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Problems for lecture 13
February 25, 2014
1. Assume that f : R R and g : R R are dierentiable at a point c A.
(a) Prove that for any k R, the function kf (x) is dierentiable at c and its derivative at c
is kf (c).
(b) Suppose g(c) = 0. Use the fact that g
Problems for lecture 14
March 4, 2014
1. Recall that a function f : A R is Lipschitz if there is a number M > 0 so that
f (y) f (x)
<M
yx
for all x, y A x = y.
(a) Consider the function f (x) = x on (0, 1). In a previous homework, we showed that
this func
Problems for lecture 16
March 18, 2014
1. Let f and g be continuous on [a, b] and dierentiable on (a, b). Prove that there is a point
c (a, b) where
[f (b) f (a)]g (c) = [g(b) g(a)]f (c).
Hint: Apply the Mean Value Theorem to the function h(x) = [f (b) f
Problems for lecture 17
March 21, 2014
1. Recall that a function f : R R is uniformly dierentiable if it is dierentiable and for every
> 0 we can nd > 0 so that if 0 < |y x| < then
f (y) f (x)
f (x) < .
yx
In a previous homework we show that if f is unif
Problems for lecture 14
March 6, 2014
1. We say a function f has the intermediate value property on [a, b] if for all x < y in [a, b]
and all L between f (x) and f (y), it is always possible to nd a point c (x, y) where
f (c) = L. The Intermediate Value T
Problems for lecture 18
April 1, 2014
1. Consider the function h(x) = |x| on [1, 1] and extend the denition of h(x) to all of R by
requiring that h(x + 2) = h(x). We dene the function
f (x) =
1
h(2n x).
2n
n=0
Prove that this function is not dierentiable
Problems for lecture 19
April 1, 2014
1. Let h(x) = |x| when x [1, 1] and extend h(x) to x R by requiring that h(x + 2) = h(x).
We dene the functions
f (x)
1
h(2n x),
2n
n=0
=
m
fm (x)
1
h(2m x).
2n
n=0
=
Suppose x is not a dyadic number. Let ym be a dyad
Problems for lecture 20
April 9, 2014
1. We assume the theorem that in general, if fn
f and fn (x) is continuous on A for each n
then f (x) is continuous on A. Consider the sequence of functions
fn (x) =
x
.
1 + xn
(a) Find a function f (x) so that fn f o
Problems for lecture 22
April 10, 2014
1. Given a sequence of dierentiable function fn (x) on [a, b]. A common question is whether
the derivative of a limit is the limit of derivatives (i.e., will we have (limn fn (x) =
limn fn (x)?). This is certainly no
Problems for lecture 21
April 16, 2014
1. Prove that if fn
f on A then for every > 0, we can nd N N such that |fm (x)fn (x)| <
for all m, n N and all x A.
2. Consider the sequence of functions
1
fn (x) = x1+ 2n1
on [1, 1]. Show that fn (x) converges unifo
Problems for lecture 22
April 17, 2014
1. Recall that a series of numbers
ak is convergent if and only if the series is Cauchy (i.e., for
every > 0, we can nd an N so that |am+1 + am+2 + + an | < for all n > m N ).
We want to have a similar result for a s
Problems for lecture 25
April 29, 2014
1. We know that power the series k=0 xk converges pointwise on (1, 1) to f (x) = 1/(1 x).
As a consequence of the theorems on power series, we can say that this series converges
absolutely on (1, 1) and uniformly on
Problems for lecture 22
April 24, 2014
This homework set is about the summation by parts formula
n
n
ak .bk = sn bn+1 sm bm+1 +
sk (bk bk+1 )
k=m+1
k=m+1
where sn = a0 + a1 + a2 + + an . You can see this formula in Exercise 2.7.12 on page 68 in the
book.
Math 4606, Summer 2002: The Intermediate Value Theorem and two proofs; recursion
Page 1 of 2
The Intermediate Value Theorem: If f (x) is continuous on [a, b], and there exists a real number y such
that f (a) < y < f (b), then there exists xo , where a < x
MATH 401 - NOTES
Sequences of functions
Pointwise and Uniform Convergence
Fall 2005
Previously, we have studied sequences of real numbers. Now we discuss
the topic of sequences of real valued functions. A sequence of functions cfw_fn
is a list of functio
Math 341 Lecture #32
6.5: Power Series, Part II
For a power series with radius of convergence R > 0, we have shown that the power series
is continuous on (R, R).
We saw in an example that the interval of convergence could be (R, R], and the question
is th
Summation by Parts
An important technique of calculus is integration by parts:
b
b
a u(x)v'(x)dx = u(b)v(b) u(a)v(a) a u'(x)v(x)dx
This is useful, obviously, when u'(x)v(x) is easier to integrate than
u(x)v'(x), e.g., if u(x) = x and v(x) = ex.
An analog
Problems for lecture 10
February 19, 2014
1. Let f :A R be a continuous function and K be a compact subset of A. Show that minf (K)
exists.
2. Consider the function
f (x) =
1
.
x2
(a) Show that this function is uniformly continuous on [1, ).
(b) Show that
Problems for lecture 12
February 20, 2014
1. A function f : A R is called Lipschitz if there is a number M > 0 so that
f (x) f (y)
<M
xy
for all x, y A, x = y.
(a) Show that if f : A R is Lipschitz then it is uniformly continuous on A.
(b) Consider the fu
b
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Problems for Lecture 1
January 14, 2014
1. Using the -denition of a limit, prove that
lim
3n + 1
3
= .
2n + 5
2
2. Using the -denition of a limit, prove that
lim
10
= 0.
(n + 1)(n + 2)(n + 3)
3. We learned in class that if a sequence is convergent then it
Problems for Lecture 3
January 29, 2014
1. Suppose that the sequence an is decreasing and bounded. Show that the sequence an converges. (Do not say that this is a consequence of the Monotone Convergence Theorem, we
are trying to prove this theorem.)
2. We
Problems for Lecture 2
January 16, 2014
1. Let c be a xed real number. Prove that if lim an = a then
lim c.an = c.a.
n
2. Suppose that an is a positive sequence and lim an = a = 0. Prove that
lim an = a.
Hint:
|an a|
|a a|
n
| an a| =
.
an + a
a
3. Cons