Explanation Given info. Equation of space curve r ( t ) = 〈 cos t . sin t 〉 given point is t = π 4 Formula used: O Q → = r → ( t 0 ) + 1 k ( t 0 ) 〈 cos t , sin t 〉 where O Q → is the center of circle T ∧ ( t ) = r → ( t ) ‖ r → ( t ) ‖ N ∧ ( t ) = T ∧ ' ( t ) ‖ T ∧ ' ( t ) ‖ K ( t ) = ‖ r → ' ( t ) × r → " ( t ) ‖ ‖ r → ' ( t ) ‖ 3 Equation of circle C → ( t ) = O Q → + 1 K ( t 0 ) 〈 cos t , sin t 〉 Calculation: r ( t ) = 〈 cos t , sin t 〉 r ' ( t ) = d d t 〈 cos t , sin t 〉 = 〈 d cos t d t , d sin t d t 〉 = 〈 − sin t , cos t 〉 r ' ( π 4 ) = 〈 − 1 2 , 1 2 〉 r " ( t ) = d d t ( r ' ( t ) ) = d d t 〈 − sin t , cos t 〉 = 〈 − d sin t d t , d cos t d t 〉 = 〈 − cos t , − sin t 〉 r " ( π 4 ) = 〈 − 1 2 , − 1 2 〉 T ∧ ( t ) = r → ' ( t ) ‖ r → ' ( t ) ‖ = 〈 − sin t , cos t 〉 sin 2 t + cos 2 t = 〈 − sin t , cos t 〉 ( ∵ sin 2 t + cos 2 t = 1 ) T ∧ ' ' ( t ) = d d t 〈 − sin t , cos t 〉 = 〈 − d d t ( sin t ) , d d t ( cos t ) 〉 = 〈 − cos t , − sin t 〉 ( d d t ( sin t ) = cos t , d d t ( cos t ) = − sin t ) N ∧ ( t ) = T ∧ ' ( t ) ‖ T ∧ ' ( t ) ‖ = 〈 − cos t , − sin t 〉 ( − cos t ) 2 + ( − sin t ) 2 = 〈 − cos t , − sin t 〉 Putting t = π 4 , both side...

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Author: Rogawski

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Chapter 14.4, Problem 65E

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The parameterization of the osculating circle at the point indicated. $r (t) = 〈 \cos t, \sin t 〉, t = \frac{π}{4}$

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Chapter 14.4, Problem 64E

Chapter 14.4, Problem 66E

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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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The parameterization of the osculating circle at the point indicated. r ( t ) = 〈 cos t , sin t 〉 , t = π 4

Solution Summary: The author explains how to determine the parameterization of the osculating circle at the point indicated.

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The parameterization of the osculating circle at the point indicated. $r (t) = 〈 \cos t, \sin t 〉, t = \frac{π}{4}$

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Chapter 14 Solutions

The parameterization of the osculating circle at the point indicated. r ( t ) = 〈 cos t , sin t 〉 , t = π 4

Solution Summary: The author explains how to determine the parameterization of the osculating circle at the point indicated.

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The parameterization of the osculating circle at the point indicated.r(t)=〈cost,sint〉,t=π4

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Chapter 14 Solutions

The parameterization of the osculating circle at the point indicated. $r (t) = 〈 \cos t, \sin t 〉, t = \frac{π}{4}$