This preview shows page 1. Sign up to view the full content.
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 ....
View
Full
Document
This note was uploaded on 06/19/2011 for the course MATH 600 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
 Spring '11
 NA

Click to edit the document details