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