3.2_notes - 3.2 Differentiability Things we discovered...

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Calculus Section 2.1 Page 8 3.2 Differentiability Things we discovered yesterday: Definition of the Derivative: f x ( ) = lim h 0 f x + h ( ) - f x ( ) h The Derivative at a Point (2 formulas) f a ( ) = lim h 0 f a + h ( ) - f a ( ) h f a ( ) = lim x a f x ( ) - f a ( ) x - a Right-Hand Derivative at a Left-Hand Derivative at a f a ( ) = lim h 0 + f a + h ( ) - f a ( ) h f a ( ) = lim h 0 - f a + h ( ) - f a ( ) h Some Formulas f x ( ) = x 2 f x ( ) = f x ( ) = x 3 f x ( ) = f x ( ) = x 4 f x ( ) = f x ( ) = 3 x f x ( ) = f x ( ) = mx f x ( ) = f x ( ) = 3 f x ( ) = f x ( ) = a f x ( ) = f x ( ) = 1 x f x ( ) = f x ( ) = x f x ( ) = Notation for the Derivative: f x ( ) y dy dx df dx d dx f x ( ) We also learned to graph the derivative function f from the original function f by looking at slopes of tangent lines How f a ( ) Might Fail to Exist A function will not have a derivative at a point P a , f a ( ) ( ) where the slopes of th secant lines, f x ( ) - f a ( ) x - a fail to approach a limit as x approaches a .
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Calculus Section 2.1 Page9 1. A corner (where the one-sides derivatives differ) y = x 2. A cusp (where the slopes of the secant lines approach from one side and
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