2. Is it true that if
f
+
g
is integrable on [
a, b
] then both
f
and
g
must be integrable there?
3. Show that the following function is integrable on [0
,
2]:
f
(
x
) =
(
x
if
x
6
= 1
4
if
x
= 1
Hint: this is almost continuous, except at
x
= 1.
4. More general version: show that if
f
is an integrable function on [
a, b
] and
g
is equal
to
f
except for maybe finitely many points in [
a, b
] then
g
is also integrable.
Hint: take the difference.
5. Compute the following integrals:
(a)
R
4
1
(
x
4

3
x
2
+ 2
x

1)
dx
(b)
R
3

5

x
2

4

dx
(c)
R
7
2
(
√
x
+ 2

x
1
/
3
)
dx
(d)
R
2
π/
3
π/
2
cos(2
x
+
π/
6)
dx
(e)
R
13
π/
6
0

sin
x

dx
2
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6. Suppose that
f
is strictly increasing and continuous on [0
,
∞
) with range [0
,
∞
) (which
also means
f
(0) = 0). Let
g
(
x
) be the inverse function of
f
and
F
(
x
) =
R
x
0
f
(
y
)
dy
.
Find an expression for
R
x
0
g
(
y
)
dy
in terms of
F, g
.
Hint: copy the proof of the integral of
x
1
/p
.
7. Find lim
x
→
0
sin(2
x
)+cos(
x
3
)
x
2
x
.
8. Find lim
x
→∞
√
x
(
√
x
+ 2

√
x
).
9. Show that if lim
x
→
a
f
(
x
) = 0 then lim
x
→
a
1

f
(
x
)

=
∞
.
10. Show that if lim
x
→∞
f
(
x
) = 0 then lim
x
→∞
1
f
(
x
)
might not exists (not even in the
infinite sense).
11. Is there a function
f
which is continuous on
R
and
f
(1
/n
) = (

1)
n
for all positive
integer
n
?
12. Is there a function
f
which is continuous on
R
and
f
(1
/n
) = (

1)
n
/n
for all positive
integer
n
?
13. We define the positive part of a number
x
as
x
+
=
(
x
if
x
≥
0
0
if
x <
0
(a) Show that the function
x
→
x
+
is continuous everywhere.
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 Fall '08
 Staff
 Math, Calculus, Continuous function, Value Theorem, basic limit laws

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