Continuous Everywhere but Differentiable Nowhere

Today in calculus I wanted to check if students really understood what they were doing when they were finding the instantaneous rate of change. (We haven’t learned the word *derivative *yet, but this is the formal definition of the derivative.)

So I handed out this worked out problem.

And I had them next to each of the letters write a note answering the following *individually* (not as a group):

**A:** write what the expression represents graphically and conceptually

**B:** write what the notation $latex \lim_{h\rightarrow0}$ actually means. Why does it need to be there to calculate the instantaneous rate of change. (Be sure to address with *h* means.)

**C:** write what mathematical simplification is happening, and why were are allowed to do that

**D:** write what the reasoning is behind why were are allowed to make this mathematical move

**E:** explain what this number (-1) means, both conceptually and graphically

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