I need help with my assignment. Especially question no.3 and 2
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
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,
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
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
1 + s2 0 0 ?x x sin t2 dt. 1 (d) lim n!1 ?1 sin(nt) dt. 0 1
x!1 x ?x
0 p t2
1 + t2 2
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