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Unformatted text preview: Sections 2.8, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6: Review Theory Instructor: Ms. Hoa Nguyen (email@example.com) 2.8 The graphs of a function f ( x ) and its derivative f ( x ): f ( x ) > 0 on ( a, b ) f increases on ( a, b ). f ( x ) < 0 on ( a, b ) f decreases on ( a, b ). f ( x ) = 0 at x = c the tangent at the point ( c, f ( c )) is horizontal. Differentiability and Continuity If f is differentiable at a , then f is continuous at a . In other words, if f is NOT continuous at a , then f is NOT differentiable at a . How can a function fail to be differentiable at a ? Three possibilities for a function NOT to have a derivative (i.e., not differentiable) at a : 3.1 The constant multiple rule : d dx [ cf ] = c df dx ( cf ) = cf ( c is a constant) The sum or difference rule : d dx [ f g ] = df dx dg dx ( f g ) = f g 3.2 The Product Rule d dx [ fg ] = df dx g + f dg dx ( fg ) = f g + fg ....
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