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The Derivative

Problems

Geometric Definition of Derivative

Limit Definition of the Derivative

Problem : For which values of x does the function f (x) , with graph shown below, fail to be continuous? differentiable?

Figure %: Locate the Discontinuities

The graph is not continuous at x = 4 and not differentiable at x = 1, 4, 5 .

Problem : For which values of x does the function f (x) with the following graph have f'(x) = 0 ?

Figure %: Locate the Points for which f'(x) = 0

The derivative f'(x) = 0 for x = - 2, 0 . Note that the derivative is not defined at x = 2 , because there is no well-defined tangent line to the graph at that point.

Problem : Let f (x) = x 1/3 , a function defined on the entire real line. Is f differentiable at x = 0 ?

No, the tangent to the graph at x = 0 is vertical, with undefined slope.

Problem : Let f (x) = sin(x) . For which values of x is f'(x) = 0 ?

We have f'(x) if and only if the tangent line to the graph of f (x) at x is a horizontal line. This occurs whenever sin(x) = ±1 . Thus f'(x) = 0 for x = Π/2 + , where k is any integer.

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