Download X Z b X X Z b ! Z b Z x 2 Z 1 Z x Z b

MATH 680
Analysis I
? Due: Wednesday, March 15th ?
Winter 2000
Instructions : Do any ve problems.
1.
Let f be Riemann integrable on [a; b]: Assume the result that if the limit
n X
b
,
a
lim
f a+k n
n!1
exists, then its value is the integral
lim
n!1
2.
n
X
n
= ;
2 + n2
k
4
k=1
lim
n!1
n
X
k=1
k=1
Z
b
a
f (x) dx: Deduce the following limits:
p
p 1 = ln(1 + 2):
n +k
2
2
Assume the result of dierentiating under the integral sign given as
!
d Z b g (t; x) dt = Z b @ g (t; x) dt
dx
@x
a
a
where g (t; x) is continuous for (t; x) 2 [a; b] [c; d] and continuously dierentiable in x for
x 2 [c; d]. (This is not dicult to prove!)
Dene
Z x 2 2
Z 1 ,x2 (t2 +1)
f (x) =
e,t dt ; g(x) = e t2 + 1 dt:
0
0
(a) Show that g 0 (x) + f 0 (x) = 0 for all x and deduce that g (x) + f (x) = =4:
(b) Use part (a) to prove that
3.
lim
x!1
Z
x
0
p
e,t2 dt = 21 :
(a) Let f be a positive continuous function in [a; b]: Let M denote the maximum value of f on
[a; b]: Suppose f (c) = M; where c 2 (a; b): Show that there exists a > 0 so that f (x) > 21 M; for
all x 2 (c , ; c + ):
(b) Let f be a positive continuous function in [a; b]: Let M denote the maximum value of f on
[a; b]: Show that
!1=n
Zb
n
lim
[f (x)] dx
= M:
n!1
a
4.
Show that 2 is irrational.
Let f (x) = xn (1 , x)n =n!.
(a) Show that 0 < f (x) < 1=n!, for 0 < x < 1.
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(b) Each k-th derivative f (k) (0) and f (k) (1) is an integer.
Now assume that 2 = a=b; where a and b are positive integers, and let
n
X
F (x) = bn (,1)k f (2k) (x) 2n,2k :
k=0
Show the following parts:
(c) F (0) and F (1) are both integers.
(d) 2 an f (x) sin(x) = d (F 0 (x) sin(x) , F (x) cos(x)).
dx
Z1
(e) F (1) + F (0) = an f (x) sin(x) dx.
0
(f) Use (a) in (e) to deduce that 0 < F (1) + F (0) < 1 for n suciently large. This contradicts
(c) and shows that 2 (and hence ) is irrational.
5.
(a) Let f be continuous on [a; b]: If c 2 (a; b) and f (c) > 0, show that there exists a > 0 so that
f (x) > 0 for all x 2 (c , ; c + ):
(b) Left f be continuous on [a; b] and let f (x) = 0 whenever x is rational. Prove that f (x) = 0 for
every x 2 [a; b]:
6.
Let f be dened on all of the real line R: and assume that f is continuous at x = 0: Further,
suppose also that for all x; y 2 R; that f satises the functional equation
f (x + y) = f (x) + f (y ):
Prove that there exists a constant c so that f (x) = cx; for all x:
Hint: Show that f (n) = nf (1); for all positive integers n; by using induction. Next, determine
the values of f (1=2); f (1=3); f (1=4); : : :: Generalize this method to determine f (1=m): After this,
determine the value of f (r) for any rational number r: Finally, use continuity of f to determine
f (x) for any real x:
7.
(a) Suppose that f is dened on a subset S of Rn : If f is uniformly continuous on S and if S is
bounded, prove that f is bounded on S:
(b) Prove that f (x) = x2 is NOT uniformly continuous on R:
8.
(a) Suppose that f : (X; dX ) ! (Y; dY ) is a uniformly continuous function between two metric
spaces. Show that f preserves Cauchy sequences; that is, if fxn g is a Cauchy sequence in X , then
ff (xn )g must be a Cauchy sequence in Y:
(b) Show that the result of (a) is false if f is simply assumed to be continuous.
9.
(a) Show: if A is a dense subset of a metric space S and S is a dense subset of a metric space T;
then A is a dense subset of the metric space T:
(b) Consider Q to be the metric space of rational numbers with metric d(x; y ) = jx , y j: Find an
example of a closed and bounded subset of Q which is NOT compact!
2
10.
Let fan g be a Cauchy sequence in a metric space (X; dX ):
(a) Show that fan g must be a bounded sequence.
(b) Show that fan g has a convergent subsequence with limit L, then the whole sequence itself must
converge to L:
11.
Consider the function
x ); x 6= 0, :
f (x) = x sin(
0;
x=0
(a) If = 2 and = 1, show that f is dierentiable at x = 0, while if = 1 and = 1, f is not
dierentiable at x = 0.
(b) With = 2 and = 1, show that f 0 is not continuous at x = 0.
12.
Let f; f 0 and f 00 be all continuous.
(a) Show that
f (x + h) , f (x , h) = 2hf (0) + O(h3 ):
(b) Show that
f (x + h) , f (x , h) = f 0 (x):
lim
h!0
2h
(c) Give an example where the limit lim h!0 g(x+h)2,hg(x,h) exists for a function g (x) but g fails to
be dierentiable at x.
13.
Let f and g be two real-valued bounded functions on [a; b].
(a) Assume that f + g is Riemann-integrable, EITHER prove that both f and g must be Riemannintegrable OR give an example where neither f nor g are Riemann-integrable.
(b) Assume that f g is Riemann-integrable, EITHER prove that both f and g must be Riemannintegrable OR give an example where neither f nor g are Riemann-integrable.
(c) Assume that f 2 is Riemann-integrable, EITHER prove that f must be Riemann-integrable OR
give an example where f is not Riemann-integrable.
14.
Assume that f; f 0 ; f 00 are all continuous on the whole real line. For notational convenience, let F
be an antiderivative of f so F 0 = f . Assume h > 0. By using appropriate Taylor expansions, show
that
F (h) , F (0) = 21 (f (0) + f (h)) + O(h3 ):
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