MIT18_014F10_ex1

MIT18_014F10_ex1 - f is defned For all x ( 1 , 1) and that...

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EXAM 1 3 (1) (10 points) Find 2 2 x 2 [ | x | ] dx . (Here, as usual, [ x ] denotes the largest integer x .) (2) (10 points) Let f be an integrable function on [ a,b ] and a < d < b . Further suppose that ± b + d ± d f ( x d ) dx = 4 , f ( x ) dx = 7 . a + d a Find ± b 2 f ( x ) dx. d 1
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2 EXAM 1 (3) (10 points) Suppose A,B are inductive sets. Prove A B is an inductive set. Give an example of inductive sets A,B such that A B is not an inductive set. (4) (15 points) Let f be a bounded, integrable function on [0 , 1]. Suppose there exists C R such that f ( x ) C > 0 for all x [0 , 1]. Prove that g ( x ) = 1 /f ( x ) is integrable on [0 , 1].
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3 EXAM 1 (5) (15 points) Suppose
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Unformatted text preview: f is defned For all x ( 1 , 1) and that lim x f ( x ) = A . Show there exists a constant c &lt; 1 such that f ( x ) is bounded For all x ( c,c ). 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_ex1 - f is defned For all x ( 1 , 1) and that...

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