Explanation Given: The parameterized curve r ( t ) = 〈 t , ln cos t 〉 . Formula used: Suppose r is a smooth parameterized curve and s is the arc length. The unit tangent vector T is r ′ | r ′ | . The principal unit normal vector is N ( t ) = T ′ ( t ) | T ′ ( t ) | . Calculation: Differentiate r ( t ) to compute r ′ ( t ) . r ′ ( t ) = 〈 t ′ , ( ln cos t ) ′ 〉 = 〈 1 , 1 cos t ( − sin t ) 〉 = 〈 1 , − tan t 〉 Use magnitude formula to obtain the value of | r ′ ( t ) | . | r ′ ( t ) | = | 〈 1 , − tan t 〉 | = 1 2 + ( − tan 2 t ) 2 = 1 + tan 2 t = sec 2 t = sec t Use unit tangent vector formula to compute T ( t ) . T ( t ) = 〈 1 , − tan t 〉 sec t = 〈 1 sec t , − tan t sec t 〉 = 〈 cos t , − sin t 〉 Thus, the unit tangent vector T ( t ) is 〈 cos t , − sin t 〉 . Use magnitude formula to compute the value of | T ( t ) | . | T ( t ) | = | 〈 cos t , − sin t 〉 | = ( cos t ) 2 + ( − sin t ) 2 = cos 2 t + sin 2 t = 1 = 1 (1) Differentiate T ( t ) , to compute T ′ ( t )

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Author: Briggs

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Chapter 14.5, Problem 34E

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To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈t,lncost〉 and to verify |T|=|N|=1 and T⋅N=0 .

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Chapter 14.5, Problem 33E

Chapter 14.5, Problem 35E

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Chapter 14 Solutions

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The unit tangent vector T and the principal unit normal vector N for the parameterized curve r ( t ) = 〈 t , ln cos t 〉 and to verify | T | = | N | = 1 and T ⋅ N = 0 .

Solution Summary: The author explains the unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=langle t.

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To find: The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t)=〈t,lncost〉 and to verify |T|=|N|=1 and T⋅N=0 .

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