Math Problem Statement
Write the line segment given by
r
(
t
)
e 3 t
i − 8 e 3 t
j − 4 e 3 t
k
as a function
r
(
s
)
parameterized by the arc length
s
from the point where
t
0 .
Solution
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Arc Length Parameterization
Exponential Functions
Formulas
r(t) = e^{3t} i − 8 e^{3t} j − 4 e^{3t} k
r'(t) = 3e^{3t} i − 24 e^{3t} j − 12 e^{3t} k
Arc length s = ∫ |r'(u)| du
t = (1/3) ln(s/9 + 1)
Theorems
Arc Length Parameterization
Chain Rule in Differentiation
Suitable Grade Level
Undergraduate Calculus
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