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Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/20/09
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 differential 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.
Determine the following integrals.
Be able to use the following methods and
combinations of them to determine the integrals:
(1) Substitution, (2) Integration
by Parts, (3) Trigonometric Integration, (4) Partial Fractions, and (5)
Trigonometric Substitution.
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 Spring '07
 JURY
 Calculus

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