Math 317, Section 202, Homework no. 2
(due Friday (!) January 22, 2010)
[7 problems, 36 points]
Problem 1
(6 points)
.
Suppose the curve
C
is the part of the intersection of the cylinder
x
2
+
y
2
= 1 with the plane
x
+
y
+
z
= 1 for which
y
≥
0.
1. (3 points) Parameterize
C
by a vector function
r
(
t
) such that the parameter
t
is
(a) the polar angle in the
xy
plane,
(b)
t
=
x
,
(c)
t
=

x
.
2. (3 points) For each of the three parameterizations, compute the tangent vector to
C
,
the unit tangent vector and a parameterization of the tangent line, all at the point
(
1
√
2
,
1
√
2
,
1

√
2).
Problem 2
(4 points)
.
Just answer Yes or No (Y/N) to the following questions; no explana
tion required.
Consider the curve
C
parameterized by
r
(
t
) =
t
3
,
2
t
3

1
,

t
3
+ 2 ,
∞
< t <
∞
.
(a) (1 point) Is the parameterization smooth?
(b) (1 point) Is the curve smooth?
Consider the curve
˜
C
parameterized by ˜
r
(
t
) =
1

t
2
,
√
1

t
2
, 0
≤
t
≤
1
2
.
(c) (1 point) Is the parameterization smooth?
(d) (1 point) Is the curve smooth?
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 Spring '08
 BEHREND
 Math, Derivative, Osculating circle, ϑ. f

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