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Unformatted text preview: x 2 x > 1 . 6? ± What do you note about continuity and di±erentiability of f ( x ) at x = 1? Theorem: If f is di±erentiable at x = a , then f is continuous at x = a . When is a function not di±erentiable at a point? 6? ± 6? ± 6? ± Def. The graph of a function f ( x ) has a vertical tangent line at x = a if ex. Find f ( ± 1) if f ( x ) = ( x + 1) 1 = 3 . ex. Let f ( x ) = j x j . Use the limit de±nition to show that f ( x ) is not di²erentiable at x = 0. ex. Let g ( x ) = x j x j . Show that g ( x ) is both continuous and di±erentiable at x = 0. Higher Derivatives If y = f ( x ) is di±erentiable, we can ²nd its derivative, a new function called f 00 ( x ). The limit de²nition: In the same way, the derivative of f 00 ( x ) is f 000 ( x ), and in general, we denote the n th derivative of f as f n ( x ). Other notation:...
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