AFunctionalBestiary(Powerpoint).ppt

Continuous but not differentiable: corners (e.g., absolute value, ); points of verticality (e.g., cube root); cusps (e.g., cube root of absolute value) A warm-up: Find a function that is continuous for all real numbers but non-differentiable at every integer. One of the strangest beasts in function history is Karl Weierstrass’s function that is continuous everywhere but differentiable nowhere! Since his time, simpler functions with this property have been constructed. Also, we now know that “most” functions that are continuous everywhere are, in fact, differentiable nowhere! Define G(x) to be the distance from x to the nearest integer to x. Note that each of these functions has half the amplitude and half the period of its predecessor. Now add them all up: This function is continuous everywhere and differentiable nowhere. Thanks to Tom Vogel’s Gallery of Calculus Pathologies! Some miscellaneous functions from the bestiary… This is a pretty normal-looking quadratic function until you carefully consider a table of values… …which we do in the next slide. All these function values are primes! 1601 39 911 29 421 19 131 9 1523 38 853 28 383 18 113 8 1447 37 797 27 347 17 97 7 1373 36 743 26 313 16 83 6 1301 35 691 25 281 15 71 5 1231 34 641 24 251 14 61 4 1163 33 593 23 223 13 53 3 1097 32 547 22 197 12 47 2 1033 31 503 21 173 11 43 1 971 30 461 20 151 10 41 0 y x y x y x y x * A Functional Bestiary Dan Kennedy Baylor School Chattanooga, TN A bestiary, or Bestiarum vocabulum is a compendium of beasts. Bestiaries were made popular in the Middle Ages in illustrated volumes that described various animals, birds and even rocks. The natural history and illustration of each beast was usually accompanied by a moral lesson. From WikipediA, the free encyclopedia: Most functions we encounter in the real world are fairly tame and domesticated. They are continuous. They are differentiable. They are smooth. Have a nice day! Polynomials Trig Functions Rational Func