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Unit Two Notes--The Derivative-1

Unit Two Notes--The Derivative-1 - Unit Two NotesThe...

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Unit Two Notes—The Derivative 2.1 CONTINUITY D A function f is continuous at x = c if the following three conditions are satisfied: 1. f(c) is defined 2. ) ( lim x f c x exists 3. ) ( ) ( lim c f x f c x = D Continuity on a closed interval -- A function is continuous on a closed interval [a, b] if the following conditions are met: 1. f is continuous on the open interval (a, b) 2. f is continuous from the right at x = a, and 3. f is continuous from the left at x = b D Types of discontinuities—1) Removable 2) Non-Removable Ex 1 (removable discontinuity): ( 29 2 2 2 - - - = x x x x f Ex 2 (non-removable discontinuity): ( 29 2 2 2 + - - = x x x x f Ex 3: Determine if f(x) is continuous on [2, 4]: ( 29 + - < - = 3 , 8 3 , 4 2 x x x x x f Ex 4: Determine all values x = a where f(x) is discontinuous. Classify what types of discontinuities exist, if any. ( 29 ( 29 ( 29 6 3 1 2 2 + + - = x x x x f
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2.2 Amount of Change, Percentage Change, and Average Rate of Change (Over an Interval) Describing Change is the underlying theme of the first four units. Three ways that we may consider describing numeric change that is important for Calculus can be done in the following ways: Type of Change How Change is Calculuated Units of Change Change output units Percentage Change percentage % Average Rate of Change units of output (A.R.O.C.) per unit input Ex 1: The number of Facebook users in the Eastern US grew from 0.9 million in 2007 to 12.0 million in 2010.
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