This preview shows pages 1–2. Sign up to view the full content.
Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/27/10
1.
Suppose that
f
is an increasing function on the interval
[
a
,
b
]
.
Subdivide
[
a
,
b
]
into
n
equal subintervals.
Let
S
n
(
f
,
a
,
b
)
be the upper sum and let
S
n
(
f
,
a
,
b
)
be the lower
sum.
Show that
S
n
(
f
,
a
,
b
)
!
S
n
(
f
,
a
,
b
)
=
(
f
(
b
)
!
f
(
a
))
"
(
b
!
a
)
n
.
Use this to show that
lim
n
!"
S
n
(
f
,
a
,
b
)
#
S
n
(
f
,
a
,
b
)
$
%
'
=
0
.
2.
Suppose that
F
(
x
)
and
G
(
x
)
are differentiable functions such that
!
F
(
x
)
"
!
G
(
x
)
on
an interval.
Show that there is a constant
C
such that
F
(
x
)
!
G
(
x
)
+
C
.
3.
Suppose that
f
(
x
)
is a continuous function and that
F
(
x
)
=
f
(
t
)
dt
a
x
!
.
Show that
!
F
(
x
)
"
f
(
x
)
.
4.
Suppose that
f
(
x
)
is a continuous function and that
G
(
x
)
is a differentiable
function having the property that
!
G
(
x
)
"
f
(
x
)
on the interval
[
a
,
b
]
.
Show that
f
(
x
)
dx
a
b
!
=
G
(
b
)
"
G
(
a
)
.
5.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 07/14/2011 for the course MAC 3472 taught by Professor Jury during the Spring '07 term at University of Florida.
 Spring '07
 JURY
 Calculus

Click to edit the document details