Miller, Kierste – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert
1
This
printout
should
have
20
questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
The due time is Central
time.
001
(part 1 oF 1) 10 points
±ind all values oF
t
in (0
, π
) For which the
tangent line to the graph oF
x
(
t
) =
t
+ cos 2
t,
y
(
t
) =
t

cos 2
t,
is horizontal.
1.
t
=
π
4
,
3
π
4
2.
t
=
π
12
,
5
π
12
3.
t
=
π
3
,
2
π
3
4.
t
=
5
π
12
,
7
π
12
5.
t
=
π
6
,
5
π
6
6.
t
=
7
π
12
,
11
π
12
correct
Explanation:
AFter di²erentiating, we see that
x
0
(
t
) = 1

2 sin 2
t,
y
0
(
t
) = 1 + 2 sin 2
t.
Thus
dy
dx
=
y
0
(
t
)
x
0
(
t
)
=
1 + 2 sin 2
t
1

2 sin 2
t
.
Now the tangent line will be horizontal when
y
0
(
t
) = 1 + 2 sin 2
t
= 0
,
hence when sin 2
t
=

1
/
2
.
±or
t
in (0
, π
),
thereFore, the tangent line will be horizontal
when
t
=
7
π
12
,
11
π
12
.
keywords:
derivative, tangent line, vertical,
horizontal, trig Function, parametric curve
002
(part 1 oF 1) 10 points
Locate the points given in polar coordinates
by
P
‡
4
,
1
3
π
·
,
Q
‡
4
,
5
6
π
·
,
R
‡
1
,
1
2
π
·
among
2
4

2

4
2
4

2

4
1.
P
:
Q
:
R
:
2.
P
:
Q
:
R
:
3.
P
:
Q
:
R
:
4.
P
:
Q
:
R
:
5.
P
:
Q
:
R
:
correct
6.
P
:
Q
:
R
:
Explanation:
To convert From polar coordinates to Carte
sian coordinates we use
x
=
r
cos
θ ,
y
=
r
sin
θ .
±or then the points
P
‡
4
,
1
3
π
·
,
Q
‡
4
,
5
6
π
·
,
R
‡
1
,
1
2
π
·