Explanation Given: The function r ( t ) = 〈 cos t , sin t 〉 and the point t = π 4 . Calculation: Find the first and the second derivative of the function r ( t ) = 〈 cos t , sin t 〉 . The first and the second derivative of the given function are r ′ ( t ) = 〈 − sin t , cos t 〉 and r ″ ( t ) = 〈 − cos t , − sin t 〉 respectively. Then, ‖ r ′ ( t ) ‖ = ( − cos t ) 2 + ( − sin t ) 2 = 1 , and r ′ ( t ) × r ″ ( t ) = 〈 − sin t , cos t 〉 × 〈 − cos t , − sin t 〉 = 〈 − sin t , cos t , 0 〉 × 〈 − cos t , − sin t , 0 〉 = 〈 0 , 0 , sin 2 t + cos 2 t 〉 = 〈 0 , 0 , 1 〉 Therefore, ‖ r ′ ( t ) × r ″ ( t ) ‖ = 0 + 0 + ( 1 ) 2 = 1 . So, the curvature at t = π 4 is, k ( π 4 ) = ‖ r ′ ( π 4 ) × r ″ ( π 4 ) ‖ ‖ r ′ ( π 4 ) ‖ 3 = 1 1 3 = 1 . Thus, the radius of the osculating circle at t = π 4 is 1

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 13.4, Problem 65E

To determine

The equation of the osculating circle of the curve r(t)=〈cost,sint〉 at the point t=π4 or state that the osculating circle does not exist.

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Chapter 13.4, Problem 64E

Chapter 13.4, Problem 66E

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Chapter 13 Solutions

Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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The equation of the osculating circle of the curve r ( t ) = 〈 cos t , sin t 〉 at the point t = π 4 or state that the osculating circle does not exist.

Solution Summary: The author states that the osculating circle of the curve r(t)=langle mathrm

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The equation of the osculating circle of the curve r(t)=〈cost,sint〉 at the point t=π4 or state that the osculating circle does not exist.

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Chapter 13 Solutions