quizsoln8

quizsoln8 - x = 2 x c Use the Slope Criterion to determine...

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Name:___________________________ Math 155 Quiz 8 17 October 2006 Instructions : This quiz is closed book and closed notes. You may NOT use a calculator. Label your answers and show all work. You need not simplify your answers. 1. Find the derivatives of the following functions (2 pts each) a) f ( x ) = 3 1- ln( x ) REWRITE: f ( x ) = 3 1- ln( x ) ( ) 1 2 f '( x ) = 3 1 2 1- ln( x ) ( ) - 1 2 - 1 x b) g ( x ) = e cos( x ) g '( x ) = e cos( x ) - sin( x ) ( ) 2. Answer the following questions about the discrete-time dynamical system x t +1 = x t 2 a) Find the equilibria (2 pts) x * = x * ( ) 2 x *(1- x *) = 0 x * = 0 and x * =1 b) Write as an updating function and find the derivative (2 pts) f ( x ) = x 2 Updating Function Notation f

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Unformatted text preview: '( x ) = 2 x c) Use the Slope Criterion to determine the stability of the equilibria (2 pts) f '(0) = 0 = 0 <1 So x* = 0 is STABLE f '(1) = 2 = 2 >1 So x* = 1 is UNSTABLE EXTRA CREDIT (2 pts) Find the derivative of the following function at x=2 f ( x ) = x cos( px ) x 2 METHOD 1: Simplify: f ( x ) = cos( px ) x f '( x ) = x (- sin( px ) p ) - cos( px ) x 2 f '(2) = 2(0) - 1 4 = -1 4 METHOD 2: (More complicated, but works out) f '( x ) = x 2 x- sin( px ) ( ) p + cos( px ) [ ]-x cos( px ) 2 x x 4 f '(2) = 2 2 2(0) p +1 [ ]- 2(1) 2(2) 2 4 = 4 - 8 16 = -1 4...
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quizsoln8 - x = 2 x c Use the Slope Criterion to determine...

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