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It would be natural to think that every function has a rate of change at all but a few moments. This, however, is not true. The Blancmange Function and the Weierstrass Function do not have a rate of change at a moment for

The Blancmange and Weierstrass functions are like the sound wave function of Section 4.7, except that both are what we will call “infinitely wrinkly”, meaning that no matter how much you zoom in on their graphs, they remain just as wrinkly as before zooming. A consequence of these functions being infinitely wrinkly is that for any value of their independent variable, you cannot

The animations in Figure 4.8.1 are approximations of the Blancmange and Weierstrass functions. The actual function definitions are in terms of the value of

2. Define the Weierstrass functions in GC. Press ctrl-shift-S to create a summation. Zoom in repeatedly on each graph.

a) Examine the definition of the Weierstrass
function. What about it makes its graph so wrinkly? (Section
4.7 is highly related to this question.)

b) The animations in Figure 4.8.1 focused on a particular value of x. Reset your graph from Part (a). Zoom in repeatedly around another value of x. Then reset the graph and try another point. Skip around the graph which point you expand around. Does it make a difference regarding which point you choose as to whether this function has a rate of change at any moment?

b) The animations in Figure 4.8.1 focused on a particular value of x. Reset your graph from Part (a). Zoom in repeatedly around another value of x. Then reset the graph and try another point. Skip around the graph which point you expand around. Does it make a difference regarding which point you choose as to whether this function has a rate of change at any moment?