L10 - Lecture 10 The Derivative Function In the last...

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Lecture 10 — The Derivative Function In the last examples of Lecture 9, we saw that, from a function f , we could derive a new function that represents the slope (or instantaneous rate of change) of f at each x = a . This function is called the derivative of f , and we give it the special nota- tion f 0 . Thus, f 0 ( a ) = or equivalently, if we let h = x - a , f 0 ( a ) = Other notations Interpretations
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The process of finding the derivative of a function is called differentiation . Note the derivative is defined as a limit, which may or may not exist. If f 0 ( a ) exists, we say that f is differentiable at a . We say it is differentiable on an interval I if it is differentiable at each point in I . On what interval(s) is the function V ( t ) = ( t 2 + t ) 1 3 differentiable?
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f is differentiable at a , then it is continuous at a . Beyond calculus: The world is full of wild functions that are continuous, but nowhere differentiable, like the stock market. They have NO derivative and cannot be drawn—calculus cannot handle them. Some causes for functions to be not differentiable:
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L10 - Lecture 10 The Derivative Function In the last...

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