FUNCTIONS This slide show consists of graphs of various functions. When viewing the slides, the following keys are operational: HOME takes you to the first slide in the sequence you selected END takes you to the last slide in the sequence you selected UP ARROW takes you to the previous slide in the sequence you selected F9 immediately quit the program These keys do NOT operate like that while you are reading this document. A. sin(1/x) This function is first graphed with -100 < x < 100, and -1.5 < y < 1.5. Notice how the local behavior is hidden and what happens for x large. Then we zoom in and look at -10 < x < 10, -1 < x < 1, -.1 < x < .1, -.01 < x < .01, -.001 < x < .001. The zoomed regions are first indicated by a box. The y range remains constant throughout this. Notice how the computer has trouble plotting as the scale gets smaller. B. x sin(1/x) The same procedure is used as in the last case, except that the following ranges are used -100 < x < 100, and -1.5 < y < 1.5 -10 < x < 10, and -1.5 < y < 1.5 -1 < x < 1, and -1.5 < y < 1.5 0 < x < 1, and -1 < y < 1 0 < x < .1, and -.1 < y < .1 0 < x < .01, and -.01 < y < .01 0 < x < .001, and -.001 < y < .001 Notice the y values have been adjusted. C. Continuous, but not differentiable Here it is, a function which is continuous everywhere, and differentiable nowhere. The first slide shows x between 0 and 1, then 0 and 1/10, then 0 and 1/100, then 0 and 1/1,000, then 0 and 1/10,000, and, finally, between 0 and 1/100,000, a magnification of 100,000 over the initial slide. The non- differentiable behavior is still apparent. The zoomed regions are first indicated by a box. The function being drawn is the sum k k ä cos (3 ãx) / 2 from k = 0 to infinity. D. x and ³x³ Here the function x is plotted from -25 to 25, then the same is done for the function ³x³. Notice how you can obtain the second graph from the first (but not the first from the second). E. sin x and ³sin x³ Here the function sin x is plotted from -25 to 25, then the same is done for function ³sin x³. Notice how you can obtain the second graph from the first (but not the first from the second). F. sin x/x This function is first graphed with -100 < x < 100, and -.4 < y < 1.2. Notice how the local behavior is hidden and what happens for x large. Then we zoom in and look at -25 < x < 25, -10 < x < 10, -1 < x < 1, -.1 < x < .1, The zoomed regions are first indicated by a box. The y range remains constant throughout this. G. (1 - cos x)/x This function is first graphed with -100 < x < 100, and -.8 < y < .8. Notice how the local behavior is hidden and what happens for x large. Then we zoom in and look at -25 < x < 25, -10 < x < 10, -1 < x < 1, -.1 < x < .1, and -.1 < y < .1 The zoomed regions are first indicated by a box. The y range remains constant throughout this. H. a^x and its derivative These functions are displayed for various values of "a". The object is to show that there is a special value of "a" for which the two graphs coincide. The first slide has a = 10, then 9, 8, 7, 6, 5, 4, 3.5, 3, 2.5, 2.7, and, finally 2.718... . The solid line is the graph of a^x. Notice that between 3 and 2.5 the graphs "change sides". I. sin 2ãx + sin 2ãax The function is shown for various values of "a", viz. 2, 3, 4, 5/2, 5/3 5/4, and û2, with 0 < x < 6. Notice the periodicity, or lack of it. Can you see the pattern? When you have finished reading this document, press Q to quit.