Math 301 Problem Set 12 Solutions

Math 301 Problem Set 12 Solutions - Math 301/ENAS 513:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 301/ENAS 513: Homework 12 Solution Triet M. Le Page 116: 2: Let f ( x ) = , x e- 1 /x , x > . To show f is infinitely differentiable at x = 0. Clearly, f is infinitely differentiable on (- , 0) (0 , ), with f ( n ) ( x ) = , x < P 2 n (1 /x ) e- 1 /x , x > , where P 2 n ( x ) is some polynomial of degree 2 n . (This can be shown by induction.) Note that for any m N , y m e- y 0 as y . Hence, for all integer n 0, P 2 n (1 /x ) e- 1 /x , as x . If x n is any sequence converging to 0, then | f ( x n ) | e- 1 /x n 0 = f ( x ) , as n . Hence f is continuous at 0. Now suppose f ( n ) (0) exists and is = 0 for some n 0, then lim x - f ( n ) (0)- f ( n ) ( x )- x = 0 , and lim x + f ( n ) (0)- f ( n ) ( x )- x = 1 x P 2 n (1 /x ) e- 1 /x = P 2( n +1) (1 /x ) e- 1 /x , as x . This implies, f ( n +1) (0) = lim x f ( n ) (0)- f ( n ) ( x )- x = 0. So by induction f ( n ) exists and is equal to 0 for all n 0. The Taylor polynomial T n at a = 0 is identically 0 for all n . Page 116: 3: Suppose f is a bounded real-valued on R , and that f and f 00 are also bounded and continuous. To show sup x R | f ( x ) | 4 sup x R | f ( x ) | sup x R | f 00 ( x ) | . (1) Pick any x, a R , x 6 = a , Taylor theorem implies f ( x ) = f ( a ) + f ( a )( x- a ) + f 00 ( c ) 2 ( x- a ) 2 f ( a ) = f ( x )- f ( a ) x- a- f 00 ( c ) 2 ( x- a ) , 1 for some c between a and x . Now, let h > 0 and x = a + h . We have f ( a ) = f ( a + h )- f ( a ) h- f 00 ( c ) 2 ( h ) | f ( a ) | 2 h sup x R | f ( x ) | + h 2 sup x R | f 00 ( x ) | ....
View Full Document

Page1 / 5

Math 301 Problem Set 12 Solutions - Math 301/ENAS 513:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online