Someone asked a question after class about the derivative of the absolute value. Here's a little more verbage for clarification:
The derivative of f(x)=|x| at any point to the right of 0 is 1 (eg., the slope of the line is 1). The derivative of |x| to the left of 0 is -1. So the derivative f'(x) is a piecewise function. (This is what I forgot to write down.) It is
{ 1, x >= 0
{ -1, x < 0
So as we found, the limit of f'(x) does not exit at 0 (left and right hand limits are not equal). But this is made more clear by the jump from -1 to 1 in the graph of f'(x).
-Jesse
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2 comments:
How do you decide which equation to use when finding derivatives? (ex. when x approaches a or when h approaches 0) Is it necessary to use a certain one for a certain problem? And is it possible to use the "as h approaches 0" and then once the derivative is found, plug in x and solve for f(x)? will you get the same answer?
Kaitlin,
Good question. The book does it both ways, which can be confusing. I recommend using the "as h->0" method, and then plugging in a specific value for x. Eventually the book settles on "as h->0". That's how the derivative is "technically" defined, too. But to answer your other question, they both work.
-Jesse
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