Quiz III, Math 448, 5 March, 2014
Name:
Solutions
1. (10 points): Let X be a metric space and K X a compact set. Suppose that
f : K R cfw_ is lower semi-continuous. Prove that f attains its inmum on
K.
Proof: (S1) There exists a minimizing sequence, i.e.,
Quiz II, Math 448, 5 February, 2014
Name:
Solutions
1. (10 points): Verify the Implicit Function Theorem for the function
f (x, y, z ) = 1 x2 y 2 z 2 ,
where (x0 , y0 , z0 ) =
f (x, y, z ) = f (x, z ).
1
1
, 1 , 2
2
2
. For this problem, write x = (x, y )
Quiz I, Math 448, 22 January, 2014
Name:
Solutions
1. (10 points): Suppose Vi , i = 1, . . . , d are Banach spaces and V := V1 Vd
is open. Suppose that all of the partial derivatives of f : R exist and are
bounded in . Prove that f is continuous in .
Proo
Midterm Exam I, Math 448, 19 February, 2014
Name:
Solutions
1. (10 points): Verify the Implicit Function Theorem for the function
f (x, y, z ) = 4 x2 y 2 z 2 ,
where (x0 , y0 , z0 ) = 0, 2, 2 .
Solution: (S1) Set x0 = (x0 , y0 ) and observe that f (x0 , z
Homework 10, Math 448
12 March, 2014
Name:
1. Let W := [a, b] [c, d] and suppose that t : W R is an elementary function.
Prove that
d
t(x, y ) dy
T (x) =
c
is an elementary function.
2. Suppose that f Cc (Rd ) with supp(f ) W , W a cuboid in Rd . If t : W
Homework 09, Math 448
5 March, 2014
Name:
1. Let X be a metric space and f : X R cfw_+ be a function. Dene
f() (x) := sup cfw_ g (x) | g f, g : X R cfw_+ is l.s.c.
and
f(s) (x) := sup cfw_ g (x) | g f, g : X R is continuous .
(a) Prove that f() and f(s) a
Homework 08, Math 448
26 February, 2014
Name:
1. Let X be a metric space, and suppose that f, g : X R cfw_ are lower semicontinuous.
(a) Prove that inf(f, g ), sup(f, g ), and f + g are lower semi-continuous.
(b) Suppose additionally that f, g 0. Prove th
Homework 07, Math 448
19 February, 2014
Name:
1. Let : [a, b] Rd be continuously dierentiable, with (t) = 0 for all t [a, b].
(a) Prove that the arc length function
s(t) := L |[a,t] ,
t [a, b],
is invertible and its inverse function t : [0, L( )] [a, b]
Homework 06, Math 448
12 February, 2014
Name:
1. Suppose that f : [a, b] Rd is continuous. Prove that
b
b
f (t)dt
a
f (t)
2
dt.
a
2
2. Suppose that : [a, b] Rd is a continuously dierentiable curve. Prove that is
rectiable and
b
L( ) =
( ) d,
2
a
and
s(t)
Homework 05, Math 448
5 February, 2014
Name:
1. Suppose that Rd is open and : Rk is continuously dierentiable in ,
with x0 and (x0 ) =: y0 Rk . Assume that d k and
rank D[x0 ] = d.
In fact, to make things simple, assume that the upper left d d block of th
Homework 03, Math 448
22 January, 2014
Name:
1. Let a Rn and r > 0 be given. Suppose that f : U (a, r) Rn , fi C 2 (U (a, r), for
each i = 1, . . . , n. Suppose that there is a point U (a, r), such that f ( ) = 0, and
that the Jacobian matrix Jf (x) is in
Homework 03, Math 448
22 January, 2014
Name:
1. Dene f : R2 R via
xy
r
f (x, y ) :=
1
r
sin
0
for x = 0
,
for x = 0
where x = (x, y ) and r = x 2 .
(a) Where in R2 do the partials
f
x
and
f
y
exist?
(b) Where are the partials continuous?
(c) For which (a,
Homework 02, Math 448
15 January, 2014
Name:
1. (Taylors Theorem): Let Rd be open. Suppose that f : R is twice continuously dierentiable, i.e., f C 2 (). Let x0 , and suppose that U (x0 , r) ,
for some r > 0. Starting with the one dimensional Taylor Theor
Homework 01, Math 448
8 January, 2014
Name:
1. On Rd consider f1 (x) = x 1 , f2 (x) = x 2 , and f (x) = x . Where are these
functions dierentiable? Where are they not? Compute the derivatives where they
exist.
2. Denote x = (x1 , x2 ) R2 . Dene, for x = 0