Calculus Group I (1)
Two straight hallways meet at right angles. See figure below. The first hallway has
width
A
, and the second hallway has width
B
. A ladder of negligible thickness
is to be taken horizontally from one hallway to the other. Find the length of the
longest such ladder.
Calculus Group I (2)
Suppose that the points
P
,
Q
,
R
have coordinates
(
0, 5
)
,
(
0, 3
)
,
(
x
, 0
)
respectively.
Find the value or values of
x
which maximize the size of angle
PRQ
.
Calculus Group I (3)
Let
f
(
x
) =
Z
x
1
1
1
+
t
6
dt
.
Evaluate the integral
Z
1
0
x f
(
x
)
dx
.
Calculus Group I (4)
A baseball, hit by a baseball player at a 25
◦
angle from 3 ft above the ground, just
cleared the leftfield wall. This wall is 35 ft high and 320 ft from home plate. Use
the Ideal Projectile Motion Equation (in vector form)
r
(
t
) = ((
v
0
cos
α
)
t
+
x
0
)
i
+

1
2
gt
2
+ (
v
0
sin
α
)
t
+
y
0
j
to answer the following questions. Note that
g
=
32 ft/sec
2
, and that
i
and
j
are
the standard unit vectors:
i
=
h
1, 0
i
,
j
=
h
0, 1
i
.
(a) How long did it take the ball to reach the wall?
(b) What was the initial speed of the ball?
(c) Would the same hit have cleared the right field wall which is 5 ft high and 420
ft from home plate? (show work)
(d) For this hit, what is the position vector of the ball at its maximum height?
Calculus Group I (5)
A curve in the
xy
plane is defined by the parametric equations
x
(
t
) =
t
2
,
y
(
t
) =
t
3

3
t
.
This curve intersects itself at the point
(
3, 0
)
, and so it makes a selfenclosed loop
as
t
ranges from

√
3 to
√
3.
(a) Find all points on this loop that have either horizontal or vertical tangent lines.
(b) Calculate the area enclosed by the loop.
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Abstract Algebra Group I (6)
The general linear group
GL
2
(
R
)
is the group of 2
×
2 invertible matrices with real
entries. The group operation is ordinary multiplication of matrices. Let
I
denote
the 2
×
2 identity matrix and let
A
=
0
1
1
0
B
=

1/2

√
3/2
√
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 Spring '11
 crank
 Linear Algebra, Vector Space, Calculus Group

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