Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/22/08
1.
State the Mean Value Theorem.
2.
Suppose that
!
f
(
x
)
<
1
for all
x
in
x
0
!
"
,
x
0
+
"
[
]
and that
f
(
x
0
)
=
x
0
.
Let
x
1
!
x
0
"
#
,
x
0
+
#
[
]
with
x
1
!
x
0
.
Show that
f
(
x
1
)
!
x
0
<
x
1
!
x
0
. This says that
f
(
x
1
)
is
closer to
x
0
than
x
1
.
3.
Let
f
(
x
)
=
0
be an equation to be solved numerically.
Let
g
(
x
)
=
x
!
f
(
x
)
"
f
(
x
)
.
Show
that the set of values where
f
(
x
)
=
0
is precisely the set of values where
g
(
x
)
=
x
.
4.
Solve the following equations using Newton’s Method.
(a)
x
3
!
5
=
0
(b)
x
9
+
sin
x
=
10
(c)
tan
x
=
2
5.
Find the right circular cylinder of greatest volume that can be inscribed in a
sphere of radius
R
.
Find the right circular cylinder having greatest surface area
that can be inscribed in a sphere of radius 10.
6.
Find the right circular cone of greatest volume that can be inscribed in a sphere of
radius
R
.
Find the one with greatest surface area for
R
=
10
.
7.
A wire of length 100 cm is cut into two pieces. One piece is bent into a circle, the
other into a square. Where should the cut be made to maximize the sum of the
areas of the square and the circle? To minimize that sum?
8.
Determine the numbers
x
and
y
such that
x
!
y
is maximum subject to
x
+
y
=
10
and
x
!
0
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 Spring '07
 JURY
 Calculus, Mean Value Theorem, dx

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