Explanation Given: Bernoulli spiral: r ( t ) = 〈 e t cos 4 t , e t sin 4 t 〉 And, the integral is: s ( t ) = ∫ − ∞ t ‖ r ' ( u ) ‖ d u Proof: Consider the Bernoulli spiral: r ( t ) = 〈 e t cos 4 t , e t sin 4 t 〉 Differentiating, r ' ( t ) = 〈 e t ( cos 4 t − 4 sin 4 t ) , e t ( sin 4 t + 4 cos 4 t ) 〉 And ‖ r ' ( t ) ‖ = ‖ 〈 e t ( cos 4 t − 4 sin 4 t ) , e t ( sin 4 t + 4 cos 4 t ) 〉 ‖ = e t ( cos 4 t − 4 sin 4 t ) 2 + ( sin 4 t + 4 cos 4 t ) 2 = e t cos 2 4 t − 8 sin 4 t cos 4 t + 16 sin 2 4 t + 8 sin 4 t cos 4 t + 16 cos 2 4 t = e t cos 2 4 t + sin 2 4 t + 16 sin 2 4 t + 16 cos 2 4 t ‖ r ' ( t ) ‖ = 17 e t Now, the unit tangent vector is: T ( t ) = r ' ( t ) ‖ r ' ( t ) ...

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Chapter 14.4, Problem 33E

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To prove:

The radius ofcurvature is proportional to $s (t)$ .

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Chapter 14.4, Problem 32E

Chapter 14.4, Problem 34E

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Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY

To prove: The radius ofcurvature is proportional to s ( t ) .

Solution Summary: The author explains that the radius ofcurvature is proportional to s (t).

Concept explainers

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To prove:

The radius ofcurvature is proportional to $s (t)$ .

Want to see the full answer?

Chapter 14 Solutions

To prove: The radius ofcurvature is proportional to s ( t ) .

Solution Summary: The author explains that the radius ofcurvature is proportional to s (t).

Concept explainers

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To prove: The radius ofcurvature is proportional to s(t).

Want to see the full answer?

Chapter 14 Solutions

To prove:

The radius ofcurvature is proportional to $s (t)$ .