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Homework #1
Due: Tuesday, February 2, 2010
1.
(2pt) Compute the following integrals:
∫
1
0
)
log(
dx
x
where
log(x)
denotes the natural logarithm of
x
, and
dx
x
a
∫
∞
1
1
for a fixed parameter
a>1
. What happens for
a=1
?
2.
(5pt) Consider the following function:
⎩
⎨
⎧
=
≠
=
→
0
0
0
)
log(
)
(
,
:
2
x
x
x
x
x
f
R
R
f
a.
Is this function continuous on
R
?
b.
Does this function have a derivative at every point
x
in
R
?
c.
Is this function differentiable on
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Unformatted text preview: R ? d. Is this function of class C 1 ? e. What is the largest integer k so that f is of class C k ? 3. (3pt) Consider the following functions: ∫ = → ⎪ ⎩ ⎪ ⎨ ⎧ = ≠ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = → x dt t g x f R R f x x x x g R R g ) ( ) ( , : if , if , 1 sin ) ( , : Prove that f is differentiable but is not of class C 1 . Total: 10 pts...
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This note was uploaded on 05/05/2010 for the course MATH 464 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Integrals

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