Explanation Given info: r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ] Concept used: Are length ( s ) = ∫ a b ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 dt Calculation: r ( t ) = r ( t o ) + 1 k N T ( sin k s − cos k s ) = r ( t o ) + R N + R t < cos t , sin t > An are length parameterization of the osculating circle r ' ( t ) = 0 + R 0 + R < − sin t , cos t > ‖ r ' t ‖ = R 2 [ ( − sin t ) 2 + ( cos t ) 2 ] = R sin 2 t + cos 2 t = R . Now, are length at point t can be calculated as s = ∫ 0 t ‖ r ' ( t ) ‖ d t = ∫ 0 t R d t = R | t | 0 t = R t . Now, the inverse of the are length function can be calculated as s = g ( t ) is t = g − 1 ( s ) = S R The are length parameterization of the osculating circle is r ( s ) = r ( g 1 ( x ) ) = r × ( S R ) Now, r ( t ) = r ( t o ) + 1 k N + 1 k T ( cos k s , sin k s )

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ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 14.4, Problem 80E

To determine

An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$

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Chapter 14.4, Problem 79E

Chapter 14.4, Problem 81E

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CALCULUS W/SAPLING ACCESS >IC<

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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An are length of the osculating circle at r ( t o ) the function r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ]

Solution Summary: The author calculates the length of the osculating circle for the function r ( t ) = a b.

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An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$

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Chapter 14 Solutions

An are length of the osculating circle at r ( t o ) the function r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ]

Solution Summary: The author calculates the length of the osculating circle for the function r ( t ) = a b.

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An are length of the osculating circle at r(to) the function r(s)=r(to)+1kN+1k[(sinks)T−(cosks)N]

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Chapter 14 Solutions

An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$