HW1SOLUTIONS - ENGRD 270 Summer 2007 PROBLEM SET 1 –...

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: ENGRD 270 Summer 2007 PROBLEM SET 1 – Solutions 1. Since the function F ( a ) = n X i =1 ( x i- a ) 2 is twice continuously differentiable, if a minimizes F ( a ), then F ( a ) must be 0. Now F ( a ) = d da " n X i =1 ( x i- a ) 2 # = n X i =1 d da ( x i- a ) 2 =- 2 n X i =1 ( x i- a ) . So F ( a ) = 0 ⇒ n X i =1 ( x i- a ) = 0 ⇒ n X i =1 x i = na ⇒ a = 1 n n X i =1 x i = x. Also note that F 00 ( a ) = d da "- 2 n X i =1 ( x i- a ) # = d da (2 na ) = 2 n > . Hence x will minimize F ( a ). 2.1 For λ ∈ (0 , 1], the series n X i =1 i- λ will not converge, as n X i =1 i- λ ≥ n X i =1 1 i ≥ n X i =1 Z i +1 i 1 x dx as x ∈ ( i, i + 1) ⇒ 1 x < 1 i ⇒ Z i +1 i 1 x dx ≤ 1 i = Z n +1 1 1 x dx = log( n + 1) , 1 and consequently lim n →∞ n X i =1 i- λ ≥ lim n →∞ log( n + 1) = ∞ . 2.2 For λ > 1, the series n X i =1 i- λ will converge, as n X i =1 i- λ = 1 + n X i =2 i- λ ≤ 1 + n X i =2 Z i i- 1 1 x λ dx as x ∈ ( i- 1 , i ) ⇒ 1 x > 1 i ⇒ Z i i- 1 1 x...
View Full Document

This note was uploaded on 04/17/2008 for the course ENGRD 2700 taught by Professor Staff during the Summer '05 term at Cornell University (Engineering School).

Page1 / 11

HW1SOLUTIONS - ENGRD 270 Summer 2007 PROBLEM SET 1 –...

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