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MIT18_014F10_pset4

# MIT18_014F10_pset4 - by ± g x = f x a x = x = is not...

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Limits and Continuity - Week 2 Pset 4 Due October 8 (1) Page 138: 17, 18, 21 (1,1,and 2 points respectively) (2) Let A ( x ) = x 2 f ( t ) dt where f ( t ) = 1 if t < 0 and f ( t ) = 1 if t 0. Graph y = A ( x ) for x [ 2 , 2]. Using �, δ , show that lim x 0 A ( x ) exists and find its value. (You may want to draw yourself a picture of | A ( x ) A (0) | by considering the appropriate regions on a t y coordinate plane that contains the graph of y = f ( t ). This will help you see geometrically how to write δ in terms of .) (3) Notes F.2:2 (4) Suppose that g, h are two continuous functions on [ a, b ]. Suppose there exists c ( a, b ) such that g ( c ) = h ( c ). Define f ( x ) such that f ( x ) = g ( x ) for x < c and f ( x ) = h ( x ) for x c . Prove that f is continuous on [ a, b ]. (5) Let f ( x ) = sin(1 /x ) for x R , x = 0. Show that for any a R , the function g ( x ) defined

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Unformatted text preview: by ± g ( x ) = f ( x ) a : x = : x = is not continuous at x = 0. (6) page 145:5 Bonus: Let f be a bounded function that is integrable on [ a,b ]. Prove that there b c exists c ∈ R with a ≤ c ≤ b such that f ( x ) dx = 2 f ( x ) dx . a a 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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