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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
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 (
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
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)
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 p
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
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
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
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
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
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
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
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 t
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
tha
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 function
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
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 tha
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 f
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 th
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 le
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 Convergen
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 =