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[answered] Math 325 Elementary Analysis Fall 2016 University of Nebras


I need help with my assignment. Especially question no.3 and 2


Math 325

 

Elementary Analysis Fall 2016

 

University of Nebraska-Lincoln Assignment #8

 

Hard deadline: Tuesday, December 6 at 10:30am

 

Will be graded out of 60 points (20 points for each problem) 1 Definitions Given a bounded function f : [a, b] ! R, the total variation of f over [a, b] is defined by

 

( n

 

)!

 

X

 

V[a,b] (f ) = lub

 

|f (xk ) f (xk 1 )| : a = x0 < x1 < x2 < ? ? ? < xn = b

 

.

 

k=1 A function f : [a, b] ! R has bounded variation over [a, b], if V[a,b] (f ) < 1, and the space of functions

 

of bounded variation (BV functions) is

 

BV([a, b]) = f : [a, b] ! R : V[a,b] (f ) < 1 . 2 Problem Set

 

1. For this problem, define f, F : R ! R by

 

f (x) = ( x,

 

1 x<0

 

x, x and 0 F (x) = ?x f (t) dt. 1 (a) Produce a piecewise algebraic formula for F (x), for x 2 R. (b) At what points if F discontinuous? At what points is F not di?erentiable? At what points

 

does F 0 (x) 6= f (x)?

 

2. For this problem, let f : [a, b] ! R be continuously di?erentiable on [a, b].

 

(a) Use the Fundamental Theorem of Calculus to verify that

 

V[a,b] (f ) ?b

 

a |f 0 (x)| dx. (b) Use the Mean Value Theorem to show that

 

?b V[a,b] (f ) a |f 0 (x)| dx. Parts (a) and (b) establish the following

 

Theorem: If f : [a, b] ! R is continuously di?erentiable on [a, b], then f 2

 

BV([a, b]) and

 

?b

 

V[a,b] (f ) = |f 0 (x)| dx.

 

a 1 3. Determine the indicated function or quantity. Provide justification for your answer.

 

?x (a) F 0 (x), with F (x) = 2 cos(t) dt. x4

 

2 (b) F 0 (x), with F (x) = (c) F 00 (x), with F (x) = ?x ?t 1

 

ds dt.

 

1 + s2 0 0 ?x x sin t2 dt. 1 (d) lim n!1 ?1 sin(nt) dt. 0 1

 

(e) lim

 

x!1 x ?x

 

0 p t2

 

dt.

 

1 + t2 2

 


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