{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 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 "" -1 285 "" 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 "" -1 295 "" 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 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 "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 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 "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 0 0 2 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 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 311 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 0 2 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 0 0 1 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 "" -1 321 "" 0 1 0 0 0 0 1 0 0 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 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 327 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 0 0 2 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 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 331 "" 0 1 0 0 0 0 1 0 0 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 1 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 338 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 0 0 2 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 0 0 1 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 0 0 1 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 0 0 1 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 0 0 1 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 0 1 0 0 0 0 1 0 0 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 "" -1 372 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 373 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 0 1 0 0 0 0 1 0 0 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 0 1 0 0 0 0 1 0 0 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 "" -1 381 "" 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 1 0 1 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 Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 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" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Parametric curves " }}{PARA 0 "" 0 "" {TEXT -1 58 "by Peter Stone, School of Life and Physical Sciences , RMIT" }}{PARA 0 "" 0 "" {TEXT -1 23 "peter.stone@rmit.edu.au" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: \+ 15.8.2004 " }}{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 57 "Parametric curves and th e gradient of a parameteric curve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 21 "A pair of equations : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = \+ f(t) ,``],[y = g(t) ,``])" "6#-%*PIECEWISEG6$7$/%\"xG-%\"fG6#%\"tG%!G7 $/%\"yG-%\"gG6#F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "ca n be used to define a curve in two dimensions." }}{PARA 0 "" 0 "" {TEXT -1 26 "Such equations are called " }{TEXT 259 20 "parametric equ ations" }{TEXT -1 15 " involving the " }{TEXT 259 9 "parameter" } {TEXT -1 1 " " }{TEXT 268 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "For example, the equations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 32 "Given a value for the parameter \+ " }{TEXT 275 1 "t" }{TEXT -1 41 ", we compute the corresponding values of " }{TEXT 269 1 "x" }{TEXT -1 5 " and " }{TEXT 270 1 "y" }{TEXT -1 46 " from these equations, and then plot the point" }{XPPEDIT 18 0 " ` `(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 14 " on the curve." }}{PARA 15 "" 0 "" {TEXT -1 49 "It is often convenient to think of the parameter \+ " }{TEXT 271 1 "t" }{TEXT -1 56 " as being time, so that curve is trac ed out by the point" }{XPPEDIT 18 0 " ``(f(t), g(t))" "6#-%!G6$-%\"fG6 #%\"tG-%\"gG6#F)" }{TEXT -1 20 " as time progresses." }}{PARA 15 "" 0 "" {TEXT -1 52 "Sometimes it is possible to eliminate the parameter " }{TEXT 272 1 "t" }{TEXT -1 104 " from the two parametric equations to \+ give a single equation for the curve involving only the variables " } {TEXT 273 1 "x" }{TEXT -1 5 " and " }{TEXT 274 1 "y" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "For exam ple, the equations:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=cos*t,``],[y=sin*t,``])" "6#-%*PIECEWISEG6$7$/%\"xG*&% $cosG\"\"\"%\"tGF+%!G7$/%\"yG*&%$sinGF+F,F+F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 74 "describe a circle with its centre at the \+ origin and having a radius of 1. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 503 265 265 {PLOTDATA 2 "64-%'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$$\"3?goz=42`')F/$\"3j9NXR4U 7]F/7$$\"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*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'y I2&oN')F/7$$!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*z 7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[g F/$!3)4j2yeGL'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/$!3 6.2<#>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< bQ38'F/7$$\"3%*R,@;vLu')F/$!3e>)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2nd o*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4 O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICK NESSG6#F)-F$6$7$7$F+F+7$$\"3w**************fF/$\"3U+++++++!)F/-Fjz6&F \\[lF*F*F*-F$6$7$7$Fh[lF+Fg[lF\\\\l-F$6$7S7$$\"3++++++++DF/F+7$$\"3yNT :T$*[*\\#F/$\"3-I#=j9bF0&Fas7$$\"3K2\\8[U@)\\#F/$\"35;b/0P_Z%*Fas7$$\" 3/v!\\0Bde\\#F/$\"3,YMHQUjQ9!#>7$$\"3Ed%HVe)\\#\\#F/$\"3cVvY,6AN>Fg]l7 $$\"31$3#p(*\\<)[#F/$\"3-B;(R\\'oGCFg]l7$$\"3]?\\%\\v$H$[#F/$\"3#o'>8i WM&)GFg]l7$$\"3GC-:;kNxCF/$\"3G8>PD$frN$Fg]l7$$\"3#Q#G&o=t-Z#F/$\"3pt3 (=>JQ%QFg]l7$$\"3)R!z6,8EiCF/$\"37G%[<8iuK%Fg]l7$$\"31xQ/GB.`CF/$\"3;0 ]'R3)>B[Fg]l7$$\"3Dmc0!owSW#F/$\"3S!>-P'QAe_Fg]l7$$\"3p0+%y%=2LCF/$\"3 +!*esNr'fu&Fg]l7$$\"3R'[/yeU5U#F/$\"3x$p'=g-SLiFg]l7$$\"3W4*H'[i_3CF/$ \"3?.Cf**fv+nFg]l7$$\"3%R:6IayjR#F/$\"3q)y?E4!*H7(Fg]l7$$\"3;[$Gyuq4Q# F/$\"3W?**pSK>AwFg]l7$$\"3i\\Q*y_/rO#F/$\"3bc90N\")[U!)Fg]l7$$\"3'Hybj uT+N#F/$\"3r5$>fx5#G&)Fg]l7$$\"3yoJ$yc1TL#F/$\"3]$RbCtY^&*)Fg]l7$$\"3% *eT!37JdJ#F/$\"3nB)>>pM,U*Fg]l7$$\"3'=d3IKstH#F/$\"3K!)HsBQTf)*Fg]l7$$ \"3IH\">^tGtF#F/$\"3p+Lp?TRJ5F/7$$\"3+%[\\JGI\"eAF/$\"3aI%eNmvF2\"F/7$ $\"3i$*37r%zlB#F/$\"3`f)efT5q6\"F/7$$\"3MVh&o'pF8AF/$\"3ISfg\"\\*\\i6F /7$$\"3?T45%eQA>#F/$\"3F$R-V&Hq,7F/7$$\"3?`!QC&oto@F/$\"3yj<<$)ehV7F/7 $$\"39@m=BfhV@F/$\"3=Bc$QNLkG\"F/7$$\"3$*o$*z&*>A=@F/$\"3\\m;7b6$yK\"F /7$$\"3ucDIv5*G4#F/$\"3-T&)4W)3uO\"F/7$$\"3o!yGGa**Q1#F/$\"3]9AbXly59F /7$$\"3=-G`[A3P?F/$\"3!>@31xR#\\9F/7$$\"3.=Usf&ew+#F/$\"3!Q&G&\\!Rt*[ \"F/7$$\"3I'*[[=4I!)>F/$\"3;7.TvA\"f_\"F/7$$\"3s.`3Lik\\>F/$\"3i&Q\\Pj *)[c\"F/7$$\"3f$)>&f#>5?>F/$\"35kqirB+,;F/7$$\"3)QFiH&[]))=F/$\"3N#[ZC Fa\"Q;F/7$$\"3k+ua+!=p&=F/$\"3c@'Q'4W(Qn\"F/7$$\"3Zv^R'*e7B=F/$\"3T$)= UAuh5SF*)*y\"F/$\"3w=id,2OX!*3>F/7$$\"3+c>0.)Rmd\"F/$\"3u\\4K>k:S>F/7$$\"3EfC[u\"*)* R:F/$\"3I!*4GXQPp>F/7$$\"3gCK?++++:F/$\"3Y\"eZ)********>F/F\\\\l-F$6$7 %7$$\"3;+++++++_F/F+7$F]\\m$\"3=+++++++!)Fg]l7$Fh[lF`\\mF\\\\l-F$6&7#F g[l-%'SYMBOLG6#%'CIRCLEGF\\\\l-%&STYLEG6#%&POINTG-F$6&Fe\\m-Fg\\m6#%(D IAMONDGF\\\\lFj\\m-F$6&Fe\\m-Fg\\m6#%&CROSSGF\\\\lFj\\m-%%TEXTG6%7$$\" #$*!\"#F\\^mQ1P(~cos~t,sin~t~)6\"F\\\\l-Fi]m6%7$$\"#AF^^m$\"#aF^^mQ\"1 F`^mF\\\\l-Fi]m6%7$$\"$:\"F^^m$!\"%F^^mQ\"xF`^mF\\\\l-Fi]m6%7$$!\"&F^^ mF\\_mQ\"yF`^mF\\\\l-Fi]m6%7$$!\"'F^^m$!\"*F^^mQ\"OF`^mF\\\\l-Fi]m6%7$ $\"#:F^^m$\"\"*F^^mQ\"tF`^mF\\\\l-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q! F`^mF]am-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$FaamF aam" 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" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "If P is the point" } {XPPEDIT 18 0 "``(cos*t,sin*t)" "6#-%!G6$*&%$cosG\"\"\"%\"tGF(*&%$sinG F(F)F(" }{TEXT -1 33 ", we can visualise the parameter " }{TEXT 285 1 "t" }{TEXT -1 56 " as the angle which the line OP makes with the posit ive " }{TEXT 286 1 "x" }{TEXT -1 12 " direction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The following animation g ives the location of the point " }{XPPEDIT 18 0 "P(cos*t,sin*t)" "6#-% \"PG6$*&%$cosG\"\"\"%\"tGF(*&%$sinGF(F)F(" }{TEXT -1 18 " as the param eter " }{TEXT 376 1 "t" }{TEXT -1 21 " increases from 0 to " } {XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "rng := 0..2*Pi: # parameter range\npt := [.6,1]: # location to di splay values of parameter\nt := 't':\nf := t -> cos(t);\ng := t -> sin (t);\np1 := plot([f(t),g(t),t=rng],labels=[`x`,`y`]):\nnumframes := 61 :\nfrms := []:\naa := op(1,rng): bb := op(2,rng):\nhh := evalf(abs(bb- aa)/(numframes-1)):\nfor i from 0 to numframes-1 do \n tt := aa+hh*i ;\n p2 := plot([[[f(tt),g(tt)]]$3],style=point,symbol=[circle,diamon d,cross],\n color=black);\n t1 := plots[textplot]([op (1,pt),op(2,pt),t=tt],color=blue);\n frms := [op(frms),plots[display ]([p1,p2,t1])];\nend do:\nplots[display](frms,insequence=true,scaling= constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We can el iminate the parameter " }{TEXT 287 1 "t" }{TEXT -1 32 " by squaring t he two equations " }{XPPEDIT 18 0 "x = cos*t,y = sin*t;" "6$/%\"xG*&%$ cosG\"\"\"%\"tGF'/%\"yG*&%$sinGF'F(F'" }{TEXT -1 11 " to give: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x^2=cos^2* t,``],[y^2=sin^2*t,``])" "6#-%*PIECEWISEG6$7$/*$%\"xG\"\"#*&%$cosGF*% \"tG\"\"\"%!G7$/*$%\"yGF**&%$sinGF*F-F.F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 49 "and then adding the resulting equations to give: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=cos^2*t+s in^2*t" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(,&*&%$cosGF'%\"tGF(F(*&%$si nGF'F.F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$ %\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Suppose that the parametr ic equations:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = f(t)" "6#/%\"xG-%\"fG6#%\"tG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = g(t)" "6#/%\"yG-%\"gG6#%\"tG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "describe a curve in the \+ " }{TEXT 277 1 "x" }{TEXT -1 1 "-" }{TEXT 278 1 "y" }{TEXT -1 7 " plan e." }}{PARA 0 "" 0 "" {TEXT -1 17 "If the functions " }{XPPEDIT 18 0 " f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t);" "6 #-%\"gG6#%\"tG" }{TEXT -1 42 " are differentiable functions we can wri te" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `f '`(t );" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$f~'G6#%\"tG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt = `g '`(t);" "6 #/*&%#dyG\"\"\"%#dtG!\"\"-%$g~'G6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We now investigate w hether we can determine " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dx G!\"\"" }{TEXT -1 7 " from " }{XPPEDIT 18 0 "dx/dt;" "6#*&%#dxG\"\"\" %#dtG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/dt;" "6#*&%#dyG\"\" \"%#dtG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "In some c ases we have seen that the parameter " }{TEXT 282 1 "t" }{TEXT -1 49 " can be eliminated to form an equation involving " }{TEXT 280 1 "x" } {TEXT -1 5 " and " }{TEXT 281 1 "y" }{TEXT -1 59 " for the curve. In s uch cases it would be possible to find " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 54 " by implicit differentiation. Then , by the chain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt=dy/dx" "6#/*&%#dyG\"\"\"%#dtG!\"\"*&F%F&%#dxGF(" }{TEXT -1 1 " " }{TEXT 276 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG \"\"\"%#dtG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/d t)/``(dx/dt);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6# *&F'F&F.F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "Under suitable conditions, this formula s till holds whether or not the parameter " }{TEXT 279 1 "t" }{TEXT -1 28 " can be formally eliminated." }}{PARA 0 "" 0 "" {TEXT -1 65 "For t he example of the circle given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=cos*t,``],[y=s in*t,``])" "6#-%*PIECEWISEG6$7$/%\"xG*&%$cosG\"\"\"%\"tGF+%!G7$/%\"yG* &%$sinGF+F,F+F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "we have : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/d t = -sin*t, ``],[dy/dt = cos*t, ``]);" "6#-%*PIECEWISEG6$7$/*&%#dxG\" \"\"%#dtG!\"\",$*&%$sinGF*%\"tGF*F,%!G7$/*&%#dyGF*F+F,*&%$cosGF*F0F*F1 " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the relation \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/dt )/``(dx/dt);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6#* &F'F&F.F(F(" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 13 "we see \+ that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -co s*t/(sin*t);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*(%$cosGF&%\"tGF&*&%$sinGF& F,F&F(F(" }{XPPEDIT 18 0 "``=-cot*t" "6#/%!G,$*&%$cotG\"\"\"%\"tGF(!\" \"" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 69 "This agrees with the result obtained by d ifferentiating the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "implicitly to give: " } }{PARA 256 "" 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 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 7 "whence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=-x/y" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"xGF&%\"yG F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " } {XPPEDIT 18 0 "x=cos*t" "6#/%\"xG*&%$cosG\"\"\"%\"tGF'" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "y =sin*t" "6#/%\"yG*&%$sinG\"\"\"%\"tGF'" } {TEXT -1 26 " in this expression gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -cos*t/(sin*t)" "6#/*&%#dyG\"\"\"%#dxG! \"\",$*(%$cosGF&%\"tGF&*&%$sinGF&F,F&F(F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 65 "The fo rmula (i) can also be verified from the following picture. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 352 299 299 {PLOTDATA 2 "67-%'CU RVESG6%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*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$$!3)>b`s YlSn*F/$!3UNbFGIGKDF/7$$!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$$!3g@UD=%)o!*\\F/$!3I_Mh 6Mil')F/7$$!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$$ \"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGB G$\"*++++\"!\")F+F+-%*THICKNESSG6#F)-F$6$7$7$F+F+7$$\"3w************** fF/$\"3U+++++++!)F/-Fjz6&F\\[lF*F*F*-F$6$7$7$Fh[lF+Fg[lF\\\\l-F$6$7S7$ $\"3++++++++DF/F+7$$\"3yNT:T$*[*\\#F/$\"3-I#=j9bF0&Fas7$$\"3K2\\8[U@) \\#F/$\"35;b/0P_Z%*Fas7$$\"3/v!\\0Bde\\#F/$\"3,YMHQUjQ9!#>7$$\"3Ed%HVe )\\#\\#F/$\"3cVvY,6AN>Fg]l7$$\"31$3#p(*\\<)[#F/$\"3-B;(R\\'oGCFg]l7$$ \"3]?\\%\\v$H$[#F/$\"3#o'>8iWM&)GFg]l7$$\"3GC-:;kNxCF/$\"3G8>PD$frN$Fg ]l7$$\"3#Q#G&o=t-Z#F/$\"3pt3(=>JQ%QFg]l7$$\"3)R!z6,8EiCF/$\"37G%[<8iuK %Fg]l7$$\"31xQ/GB.`CF/$\"3;0]'R3)>B[Fg]l7$$\"3Dmc0!owSW#F/$\"3S!>-P'QA e_Fg]l7$$\"3p0+%y%=2LCF/$\"3+!*esNr'fu&Fg]l7$$\"3R'[/yeU5U#F/$\"3x$p'= g-SLiFg]l7$$\"3W4*H'[i_3CF/$\"3?.Cf**fv+nFg]l7$$\"3%R:6IayjR#F/$\"3q)y ?E4!*H7(Fg]l7$$\"3;[$Gyuq4Q#F/$\"3W?**pSK>AwFg]l7$$\"3i\\Q*y_/rO#F/$\" 3bc90N\")[U!)Fg]l7$$\"3'HybjuT+N#F/$\"3r5$>fx5#G&)Fg]l7$$\"3yoJ$yc1TL# F/$\"3]$RbCtY^&*)Fg]l7$$\"3%*eT!37JdJ#F/$\"3nB)>>pM,U*Fg]l7$$\"3'=d3IK stH#F/$\"3K!)HsBQTf)*Fg]l7$$\"3IH\">^tGtF#F/$\"3p+Lp?TRJ5F/7$$\"3+%[\\ JGI\"eAF/$\"3aI%eNmvF2\"F/7$$\"3i$*37r%zlB#F/$\"3`f)efT5q6\"F/7$$\"3MV h&o'pF8AF/$\"3ISfg\"\\*\\i6F/7$$\"3?T45%eQA>#F/$\"3F$R-V&Hq,7F/7$$\"3? `!QC&oto@F/$\"3yj<<$)ehV7F/7$$\"39@m=BfhV@F/$\"3=Bc$QNLkG\"F/7$$\"3$*o $*z&*>A=@F/$\"3\\m;7b6$yK\"F/7$$\"3ucDIv5*G4#F/$\"3-T&)4W)3uO\"F/7$$\" 3o!yGGa**Q1#F/$\"3]9AbXly59F/7$$\"3=-G`[A3P?F/$\"3!>@31xR#\\9F/7$$\"3. =Usf&ew+#F/$\"3!Q&G&\\!Rt*[\"F/7$$\"3I'*[[=4I!)>F/$\"3;7.TvA\"f_\"F/7$ $\"3s.`3Lik\\>F/$\"3i&Q\\Pj*)[c\"F/7$$\"3f$)>&f#>5?>F/$\"35kqirB+,;F/7 $$\"3)QFiH&[]))=F/$\"3N#[ZCFa\"Q;F/7$$\"3k+ua+!=p&=F/$\"3c@'Q'4W(Qn\"F /7$$\"3Zv^R'*e7B=F/$\"3T$)=UAuh5SF*)*y\"F/$\"3w=id,2OX!*3>F/7$$\"3+c>0.)Rmd\"F/$\"3u \\4K>k:S>F/7$$\"3EfC[u\"*)*R:F/$\"3I!*4GXQPp>F/7$$\"3gCK?++++:F/$\"3Y \"eZ)********>F/F\\\\l-F$6$7%7$$\"3;+++++++_F/F+7$F]\\m$\"3=+++++++!)F g]l7$Fh[lF`\\mF\\\\l-F$6&7#Fg[l-%'SYMBOLG6#%'CIRCLEGF\\\\l-%&STYLEG6#% &POINTG-F$6&Fe\\m-Fg\\m6#%(DIAMONDGF\\\\lFj\\m-F$6&Fe\\m-Fg\\m6#%&CROS SGF\\\\lFj\\m-F$6%7$7$$\"3/+++++++5F/$\"3/++++++v6Fdo7$$\"33+++++++6Fd o$\"3))************\\UF/-Fjz6&F\\[lF+F][lF+-Fa[l6#\"\"#-%%TEXTG6%7$$\" #$*!\"#F^_mQ1P(~cos~t,sin~t~)6\"F\\\\l-F[_m6%7$$\"#AF`_m$\"#aF`_mQ\"1F b_mF\\\\l-F[_m6%7$$\"$:\"F`_m$!\"%F`_mQ\"xFb_mF\\\\l-F[_m6%7$$!\"&F`_m F^`mQ\"yFb_mF\\\\l-F[_m6%7$$!\"'F`_m$!\"*F`_mQ\"OFb_mF\\\\l-F[_m6%7$$ \"$<\"F`_m$\"#`F`_mQ\"BFb_mF\\\\l-F[_m6%7$$Fi^m!\"\"$\"$F\"F`_mQ\"AFb_ mF\\\\l-F[_m6%7$$\"#:F`_m$\"\"*F`_mQ\"tFb_mF\\\\l-%*AXESTICKSG6$F*F*-% +AXESLABELSG6%Q!Fb_mF_cm-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINED G-%%VIEWG6$FccmFccm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 25 "The line OP has gradient " }{XPPEDIT 18 0 "tan*t" "6#*&%$tanG\"\"\"%\"tGF%" }{TEXT -1 83 " so the tangent line AB at P is perpendicular to OP and consequently has gradient " } {XPPEDIT 18 0 "-1/(tan*t)=-cos*t/(sin*t)" "6#/,$*&\"\"\"F&*&%$tanGF&% \"tGF&!\"\"F*,$*(%$cosGF&F)F&*&%$sinGF&F)F&F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 16 ": The procedure " }{TEXT 0 4 "plot" }{TEXT -1 54 " cab be used to plot the graph of parameterized curve." }}{PARA 0 "" 0 "" {TEXT -1 24 "The standard syntax is " }{TEXT 0 25 "plot([f(t),g(t),t= a..b]);" }{TEXT -1 8 " where " }{TEXT 262 6 "t=a..b" }{TEXT -1 97 " i s the range of the parameter to be used in plotting the graph. Note th e use of square brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{MPLTEXT 1 0 9 "eliminate" } {TEXT -1 47 " can be used to try to eliminate the parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "el iminate(\{x=cos(t),y=sin(t)\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 $<#/%\"tG-%'arctanG6$%\"yG%\"xG<#,(*$)F*\"\"#\"\"\"F1*$)F+F0F1F1F1!\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 257 "" 0 "" {TEXT 258 8 "Question " }{TEXT 308 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 78 "This question is co ncerned with the curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ x = t*cos*t,``] ,[ y = t*sin*t,``])" "6#-%*PIECEWISEG6$7$/%\"xG*(%\"tG\"\"\"%$cosGF+F* F+%!G7$/%\"yG*(F*F+%$sinGF+F*F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) Sketch the graph of the curve. " }}{PARA 0 "" 0 "" {TEXT -1 23 "(b) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in terms of the parameter " }{TEXT 309 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "(c) Find the equation of the tangent line to the curve at the point given by " } {XPPEDIT 18 0 "t = 9*Pi/4;" "6#/%\"tG*(\"\"*\"\"\"%#PiGF'\"\"%!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 310 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following animation gives the lo cation of the point" }{XPPEDIT 18 0 "``(cos*t,sin*t);" "6#-%!G6$*&%$co sG\"\"\"%\"tGF(*&%$sinGF(F)F(" }{TEXT -1 18 " as the parameter " } {TEXT 375 1 "t" }{TEXT -1 21 " increases from 0 to " }{XPPEDIT 18 0 "8 *Pi;" "6#*&\"\")\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "rng := 0..8 *Pi: # parameter range\npt := [16.5,18]: # location to display values \+ of parameter\nt := 't':\nf := t -> t*cos(t);\ng := t -> t*sin(t);\np1 \+ := plot([f(t),g(t),t=rng],labels=[`x`,`y`]):\nnumframes := 161:\nfrms \+ := []:\naa := op(1,rng): bb := op(2,rng):\nhh := evalf(abs(bb-aa)/(num frames-1)):\nfor i from 0 to numframes-1 do \n tt := aa+hh*i;\n p2 := plot([[[f(tt),g(tt)]]$3],style=point,symbol=[circle,diamond,cross] ,\n color=black);\n t1 := plots[textplot]([op(1,pt),o p(2,pt),t=tt],color=black);\n frms := [op(frms),plots[display]([p1,p 2,t1])];\nend do:\nplots[display](frms,insequence=true,scaling=constra ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "(a) The curve is a \+ " }{TEXT 259 6 "spiral" }{TEXT -1 41 " such that the line joining the \+ origin to" }{XPPEDIT 18 0 "``(t*cos*t,t*sin*t);" "6#-%!G6$*(%\"tG\"\" \"%$cosGF(F'F(*(F'F(%$sinGF(F'F(" }{TEXT -1 13 ", has length " }{TEXT 263 1 "t" }{TEXT -1 24 ", and makes an angle of " }{TEXT 264 1 "t" } {TEXT -1 27 " radians with the positive " }{TEXT 265 1 "x" }{TEXT -1 11 " direction." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 458 462 462 {PLOTDATA 2 "6/-%'CURVESG6%7[r7$$\"\"!F)F(7$$\"3B)z;Cy_<-\"!#= $\"3eQde'47$$\"37)\\0ZDO6,#F-$\"3u4:xMNm!>%F07$$\"35c#*)e+]j$H F-$\"3u1!Rl'e2Y$*F07$$\"3[#)=?EIsmPF-$\"3j;[yKF-7$$\"3Sr%oTG-\\_&F-$\"3\"4+nkvl(R`F-7$$\"3%Qz$y#Qg[[& F-$\"3!pT9A`fG*zF-7$$\"3WAM)o**R?c%F-$\"3a#Hvu$f#y2\"!#<7$$\"3\"o8&om< &4q#F-$\"3!)fk.]d\"fM\"FT7$$!3i3*eG0Q+y'!#?$\"3a%e'3ei3v:FT7$$!3y:;F(R F%GOF-$\"3Hl))R'=T!R6@ \"FT$\"3)Hiaa[iSz\"FT7$$!3'f<]-X$[a;FT$\"31_Z7A'H2n\"FT7$$!3s<\\*=v^X5 #FT$\"34eHwxWSI9FT7$$!36=G]fD'z^#FT$\"3/Ym?G*R`2\"FT7$$!3#**)))=7z=xGF T$\"3?.lX\"GhD%fF-7$$!3nzY]u!Qy8$FT$\"3A,mr\">s7<\"F07$$!3#HLk\">latKF T$!3+)ov5Ej\\['F-7$$!3Itjq5'=VE$FT$!3%3W[`0X/O\"FT7$$!3)Q\\dT?C#)3$FT$ !3#e:U#pNA7@FT7$$!3]&4MZJqct#FT$!3sZ`'Q,G#FT$!3*HxE T4++W$FT7$$!3m1%=&H\\O'p\"FT$!3A-?@>%Q&fRFT7$$!3#eAIz_lv-*F-$!3N_)GeBW '>WFT7$$\"3zR6**>6gW%)Fgn$!3pW$HptsTr%FT7$$\"3mEWZORc05FT$!3&p55\"o;S9 [FT7$$\"3!oe#oVS$>/#FT$!3W1dI#3Fyp%FT7$$\"3+jT\"\\*y$G.$FT$!3O!*f2XQwp VFT7$$\"3'*p$y:'p$*pRFT$!3n&*y:\"HL$HQFT7$$\"3&ogn,>-dt%FT$!3aISX+mBiJ FT7$$\"3!>*)GW7\" FT7$$\"3/ihQ?g=(H'FT$\"3jV&\\ez\\WC*F07$$\"3[G2_1Z^bjFT$\"3?[lU=\"4FD \"FT7$$\"3g*)f\">^!='>'FT$\"3#4\">67CrNCFT7$$\"3urc:()o\\HdFT$\"3!*R-[ peO&y$FT7$$\"3CrL22]yk\\FT$\"3Jq$zF>kB/&FT7$$\"3M(=kG!QSaSFT$\"3+ilSQN MCgFT7$$\"375t%e\"oi[HFT$\"3,c!*H'[S#QoFT7$$\"3+]`P!=8'[:FT$\"3op-Qh=w \"\\(FT7$$\"37u6PPX*HY%Fgn$\"3kr(z&y?T`yFT7$$!3$G!>\"y$HQV:FT$\"3u9nF! [zv*yFT7$$!3$G87]@nm5$FT$\"3&pq\\(e&)\\KwFT7$$!3klg=;Hk'o%FT$\"39h@(pe '=AqFT7$$!3>=d>31]VhFT$\"3Qx1I(H[93'FT7$$!3XQ(*\\>!yJJ(FT$\"35rr()4.A[ \\FT7$$!3m!3jW**e.F)FT$\"37J%*pr'G')e$FT7$$!3w&H#)*G)=U,*FT$\"3]D42pD> :>FT7$$!3jk4y+l1:%*FT$\"3a.HSk!z*f()F07$$!3#z`tN5@VV*FT$!3%)[@sMqe(*=F T7$$!3$pX=(*Q#yI!*FT$!3W?-=`n=&)QFT7$$!3tDi7$y$4N$)FT$!3_*=nv'zjYbFT7$ $!3nZsY^vPJtFT$!3wuGQ]\\G\"3(FT7$$!3=)G,?[/)GfFT$!3<=*Hm$>;I&)FT7$$!3o DokEcRQUFT$!3_+v#**R/yp*FT7$$!3U;oB%\\x/D#FT$!3Oc`D0kza5!#;7$$!3\\s(yl cbm?*F0$!3-Ho!3!4o)4\"Fe]l7$$\"35w3#z$>2-@FT$!3)>l6\"*4L&)4\"Fe]l7$$\" 35^G:x4h!H%FT$!35#QJB%=Ca5Fe]l7$$\"3)Q8f)[&\\'=jFT$!3O@f&**Hwfp*FT7$$ \"3M*o7iqjv<)FT$!37X'4BkKrX)FT7$$\"31-KSZ-,^**FT$!3[EJX6A*Rm'FT7$$\"3 \"on;dx4G8\"Fe]l$!31e$4aUEz\\%FT7$$\"3B(fo%*3Li@\"Fe]l$!3%Rw3D#*HfI#FT 7$$\"3W2ex))H&pD\"Fe]l$\"3A8MICZ@`SF07$$\"3k,FG&>8,D\"Fe]l$\"3OY*edsz< i#FT7$$\"3Ea%zb6#***=\"Fe]l$\"35y\\eD'H`<&FT7$$\"3sh%[!po>!4\"Fe]l$\"3 kWhfTH'GP(FT7$$\"3wWP%*\\PY'\\*FT$\"3eVBMQ5Tw$*FT7$$\"3xJ%[!*3*zQvFT$ \"3O_d:SvbD6Fe]l7$$\"3uLSAB$)o3_FT$\"3_dk#HFvBF\"Fe]l7$$\"3KbI8i2f`FFT $\"3eW44X-Om8Fe]l7$$\"3I(f;i=DeH\"F-$\"3Z2CJ\\+u79Fe]l7$$!3mH\"H.cEJp# FT$\"3Gb)oLm'329Fe]l7$$!3/7(flDCq[&FT$\"3*4&Go&)[#[M\"Fe]l7$$!39syGayU \"3)FT$\"3&G*>_!QJ,B\"Fe]l7$$!3EPBOP,bV5Fe]l$\"3SfggYADl5Fe]l7$$!3-w\" oH*zn`7Fe]l$\"3:ygQrP)QW)FT7$$!3>UpoOb\"pT\"Fe]l$\"3L)4r40&y?eFT7$$!3z Uu@tb;A:Fe]l$\"3G!>?=o1d*HFT7$$!3UsYMZ'44d\"Fe]l$!3aR=x_wH'z\"F07$$!3I gk^cW$Gd\"Fe]l$!373!R5w1af\"FT7$$!3IS'>!G(f)e:Fe]l$!3!4xE_Hqn<$FT7$$!3 !G#4o]z#*G:Fe]l$!3W?,wI.(fu%FT7$$!3ieRm`[9$[\"Fe]l$!3/t%H6'*\\oG'FT7$$ !3:*p4^J^BU\"Fe]l$!3ee1*FT7$$! 3I*peg\"Gjc7Fe]l$!3eCrgg>wa5Fe]l7$$!3#*p')e.S,`6Fe]l$!3;T4ls(G5=\"Fe]l 7$$!3a]#H[dN9I*FT$!3u#4&FT$!3m1!)=RPA>;Fe]l7$$!3)H*GY[*\\!)R$FT$!3AIArC#*pt;Fe ]l7$$!37$>fP+#fX;FT$!37h?]%G'Q5$*Q$FT$!35`u>r1@9q7Fe]l$!3ye Yk9*yOG\"Fe]l7$$\"3om40&otQR\"Fe]l$!3_oZ8S57j6Fe]l7$$\"3%e^e=6og]\"Fe] l$!3$3>!R8.iI5Fe]l7$$\"3ee<;>FT7$$\"3/+++ #fb\\)=Fe]l$!36^viPF\\+H!#D-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THIC KNESSG6#\"\"#-F$6&7#7$$\"3A+++J9>lOFT$\"3k*****Hx(H[jFT-%'SYMBOLG6#%'C IRCLEG-Fa^m6&Fc^mF)F)F)-%&STYLEG6#%&POINTG-F$6&F]_m-Fd_m6#%(DIAMONDGFg _mFi_m-F$6&F]_m-Fd_m6#%&CROSSGFg_mFi_m-F$6$7$F'F^_mFg_m-F$6%7S7$$\"3/+ ++Fjzq:FT$\"3#p&zm[Gq)G\"!#E7$$\"3%*H)G#)=C-f\"FT$\"3S*[9$))*HFc#F-7$$ \"3+#e-&)[$es:FT$\"3@%3LZl8S%[F-7$$\"3y!z;a_#38:FT$\"3=iB#\\hcaS(F-7$$ \"32:#\\')[/0T\"FT$\"3^C]XG#)o6**F-7$$\"3qG\"pu__nE\"FT$\"34&olSb0jA\" FT7$$\"3aP.]IJ6)4\"FT$\"3NdlPF-$\"3o813Eb9y=FT 7$$\"3Oq7Tm2>XqF0$\"3GAh\"Hp6K&>FT7$$!3s:=+9\"eJ2#F-$\"3/\"*[`l@wy>FT7 $$!3%)>afR**Rl_F-$\"3=8Kf=Dhf>FT7$$!3#R)eN&o=%f%)F-$\"3s\")o*Gn[z)=FT7 $$!3G?iez(\\X9\"FT$\"3#\\cw)>J4p;FT7 $$!3Ir>BQjK!o\"FT$\"3[sH&z$Gp$R\"FT7$$!3A'3ciK(3))=FT$\"3?c;fm#QS;\"FT 7$$!3g\"3TI(4V*3#FT$\"3gS:5v%**4e)F-7$$!3X*pk4cUfA#FT$\"3_4z+MVAzbF-7$ $!3O6f%R![RDBFT$\"3t>GG[`jR?F-7$$!3s)QN*4\\CnBFT$!3!3t)*o*>w%\\\"F-7$$ !3ia7Z\"**QIN#FT$!3uZvz)e8`E&F-7$$!3OE])>z1mG#FT$!3\\&3;A+<;s)F-7$$!3C 3M.Uqed@FT$!3t7[2VjHN7FT7$$!3)Q;37:r?'>FT$!3)y#z@(GR=f\"FT7$$!3#=[;NE3 Mu\"FT$!3-@A*Q#4,x=FT7$$!3ib'*e\\Sbg9FT$!3uw[p$z&p]@FT7$$!3\"RfNi(e2B6 FT$!3#Hn`/P&4)Q#FT7$$!3%[20[:;Rb(F-$!3_Z\"oMf;'oDFT7$$!3hn>6([%)*=PF-$ !3'**z?ujQ*)o#FT7$$\"3qeiK@A')*e(F0$!3__%4CYMZv#FT7$$\"3)[(G`#H.U)[F-$ !3w$*\\)3Y+)\\FFT7$$\"3Eq2aHT7(G*F-$!3+t*eGKDen#FT7$$\"3P;897/7=8FT$!3 oFF^\\>XZDFT7$$\"3<@l`#G*)Hs\"FT$!3nS,$>eP>M#FT7$$\"3cy>!)38`u?FT$!3i \\hE\\QQ*3#FT7$$\"3J?n!pv.>S#FT$!3m'4yLe\"yo(=wV.)*F-7$$\"3eHBqqGC^IFT$!3Eh k'36vhO&F-7$$\"3v5Z1][ROJFT$!3@jG7a$\\'ReF07$$\"3!yV()Q&)[j9$FT$\"3UI# H)\\1u\"H%F-7$$\"3k]5Fy()G)3$FT$\"3/y=%p!f])y)F-7$$\"3B'G\"fEkJUHFT$\" 373*4'oI%QQ\"FT7$$\"3'olB*opLTFFT$\"3;Z)frWX^\"=FT7$$\"3g=:g/p 7$$\"37)\\0ZDO6,#F-$\"3u4:xMNm!>%F07$$\"35c#*)e+]j$HF-$\"3u1!Rl'e2Y$*F 07$$\"3[#)=?EIsmPF-$\"3j;[yKF-7 $$\"3Sr%oTG-\\_&F-$\"3\"4+nkvl(R`F-7$$\"3%Qz$y#Qg[[&F-$\"3!pT9A`fG*zF- 7$$\"3WAM)o**R?c%F-$\"3a#Hvu$f#y2\"!#<7$$\"3\"o8&om<&4q#F-$\"3!)fk.]d \"fM\"FT7$$!3i3*eG0Q+y'!#?$\"3a%e'3ei3v:FT7$$!3y:;F(RF%GOF-$\"3Hl))R'= T!R6@\"FT$\"3)Hiaa[iSz \"FT7$$!3'f<]-X$[a;FT$\"31_Z7A'H2n\"FT7$$!3s<\\*=v^X5#FT$\"34eHwxWSI9F T7$$!36=G]fD'z^#FT$\"3/Ym?G*R`2\"FT7$$!3#**)))=7z=xGFT$\"3?.lX\"GhD%fF -7$$!3nzY]u!Qy8$FT$\"3A,mr\">s7<\"F07$$!3#HLk\">latKFT$!3+)ov5Ej\\['F- 7$$!3Itjq5'=VE$FT$!3%3W[`0X/O\"FT7$$!3)Q\\dT?C#)3$FT$!3#e:U#pNA7@FT7$$ !3]&4MZJqct#FT$!3sZ`'Q,G#FT$!3*HxET4++W$FT7$$!3m1%= &H\\O'p\"FT$!3A-?@>%Q&fRFT7$$!3#eAIz_lv-*F-$!3N_)GeBW'>WFT7$$\"3zR6**> 6gW%)Fgn$!3pW$HptsTr%FT7$$\"3mEWZORc05FT$!3&p55\"o;S9[FT7$$\"3!oe#oVS$ >/#FT$!3W1dI#3Fyp%FT7$$\"3+jT\"\\*y$G.$FT$!3O!*f2XQwpVFT7$$\"3'*p$y:'p $*pRFT$!3n&*y:\"HL$HQFT7$$\"3&ogn,>-dt%FT$!3aISX+mBiJFT7$$\"3!>*)GW7\"FT7$$\"3/ihQ?g=(H 'FT$\"3jV&\\ez\\WC*F07$$\"3[G2_1Z^bjFT$\"3?[lU=\"4FD\"FT7$$\"3g*)f\">^ !='>'FT$\"3#4\">67CrNCFT7$$\"3urc:()o\\HdFT$\"3!*R-[peO&y$FT7$$\"3CrL2 2]yk\\FT$\"3Jq$zF>kB/&FT7$$\"3M(=kG!QSaSFT$\"3+ilSQNMCgFT7$$\"375t%e\" oi[HFT$\"3,c!*H'[S#QoFT7$$\"3+]`P!=8'[:FT$\"3op-Qh=w\"\\(FT7$$\"37u6PP X*HY%Fgn$\"3kr(z&y?T`yFT7$$!3$G!>\"y$HQV:FT$\"3u9nF![zv*yFT7$$!3$G87]@ nm5$FT$\"3&pq\\(e&)\\KwFT7$$!3klg=;Hk'o%FT$\"39h@(pe'=AqFT7$$!3>=d>31] VhFT$\"3Qx1I(H[93'FT7$$!3XQ(*\\>!yJJ(FT$\"35rr()4.A[\\FT7$$!3m!3jW**e. F)FT$\"37J%*pr'G')e$FT7$$!3w&H#)*G)=U,*FT$\"3]D42pD>:>FT7$$!3jk4y+l1:% *FT$\"3a.HSk!z*f()F07$$!3#z`tN5@VV*FT$!3%)[@sMqe(*=FT7$$!3$pX=(*Q#yI!* FT$!3W?-=`n=&)QFT7$$!3tDi7$y$4N$)FT$!3_*=nv'zjYbFT7$$!3nZsY^vPJtFT$!3w uGQ]\\G\"3(FT7$$!3=)G,?[/)GfFT$!3<=*Hm$>;I&)FT7$$!3oDokEcRQUFT$!3_+v#* *R/yp*FT7$$!3U;oB%\\x/D#FT$!3Oc`D0kza5!#;7$$!3\\s(ylcbm?*F0$!3-Ho!3!4o )4\"Fe]l7$$\"35w3#z$>2-@FT$!3)>l6\"*4L&)4\"Fe]l7$$\"35^G:x4h!H%FT$!35# QJB%=Ca5Fe]l7$$\"3)Q8f)[&\\'=jFT$!3O@f&**Hwfp*FT7$$\"3M*o7iqjv<)FT$!37 X'4BkKrX)FT7$$\"31-KSZ-,^**FT$!3[EJX6A*Rm'FT7$$\"3\"on;dx4G8\"Fe]l$!31 e$4aUEz\\%FT7$$\"3B(fo%*3Li@\"Fe]l$!3%Rw3D#*HfI#FT7$$\"3W2ex))H&pD\"Fe ]l$\"3A8MICZ@`SF07$$\"3k,FG&>8,D\"Fe]l$\"3OY*edsz!4\"Fe]l$\"3kWhfTH'GP(FT7$$\"3wW P%*\\PY'\\*FT$\"3eVBMQ5Tw$*FT7$$\"3xJ%[!*3*zQvFT$\"3O_d:SvbD6Fe]l7$$\" 3uLSAB$)o3_FT$\"3_dk#HFvBF\"Fe]l7$$\"3KbI8i2f`FFT$\"3eW44X-Om8Fe]l7$$ \"3I(f;i=DeH\"F-$\"3Z2CJ\\+u79Fe]l7$$!3mH\"H.cEJp#FT$\"3Gb)oLm'329Fe]l 7$$!3/7(flDCq[&FT$\"3*4&Go&)[#[M\"Fe]l7$$!39syGayU\"3)FT$\"3&G*>_!QJ,B \"Fe]l7$$!3EPBOP,bV5Fe]l$\"3SfggYADl5Fe]l7$$!3-w\"oH*zn`7Fe]l$\"3:ygQr P)QW)FT7$$!3>UpoOb\"pT\"Fe]l$\"3L)4r40&y?eFT7$$!3zUu@tb;A:Fe]l$\"3G!>? =o1d*HFT7$$!3UsYMZ'44d\"Fe]l$!3aR=x_wH'z\"F07$$!3Igk^cW$Gd\"Fe]l$!373! R5w1af\"FT7$$!3IS'>!G(f)e:Fe]l$!3!4xE_Hqn<$FT7$$!3!G#4o]z#*G:Fe]l$!3W? ,wI.(fu%FT7$$!3ieRm`[9$[\"Fe]l$!3/t%H6'*\\oG'FT7$$!3:*p4^J^BU\"Fe]l$!3 ee1*FT7$$!3I*peg\"Gjc7Fe]l$!3e Crgg>wa5Fe]l7$$!3#*p')e.S,`6Fe]l$!3;T4ls(G5=\"Fe]l7$$!3a]#H[dN9I*FT$!3 u#4&FT$! 3m1!)=RPA>;Fe]l7$$!3)H*GY[*\\!)R$FT$!3AIArC#*pt;Fe]l7$$!37$>fP+#fX;FT$ !37h?]%G'Q5$*Q$FT$ !35`u>r1@9q7Fe]l$!3yeYk9*yOG\"Fe]l7$$\"3o m40&otQR\"Fe]l$!3_oZ8S57j6Fe]l7$$\"3%e^e=6og]\"Fe]l$!3$3>!R8.iI5Fe]l7$ $\"3ee<;>FT7$$\"3/+++#fb\\)=Fe]l$!36^viPF \\+H!#D-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"\"-F$6&7 #7$$\"3I+++*HV#)*\\FT$\"3w*****RJV#)*\\FT-%'SYMBOLG6#%'CIRCLEG-Fa^m6&F c^mF)F)F)-%&STYLEG6#%&POINTG-F$6&F]_m-Fd_m6#%(DIAMONDGFg_mFi_m-F$6&F]_ m-Fd_m6#%&CROSSGFg_mFi_m-F$6%7S7$$!\"$F)$\"3IO)*=FPCj:Fe]l7$$!32++]i(f Aa#FT$\"3nHpKKTQ-:Fe]l7$$!3$)**\\PHC)R9#FT$\"3))*>^U_I%\\9Fe]l7$$!3%** **\\P\"z2'p\"FT$\"39h]s`'y)*Q\"Fe]l7$$!3!****\\iR/_C\"FT$\"3lWXuM?$*H8 Fe]l7$$!3U***\\(=!QZ'zF-$\"3en)z:Lq-F\"Fe]l7$$!3:.+v$*4V/QF-$\"3%og;bH c\\@\"Fe]l7$$\"3<.+]i&fK.&F0$\"3#)4r3:=od6Fe]l7$$\"3]++DcVTe\\F-$\"3/L \\\\l%[%)4\"Fe]l7$$\"3o(**\\7B:#*R*F-$\"3E>h+y]SR5Fe]l7$$\"3Q++]xCr'R \"FT$\"3%zK!>mpr'y*FT7$$\"3Q++v)>a!*z\"FT$\"3/%=aB,xD#FT$\"3Se\"=XKh&\\')FT7$$\"3v****\\()pz1FFT$\"3L&=37)G([/)FT7$$\"3+ ++]OVpH$pFT7$$\"35++] 2RN;SFT$\"3**R!e?EIPI'FT7$$\"3`+++IFFFF7FT7$$\"3***\\i S!yi55Fe]l$!3#fC]@.FKz\"FT7$$\"3'***\\(y3\"*y0\"Fe]l$!3Mtc@zOi@CFT7$$ \"3&)****\\]#f.5\"Fe]l$!3O@Rn:gE')HFT7$$\"33++vyIqX6Fe]l$!3YimLIA9*e$F T7$$\"3#**\\ilK\"z'=\"Fe]l$!3Ht\\%eoPa8%FT7$$\"3.++DM;rJ7Fe]l$!3`AVD9H oKZFT7$$\"3y*\\iI9yRF\"Fe]l$!33!\\y(HTk%H&FT7$$\"3;+DJB)e\"=8Fe]l$!33O `w@b0#)eFT7$$\"3y***\\P^`8O\"Fe]l$!3!\\ifk_djX'FT7$$\"3\")*\\i:NulS\"F e]l$!3Ep[$=])fdqFT7$$\"3++++$RF,X\"Fe]l$!34+iI?]mOwFT7$$\"3%***\\Pomm% \\\"Fe]l$!3)Hct=h\"))FT7$$\"3-+ +]3_Uz:Fe]l$!3C8w&Q2ldN*FT7$$\"3(***\\()>P%fi\"Fe]l$!3o!\\>$y%fU(**FT7 $$\"35+++J/bn;Fe]l$!31_&zY#[u_5Fe]l7$$\"3'**\\iI'=\">r\"Fe]l$!3,5*y`GE <6\"Fe]l7$$\"3=+v$RTrVv\"Fe]l$!3i.I8p!z\"o6Fe]l7$$\"#=F)$!3!\\Di\\HX)G 7Fe]l-Fa^m6&Fc^mF(Fd^mF(-Fh^m6#\"\"#-%%TEXTG6$7$$\"#AF)$Ff^m!\"\"Q\"x6 \"-Fe`n6$7$Fj`n$\"# " 0 "" {MPLTEXT 1 0 69 "9*Pi/(4*sqrt(2))+((4+9*Pi)/( 4-9*Pi))*(x-9*Pi/(4*sqrt(2)));\nevalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**\"\"*\"\"\"\"\")!\"\"%#PiGF&\"\"##F&F*F&*(,&\"\"%F& *&F%F&F)F&F&F&,&F.F&*&F%F&F)F&F(F(,&%\"xGF&**F%F&F'F(F)F&F*F+F(F&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+nQPk6!\")\"\"\"*&$\"+,icH8!\"*F '%\"xGF'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 8 ": It is " }{TEXT 259 3 "not" }{TEXT -1 46 " possible to formally eliminate the parameter " }{TEXT 266 1 "t " }{TEXT -1 18 " in this example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 314 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 78 "This question is concerned with the curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 1/(1+t^2) ,``],[ y = t/(1+t^2),``]) " "6#-%*PIECEWISEG6$7$/%\"xG*&\"\"\"F*,&F*F**$%\"tG\"\"#F*!\"\"%!G7$/% \"yG*&F-F*,&F*F**$F-F.F*F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in terms of the parameter " }{TEXT 315 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 90 "(b) Find the coordinates of the points on the curve where the tangent line is hori zontal. " }}{PARA 0 "" 0 "" {TEXT -1 54 "(c) Show that the cartesian e quation of the curve is " }{XPPEDIT 18 0 "x^2+y^2 = x" "6#/,&*$%\"xG \"\"#\"\"\"*$%\"yGF'F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(d) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%# dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 316 1 "x" }{TEXT -1 5 " \+ and " }{TEXT 317 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 318 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following animation gives the location of the point" }{XPPEDIT 18 0 "``(1/(1+t^2),t/(1+t^2));" "6#-%!G6$*&\"\"\"F',&F'F'*$%\"tG\"\"#F '!\"\"*&F*F',&F'F'*$F*F+F'F," }{TEXT -1 18 " as the parameter " } {TEXT 374 1 "t" }{TEXT -1 27 " increases from -20 to 20. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "rng \+ := -20..20: # parameter range\npt := [.5,.2]: # location to display va lues of parameter\nt := 't':\nf := t -> 1/(1+t^2);\ng := t -> t/(1+t^2 );\np1 := plot([f(t),g(t),t=rng],labels=[`x`,`y`]):\nnumframes := 161: \nfrms := []:\naa := op(1,rng): bb := op(2,rng):\nhh := evalf(abs(bb-a a)/(numframes-1)):\nfor i from 0 to numframes-1 do \n tt := aa+hh*i; \n p2 := plot([[[f(tt),g(tt)]]$3],style=point,symbol=[circle,diamond ,cross],\n color=black);\n t1 := plots[textplot]([op( 1,pt),op(2,pt),t=tt],color=black);\n frms := [op(frms),plots[display ]([p1,p2,t1])];\nend do:\nplots[display](frms,insequence=true,scaling= constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Since " }{XPPEDIT 18 0 "x=(1+t^2)^(-1)" "6#/%\"xG),&\"\"\"F'*$%\"tG\"\"#F', $F'!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt = (-1)*(1+t^2)^(-2)*(2*t)" "6#/*&%#dxG\"\"\"%#dtG !\"\"*(,$F&F(F&),&F&F&*$%\"tG\"\"#F&,$F/F(F&*&F/F&F.F&F&" }{XPPEDIT 18 0 "`` = -2*t/((1+t^2)^2);" "6#/%!G,$*(\"\"#\"\"\"%\"tGF(*$,&F(F(*$F )F'F(F'!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "By the quotient rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dt = (1*`.`*(1+t^2)-t*`.`*2*t)/(1+t^2)^2" "6#/*&%#dyG\"\"\"%#dtG!\" \"*&,&*(F&F&%\".GF&,&F&F&*$%\"tG\"\"#F&F&F&**F/F&F,F&F0F&F/F&F(F&*$,&F &F&*$F/F0F&F0F(" }{XPPEDIT 18 0 "``=(1-t^2)/(1+t^2)^2" "6#/%!G*&,&\"\" \"F'*$%\"tG\"\"#!\"\"F'*$,&F'F'*$F)F*F'F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/dt)/``(dx/dt);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6 #*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{XPPEDIT 18 0 "`` = ``((1-t^2)/((1 +t^2)^2))/``(-2*t/((1+t^2)^2));" "6#/%!G*&-F$6#*&,&\"\"\"F**$%\"tG\"\" #!\"\"F**$,&F*F**$F,F-F*F-F.F*-F$6#,$*(F-F*F,F**$,&F*F**$F,F-F*F-F.F.F ." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-t^2)/(-2*t);" "6#/%!G*&,&\"\" \"F'*$%\"tG\"\"#!\"\"F',$*&F*F'F)F'F+F+" }{XPPEDIT 18 0 "`` = (t^2-1)/ (2*t);" "6#/%!G*&,&*$%\"tG\"\"#\"\"\"F*!\"\"F**&F)F*F(F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " (b) The tangent line to the curve is horizontal when " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 20 ". This occur s when " }{XPPEDIT 18 0 "t^2=1" "6#/*$%\"tG\"\"#\"\"\"" }{TEXT -1 9 " , giving " }{XPPEDIT 18 0 "t=``" "6#/%\"tG%!G" }{TEXT 319 1 "+" } {TEXT -1 3 " 1." }}{PARA 0 "" 0 "" {TEXT -1 41 "The corresponding poin ts on the curve are" }{XPPEDIT 18 0 " ``(1/2,1/2)" "6#-%!G6$*&\"\"\"F' \"\"#!\"\"*&F'F'F(F)" }{TEXT -1 17 " given by taking " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" }{TEXT -1 5 ", and" }{XPPEDIT 18 0 " ``(1/2,-1/ 2)" "6#-%!G6$*&\"\"\"F'\"\"#!\"\",$*&F'F'F(F)F)" }{TEXT -1 17 " given \+ by taking " }{XPPEDIT 18 0 "t=-1" "6#/%\"tG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "(c ) We can eliminate the parameter using the fact that " }{XPPEDIT 18 0 "y = t*x;" "6#/%\"yG*&%\"tG\"\"\"%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "x=1/(1+t^2)" "6#/% \"xG*&\"\"\"F&,&F&F&*$%\"tG\"\"#F&!\"\"" }{TEXT -1 15 " implies that \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*(1+t^2)=1" "6#/ *&%\"xG\"\"\",&F&F&*$%\"tG\"\"#F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+t^2*x=1" "6#/,&%\"xG\"\"\"*&%\"tG\"\"#F%F&F&F&" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 26 "Multiplying both sides by " }{TEXT 320 1 "x" }{TEXT -1 9 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2+t^2*x^2=x" "6#/,&*$%\"xG\"\"#\"\"\"*&%\"tGF'F&F'F(F &" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 23 "and then substitutin g " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "t^2*x^2" "6#*&%\"tG\"\"#%\"xGF%" }{TEXT -1 8 " gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=x" "6#/,& *$%\"xG\"\"#\"\"\"*$%\"yGF'F(F&" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Alternatively, since we are given this last equation in the question, we can check t hat the coordinates of any point" }{XPPEDIT 18 0 " ``(1/(1+t^2), t/(1+ t^2))" "6#-%!G6$*&\"\"\"F',&F'F'*$%\"tG\"\"#F'!\"\"*&F*F',&F'F'*$F*F+F 'F," }{TEXT -1 22 " satisfy equation (i)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=1/(1+t^2)^2+t^2/(1+t^2)^2" "6#/,&*$% \"xG\"\"#\"\"\"*$%\"yGF'F(,&*&F(F(*$,&F(F(*$%\"tGF'F(F'!\"\"F(*&F0F'*$ ,&F(F(*$F0F'F(F'F1F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+t^2)/(1+t^2)^2" "6#/%!G*&,&\"\"\"F'*$%\"tG\" \"#F'F'*$,&F'F'*$F)F*F'F*!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(1+t^2)" "6#/%!G*&\"\"\"F&,&F&F&*$ %\"tG\"\"#F&!\"\"" }{XPPEDIT 18 0 "``=x" "6#/%!G%\"xG" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 18 "for all values of " }{TEXT 321 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "(c) Differentiating the equation " }{XPPEDIT 18 0 "x^2+y ^2 = x" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F&" }{TEXT -1 17 " with res pect to " }{TEXT 322 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 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=1" "6#/*&%#dyG\"\" \"%#dxG!\"\"F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\"\"\" %\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1-2*x" "6#/*&%#dyG\"\" \"%#dxG!\"\",&F&F&*&\"\"#F&%\"xGF&F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(1-2*x)/(2*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*&\"\"#F&%\"x GF&F(F&*&F,F&%\"yGF&F(" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 3 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "x=1/( 1+t^2)" "6#/%\"xG*&\"\"\"F&,&F&F&*$%\"tG\"\"#F&!\"\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "y=t/(1+t^2)" "6#/%\"yG*&%\"tG\"\"\",&F'F'*$F&\"\"# F'!\"\"" }{TEXT -1 16 " in (ii) gives: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=``(1-2/(1 +t^2))/``(2*t/(1+t^2))" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6#,&F&F&*&\" \"#F&,&F&F&*$%\"tGF/F&F(F(F&-F+6#*(F/F&F2F&,&F&F&*$F2F/F&F(F(" } {XPPEDIT 18 0 "``= ``(1-2/(1+t^2))*`.`*``((1+t^2)/(2*t))" "6#/%!G*(-F$ 6#,&\"\"\"F)*&\"\"#F),&F)F)*$%\"tGF+F)!\"\"F/F)%\".GF)-F$6#*&,&F)F)*$F .F+F)F)*&F+F)F.F)F/F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+t^2-2)/(2*t)" "6#/%!G*&,(\"\"\"F'*$%\"tG\"\" #F'F*!\"\"F'*&F*F'F)F'F+" }{XPPEDIT 18 0 "``=(t^2-1)/(2*t)" "6#/%!G*&, &*$%\"tG\"\"#\"\"\"F*!\"\"F**&F)F*F(F*F+" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The curve given by equation (i) is a ci rcle. To see this write the equation in the form: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-x+y^2=0" "6#/,(*$%\"xG\"\"#\"\"\" F&!\"\"*$%\"yGF'F(\"\"!" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 6 "Addng " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT -1 22 " to the pair of terms " }{XPPEDIT 18 0 "x^2-x" "6#,&*$%\"xG\"\"#\"\" \"F%!\"\"" }{TEXT -1 49 " completes the square to form the perfect squ are " }{XPPEDIT 18 0 "x^2-x+1/4=(x-1/2)^2" "6#/,(*$%\"xG\"\"#\"\"\"F&! \"\"*&F(F(\"\"%F)F(*$,&F&F(*&F(F(F'F)F)F'" }{TEXT -1 33 ", so the equa tion can be written " }}{PARA 0 "" 0 "" {TEXT -1 13 "in the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1/2)^2+y^2 = 1/4" "6#/,&*$,&%\"xG\"\"\"*&F(F(\"\"#!\"\"F+F*F(*$%\"yGF*F(*&F(F(\"\"%F+" } {TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 57 "This is \+ the equation of a circle with centre at the point" }{XPPEDIT 18 0 "``( 1/2,0)" "6#-%!G6$*&\"\"\"F'\"\"#!\"\"\"\"!" }{TEXT -1 24 " and having \+ a radius of " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 " Actually, the curve given by the parametric equations is not the whole circle because the point" }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 17 " must be removed." }}{PARA 0 "" 0 "" {TEXT -1 17 "As the parameter " }{TEXT 323 1 "t" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "-inf inity" "6#,$%)infinityG!\"\"" }{TEXT -1 11 ", the point" }{XPPEDIT 18 0 "``(1/(1+t^2),t/(1+t^2));" "6#-%!G6$*&\"\"\"F',&F'F'*$%\"tG\"\"#F'! \"\"*&F*F',&F'F'*$F*F+F'F," }{TEXT -1 21 " approaches the point" } {XPPEDIT 18 0 "``(0,0);" "6#-%!G6$\"\"!F&" }{TEXT -1 42 ", but no para meter value gives this point." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 310 266 266 {PLOTDATA 2 "61-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F* 7$$\"3WY0)p=\"=`**!#=$\"3b@$)y74REo!#>7$$\"3=d:T\\5!p$)*F/$\"3UiuzzWkm 7F/7$$\"3'p^I%RKHC'*F/$\"3!o;GC;b:!>F/7$$\"3lI%)Rfa`E$*F/$\"3Kdnsp/@1D F/7$$\"39FvV;g7_*)F/$\"3Ow9,NGziIF/7$$\"3Jv'>V3(GT&)F/$\"3S\"=Y/Nr(HNF /7$$\"3gk'fJ(yAe!)F/$\"3W21wC&fc&RF/7$$\"3x9P:ydQ0vF/$\"3^'f%QpI,FVF/7 $$\"3y6aLA)p+\"pF/$\"3F;+`bFy?YF/7$$\"39L26QJpiiF/$\"3'GpSs1Mz$[F/7$$ \"3ShvI6%oDn&F/$\"3/*zr=oeX&\\F/7$$\"3')yH6lW,(*\\F/$\"3uk!e'3\"***** \\F/7$$\"3'4Dz.#)e(=VF/$\"3yM(4QktL&\\F/7$$\"31A9#[#*3pn$F/$\"3>h4fuiw @[F/7$$\"3D5yYMbW8JF/$\"3l*Q6zEL/j%F/7$$\"3G^()=jB%)yCF/$\"3P@$Rl`UyJ% F/7$$\"3wo>#=Eb-)>F/$\"3S\"3*>>]6&)RF/7$$\"3[xyhE=Mb9F/$\"3hzP=IgQENF/ 7$$\"334PA(*y*y/\"F/$\"3Qae:#pBG1$F/7$$\"3>'f[4ImMq'F2$\"37Zmu/-#3]#F/ 7$$\"3![tF/7$$\"3Yl/)HX#)=o\"F2$\"3%oM2L\">#fG \"F/7$$\"35a6:H!*\\[Y!#?$\"3H:-ED]7-oF27$$\"3B!H(*p$e`/C!#B$\"3Q7hC=Zl ]:Fir7$$\"3)RPZQ1*4yXFir$!3aja*4Gf1v'F27$$\"32SAtjEnH;F2$!3snx89:9m7F/ 7$$\"3WRU$[$*)o$o$F2$!3%[Df\"oF/$!3]:Q!RHk;)RF/7$$\"3[*)G(3zbY]#F/$!3:En!eq6GL%F/7$$\"3m *pR\"f#4O6$F/$!3Y_w1q**\\IYF/7$$\"3ChPN4sm&p$F/$!3frf)zMuo#[F/7$$\"31% RzBh;UN%F/$!3e'>Ml*47e\\F/7$$\"3]gJ&G38Z)\\F/$!3A.='4jw***\\F/7$$\"3p: h\"oihQk&F/$!3KtBaA4Pe\\F/7$$\"39m>&Gn?vF'F/$!3rTUt9)RS$[F/7$$\"3QU*yF >**y\"pF/$!3b^`3'fQvh%F/7$$\"3u1)[!)G];](F/$!3)[,y_!Q vY0)F/$!3-xW#fk.%eRF/7$$\"3/-QB$>]'\\&)F/$!3\\&G%HI5O@NF/7$$\"3uH4'o>* 3]*)F/$!37`Y'eF>a1$F/7$$\"3(*p]5e(orL*F/$!3y4**G'[jx[#F/7$$\"3&p\"zgw/ ]7'*F/$!3\"yf`$)G%)*H>F/7$$\"3fL9hMWMF)*F/$!3#*Hv$)>>f-8F/7$$\"3;!z#)z wxM&**F/$!3;q[tIl$[!oF27$F($\"30BmIq&o?5%!#F-%'COLOURG6&%$RGBG$\"*++++ \"!\")F+F+-F$6&7$7$$\"3++++++++]F/Fd[l7$Fd[l$!3++++++++]F/-%'SYMBOLG6# %'CIRCLEG-Fjz6&F\\[lF*F*F*-%&STYLEG6#%&POINTG-F$6&Fb[l-Fj[l6#%(DIAMOND GF]\\lF_\\l-F$6&Fb[l-Fj[l6#%&CROSSGF]\\lF_\\l-F$6%7$7$$\"3/+++++++5F/F d[l7$$\"3A+++++++!*F/Fd[l-Fjz6&F\\[lF+F][lF+-%*THICKNESSG6#\"\"#-F$6%7 $7$Fa]lFg[l7$Fd]lFg[lFf]lFh]l-F$6&7#7$F+F+-Fj[l6$%(DEFAULTG\"#;FizF_\\ l-F$6&Fc^l-Fj[l6$Fg^l\"#?FizF_\\l-%%TEXTG6$7$$\"#6!\"\"$!\"$!\"#Q\"x6 \"-F__l6$7$Fe_l$\"\"'Fd_lQ\"yFi_l-F__l6$7$$\"\"&Fd_lF]`lQ(t~=~1/2Fi_l- F__l6$7$Fc`l$!\"'Fd_lQ)t~=~-1/2Fi_l-%*AXESTICKSG6$\"\"%7&/$!\"%Fd_l%%- 0.4G/$Fg_lFd_l%%-0.2G/$F[^lFd_l%$0.2G/$F_alFd_l%$0.4G-%+AXESLABELSG6%% !GFabl-%%FONTG6#Fg^l-%%VIEWG6$Fg^lFg^l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 2 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eliminate(\{x=1/(1+t^2),y=t /(1+t^2)\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<#/%\"tG*&%\"yG\" \"\"%\"xG!\"\"<#,(*$)F*\"\"#F)F)F*F+*$)F(F0F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 324 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 77 "This question is concerned w ith the curve given by the parametric equations: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 3*t/(1+t^3) ,``],[y = \+ 3*t^2/(1+t^3) ,``])" "6#-%*PIECEWISEG6$7$/%\"xG*(\"\"$\"\"\"%\"tGF+,&F +F+*$F,F*F+!\"\"%!G7$/%\"yG*(F*F+*$F,\"\"#F+,&F+F+*$F,F*F+F/F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for \+ " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in terms of the parameter " }{TEXT 325 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "(b) Find the equation of the tangent line to the \+ curve at the point given by " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "(c) Show that the carte sian equation of the curve is " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/ ,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(*(F'F(F&F(F*F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "(d) Find a formula for " }{XPPEDIT 18 0 " dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " } {TEXT 326 1 "x" }{TEXT -1 5 " and " }{TEXT 327 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "The curve is called " }{TEXT 259 23 "th e folium of Descartes" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 328 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following animation gives the location of the point" }{XPPEDIT 18 0 "``(3*t/(1+t^3),3*t^2/(1+t^3));" "6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F (F(*$F)F'F(!\"\"*(F'F(*$F)\"\"#F(,&F(F(*$F)F'F(F," }{TEXT -1 18 " as t he parameter " }{TEXT 377 1 "t" }{TEXT -1 28 " increases from -0.5 to \+ 20. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 636 "rng := -.5..20: # parameter range\npt := [-.5,1]: # \+ location to display values of parameter\nt := 't':\nf := t -> 3*t/(1+t ^3);\ng := t -> 3*t^2/(1+t^3);\np1 := plot([f(t),g(t),t=rng],labels=[` x`,`y`]):\nnumframes := 161:\nfrms := []:\naa := op(1,rng): bb := op(2 ,rng):\nhh := evalf(abs(bb-aa)/(numframes-1)):\nfor i from 0 to numfra mes-1 do \n tt := aa+hh*i;\n p2 := plot([[[f(tt),g(tt)]]$3],style= point,symbol=[circle,diamond,cross],\n color=black);\n \+ t1 := plots[textplot]([op(1,pt),op(2,pt),t=tt],color=black);\n frm s := [op(frms),plots[display]([p1,p2,t1])];\nend do:\nplots[display](f rms,insequence=true,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 55 "The following ani mation gives the location of the point" }{XPPEDIT 18 0 "``(3*t/(1+t^3) ,3*t^2/(1+t^3));" "6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F(F(*$F)F'F(!\"\"*(F' F(*$F)\"\"#F(,&F(F(*$F)F'F(F," }{TEXT -1 18 " as the parameter " } {TEXT 378 1 "t" }{TEXT -1 29 " increases from -10 to -1.5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 637 "r ng := -10..-1.5: # parameter range\npt := [1,.3]: # location to displa y values of parameter\nt := 't':\nf := t -> 3*t/(1+t^3);\ng := t -> 3* t^2/(1+t^3);\np1 := plot([f(t),g(t),t=rng],labels=[`x`,`y`]):\nnumfram es := 161:\nfrms := []:\naa := op(1,rng): bb := op(2,rng):\nhh := eval f(abs(bb-aa)/(numframes-1)):\nfor i from 0 to numframes-1 do \n tt : = aa+hh*i;\n p2 := plot([[[f(tt),g(tt)]]$3],style=point,symbol=[circ le,diamond,cross],\n color=black);\n t1 := plots[text plot]([op(1,pt),op(2,pt),t=tt],color=black);\n frms := [op(frms),plo ts[display]([p1,p2,t1])];\nend do:\nplots[display](frms,insequence=tru e,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "(a) \+ By the quotient rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=(3*`.`*(1+t^3)-(3*t)*`.`*3*t^2)/(1+t^3)^2" "6#/*&%#dxG\"\" \"%#dtG!\"\"*&,&*(\"\"$F&%\".GF&,&F&F&*$%\"tGF,F&F&F&*,F,F&F0F&F-F&F,F &F0\"\"#F(F&*$,&F&F&*$F0F,F&F2F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ (3+3*t^3-9*t^3)/((1+t^3)^2);" "6#/%!G*&,(\"\"$\"\"\"*&F'F(*$%\"tGF'F(F (*&\"\"*F(*$F+F'F(!\"\"F(*$,&F(F(*$F+F'F(\"\"#F/" }{XPPEDIT 18 0 "``=( 3-6*t^3)/(1+t^3)^2" "6#/%!G*&,&\"\"$\"\"\"*&\"\"'F(*$%\"tGF'F(!\"\"F(* $,&F(F(*$F,F'F(\"\"#F-" }{XPPEDIT 18 0 " ``=3*(1-2*t^3)/(1+t^3)^2" "6# /%!G*(\"\"$\"\"\",&F'F'*&\"\"#F'*$%\"tGF&F'!\"\"F'*$,&F'F'*$F,F&F'F*F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt =(6*t*`.`*(1+t^3)-3*t^2*`.`*3*t^2)/(1+t^3)^2" "6#/*&%#dyG\"\"\"%#dtG! \"\"*&,&**\"\"'F&%\"tGF&%\".GF&,&F&F&*$F-\"\"$F&F&F&*,F1F&*$F-\"\"#F&F .F&F1F&F-F4F(F&*$,&F&F&*$F-F1F&F4F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(6*t+6*t^4-9*t^4)/(1+t^3)^2" "6#/ %!G*&,(*&\"\"'\"\"\"%\"tGF)F)*&F(F)*$F*\"\"%F)F)*&\"\"*F)*$F*F-F)!\"\" F)*$,&F)F)*$F*\"\"$F)\"\"#F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=(6*t- 3*t^4)/(1+t^3)^2" "6#/%!G*&,&*&\"\"'\"\"\"%\"tGF)F)*&\"\"$F)*$F*\"\"%F )!\"\"F)*$,&F)F)*$F*F,F)\"\"#F/" }{XPPEDIT 18 0 " ``=(3*t*(2-t^3))/(1+ t^3)^2" "6#/%!G**\"\"$\"\"\"%\"tGF',&\"\"#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 7 " Hence " }}{PARA 256 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "dy/dx = ``(dy/dt)/``(dx/dt);" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{XPPEDIT 18 0 " ``=(3*t*( 2-t^3))/(3*(1-2*t^3))" "6#/%!G**\"\"$\"\"\"%\"tGF',&\"\"#F'*$F(F&!\"\" F'*&F&F',&F'F'*&F*F'*$F(F&F'F,F'F," }{XPPEDIT 18 0 " ``= t*(2-t^3)/((1 -2*t^3))" "6#/%!G*(%\"tG\"\"\",&\"\"#F'*$F&\"\"$!\"\"F',&F'F'*&F)F'*$F &F+F'F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) The par ameter value " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" }{TEXT -1 8 ", gi ves " }{XPPEDIT 18 0 "x=3/2" "6#/%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3/2" "6#/%\"yG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "The gradient of the ta ngent line at this point is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "eval(dy/dx,t=1)=(1*`.`*(2-1))/(1-2)" "6#/-%%evalG6$*&%# dyG\"\"\"%#dxG!\"\"/%\"tGF)**F)F)%\".GF),&\"\"#F)F)F+F),&F)F)F1F+F+" } {XPPEDIT 18 0 "``=-1" "6#/%!G,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "The equation of he tangent line at the point" } {XPPEDIT 18 0 " ``(3/2,3/2)" "6#-%!G6$*&\"\"$\"\"\"\"\"#!\"\"*&F'F(F)F *" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-3/2=(-1)*(x-3/2)" "6#/,&%\"yG\"\"\"*&\"\"$F&\"\"#!\"\"F**&,$F&F*F &,&%\"xGF&*&F(F&F)F*F*F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+y=3" "6#/ ,&%\"xG\"\"\"%\"yGF&\"\"$" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 363 323 323 {PLOTDATA 2 "60-%'CURVESG6$7ip7$$!3+c$ QQ%=cuG!#<$\"3Z)pr`Vcf#>F*7$$!3wYbjQJjkDF*$\"3(Gn5'[tyH;F*7$$!3;W[\">+ ?FI#F*$\"37'*3me#oQQ\"F*7$$!3mH['Q?io2#F*$\"3q!**>I__k<\"F*7$$!3w,bs\\ ?fy=F*$\"3GxIY$GDI***!#=7$$!3;.aEOVt,<(Gd\\FA7$$!3kF]![T#eI6F*$\"3/IV[h`(H1%FA7$$!3'3N<%GO ;45F*$\"3#=ZqzVD$yKFA7$$!3HK&G98r(G*)FA$\"3[Q,3A1R#f#FA7$$!3bc*fWg$\\0 yFA$\"3)RKI;kao*>FA7$$!3vrlyU^qScFA$\"3A9csBAn`5FA7$$!3#*eaxeL&)QNFA$ \"3cXk>AdnnT!#>7$$!3IIX'HUwBY\"FA$\"3&4#))H$Gdw7(!#?7$$\"3KRNutp`'3'F^ p$\"3_mb!Rsu[B\"Fdp7$$\"3'R[L$[nhxEFA$\"3mlYSM+e\"R#F^p7$$\"3#3)ep4YgJ ZFA$\"31/;b`KK#\\(F^p7$$\"3OV.EUH+UnFA$\"31AOdgV'H`\"FA7$$\"3)zEGV9Ukm )FA$\"3OMKlz(o(oDFA7$$\"3/S'yJO)GX5F*$\"3=)oLo)z#)>QFA7$$\"3$HV()f(*)f /7F*$\"3(=IJMTANB&FA7$$\"3*flbEj3NK\"F*$\"3Q:(=n\\g]a'FA7$$\"3r&z'>)p> 8U\"F*$\"3-A7LHrU#)yFA7$$\"3)z()o(\\BX'\\\"F*$\"3ki__*\\&*y>*FA7$$\"3o KF\\AW][:F*$\"3LG.A3*)yW5F*7$$\"3-&pC\\YN#y:F*$\"3+i()p3,kf6F*7$$\"3Ib ^%4$yR(e\"F*$\"3+s;SkSrh7F*7$$\"3eK/t-iYy:F*$\"3')=K**)>?%\\8F*7$$\"3U e^X4'HVb\"F*$\"34)4**)y:9A9F*7$$\"3Xm!)R/G+=:F*$\"3c@O+Lz2![\"F*7$$\"3 OLuHPWRs9F*$\"3p60))[M/C:F*7$$\"3Dh.LJu9?9F*$\"3)*)H!\\*4g_b\"F*7$$\"3 1Xa3V$\\NO\"F*$\"3CEC,\"Hu^d\"F*7$$\"3Cfvw;(e\"F*7$$\"39#)[U0by%=\"F*$\"3WHKRGl9#e\"F* 7$$\"3^&3@\"e@k!p$o*)))FA$\"3[9P&)RGp! [\"F*7$$\"3Iq:yz*Gd.(FA$\"33e'zA]&)>O\"F*7$$\"3wza^o()QRcFA$\"3=J>X@l/ W7F*7$$\"3%R+y/(=X!f%FA$\"3n(oGWYxm8\"F*7$$\"3k#z7Iv*o*y$FA$\"3')3ae$* RYT5F*7$$\"3g#)Rc5H%Q<$FA$\"3GPtpd/S&e*FA7$$\"3YYhZ\"o&f#p#FA$\"3]&\\b 9@)Gk))FA7$$\"3N*4I)>=o5BFA$\"3R\\y(4JMaB)FA7$$\"3eukt(=vXv\"FA$\"3i!) =rG+H.sFA7$$\"3[;(R0`[cP\"FA$\"3MjR(oBbBR'FA7$$\"3h7vl\"*GXB6FA$\"3yg] W/yK%y&FA7$$\"3jY\"z`\"*>RM*F^p$\"3%*\\iZy3*)z_FA7$$\"3WB5mH8l!o'F^p$ \"3;D8KdpOpWFA7$$\"3!**\\]SJd\\'\\F^p$\"3ogMqiCFbQFA7$$\"3Bv^?nr\\OQF^ p$\"3c)H,UO3,R$FA7$$\"3vk=H#enS.$F^p$\"3zG-\\s-X:IFA7$$\"3vh+IDdt=DF^p $\"3IIb4jwzZFFA7$$\"3)z&eOpeK\"3#F^p$\"3%oLY>$H2)\\#FA7$$\"3GcQ2`jXZFA7$$\"3&H)**H%=4X8\"F^p$\"3u_*>'H1lW=FA7$$\"3ozSOP*3 2,\"F^p$\"31mpzT+8TJX%f!*)Fdp$\"37uusioUM;FA7$$\"3M[f&*G2 x7!)Fdp$\"3/$*>MZBK]:FA7$$\"3m/)e(Ga0zrFdp$\"3m6K'otouY\"FA7$$\"3cud7# )*e/]'Fdp$\"3#>qbT(GS'R\"FA7$$\"3IwwgQB2!*eFdp$\"3$\\%Fdp$\"3j.Y6%3#HA6FA7$$ \"3%=W![V;A()QFdp$\"3'=C#[/n')z5FA7$$\"3/6^GrN[+OFdp$\"3RxMAt'y#R5FA7$ $\"3kdOQKBg\\LFdp$\"3A#o)>z'=C+\"FA7$$\"3P%>1.Z858$Fdp$\"3OXznClg\"p*F ^p7$$\"3fF%o(*3oC\"HFdp$\"35VzEjpEZ$*F^p7$$\"3#=E<$*HR]t#Fdp$\"3A=T<2f 3e!*F^p7$$\"3-.Uc7k\"Hc#Fdp$\"3oGwU\\=Wo()F^p7$$\"3a`+)[%Hg?CFdp$\"34a 10\"4F:_)F^p7$$\"3A%3?$[\"o\"yAFdp$\"37!\\J#p'>qE)F^p7$$\"3K1MI/TKb@Fd p$\"3j0RYc&\\5/)F^p7$$\"3^hm)4!=CP?Fdp$\"3zk=t=Lo Fdp$\"3C90d;8)4h(F^p7$$\"3rOMwQM,TvOqF^p7$$\"3?2>+sO- r:Fdp$\"3OeqjE'F^p7$$\"3m*o4a5LYD\"Fdp$\"3DtaNpj.NhF^p7$$\" 3c*zw+S!***>\"Fdp$\"33(*RQ+?&***fF^p-%'COLOURG6&%$RGBG$\"*++++\"!\")$ \"\"!F^ilF]il-F$6$7`o7$$\"3o+o2+'4+?\"Fdp$!3V.SQ+![++'F^p7$$\"3y1c7)Q5 CD\"Fdp$!3*o(\\Uw=lHhF^p7$$\"3%z\"z5kx%3I\"Fdp$!3mDuFL@1ZiF^p7$$\"3#yG c5*\\ve8Fdp$!3Q$*[+kTf%Q'F^p7$$\"3Y)R[+xS5U\"Fdp$!3tg\\jcEHHlF^p7$$\"3 S1Ui(y\"Q([\"Fdp$!3!*y*))[.m*zmF^p7$$\"3gKSDcy7`:Fdp$!3wlFT]z+EoF^p7$$ \"3fCr\"*zk!fi\"Fdp$!3+,RuS76%)pF^p7$$\"3OoS8=,p12qI'*3)>Fdp$!3j)>5L=\\*3xF^p7$$\"3M!*[JMe%=4#Fdp$!3o$Gs( \\>!>#zF^p7$$\"3-xmpl5!H@#Fdp$!31KF.r`!z9)F^p7$$\"38*))e`.%yRBFdp$!3_1 cF?'[#y$)F^p7$$\"3EV$)zbTqkCFdp$!3'oB=;x.!*f)F^p7$$\"3=I*f(**frEEFdp$! 3AC'z9wMr())F^p7$$\"3eiqiy[zwFFdp$!32/K'yeAs7*F^p7$$\"3I<0PPq8oHFdp$!3 phvQabYO%*F^p7$$\"3'>Tuy)R]aJFdp$!33\"o,L@C#G(*F^p7$$\"3y!)GR?C\"*zLFd p$!3ytC8j;)p+\"FA7$$\"3mko5*R(y1$*>$>6FA7$$\"3TNbf=LK;X Fdp$!3-(oe9UMS;\"FA7$$\"3j69A[oy:\\Fdp$!3N)QhNQFW@\"FA7$$\"3R&)pYtnI3` Fdp$!3Qq5I]n)>E\"FA7$$\"3ipUB*z\"[(y&Fdp$!3/JAuW@s<8FA7$$\"3I;GXrp,ajF dp$!3+X?.[6s!Q\"FA7$$\"3I\"*y')G+x#*pFdp$!3c=^M*fq%[9FA7$$\"3$e<(>N>52 xFdp$!3Fkks_\"o1_\"FA7$$\"3K*fQyNPkj)Fdp$!3%>))3!z7w4;FA7$$\"3q[biiQuA '*Fdp$!3c%Rkco>#*p\"FA7$$\"3/Xw?>#pu3\"F^p$!34Lga!p5k!=FA7$$\"3bF&RR7) GB7F^p$!3)\\)G-k$Qf\">FA7$$\"3g[lSY:3.9F^p$!3/U<3-C(>0#FA7$$\"3Y?\"=lZ v5h\"F^p$!3)QF'fzC*))>#FA7$$\"3'zf?#4$Q@)=F^p$!3KL(Qq!)3oP#FA7$$\"3x_E N\\f;>AF^p$!3cn]3WJ.\"e#FA7$$\"3C#o1>Q$=zEF^p$!3AT`@DHDOGFA7$$\"3kF-%= ioAF$F^p$!3RExKQ8'\\8$FA7$$\"3)4X1l'FA7$$\"31/F)*ff0g =FA$!3oHw6A81FvFA7$$\"39TwDC*prX#FA$!3!Rom%o)FA7$$\"3s8Q>dR+`GFA$ !3k6.JX2G%Q*FA7$$\"3yu5$yi'4]&FA$! 3q;!4)>dPK8F*7$$\"3![&H05SV:F*7$$\"3EW8t&RL))H)FA$!3-'G8.\"=4#o\"F*7$$\"3O$)oq&)z]$**)FA$!3 cr!*GX=mjF*7$$\"3+D'QhZ`)z6F*$!3-Q!3[^813#F*7$$\"3L6(4B>#HT7F*$!3=ffl V-3[@F*7$$\"3s_4Mp8[38F*$!35js3nBE@AF*7$$\"3%*oZt62E#Q\"F*$!3$Ql4Of*)4 I#F*7$$\"3e4h\\hMkj9F*$!3%>%=)o=k#)Q#F*7$$\"3;:j7mR'Qb\"F*$!3')=6-V%3V [#F*7$$\"3euZRT=Wa;F*$!31(3'=a(G1f#F*7$$\"3s\"\\ea!*psw\"F*$!3Ww_?MM54 FF*7$$\"3/j_5Uot%*=F*$!3p%*y:j_5UGF*Ffhl-F$6%7S7$$!3++++++++]FA$\"3+++ +++++NF*7$$!3PLLLLQ6GTFA$\"3cLLL$Q6GT$F*7$$!3immmT.\\pLFA$\"3bmm;M!\\p L$F*7$$!3LLLL$))Qj^#FA$\"37LLL))Qj^KF*7$$!3ULLL$=Kvl\"FA$\"3ALLL=KvlJF *7$$!3gqmm;C2G!)F^p$\"3wmm;C2G!3$F*7$$!3phLLL3yO5Fdp$\"39LL$3yO5+$F*7$ $\"3t$*****\\Kd,\")F^p$\"3&*****\\nU)*=HF*7$$\"3-mmm\"fX(e;FA$\"3iLL$3 WDT$GF*7$$\"3.*****\\U7Y]#FA$\"3))****\\d(Q&\\FF*7$$\"3'QLLLV!puLFA$\" 3gmmmc4`iEF*7$$\"3xmmm;c0TTFA$\"3KLLLQW*ee#F*7$$\"3B*******H,Q+&FA$\"3 3+++q)>'*\\#F*7$$\"3u*******\\*3qeFA$\"3.+++]5*HT#F*7$$\"3!********p= \\q'FA$\"3z******H\"3&HBF*7$$\"3_mmm\"fBIY(FA$\"3OLL$3k(p`AF*7$$\"3yKL LLO[k$)FA$\"3%pmmmj^N;#F*7$$\"3.KLLL&Q\"G\"*FA$\"3!ommm9'=(3#F*7$$\"3! *****\\s]k,5F*$\"35++]F\\N)*>F*7$$\"39LLL`dF!3\"F*$\"3&ommmCC(>>F*7$$ \"33++]sgam6F*$\"3\"*****\\FRXL=F*7$$\"3/++]]\"F*7$$\"3Mmm;f`@'e\"F*$\"3mLL$3k%y8 9F*7$$\"3y****\\nZ)Hm\"F*$\"3?++]K_,P8F*7$$\"3YmmmJy*eu\"F*$\"3aLLLo@5 a7F*7$$\"3')******R^bJ=F*$\"39+++g[Wo6F*7$$\"3f*****\\5a`\">F*$\"3U+++ &*ek%3\"F*7$$\"3o****\\7RV'*>F*$\"3J++](3mN+\"F*7$$\"3k*****\\@fk3#F*$ \"3b.++]ySN\"*FA7$$\"3/LLL`4Nn@F*$\"3[pmmm/\\E$)FA7$$\"3#*******\\,s`A F*$\"3y++++&)ziuFA7$$\"3[mm;zM)>L#F*$\"3;NLL3_;!o'FA7$$\"3$*******pfa< CF*$\"3n++++.aCeFA7$$\"3#HLLeg`!)\\#F*$\"3(3nm;%RY>]FA7$$\"3w****\\#G2 Ae#F*$\"3W-++vr#z<%FA7$$\"3;LLL$)G[kEF*$\"3Qommm6Fj`o$\"#=Fj`oQ&t~=~1F^ao-%*AXESTICKSG6$\"\"%F^bo-%+AX ESLABELSG6%%!GFbbo-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VI EWG6$;F^`oFh`oF^co" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "(c) We show that the cartesian equation of the \+ curve is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(*(F'F(F&F(F*F(" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=3*t/(1+t^3)" "6#/%\"xG*(\"\"$\"\"\"%\"tGF',&F'F'*$F(F&F'!\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=3*t^2/(1+t^3)" "6#/%\"yG*(\"\" $\"\"\"*$%\"tG\"\"#F',&F'F'*$F)F&F'!\"\"" }{TEXT -1 27 ", the left sid e of (i) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3+ y^3=27*t^3/(1+t^3)^3+27*t^6/(1+t^3)^3" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yG F'F(,&*(\"#FF(*$%\"tGF'F(*$,&F(F(*$F/F'F(F'!\"\"F(*(F-F(*$F/\"\"'F(*$, &F(F(*$F/F'F(F'F3F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(27*t^3+27*t^6)/( 1+t^3)^2" "6#/%!G*&,&*&\"#F\"\"\"*$%\"tG\"\"$F)F)*&F(F)*$F+\"\"'F)F)F) *$,&F)F)*$F+F,F)\"\"#!\"\"" }{XPPEDIT 18 0 "``=27*t^3*(1+t^3)/(1+t^3)^ 3" "6#/%!G**\"#F\"\"\"*$%\"tG\"\"$F',&F'F'*$F)F*F'F'*$,&F'F'*$F)F*F'F* !\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=27*t^3/(1+t^3)^2" "6#/%!G*(\"#F \"\"\"*$%\"tG\"\"$F'*$,&F'F'*$F)F*F'\"\"#!\"\"" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 45 "On the other hand, the right s ide of (i) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3* x*y=3*``(3*t/(1+t^3))*``(3*t^2/(1+t^3))" "6#/*(\"\"$\"\"\"%\"xGF&%\"yG F&*(F%F&-%!G6#*(F%F&%\"tGF&,&F&F&*$F.F%F&!\"\"F&-F+6#*(F%F&*$F.\"\"#F& ,&F&F&*$F.F%F&F1F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "``=27*t^3/(1+t^3)^2" "6#/%!G*(\"#F\"\"\"*$%\"tG\"\"$ F'*$,&F'F'*$F)F*F'\"\"#!\"\"" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Since the expre ssions (ii) and (iii) are identical, the point" }{XPPEDIT 18 0 " ``(3* t/(1+t^3),3*t^2/(1+t^3))" "6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F(F(*$F)F'F(! \"\"*(F'F(*$F)\"\"#F(,&F(F(*$F)F'F(F," }{TEXT -1 26 " always lies on t he curve " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/,&*$%\"xG\"\"$\"\"\"*$ %\"yGF'F(*(F'F(F&F(F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "(d) Differentiating the equation \+ " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F( *(F'F(F&F(F*F(" }{TEXT -1 17 " with respect to " }{TEXT 332 1 "x" } {TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2+3*y^2" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&F%F&*$%\"yGF)F&F& " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 3*y + 3*x" "6#/*&%#dyG\"\"\" %#dxG!\"\",&*&\"\"$F&%\"yGF&F&*&F+F&%\"xGF&F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y + x" "6#/*&%#dyG\"\"\"%#dxG! \"\",&%\"yGF&%\"xGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#dxG!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "Henc e " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2" "6#*$%\"yG \"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx - x" "6#,&*&%#dyG\"\"\"%# dxG!\"\"F&%\"xGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = y - x^2" "6 #/*&%#dyG\"\"\"%#dxG!\"\",&%\"yGF&*$%\"xG\"\"#F(" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 " ``( y^2-x)=y-x^2" "6#/-%!G6#,&*$%\"yG\"\"#\"\"\"%\"xG!\"\",&F)F+*$F,F*F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(y-x^2)/(y^2-x)" "6#/*&%#d yG\"\"\"%#dxG!\"\"*&,&%\"yGF&*$%\"xG\"\"#F(F&,&*$F+F.F&F-F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x^2-y)/(x-y^2)" "6#/%!G*&,&*$%\"xG\"\"#\" \"\"%\"yG!\"\"F*,&F(F**$F+F)F,F," }{TEXT -1 16 " ------- (ii). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Alternati vely, substituting " }{XPPEDIT 18 0 "x = 3*t/(1+t^3)" "6#/%\"xG*(\"\"$ \"\"\"%\"tGF',&F'F'*$F(F&F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " y = 3*t^2/(1+t^3)" "6#/%\"yG*(\"\"$\"\"\"*$%\"tG\"\"#F',&F'F'*$F)F&F'! \"\"" }{TEXT -1 17 " in (ii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(9*t^2/(1+t^3)^2-3*t^2/(1+t^3))/(3*t/(1+t^3) -9*t^4/(1+t^3)^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&*(\"\"*F&*$%\"tG\"\" #F&*$,&F&F&*$F.\"\"$F&F/F(F&*(F3F&*$F.F/F&,&F&F&*$F.F3F&F(F(F&,&*(F3F& F.F&,&F&F&*$F.F3F&F(F&*(F,F&*$F.\"\"%F&*$,&F&F&*$F.F3F&F/F(F(F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (9*t^2-3*t^2*(1+t^3))/(3*t*(1+t^3) -9*t^4);" "6#/%!G*&,&*&\"\"*\"\"\"*$%\"tG\"\"#F)F)*(\"\"$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)F2F 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(6*t^2-3*t^4)/(3*t-6*t^4) " "6#/%! G*&,&*&\"\"'\"\"\"*$%\"tG\"\"#F)F)*&\"\"$F)*$F+\"\"%F)!\"\"F),&*&F.F)F +F)F)*&F(F)*$F+F0F)F1F1" }{XPPEDIT 18 0 "`` = 3*t^2*(2-t^2)/(3*t*(1-2* t^3));" "6#/%!G**\"\"$\"\"\"*$%\"tG\"\"#F',&F*F'*$F)F*!\"\"F'*(F&F'F)F ',&F'F'*&F*F'*$F)F&F'F-F'F-" }{XPPEDIT 18 0 "`` = t*(2-t^2)/(1-2*t^3); " "6#/%!G*(%\"tG\"\"\",&\"\"#F'*$F&F)!\"\"F',&F'F'*&F)F'*$F&\"\"$F'F+F +" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "The parametric descript ion of the curve misses out the point" }{XPPEDIT 18 0 " ``(0,0)" "6#-% !G6$\"\"!F&" }{TEXT -1 69 ", which certainly lies on the curve given b y the cartesian equation. " }}{PARA 0 "" 0 "" {TEXT -1 17 "As the para meter " }{TEXT 329 1 "t" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "inf inity" "6#%)infinityG" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "-infinity " "6#,$%)infinityG!\"\"" }{TEXT -1 11 ", the point" }{XPPEDIT 18 0 "`` (3*t/(1+t^3),3*t^2/(1+t^3));" "6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F(F(*$F)F 'F(!\"\"*(F'F(*$F)\"\"#F(,&F(F(*$F)F'F(F," }{TEXT -1 9 " tends to" } {XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 17 "As the parameter " }{TEXT 330 1 "t" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 25 " from \+ the left, the point" }{XPPEDIT 18 0 "``(3*t/(1+t^3),3*t^2/(1+t^3));" " 6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F(F(*$F)F'F(!\"\"*(F'F(*$F)\"\"#F(,&F(F( *$F)F'F(F," }{TEXT -1 9 " tends to" }{XPPEDIT 18 0 "``(infinity,-infin ity);" "6#-%!G6$%)infinityG,$F&!\"\"" }{TEXT -1 23 ", and as the param eter " }{TEXT 331 1 "t" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 26 " from the right, the point" }{XPPEDIT 18 0 "``(3*t/(1+t^3),3*t^2/(1+t^3));" "6#-%!G6$*(\"\"$\"\"\"%\"tGF(,&F (F(*$F)F'F(!\"\"*(F'F(*$F)\"\"#F(,&F(F(*$F)F'F(F," }{TEXT -1 9 " tends to" }{XPPEDIT 18 0 "``(-infinity,infinity);" "6#-%!G6$,$%)infinityG! \"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "For a point" }{XPPEDIT 18 0 " ``(x,y)" "6#-%!G6$%\"x G%\"yG" }{TEXT -1 15 " on the curve, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+y=3*t/(1+t^3)+3*t^2/(1+t^3)" "6#/,&%\"xG\"\"\"% \"yGF&,&*(\"\"$F&%\"tGF&,&F&F&*$F+F*F&!\"\"F&*(F*F&*$F+\"\"#F&,&F&F&*$ F+F*F&F.F&" }{XPPEDIT 18 0 "`` = (3*t+3*t^2)/(1+t^3)" "6#/%!G*&,&*&\" \"$\"\"\"%\"tGF)F)*&F(F)*$F*\"\"#F)F)F),&F)F)*$F*F(F)!\"\"" }{XPPEDIT 18 0 "``=(3*t*(1+t))/((1+t)*(1-t+t^3))" "6#/%!G**\"\"$\"\"\"%\"tGF',&F 'F'F(F'F'*&,&F'F'F(F'F',(F'F'F(!\"\"*$F(F&F'F'F-" }{XPPEDIT 18 0 "``=3 *t/(1-t+t^2)" "6#/%!G*(\"\"$\"\"\"%\"tGF',(F'F'F(!\"\"*$F(\"\"#F'F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "which tends to " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 " t-> -1" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\",$\"\"\"!\"\"F*F*F* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Hence the line " } {XPPEDIT 18 0 "x+y=-1" "6#/,&%\"xG\"\"\"%\"yGF&,$F&!\"\"" }{TEXT -1 32 " is an asymptote for the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "[Limit(3*t/(1+t^3),t=-1,le ft),Limit(3*t^2/(1+t^3),t=-1,left)];\nmap(value,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$-%&LimitG6%,$*(\"\"$\"\"\"%\"tGF*,&F*F**$)F+F)F*F*! \"\"F*/F+F/%%leftG-F%6%,$*(F)F*F+\"\"#F,F/F*F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%)infinityG,$F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "[Limit(3*t/(1+t^3),t=-1, right),Limit(3*t^2/(1+t^3),t=-1,right)];\nmap(value,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$-%&LimitG6%,$*(\"\"$\"\"\"%\"tGF*,&F*F**$)F+F) F*F*!\"\"F*/F+F/%&rightG-F%6%,$*(F)F*F+\"\"#F,F/F*F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$%)infinityG!\"\"F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Limit(3*t/(1+t^3)+ 3*t^2/(1+t^3),t=-1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%& LimitG6$,&*(\"\"$\"\"\"%\"tGF),&F)F)*$)F*F(F)F)!\"\"F)*(F(F)F*\"\"#F+F .F)/F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "If we use the parametric description to draw the curve, it is necessary to draw the graph in t wo pieces, since we want to avoid values of " }{TEXT 267 1 "t" }{TEXT -1 10 " close to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot([[3*t/(1+t^3),3*t^2/(1+t^3),t=-0.67..50],\n \+ [3*t/(1+t^3),3*t^2/(1+t^3),t=-50..-1.5]],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 208 200 200 {PLOTDATA 2 "6'-%'CURVESG6$7ip7$$!3+c$QQ%=c uG!#<$\"3Z)pr`Vcf#>F*7$$!3wYbjQJjkDF*$\"3(Gn5'[tyH;F*7$$!3;W[\">+?FI#F *$\"37'*3me#oQQ\"F*7$$!3mH['Q?io2#F*$\"3q!**>I__k<\"F*7$$!3w,bs\\?fy=F *$\"3GxIY$GDI***!#=7$$!3;.aEOVt,<(Gd\\FA7$$!3kF]![T#eI6F*$\"3/IV[h`(H1%FA7$$!3'3N<%GO;45F*$ \"3#=ZqzVD$yKFA7$$!3HK&G98r(G*)FA$\"3[Q,3A1R#f#FA7$$!3bc*fWg$\\0yFA$\" 3)RKI;kao*>FA7$$!3vrlyU^qScFA$\"3A9csBAn`5FA7$$!3#*eaxeL&)QNFA$\"3cXk> AdnnT!#>7$$!3IIX'HUwBY\"FA$\"3&4#))H$Gdw7(!#?7$$\"3KRNutp`'3'F^p$\"3_m b!Rsu[B\"Fdp7$$\"3'R[L$[nhxEFA$\"3mlYSM+e\"R#F^p7$$\"3#3)ep4YgJZFA$\"3 1/;b`KK#\\(F^p7$$\"3OV.EUH+UnFA$\"31AOdgV'H`\"FA7$$\"3)zEGV9Ukm)FA$\"3 OMKlz(o(oDFA7$$\"3/S'yJO)GX5F*$\"3=)oLo)z#)>QFA7$$\"3$HV()f(*)f/7F*$\" 3(=IJMTANB&FA7$$\"3*flbEj3NK\"F*$\"3Q:(=n\\g]a'FA7$$\"3r&z'>)p>8U\"F*$ \"3-A7LHrU#)yFA7$$\"3)z()o(\\BX'\\\"F*$\"3ki__*\\&*y>*FA7$$\"3oKF\\AW] [:F*$\"3LG.A3*)yW5F*7$$\"3-&pC\\YN#y:F*$\"3+i()p3,kf6F*7$$\"3Ib^%4$yR( e\"F*$\"3+s;SkSrh7F*7$$\"3eK/t-iYy:F*$\"3')=K**)>?%\\8F*7$$\"3Ue^X4'HV b\"F*$\"34)4**)y:9A9F*7$$\"3Xm!)R/G+=:F*$\"3c@O+Lz2![\"F*7$$\"3OLuHPWR s9F*$\"3p60))[M/C:F*7$$\"3Dh.LJu9?9F*$\"3)*)H!\\*4g_b\"F*7$$\"31Xa3V$ \\NO\"F*$\"3CEC,\"Hu^d\"F*7$$\"3Cfvw;(e\"F*7$$\"39#)[U0by%=\"F*$\"3WHKRGl9#e\"F*7$$\"3 ^&3@\"e@k!p$o*)))FA$\"3[9P&)RGp![\"F*7 $$\"3Iq:yz*Gd.(FA$\"33e'zA]&)>O\"F*7$$\"3wza^o()QRcFA$\"3=J>X@l/W7F*7$ $\"3%R+y/(=X!f%FA$\"3n(oGWYxm8\"F*7$$\"3k#z7Iv*o*y$FA$\"3')3ae$*RYT5F* 7$$\"3g#)Rc5H%Q<$FA$\"3GPtpd/S&e*FA7$$\"3YYhZ\"o&f#p#FA$\"3]&\\b9@)Gk) )FA7$$\"3N*4I)>=o5BFA$\"3R\\y(4JMaB)FA7$$\"3eukt(=vXv\"FA$\"3i!)=rG+H. sFA7$$\"3[;(R0`[cP\"FA$\"3MjR(oBbBR'FA7$$\"3h7vl\"*GXB6FA$\"3yg]W/yK%y &FA7$$\"3jY\"z`\"*>RM*F^p$\"3%*\\iZy3*)z_FA7$$\"3WB5mH8l!o'F^p$\"3;D8K dpOpWFA7$$\"3!**\\]SJd\\'\\F^p$\"3ogMqiCFbQFA7$$\"3Bv^?nr\\OQF^p$\"3c) H,UO3,R$FA7$$\"3vk=H#enS.$F^p$\"3zG-\\s-X:IFA7$$\"3vh+IDdt=DF^p$\"3IIb 4jwzZFFA7$$\"3)z&eOpeK\"3#F^p$\"3%oLY>$H2)\\#FA7$$\"3GcQ2`jXZFA7$$\"3&H)**H%=4X8\"F^p$\"3u_*>'H1lW=FA7$$\"3ozSOP*32,\"F^p $\"31mpzT+8TJX%f!*)Fdp$\"37uusioUM;FA7$$\"3M[f&*G2x7!)Fdp $\"3/$*>MZBK]:FA7$$\"3m/)e(Ga0zrFdp$\"3m6K'otouY\"FA7$$\"3cud7#)*e/]'F dp$\"3#>qbT(GS'R\"FA7$$\"3IwwgQB2!*eFdp$\"3$\\%Fdp$\"3j.Y6%3#HA6FA7$$\"3%=W! [V;A()QFdp$\"3'=C#[/n')z5FA7$$\"3/6^GrN[+OFdp$\"3RxMAt'y#R5FA7$$\"3kdO QKBg\\LFdp$\"3A#o)>z'=C+\"FA7$$\"3P%>1.Z858$Fdp$\"3OXznClg\"p*F^p7$$\" 3fF%o(*3oC\"HFdp$\"35VzEjpEZ$*F^p7$$\"3#=E<$*HR]t#Fdp$\"3A=T<2f3e!*F^p 7$$\"3-.Uc7k\"Hc#Fdp$\"3oGwU\\=Wo()F^p7$$\"3a`+)[%Hg?CFdp$\"34a10\"4F: _)F^p7$$\"3A%3?$[\"o\"yAFdp$\"37!\\J#p'>qE)F^p7$$\"3K1MI/TKb@Fdp$\"3j0 RYc&\\5/)F^p7$$\"3^hm)4!=CP?Fdp$\"3zk=t=LoFdp$\"3 C90d;8)4h(F^p7$$\"3rOMwQM,TvOqF^p7$$\"3?2>+sO-r:Fdp$ \"3OeqjE'F^p7$$\"3m*o4a5LYD\"Fdp$\"3DtaNpj.NhF^p7$$\"3c*zw+ S!***>\"Fdp$\"33(*RQ+?&***fF^p-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F^ilF] il-F$6$7`o7$$\"3o+o2+'4+?\"Fdp$!3V.SQ+![++'F^p7$$\"3y1c7)Q5CD\"Fdp$!3* o(\\Uw=lHhF^p7$$\"3%z\"z5kx%3I\"Fdp$!3mDuFL@1ZiF^p7$$\"3#yGc5*\\ve8Fdp $!3Q$*[+kTf%Q'F^p7$$\"3Y)R[+xS5U\"Fdp$!3tg\\jcEHHlF^p7$$\"3S1Ui(y\"Q([ \"Fdp$!3!*y*))[.m*zmF^p7$$\"3gKSDcy7`:Fdp$!3wlFT]z+EoF^p7$$\"3fCr\"*zk !fi\"Fdp$!3+,RuS76%)pF^p7$$\"3OoS8=,p12qI'*3)>Fdp$!3j)>5L=\\*3xF^p7$$\"3M!*[JMe%=4#Fdp$!3o$Gs(\\>!>#zF^p7 $$\"3-xmpl5!H@#Fdp$!31KF.r`!z9)F^p7$$\"38*))e`.%yRBFdp$!3_1cF?'[#y$)F^ p7$$\"3EV$)zbTqkCFdp$!3'oB=;x.!*f)F^p7$$\"3=I*f(**frEEFdp$!3AC'z9wMr() )F^p7$$\"3eiqiy[zwFFdp$!32/K'yeAs7*F^p7$$\"3I<0PPq8oHFdp$!3phvQabYO%*F ^p7$$\"3'>Tuy)R]aJFdp$!33\"o,L@C#G(*F^p7$$\"3y!)GR?C\"*zLFdp$!3ytC8j;) p+\"FA7$$\"3mko5*R(y1$*>$>6FA7$$\"3TNbf=LK;XFdp$!3-(oe9 UMS;\"FA7$$\"3j69A[oy:\\Fdp$!3N)QhNQFW@\"FA7$$\"3R&)pYtnI3`Fdp$!3Qq5I] n)>E\"FA7$$\"3ipUB*z\"[(y&Fdp$!3/JAuW@s<8FA7$$\"3I;GXrp,ajFdp$!3+X?.[6 s!Q\"FA7$$\"3I\"*y')G+x#*pFdp$!3c=^M*fq%[9FA7$$\"3$e<(>N>52xFdp$!3Fkks _\"o1_\"FA7$$\"3K*fQyNPkj)Fdp$!3%>))3!z7w4;FA7$$\"3q[biiQuA'*Fdp$!3c%R kco>#*p\"FA7$$\"3/Xw?>#pu3\"F^p$!34Lga!p5k!=FA7$$\"3bF&RR7)GB7F^p$!3) \\)G-k$Qf\">FA7$$\"3g[lSY:3.9F^p$!3/U<3-C(>0#FA7$$\"3Y?\"=lZv5h\"F^p$! 3)QF'fzC*))>#FA7$$\"3'zf?#4$Q@)=F^p$!3KL(Qq!)3oP#FA7$$\"3x_EN\\f;>AF^p $!3cn]3WJ.\"e#FA7$$\"3C#o1>Q$=zEF^p$!3AT`@DHDOGFA7$$\"3kF-%=ioAF$F^p$! 3RExKQ8'\\8$FA7$$\"3)4X1l'FA7$$\"31/F)*ff0g=FA$!3oHw6 A81FvFA7$$\"39TwDC*prX#FA$!3!Rom%o)FA7$$\"3s8Q>dR+`GFA$!3k6.JX2G% Q*FA7$$\"3yu5$yi'4]&FA$!3q;!4)>dP K8F*7$$\"3![&H05SV:F* 7$$\"3EW8t&RL))H)FA$!3-'G8.\"=4#o\"F*7$$\"3O$)oq&)z]$**)FA$!3cr!*GX=mj F*7$$\"3+D'QhZ`)z6F*$!3-Q!3[^813#F*7$$\"3L6(4B>#HT7F*$!3=fflV-3[@F*7$ $\"3s_4Mp8[38F*$!35js3nBE@AF*7$$\"3%*oZt62E#Q\"F*$!3$Ql4Of*)4I#F*7$$\" 3e4h\\hMkj9F*$!3%>%=)o=k#)Q#F*7$$\"3;:j7mR'Qb\"F*$!3')=6-V%3V[#F*7$$\" 3euZRT=Wa;F*$!31(3'=a(G1f#F*7$$\"3s\"\\ea!*psw\"F*$!3Ww_?MM54FF*7$$\"3 /j_5Uot%*=F*$!3p%*y:j_5UGF*-Fghl6&FihlF]ilFjhlF]il-%*THICKNESSG6#\"\"# -%+AXESLABELSG6$Q!6\"F__n-%%VIEWG6$%(DEFAULTGFd_n" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "elimi nate(\{x=3*t/(1+t^3),y=3*t^2/(1+t^3)\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<#/%\"tG,$*&*$)%\"xG\"\"#\"\"\"F-,&F+!\"$*$)%\"yGF,F- F-!\"\"F3<#,(*$)F+\"\"$F-F-*&F2F-F+F-F/*$)F2F8F-F-" }}}{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 257 "" 0 "" {TEXT 258 8 "Question" } {TEXT 335 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 78 "This question is conce rned with the curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=t^2,``],[y=t^3-2*t ,``])" "6#-%*PIECEWISEG6$7$/%\"xG*$%\"tG\"\"#%!G7$/%\"yG,&*$F*\"\"$\" \"\"*&F+F3F*F3!\"\"F," }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%# dxG!\"\"" }{TEXT -1 27 " in terms of the parameter " }{TEXT 345 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Find the equations of the two tangent lines at the point of self-intersection of the cur ve. " }}{PARA 0 "" 0 "" {TEXT -1 54 "(c) Show that the cartesian equat ion of the curve is " }{XPPEDIT 18 0 "y^2=x*(x-2)^2" "6#/*$%\"yG\"\"# *&%\"xG\"\"\"*$,&F(F)F&!\"\"F&F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(d) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#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 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 336 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " The following animation gives the location of the point" }{XPPEDIT 18 0 "``(t^2,t^3-2*t);" "6#-%!G6$*$%\"tG\"\"#,&*$F'\"\"$\"\"\"*&F(F,F'F,! \"\"" }{TEXT -1 18 " as the parameter " }{TEXT 379 1 "t" }{TEXT -1 25 " increases from -2 to 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 597 "rng := -2..2: # parameter range\np t := [2,3]: # location to display values of parameter\nt := 't':\nf := t ->t^2;\ng := t -> t^3-2*t;\np1 := plot([f(t),g(t),t=rng],labels=[`x `,`y`]):\nnumframes := 161:\nfrms := []:\naa := op(1,rng): bb := op(2, rng):\nhh := evalf(abs(bb-aa)/(numframes-1)):\nfor i from 0 to numfram es-1 do \n tt := aa+hh*i;\n p2 := plot([[[f(tt),g(tt)]]$3],style=p oint,symbol=[circle,diamond,cross],\n color=black);\n \+ t1 := plots[textplot]([op(1,pt),op(2,pt),t=tt],color=black);\n frms := [op(frms),plots[display]([p1,p2,t1])];\nend do:\nplots[display](fr ms,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/dt = 2* t ,``],[dy/dt = 3*t^2-2 ,``])" "6#-%*PIECEWISEG6$7$/*&%#dxG\"\"\"%#dtG !\"\"*&\"\"#F*%\"tGF*%!G7$/*&%#dyGF*F+F,,&*&\"\"$F**$F/F.F*F*F.F,F0" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/dt)/``(dx/dt);" "6#/*&%#d yG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(3*t^2-2)/(2*t)" "6#*&,&*&\"\"$\"\"\"*$%\"tG\" \"#F'F'F*!\"\"F'*&F*F'F)F'F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "(b) If the curve has a point of self-intersection, it must be given by two distinct values of " } {TEXT 346 1 "t" }{TEXT -1 6 ", say " }{XPPEDIT 18 0 "t[1];" "6#&%\"tG6 #\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t[2]" "6#&%\"tG6#\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "In order to obtain the \+ same " }{TEXT 347 1 "x" }{TEXT -1 33 " coordinate at the two value of \+ " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "t=t[2]" "6#/%\"tG&F$6#\"\"#" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "t[2]^2 = t[1]^2;" "6#/*$&%\"tG6#\"\"#F(*$&F&6#\"\"\"F( " }{TEXT -1 9 " , so if " }{XPPEDIT 18 0 "t[1]<>t[2]" "6#0&%\"tG6#\"\" \"&F%6#\"\"#" }{TEXT -1 15 ", we must have " }{XPPEDIT 18 0 "t[2]=-t[1 ]" "6#/&%\"tG6#\"\"#,$&F%6#\"\"\"!\"\"" }{TEXT -1 29 ", that is, the t wo values of " }{TEXT 348 1 "t" }{TEXT -1 61 " must have the same magn itude, but opposite signs. Since the " }{TEXT 349 1 "y" }{TEXT -1 42 " coordinates must also coincide, we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[1]^3-2*t[1]=t[2]^3-2*t[2]" "6#/,&*$&%\"tG6# \"\"\"\"\"$F)*&\"\"#F)&F'6#F)F)!\"\",&*$&F'6#F,F*F)*&F,F)&F'6#F,F)F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[1]^3-2*t[2]=-t[1]^3+2*t[1]" "6 #/,&*$&%\"tG6#\"\"\"\"\"$F)*&\"\"#F)&F'6#F,F)!\"\",&*$&F'6#F)F*F/*&F,F )&F'6#F)F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gi ves: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*t[1]^3-4*t [1]=0" "6#/,&*&\"\"#\"\"\"*$&%\"tG6#F'\"\"$F'F'*&\"\"%F'&F*6#F'F'!\"\" \"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[1]*(t[1]^2-2)=0" "6#/*&&% \"tG6#\"\"\"F(,&*$&F&6#F(\"\"#F(F-!\"\"F(\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "t[1]=0" "6#/&%\"tG6#\"\"\"\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "t[1]=sqrt(2)" "6#/&%\"tG6#\"\"\"-%%sqrtG6#\"\"#" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "t[1]=-sqrt(2)" "6#/&%\"tG6#\"\"\",$- %%sqrtG6#\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "T he two values of " }{TEXT 356 1 "t" }{TEXT -1 43 " giving the point of self-intersection are " }{XPPEDIT 18 0 "t= ``" "6#/%\"tG%!G" }{TEXT 350 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"# " }{TEXT -1 59 ", and the coordinates of the point of self-intersectio n are" }{XPPEDIT 18 0 "``(2,0)" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "t=sqrt(2)" "6#/%\"tG-%%sqrtG6#\"\"#" }{TEXT -1 52 ", th e gradient of the corresponding tangent line is " }{XPPEDIT 18 0 "dy/d x = (6-2)/(2*sqrt(2));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&\"\"'F&\"\"#F(F &*&F,F&-%%sqrtG6#F,F&F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(2);" " 6#-%%sqrtG6#\"\"#" }{TEXT -1 11 ", and when " }{XPPEDIT 18 0 "t = -sqr t(2);" "6#/%\"tG,$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 52 ", the gradient o f the corresponding tangent line is " }{XPPEDIT 18 0 "dy/dx = -sqrt(2) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$-%%sqrtG6#\"\"#F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 26 "The two tangent lines are " }{XPPEDIT 18 0 "y=sqrt(2)*(x-2)" "6#/%\"yG*&-%%sqrtG6#\"\"#\"\"\",&%\"xGF*F)!\"\"F* " }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "y=-sqrt(2)*(x-2)" "6#/%\"yG,$*& -%%sqrtG6#\"\"#\"\"\",&%\"xGF+F*!\"\"F+F." }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot ([[t^2,t^3-2*t,t=-2..2],sqrt(2)*(x-2),-sqrt(2)*(x-2)],\n x=0..4,colo r=[red,green,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 293 254 254 {PLOTDATA 2 "6'-%'CURVESG6$7fn7$$\"\"%\"\"!$!\"%F*7$$\"1HdQIK_FQ!#:$!1 $pz&\\oPvNF07$$\"1!e4#)QZ)eOF0$!1a+Qn@1tJF07$$\"1WC;(zv^^$F0$!1qo([6t2 %GF07$$\"1\\e7aKF0$!1XE'o'e5)=#F07 $$\"1_xvy7AoIF0$!1Z36#fK6(=F07$$\"1#*G?fOj>HF0$!1E$3\\5s8d\"F07$$\"1dy Q%yLZx#F0$!1Ocp%H:0H\"F07$$\"1Bb'*)3%=MEF0$!1kxGR:HH5F07$$\"1DF!#<7$$\"1O(y=p!*)zun#Rm#RvFjn7$$\"1lK?Jn;z6F0$\"1TM&Q \"GQ8*)Fjn7$$\"1r\"))\\%))R#***Fjn$\"1y%elpz.+\"F07$$\"1D5!QdEbL)Fjn$ \"1]#3T)f&\\1\"F07$$\"1p4Oxt$3)oFjn$\"1?rU0_C)3\"F07$$\"1Bt(zL,1o&Fjn$ \"1[=(*)p\\#z5F07$$\"1@,BXx+.WFjn$\"1))Hxt4%\\.\"F07$$\"1$[s$3d(yW$Fjn $\"1$\\[>3y\">(*Fjn7$$\"1.`jPjd$[#Fjn$\"1XqJWgRH()Fjn7$$\"15!*RKWoh6\"Fjn$\"1/qi`gI)H'Fjn7$$\"1H$*>9#z`J'Fe o$\"19)R*GbPn[Fjn7$$\"1sw$4j*)>u#Feo$\"1g\"4FN'QmKFjn7$$\"1a/+x'ova(!# =$\"17IS,y(4t\"Fjn7$$\"1CHSG34)*Q!#@$\"1H'Q/(*3([RFdt7$$\"1xsBc\")3LuF dt$!1_Ao>()*yr\"Fjn7$$\"1**GqVMScEFeo$!1%=M--+k@$Fjn7$$\"1Bo$=Oul/'Feo $!1OY(3os#pZFjn7$$\"1&>/'3\")G*4\"Fjn$!1*R^sD#pFjn$!1&3/!)*32)3\"F07$$\"13Cjqg!*=%)Fjn$!1q#3j g=E1\"F07$$\"1z\"p6+56'**Fjn$!1H(*=O]$>+\"F07$$\"1MgH-E@QK!*\\(Fjn7$$\"1&GUH\"\\/k:F0$!1qOCt>8_a Fjn7$$\"1uR@7\\UyWrR?$F0$\"1)3ggkl]:#F07$$\"1\"*)RI u/qN$F0$\"1#>U'H=K'[#F07$$\"1[D-%z?o]$F0$\"17P(\\2\\<#GF07$$\"1')3\"\\ D2*fOF0$\"1k^*oK\\b<$F07$$\"1AxA@_1GQF0$\"1g+n7/pwNF07$F(F(-%'COLOURG6 &%$RGBG$\"*++++\"!\")F*F*-F$6$7S7$F*$!1!>YZ7F%GGF07$$\"1mmmm;')=()Feo$ !1T;d+Q70FF07$$\"1LLLe'40j\"Fjn$!1em]P#Qyf#F07$$\"1nmm;6m$[#Fjn$!1?L>, W=xCF07$$\"1nmm;yYULFjn$!1KO$HzIdN#F07$$\"1LLLeF>(>%Fjn$!1wKZKW&[B#F07 $$\"1mmm\">K'*)\\Fjn$!1^`2tlyA@F07$$\"1*****\\Kd,\"eFjn$!1_D*ezYn+#F07 $$\"1mmm\"fX(emFjn$!1%[LBHQn)=F07$$\"1*****\\U7Y](Fjn$!1EtcdY6nk'oN:F07$$\"1+++ I,Q+5F0$!1I;Pxfn89F07$$\"1+++]*3q3\"F0$!1`_BDV;\"H\"F07$$\"1+++q=\\q6F 0$!1WZaZ;5t6F07$$\"1nm;fBIY7F0$!11^fD%*)e1\"F07$$\"1LLLj$[kL\"F0$!1W') 3Rs.%Q*Fjn7$$\"1LLL`Q\"GT\"F0$!1wf`Ah1/$)Fjn7$$\"1++]s]k,:F0$!12'fM(H! y/(Fjn7$$\"1LLL`dF!e\"F0$!11$G4A(zNfFjn7$$\"1++]sgam;F0$!1Z2(pm]dr%Fjn 7$$\"1++]F0$!1xJb)pA'G7Fjn7$$\"1nmmTc-)*>F0$!1#R%)>Lk@z#Fdt7$$\"1mm;f`@ '3#F0$\"1rO=-$p#>7Fjn7$$\"1++]nZ)H;#F0$\"1ase'o_\\I#Fjn7$$\"1mmmJy*eC# F0$\"1h8,&[?vZ$Fjn7$$\"1+++S^bJBF0$\"1,^icx*))o%Fjn7$$\"1+++0TN:CF0$\" 1!Q$y%3%*R(eFjn7$$\"1++]7RV'\\#F0$\"1zVz=dj?qFjn7$$\"1+++:#fke#F0$\"1% y;jv&y$H)Fjn7$$\"1LLL`4NnEF0$\"1Thm!pnxV*Fjn7$$\"1+++],s`FF0$\"1%)QOe7 #f1\"F07$$\"1mm;zM)>$GF0$\"1`y!*>Bgw6F07$$\"1+++qfad:bq#F07$F($\"1!>YZ7F%GGF0-Ff]l6&Fh]lF*Fi]lF* -F$6$7S7$F*F^]m7$Fc^l$\"1T;d+Q70FF07$Fh^l$\"1em]P#Qyf#F07$F]_l$\"1?L>, W=xCF07$Fb_l$\"1KO$HzIdN#F07$Fg_l$\"1wKZKW&[B#F07$F\\`l$\"1^`2tlyA@F07 $Fa`l$\"1_D*ezYn+#F07$Ff`l$\"1%[LBHQn)=F07$F[al$\"1EtcdY6nk'oN:F07$Fjal$\"1I;Pxfn89F07$F_bl$\"1`_BDV;\" H\"F07$Fdbl$\"1WZaZ;5t6F07$Fibl$\"11^fD%*)e1\"F07$F^cl$\"1W')3Rs.%Q*Fj n7$Fccl$\"1wf`Ah1/$)Fjn7$Fhcl$\"12'fM(H!y/(Fjn7$F]dl$\"11$G4A(zNfFjn7$ Fbdl$\"1Z2(pm]dr%Fjn7$Fgdl$\"1MesJy(Rb$Fjn7$F\\el$\"1Z97c(*yTBFjn7$Fae l$\"1xJb)pA'G7Fjn7$Ffel$\"1#R%)>Lk@z#Fdt7$F[fl$!1rO=-$p#>7Fjn7$F`fl$!1 ase'o_\\I#Fjn7$Fefl$!1h8,&[?vZ$Fjn7$Fjfl$!1,^icx*))o%Fjn7$F_gl$!1!Q$y% 3%*R(eFjn7$Fdgl$!1zVz=dj?qFjn7$Figl$!1%y;jv&y$H)Fjn7$F^hl$!1Thm!pnxV*F jn7$Fchl$!1%)QOe7#f1\"F07$Fhhl$!1`y!*>Bgw6F07$F]il$!1xZ([&fg(H\"F07$Fb il$!1r[MX4Y69F07$Fgil$!1J/@;AZI:F07$F\\jl$!1E8w'[Fok\"F07$Fajl$!1sGyr1 kod:bq#F07$F(F`^l-Ff]l6&Fh]lF*F*Fi]l-%+AXESLABELS G6$Q\"x6\"%!G-%%VIEWG6$;F*F(%(DEFAULTG" 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" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(c) Since " }{XPPEDIT 18 0 "y = t^3-2*t" "6#/%\"yG,&*$%\"tG\"\"$\"\"\"*&\"\"#F)F'F)!\"\"" }{XPPEDIT 18 0 "``=t*(t^2-2)" "6#/%!G*&%\"tG\"\"\",&*$F&\"\"#F'F*!\"\"F'" } {TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "y^2=t^2*(t^2-2)^2" "6 #/*$%\"yG\"\"#*&%\"tGF&,&*$F(F&\"\"\"F&!\"\"F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "x=t^2" "6# /%\"xG*$%\"tG\"\"#" }{TEXT -1 29 " in the last equation gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2=x*(x-2)^2" "6#/*$ %\"yG\"\"#*&%\"xG\"\"\"*$,&F(F)F&!\"\"F&F)" }{TEXT -1 15 " ------- (i) . " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 76 ": This means t hat the curve can be drawn by fitting together the graphs of " } {XPPEDIT 18 0 "y=(x-2)*sqrt(x)" "6#/%\"yG*&,&%\"xG\"\"\"\"\"#!\"\"F(-% %sqrtG6#F'F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "(2-x)*sqrt(x)" "6#*& ,&\"\"#\"\"\"%\"xG!\"\"F&-%%sqrtG6#F'F&" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot( [(x-2)*sqrt(x),(2-x)*sqrt(x)],x=0..4,color=red,numpoints=100);" }} {PARA 13 "" 1 "" {GLPLOT2D 332 237 237 {PLOTDATA 2 "6'-%'CURVESG6#7`q7 $$\"\"!F)F(7$$\"3(*oooooKFU!#>$!3Nb(*HJq/7F0$!3mB\"oc#*QC_'F07$$\"3wcccc/f?;F0$!39Oz[] w\"*)R(F07$$\"3Z\\\\\\\\D+N?F0$!3?#Gw[X!>/\")F07$$\"3/3333c@>CF0$!3W0< .N1?Z')F07$$\"34wvvvf/\"*F07$$\"3S6666F[GKF0$!38%G4sI K&H&*F07$$\"3ijjjj**fQOF0$!3g.[\\=)G$p)*F07$$\"3?onnnfXgSF0$!3@I%Gw&\\ p:5!#<7$$\"3%eccc'p-KWF0$!3>nE_#*RTO5Fhn7$$\"3q....zK][F0$!3%QoZ5C)3b5 Fhn7$$\"3sCCCCkMq_F0$!35u6/%*3Lp5Fhn7$$\"3w(ppp47^n&F0$!33$y$4\\;9z5Fh n7$$\"3<0000\"yE/'F0$!3#pXC%Qx'\\3\"Fhn7$$\"37TSSS'\\(zkF0$!3/4NeM\"Q$ )3\"Fhn7$$\"31ihhhl+]oF0$!3M+lVNhN)3\"Fhn7$$\"37#===Q.2G(F0$!37*>&[.** H&3\"Fhn7$$\"3_YYYYI%>m(F0$!3)*o@u'G\")*z5Fhn7$$\"3Y$===QB-3)F0$!3c.a@ o%p92\"Fhn7$$\"3Yxvvv^_y%)F0$!3O;$)o7[)31\"Fhn7$$\"3K988886%*))F0$!3o< ;Pr2QZ5Fhn7$$\"3e?>>>\"\\dF*F0$!3E/!R#3+'G.\"Fhn7$$\"33^]]]qR(o*F0$!3 \\yK)pc8],\"Fhn7$$\"3.'fffN)\\65Fhn$!3Q4H\"os%oT**F07$$\"3CFFF())>([5F hn$!3D?3iwmxT(*F07$$\"3'4333;?*)3\"Fhn$!3DF\"=e'*Rs]*F07$$\"35(ppp(4XI 6Fhn$!3o@%4XAx_C*F07$$\"3A===y23r6Fhn$!3yL1=8DFq*)F07$$\"3.++++AR57Fhn $!3U#o68M$4(o)F07$$\"3m^^^J3/a7Fhn$!3_x5a2la`$)F07$$\"3GGGGo5E$H\"Fhn$ !3%[R\"\\73:P!)F07$$\"3eUUUUq8N8Fhn$!3yk@ynrO#o(F07$$\"35**)*)*)*G3t8F hn$!3y***ej2GhM(F07$$\"3>...Vxc99Fhn$!3#o7jIztG'pF07$$\"3GGGG[pyswe'F07$$\"31\"444`.W\\\"Fhn$!3wDHJ?Wq!='F07$$\"3VJJJJZHM:Fhn$ !3IMc;9_`odF07$$\"3E@@@@r0w:Fhn$!3v&Ql>6EAK&F07$$\"3TEEEm#zih\"Fhn$!3A 48&Qub$y[F07$$\"3A**)*)*=ATd;Fhn$!3Gq*H/+$\\5WF07$$\"3W555qX?)p\"Fhn$! 3wc`RED&G$RF07$$\"3QCCC/&)oN===qU!e=Fhn$!3# =GU+]?]$>F07$$\"3sXXX&)\\D(*=Fhn$!36+6il)=_T\"F07$$\"3U222(=&zS>Fhn$!3 q)y1X3mzC)F-7$$\"3cUUU-y\\y>Fhn$!3@VOfB%zW-$F-7$$\"3S33331H@?Fhn$\"3eI wzl/$p-$F-7$$\"3%\\[[[g\"Hg?Fhn$\"3UG&)yyc3a')F-7$$\"3a111mD&H5#Fhn$\" 3kD**=kG(H\\\"F07$$\"3+\"333wh(R@Fhn$\"3$p,-%*[CW/#F07$$\"3S(ppp8$o\"= #Fhn$\"3cms$>taNo#F07$$\"3:xww;rJ$F07$$\"3#HGGGkWE E#Fhn$\"3#>5Q=HC2&RF07$$\"3@;;;'HjHI#Fhn$\"31mQ-&yCwf%F07$$\"3!*zzz**o pTBFhn$\"3TKlbaB&)G_F07$$\"35666^*oNQ#Fhn$\"3,<56m'[=#fF07$$\"3?777_gZ BCFhn$\"3KxP)z!4Z#f'F07$$\"3;xww'RwaY#Fhn$\"376)3]$Q$)3tF07$$\"3$GEEE# o\\.DFhn$\"3CL%y[X\\l'zF07$$\"3'4222^;ba#Fhn$\"3$3(pIz&QNq)F07$$\"3#)p pp\\Sw&e#Fhn$\"3;`bW[\"o#>%*F07$$\"3I000DO\"fi#Fhn$\"3&y)**QPCF95Fhn7$ $\"3_7777+&ym#Fhn$\"3#[@MUXO34\"Fhn7$$\"3ibbbb\\Z1FFhn$\"3M`?k=zCi6Fhn 7$$\"3;,,,\"=Kgu#Fhn$\"3Nx7%3ehiB\"Fhn7$$\"3c===e!3(*y#Fhn$\"3x'))H-]. !>8Fhn7$$\"3jZZZ()HDHGFhn$\"37Ju,@`$[R\"Fhn7$$\"3C&[[[3&ppGFhn$\"3KGrl S'yKZ\"Fhn7$$\"3w;;;cO#3\"HFhn$\"3Sc8nP'oRb\"Fhn7$$\"37dcc;)G'[HFhn$\" 3wyFR$zW*G;Fhn7$$\"3'QLLL*y#*))HFhn$\"3*4K)o\"e4(4Fhn7$$\"3)*))))))3s+`JFhn$\"3KCA-p8OZ?Fhn7$$\"3 Y<<<(z8B>$Fhn$\"38'Q3Fq7.8#Fhn7$$\"38srr6c?JKFhn$\"3sFa&GOgJ@#Fhn7$$\" 3SYYY1o-tKFhn$\"3m6o%>`(4.BFhn7$$\"3(olll<<]J$Fhn$\"3cvP*zdwUR#Fhn7$$ \"3!efff>AKN$Fhn$\"3U2;5hT*zZ#Fhn7$$\"3!*\\\\\\HUz$R$Fhn$\"3'[E;#eIonD Fhn7$$\"3@*)))))o&)QLMFhn$\"399[^fP)fl#Fhn7$$\"3e]]]]#4jZ$Fhn$\"3-fy\" oSeDv#Fhn7$$\"3\"RLLLdZM^$Fhn$\"3X:r#\\GNo$GFhn7$$\"3[FFF(G-hb$Fhn$\"3 =fD0)>SW$HFhn7$$\"3-\"333'**>'f$Fhn$\"3M)=YCjxp-$Fhn7$$\"3([XXXL5fj$Fh n$\"3i(Hq;;k$>JFhn7$$\"3E===)H_cn$Fhn$\"3+`7GQ!fD@$Fhn7$$\"3sXXX&e'=;P Fhn$\"3j/\")F07$FF$\"3W0<.N1?Z')F07$FK$\"3qca#=5***>\"*F07$FP$\"38% G4sIK&H&*F07$FU$\"3g.[\\=)G$p)*F07$FZ$\"3@I%Gw&\\p:5Fhn7$Fjn$\"3>nE_#* RTO5Fhn7$F_o$\"3%QoZ5C)3b5Fhn7$Fdo$\"35u6/%*3Lp5Fhn7$Fio$\"33$y$4\\;9z 5Fhn7$F^p$\"3#pXC%Qx'\\3\"Fhn7$Fcp$\"3/4NeM\"Q$)3\"Fhn7$Fhp$\"3M+lVNhN )3\"Fhn7$F]q$\"37*>&[.**H&3\"Fhn7$Fbq$\"3)*o@u'G\")*z5Fhn7$Fgq$\"3c.a@ o%p92\"Fhn7$F\\r$\"3O;$)o7[)31\"Fhn7$Far$\"3o<;Pr2QZ5Fhn7$Ffr$\"3E/!R# 3+'G.\"Fhn7$F[s$\"3\\yK)pc8],\"Fhn7$F`s$\"3Q4H\"os%oT**F07$Fes$\"3D?3i wmxT(*F07$Fjs$\"3DF\"=e'*Rs]*F07$F_t$\"3o@%4XAx_C*F07$Fdt$\"3yL1=8DFq* )F07$Fit$\"3U#o68M$4(o)F07$F^u$\"3_x5a2la`$)F07$Fcu$\"3%[R\"\\73:P!)F0 7$Fhu$\"3yk@ynrO#o(F07$F]v$\"3y***ej2GhM(F07$Fbv$\"3#o7jIztG'pF07$Fgv$ \"3Yo%>pyswe'F07$F\\w$\"3wDHJ?Wq!='F07$Faw$\"3IMc;9_`odF07$Ffw$\"3v&Ql >6EAK&F07$F[x$\"3A48&Qub$y[F07$F`x$\"3Gq*H/+$\\5WF07$Fex$\"3wc`RED&G$R F07$Fjx$\"3k+iAYe=#[$F07$F_y$\"3gLDs7d1_HF07$Fdy$\"3S6*Q%QA#eY#F07$Fiy $\"3#=GU+]?]$>F07$F^z$\"36+6il)=_T\"F07$Fcz$\"3q)y1X3mzC)F-7$Fhz$\"3@V OfB%zW-$F-7$F][l$!3eIwzl/$p-$F-7$Fb[l$!3UG&)yyc3a')F-7$Fg[l$!3kD**=kG( H\\\"F07$F\\\\l$!3$p,-%*[CW/#F07$Fa\\l$!3cms$>taNo#F07$Ff\\l$!3&)=ah%H ()>J$F07$F[]l$!3#>5Q=HC2&RF07$F`]l$!31mQ-&yCwf%F07$Fe]l$!3TKlbaB&)G_F0 7$Fj]l$!3,<56m'[=#fF07$F_^l$!3KxP)z!4Z#f'F07$Fd^l$!376)3]$Q$)3tF07$Fi^ l$!3CL%y[X\\l'zF07$F^_l$!3$3(pIz&QNq)F07$Fc_l$!3;`bW[\"o#>%*F07$Fh_l$! 3&y)**QPCF95Fhn7$F]`l$!3#[@MUXO34\"Fhn7$Fb`l$!3M`?k=zCi6Fhn7$Fg`l$!3Nx 7%3ehiB\"Fhn7$F\\al$!3x'))H-].!>8Fhn7$Faal$!37Ju,@`$[R\"Fhn7$Ffal$!3KG rlS'yKZ\"Fhn7$F[bl$!3Sc8nP'oRb\"Fhn7$F`bl$!3wyFR$zW*G;Fhn7$Febl$!3*4K) o\"e4(4Fhn7$Ficl$!3KCA-p8OZ?Fhn7$F^dl$!38'Q3Fq7.8#Fhn7$Fcdl$!3sFa&GOgJ@ #Fhn7$Fhdl$!3m6o%>`(4.BFhn7$F]el$!3cvP*zdwUR#Fhn7$Fbel$!3U2;5hT*zZ#Fhn 7$Fgel$!3'[E;#eIonDFhn7$F\\fl$!399[^fP)fl#Fhn7$Fafl$!3-fy\"oSeDv#Fhn7$ Fffl$!3X:r#\\GNo$GFhn7$F[gl$!3=fD0)>SW$HFhn7$F`gl$!3M)=YCjxp-$Fhn7$Feg l$!3i(Hq;;k$>JFhn7$Fjgl$!3+`7GQ!fD@$Fhn7$F_hl$!3j " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "(d) The equation (i) can be written in the form: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2=x^3-4*x^2+4*x" "6 #/*$%\"yG\"\"#,(*$%\"xG\"\"$\"\"\"*&\"\"%F+*$F)F&F+!\"\"*&F-F+F)F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*y" "6#*&\"\"#\"\"\"%\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=3 *x^2-8*x+4" "6#/*&%#dyG\"\"\"%#dxG!\"\",(*&\"\"$F&*$%\"xG\"\"#F&F&*&\" \")F&F-F&F(\"\"%F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so \+ that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(3*x^2 -8*x+4)/(2*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,(*&\"\"$F&*$%\"xG\"\"#F&F &*&\"\")F&F.F&F(\"\"%F&F&*&F/F&%\"yGF&F(" }{TEXT -1 1 "," }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(3*x-2)*(x-2)/(2*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(,&*&\"\" $F&%\"xGF&F&\"\"#F(F&,&F-F&F.F(F&*&F.F&%\"yGF&F(" }{TEXT -1 15 " ----- -- (ii). " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 15 ": Substi tuting " }{XPPEDIT 18 0 "x=t^2" "6#/%\"xG*$%\"tG\"\"#" }{TEXT -1 6 " a nd " }{XPPEDIT 18 0 "y=t*(t^2-2)" "6#/%\"yG*&%\"tG\"\"\",&*$F&\"\"#F' F*!\"\"F'" }{TEXT -1 16 " in (ii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(3*t^2-2)*(t^2-2)/(2*t*(t^2-2))" "6#/*& %#dyG\"\"\"%#dxG!\"\"*(,&*&\"\"$F&*$%\"tG\"\"#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 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(3*t^2-2)/(2*t)" "6#/%!G*&,&*&\"\"$ \"\"\"*$%\"tG\"\"#F)F)F,!\"\"F)*&F,F)F+F)F-" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "t<>2" "6#0%\"tG\"\"#" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 56 "which may be compared with the formula obtained in (a). " }}{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 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 337 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 77 "This question is concerned with the curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 1-cos*t-sin*t, ``],[y = 1-cos*t+sin* t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,(\"\"\"F**&%$cosGF*%\"tGF*!\"\"*& %$sinGF*F-F*F.%!G7$/%\"yG,(F*F**&F,F*F-F*F.*&F0F*F-F*F*F1" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in te rms of the parameter " }{TEXT 351 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 91 "(b) Find the coordinates of the points on the curve \+ where the tangent lines are horizontal." }}{PARA 0 "" 0 "" {TEXT -1 89 "(c) Find the coordinates of the points on the curve where the tang ent lines are vertical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT 261 8 "Solution" }{TEXT 338 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following animation g ives the location of the point" }{XPPEDIT 18 0 "``(1-cos*t-sin*t,1-cos *t+sin*t);" "6#-%!G6$,(\"\"\"F'*&%$cosGF'%\"tGF'!\"\"*&%$sinGF'F*F'F+, (F'F'*&F)F'F*F'F+*&F-F'F*F'F'" }{TEXT -1 18 " as the parameter " } {TEXT 380 1 "t" }{TEXT -1 21 " increases from 0 to " }{XPPEDIT 18 0 "2 *Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 639 "rng := 0..2 *Pi: # parameter range\npt := [1,1]: # location to display values of p arameter\nt := 't':\nf := t ->1-cos(t)-sin(t);\ng := t -> 1-cos(t)+sin (t);\np1 := plot([f(t),g(t),t=rng],labels=[`x`,`y`]):\nnumframes := 16 1:\nfrms := []:\naa := op(1,rng): bb := op(2,rng):\nhh := evalf(abs(bb -aa)/(numframes-1)):\nfor i from 0 to numframes-1 do \n tt := aa+hh* i;\n p2 := plot([[[f(tt),g(tt)]]$3],style=point,symbol=[circle,diamo nd,cross],\n color=black);\n t1 := plots[textplot]([o p(1,pt),op(2,pt),t=tt],color=black);\n frms := [op(frms),plots[displ ay]([p1,p2,t1])];\nend do:\nplots[display](frms,insequence=true,scalin g=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ dx/dt = s in*t-cos*t,``],[ dy/dt = sin*t+cos*t,``])" "6#-%*PIECEWISEG6$7$/*&%#dx G\"\"\"%#dtG!\"\",&*&%$sinGF*%\"tGF*F**&%$cosGF*F0F*F,%!G7$/*&%#dyGF*F +F,,&*&F/F*F0F*F**&F2F*F0F*F*F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx = ``(dy/dt)/``(dx/dt);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%# dtGF(F&-F+6#*&F'F&F.F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(sin*t+cos *t)/(sin*t-cos*t);" "6#*&,&*&%$sinG\"\"\"%\"tGF'F'*&%$cosGF'F(F'F'F',& *&F&F'F(F'F'*&F*F'F(F'!\"\"F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "(b) The tangent lines are horizontal " }{XPPEDIT 18 0 "dy /dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 17 ", that is, when " }{XPPEDIT 18 0 "sin*t+cos*t = 0;" "6#/,&*&%$sinG\"\"\"%\"tGF'F'*&% $cosGF'F(F'F'\"\"!" }{TEXT -1 14 ". This gives " }{XPPEDIT 18 0 "tan* t = -1;" "6#/*&%$tanG\"\"\"%\"tGF&,$F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "This equation is satisfied when " }{XPPEDIT 18 0 "t = n*Pi-Pi/4" "6#/%\"tG,&*&%\"nG\"\"\"%#PiGF(F(*&F)F(\"\"%!\"\"F, " }{TEXT -1 6 ", for " }{TEXT 333 1 "n" }{TEXT -1 25 " an integer, whi ch gives " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y=1+sqrt(2)" "6#/%\"yG,&\"\"\"F&-%%sqrtG6#\"\"#F&" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=1-sqrt(2)" "6#/%\"yG,&\"\"\"F&-%% sqrtG6#\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "The \+ tangent lines are horizontal at the points " }{XPPEDIT 18 0 "P(1,1+sq rt(2));" "6#-%\"PG6$\"\"\",&F&F&-%%sqrtG6#\"\"#F&" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "Q(1,1-sqrt(2));" "6#-%\"QG6$\"\"\",&F&F&-%%sqrtG6#\" \"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 40 "(c) The tangent lines are vertical when " } {XPPEDIT 18 0 "dx/dy=0" "6#/*&%#dxG\"\"\"%#dyG!\"\"\"\"!" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dx/dy = ``(dx/dt)/``(dy/dt);" "6#/*&%#dxG\" \"\"%#dyG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{TEXT -1 3 " \+ = " }{XPPEDIT 18 0 "(sin*t-cos*t)/(sin*t+cos*t);" "6#*&,&*&%$sinG\"\" \"%\"tGF'F'*&%$cosGF'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 43 "Hence the tangent lines are vertical when " }{XPPEDIT 18 0 "sin*t -cos*t = 0;" "6#/,&*&%$sinG\"\"\"%\"tGF'F'*&%$cosGF'F(F'!\"\"\"\"!" } {TEXT -1 16 ", that is when " }{XPPEDIT 18 0 "tan*t = 1;" "6#/*&%$tan G\"\"\"%\"tGF&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "This \+ equation is satisfied when " }{XPPEDIT 18 0 "t = n*Pi+Pi/4" "6#/%\"tG, &*&%\"nG\"\"\"%#PiGF(F(*&F)F(\"\"%!\"\"F(" }{TEXT -1 6 ", for " } {TEXT 334 1 "n" }{TEXT -1 25 " an integer, which gives " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x=1+sqrt( 2)" "6#/%\"xG,&\"\"\"F&-%%sqrtG6#\"\"#F&" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x=1-sqrt(2" "6#/%\"xG,&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The tangent lines are v ertical at the points " }{XPPEDIT 18 0 "R(1+sqrt(2),1);" "6#-%\"RG6$, &\"\"\"F'-%%sqrtG6#\"\"#F'F'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "S(1 -sqrt(2),1);" "6#-%\"SG6$,&\"\"\"F'-%%sqrtG6#\"\"#!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 " Note" }{TEXT -1 33 ": We can eliminate the parameter " }{TEXT 343 1 "t " }{TEXT -1 24 " between the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 1-cos*t-sin*t, ``],[y = 1-cos* t+sin*t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,(\"\"\"F**&%$cosGF*%\"tGF*! \"\"*&%$sinGF*F-F*F.%!G7$/%\"yG,(F*F**&F,F*F-F*F.*&F0F*F-F*F*F1" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "These two equations can be written in the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([ cos*t+sin*t=1-x,``],[cos*t-sin*t=1-y ,``]) " "6#-%*PIECEWISEG6$7$/,&*&%$cosG\"\"\"%\"tGF+F+*&%$sinGF+F,F+F+,&F+F+ %\"xG!\"\"%!G7$/,&*&F*F+F,F+F+*&F.F+F,F+F1,&F+F+%\"yGF1F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Squaring these two equations give s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ co s^2*t+2*cos*t*sin*t +sin^2*t=(x-1)^2,``],[cos^2*t-2*cos*t*sin*t +sin^2 *t=(y-1)^2,``])" "6#-%*PIECEWISEG6$7$/,(*&%$cosG\"\"#%\"tG\"\"\"F-*,F+ F-F*F-F,F-%$sinGF-F,F-F-*&F/F+F,F-F-*$,&%\"xGF-F-!\"\"F+%!G7$/,(*&F*F+ F,F-F-*,F+F-F*F-F,F-F/F-F,F-F4*&F/F+F,F-F-*$,&%\"yGF-F-F4F+F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Adding the last two equations gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*cos^2*t +2*sin^2*t = (x-1)^2+(y-1)^2;" "6#/,&*(\"\"#\"\"\"*$%$cosGF&F'%\"tGF'F '*(F&F'*$%$sinGF&F'F*F'F',&*$,&%\"xGF'F'!\"\"F&F'*$,&%\"yGF'F'F2F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1)^2+(y-1)^2=2" "6#/,&*$,&% \"xG\"\"\"F(!\"\"\"\"#F(*$,&%\"yGF(F(F)F*F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This is t he equation of a circle with centre at the point" }{XPPEDIT 18 0 "``(1 ,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 12 " and radius " }{XPPEDIT 18 0 "sq rt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 481 481 481 {PLOTDATA 2 "63-%'CURVESG6$7S7$$\"\"!F )F(7$$!33c(=lbS;F\"!#=$\"3aslf3e\"*e9F-7$$!35Q!=%e542AF-$\"3e6=xgo[fGF -7$$!3iotr.op^IF-$\"3e)H&*f%Q_aXF-7$$!3#[P]#e=\\lOF-$\"3Waml?+NfjF-7$$ !383!)*GqP)HSF-$\"3L(*y9POL@#)F-7$$!3K7<`po6UTF-$\"3G8IDK&op(**F-7$$!3 5W0%ezux-%F-$\"3c)=?.L'[z6!#<7$$!3Y@m2&p(zkOF-$\"3cwhCeaKk8FN7$$!35c3t b^qhIF-$\"3!4#*QmeU@a\"FN7$$!3*>&Gq5WD,AF-$\"3$=*f#e=[]r\"FN7$$!3)3sej =aUD\"F-$\"3_ZG6a!)Rc=FN7$$\"3u\"z7zX_G(f!#@$\"31(4S#FN7$$\"3po'R'GWl.5FN$\"3K!=82%)3UT#FN7$$\"3_(3C@ oby<\"FN$\"31Hk)*e^)HS#FN7$$\"3U)pJIjmdO\"FN$\"3]d.$\\r%4mBFN7$$\"3Ny( =`L=&Q:FN$\"3.7&3$>*owI#FN7$$\"3V@*y#oz<4!pW-buE@FN7$$\"3yL=\\\")Q*o*>FN$\"3B=[O+l4.?FN7$$\"3'[@ #\\()p&e7#FN$\"35'R_ihIe&=FN7$$\"3\"*3Hb\\[j?AFN$\"3r,y*QCyTr\"FN7$$\" 38m^$\\d[II#FN$\"3Ik9nnQg\\:FN7$$\"3W!=b=q(*\\O#FN$\"3+vWcC&R)p8FN7$$ \"3jm4./7t,CFN$\"3-=]A]B[(=\"FN7$$\"3#\\&f\"\\P\">9CFN$\"3h5B]7=#z+\"F N7$$\"3PZDz,:>,CFN$\"3%G@5B%yD&3)F-7$$\"3cnn)H=JcO#FN$\"3wo2k1]1DjF-7$ $\"3M!f&=U\"yLI#FN$\"3l%H&eT:y6XF-7$$\"3=UksF9CEAFN$\"3KM0K&)o\"\\&HF- 7$$\"3ig4$o(3y?@FN$\"3i_/(\\\\]HF-$!3!f#z*)3F()fAF-7$$\"3#[ R\"=q<,a9F-$!3C`D6UG#zE\"F-7$$!36YKhSr8/#)!#F$\"36YKhSr8/#)Ffz-%'COLOU RG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7&7$$\"\"\"F)$\"3/+++iN@9CFN7$Fd[l$!3 !)******>c8UTF-7$Ff[lFd[l7$Fi[lFd[l-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[lF)F )F)-%&STYLEG6#%&POINTG-F$6&Fb[l-F^\\l6#%(DIAMONDGFa\\lFc\\l-F$6&Fb[l-F ^\\l6#%&CROSSGFa\\lFc\\l-F$6%7$7$F(Ff[l7$$\"\"#F)Ff[l-Fjz6&F\\[lF(F][l F(-%*THICKNESSG6#Fg]l-F$6%7$7$F(Fi[l7$Ff]lFi[lFh]lFj]l-F$6%7$7$Ff[lF(7 $Ff[lFf]l-Fjz6&F\\[lF(F(F][lFj]l-F$6%7$7$Fi[lF(7$Fi[lFf]lFg^lFj]l-%%TE XTG6$7$$\"$`#!\"#$Fd_l!\"\"Q\"x6\"-F__l6$7$Fe_lFb_lQ\"yFh_l-F__l6$7$Fd [l$\"#EFf_lQ\"PFh_l-F__l6$7$Fd[l$!\"'Ff_lQ\"QFh_l-F__l6$7$F``lFd[lQ\"R Fh_l-F__l6$7$Ff`lFd[lQ\"SFh_l-%*AXESTICKSG6$\"\"$Fdal-%+AXESLABELSG6%% !GFhal-%%FONTG6#%(DEFAULTG-%%VIEWG6$F\\blF\\bl" 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" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur ve 11" "Curve 12" "Curve 13" "Curve 14" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 67 ": The cartesian equation of the \+ circle can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x^2+y^2=2*x+2*y" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F( ,&*&F'F(F&F(F(*&F'F(F*F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Differentiating implicitly gives: " }}{PARA 256 "" 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 = 2+2" "6#/*&%#dyG\"\"\" %#dxG!\"\",&\"\"#F&F*F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&% #dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x + y" "6# ,&%\"xG\"\"\"%\"yGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1+dy/dx" " 6#/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*&F%F&F'F(F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 352 1 "y" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx- dy/dx = 1-x " "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&F&F'F(F)F),&F'F'%\"xGF)" }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 " ``(y-1) = 1- x" "6#/-%!G6#,&%\"yG\"\"\"F)!\"\",&F)F)%\"xGF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(1-x)/(y-1)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F &F&%\"xGF(F&,&%\"yGF&F&F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "x=1-cos*t-sin*t" "6#/%\"xG,(\"\" \"F&*&%$cosGF&%\"tGF&!\"\"*&%$sinGF&F)F&F*" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "y=1-cos*t+sin*t" "6#/%\"yG,(\"\"\"F&*&%$cosGF&%\"tGF&! \"\"*&%$sinGF&F)F&F&" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=(sin*t+cos*t)/(sin*t-cos*t)" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&,&*&%$sinGF&%\"tGF&F&*&%$cosGF&F-F&F&F&,&*&F,F&F-F& F&*&F/F&F-F&F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 "as be fore. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "eliminate(\{x=1-cos(t)-s in(t),y=1-cos(t)+sin(t)\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<#/ %\"tG-%'arctanG6$,&%\"yG#\"\"\"\"\"#%\"xG#!\"\"F.,(F/F0F-F-F+F0<#,**$) F+F.F-F-F+!\"#F/F7*$)F/F.F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 341 1 "x" }{TEXT -1 1 "-" } {TEXT 342 1 "y" }{TEXT -1 27 " equation of the curve is " }{XPPEDIT 18 0 "y^2-2*y-2*x+x^2=0" "6#/,**$%\"yG\"\"#\"\"\"*&F'F(F&F(!\"\"*&F'F( %\"xGF(F**$F,F'F(\"\"!" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "(x- 1)^2+(y-1)^2=2" "6#/,&*$,&%\"xG\"\"\"F(!\"\"\"\"#F(*$,&%\"yGF(F(F)F*F( F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "This is a circle wi th centre at" }{XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 12 " and radius " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "student[completesquare](y^2-2*y-2*x+x^2=0,\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$),&%\"xG\"\"\"!\"\"F)\"\"#F)F)! \"#F)*$),&%\"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 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 339 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 77 "This question is concerned with the curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=t^3,``],[y=t^2-1,``])" "6#-%*PIECEWISE G6$7$/%\"xG*$%\"tG\"\"$%!G7$/%\"yG,&*$F*\"\"#\"\"\"F3!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in te rms of the parameter " }{TEXT 353 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 87 "(b) Find the coordinates of the point on the curve w here the tangent line has gradient " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "(c) State any val ues of the parameter " }{TEXT 344 1 "t" }{TEXT -1 11 " for which " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 16 " is no t defined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 340 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "The following animation gives the locatio n of the point" }{XPPEDIT 18 0 "``(t^3,t^2-1);" "6#-%!G6$*$%\"tG\"\"$, &*$F'\"\"#\"\"\"F,!\"\"" }{TEXT -1 18 " as the parameter " }{TEXT 381 1 "t" }{TEXT -1 25 " increases from -2 to 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "rng := -2.. 2: # parameter range\npt := [6,1]: # location to display values of par ameter\nt := 't':\nf := t ->t^3;\ng := t -> t^2-1;\np1 := plot([f(t),g (t),t=rng],labels=[`x`,`y`]):\nnumframes := 161:\nfrms := []:\naa := o p(1,rng): bb := op(2,rng):\nhh := evalf(abs(bb-aa)/(numframes-1)):\nfo r i from 0 to numframes-1 do \n tt := aa+hh*i;\n p2 := plot([[[f(t t),g(tt)]]$3],style=point,symbol=[circle,diamond,cross],\n \+ color=black);\n t1 := plots[textplot]([op(1,pt),op(2,pt),t=tt],c olor=black);\n frms := [op(frms),plots[display]([p1,p2,t1])];\nend d o:\nplots[display](frms,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "(a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/dt = 3*t^2 ,``],[dy/dt = 2*t ,``])" "6#-%*PIECEWISEG6$7 $/*&%#dxG\"\"\"%#dtG!\"\"*&\"\"$F**$%\"tG\"\"#F*%!G7$/*&%#dyGF*F+F,*&F 1F*F0F*F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/dt)/``(dx/dt) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F( " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*t/(3*t^2);" "6#*(\"\"#\"\"\"%\"t GF%*&\"\"$F%*$F&F$F%!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/(3*t)" "6#*&\"\"#\"\"\"*&\"\"$F%%\"tGF%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) The gradient is equal to " }{XPPEDIT 18 0 "-1" "6# ,$\"\"\"!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "2/(3*t) = -1;" "6# /*&\"\"#\"\"\"*&\"\"$F&%\"tGF&!\"\",$F&F*" }{TEXT -1 13 ". This gives \+ " }{XPPEDIT 18 0 "t = -2/3;" "6#/%\"tG,$*&\"\"#\"\"\"\"\"$!\"\"F*" } {TEXT -1 47 ", and the corresponding point on the curve is " } {XPPEDIT 18 0 "P(-8/27,-5/9);" "6#-%\"PG6$,$*&\"\")\"\"\"\"#F!\"\"F+,$ *&\"\"&F)\"\"*F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) The derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 21 " is not defined when " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\" !" }{TEXT -1 23 ", that is, at the point" }{XPPEDIT 18 0 "``(0,-1)" "6 #-%!G6$\"\"!,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 481 481 481 {PLOTDATA 2 "6--%'CURVESG6$7W7$$!3++++ +++vL!#<$\"3+++++++]7F*7$$!3]#=g2Tz!fJF*$\"3Ms>4<=)H:\"F*7$$!3/g4b1-d_ HF*$\"3i))y'el,\"e5F*7$$!3c\"\\rRL\"Q!y#F*$\"3)GuQ!fQ'Gx*!#=7$$!3\\X(Q 9e?]h#F*$\"3k+z&pO(*3)*)F<7$$!319!yb:DtE#F*$\"3-gB,$pW(esF<7$$!3Wr%=zT :*\\>F*$\"3,)p\"=Pv(yg&F<7$$!3uIfNKP*[m\"F*$\"39ulM:.CZSF<7$$!3Eq![:c# yE9F*$\"3GVl&Hs^Pn#F<7$$!3$)*R]IKf`?\"F*$\"3>w`pw%HgK\"F<7$$!3)p.!)3I$ y,5F*$\"39\"*))y\"=9&)=\"!#?7$$!3#QU7%3#G1B)F<$!3dGw'f_Au@\"F<7$$!3K8U YPpAGmF<$!31FK?$)\\$zR#F<7$$!39zI%)[4!>S&F<$!3'>j\")p8(=nLF<7$$!3q@*o# 33%R@%F<$!3QbTp\\cFzVF<7$$!3V?$=k#*y0@$F<$!3JBpB-jE6`F<7$$!3yP$zkQSzS# F<$!31bWBv*G&HhF<7$$!3zlP*p$!3_y//9*p01)F<7$$!3eu*\\eWd:A&Ffq$!3'=9! 35$))Hg)F<7$$!3i$)))fv^V>JFfq$!3pcbx1D04!*F<7$$!3=oN2pk>k:Ffq$!31*)32! pZXP*F<7$$!3iolAd))\\&p'Fjn$!3$yG,#>*fZk*F<7$$!3\"*fALE:]:>Fjn$!3[c(fK 3jd%)*F<7$$!3YAUNR*oiw#!#@$!3[s=>E\\ad**F<7$$!3#HV!RMi%oC$!#E$!3s*3RK2 y*****F<7$$\"3r`sz-1d.FFes$!3![;@\"z))=e**F<7$$\"3(3=&\\C<_E=Fjn$!3C6< H1td])*F<7$$\"3qV6NWUgsiFjn$!39ym9#>!))f'*F<7$$\"3*=l?ZJFw`\"Ffq$!3sS, \"*Q/l\"Q*F<7$$\"3^e)HD53I-$Ffq$!3ME0'>W!eH!*F<7$$\"379A*e429;&Ffq$!3& enu\"4xt8')F<7$$\"3)[vg/gm$4&)Ffq$!3%*39#*Q*o`1)F<7$$\"3)pH'pkI&QD\"F< $!3)*RO5;_'[\\(F<7$$\"3f^(H<,,k!=F<$!3ett4()3Z/oF<7$$\"3$f,9X]h&HCF<$! 3mDZYLYR1hF<7$$\"3_*)>^w=()eKF<$!3kn#p_LlVE&F<7$$\"3iJLHlQ;%>%F<$!3oNB MCc(oR%F<7$$\"3SGr\"p4hqM&F<$!38.[37T:7MF<7$$\"37&>=>/q;m'F<$!3oPK8J`Q sBF<7$$\"3#fdcnhp>D)F<$!3**['*pktC-7F<7$$\"3W3rOpga+5F*$\"3\"p%Qg.78SO Fes7$$\"3w0#)3cL!R?\"F*$\"38%\\-932pJ\"F<7$$\"3X%=\"3;K4J9F*$\"3?2I'=' eE*p#F<7$$\"3'pw@BccNm\"F*$\"3oU&ew)yrRSF<7$$\"3q7i0r8Xf>F*$\"3y%>2pXA (ecF<7$$\"3OME?$pT@D#F*$\"3Y*o[\"H!3;=(F<7$$\"3Uu@!=!o%[f#F*$\"3o$Q*fz ;:$)))F<7$$\"3'[`zGXu/x#F*$\"3oKoPmp'es*F<7$$\"3cA?6;K&Q&HF*$\"3ett)3$ ype5F*7$$\"3UW#f(R/vfJF*$\"3M$fSpo'G`6F*7$$\"3+++++++vLF*F+-%'COLOURG6 &%$RGBG$\"*++++\"!\")$\"\"!Ff\\lFe\\l-F$6&7#7$$!3!G'H'H'H'H'HF<$!3!ebb bbbbb&F<-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\lFf\\lFf\\lFf\\l-%&STYLEG6#%& POINTG-F$6&Fi\\l-F`]l6#%(DIAMONDGFc]lFe]l-F$6&Fi\\l-F`]l6#%&CROSSGFc]l Fe]l-F$6%7$7$$!3s.Pq.Pq. " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "The cycloid" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 152 "The curve traced o ut in two dimensions by a point on the rim of a rigid wheel is it roll s in a straight line over a flat horizontal surface is called a " } {TEXT 259 7 "cycloid" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 116 " It provides an interesting example of a curve with a natural descripti on by means of a pair of parametric equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The following animation i llustrates this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 648 "p1 := plot([[-1,0],[20,0]],color=black):\nnum frames := 100:\nf := t -> t-sin(t):\ng := t -> 1-cos(t):\nhh := evalf( 6*Pi/(numframes-1)):\np2 := plot([cos(t),1+sin(t),t=0..2*Pi],color=nav y,thickness=2):\nfrms := [plots[display]([p1,p2])]:\nfor i from 1 to n umframes-1 do \n tt := hh*i;\n p2 := plot([tt+cos(t),1+sin(t),t=0. .2*Pi],color=navy,thickness=2);\n p3 := plot([[[f(tt),g(tt)]]$3],sty le=point,symbol=[circle,diamond,cross],\n color=black); \n p4 := plot([f(t),g(t),t=0..tt],adaptive=false,numpoints=2*i+2,thi ckness=2);\n frms := [op(frms),plots[display]([p1,p2,p3,p4])];\nend \+ do:\nplots[display](frms,insequence=true,tickmarks=[0,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Suppose that the point whose motion we \+ follow is initially at the origin in the " }{TEXT 367 1 "x" }{TEXT -1 1 "-" }{TEXT 368 1 "y" }{TEXT -1 70 " plane, with the rolling wheel ha ving its vertical diameter along the " }{TEXT 369 1 "y" }{TEXT -1 7 " \+ axis. " }}{PARA 0 "" 0 "" {TEXT -1 56 "For simplicity, we take the rad ius of the wheel to be 1." }}{PARA 0 "" 0 "" {TEXT -1 60 "Suppose that the wheel now rolls without slipping along the " }{TEXT 370 1 "x" } {TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 17 "We use the angle " }{TEXT 371 1 "t" }{TEXT -1 139 ", in radians, through wich the wheel h as rotated as the parameter. The first picture shows two positions of \+ the wheel for parameter values " }{XPPEDIT 18 0 "t=t[1]" "6#/%\"tG&F$6 #\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=t[2]" "6#/%\"tG&F$6#\" \"#" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 657 219 219 {PLOTDATA 2 "67-%'CURVESG6%7U7$$\"\"!F)F(7$$\"3Dsd6sK%Q#)) !#@$\"3L?C\\JB_;:!#>7$$\"3o!)*)[PN4\\d!#?$\"3'oqb?`q,F&F07$$\"3&o-;#=$ )p@LF<7$$\"3eX^!*=hxv:F<$\"3Kkmh\\,azXF<7$$\"3/N$HE/!> WCF<$\"3*GrpuYP_-'F<7$$\"3(45/O.^=g$F<$\"3bc%3$R'HAj(F<7$$\"3=57@)oEO. &F<$\"3!pDCY-I=I*F<7$$\"3P'44ZSfN!oF<$\"3re)4ph`R5\"!#<7$$\"3+leI(p(f6 ')F<$\"3#*)\\))3Q\"ea7Fjn7$$\"3#eeAdZz<4\"Fjn$\"3e?on=!QoT\"Fjn7$$\"3o <(*)=g00N\"Fjn$\"3'3/?yT)Hn:Fjn7$$\"3I#RKza.Ki\"Fjn$\"3u72Ms(pip\"Fjn7 $$\"3q2Qu5VA))=Fjn$\"31iYcyep'z\"Fjn7$$\"3JL&\\i\\t6A#Fjn$\"3%fBOn7e@* =Fjn7$$\"31K'z<\"R*[^#Fjn$\"3Obe&)zp\\]>Fjn7$$\"3CT9u8\"Ga'GFjn$\"3#>L &pU9X!*>Fjn7$$\"3WJ#3SA4&zJFjn$\"3s'4GjG?)**>Fjn7$$\"3Xmy,MLUBNFjn$\"3 I3r#*Q*><)>Fjn7$$\"3%)RTK^5tXQFjn$\"3c@aKr%ot$>Fjn7$$\"3+fN$[%3'><%Fjn $\"3s[<))f\\?k=Fjn7$$\"33D&)QE9leWFjn$\"3_rsM&*zju+T^\"Fjn7$$\"3A+1][R B__Fjn$\"3y!f?xHuoP\"Fjn7$$\"3=X[ahMcnaFjn$\"3l2GB,'3)=7Fjn7$$\"3abf`I j*=m&Fjn$\"3?BJ:e(e#\\5Fjn7$$\"3$pPb@l;P#eFjn$\"3g+-rVkc>))F<7$$\"3ci. s@3z`fFjn$\"3u&*4\"pC4%F07$$\"3cT%pVE1HG'Fjn$\"3=:99?r&\\/(F47$ $\"3!)yUV1s<$G'Fjn$\"3i\">'R)fP(emF-7$$\"33wM+(H,KG'Fjn$\"3i3,mBAuZ5F4 7$$\"3K?%pdcnNG'Fjn$\"3'[u$4xZA)o)F47$$\"3#4Y``54\\G'Fjn$\"3+$)\\9NF*y O#F07$$\"3?l:6H2b#H'Fjn$\"3)43WOU)R\"G(F07$$\"3*>j**GPQ5J'Fjn$\"3mS+2O v$Q\\\"F<7$$\"3l3W!o_vUM'Fjn$\"3Y#QsJ/TX\\#F<7$$\"31%>zm*=>\"R'Fjn$\"3 *Gd()fe!y,OF<7$$\"3C;%3%H)[wY'Fjn$\"3WI$)*4D$zc]F<7$$\"3y9`euS-flFjn$ \"39[[um&=3\\'F<7$$\"3qU:/d!QBo'Fjn$\"3Fw0x`jx:\")F<7$$\"3hZd2WLcEoFjn $\"3-P$[dj()>s*F<7$$\"3b=mP`Z?Fjn$\"\" \"F)7$$\"3H4hR)yLy.#Fjn$\"3KmdD=y_O6Fjn7$$\"3W6B)3wxX,#Fjn$\"3g#\\ff*) GLD\"Fjn7$$\"3S.h)))>c?(>Fjn$\"3OLc[K5J!Q\"Fjn7$$\"38'ozGk/D\">Fjn$\"3 Y^`%R4U7]\"Fjn7$$\"3V0vGaFiP=Fjn$\"3F&H-qceDh\"Fjn7$$\"33NR'y'\\XbjDJ%)e;Fjn$\"3[@@&\\!>8\"z\"Fjn7$$\"3'HuI mqu#[:Fjn$\"3I>p(Qh-a'=Fjn7$$\"3O#3na^6#H9Fjn$\"3D.g5^l:C>Fjn7$$\"3iY@ iyht*H\"Fjn$\"3YQ\"[M\"oen>Fjn7$$\"3G7:EL7r\"=\"Fjn$\"3!)fVPO<\"4*>Fjn 7$$\"3y&fASW+m/\"Fjn$\"3%GhJ<#)*****>Fjn7$$\"3(>]e2:$\\4\"*F<$\"31Z>wG Zn!*>Fjn7$$\"3=\\D`V'>Fjn7$$\"3a?c$*yl'))p'F<$\"3#yF #e`m3E>Fjn7$$\"3h-vPO-mHaF<$\"3GkyI2&oN'=Fjn7$$\"3cPRkLg[KWF<$\"3G;)RQ +Bqz\"Fjn7$$\"3+bdBj\"fEQ$F<$\"3\"evOg?x_q\"Fjn7$$\"3?=uW/8xnDF<$\"3wq 6VQZc7;Fjn7$$\"3G>(*=q(oE\"=F<$\"3aH$\\4/k,]\"Fjn7$$\"3+ZN'oBR5C\"F<$ \"3%o'[*>HvXQ\"Fjn7$$\"3MJ4'f+?N3)F0$\"3Ep9m#Q%=d7Fjn7$$\"3EJ-$e!\\X\\ cF0$\"3R/_]+D/O6Fjn7$$\"3))*Rnr+O-s%F0$\"3B#\\O%485.5Fjn7$$\"3Bv%pF\" \\PNcF0$\"3I24!Q9o)\\')F<7$$\"3c![kuU+\"zzF0$\"39lWsrprnuF<7$$\"3$z%o' pH8(37F<$\"3&3\\\"ojkxKiF<7$$\"35D79nO?F<7$$\"3)*ydu\"4(G\"[&F<$\"3rZlQ)ewVL\"F<7$$\"3O *Rz#GS>*p'F<$\"3!3&pk)f++R(F07$$\"3aAvqG*4L'yF<$\"3=o0GSI^iMF07$$\"3;) yeZt3/=*F<$\"3K%ogJp+eP)F47$$\"35K1dn,9W5Fjn$\"3kw[NRw!Qn%!#B7$$\"39BK Ow)pf<\"Fjn$\"3iM`_\"\\:eK)F47$$\"3B$Rqbo,FI\"Fjn$\"3tl^J0P?>LF07$$\"3 [)yb&*Qx2V\"Fjn$\"3#*oHHy!G#\\wF07$$\"3Oh(4'3w_Z:Fjn$\"3DqRW*Q_;M\"F<7 $$\"3#\\#)[4fK\"e;Fjn$\"3%f/^\"3F>$3#F<7$$\"3Ugnk*eFrv\"Fjn$\"3-H9TRzF dHF<7$$\"3&f=s.R:s$=Fjn$\"3u$pq#[9;pQF<7$$\"3+95i-8j9>Fjn$\"3()z,UFIZC ]F<7$$\"3S$e@jk(pp>Fjn$\"3Q/GHB9.ShF<7$$\"3s'GAzVmE,#Fjn$\"3gR\\Kgh\"[ R(F<7$$\"3/elf/J*y.#Fjn$\"3(f-`QpK!R')F<7$Fc^l$\"3u8/#3++++\"FjnF_]l-F $6$7$7$$\"3++++^v>Z5FjnF(7$Fj]mFe^lF_]l-F$6$7$F\\^mFa\\lF_]l-F$6$7S7$$ \"3-+++/-z)=&FjnFe^l7$$\"3I4hRTkUz^FjnFj^l7$$\"3-6B)QTqh:&FjnF__l7$$\" 3T.h)=&)[O6&FjnFd_l7$$\"3q&ozeH(4a]FjnFi_l7$$\"3!f](G2a@z\\FjnF^`l7$$ \"33NR'3iZq*[FjnFc`l7$$\"3\\K>jydV+[FjnFh`l7$$\"3_U2jft')*o%FjnF]al7$$ \"3#G3n%oT!3d%FjnFbal7$$\"3kY@iJ)G8W%FjnFgal7$$\"3I7:E')QIBVFjnF\\bl7$ $\"3M&fAq4$>)=%FjnFabl7$$\"3K]e2o>a_SFjnFfbl7$$\"3a%Gk*))>$FjnFhel7$ $\"3Au;28]z)=$FjnF]fl7$$\"3k%pF@SYz>$FjnFbfl7$$\"3#[kusl$Q@KFjnFgfl7$$ \"3![o'p#)RYiKFjnF\\gl7$$\"3Ts,z!fr8K$FjnFagl7$$\"3!z+soU$=%R$FjnFfgl7 $$\"3On3H5OtxMFjnF[hl7$$\"3k:#))4cJRe$FjnF`hl7$$\"3!zdu@O@(*o$FjnFehl7 $$\"31Sz#e07:\"QFjnFjhl7$$\"3F_2(ekBz#RFjnF_il7$$\"3syeZENjfSFjnFdil7$ $\"3cK1d?Gt&=%FjnFiil7$$\"3;BKOHDch^F%zF<7$$\"3S+++[Y' >v(F<$\"39+++G'ytz(F<7$$\"3%******\\io4(yF<$\"3]*****R(G/ewF<7$$\"3w** ***zx,r*zF<$\"3V+++ji7DvF<7$$\"37+++\"R=+8)F<$\"3U+++3J**)R(F<7$$\"3u* ****f9a$p#)F<$\"3y*****48*)*zsF<7$$\"3Z+++Jrs9%)F<$\"3%)*****p^S%orF<7 $$\"3-+++$*)Qdc)F<$\"3Q+++4IlkqF<7$$\"3$******R^v>s)F<$\"3C+++&36*opF< 7$$\"37+++m(3I)))F<$\"3W+++jrZ\")oF<7$$\"31+++msR[!*F<$\"3D+++&*3f-oF< 7$$\"3\\*****>p(o<#*F<$\"3-+++1&oCt'F<7$$\"3'******f.;/R*F<$\"3K+++\"> -8n'F<7$$\"3o*****p&)3hc*F<$\"3#)*****pgf#>mF<7$$\"3N+++)f%GW(*F<$\"3? +++'R$[wlF<7$$\"3X+++()[XC**F<$\"3Y+++234VlF<7$$\"3&*******)e71,\"Fjn$ \"3_*****fOt\">lF<7$$\"3&******p'*z)G5Fjn$\"3D+++Gmz/lF<7$$\"33+++_v>Z 5Fjn$\"3A+++++++lF\"Fjn7$$\"3%)*******4l\">WFjn$ \"3(******z[Iu?\"Fjn7$$\"3$)*****fI$*zS%Fjn$\"3/+++-J?>7Fjn7$$\"3)**** **>p?iR%Fjn$\"30+++'*[PI7Fjn7$$\"3A+++D&zQQ%Fjn$\"33+++[_\"4C\"Fjn7$$ \"3')*****>j.5P%Fjn$\"33+++o_z]7Fjn7$$\"3U+++0$GwN%Fjn$\"31+++wy)*f7Fj n7$$\"3E+++/-zVVFjn$\"3-+++vyYo7Fjn7$$\"3!)******es_HVFjn$\"3%******>- 7iF\"Fjn7$$\"3s*****Hcy[J%Fjn$\"3/+++#4*>$G\"Fjn7$$\"3c*****zE%))*H%Fj n$\"35+++K*4%*G\"Fjn7$$\"3%)*****>Z&e%G%Fjn$\"3!*******>v#[H\"Fjn7$$\" 3)******z5C!pUFjn$\"3++++1qV*H\"Fjn7$$\"3K+++GGC`UFjn$\"3,+++cdA.8Fjn7 $$\"3f*****z)[GPUFjn$\"31+++'Q$=18Fjn7$$\"3@+++GS>@UFjn$\"33+++)y,$38F jn7$$\"3!*******\\V,0UFjn$\"3'******f:v&48Fjn7$Fhgm$\"30++++++58Fjn7$$ \"3/+++dgcsTFjnF_en7$$\"3#)******zjQcTFjnFjdn7$$\"3c******>bHSTFjnFedn 7$$\"3t******zvLCTFjnF`dn7$$\"30++++jb3TFjnF[dn7$$\"35+++N\\*H4%FjnFfc n7$$\"3e******RhpxSFjnFacn7$$\"3A+++W=qiSFjnF\\cn7$$\"3C+++\\J0[SFjnFg bn7$$\"3x*****R?!zLSFjnFbbn7$$\"3h*****H5_*>SFjnF]bn7$$\"3<+++wnd1SFjn Fhan7$$\"3=+++#)3q$*RFjnFcan7$$\"31+++;(f8)RFjnF^an7$$\"3?+++-repRFjnF i`n7$$\"3=+++3`TeRFjnFd`n7$$\"3;+++c\\(y%RFjnF_`n7$$\"3%******f$\\*z$R FjnFj_n7$$\"3=+++GB!)GRFjnFe_n7$$\"3@+++HBK?RFjnF`_n7$$\"3@+++\"=yD\"R Fjn$\"3#******R0P29\"Fjn7$$\"3?+++76f0RFjn$\"3$*******e$)3E6Fjn7$$\"39 +++s-Q**QFjn$\"3*******R1%466Fjn7$$\"3))*****RoiR*QFjn$\"3/+++o_z&4\"F jn7$$\"3y*****z>`$*)QFjn$\"3'******R!RB!3\"Fjn7$$\"3A+++[Wc&)QFjn$\"31 +++CEXk5Fjn7$$\"3=+++=og#)QFjn$\"3-+++%o%\\[5Fjn7$$\"3%******fT)[!)QFj n$\"36+++BQSK5Fjn7$$\"3#)*****z/:#zQFjn$\"36+++YTA;5Fjn7$$\"3'******R? !zyQFjn$\"2'*****f*********Fjn7$Fi[o$\"3Z*****H`ex$)*F<7$Fd[o$\"3I+++g <'fn*F<7$F_[o$\"3V+++`J0:&*F<7$Fjjn$\"3++++bPZb$*F<7$Fejn$\"3#******\\ &4m(>*F<7$F`jn$\"3<+++9t/U!*F<7$F[jn$\"3a+++_$f!*)))F<7$Ffin$\"3B+++/k 6R()F<7$$\"3&)*****>=yD\"RFjn$\"3u*****pWHEf)F<7$F^in$\"3W+++&*****\\% )F<7$F[in$\"3@+++))*=;J)F<7$Fhhn$\"3%******Rrly<)F<7$Fehn$\"3w*****\\y 1\"\\!)F<7$Fbhn$\"32+++<^pDzF<7$F_hn$\"3R+++w*oz!yF<7$F\\hn$\"3U+++Q5D 'p(F<7$$\"3#)*****H)3q$*RFjn$\"3'******f^Z3f(F<7$Ffgn$\"3)******\\JZ? \\(F<7$$\"3p*****R5_*>SFjn$\"3;+++O77+uF<7$F`gn$\"3W*****fC@`J(F<7$$\" 3K+++]J0[SFjn$\"3'******HxzyB(F<7$$\"3K+++X=qiSFjn$\"3]******y!4!orF<7 $Fgfn$\"3')*****fn+f5(F<7$$\"3=+++O\\*H4%Fjn$\"3u*****zzC<0(F<7$Fafn$ \"3q*****p$*Hc+(F<7$F^fn$\"3i*****pVUx'pF<7$F[fn$\"3w*****R9m\"QpF<7$$ \"3#******4Q'QcTFjn$\"3c*****R7#)p\"pF<7$$\"37+++egcsTFjn$\"3a+++U%[U! pF<7$$\"35+++0-z)=%Fjn$\"3Y**************oFFecoQ\"yFgc oFhco-F_co6%7$$\"#\"*!\"#$\"#$)FhdoQ\"tFgcoFhco-F_co6%7$$\"$0%Fhdo$\"$ 2\"FhdoF[eoFhco-F_co6%7$$\"#)*Fhdo$\"#wFhdoQ\"1Fgco-Fico6$F[doFcco-F_c o6%7$$\"$8%FhdoFe^lQ\"2FgcoF[fo-%*AXESTICKSG6$F)F)-%+AXESLABELSG6%Q!Fg coFifo-Fico6#%(DEFAULTG-%%VIEWG6$F\\goF\\go" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "After the wheel has rolled through an angle " }{TEXT 372 1 "t" }{TEXT -1 32 ", \+ its point of contact with the " }{TEXT 373 1 "x" }{TEXT -1 80 " axis T is at a distance OT = arcPT from the origin, since there is no slippi ng." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 370 353 353 {PLOTDATA 2 "6?-%'CURVESG6%7S7$$\"\"!F)F(7$$ \"32)QBax\"=LI!#A$\"3_E>&Q2iag\"!#?7$$\"3#\\Q*4M\\'H)>!#@$\"3IiV&Qoc4h &F07$$\"32]4*f2aK+(F4$\"3#[yYNC$G+8!#>7$$\"3YUzln8;0Aj``$3N#F <7$$\"3%['H&Q=-/V8_F <7$$\"3lPRu\"HNO\"*)F0$\"3%)3Oo>P+ZqF<7$$\"398r`ULuQ8F<$\"3*y5v#=%z7A* F<7$$\"3l'**oG!fj6>F<$\"3aww_.\"Qj;\"!#=7$$\"3%[yi(yc&)[EF<$\"3U<0eVWQ X9Fen7$$\"3F\"*G!)Qk&[V$F<$\"3h@T)*))y&Qr\"Fen7$$\"3kE\"=ss*R'[%F<$\"3 &=Z1$zvlS?Fen7$$\"3v&*Ro;WyLdF<$\"3C;!*p>f(RR#Fen7$$\"3&y;r(fUVIrF<$\" 3/wZ.B&*GdFFen7$$\"3_acZz-Gu&)F<$\"3W-`1#)[r0JFen7$$\"3'\\*)R**\\p?0\" Fen$\"3A(=D#>$QFen$\"3p4mpN&[aJ&Fen7$$\"3UAWpJ'=UH#Fen$\"3-KszC*fOz&Fen7 $$\"3?uT8zy?JEFen$\"3$=BZRG%Q0jFen7$$\"3A`Q5OW0mHFen$\"3IQ%**))fWay'Fe n7$$\"3W.j'3lD]N$Fen$\"3l.0cg#[EJ(Fen7$$\"3'>)HIZj6!z$Fen$\"37rRQ[i!*o yFen7$$\"3i;GC)Q\")\\>%Fen$\"3]ATD,\\*)e$)Fen7$$\"39DJ&ySO)fYFen$\"3'e \\qlK[E*))Fen7$$\"3/1FbUpQq^Fen$\"3KO^5JnUZ%*Fen7$$\"3snO28]!)*p&Fen$ \"39FrX:%Q=***Fen7$$\"3*ee.&\\!y-C'Fen$\"3cH\"p*QC'=0\"!#<7$$\"3g)[n$4 i'G(oFen$\"3;)\\qU-y,6\"F_u7$$\"3]EMR#y-.Z(Fen$\"39wGF,7Ei6F_u7$$\"3:C O%*)o$HQ\")Fen$\"3!*GMIxCP<7F_u7$$\"3Md7p0[;q()Fen$\"3+@kJFYsm7F_u7$$ \"3_UC$4Y'[*[*Fen$\"3e%3cIU&))>8F_u7$$\"38.:zNvI>5F_u$\"3%GmaiD0!p8F_u 7$$\"3'Q_'=@?d&4\"F_u$\"3Hd88qvE>9F_u7$$\"3gnqJ([gF<\"F_u$\"37?0GMn>n9 F_u7$$\"31@$f@=xiD\"F_u$\"3j]_JV0%f^\"F_u7$$\"3Pp\"3#pDDR8F_u$\"3)>[Z# [)f8c\"F_u7$$\"3MEPsRUfE9F_u$\"3fr]&Q%*=hg\"F_u7$$\"3KJ$Gb+,c^\"F_u$\" 3S#Huy%zo[;F_u7$$\"3uvQu8xR*f\"F_u$\"3SgtA4u4'o\"F_u7$$\"3YkD8K=p(p\"F _u$\"3,Jakyq$os\"F_u7$$\"3!RGkF,Tvy\"F_u$\"3s(o)[opBhskTxo&z\"F_u7$$\"3Yz[f(QY/)>F_u$\"35j@)\\G:k#=F_u7$$\"3#e` y\"G')\\%3#F_u$\"3KZ*oLv))o&=F_u-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-% *THICKNESSG6#\"\"#-F$6$7S7$$\"3++++^v>Z?F_u$\"\"\"F)7$$\"3H4hR)yLy.#F_ u$\"3KmdD=y_O6F_u7$$\"3W6B)3wxX,#F_u$\"3g#\\ff*)GLD\"F_u7$$\"3S.h)))>c ?(>F_u$\"3OLc[K5J!Q\"F_u7$$\"38'ozGk/D\">F_u$\"3Y^`%R4U7]\"F_u7$$\"3V0 vGaFiP=F_u$\"3F&H-qceDh\"F_u7$$\"33NR'y'\\XbjDJ%)e;F_u$\"3[@@&\\!>8\"z\"F_u7$$\"3'HuImqu#[:F_u$\"3I>p(Qh-a' =F_u7$$\"3O#3na^6#H9F_u$\"3D.g5^l:C>F_u7$$\"3iY@iyht*H\"F_u$\"3YQ\"[M \"oen>F_u7$$\"3G7:EL7r\"=\"F_u$\"3!)fVPO<\"4*>F_u7$$\"3y&fASW+m/\"F_u$ \"3%GhJ<#)*****>F_u7$$\"3(>]e2:$\\4\"*Fen$\"31Z>wGZn!*>F_u7$$\"3=\\D`V'>F_u7$$\"3a?c$*yl'))p'Fen$\"3#yF#e`m3E>F_u7$$\"3h- vPO-mHaFen$\"3GkyI2&oN'=F_u7$$\"3cPRkLg[KWFen$\"3G;)RQ+Bqz\"F_u7$$\"3+ bdBj\"fEQ$Fen$\"3\"evOg?x_q\"F_u7$$\"3?=uW/8xnDFen$\"3wq6VQZc7;F_u7$$ \"3G>(*=q(oE\"=Fen$\"3aH$\\4/k,]\"F_u7$$\"3+ZN'oBR5C\"Fen$\"3%o'[*>HvX Q\"F_u7$$\"3MJ4'f+?N3)F<$\"3Ep9m#Q%=d7F_u7$$\"3EJ-$e!\\X\\cF<$\"3R/_]+ D/O6F_u7$$\"3))*Rnr+O-s%F<$\"3B#\\O%485.5F_u7$$\"3Bv%pF\"\\PNcF<$\"3I2 4!Q9o)\\')Fen7$$\"3c![kuU+\"zzF<$\"39lWsrprnuFen7$$\"3$z%o'pH8(37Fen$ \"3&3\\\"ojkxKiFen7$$\"35D79nO?Fen7$$\"3)*ydu\"4(G\"[&Fen$\"3rZlQ)ewVL\"Fen7$$\" 3O*Rz#GS>*p'Fen$\"3!3&pk)f++R(F<7$$\"3aAvqG*4L'yFen$\"3=o0GSI^iMF<7$$ \"3;)yeZt3/=*Fen$\"3K%ogJp+eP)F07$$\"35K1dn,9W5F_u$\"3kw[NRw!Qn%!#B7$$ \"39BKOw)pf<\"F_u$\"3iM`_\"\\:eK)F07$$\"3B$Rqbo,FI\"F_u$\"3tl^J0P?>LF< 7$$\"3[)yb&*Qx2V\"F_u$\"3#*oHHy!G#\\wF<7$$\"3Oh(4'3w_Z:F_u$\"3DqRW*Q_; M\"Fen7$$\"3#\\#)[4fK\"e;F_u$\"3%f/^\"3F>$3#Fen7$$\"3Ugnk*eFrv\"F_u$\" 3-H9TRzFdHFen7$$\"3&f=s.R:s$=F_u$\"3u$pq#[9;pQFen7$$\"3+95i-8j9>F_u$\" 3()z,UFIZC]Fen7$$\"3S$e@jk(pp>F_u$\"3Q/GHB9.ShFen7$$\"3s'GAzVmE,#F_u$ \"3gR\\Kgh\"[R(Fen7$$\"3/elf/J*y.#F_u$\"3(f-`QpK!R')Fen7$Fi[l$\"3u8/#3 ++++\"F_u-F[[l6&F][lF)F)F)-F$6$7$7$$\"3++++^v>Z5F_uF(7$Fb[mF[\\lF\\[m- F$6$7$Fd[m7$$\"3.+++t9s6=Fen$\"3#)*****z*******\\FenF\\[m-F$6$7$Fh[m7$ Fb[mF[\\mF\\[m-F$6%7$Fd[m7$$\"3'******4b(>Z7F_uF[\\lF\\[m-%*LINESTYLEG Fc[l-F$6%7$F`\\m7$Fe\\mF[\\mF\\[mFg\\m-F$6%7$Fh[m7$Fi[m$\"3E+++)****** *HFenF\\[mFg\\m-F$6&7#Fh[m-%'SYMBOLG6#%'CIRCLEGF\\[m-%&STYLEG6#%&POINT G-F$6&Fe]m-Fg]m6#%(DIAMONDGF\\[mFj]m-F$6&Fe]m-Fg]m6#%&CROSSGF\\[mFj]m- F$6&7$7$$\"+^v>(>\"!\"*$\"+*******p'!#57$F\\_m$\"+)*******\\Fa_m7%7$$ \"+^v>A7F^_m$\"+)****4R&Fa_mFb_m7$$\"+^v>s6F^_mFi_m-F[^m6#%,PATCHNOGRI DGF\\[m-F$6&7$7$F\\_m$\"+*******H)Fa_m7$F\\_m$\"+++++5F^_m7%7$F\\`m$\" ++++4'*Fa_mFg`m7$Fg_mF\\amF^`mF\\[m-F$6&7$7$$\"+#\\[=W%Fa_m$\"+)****** \\$Fa_m7$$\"+t9s6=Fa_mFeam7%7$$\"+&)zj`@Fa_m$\"+)*****\\KFa_mFgam7$F\\ bm$\"+)*****\\PFa_mF^`mF\\[m-F$6&7$7$$\"+#\\[=%yFa_mFeam7$$\"+^v>Z5F^_ mFeam7%7$$\"++f+85F^_mFabmFibm7$F^cmF^bmF^`mF\\[m-F$6$777$$\"3')*****R i>nc)Fen$\"3!)*****f*******))Fen7$$\"3:+++i-!pi)Fen$\"3Z*****47%z,))Fe n7$$\"3g*****\\7Q@p)Fen$\"3))*****>Wsoq)Fen7$$\"3!******fRaAw)Fen$\"3+ +++Q^\\:')Fen7$$\"3b*****f*o0P))Fen$\"3D+++kE\"z_)Fen7$$\"3[*****HfSj \"*)Fen$\"3I+++!3lVW)Fen7$$\"3_*****z<)))***)Fen$\"3G+++#Q\"3l$)Fen7$$ \"3Y+++`1Z(3*Fen$\"3_*****>))y-H)Fen7$$\"3M+++dz%)y\"*Fen$\"3M+++6E;?# )Fen7$$\"3#******fjpPF*Fen$\"3y*****zuC\\:)Fen7$$\"3M+++7b(>P*Fen$\"3] ******4Tu%4)Fen7$$\"3C+++9k>t%*Fen$\"3[+++XcyR!)Fen7$$\"3%*******)*[:x &*Fen$\"3))*****4***>!*zFen7$$\"3!******R-mNo*Fen$\"3)******41Bh%zFen7 $$\"3c*****f7Q@z*Fen$\"3-+++jcn2zFen7$$\"3!)*****RhtD!**Fen$\"3#)***** 4=j\\(yFen7$$\"33+++zpX,5F_u$\"3g*****zFv![yFen7$$\"3&*******o>y75F_u$ \"3n******\\c3FyFen7$$\"32+++*G,U-\"F_u$\"3q******H=07yFen7$$\"35+++TO oN5F_u$\"3]*****H-:I!yFen7$Fb[m$\"3E+++++++yFenF\\[m-%%TEXTG6$7$$\"$/ \"!\"#$\"$3\"FajmQ\"C6\"-F\\jm6$7$$\"$8\"Fajm$\"#VFajmQ\"QFejm-F\\jm6$ 7$$Fd[l!\"\"$\"#lFajmQ\"PFejm-F\\jm6$7$$\"$0\"Fajm$F`[lFajmQ\"TFejm-F \\jm6$7$F[\\n$Fb[nFb[nQ\"OFejm-F\\jm6$7$$\"#AFb[n$!\"'FajmQ\"xFejm-F\\ jm6$7$Fg\\n$\"$&>FajmQ\"yFejm-F\\jm6$7$F\\_m$\"+*******\\(Fa_mQ&cos~tF ejm-F\\jm6$7$$\"+#\\[=9'Fa_mFeamQ&sin~tFejm-F\\jm6$7$$\"#)*Fajm$\"#))F ajmQ\"tFejm-%*AXESTICKSG6$F)F)-%+AXESLABELSG6%Q!FejmFj^n-%%FONTG6#%(DE FAULTG-%%VIEWG6$F^_nF^_n" 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" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "Because the radius of the circle is 1, OT = arcPT is the radian measure of the an gle PCQ" }{XPPEDIT 18 0 "``= t" "6#/%!G%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Now PQ" }{XPPEDIT 18 0 "``=sin*t" "6#/%!G*&%$sin G\"\"\"%\"tGF'" }{TEXT -1 7 " and CQ" }{XPPEDIT 18 0 "`` = cos*t" "6#/ %!G*&%$cosG\"\"\"%\"tGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 139 "Thus the coordinates of the point P, which is the new position of the point on the rim of the wheel, which was initially at the origin, are" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = ``" "6#/% \"xG%!G" }{TEXT -1 11 "OT - PQ = " }{XPPEDIT 18 0 "t-sin*t;" "6#,&%\" tG\"\"\"*&%$sinGF%F$F%!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ``" "6#/%\"yG%!G" }{TEXT -1 10 "TC \+ - QC = " }{XPPEDIT 18 0 "1-cos*t;" "6#,&\"\"\"F$*&%$cosGF$%\"tGF$!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The parametric equat ions of the cycloid are therefore: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([x = t-sin*t ,``],[y = 1-cos*t ,``])" "6#- %*PIECEWISEG6$7$/%\"xG,&%\"tG\"\"\"*&%$sinGF+F*F+!\"\"%!G7$/%\"yG,&F+F +*&%$cosGF+F*F+F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The argument just given applies when " }{TEXT 288 1 "t" }{TEXT -1 89 " is an acute angle, but one can check t hat these parametric equations work for any angle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We now investigate the " }{TEXT 259 8 "gradient" }{TEXT -1 26 " of this parametric curve." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/dt = 1- cos*t ,``],[dy/dt = sin*t ,``])" "6#-%*PIECEWISEG6$7$/*&%#dxG\"\"\"%#d tG!\"\",&F*F**&%$cosGF*%\"tGF*F,%!G7$/*&%#dyGF*F+F,*&%$sinGF*F0F*F1" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = ``(dy/dt)/``(dx/dt);" "6#/*&%#d yG\"\"\"%#dxG!\"\"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sin*t/(1-cos*t);" "6#*(%$sinG\"\"\"%\"tGF%,&F% F%*&%$cosGF%F&F%!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Here are the graphs of the paramet ric function given by the cycloid curve and its derivative." }}{PARA 0 "" 0 "" {TEXT -1 27 "Notice that the derivative " }{XPPEDIT 18 0 "dy /dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 12 " approaches " } {XPPEDIT 18 0 "+infinity" "6#%)infinityG" }{TEXT -1 5 " as " } {XPPEDIT 18 0 "x -> 2*n*Pi" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*( \"\"#\"\"\"%\"nGF-%#PiGF-F*F*F*" }{TEXT -1 16 " from the left (" } {TEXT 290 1 "n" }{TEXT -1 19 " an integer), and, " }{XPPEDIT 18 0 "dy/ dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "-infinity;" "6#,$%)infinityG!\"\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> 2*n*Pi" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*(\"\"#\"\" \"%\"nGF-%#PiGF-F*F*F*" }{TEXT -1 17 " from the right (" }{TEXT 289 1 "n" }{TEXT -1 49 " an integer). The derivative is not defined when " } {XPPEDIT 18 0 "x = 2*n*Pi" "6#/%\"xG*(\"\"#\"\"\"%\"nGF'%#PiGF'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "d y/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " is given as a functio n of " }{TEXT 291 1 "t" }{TEXT -1 13 " rather than " }{TEXT 292 1 "x" }{TEXT -1 60 ", we can give the gradient function in parametric form t hus:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ x = t-sin*t,``],[dy/dx = sin*t/(1-cos*t) ,``])" "6#-%*PIECEWISEG6$7$/% \"xG,&%\"tG\"\"\"*&%$sinGF+F*F+!\"\"%!G7$/*&%#dyGF+%#dxGF.*(F-F+F*F+,& F+F+*&%$cosGF+F*F+F.F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "p1 := plot([t-sin(t),1-cos(t),t=0..4*Pi],color=red): \np2 := plot([t-sin(t),sin(t)/(1-cos(t)),t=0..4*Pi],discont=true,color =blue):\np3 := plot([[[2*Pi,-3.6],[2*Pi,3.6]],[[4*Pi,-3.6],[4*Pi,3.6]] ],color=black,\n linestyle=3):\nplots[display]([p1,p2,p3],view= [0..4*Pi,-3.6..3.6],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 568 320 320 {PLOTDATA 2 "6(-%'CURVESG6$7^q7$$\"\"!F)F(7$$\"3#)*z(3#G4B T$!#?$\"3k^V*Q.pzs$!#>7$$\"3hqOaU&**3@#F0$\"3oq-O\"f5NG\"!#=7$$\"3=DAk*fX](F67$$\"3Lp8OM*) 47XF6$\"3zO01Q#Qhs)F67$$\"3-ffK*)[WvcF6$\"3GKT8M'Gu'**F67$$\"3#fH.`E\" 3VqF6$\"3jXaJK/ID6!#<7$$\"3y.az05Iv&)F6$\"3UV'GEQzF^o7$$\"3wYfRY\\;0EF^o$\"3A# e7)Q@\"Q'>F^o7$$\"3eGx;JWzQFF^o$\"3;[.YK,lz>F^o7$$\"3!Gs*\\;^MtGF^o$\" 3O`D\"*y=*4*>F^o7$$\"3yp>Gc/^3IF^o$\"3/2$pNK&y(*>F^o7$$\"3nU#eL3\")R9$ F^o$\"3E?`#pG*****>F^o7$$\"3aZ!yF--+G$F^o$\"3Uu[D*R/w*>F^o7$$\"3Cf(*Q_ Gq:MF^o$\"3i73C(>$f!*>F^o7$$\"3iE.xp/x]NF^o$\"3k`v^C\")**y>F^o7$$\"3#e \"Gi-Z*[o$F^o$\"3_q;7OG(G'>F^o7$$\"3m+#=m+;'RRF^o$\"3Wc0\"Qj1$>>F^o7$$ \"3'QX'\\O&zv=%F^o$\"3#)Q_%**Ra*f=F^o7$$\"3&pNzb)=@0WF^o$\"3Q\">NHMJKz \"F^o7$$\"3&>ysI]>Uh%F^o$\"3:sa,zHF:C&F^o$\"3sr 98kQv$Q\"F^o7$$\"3AbsrL!37S&F^o$\"3+S&RfL\"\\q7F^o7$$\"3_kz!\\i2!pbF^o $\"3pdE9TO'R8\"F^o7$$\"3$**\\E`xVvr&F^o$\"3'Q+%3K(=$[**F67$$\"3!yxJ=%> !G$eF^o$\"3mmG$)[!Rnr)F67$$\"3BU%)[b+&H$fF^o$\"3q.@3g6r/vF67$$\"3%)*e_ yEue-'F^o$\"3gF59fcL>iF67$$\"3$o-8*pO#=5'F^o$\"3u!QRFM\"G.]F67$$\"3w#* \\!z>j$fhF^o$\"3+V0*)Qk:IRF67$$\"3Av7QPzo.iF^o$\"3uPI%=4jz&HF67$$\"32+ amDXPPiF^o$\"3]>^Q,$Q'o?F67$$\"37D7()Hq.giF^o$\"3!G)4eCh(GK\"F67$$\"3o )pxSR4)ziF^o$\"3,>C!#A7$$\"3#3n5'R6d$G'F^o$\"3t`'y$pdCU()F-7 $$\"3s%)4\"3*[['G'F^o$\"3]ZZ19@rXOF07$$\"3-Zii?wE0jF^o$\"3o$*flx9\\#G \"F67$$\"3XSzH04WEjF^o$\"3o@qvEri#*>F67$$\"3!fV&fT0wdjF^o$\"3eYL>]YRQG F67$$\"3w\"3^sx\\DS'F^o$\"3<%pC$4cORQF67$$\"3eLXIvxdhkF^o$\"3#f@p#4!*p ^\\F67$$\"3![7yO'oTMlF^o$\"3K;ddDrNGhF67$$\"3JSK]Y$3Ki'F^o$\"3)Q&)\\\" =9+stF67$$\"3P6dn*3.]s'F^o$\"3uSFD_\"*z=')F67$$\"3`n#)G)>TG%oF^o$\"3K& 3=eVqz))*F67$$\"3o@NWy\\l#*pF^o$\"3Mh5d11%)H6F^o7$$\"3#*Q$*H[OEirF^o$ \"3m?&Rk/#Ho7F^o7$$\"3vG2_zI4JtF^o$\"3ciOVmf?)Q\"F^o7$$\"3Y\\?pO;y9vF^ o$\"3B'=XXeg=]\"F^o7$$\"3m@97&*eFExF^o$\"3p[lk7cB9;F^o7$$\"3UN\"3wTsA& zF^o$\"3Q2![(*RA`r\"F^o7$$\"3Ce^S)Q*>o\")F^o$\"31tcn5Wh&z\"F^o7$$\"3!z I01YqKR)F^o$\"3-!)f7zs*Q'=F^o7$$\"3IBfqVN(zk)F^o$\"3'RAs_U'fB>F^o7$$\" 3<&3F*=7f4*)F^o$\"3S*=\\g5Pm'>F^o7$$\"3!po!eHRTM!*F^o$\"3E#)G3$4\"*3)> F^o7$$\"3MGcMUU,g\"*F^o$\"3C79b$oC7*>F^o7$$\"3sXEfzG9'G*F^o$\"3)*RaYwl f(*>F^o7$$\"3Hq25_%[DT*F^o$\"33k9$[I\")***>F^o7$$\"3`)yQQu+Za*F^o$\"3w XezwF^o7$$\"3E&)*Q>F\"fw'*F^o$\"3kcbgbK1#*>F^o7$$\"3-&QRPrKz!)*F^ o$\"3arbBZDf\")>F^o7$$\"3!e;`V9S%Q**F^o$\"3m**Qlv`$o'>F^o7$$\"317q#QH: !>5!#;$\"3#R$G]Nm&e#>F^o7$$\"3b`T'\\IcN/\"Fbdl$\"3gl9(GFN%p=F^o7$$\"3% [k,*pCOo5Fbdl$\"33In9!Q[[z\"F^o7$$\"35&3%*H\\f?4\"Fbdl$\"3)*RSg7It0![T9R8\"Fbdl$\"3+%>;S mR$*\\\"F^o7$$\"3E$RT&\\p5`6Fbdl$\"3?z'H\"Gbzz8F^o7$$\"3!4R+K7f1<\"Fbd l$\"3E>(=1v;ND\"F^o7$$\"3%[34l/sj=\"Fbdl$\"3df>o$fPQ7\"F^o7$$\"3H@$*[o VN+7Fbdl$\"3eIe^pn)*>**F67$$\"3\"G1!f<\\m67Fbdl$\"33Y+L/_?4()F67$$\"3e -dz^[^@7Fbdl$\"3^whzMiV'F67$$\" 37RL'y#3!)Q7Fbdl$\"3e9\"3$*pt6&\\F67$$\"3j`bUPEJW7Fbdl$\"32h2oh0Y=RF67 $$\"3yt&\\D&ef[7Fbdl$\"3)z]VfXr)zHF67$$\"3[vKgA(HUD\"Fbdl$\"3+rIU[lRd8 F67$$\"3e`dtk!*Hc7Fbdl$\"3W>7G6RY/PF07$$\"3E**fwb\"Fa[m7$$\"3IV6whC/i$)Fe[m$\"3[;@y'zK#o6Fa[m7$$ \"3^h0-lb8AG!#B$\"3i(pwpx#)zy(Fbdl7$$\"3oqdxCDL*o'Fe\\m$\"3xKH6J\"Q2%e Fbdl7$$\"3'RVP%)[$[dAF][l$\"3MxzvE.N$*QFbdl7$$\"3m07M?@_]`F][l$\"3_V5b XN^>HFbdl7$$\"3gnbI;Fbdl7$$\"3'[))[*GnSxUFj]m$\" 3k@$**GHW!e9Fbdl7$$\"3_Zt'Q5QGE\"F-$\"32w:(HiQV,\"Fbdl7$$\"39DvAhE'4z# F-$\"3#HN.->5hw(F^o7$$\"3,'34mHW:#)*F-$\"3su,zhYFh]F^o7$$\"3evmITC:zBF 0$\"3t02-D&p8s$F^o7$$\"3A5y\">>$\\tYF0$\"3J.Y$)eO'G#HF^o7$$\"3wd)e@YK: y(F0$\"3&*\\PQuZy>CF^o7$$\"3/&*4EqZD:7F6$\"3])Hx@()Qr.#F^o7$$\"3`!f50- 2b!=F6$\"3_x*)Ra>aMD,#G>'\\WF6$\"3smxE+!>\\9\"F^o7$$\" 3sM?_#=ORr&F6$\"3!f/f,lIS***F67$$\"3_IO)pV\\z;(F6$\"3FYV%4)H#)=()F67$$ \"3lH$=p\"3^U()F6$\"3v)f`oJZci(F67$$\"3$Q\\k3_+;.\"F^o$\"3?zAL^^(Qs'F6 7$$\"3\\J[4z'>dB\"F^o$\"3``=N#4G4u&F67$$\"3`[GQ8)>AU\"F^o$\"3iOM#=[H\" p\\F67$$\"3?*zwLO,Nl\"F^o$\"3P1l!)Ql+FTF67$$\"3!*op'\\hE(p=F^o$\"32czA &yX8U$F67$$\"3TXA37/k<@F^o$\"3MwAE%320o#F67$$\"3ayj%ynpAO#F^o$\"3)y*Rq STx**>F67$$\"3>SP:H3ICEF^o$\"30zx**y&>zI\"F67$$\"3!>')HN[%3pGF^o$\"3u( e#H]B*Q$oF07$$\"3yMpLH+RNJF^o$\"3)>Fe*[&e1b\"F-7$$\"3)z=,wgK?T$F^o$!3S %4)3nlq\"y'F07$$\"3'zU(\\Go$3l$F^o$!3o!>VnIF67$$\"3GpIRMt(*fTF^o$!3_U>*)**)GXm#F67$$\"3hszGyP:,WF^ o$!3+Q[0z)\\IQ$F67$$\"3g*))fd*R_CYF^o$!3'zu*e[,O4TF67$$\"3#QWus[L\"f[F ^o$!3Q)Q'=QC,i\\F67$$\"35#4eZTFk0&F^o$!3I&fEPsq1y&F67$$\"371eG%[L;D&F^ o$!3kF]')p=8CnF67$$\"3x!f=)4T%QT&F^o$!31Wrv&fRkl(F67$$\"3EW'pfo%\\ubF^ o$!35yR&RO*)>y)F67$$\"3tK$o<\"pK4dF^o$!31[cM&ps%p**F67$$\"3DCppcS>LeF^ o$!3t(Q9)p\"\\#Q6F^o7$$\"3;\"fQs/mv$fF^o$!3'[T'pavg)H\"F^o7$$\"3My=^Mi cHgF^o$!3ak!e7&*y\")\\\"F^o7$$\"3nW'>Bn.A5'F^o$!3(H[-6O8Gt\"F^o7$$\"3E d9_T3JhhF^o$!3>F=sW\"H[.#F^o7$$\"3=rVX\\+11iF^o$!3>,?\"R3RzU#F^o7$$\"3 O+B-G4JOiF^o$!3^SHe1\\p>HF^o7$$\"3;BCTJG'*fiF^o$!3#R0j5/QKv$F^o7$$\"3a -E)4)R*GF'F^o$!3ke?5%f51)\\F^o7$$\"3o=+7-Y9!G'F^o$!3@hEteLXWvF^o7$$\"3 Y9/CXQ&=G'F^o$!3Y3CG<,lk**F^o7$$\"3M(=E#G;w#G'F^o$!3MH\"=.C1FY\"Fbdl7$ $\"3)zB:Z\\1IG'F^o$!31``QbGg^>Fbdl7$$\"3)Rjn-JKJG'F^o$!33tg`bl#)GHFbdl 7$$\"3#3PCo%H;$G'F^o$!32JC=jbw0RFbdl7$$\"3Q?WS\"oyJG'F^o$!3'3N*[O(e$fe Fbdl7$$\"3Aj%e=^#=$G'F^o$!3o)4*[;#4G\"yFbdl7$$\"3QC1#*yW=$G'F^o$!3KrST cm&><\"Fa[m7$$\"3ch`vd\\=$G'F^o$!3[:9h%4DEc\"Fa[m7$$\"3?kek._=$G'F^o$! 3$Rc6GIaRM#Fa[m7$$\"3Q!yPUH&=$G'F^o$!3)\\Dx_CBzo%Fa[m7$$\"3C'ezrI&=$G' F^o$!3av71A'Rq4\"Fjjl7ao7$F``n$\"3a!3HJ>?Bv(!#57$$\"37lcC?`=$G'F^o$\"3 1]*R5W8In%Fa[m7$$\"33!)pq6a=$G'F^o$\"3NS#*))oI]OBFa[m7$$\"31io&*fc=$G' F^o$\"31e:F!Rewb\"Fa[m7$$\"3KkdQVh=$G'F^o$\"3OE2,(*=Bo6Fa[m7$$\"3o'[>$ H\")=$G'F^o$\"3w(Qv$)yyzy(Fbdl7$$\"3w(z>l*>>$G'F^o$\"3lIHFbdl7$$\"33%))=W$eO$G'F^o$\"3ud1D<4RX>Fbdl7$$\"3I)Rrz/8OG'F^o$\"3/ ?PFdT/e9Fbdl7$$\"3o(HBb9[WG'F^o$\"3xxqDe&QV,\"Fbdl7$$\"3-S$R-FwfG'F^o$ \"3%\\0nc\")4hw(F^o7$$\"39eMN_o+$H'F^o$\"3kA'=k]u71&F^o7$$\"3Q>;4Lo(pI 'F^o$\"3\"oY(RU%p8s$F^o7$$\"3dvrJT-#*HjF^o$\"3ljhK3O'G#HF^o7$$\"3+eBuM 1+hjF^o$\"3[Qy#)RZy>CF^o7$$\"3cS#=zy5ZS'F^o$\"3ai8cZ)Qr.#F^o7$$\"3E+5t 8gtjkF^o$\"32[(>j$>aMRH0&y`'F^o$\"3_.h%[rKa\\\"F^o7$$\"3o AUH-=4JmF^o$\"3W\"GFNR(\\%H\"F^o7$$\"3^.;#=CZ\"GnF^o$\"3Fy$p;**=\\9\"F ^o7$$\"3=C\"HB$*yX&oF^o$\"3e)*zE\"eIS***F67$$\"3)y[%3e-)***pF^o$\"3))y b7DH#)=()F67$$\"3%p%3A'ROu:(F^o$\"3A+%)*3FZci(F67$$\"3V>XQNey9tF^o$\"3 A#*pj7^(Qs'F67$$\"3#R3HM*\\!*=vF^o$\"39!*>tg!G4u&F67$$\"3mtDTF^S0xF^o$ \"3gym>b%H\"p\\F67$$\"3eP'*)on'oOzF^o$\"3^517\"H:) F^o$\"3'>^@twX8U$F67$$\"3\"\\Z?UsD3S)F^o$\"3C[--qq]!o#F67$$\"3%Q+K\"*) \\XX')F^o$\"3#)f9gHTx**>F67$$\"3Hl+YRh[2*)F^o$\"3f/7\"4d>zI\"F67$$\"37 s&yGzpA:*F^o$\"3L4ob&H#*Q$oF07$$\"3,&3;wLv&=%*F^o$\"3eog1u#e1b\"F-7$$ \"3+\\*eZ\"z@&p*F^o$!3g>u`mlq\"y'F07$$\"3*H4&pM@-M**F^o$!3kuj?/t6(G\"F 67$$\"3!4&)=?dv(=5Fbdl$!3Sn>A')=lb>F67$$\"3W1R'QE;V/\"Fbdl$!3q6'z9*)GX m#F67$$\"3O8#p\"3RVo5Fbdl$!3IY@Ln)\\IQ$F67$$\"3aqd%)H4x!4\"Fbdl$!3hi;G L,O4TF67$$\"3YxJ$*y=B96Fbdl$!3+U&)H=C,i\\F67$$\"3r.#Q;FhR8\"Fbdl$!3IsT +*pq1y&F67$$\"3Uv/cy=[`6Fbdl$!3=v!p*Q=8CnF67$$\"3WX05THqp6Fbdl$!3k_>1e &Rkl(F67$$\"3s>or)*zw&=\"Fbdl$!3aBP/<$*)>y)F67$$\"3mN@J@7D*>\"Fbdl$!3% 4nMujs%p**F67$$\"30a\\$e$zj67Fbdl$!3PQG`i\"\\#Q6F^o7$$\"3Dq0$\\8v?A\"F bdl$!3k%osaa2')H\"F^o7$$\"3())37P:v7B\"Fbdl$!3g.%H#R*y\")\\\"F^o7$$\"3 cvIb(*)Q&Q7Fbdl$!362w?XL\"Gt\"F^o7$$\"3F9,a9'\\WC\"Fbdl$!3$=#=#H7H[.#F ^o7$$\"3)>&=SNX#*[7Fbdl$!3gcy#G0RzU#F^o7$$\"3]M0KB'\\>D\"Fbdl$!3e\">-8 '[p>HF^o7$$\"3C'oDP\"[Ja7Fbdl$!3El$*3lzB`PF^o7$$\"3GfUtGzgb7Fbdl$!3q(Q WwX51)\\F^o7$$\"3-(e\"*3*HLc7Fbdl$!3,z[\"p.`Wa(F^o7$$\"3-F$>_\"R]c7Fbd l$!3!4GmD \"Fbdl$!3]a[uIEg^>Fbdl7$$\"3-]>uhyOmD\"Fbdl$!39!*GIIb!G\"yFbdl7$$\"3oW)4'ypjc7Fbdl$!33_^nCe&><\"F a[m7$$\"3[&R$\\Eqjc7Fbdl$!3g^O(QhBEc\"Fa[m7$$\"3#3]#3^qjc7Fbdl$!3[BxVm 4&RM#Fa[m7$$\"3mD<9gqjc7Fbdl$!3-#=r4))4zo%Fa[m7$F[jl$!37.&3/[(f8tFh`n- F^jl6&F`jlF(F(Fajl-F$6%7$7$F``n$!33+++++++OF^o7$F``n$\"33+++++++OF^o-F ^jl6&F`jlF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F[jlFafo7$F[jlFdfoFffoFhfo -%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"0#fVhqjc 7!#8;$!#O!\"\"$\"#OFdho" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "The speed of a point moving along a curve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$% \"xG%\"yG" }{TEXT -1 25 " is the position at time " }{TEXT 297 1 "t" } {TEXT -1 25 " of a point in the plane." }}{PARA 0 "" 0 "" {TEXT -1 24 "The parametric equations" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "x = f(t)" "6#/%\"xG-%\"fG6#%\"tG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = g(t)" "6#/%\"yG- %\"gG6#%\"tG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "describe \+ the motion of the point along a curve in the " }{TEXT 295 1 "x" } {TEXT -1 1 "-" }{TEXT 296 1 "y" }{TEXT -1 7 " plane." }}{PARA 0 "" 0 " " {TEXT -1 15 "The derivatives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt = `f '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$f~'G6# %\"tG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt = `g '`(t);" "6#/*&%#dyG\"\"\"%#dtG!\"\"-%$g~'G6#%\"tG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "give the " }{TEXT 259 26 "components of the velocity" }{TEXT -1 26 " of the point parallel to \+ " }{TEXT 298 1 "x" }{TEXT -1 5 " and " }{TEXT 299 1 "y" }{TEXT -1 20 " axes respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "The actual velocity has both " }{TEXT 259 23 "magnitude and direction" }{TEXT -1 19 " determined by the " }{TEXT 259 10 "vect or sum" }{TEXT -1 63 " of these two components as indicated in the fol lowing picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 245 166 166 {PLOTDATA 2 "6)-%'CURVESG6'7$7$\"\"!F(7$F(\"\"\" 7$F'7$F*F(7%7$$!\"$!\"#$\"#bF17$F($\"\"'!\"\"7$$\"\"$F1F27%7$F2F/7$F5F (7$F2F9-%&COLORG6&%$RGBGF(F5F(-F$6%7$F'7$F*F*7%7$F2F57$F5F57$$\"#fF1$ \"#aF1-F@6&FBF(F($\"\")F7-F$6%7$F)FF7$F,FF-%*LINESTYLEG6#F:-%%TEXTG6%7 $$\"\"%F7FQ%:Resultant~velocity~vectorGFO-Fen6%7$$F7F7$\"\"&F7%&g'(t)G F?-Fen6%7$F_oF^o%&f'(t)GF?-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "In particular, the " }{TEXT 259 25 "magnitude o f the velocity" }{TEXT -1 82 ", or speed of motion of the point, is th e length of the resultant vector, which is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(`f '`(t)^2+`g '`(t)^2);" "6#-%%sqrtG6#,& *$)-%$f~'G6#%\"tG\"\"#\"\"\"F.*$)-%$g~'G6#%\"tGF-F.F." }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 366 1 "s" }{TEXT -1 97 " is the distance along the curve, measur ed from some fixed starting point on the curve, then the " }{TEXT 259 5 "speed" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dt = sqrt(``(dx/dt)^2+`` (dy/dt)^2);" "6#/*&%#dsG\"\"\"%#dtG!\"\"-%%sqrtG6#,&*$-%!G6#*&%#dxGF&F 'F(\"\"#F&*$-F/6#*&%#dyGF&F'F(F3F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "The speed of a point on the rim of rolling wheel" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 125 "The parametric equ ations of the cycloid obtained as the locus of a point on the rim of a wheel of radius 1 rolling along the " }{TEXT 304 1 "x" }{TEXT -1 9 " \+ axis are" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x = t-sin*t;" "6#/%\"xG,& %\"tG\"\"\"*&%$sinGF'F&F'!\"\"" }{TEXT -1 2 " ," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1-cos*t;" "6#/%\"yG,&\"\"\"F&*&%$co sGF&%\"tGF&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "where the parameter " }{TEXT 305 1 "t" }{TEXT -1 83 " is the angle through \+ which the wheel has rolled since the point was at the origin." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Now suppo se that the wheel is rotating with a constant angular velocity of 1 ra dian per second." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then the parameter " }{TEXT 306 1 "t" }{TEXT -1 75 " represents the time which has ellapsed since the point was at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivatives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = 1-cos*t;" "6#/*&%#dxG\"\"\"%#dt G!\"\",&F&F&*&%$cosGF&%\"tGF&F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt = sin*t;" "6#/*&%#dyG\"\"\"%#dtG! \"\"*&%$sinGF&%\"tGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 102 "represent the horizontal and vertical components of the velocity \+ of the point on the rim of the wheel." }}{PARA 0 "" 0 "" {TEXT -1 82 " The speed of the point is the magnitude of the resultant velocity vect or, which is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dt = sqrt(``(dx/dt)^2+``(dy/dt)^2); " "6#/*&%#dsG\"\"\"%#dtG!\"\"-%%sqrtG6#,&*$-%!G6#*&%#dxGF&F'F(\"\"#F&* $-F/6#*&%#dyGF&F'F(F3F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "f := t -> t-sin(t); \ng := t -> 1-cos(t);\nsqrt(Diff(f(t),t)^2+Diff(g(t),t)^2);\nvalue(%); \nsimplify(%);\nspd := unapply(%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$sinG6#F-!\"\"F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operato rG%&arrowGF(,&\"\"\"F--%$cosG6#9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)-%%DiffG6$,&%\"tG\"\"\"-%$sinG6#F.!\" \"F.\"\"#F/F/*$)-F+6$,&F/F/-%$cosGF2F3F.F4F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$),&\"\"\"F+-%$cosG6#%\"tG!\"\"\"\"#F+F+ *$)-%$sinGF.F1F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,& \"\"#\"\"\"-%$cosG6#%\"tG!\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ spdGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*$-%%sqrtG6#,&\"\"#\"\"\"-%$cos G6#9$!\"#F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 23 "Thus the speed at time " }{TEXT 302 1 "t " }{TEXT -1 4 " is " }{XPPEDIT 18 0 "ds/dt = sqrt(2-2*cos*t);" "6#/*&% #dsG\"\"\"%#dtG!\"\"-%%sqrtG6#,&\"\"#F&*(F-F&%$cosGF&%\"tGF&F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "We can plot the graph o f the speed as a function of " }{TEXT 303 1 "x" }{TEXT -1 39 " by mean s of the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "x = t-sin*t;" "6#/%\"xG,&%\"tG\"\"\"*&%$sinGF'F&F'!\"\" " }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " ds/dt = sqrt(2-2*cos*t);" "6#/*&%#dsG\"\"\"%#dtG!\"\"-%%sqrtG6#,&\"\"# F&*(F-F&%$cosGF&%\"tGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 95 "In the following graph, the cycloid, or path of a point on the \+ rim of the wheel, is plotted in " }{TEXT 260 3 "red" }{TEXT -1 54 ", a nd the speed of the point on the rim is plotted in " }{TEXT 256 4 "blu e" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot([[t-sin(t),1-cos(t),t=0..4*Pi], [t-si n(t),sqrt(2-2*cos(t)),t=0..4*Pi]],color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 844 170 170 {PLOTDATA 2 "6&-%'CURVESG6$7^q7$\"\"!F(7$$\"1 +y3#G4BT$!#=$\"1_V*Q.pzs$!#<7$$\"1rOaU&**3@#F/$\"1r-O\"f5NG\"!#;7$$\"1 DAk*fX](F57$$\"1p8OM*)47XF5$\"1P01Q# Qhs)F57$$\"1ffK*)[WvcF5$\"1KT8M'Gu'**F57$$\"1'H.`E\"3VqF5$\"1YaJK/ID6! #:7$$\"1/az05Iv&)F5$\"1V'GEQzF]o7$$\"1ZfRY\\;0EF]o$\"1#e7)Q@\"Q'>F]o7$$\"1Hx;JWzQFF]o$\"1[.YK,lz >F]o7$$\"1B(*\\;^MtGF]o$\"1`D\"*y=*4*>F]o7$$\"1q>Gc/^3IF]o$\"12$pNK&y( *>F]o7$$\"1V#eL3\")R9$F]o$\"1?`#pG*****>F]o7$$\"1[!yF--+G$F]o$\"1u[D*R /w*>F]o7$$\"1f(*Q_Gq:MF]o$\"183C(>$f!*>F]o7$$\"1F.xp/x]NF]o$\"1av^C\") **y>F]o7$$\"1;Gi-Z*[o$F]o$\"1r;7OG(G'>F]o7$$\"1,#=m+;'RRF]o$\"1c0\"Qj1 $>>F]o7$$\"1ak\\O&zv=%F]o$\"1R_%**Ra*f=F]o7$$\"1d$zb)=@0WF]o$\"1\">NHM JKz\"F]o7$$\"1#ysI]>Uh%F]o$\"1sa,zHF:C&F]o$\"1s98kQv $Q\"F]o7$$\"1bsrL!37S&F]o$\"1S&RfL\"\\q7F]o7$$\"1lz!\\i2!pbF]o$\"1eE9T O'R8\"F]o7$$\"1+lKvPa!G$eF]o$\"1nG$)[ !Rnr)F57$$\"1U%)[b+&H$fF]o$\"1/@3g6r/vF57$$\"1!f_yEue-'F]o$\"1G59fcL>i F57$$\"1FI\"*pO#=5'F]o$\"1\"QRFM\"G.]F57$$\"1$*\\!z>j$fhF]o$\"1V0*)Qk: IRF57$$\"1v7QPzo.iF]o$\"1QI%=4jz&HF57$$\"1+amDXPPiF]o$\"1?^Q,$Q'o?F57$ $\"1D7()Hq.giF]o$\"1$)4eCh(GK\"F57$$\"1*pxSR4)ziF]o$\"1>C!#?7$$\"1 r1hR6d$G'F]o$\"1a'y$pdCU()F,7$$\"1&)4\"3*[['G'F]o$\"1ZZ19@rXOF/7$$\"1Z ii?wE0jF]o$\"1%*flx9\\#G\"F57$$\"1SzH04WEjF]o$\"1AqvEri#*>F57$$\"1OafT 0wdjF]o$\"1ZL>]YRQGF57$$\"1#3^sx\\DS'F]o$\"1%pC$4cORQF57$$\"1MXIvxdhkF ]o$\"1<#p#4!*p^\\F57$$\"1D\"yO'oTMlF]o$\"1;ddDrNGhF57$$\"1SK]Y$3Ki'F]o $\"1a)\\\"=9+stF57$$\"16dn*3.]s'F]o$\"1TFD_\"*z=')F57$$\"1o#)G)>TG%oF] o$\"1&3=eVqz))*F57$$\"1ANWy\\l#*pF]o$\"1h5d11%)H6F]o7$$\"1R$*H[OEirF]o $\"1@&Rk/#Ho7F]o7$$\"1H2_zI4JtF]o$\"1jOVmf?)Q\"F]o7$$\"1\\?pO;y9vF]o$ \"1'=XXeg=]\"F]o7$$\"1A97&*eFExF]o$\"1\\lk7cB9;F]o7$$\"1N\"3wTsA&zF]o$ \"12![(*RA`r\"F]o7$$\"1e^S)Q*>o\")F]o$\"1tcn5Wh&z\"F]o7$$\"13`gg/F$R)F ]o$\"1!)f7zs*Q'=F]o7$$\"1BfqVN(zk)F]o$\"1CAFDkfB>F]o7$$\"1&3F*=7f4*)F] o$\"1*=\\g5Pm'>F]o7$$\"1(o!eHRTM!*F]o$\"1#)G3$4\"*3)>F]o7$$\"1GcMUU,g \"*F]o$\"179b$oC7*>F]o7$$\"1YEfzG9'G*F]o$\"1SaYwlf(*>F]o7$$\"1q25_%[DT *F]o$\"1k9$[I\")***>F]o7$$\"1*yQQu+Za*F]o$\"1YezwF]o7$$\"1&)*Q>F \"fw'*F]o$\"1dbgbK1#*>F]o7$$\"1&QRPrKz!)*F]o$\"1sbBZDf\")>F]o7$$\"1mJN W,WQ**F]o$\"1+Rlv`$o'>F]o7$$\"17q#QH:!>5!#9$\"1MG]Nm&e#>F]o7$$\"1aT'\\ IcN/\"Fadl$\"1m9(GFN%p=F]o7$$\"1X;!*pCOo5Fadl$\"1Hn9!Q[[z\"F]o7$$\"1&3 %*H\\f?4\"Fadl$\"1SSg7It0![T9R8\"Fadl$\"1%>;SmR$*\\\"F]o7$$\"1%RT&\\p5`6Fadl$\"1x'H\"Gbzz8F ]o7$$\"1\"R+K7f1<\"Fadl$\"1>(=1v;ND\"F]o7$$\"1&34l/sj=\"Fadl$\"1c>o$fP Q7\"F]o7$$\"1@$*[oVN+7Fadl$\"1Je^pn)*>**F57$$\"1j+f<\\m67Fadl$\"1Y+L/_ ?4()F57$$\"1.dz^[^@7Fadl$\"1xhzMiV'F57$$\"1RL'y#3!)Q7Fadl$\"1:\"3$*pt6&\\F57$$\"1abUPEJW7Fadl$\"1u2oh0Y =RF57$$\"1u&\\D&ef[7Fadl$\"13N%fXr)zHF57$$\"1vKgA(HUD\"Fadl$\"1rIU[lRd 8F57$$\"1adtk!*Hc7Fadl$\"1>7G6RY/PF/7$$\"1$\"1M*[Z[&[o8F57$F*$ \"1[!zfOc0t#F57$$\"1G'\\uKKW+\"F/$\"1./H9u\\0RF57$F1$\"1urw?zdm]F57$F7 $\"1=$QUp]1N'F57$F<$\"1-(4?l?ig(F57$FA$\"1G92-GiN))F57$FF$\"1H*z\")=%[ -5F]o7$FK$\"1VG!R,:j6\"F]o7$FP$\"1ZjLMr6D7F]o7$FU$\"1gTWW12@8F]o7$FZ$ \"1n=`S&3>T\"F]o7$F_o$\"1>'f-\"QE#e\"F]o7$Fio$\"1$\\(3G_!3t\"F]o7$Fcp$ \"1$p)\\-JJ[=F]o7$F]q$\"1qx5FOF]o7$Fbq$\"1;d8%f^?'>F]o7$Fgq$\"1z;(G ZB=)>F]o7$Far$\"1!>ij&3\\&*>F]o7$F[s$\"1GEYV'*****>F]o7$Fes$\"1T)z]0\" H&*>F]o7$F_t$\"1\"zvtX\\8)>F]o7$Fdt$\"1:UKIzBf>F]o7$Fit$\"1^m_4lqG>F]o 7$Fcu$\"18Vp1L<_=F]o7$F]v$\"1\\=%R,Prs\"F]o7$Fgv$\"1lj#o+YSf\"F]o7$Faw $\"1[*35Ta0T\"F]o7$Ffw$\"1i0/q*e.K\"F]o7$F[x$\"1gKhv%H^A\"F]o7$F`x$\"1 x*>V\"zG:6F]o7$Fex$\"1XbX!3G.+\"F]o7$Fjx$\"16bvc0%e'))F57$F_y$\"1R_vBe ]\"p(F57$Fdy$\"1(>#>6l;KkF57$Fiy$\"1SB]NwoV^F57$$\"1]$QU)=(GF'F]o$\"1R (eN,&zRRF57$F^z$\"182:\"**\\3s#F57$Fcz$\"1],SXt`%R\"F57$Fhz$\"1Y<'\\(o h-iF,7$F^[l$\"1p@(3O*GA8F57$Fc[l$\"1E$pItj-q#F57$$\"1q$Qyh0JH'F]o$\"1l \")pSF[*)QF57$Fh[l$\"1DfZxgck]F57$F]\\l$\"1:Ai?q)GJ'F57$Fb\\l$\"1f8W$4 ZW`(F57$Fg\\l$\"1\"ovT*p$Gw)F57$F\\]l$\"1*RJJz\"e^**F57$Fa]l$\"1\\nXe/ 526F]o7$Ff]l$\"10_sb)[U@\"F]o7$F[^l$\"1\"***GW.#HJ\"F]o7$F`^l$\"1]()=k &piS\"F]o7$Fj^l$\"1pf7>dm#f\"F]o7$Fd_l$\"1$)Ql$oCJt\"F]o7$F^`l$\"1zf1* )**>_=F]o7$Fh`l$\"1fJ$*R(\\2$>F]o7$F]al$\"1/wnGiUh>F]o7$Fbal$\"1*\\!** )R[K)>F]o7$F\\bl$\"1R=I=vg&*>F]o7$Ffbl$\"1))QR_1****>F]o7$F`cl$\"1Yr(H oFg*>F]o7$Fjcl$\"1Y+io$[L)>F]o7$F_dl$\"1QMN?#yD'>F]o7$Fedl$\"1e^3DfhL> F]o7$F_el$\"1ps=Pa,Z=F]o7$Fiel$\"18F]o7$Fggl$\"15x^2x;E7F]o7$F\\h l$\"1&y%*HQJL6\"F]o7$Fahl$\"1R\"QL\"R0^**F57$Ffhl$\"1PPQf'QE&))F57$F[i l$\"1=<%)=r$*>xF57$$\"1EGT`>&>D\"Fadl$\"13c\"yj:%zkF57$F`il$\"1)H^Bkn. @&F57$$\"1fW-o#ybD\"Fadl$\"1@vz3d;uRF57$Feil$\"1_r_tX$>s#F57$$\"1,mb$p %fc7Fadl$\"1S\"*>4T9k8F5Fiil-F]jl6&F_jlF(F(F`jl-%+AXESLABELSG6$%!GFgjm -%%VIEWG6$%(DEFAULTGF[[n" 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 81 "Notice that the speed is instantaneously \+ 0 when the point is in contact with the " }{TEXT 300 1 "x" }{TEXT -1 80 " axis, and that the speed is at its maximum when the point is furt hest from the " }{TEXT 301 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ta sks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 77 "This question i s concerned with the curve given by the parametric equations: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = t^3-3* t, ``],[y = 2/(1+t^2), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,&*$%\"tG\"\"$ \"\"\"*&F,F-F+F-!\"\"%!G7$/%\"yG*&\"\"#F-,&F-F-*$F+F5F-F/F0" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in te rms of the parameter " }{TEXT 357 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 104 "(b) Find the equation of the two tangent lines at t he point of self-intersection of the curve given by " }{XPPEDIT 18 0 "t=``" "6#/%\"tG%!G" }{TEXT 361 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt(3)" "6#-%%sqrtG6#\"\"$" }{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=4*t/(3*(1+t^2)^2*t*(1-t^2)" "6 #/*&%#dyG\"\"\"%#dxG!\"\"*(\"\"%F&%\"tGF&**\"\"$F&*$,&F&F&*$F+\"\"#F&F 1F&F+F&,&F&F&*$F+F1F(F&F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "(b) The point of self-intersection is" }{XPPEDIT 18 0 "``(0,1/2 )" "6#-%!G6$\"\"!*&\"\"\"F(\"\"#!\"\"" }{TEXT -1 11 " given by " } {XPPEDIT 18 0 "t=``" "6#/%\"tG%!G" }{TEXT 359 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT -1 52 ". The tangen t lines to the curve at this point are " }{XPPEDIT 18 0 "y=``" "6#/% \"yG%!G" }{TEXT 360 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/24; " "6#*&-%%sqrtG6#\"\"$\"\"\"\"#C!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x+1/2" "6#,&%\"xG\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 244 188 188 {PLOTDATA 2 "6'-%'CU RVESG6$7_o7$$!$)[\"\"!$\"35xI#p2Bp2$!#>7$$!3;,**y\"*4i\\U!#:$\"3g#QM2U )**eLF-7$$!3_/nb!>\\mu$F1$\"3piaF1$\"35l:uCB9)R&F-7$$!33Z`!Ri'He;F1$\"3Ff#G+=Z6-'F-7$ $!3/M-iggG-m)!#;$\"3:=c\"p'eIS))F-7$$!3yj/!Q[5=*oFin$\"3\"3OrqN6n+\"!#=7$ $!3OWd-M/;$>&Fin$\"3]`t;TGJx6Fao7$$!3e+sq\")y(\\x$Fin$\"3/Id(>I2]R\"Fa o7$$!3YLkE$*\\_dEFin$\"3/N7a![q`m\"Fao7$$!3fQoCgXpN=Fin$\"3iBUPhWO#)>F ao7$$!3M7i=`ndt5Fin$\"3wDu!Q8ug[#Fao7$$!3m\"*H'))G`3\"f!#<$\"3!>=T>d%3 pIFao7$$!3HE4P1p.T>F^q$\"3#3P)pbL8@SFao7$$\"3L-O!)RE#R/$Fao$\"3o-f!== \"RP_Fao7$$\"3q2[pA;)45\"F^q$\"3H9z'H4uy6'Fao7$$\"3zPG@`J]G;F^q$\"3SK \"*>`nm'>(Fao7$$\"3M&)z%zd@*3>F^q$\"3s*Hzx#GxY%)Fao7$$\"3s7[@:#=***>F^ q$\"3#yb<&p(oz%**Fao7$$\"3M$\\l&y=3\")>F^q$\"3Y%eL9\\EP3\"F^q7$$\"3`)* [t_?r@>F^q$\"332!)3V2!)z6F^q7$$\"3@#)=M5seD=F^q$\"3;Q@rp%pBG\"F^q7$$\" 3y(H0S#e['p\"F^q$\"3sjWF^q7$$\"35L,9W2=pBF-$\"3M.H()o_()**>F^q7$$!3\\2z]PWj1]Fao$\"3 ,:U8)p%zW>F^q7$$!3:n6LE9qN**Fao$\"3Cjgp`GU(y\"F^q7$$!3u*oAj8Q**=\"F^q$ \"3eD&4pr?\")p\"F^q7$$!35c*4`t'Rr8F^q$\"3I\")\\a:q/-;F^q7$$!3MC_F^q$\"3=_kZ7\"4&)>\"F^q7$$!3U?\\ ?n`Qr>F^q$\"3B0'epFFU5\"F^q7$$!3qgVE%o'>**>F^q$\"3`R1>xKa;5F^q7$$!3%yT \"=Y`HC>F^q$\"3I^N&[@'op&)Fao7$$!3Cl*3W4Cgk\"F^q$\"3k=)>$e&o$\\sFao7$$ !3aHc!*[*3![6F^q$\"3U@;i#Q)3*='Fao7$$!3UE:*>0SB)RFao$\"38Tomv\"=(=`Fao 7$$\"3)=D%px(QG(=F^q$\"3#*p;4qi+YSFao7$$\"3!G*[i9P^reF^q$\"3il2O*)p_vI Fao7$$\"3zym9%R?85\"Fin$\"3mp(HYM@8Y#Fao7$$\"3&p?N&eI\"f$=Fin$\"3%=$*4 _zdA)>Fao7$$\"3)))=26hato#Fin$\"3]P\"))F-7$$\"3/0C@;- y,6F1$\"3Em^6T9+&o(F-7$$\"3SJ<4Ej$yN\"F1$\"3SEy+,^1!z'F-7$$\"33'**)*fX chl\"F1$\"3c*z_b]ae-'F-7$$\"39.w9\"y:2*>F1$\"3#G4a=0*e(Q&F-7$$\"3\"48[ M,%4MBF1$\"35'3GU(4x&)[F-7$$\"3/w_w%fWBx#F1$\"3C?+*o@3V?,]bOF-7$$\"3c0H\">7M: D%F1$\"3kgF.E?/eLF-7$$\"$)[F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F[al-F $6%7S7$$!\")F*$!36tD'*=p-NxF-7$$!3OLLLLbC^wF^q$!3h8S#QN3\"=_F-7$$!3?mm mOhzZtF^q$!30$R(=o4:GIF-7$$!3LLLL`b`1qF^q$!3-!RgaZ[Jl&!#?7$$!3#HLLL(G, jmF^q$\"3'GB)e\\l%Q\">F-7$$!30nmm'*G7@jF^q$\"3q;*f`#\\A\"Q%F-7$$!3XLLL Br9/gF^q$\"3e&GiSk\"Fao7$$!3HLLL`xdVVF^q$\"3)Gm$eoFHl=Fao7$$!3I+ ++![z%)*RF^q$\"3\\15%[*fM9@Fao7$$!35++++U'>l$F^q$\"370Rdd=UkBFao7$$!3/ +++?D.=LF^q$\"3<&eZ#*G;ag#Fao7$$!3SLLLj0z9IF^q$\"3\"fW')3KiU#GFao7$$!3 !pmmma1Ul#F^q$\"3G;ku>9\\%3$Fao7$$!3=nmm'eW([BF^q$\"3]/zy+'R\\I$Fao7$$ !3S+++5(>M*>F^q$\"3/$\\zACt8c$Fao7$$!3Unmm')p*)y;F^q$\"33Nz+m/O)y$Fao7 $$!3l******4d\"QL\"F^q$\"3\"eoxfU,u.%Fao7$$!3*)******Hn@05F^q$\"33Jd#H JZXF%Fao7$$!3qkmmm;eBmFao$\"3i;)=x;%)>_%Fao7$$!3Wnmmm(p]Z$Fao$\"3Y$GX= W3#\\ZFao7$$!3?\"[LLLLu*yFcbl$\"3!f'QU=0I%*\\Fao7$$\"3c`mmmVh[MFao$\"3 1vd6/B))[_Fao7$$\"3x\"******p!R>lFao$\"3An'\\p\\'\\qaFao7$$\"3AemmmK\" f$)*Fao$\"3;v%Hl*e%)4dFao7$$\"3W******f0AE8F^q$\"3E?5lYs6dfFao7$$\"3M) *****>kTh;F^q$\"3[HQl@S-*>'Fao7$$\"3u)*****\\ct&)>F^q$\"3%)oO3l73LkFao 7$$\"3e)*****fo$eM#F^q$\"3_J\\CG>'Hp'Fao7$$\"3?KLL8QSpEF^q$\"3-mZviiZE pFao7$$\"3p*******f!)[,$F^q$\"3WD2[dE!e<(Fao7$$\"3%fmmm\"R$zK$F^q$\"3z \")4HG%HjF^q$\"3i)H,+)elg&* Fao7$$\"3?LLL8p&Qn'F^q$\"3\"[b2oNTk\")*Fao7$$\"33mmmE/'3*pF^q$\"3p(Q]O *=_/5F^q7$$\"3Q+++!H_)GtF^q$\"318.JbV\"*G5F^q7$$\"3O+++ION_wF^q$\"3kQo _`5E_5F^q7$$\"\")F*$\"3uD'*=p-Nx5F^q-Fe`l6&Fg`lF[al$\"*++++\"FaalF[al- %*THICKNESSG6#\"\"#-F$6%7S7$F`alFc`m7$Feal$\"39S#QN3\"=_5F^q7$Fjal$\"3 /u=o4:GI5F^q7$F_bl$\"3]gaZ[Jl05F^q7$Febl$\"3;w6/X`h3)*Fao7$Fjbl$\"3M3S Y2v(=c*Fao7$F_cl$\"3*3F)QZ*>JL*Fao7$Fdcl$\"3-W[3WZD'4*Fao7$Ficl$\"3Ul# *)>T)G^))Fao7$F^dl$\"3*yZF+p2rg)Fao7$Fcdl$\"3UlQ'Q!y$fN)Fao7$Fhdl$\"36 PjTJsqM\")Fao7$F]el$\"3%H**e^+ac)yFao7$Fbel$\"3*[4EC9ybj(Fao7$Fgel$\"3 #[T_2r$e%R(Fao7$F\\fl$\"33aN6zwtvrFao7$Fafl$\"3G%e`-e3b\"pFao7$Fffl$\" 31'47#*Rg]p'Fao7$F[gl$\"3(p]?xvE'QkFao7$F`gl$\"3#\\1#*R`R;@'Fao7$Fegl$ \"3=9B-u&)fifFao7$Fjgl$\"3YpU2(o_as&Fao7$F_hl$\"3Q$=\"GKe,yaFao7$Fdhl$ \"3)fra\"e:z]_Fao7$Fihl$\"3aLhd\"[*p0]Fao7$F^il$\"3QCU)ep<6v%Fao7$Fcil $\"3yK.0.N]HXFao7$Fhil$\"3GC0Z.T:!H%Fao7$F]jl$\"3H!)*[Lv#)G/%Fao7$Fbjl $\"32rhMyf(4!QFao7$Fgjl$\"3gIj\"\\t=pc$Fao7$F\\[m$\"3[o]vr!QqI$Fao7$Fa [m$\"3)RBXstBN2$Fao7$Ff[m$\"37v#>DM(>CGFao7$F[\\m$\"3w=!4K8OHF^BFao7$Fe\\m$\"3bVA.5u')=@Fao7$Fj\\m$\"3Uf#[1nOf(=Fao7$ F_]m$\"3cVR?puUQ;Fao7$Fd]m$\"3/@*)\\Ytx*Q\"Fao7$Fi]m$\"3Q'*ecDxH]6Fao7 $F^^m$\"3)zGQNC_R0*F-7$Fc^m$\"34CS+&R1_i'F-7$Fh^m$\"3A3q)**>TMR%F-7$F] _m$\"3QYW#>V'eN=F-7$Fb_m$!3@(o(Q]O*=_%Fcbl7$Fg_m$!3e08.JbV\"*GF-7$F\\` m$!3,jQo_`5E_F-7$Fa`mFbalFe`mFi`m-%+AXESLABELSG6%Q!6\"Fbjm-%%FONTG6#%( DEFAULTG-%%VIEWG6$;F`alFa`mFgjm" 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" }}{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := t -> t^3-3*t;\ng := t->2/(1+t^2);\nDiff(g(t),t)/Diff(f(t) ,t);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6 $%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"F1*&F0F1F/F1!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrow GF(,$*&\"\"#\"\"\",&F/F/*$)9$F.F/F/!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%DiffG6$,$*&\"\"#\"\"\",&F*F**$)%\"tGF)F*F*!\"\"F*F .F*-F%6$,&*$)F.\"\"$F*F**&F5F*F.F*F/F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\",&F&F&*$)%\"tG\"\"#F&F&!\"#F*F&,&*&\"\"$F&F)F&F&F /!\"\"F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "eliminate(\{x=f(t),y=g(t)\},t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$<#/%\"tG,$**\"\"#!\"\"%\"xG\"\"\"%\"yGF,,&*&F)F,F-F ,F,F,F*F*F*<#,,*&)F+F)F,)F-\"\"$F,F,*&\"#;F,F4F,F,*&\"#[F,)F-F)F,F**& \"#OF,F-F,F,\"\")F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "implicitdiff(x^2*y^3+16*y^3-48*y^2+36*y- 8,y,x);\nsubs(\{x=f(t),y=g(t)\},%);\nsimplify(%);\neval(%,t=sqrt(3)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\"\"\"$!\"\"%\"xGF&% \"yGF',**&)F)F%F&)F*F%F&F&*&\"#;F&F.F&F&*&\"#KF&F*F&F(\"#7F&F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"#;\"\"\"\"\"$!\"\",&*$)%\"tGF'F& F&*&F'F&F,F&F(F&,&F&F&*$)F,\"\"#F&F&!\"$,**(\"\"%F&F)F1F.!\"#F&*&\"#kF &F.F6F&*&F8F&F.F(F(\"#7F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*, \"\"%\"\"\"\"\"$!\"\"%\"tGF&,&*$)F)F%F&F&F&F(F(,&F&F&*$)F)\"\"#F&F&F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#C!\"\"\"\"$#\"\"\"\"\"#F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "p1 := plot([f(t),g(t),t=-8..8]):\np2 := plot([sqrt(3 )/24*x+1/2,-sqrt(3)/24*x+1/2],x=-8..8,color=green,thickness=2):\nplots [display]([p1,p2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 44 "____________ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 " " {TEXT -1 77 "This question is concerned with the curve given by the \+ parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 1+2*sin*t, ``],[y = cos^2*t, ``]);" "6#-%*PIECEWI SEG6$7$/%\"xG,&\"\"\"F**(\"\"#F*%$sinGF*%\"tGF*F*%!G7$/%\"yG*&%$cosGF, F.F*F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a for mula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in terms of the parameter " }{TEXT 358 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Find the equation of the tangent li ne to the curve at the point where " }{XPPEDIT 18 0 "t=Pi/6" "6#/%\"tG *&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "(c) Show that the cartesian equation of the curve is " }{XPPEDIT 18 0 "4*y+x^2-2*x-3=0" "6#/,**&\"\"%\"\"\"%\"yGF'F'*$%\"xG\"\"#F'*&F+F 'F*F'!\"\"\"\"$F-\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "(d) Plot the graphs of the curve and the tangent line from part (a) in the same picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "dy/dx=-sin*t" "6#/*&%#dyG \"\"\"%#dxG!\"\",$*&%$sinGF&%\"tGF&F(" }{TEXT -1 7 " (b) " } {XPPEDIT 18 0 "y=-1/2" "6#/%\"yG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x+1/2" "6#,&%\"xG\"\"\"*&F%F%\"\"#!\"\"F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eliminate(\{x=1+2*sin(t),y=cos(t)^2\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<#/%\"tG-%'arctanG6$,&*&\"\"#!\"\"%\"xG\" \"\"F/#F/F,F--%'RootOfG6#,&%\"yGF-*$)%#_ZGF,F/F/<#,**$)F.F,F/F/*&F,F/F .F/F-\"\"$F-*&\"\"%F/F5F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "implicitdiff(4*y+x^2-2*x-3,y ,x);\nsubs(\{x=1+2*sin(t),y=cos(t)^2\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\"%\"xG\"\"\"F&#F(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sinG6#%\"tG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "a := eval(1+2*sin(t),t=P i/6);\nf := x -> -x^2/4+x/2+3/4;\nf(a)+D(f)(a)*(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&#\"\"\"\"\"%F/*$)9$\"\"#F/F /!\"\"*&#F/F4F/F3F/F/#\"\"$F0F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"(\"\"%\"\"\"*&\"\"#!\"\"%\"xGF'F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot([[1+2*sin(t), cos(t)^2,t=-Pi..Pi],[x,-x^2/4+x/2+3/4,x=-1..3]],thickness=[1,2]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([-x^2/4+x/2+3/4,7/4-1/2*x],x=-1.4..3.4,thickness=[1,2]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 77 "This qu estion is concerned with the curve given by the parametric equations: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = t^ 2-1, ``],[y = t/(1+t^2), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,&*$%\"tG\" \"#\"\"\"F-!\"\"%!G7$/%\"yG*&F+F-,&F-F-*$F+F,F-F.F/" }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 27 " in terms of th e parameter " }{TEXT 362 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Find the points on the curve where the tangent lines \+ are horizontal. " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Find the points o n the curve where the tangent lines are vertical. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "dy/dx = (1-t^2)/(2*t*(1+t^2) ^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*$%\"tG\"\"#F(F&*(F-F&F,F&,&F& F&*$F,F-F&F-F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "(b) Th e tangent lines are horizontal at the points" }{XPPEDIT 18 0 " ``(0,1/ 2)" "6#-%!G6$\"\"!*&\"\"\"F(\"\"#!\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(0,-1/2)" "6#-%!G6$\"\"!,$*&\"\"\"F)\"\"#!\"\"F+" }{TEXT -1 11 " given by " }{XPPEDIT 18 0 "t = ``" "6#/%\"tG%!G" }{TEXT 363 1 "+ " }{TEXT -1 4 " 1. " }}{PARA 0 "" 0 "" {TEXT -1 45 "(c) The tangent li ne is vertical at the point" }{XPPEDIT 18 0 "``(-1, 0);" "6#-%!G6$,$\" \"\"!\"\"\"\"!" }{TEXT -1 11 " given by " }{XPPEDIT 18 0 "t = 0;" "6# /%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 310 176 176 {PLOTDATA 2 "6%-%'CURVESG6$7V7$$\"39************ *G%!#<$!3-L[CuxfcO!#=7$$\"3')or!4nD)QQF*$!3a0<8>QUnPF-7$$\"3cM9$)p)>EY $F*$!3cuwsiiJR1?rrBUF-7$$\"3]r1`\\)R(z>F*$!3\"* f!zF4]uL%F-7$$\"3-YIuu`(Gm\"F*$!3u*=Cf8d]X%F-7$$\"3)zi4+W0RN\"F*$!3)eY ]jj+Xd%F-7$$\"3Em\"R$[:)[1\"F*$!3)\\_JS`*\\)o%F-7$$\"3sm*z$G&*HtyF-$!3 Uc'Q_V%Q'z%F-7$$\"3!R=j,-![%f&F-$!3W>x)GZ$4z[F-7$$\"3VZo\\su%\\@$F-$!3 W*ef\"*)=#=&\\F-7$$\"3a_=&Q*QtB5F-$!3M)4`#p'oS*\\F-7$$!3W17S%oD4+*!#>$ !3]iMm$)[W%*\\F-7$$!3A!*\\_IZS([#F-$!3Kx@FZ!3$\\\\F-7$$!3?nJ$pDAq<%F-$ !3[Qg;\"\\ME#[F-7$$!3m#RxcP%=SaF-$!3YskT*Rhyj%F-7$$!3]7tZ$Hqar'F-$!3Z# )QiO?59VF-7$$!395B:QBD$[F\\p7$$!3igP84up,**F-$\"3_hoNf.D=)*F\\p7$$!39!=u-bzYz*F-$\"3y+ hn`A2/9F-7$$!3O\"HqXk!p['*F-$\"3?8>(=z72\"=F-7$$!3HxqkS0M+#*F-$\"3&[2; V5R%=EF-7$$!3M_4Pw9>Y&)F-$\"3ve[-N\"F*$\"3AH1rg5EvXF-7$$\"3!4u\\#)fI2m\"F*$\"3)>!e H`<(eX%F-7$$\"3/7!yq]Qd)>F*$\"3W!pN0Iz_L%F-7$$\"3:URDsO*3I#F*$\"3CDAa` CKCUF-7$$\"3MdRUN'R:o#F*$\"3y@w*[M5&)4%F-7$$\"3;M(3W&peRIF*$\"3qRoh`)o \"))RF-7$$\"3Ua-ns(Q'RMF*$\"3_3!H_v,N(QF-7$$\"35Z'>YMF-%QF*$\"3]ERu8^1 nPF-7$F($\"3-L[CuxfcOF--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F_\\lF^\\l-%+ AXESLABELSG6$Q!6\"Fc\\l-%%VIEWG6$%(DEFAULTGFh\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}{TEXT -1 3 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f := t -> t^2-1;\ng := t->t/ (1+t^2);\nDiff(g(t),t)/Diff(f(t),t);\nvalue(%);\nsimplify(%);\nsolve(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG% &arrowGF(,&*$)9$\"\"#\"\"\"F1F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F. F.*$)F-\"\"#F.F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%Di ffG6$*&%\"tG\"\"\",&F)F)*$)F(\"\"#F)F)!\"\"F(F)-F%6$,&F+F)F)F.F(F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\",&*&\"\"\"F),&F)F)*$)%\" tGF%F)F)F&F)*(F%F)F-F%F*!\"#F&F)F-F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\",&*$)%\"tGF%\"\"\"F+F+F&F+,&F+F+F(F+!\"#F*F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "tt := -1;\n``(f(tt ),g(tt));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ttG!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%!G6$\"\"!#!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eliminate(\{x=f(t),y=g(t)\},t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$<#/%\"tG,&*&\"\"#\"\"\"%\"yGF*F**&%\"xGF*F+F*F*<#,, F-!\"\"F*F0*&\"\"%F*)F+F)F*F**(F2F*F-F*F3F*F**&)F-F)F*F3F*F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "implicitdiff(-x-1+4*y^2+4*x*y^2+x^2*y^2,y,x);\nsubs(\{x=f(t),y=g(t )\},%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\" \",(\"\"\"F&*&\"\"%F()%\"yGF%F(F(*(F%F(%\"xGF(F+F(F(F(F,F&,(F*F(*&F*F( F.F(F(*$)F.F%F(F(F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#!\" \",(\"\"\"F&*(\"\"%F(%\"tGF%,&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(*$)F1F%F(F(F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\",&*$)%\"tGF%\"\"\"F+F+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 30 "plot([f(t),g(t),t=-2.3..2.3]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 77 "This qu estion is concerned with the curve given by the parametric equations: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = t^ 2*(t+1), ``],[y = t*(t+1), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%\"tG\" \"#,&F*\"\"\"F-F-F-%!G7$/%\"yG*&F*F-,&F*F-F-F-F-F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 9 "(a) Find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 307 1 "t" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Find the points on t he curve where the tangent lines are horizontal. " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Find the points on the curve where the tangent lines \+ are vertical. " }}{PARA 0 "" 0 "" {TEXT -1 54 "(d) Show that the carte sian equation of the curve is " }{XPPEDIT 18 0 "y^3=x*(x+y)" "6#/*$% \"yG\"\"$*&%\"xG\"\"\",&F(F)F%F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(e) Find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#dxG!\"\"" }{TEXT -1 13 " in terms of " }{TEXT 364 1 "x" } {TEXT -1 5 " and " }{TEXT 365 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 43 "(f) Show that the expressions obtained for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 37 " in parts (a) a nd (e) are equivalent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a ) \+ " }{XPPEDIT 18 0 "dy/dx = (2*t+1)/(t*(3*t+2));" "6#/*&%#dyG\"\"\"%#dxG !\"\"*&,&*&\"\"#F&%\"tGF&F&F&F&F&*&F-F&,&*&\"\"$F&F-F&F&F,F&F&F(" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "(b) The tangent line is horizontal at the points" }{XPPEDIT 18 0 " ``(1/8,-1/4)" "6#-%!G6$*& \"\"\"F'\"\")!\"\",$*&F'F'\"\"%F)F)" }{TEXT -1 11 " given by " } {XPPEDIT 18 0 "t = -1/2;" "6#/%\"tG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "(c) The tangent lines are ver tical at the points" }{XPPEDIT 18 0 "``(0, 0);" "6#-%!G6$\"\"!F&" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(4/27,-2/9)" "6#-%!G6$*&\"\"%\"\" \"\"#F!\"\",$*&\"\"#F(\"\"*F*F*" }{TEXT -1 11 " given by " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t=-2 /3" "6#/%\"tG,$*&\"\"#\"\"\"\"\"$!\"\"F*" }{TEXT -1 15 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "dy/dx=(2*x+y)/ (3*y^2-x)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&*&\"\"#F&%\"xGF&F&%\"yGF&F&, &*&\"\"$F&*$F.F,F&F&F-F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 236 212 212 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$!3=,+ ++++q]!#=$\"3o+++++++RF*7$$!3O=#[&RU`'f%F*$\"3m-6.>%p**e$F*7$$!31c\\PY k$\\9%F*$\"3w+W7;dj(G$F*7$$!3%3@H'yBWpPF*$\"3BZ*y8!)Q3.$F*7$$!35g'GrXi )4MF*$\"3)\\%[N@*o)zFF*7$$!30]@^.\"=T-$F*$\"3'y=-pU)e/DF*7$$!3u)RXZ:*p dEF*$\"3%zW%)**exmB#F*7$$!3+$)[sY(RzI#F*$\"3'p&e)='RVu>F*7$$!3?$)4)4&) **o(>F*$\"3O8a)o,e'>P\"F*$\"3()oZxhQvM7F*7$$!3JRq'f#\\ nG()!#>$\"3zmN9v^<;\")Fgn7$$!3k<$4\"\\\"Hr:%Fgn$\"3#fO.'3x%G+%Fgn7$$!3 3r#od7,'[Jz)Fgn$!3GpQ8QO# \\*)*Fgn7$$\"3)>*=*\\C\\X3\"F*$!3%e*)*)o6(>w7F*7$$\"3)3qawBLOC\"F*$!3z \"H0Q\"yuL:F*7$$\"3oVt$*pEqb8F*$!3!QS%3[x=`F*7$$\"3E:QkZ'f4Z\"F*$!3Q+4f/ue0@F*7$$\"3!3ZGaVA9[\"F*$!3E sX0,FFIAF*7$$\"3qfVbXwcw9F*$!3Q'RD^xG>H#F*7$$\"3%3T&481Ek9F*$!3)4&)yjA &fXBF*7$$\"3yF]XaM`Z9F*$!3]%)Hz4IV'Q#F*7$$\"3[/\\*e*[nD9F*$!3o+#RWs55U #F*7$$\"3ywuViP='R\"F*$!3eE<%Q,YK#[93\"F*$!31u!zJC0HY #F*7$$\"3&*p-g-\\TG'*Fgn$!3dsg')G!pOBtTN2 BF*7$$\"3kL\"zx(y)[=(Fgn$!3m;]^R:_*>#F*7$$\"3%efBS@t'\\fFgn$!3S/Gn?)[i 0#F*7$$\"3Sj\\***z@Dr%Fgn$!3k]w+KX**y=F*7$$\"3)*H!ejboJd$Fgn$!3[*y0YEI on\"F*7$$\"3a([=Lu1\"oDFgn$!3'Q)G0a]:a9F*7$$\"3OTHsDh2+;Fgn$!3I3<^C&>d <\"F*7$$\"3)\\F'Qt%=x%*)!#?$!3_7J(z>c`(*)Fgn7$$\"376PYS1vuMF_x$!3!ea]N %Qh7dFgn7$$\"3whz[%e)[*Q'Fbo$!3%*=I(zOp^\\#Fgn7$$\"3Y;m(fmKWo\"Fbo$\"3 ;M1.B&)>18Fgn7$$\"3R73gXZsJDF_x$\"3Q!fn[?IP:&Fgn7$$\"3tN0U?1=E#)F_x$\" 3!pp%**Q26c%*Fgn7$$\"3MK#f*37xG " 0 "" {MPLTEXT 1 0 88 "f := t -> t^2*(t+1);\ng := t-> t*(t+1);\nDiff(g(t),t)/Diff(f(t),t);\n value(%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"tG6\"6$%)operatorG%&arrowGF(*&)9$\"\"#\"\"\",&F.F0F0F0F0F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrow GF(*&9$\"\"\",&F-F.F.F.F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&- %%DiffG6$*&%\"tG\"\"\",&F)F)F(F)F)F(F)-F%6$*&)F(\"\"#F)F*F)F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*&\"\"#F%%\"tGF%F%F%,&*(F' F%F(F%,&F%F%F(F%F%F%*$)F(F'F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*(,&\"\"\"F%*&\"\"#F%%\"tGF%F%F%F(!\"\",&F'F%*&\"\"$F%F(F%F%F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eliminate(\{x=f(t),y=g(t)\},t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$<#/%\"tG*&%\"xG\"\"\"%\"yG!\"\"<#,(*$)F*\"\"$F)F)*&F(F)F*F)F+*$)F( \"\"#F)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "implicitdiff(y^3-x*y-x^2=0,y,x);\nsubs(\{x=f(t),y=g (t)\},%);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"yG \"\"\"*&\"\"#F'%\"xGF'F'F',&*&\"\"$F')F&F)F'!\"\"F*F'F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&%\"tG\"\"\",&F(F(F'F(F(F(*(\"\"#F()F'F+ F(F)F(F(F(,&*(\"\"$F(F,F()F)F+F(!\"\"*&F,F(F)F(F(F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&\"\"\"F%*&\"\"#F%%\"tGF%F%F%F(!\"\",&F'F%*&\" \"$F%F(F%F%F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "tt := -2/3; \n``(f(tt),g(tt));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ttG#!\"#\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%!G6$#\"\"%\"#F#!\"#\"\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([f(t),g(t),t=-1.3..0.5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 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 dr awing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "Code circle picture " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 623 "cs : = .6:\nsn := .8:\ndd := evalf(arctan(4/3)):\np1 := plot([cos(t),sin(t) ,t=0..2*Pi],color=red,thickness=1):\np2 := plot([[[0,0],[cs,sn]],[[cs, 0],[cs,sn]]],color=black):\np3 := plot([0.25*cos(t),0.25*sin(t),t=0..d d],color=black):\np4 := plot([[.52,0],[.52,.08],[.6,.08]],color=black) :\np5 := plot([[[cs,sn]]$3],style=point,symbol=[circle,diamond,cross], color=black):\nt1 := plots[textplot]([[.93,.93,`P( cos t,sin t )`],\n \+ [.22,.54,`1`],[1.15,-.04,`x`],[-.05,1.15,`y`],\n [-.06,-.09,`O`]], color=black):\nt2 := plots[textplot]([.15,.09,`t`],color=black):\nplot s[display]([p1,p2,p3,p4,p5,t1,t2],tickmarks=[0,0],scaling=constrained) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 716 "cs := .6:\nsn := .8:\ndd := evalf(arctan(4/3)):\np1 \+ := plot([cos(t),sin(t),t=0..2*Pi],color=red,thickness=1):\np2 := plot( [[[0,0],[cs,sn]],[[cs,0],[cs,sn]]],color=black):\np3 := plot([0.25*cos (t),0.25*sin(t),t=0..dd],color=black):\np4 := plot([[.52,0],[.52,.08], [.6,.08]],color=black):\np5 := plot([[[cs,sn]]$3],style=point,symbol=[ circle,diamond,cross],color=black):\np6 := plot([[.1,1.175],[1.1,.425] ],color=green,thickness=2):\nt1 := plots[textplot]([[.93,.93,`P( cos t ,sin t )`],\n [.22,.54,`1`],[1.15,-.04,`x`],[-.05,1.15,`y`],\n [-. 06,-.09,`O`],[1.17,.53,`B`],[.2,1.27,`A`]],color=black):\nt2 := plots[ textplot]([.15,.09,`t`],color=black):\nplots[display]([p1,p2,p3,p4,p5, p6,t1,t2],tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "Code for examples pictures" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "tt : = evalf(7*Pi/3):xx := tt*cos(tt): yy := tt*sin(tt):\np1 := plot([t*cos (t),t*sin(t),t=0..6*Pi],color=red,thickness=2):\np2 := plot([[[xx,yy]] $3],style=point,symbol=[circle,diamond,cross],color=black):\np3 := plo t([[0,0],[xx,yy]],color=black):\np4 := plot([t/4*cos(t),t/4*sin(t),t=2 *Pi..13*Pi/3],\n color=COLOR(RGB,.5,.5,.5),linestyle=1):\nt1 := pl ots[textplot]([[22,-.8,`x`],[-.8,17,`y`],[3.74,1.5,`t`],\n [8.6,6 .5,`( t cos t, t sin t )`]]):\nplots[display]([p1,p2,p3,p4,t1],tickmar ks=[5,5],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "tt := evalf(9*Pi/4):xx := tt*cos(t t): yy := tt*sin(tt):\np1 := plot([t*cos(t),t*sin(t),t=0..6*Pi],color= red,thickness=1):\np2 := plot([[[xx,yy]]$3],style=point,symbol=[circle ,diamond,cross],color=black):\np3 := plot(9*Pi/(4*sqrt(2))+((4+9*Pi)/( 4-9*Pi))*(x-9*Pi/(4*sqrt(2))),\n x=-3..18,color=green,thickness= 2):\nt1 := plots[textplot]([[22,-.8,`x`],[-.8,17,`y`]]):\nplots[displa y]([p1,p2,p3,t1],tickmarks=[5,5],labels=[``,``],\n view=[-16..22,-1 8..17]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 524 "p1 := plot([.5+.5*cos(t),.5*sin(t),t=0..2*Pi],color= red):\np2 := plot([[[.5,.5],[.5,-.5]]$3],style=point,symbol=[circle,di amond,cross],color=black):\np3 := plot([[[.1,.5],[.9,.5]],[[.1,-.5],[. 9,-.5]]],color=green,thickness=2):\np4 := plot([[[0,0]]$2],style=point ,symbol=circle,symbolsize=[16,20],color=red):\nt1 := plots[textplot]([ [1.1,-.03,`x`],[-.03,.6,`y`],\n [.5,.6,`t = 1/2`],[.5,-.6,`t = - 1/2`]]):\nplots[display]([p1,p2,p3,p4,t1],xtickmarks=4,labels=[``,``], \n ytickmarks=[-.4=`-0.4`,-.2=`-0.2`,.2=`0.2`,.4=`0.4`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "p 1 := plot([[3*t/(1+t^3),3*t^2/(1+t^3),t=-0.67..50],\n [3*t/(1+t^3), 3*t^2/(1+t^3),t=-50..-1.5]],color=red):\np2 := plot(3-x,x=-.5..3.5,col or=green,thickness=2):\np3 := plot([[[1.5,1.5]]$3],style=point,symbol= [circle,diamond,cross],color=black):\np4 := plot([[-3,2],[2,-3]],color =black,linestyle=2):\nt1 := plots[textplot]([[3.8,-.2,`x`],[-.2,3.8,`y `],[1.9,1.8,`t = 1`]]):\nplots[display]([p1,p2,p3,p4,t1],tickmarks=[4, 4],labels=[``,``],\n view=[-3..3.8,-3..3.8],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 545 "rt2 := evalf(sqrt(2)):\np1 := plot([1-cos(t)-sin(t),1-cos(t)+ sin(t),t=0..2*Pi],\n color=red):\np2 := plot([[[1,1+rt2],[1,1-rt2],[ 1+rt2,1],[1-rt2,1]]$3],\n style=point,symbol=[circle,diamond,cross], color=black):\np3 := plot([[[0,1+rt2],[2,1+rt2]],[[0,1-rt2],[2,1-rt2]] ,\n [[1+rt2,0],[1+rt2,2]],[[1-rt2,0],[1-rt2,2]]],color=[green$2,blu e$2],thickness=2):\nt1 := plots[textplot]([[2.53,-.2,`x`],[-.2,2.53,`y `],\n [1,2.6,`P`],[1,-.6,`Q`],[2.6,1,`R`],[-.6,1,`S`]]):\nplots[di splay]([p1,p2,p3,t1],tickmarks=[3,3],labels=[``,``],scaling=constraine d);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "p1 := plot([t^3,t^2-1,t=-1.5..1.5],\n color=red): \np2 := plot([[[-8/27,-5/9]]$3],style=point,symbol=[circle,diamond,cro ss],color=black):\np3 := plot([[-46/27,23/27],[23/27,-46/27]],color=gr een,thickness=2):\nt1 := plots[textplot]([[3.3,-.2,`x`],[-.2,1.7,`y`], [-.5,-.75,`P`]]):\nplots[display]([p1,p2,p3,t1],tickmarks=[3,3],labels =[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Code for cycloid picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 852 "t1 := evalf(Pi/3):\nt2 := evalf(4*Pi/3):\np1 \+ := plot([t-sin(t),1-cos(t),t=0..8],color=red,thickness=2):\np2 := plot ([[[t1-sin(t1),1-cos(t1)],[t2-sin(t2),1-cos(t2)]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\np3 := plot([[cos (t)+t1,sin(t)+1,t=0..2*Pi],[[t1,0],[t1,1]],[[t1,1],\n [t1-sin(t1), 1-cos(t1)]]],color=black):\np4 := plot([[cos(t)+t2,sin(t)+1,t=0..2*Pi] ,[[t2,0],[t2,1]],[[t2,1],\n [t2-sin(t2),1-cos(t2)]]],color=black): \np5 := plot([seq([t1+.35*cos(i*h),1+.35*sin(i*h)],i=70..90)],color=bl ack):\np6 := plot([seq([t2+.31*cos(i*h),1+.31*sin(i*h)],i=10..90)],col or=black):\ntxt1 := plots[textplot]([[7,-.1,`x`],[-.1,1.9,`y`],[.91,.8 3,`t`],\n [4.05,1.07,`t`]],font=[HELVETICA,10]):\ntxt2 := plots[te xtplot]([[.98,.76,`1`],[4.13,1.,`2`]],font=[HELVETICA,7]):\nplots[disp lay]([p1,p2,p3,p4,p5,p6,txt1,txt2],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "old " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "txt := plots[textplot]([[1.2,1.1,`C`],[1.2 ,.6,`Q`],[.2,.8,`P`],\n[1.1,-.15,`T`],[-.1,-.15,`O`],[7,-.1,`x`],[-.1, 1.9,`y`]]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1328 "t1 := evalf(Pi/3): h := evalf(Pi/60):\np1 := plo t([t-sin(t),1-cos(t),t=0..2.6],color=red,thickness=2):\np2 := plot([[c os(t)+t1,sin(t)+1,t=0..2*Pi],[[t1,0],[t1,1]],[[t1,1],\n [t1-sin(t1), 1-cos(t1)]],[[t1-sin(t1),1-cos(t1)],[t1,1-cos(t1)]]],\n color=black ):\np3 := plot([[[t1,1],[t1+.2,1]],[[t1,1-cos(t1)],[t1+.2,1-cos(t1)]], \n [[t1-sin(t1),1-cos(t1)],[t1-sin(t1),1-cos(t1)-.2]]],\n colo r=black,linestyle=2):\np4 := plot([[[t1-sin(t1),1-cos(t1)]]$3],style=p oint,\n symbol=[circle,diamond,cross],color=black):\np5 := \+ plottools[arrow]([t1+.15,1-cos(t1)/2-.08],[t1+.15,1-cos(t1)],\n \+ 0,.05,.23,arrow,color=black):\np6 := plottools[arrow]([t1+.15,1-cos (t1)/2+.08],[t1+.15,1],\n 0,.05,.23,arrow,color=black):\np7 : = plottools[arrow]([t1-sin(t1)/2-.17,1-cos(t1)-.15],[t1-sin(t1),1-cos( t1)-.15],\n 0,.05,.13,arrow,color=black):\np8 := plottools[ar row]([t1-sin(t1)/2+.17,1-cos(t1)-.15],[t1,1-cos(t1)-.15],\n 0 ,.05,.13,arrow,color=black):\np9 := plot([seq([t1+.22*cos(i*h),1+.22*s in(i*h)],i=70..90)],color=black):\ntxt := plots[textplot]([[1.04,1.08, `C`],[1.13,.43,`Q`],[.2,.65,`P`],\n [1.05,-.08,`T`],[-.08,-.1,`O`],[2 .2,-.06,`x`],[-.06,1.95,`y`],\n [t1+.15,1-cos(t1)/2,`cos t`],[t1-sin( t1)/2,1-cos(t1)-.15,`sin t`],[.98,.88,`t`]]):\nplots[display]([p1,p2,p 3,p4,p5,p6,p7,p8,p9,txt],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "Code for velocity vector picture" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "PLOT(CURVES([[0,0],[0,1]],[[0,0],[1,0]],[[-.03,.55],[0,.6],[.03,. 55]],\n[[.55,-.03],[.6,0],[.55,.03]],COLOR(RGB,0,.6,0)),\nCURVES([[0,0 ],[1,1]],[[.55,.6],[.6,.6],[.59,.54]],COLOR(RGB,0,0,.8)),\nCURVES([[0, 1],[1,1]],[[1,0],[1,1]],LINESTYLE(3)),\nTEXT([.4,0.8],'`Resultant velo city vector`',COLOR(RGB,0,0,.8)),\nTEXT([-.1,0.5],'`g'(t)`',COLOR(RGB, 0,.6,0)),\nTEXT([.5,-.1],'`f'(t)`',COLOR(RGB,0,.6,0)),\nAXESSTYLE(NONE ));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }