The arc-length function s(t) of a given path c() is defined by: s(t) = )=["'\e (2) de It measures the length of the path c(x) from x = a to x = t. (a) Compute the arclength function s(t) of c₁ (t) = (cos(t), sin(t), t) with a = 0. (b) Compute the arclength function s(t) of c₂ (t) = (cosh(t), sinh(t), t) with a = 0. The definitions of cosh and sinh are as follows: ez tez cosh(t) They are related to parametrizing the hyperbola x² - y² = 1 in the same way that cos and sin are related to paramatrizing the circle x² + y² = 1. Thus they are called the hyperbolic trigonometric functions. " 2 sinh(t) e² - e-¹ 2 =
The arc-length function s(t) of a given path c() is defined by: s(t) = )=["'\e (2) de It measures the length of the path c(x) from x = a to x = t. (a) Compute the arclength function s(t) of c₁ (t) = (cos(t), sin(t), t) with a = 0. (b) Compute the arclength function s(t) of c₂ (t) = (cosh(t), sinh(t), t) with a = 0. The definitions of cosh and sinh are as follows: ez tez cosh(t) They are related to parametrizing the hyperbola x² - y² = 1 in the same way that cos and sin are related to paramatrizing the circle x² + y² = 1. Thus they are called the hyperbolic trigonometric functions. " 2 sinh(t) e² - e-¹ 2 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The arc-length function s(t) of a given path c() is defined by:
s(t) =
)=["'e (2)|de
It measures the length of the path c(x) from x = a to x = t.
(a) Compute the arclength function s(t) of c₁ (t) = (cos(t), sin(t), t) with a = 0.
(b) Compute the arclength function s(t) of c₂ (t) = (cosh(t), sinh(t), t) with a = 0.
The definitions of cosh and sinh are as follows:
ez tez
cosh(t)
They are related to parametrizing the hyperbola x² - y² = 1 in the same way that cos and sin are related to paramatrizing the circle
x² + y² = 1. Thus they are called the hyperbolic trigonometric functions.
"
2
sinh(t)
e² - e-¹
2
=
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