$\"3#>Nn $=fz*[*FasF[\\o7$7$F`_oFb]o7$$\"3)z;zqqQwe*Fas$\"3/KW%)>s_TUF37$7$$\"3 ic0xzGw$o*Fas$\"3#QG9dG9dG%F3Fd_o7$Fj_o7$$\"3wGy>4oS`(*Fas$\"3[M:%)[8_ 4VF37$7$Fe^o$\"3)42^hc.)HVF3F``o7$F^^o7$$\"3Tf+&\\]ES+\"F3$!3[NT/`5c9R Fas7$7$$\"3E%=fz*[C75F3$!3Oq=)4XA$)*QFasFj`o7$Fd^o7$$\"3Dn*3Mrh$H)*Fas $\"3MR$zgi0z.(Fas7$7$Faao$\"3)fInxjiVm'FasFfao7$Ff`o7$$\"35$e_$*>RE#** Fas$\"3;Z5mg4&fP%F37$7$$\"3x7oCMq'R+\"F3$\"30F9dG9dGWF3F`bo7$7$$\"3*H \"oCMq'R+\"F3Fibo7$$\"3[`%\\#\\_q35F3$\"3qMI.\"F3$!3E%Q+6BPgz$Fas7$7$$\"3!)3/-^v(Q /\"F3$!3W1UTn2otPFasFico7$F\\bo7$$\"3[Z/'H+([D5F3$\"34*)HT%fE[a'Fas7$7 $F`do$\"3Nr\\N\\9l6jFasFedo7$Feco7$$\"3V$e:&o&)oD5F3$\"36.(G0A:2^%F37$ 7$$\"3GztM-\\PR5F3$\"3Gq&G9dG9d%F3F_eo7$Feeo7$$\"3=QDx,'e>/\"F3$\"3k4$ QQz%4!e%F37$7$F`do$\"3K&p>Ncyse%F3F[fo7$7$F`do$!3)e?9uw!otPFas7$$\"3M \\2^:m3i5F3$!3-=P40x[zOFas7$7$$\"3ML;3/-^v5F3$!3A*o8Nas/l$FasFhfo7$F[e o7$$\"3y3BT>U*o1\"F3$\"3(Q(>N*p%R.hFas7$7$F_go$\"3S'e@xIAR*fFasFdgo7$F afo7$$\"3]b$*o0h;ZF37$7$F_go$\"3nCUI8g`=Z F3Fjho7$F^go7$$\"3]?_Z)Rz64\"F3$!3)ygsU*QykNFas7$7$$\"3(y&G9dG926F3$!3 nXeyK/VGNFasFdio7$7$F[jo$\"3rv'3)R'3Fq&Fas7$$\"3(*o_.eH416F3$\"3J2dG9d G9dFas7$7$$\"3vo_.eH416F3FfjoFjgo7$F`io7$$\"3Y%*)p'[!f=4\"F3$\"3gNRXl$ 4Ly%F37$7$F[jo$\"3Wq)4*>gD^[F3F][p7$Fjio7$$\"3%HD`Z:6.7\"F3$!3%41!eo,% =X$Fas7$7$$\"3T#3/-^v(Q6F3$!3yi9ihnH2MFasFg[p7$7$F[jo$\"3#on3)R'3Fq&Fa s7$$\"3n.^J`lW26F3$\"3eb5G./d+dFas7$7$F^\\p$\"3aP:el)Q3N&FasFf\\p7$Fc[ p7$$\"3#[Mwv*yx26F3$\"3#*GXgK\\Fa[F37$7$$\"3`au!fS4$36F3$\"3wcG9dG9d[F 3F`]p7$Ff]p7$$\"3C?\"QT5'QC6F3$\"3v)HM;FF@#\\F37$7$F^\\p$\"3g=65?=C() \\F3F\\^p7$F]\\p7$$\"3?0j\"*e6[\\6F3$!3e@rsmPiSLFas7$7$$\"3%pIlK;3/<\" F3$!3/;A7(H@oG$FasFf^p7$F\\]p7$$\"3'H-9A%3N[6F3$\"3[$o\\aO^=G&Fas7$7$$ \"3;2`Ej\"3/<\"F3$\"3&Rr[&z&=0.&FasFb_p7$Fb^p7$$\"3k%37P\\I,9\"F3$\"3m !GT'>2)Q*\\F37$7$$\"3HNzuaFBT6F3$\"\"&F-F^`p7$7$F]_p$!3g;A7(H@oG$Fas7$ $\"3m2+;(>!py6F3$!3K%*GK%>q6B$Fas7$7$$\"3[JlK;3/-7F3$!3/q[nX\"fn;$FasF ]ap7$7$F]_pF[`p7$$\"3wUs_)\\P&)=\"F3$\"3#>-m6aTb*[Fas7$7$Fdap$\"3M(o#) *p[7PZFasFjap7$7$Fdap$!3gq[nX\"fn;$Fas7$$\"39Sri&4Tz?\"F3$!3gd>i$32O7$ Fas7$7$$\"3-cxQpMnL7F3$!3GFk8W(oo/$FasFgbp7$F`bp7$$\"3vu,\\J&y\"G7F3$ \"3e*\\69xsQ`%Fas7$7$F^cp$\"3')yg#ph!)oY%FasFccp7$F]cp7$$\"3A^_u[&RsB \"F3$!3Xk+K-;>=IFas7$7$$\"3c!)*[C71`E\"F3$!3)H:p#3*4p#HFasF]dp7$7$$\"3 y!)*[C71`E\"F3$\"33)>Y5J+R=%Fas7$$\"3#\\O<)Hhrb7F3$\"3EyUr&G9dG%Fas7$F ^ep7$$\"3CcxQpMnL7F3$\"3Szg#ph!)oY%Fas7$Fcdp7$$\"3/j4'oW&fm7F3$!3F&RK7 Tq`\"HFas7$7$$\"3%Q(R0f_%GG\"F3$!3kjG9dG9dGFasFjep7$F`fp7$$\"3#p#[(3mr _H\"F3$!3=IN44Q&=y#Fas7$7$$\"3'[?5bxQpH\"F3$!3#RCU][f&zFFasFffp7$7$Fdd pF\\ep7$$\"3gQzTt3*yE\"F3$\"3k+Oz'f$)*oTFas7$7$$\"330-^v(QpH\"F3$\"3:) e*G0;vVQFasFcgp7$F\\gp7$$\"3$\\f]%zu)GK\"F3$!3Qy!=2Q[/g#Fas7$7$$\"3jH9 dG9dG8F3$!3=2Gi>([;f#FasF_hp7$Figp7$$\"3&R@I*QRH38F3$\"3'*3H%QX+Hx$Fas 7$7$Ffhp$\"3D1Yv!e[,`$FasF[ip7$7$Ffhp$!3i1Gi>([;f#Fas7$$\"3[!H\"\\U[W] 8F3$!3HbA&\\-.kT#Fas7$7$$\"3;aEj\"3/-O\"F3$!3MZ9xX6:*R#FasFhip7$Faip7$ $\"3)o=`Q`5$[8F3$\"3)z%oP@3F%R$Fas7$7$F_jp$\"3p&z:l*3aRKFasFdjp7$7$F_j p$!3iZ9xX6:*R#Fas7$$\"3Yx0v(=^zP\"F3$!39:'[.k`+B#Fas7$7$$\"3oyQpMn$=R \"F3$!3Zq#QdV'\\,AFasFa[q7$Fjjp7$$\"3G\"of24*>)Q\"F3$\"3n9`%4bB9-$Fas7 $7$Fh[q$\"39>wA(fm*oHFasF]\\q7$7$Fh[q$!3?q#QdV'\\,AFas7$$\"3*e!*\\TREa S\"F3$!3F)3]Y)zHU?Fas7$7$$\"3B.^v(QpMU\"F3$!3A9&R\"=?1)*>FasFj\\q7$7$F a]q$\"3vyQ2pA?3EFas7$$\"3B:0A\"**)3/9F3$\"3K]G9dG9dGFas7$Fi]qFc\\q7$7$ Fa]q$!3%R^R\"=?1)*>Fas7$$\"3XECnUi\"HV\"F3$!3?kz7'e._&=Fas7$7$$\"3wFj \"3/-^X\"F3$!3u_))>tW;)y\"FasFc^q7$Ff]q7$$\"3zKB2\"[g'H9F3$\"3%\\DHW4X vd#Fas7$7$Fj^q$\"3kS2k$G]$)>#FasF__q7$Fi^q7$$\"3#Q`v0LE0Y\"F3$!3QR2=`$ RNn\"Fas7$7$$\"3I_v(QpMn[\"F3$!3kO2qg%\\5d\"FasFi_q7$Fe_q7$$\"3Oe@/^&[ @Z\"F3$\"3Cqs%)f/I(3#Fas7$7$F``q$\"3k_#y^m:?#=FasFe`q7$F_`q7$$\"3#H#*[ '=U_)[\"F3$!3N`2Jw')Q4:Fas7$7$$\"3^mUQMZF.:F3$!3;N9dG9dG9FasF_aq7$Feaq 7$$\"31T8p-6J8:F3$!3O06<9\\A+7Fas7$7$$\"3%oxQpMn$=:F3F^bqF[bq7$7$$\"3a _v(QpMn[\"F3$\"3#HDy^m:?#=Fas7$$\"3ppe\\ds=;:F3$\"3\\Q<$33Fq_\"Fas7$7$ Fbbq$\"3sWIvD1eu9FasFjbq7$Fabq7$$\"3/w_&Gej'H:F3$!3u!)Qx(*e_,^F*7$7$$ \"3Q,+++++]:F3$!3#G)Qx(*e_,^F*Fdcq7$7$F[dq$\"3g2zH>$yLo&F*7$$\"3+/5t4? E?:F3$\"3$=Ur&G9dG9Fas7$Fcdq7$Fbbq$\"3;WIvD1eu9Fas-%'COLOURG6&%$RGBG$ \"*++++\"!\")$F-F-Fceq-F$6%7S7$Fceq$\"3c'*[zEjzqDF37$$\"3YLL$e*)e&yLF* $\"3yH#y)[^A.DF37$$\"3-n;/E\\A=jF*$\"3NjSFG=VWCF37$$\"3iLL3Fo=C'*F*$\" 3gHKD!f7$yBF37$$\"3]L$e*yi?&H\"Fas$\"3vHK+r]v6BF37$$\"3vmT&Q>7ki\"Fas$ \"3/jS-))Q^XAF37$$\"3UL3FuC[L>Fas$\"3')H2%>$)*4%=#F37$$\"3-+vVjfV^AFas $\"3]'R2T8407#F37$$\"3cL3x;RE!e#Fas$\"3'*H2WVNua?F37$$\"3$**\\(o9t.3HF as$\"3o'RdQ'))=*)>F37$$\"3Enm\"Ha#>XKFas$\"36j:@=yv@>F37$$\"3)RLe90f@a $Fas$\"3')HK];XOi=F37$$\"3e++v.IZwQFas$\"3W'*[/E<]&z\"F37$$\"31++D\"of @@%Fas$\"3V'*[a!Rk$G\"\\g\"F37$$\"3/nm\"zSP(y^Fas$\"3:j:@X)[]`\"F37$$\"3 )omm;o`YZ&Fas$\"3=j:Y!fleZ\"F37$$\"3U+v$flu)=eFas$\"3Z'R2cR@qS\"F37$$ \"3_mm;W&oN7'Fas$\"3Ej:'zh#3Y8F37$$\"3M,v$4.myX'Fas$\"3I'R217B#z7F37$$ \"3!3]7GH'>wnFas$\"3S'RK#oqb:7F37$$\"3mn;/E0M3rFas$\"3-jle@#G\"\\6F37$ $\"3UnTN;EN8uFas$\"33jS_.e7)3\"F37$$\"3&QLe9O\\Bu(Fas$\"3zHK]akKA5F37$ $\"3=M3x;X3%3)Fas$\"3C(H2WB%zR&*Fas7$$\"3O+D1ufc\"Q)Fas$\"3%['R#)>8$[% *)Fas7$$\"3iM$3x4aGq)Fas$\"3L'HKD2bAI)Fas7$$\"3X++]nhxM!*Fas$\"3nk*[H$ 4TQwFas7$$\"3s+](o:(\\f$*Fas$\"37k*)>a*o*))pFas7$$\"3M,v$4T\"ot'*Fas$ \"3!H'R2Y/ggjFas7$$\"3-+D\"e%HD-5F3$\"3Al*)p^V!Hm&Fas7$$\"3ummT%\\)fL5 F3$\"3(3j:'zL*f.&Fas7$$\"31+]7el1n5F3$\"3Ik*[a5KmO%Fas7$$\"3E$3x\")f$R (4\"F3$\"3K+tS/84gPFas7$$\"3;+]Pj!\\08\"F3$\"3Ki*[/+#)p4$Fas7$$\"3w;/E sduh6F3$\"3]I1uAy/tCFas7$$\"39](o>KbV>\"F3$\"3OiRdGo&3#=Fas7$$\"3im;H< rBE7F3$\"37Lc6A4A$=\"Fas7$$\"35]PM-Whf7F3$\"3kM'R2@_n:&F*7$$\"3cLL$[pg PfsSvnV\"Fas7$$\"3C++v'z%>(Q\"F3$!3:R50nE$f.#Fas7$$\"3&p;z%)) )H:U\"F3$!3PtVj,XjAFFas7$$\"3SLL$QgRAX\"F3$!3F-xr3)GoL$Fas7$$\"33]PflD )\\[\"F3$!3/Og#R/)o\"*RFas7$$\"3B]i!zv@j^\"F3$!35Rg " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }