Math 23
Sections 110113
B. Dodson
Week 4 Homework:
13.1, 13.2 vector functions, derivatives
13.3 arc length, curvature
13.4 velocity, acceleration
Problem 13.2.9:
Find the derivative of the vector function
r
(
t
) =
< t
2
,
1

t,
√
t >
.
Solution:
We just take the derivative of the components,
r
(
t
) =
<
(
t
2
)
,
(1

t
)
,
(
√
t
)
>
=
<
2
t,

1
,
1
2
√
t
>,
where (
t
1
2
) =
1
2
t

1
2
.
Week 4 Homework:
13.3 arc length,
curvature (1st Wed)
13.4 velocity, acceleration
Problem 13.2.17:
If
r
(
t
) =
<
6
t
5
,
4
t
3
,
2
t >,
find the unit tangent vector
T
when
t
= 1
.
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2
.
Solution:
First,
r
(
t
) =
<
30
t
4
,
12
t
2
,
2
>
= 2
<
15
t
3
,
6
t
2
,
1
>,
and
r
(1) = 2
<
15
,
6
,
1
>,
so the length

r
(1)

= 2
√
225 + 36 + 1 = 2
√
252
,
so
T
(1) =
1
√
252
<
15
,
6
,
1
> .
Note that finding
T
(
t
) first, before
setting
t
= 1
,
makes the problem harder; while
setting
t
= 1 in
r
(
t
) before differentiating
changes the derivative to 0 : the above
steps are in exactly the correct order to get the
right answer with least computation.
Week 4 Homework:
13.3 curvature (2nd Wed)
Problem 13.3.16
Use formula (9) to find the curvature of
r
(
t
) =
< t
2
,
2
t,
ln
t >
.
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 Spring '06
 YUKICH
 Arc Length, Derivative, Vector Space, Acceleration, Velocity

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