Unformatted text preview: 5. Let f n ( x ) = n ( e x 2 /n1) 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 xx 2 y,y cos xe 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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 Spring '11
 NA
 Topology, Mathematical analysis, Continuous function, Metric space

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