2.1 The Derivative and the Tangent Line Problem

2.1 The Derivative and the Tangent Line Problem - The...

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The Derivative The Derivative and the Tangent and the Tangent Line Problem Line Problem Section 2-1
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Slope of Secant Line Slope of Secant Line x x ) ( ) ( c f x c f - + (c, f(c)) (c + ,f(c + ) x x x c f x c f m - + = ) ( ) ( sec
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Definition of Tangent Line with Definition of Tangent Line with Slope Slope m m If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). m x c f x c f x y x x = - + = ) ( ) ( lim lim 0 0
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Vertical Tangent Lines Vertical Tangent Lines The defn. of a tangent line to a curve does not cover the possibility of a vertical tangent line. If f is continuous at c and The vertical line x = c, passing through (c, f(c)) is a vertical tangent line to the graph of f. - = - + or x c f x c f x ) ( ) ( lim 0
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Definition of the Derivative of Definition of the Derivative of a Function a Function
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Unformatted text preview: The derivative of f at x is given by provided the limit exists. x x f x x f x f x - + = ) ( ) ( lim ) ( ' Notation for Derivatives Notation for Derivatives ) ( ' x f dx dy ' y [ ] ) ( x f dx d [ ] y D x Alternative form of the Alternative form of the Derivative Derivative c x c f x f c f c x--= ) ( ) ( lim ) ( ' Thm. 2.1 Differentiability Thm. 2.1 Differentiability Implies Continuity Implies Continuity If f is differentiable at x = c, then f is continuous at x = c. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Thus, continuity does not imply differentiability. Joke Time Joke Time What would America be called if we all drove pink cars? A pink carnation! Why should you never play cards in the jungle? Its full of cheetahs!...
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2.1 The Derivative and the Tangent Line Problem - The...

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