Midterm version Ib
Solutions
Problem 1.
Label the following statements as either true or false.
a) If g is a function defined on
[0
,
1]
, then
integraltext
1
0
g
(
x
)
dx
exists.
b) Suppose
h
is a differentiable function with derivative
h
′
. Then,
integraltext
b
a
h
′
(
x
)
dx
=
h
(
b
)
−
h
(
a
)
c)
Σ
−
1
k
=
−
3
k
2
= Σ
4
k
=1
(
k
−
1)
2
d)
integraltext
b
a
f
(
x
)
·
g
(
x
)
dx
=
integraltext
b
a
f
(
x
)
dx
·
integraltext
b
a
g
(
x
)
dx
e)
(3
3
)(3
3
) = 3
9
Solution 1.
a) False. Consider
g
(
x
) =
braceleftbigg
1
if
x
is rational
0
if
x
is irrational
.
This function is defined everywhere, but has upper and lower sums which converge to different values.
b) False. If
h
′
(
x
)
is continuous, then the integral must exist. There are, however, functions that are differ
entiable, yet whose derivatives are not continuous. Consider
h
(
x
) =
braceleftbigg
x
2
sin(
1
x
)
if
x
negationslash
= 0
0
if
x
= 0
.
The
lim
a
→
0
h
′
(
a
)
does not exist, even though the derivative is defined everywhere.
In fact, this derivative
oscillates so much near 0, that it is not integrable.
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 Spring '08
 Vershynin
 Math, Derivative, Fundamental Theorem Of Calculus, dx, 0 g, 1 0 1 1 1 1 k

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