Vector derivatives and arc length
1 .
Let r ( t ) = t
2 i + t
3 j .
a) Compute, velocity, speed, unit tangent vector and acceleration.
b) Write down the integral for arc length from t = 1 to t = 4.
(Do not compute the integral.)
Answer: a) Velocity = v =
Speed b) Arc
= | | = length
�
=
4
�
1 t 2
4
+ 9 t 4 .
Unit tangent vector = T = ds dt = dt d r dt v ds/dt
�
4
1
= � 2 t, 3 t
2
�
=
�
√
4 t 2
2 t
+ 9 t 4
,
4 t 2 + 9 t
� .
4 dt.
√
4 t
3 t
2
2 + 9 t 4
�
.
2 .
Consider the parametric curve x ( t ) = 3 t + 1 , y ( t ) = 4 t + 3 .
a.
Compute, velocity, speed, unit tangent vector and acceleration.
b.
Compute the arc length of the trajectory from t = 0 to t = 2.
d r
Answer: a) Velocity = v = = � 3 , 4 � .
dt
Speed = | v | =
√
9 + 16 = 5 .
Unit tangent vector = T = v ds/dt
� 3 4
= , .
5 5
� b) Arc length =
�
0
2 ds �
2 dt = 5 dt = 10 .
dt
0

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18.02SC Multivariable Calculus
Fall 2010
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