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To study the contour diagrams (or level curves) of surfaces . " }}{PARA 0 "" 0 "" {TEXT -1 80 "2. To use contour diagrams to fin d whether a function is continuous at a point." }}{PARA 0 "" 0 "" {TEXT -1 11 "3. To use " }{TEXT 371 5 "Maple" }{TEXT -1 52 " to draw \+ a graph and a contour diagram of a surface." }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 259 22 " Required Information:" }}{PARA 0 "" 0 "" {TEXT -1 18 "Given a function " }{TEXT 372 1 "z" }{TEXT -1 5 " = \+ " }{TEXT 373 1 "f" }{TEXT -1 1 "(" }{TEXT 374 1 "x" }{TEXT -1 2 ", " }{TEXT 375 1 "y" }{TEXT -1 5 "), a " }{TEXT 376 15 "contour diagram" } {TEXT -1 37 " is defined as the family of curves " }{TEXT 377 1 "c" } {TEXT -1 5 " = " }{TEXT 378 1 "f" }{TEXT -1 1 "(" }{TEXT 379 1 "x" } {TEXT -1 2 ", " }{TEXT 380 1 "y" }{TEXT -1 11 ") in the " }{TEXT 381 3 "x-y" }{TEXT -1 16 " plane, where " }{TEXT 382 1 "c" }{TEXT -1 96 " is a constant. This equation represents the curve obtained a s the intersection of the plane " }{TEXT 383 5 "z = c" }{TEXT -1 20 " with the surface " }{TEXT 384 1 "z" }{TEXT -1 5 " = " }{TEXT 385 1 "f" }{TEXT -1 1 "(" }{TEXT 386 1 "x" }{TEXT -1 2 ", " }{TEXT 387 1 " y" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 42 "or the curve at hei ght c for the surface " }{TEXT 388 1 "z" }{TEXT -1 5 " = " }{TEXT 389 1 "f" }{TEXT -1 1 "(" }{TEXT 390 1 "x" }{TEXT -1 2 ", " }{TEXT 391 1 "y" }{TEXT -1 3 "). " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 260 18 " Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 41 "1. D raw a graph of each of the surfaces." }}{PARA 0 "" 0 "" {TEXT -1 52 "2 . Draw a contour diagram for each of the surfaces." }}{PARA 0 "" 0 " " {TEXT -1 44 "3. Describe each graph and contour diagram." }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{TEXT 392 2 " z" }{TEXT -1 8 " = 6 + 2" } {TEXT 393 1 "x" }{TEXT -1 4 " + 3" }{TEXT 394 1 "y" }{TEXT -1 12 " \+ (b) " }{TEXT 395 1 "z" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "-x^2-y^ 2;" "6#,&*$%\"xG\"\"#!\"\"*$%\"yGF&F'" }{TEXT -1 12 " (c) " } {TEXT 396 1 "z" }{TEXT -1 8 " = exp( " }{XPPEDIT 18 0 "-x^2-y^2;" "6#, &*$%\"xG\"\"#!\"\"*$%\"yGF&F'" }{TEXT -1 19 ") (d) " } {TEXT 397 1 "z" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(x^2+y^2);" "6#- %%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*" }{TEXT -1 25 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 261 1 " " }{TEXT 262 9 "Solution:" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 263 40 " (a) The graph and conto ur diagram of " }{TEXT 398 2 "z " }{TEXT 399 11 " = 6 + 2" }{TEXT 400 1 "x" }{TEXT 401 6 " + 3" }{TEXT 402 1 "y" }{TEXT 403 1 " " }} {PARA 0 "" 0 "" {TEXT -1 33 "We activate the plotting package:" }} {PARA 0 "" 0 "" {TEXT -1 0 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}{TEXT 327 1 " " }{TEXT 328 10 "function " }{TEXT 408 1 "f " }{TEXT 409 1 "(" }{TEXT 410 1 "x" }{TEXT 411 2 ", " }{TEXT 412 1 "y " }{TEXT 413 6 ") is " }{TEXT 414 21 "continuous at a point" }{TEXT 415 1 " " }{TEXT 417 1 "(" }{XPPEDIT 416 0 "x[0],y[0];" "6$&%\"xG6#\" \"!&%\"yG6#F&" }{TEXT 418 1 ")" }{TEXT -1 15 " if the value " }{TEXT 419 1 "f" }{TEXT -1 1 "(" }{TEXT 420 1 "x" }{TEXT -1 2 ", " }{TEXT 421 1 "y" }{TEXT -1 24 ") approaches the value " }{TEXT 422 2 " f" } {TEXT 329 1 "(" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F& " }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 5 "as (" }{TEXT 423 1 " x" }{TEXT -1 2 ", " }{TEXT 424 1 "y" }{TEXT -1 15 ") approaches " } {TEXT 330 1 "(" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F& " }{TEXT -1 55 ") . This means that there is no break in the surface \+ " }{TEXT 425 1 "f" }{TEXT -1 1 "(" }{TEXT 426 1 "x" }{TEXT -1 2 ", " }{TEXT 427 1 "y" }{TEXT -1 20 ") in any direction " }}{PARA 0 "" 0 " " {TEXT -1 22 "surrounding the point " }{TEXT 331 2 " (" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 71 "). From all \+ sides surrounding the point on the surface, the change is " }}{PARA 0 "" 0 "" {TEXT -1 35 "gradual. The difference between " }{TEXT 428 1 "f" }{TEXT -1 1 "(" }{TEXT 429 1 "x" }{TEXT -1 2 ", " }{TEXT 430 1 " y" }{TEXT -1 8 ") and " }{TEXT 431 2 " f" }{TEXT 332 1 "(" } {XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 30 ") goes to zero as the point " }{TEXT 432 1 "a" }{TEXT -1 4 " = (" } {TEXT 433 1 "x" }{TEXT -1 2 ", " }{TEXT 434 1 "y" }{TEXT -1 13 ") appr oaches " }}{PARA 0 "" 0 "" {TEXT 435 1 "b" }{TEXT -1 3 " = " }{TEXT 333 1 "(" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" } {TEXT -1 41 ") . In symbols, this can be written as " }{XPPEDIT 436 1 "limit(abs(f(x,y)-f(x[0], y[0])),a = b);" "6#-%&limitG6$-%$absG6#,&- %\"fG6$%\"xG%\"yG\"\"\"-F+6$&F-6#\"\"!&F.6#F4!\"\"/%\"aG%\"bG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "If a function is continuous at \+ every point on the plane, then the function is " }{TEXT 437 10 "contin uous" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 73 "All functions in \+ Solved Example 1 are examples of continuous functions. " }}{PARA 0 " " 0 "" {TEXT -1 100 "The contour plots of discontinuous functions show abnormal changes in one or more directions or else" }}{PARA 0 "" 0 " " {TEXT -1 54 "contour curves do not exist in one or more directions. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 273 18 " Solved Exam ple 2:" }}{PARA 0 "" 0 "" {TEXT -1 25 " Consider the function " } {TEXT 274 1 "f" }{TEXT -1 14 ", given by " }{TEXT 275 4 "f = " } {XPPEDIT 18 0 "xy/(x^2+y^2);" "6#*&%#xyG\"\"\",&*$%\"xG\"\"#F%*$%\"yGF )F%!\"\"" }{TEXT -1 9 " for ( " }{TEXT 276 4 "x, y" }{TEXT -1 20 ") \+ <> (0, 0) and " }{TEXT 277 4 "f = " }{TEXT 438 1 "0" }{TEXT 439 2 " " }{TEXT -1 4 "for " }{TEXT 278 0 "" }{TEXT -1 2 " (" }{TEXT 279 4 "x, y" }{TEXT -1 10 ") = (0, 0)" }}{PARA 0 "" 0 "" {TEXT -1 44 "(the s ymbol <> standing for \"not equal to\")." }}{PARA 0 "" 0 "" {TEXT -1 43 "(a) Graph and draw a contour diagram for " }{TEXT 280 1 "f" } {TEXT -1 46 ". Describe the graph and the contour diagram." }}{PARA 0 "" 0 "" {TEXT -1 16 "(b) Show that " }{TEXT 284 1 "f" }{TEXT -1 3 "(0," }{TEXT 283 1 "y" }{TEXT -1 9 ") and " }{TEXT 285 1 "f" } {TEXT -1 1 "(" }{TEXT 286 1 "x" }{TEXT -1 52 ", 0) are each continuou s functions of one variable." }}{PARA 0 "" 0 "" {TEXT -1 9 "(c) Is \+ " }{TEXT 281 1 "f" }{TEXT -1 1 "(" }{TEXT 282 4 "x, y" }{TEXT -1 25 ") continuous at (0,0) ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 287 10 "Solution: " }}{PARA 257 " " 0 "" {TEXT 290 45 "We draw the graph for the nonzero values of " } {TEXT 440 3 "x y" }{TEXT 441 1 "." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 288 41 " (a) The graph and contour diagram of " } {XPPEDIT 18 0 "xy/(x^2+y^2);" "6#*&%#xyG\"\"\",&*$%\"xG\"\"#F%*$%\"yGF )F%!\"\"" }{TEXT -1 3 " " }{TEXT 289 13 "for nonzero " }{TEXT 442 3 "x y" }{TEXT 443 2 ". 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a6**FctFj[lFbb^l7$7$$\"3YZ&Qk!\\a6**FctFj[l7$$\"3LMD;G<(y6*Fct$!3k*=7' z.%Q)=F-7$7$$\"3eM#G&3BTK()FctFf\\lF_c^l7$7$Ffc^lF77$Fa^m$!3'e$4*ow`\" y;F-7$Fca^l7$$\"3\"o9XXzg]Uc\"p7F-7$7$Fdhs$!3VZ9(=4W5I\"F-F[i^l7$7$Fi^n$ !37:l%Gvf\"e9F-7$$\"3K]R`!\\Wew#F-Fhp7$Fhi^lFai^l7$7$Fgis$!3C'[96T)H3; F-7$$\"3ELt.;/))oKF-Fhn7$F`j^lFei^l7$F]j^l7$$\"3Yj^7TsGLNF-$!3_sQ%3Vl* *o\"F-7$7$Fcan$!3%Gqg$[@LgF-7$$\"3mc!=WTC8 x$F-F77$7$$\"35d!=WTC8x$F-Ff\\l7$$\"3Q)***********zOF-$!3i-2O[@Lg " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 291 21 "(b) Continuity of " }{TEXT 292 1 "f" }{TEXT 293 4 "(0, " }{TEXT 294 2 "y)" }{TEXT 295 8 " and " }{TEXT 296 1 "f " }{TEXT 297 1 "(" }{TEXT 298 1 "x" }{TEXT 299 5 ", 0):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 300 9 " f" }{TEXT -1 4 "(0, " }{TEXT 301 1 "y" }{TEXT -1 42 ") is a function defined by substituting " } {TEXT 302 1 "x" }{TEXT -1 10 " = 0 in " }{XPPEDIT 18 0 "xy/(x^2+y^2) " "6#*&%#xyG\"\"\",&*$%\"xG\"\"#F%*$%\"yGF)F%!\"\"" }{TEXT -1 18 " , \+ which is 0. " }}{PARA 0 "" 0 "" {TEXT -1 20 " Similarly, " } {TEXT 303 1 "f" }{TEXT -1 1 "(" }{TEXT 304 1 "x" }{TEXT -1 18 ", 0) i s also 0. " }}{PARA 0 "" 0 "" {TEXT -1 12 " As " }{TEXT 305 2 "y " }{TEXT -1 47 " approaches 0, the function remains 0. Hence " }{TEXT 306 1 "f" }{TEXT -1 4 "(0, " }{TEXT 307 1 "y" }{TEXT -1 21 ") \+ is continuous at " }{TEXT 444 1 "y" }{TEXT -1 12 " = 0 and " }} {PARA 0 "" 0 "" {TEXT 308 8 " f" }{TEXT -1 1 "(" }{TEXT 310 3 "x , " }{TEXT -1 22 "0) is continuous at " }{TEXT 445 1 "x" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 309 6 " (c) " } {TEXT 311 0 "" }{TEXT 312 16 "Continuity of " }{TEXT 313 1 "f" } {TEXT 314 1 "(" }{TEXT 315 4 "x, y" }{TEXT 316 2 "):" }}{PARA 0 "" 0 " " {TEXT -1 92 " Examining the contour diagrams, we conclude that the re is a break in the contour diagram " }}{PARA 0 "" 0 "" {TEXT -1 20 " along the lines " }{TEXT 446 5 "y = x" }{TEXT -1 8 " and " } {TEXT 447 6 "y = -x" }{TEXT -1 30 ". In fact, if we substitute " } {TEXT 448 5 "y = x" }{TEXT -1 18 " in the function." }}{PARA 0 "" 0 " " {TEXT -1 20 " " }{TEXT 317 1 "f" }{TEXT -1 1 "(" }{TEXT 318 4 "x, x" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "xx/(x^2+x^2); " "6#*&%#xxG\"\"\",&*$%\"xG\"\"#F%*$F(F)F%!\"\"" }{TEXT -1 7 " = \+ " }{XPPEDIT 18 0 "x^2/(2*x^2);" "6#*&%\"xG\"\"#*&F%\"\"\"*$F$F%F'!\"\" " }{TEXT -1 5 " = " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 31 " when we cancel the nonzero " }{TEXT 319 1 "x" }{TEXT -1 3 "'s." }}{PARA 0 "" 0 "" {TEXT -1 24 " This indicates that " } {TEXT 320 1 "f" }{TEXT -1 1 "(" }{TEXT 321 4 "x, y" }{TEXT -1 15 ") a pproaches " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 7 " as (" }{TEXT 325 4 "x, y" }{TEXT -1 1 ")" }{TEXT 323 2 " " } {TEXT -1 36 "approaches (0, 0) along the line " }{TEXT 449 5 "y = x " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 " Since the limit i s not the value of the function " }{TEXT 334 1 "f" }{TEXT -1 47 "(0, \+ 0) = 0, there is a discontinuity at (0, 0)" }{TEXT 324 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 " The substitution of " }{TEXT 335 1 "y" } {TEXT -1 4 " = -" }{TEXT 336 3 "x " }{TEXT -1 8 "gives " }{TEXT 337 1 "f" }{TEXT -1 1 "(" }{TEXT 450 1 "x" }{TEXT -1 3 ", -" }{TEXT 451 1 "x" }{TEXT -1 5 ") = -" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"# !\"\"" }{TEXT -1 1 "." }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "____________________________________________________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 338 13 " Problem 1: " }}{PARA 0 "" 0 "" {TEXT 364 65 " Draw the graphs and contour diagrams for each of the functions. " }} {PARA 0 "" 0 "" {TEXT 367 57 "Describe in your own words the graph and contour diagram:" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 339 3 "(a) " }{TEXT 347 2 " " }{TEXT 340 1 "f" }{TEXT 452 1 "(" }{TEXT 341 4 "x, y" }{TEXT 342 5 ") = " }{TEXT 343 3 "y " }{TEXT 344 8 "- sin (" } {TEXT 345 1 "x" }{TEXT 346 12 ") (b) " }{TEXT 453 2 " f" }{TEXT 454 1 "(" }{TEXT 348 8 "x,y) = " }{XPPEDIT 18 0 "x^2-y^2;" "6#,&*$%\" xG\"\"#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 17 " " }{TEXT 349 4 "(c) " }{TEXT 455 1 "f" }{TEXT 456 1 "(" }{TEXT 350 1 "x" } {TEXT 351 2 ", " }{TEXT 352 1 "y" }{TEXT 353 4 ") = " }{XPPEDIT 18 0 " exp(-(x^2+y^2));" "6#-%$expG6#,$,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+!\"\"" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 354 13 " Problem 2: \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the function " }{TEXT 355 1 "f" }{TEXT -1 14 ", given by " }{TEXT 356 4 "f = " }{XPPEDIT 18 0 "(x^2-y^2)/(x^2+y^2);" "6#*&,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(,&*$ F&F'F(*$F*F'F(F+" }{TEXT -1 3 " (" }{TEXT 365 4 "x, y" }{TEXT -1 16 " ) <> (0, 0) , " }{TEXT 366 2 "f " }{TEXT -1 11 "(0, 0) = 0." }} {PARA 0 "" 0 "" {TEXT -1 43 "(a) Graph and draw a contour diagram for " }{TEXT 357 1 "f" }{TEXT -1 46 ". Describe the graph and the conto ur diagram." }}{PARA 0 "" 0 "" {TEXT -1 16 "(b) Show that " }{TEXT 361 1 "f" }{TEXT -1 3 "(0," }{TEXT 360 1 "y" }{TEXT -1 8 ") and " } {TEXT 362 1 "f" }{TEXT -1 1 "(" }{TEXT 363 1 "x" }{TEXT -1 51 ", 0) a re not continuous functions of one variable." }}{PARA 0 "" 0 "" {TEXT -1 9 "(c) Is " }{TEXT 358 1 "f" }{TEXT -1 1 "(" }{TEXT 359 4 "x, y" }{TEXT -1 25 ") continuous at (0,0) ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 65 "___________________________________ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 172 "MSIP G rant #P120A80089-98: \"Three Urban Calculus Reform programs: Adopting the Best\" 1998-2001, MSEIP Grant #P120A010031: \"Four Colleges: Calc ulus + Enhancements\" 2001-04" }}}{MARK "12 3 22" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 1 33 1 1 }