MIT18_014F10_ex2

# MIT18_014F10_ex2 - x for every irrational x(a Prove f x is...

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EXAM 2 - OCTOBER 29, 2010 (1) (5 points each) Find the derivative of each of the following functions g ( x ) = log(cos( x 2 )) h ( x ) = e x sin x (2) (10 points) Consider the function log x g ( x ) = x 2 . Determine the behavior of g in a neighborhood of x = 1. Speci±cally, is the function increasing or decreasing? Is it convex or concave? Justify your answers. 1

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2 EXAM 2 - OCTOBER 29, 2010 (3) (10 points) Consider the functions f ( x ) = x sin x and g ( x ) = ( x + 5) cos x . Prove there exists c (0 ,π/ 2) such that f ( c ) = g ( c ). (If you are using a theorem, make sure you explain why the function or functions you are considering satisfy the hypotheses of the theorem.) (4) (15 points) DeFne f ( x ) such that f ( x ) = x for every rational value of x and f ( x ) =

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Unformatted text preview: x for every irrational x . (a) Prove f ( x ) is continuous at x = 0. (b) Set a = 0. Prove that f ( x ) is not continuous at x = a . EXAM 2- OCTOBER 29, 2010 3 (5) (15 points) Let f be continuous. Prove that x x ± t ² f ( t )( x − t ) dt = f ( u ) du dt. MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT18_014F10_ex2 - x for every irrational x(a Prove f x is...

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