Analysis2009jan - 5 Let f n x = n e x 2/n-1 for all real...

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Analysis preliminary exam Jan. 8, 2009 1. Let C be the standard Cantor set on the interval [0 , 1] and let A = C c be its comple- ment on the real line. Identify the set of all limit points A 0 of A , explaining your answer. 2. (a) Prove n X k =1 k = n ( n + 1) 2 (b) Let { a n } be a sequence with limit L . Define a sequence b n = 1 n 2 n X k =1 ka k Prove lim n →∞ b n = L/ 2 . 3. Let f be a continuous real valued function on [ a, b ] and differentiable on ( a, b ) . (a) Prove max a x b | f ( x ) | ≤ 1 b - a Z b a | f ( x ) | dx + ( b - a ) sup a<x<b | f 0 ( x ) | (b) Given any ² > 0 prove max a x b | f ( x ) | ≤ 1 ² Z b a | f ( x ) | dx + ² 2 sup a<x<b | f 0 ( x ) | 4. Suppose f ( x + 1) = f ( x ) for all real x , f is real valued, f is Riemann integrable on every compact interval, and R 1 0 f ( x ) dx = 0 . (a) Prove there exists x 0 such that F ( x ) = R x x 0 f ( t ) dt 0 for all x . (b) Show by example that F 0 ( x 0 ) = 0 need not be true.
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Unformatted text preview: 5. Let f n ( x ) = n ( e x 2 /n-1) for all real x . (a) Prove lim n →∞ f n ( x ) = x 2 for each x . (b) Prove { f n } is equicontinuous on [0 ,M ] for all positive M . (c) Prove that lim n →∞ R 1 ( f n ( x )) 1 / 3 dx exists and equals 3 5 . 6. The map ( x,y ) 7→ ( e x sin x-x 2 y,y cos x-e x + 1) maps the origin to the origin . Show that the inverse map G exists in a neighborhood of the origin and compute d dt ﬂ ﬂ t =0 f ◦ G (-t,t 2 ) and d dt ﬂ ﬂ t =0 f ◦ G (-t,t ) when f ( x,y ) = x + 2 y ....
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