Explanation Formula used: Write the expression to find unit tangent vector T ( t ) for the vector r ( t ) at a finite value of scalar parameter t . T ( t ) = r ′ ( t ) | r ′ ( t ) | (1) Here, r ′ ( t ) is the tangent vector, which is the derivative of vector r ( t ) . Find the tangent vector r ′ ( t ) by differentiating each component of the vector r ( t ) as follows. d d t [ r ( t ) ] = d d t 〈 tan − 1 t , 2 e 2 t , 8 t e t 〉 r ′ ( t ) = 〈 d d t ( tan − 1 t ) , d d t ( 2 e 2 t ) , d d t ( 8 t e t ) 〉 Use the following formula to compute the expression. d d t tan − 1 t = 1 1 + t 2 d d t [ u ( t ) v ( t ) ] = u ( t ) d d t [ v ( t ) ] + v ( t ) d d t [ u ( t ) ] d d t e t = e t d d t e 2 t = 2 e t Compute the expression r ′ ( t ) = 〈 d d t ( tan − 1 t ) , d d t ( 2 e 2 t ) , d d t ( 8 t e t ) 〉 by using the formulae as follows. r ′ ( t ) = 〈 d d t ( tan − 1 t ) , 2 d d t ( e 2 t ) , d d t ( 8 t e t ) 〉 = 〈 1 1 + t 2 , 2 ( 2 ) e 2 t , 8 [ t d d t ( e

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Question

Chapter 13.2, Problem 18E

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To find: The unit tangent vector T(t) for the vector r(t)=〈tan−1t, 2e2t, 8tet〉 at t=0.

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Chapter 13.2, Problem 17E

Chapter 13.2, Problem 19E

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The unit tangent vector T ( t ) for the vector r ( t ) = 〈 tan − 1 t , 2 e 2 t , 8 t e t 〉 at t = 0 .

Solution Summary: The author explains how to find the unit tangent vector T(t) at a finite value of scalar parameter t.

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To find: The unit tangent vector T(t) for the vector r(t)=〈tan−1t, 2e2t, 8tet〉 at t=0.

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