MAC 2311
July 8, 2003
Exam I Key
Prof. S. Hudson
1) [25pt] Short answer problems:
a) Find the area of the sector below; it’s the part of the circle of radius 2 between
θ
=
π/
6 and
θ
=
π/
3.
Answer:
The angle in between is
θ
=
π/
6, so we have 1/12 of the whole circle. The area
is
πr
2
/
12 =
π/
3.
b) Find all values of
θ
(in radians) that satisfy the equation cos(
θ
) =

1
√
2
Answer:
These can point into the second or third quadrant, so there are two groups of
answers; either
θ
= 3
π/
4 + 2
nπ
or
θ
= 5
π/
4 + 2
nπ
, where
n
is an integer.
c) Solve
x
2

4
x
+ 3
<
0.
Answer:
The parabola crosses the xaxis where 0 = (
x

3)(
x

1), where
x
= 3 or
x
= 1.
It is negative in between, where 1
< x <
3.
d) Find the slopeintercept form of the line perpendicular to
y
= 5
x
+ 9 that has
y
intercept 6.
Answer:
The perpendicular slope is
m
=

1
/
5, so
y
=

x/
5 + 6.
e) Sketch a graph of
f
(
x
) =

x

2

+ 1. Is
f
continuous?
Answer:
Draw a ”V” with a vertex at (2,1). It is continuous.
2) [10pt] Find the rate of change of the area
A
of a circle with respect to its circumference
C
.
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 Calculus, Derivative, Short Answer Problems, Prof. S. Hudson

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