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MIT18_014F10_pr_ex2

# MIT18_014F10_pr_ex2 - that f(0 = f(1 Show that for any n...

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PRACTICE EXAM 2 (1) (10 points) Find 1+ h 2 1 2 lim 0 e t dt 0 e t dt . h 0 h (3 + h 2 ) (If you’re using a theorem, state the theorem you’re using.) (2) (10 points) Find ( f 1 ) (0) where f ( x ) = 0 x cos(sin t )) dt is defined on [ π/ 2 , π/ 2]. (3) (10 points) In each case below, assume f is continuous for all x . Find f (2). x f ( x ) f ( t ) dt = x 2 (1 + x ); t 2 dt = x 2 (1 + x ) . 0 0 (4) (15 points) Give an example of a function f ( x ) defined on [ 1 , 1] such that f is continuous and differentiable on [ 1 , 1] f is not continuous for at least one value of x [ 1 , 1]. (5) (15 points) Let f ( x ) be continuous on [0

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Unformatted text preview: that f (0) = f (1). Show that for any n ∈ Z + there exists at least one x ∈ [0 , 1] such that f ( x ) = f ( x + 1 /n ). 1 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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