Explanation Given: r ( t ) = 〈 t − 1 , t , t 2 〉 Formula used: N → ( t ) = v ( t ) r ' ' ( t ) − v ' ( t ) r ' ( t ) ‖ v ( t ) r ' ' ( t ) − v ' ( t ) r ' ( t ) ‖ r → ' ( t ) = d r → ( t ) d t v ( t ) = ‖ r → ' ( t ) ‖ r ' ' ( t ) = d 2 r → ( t ) d t 2 v ' ( t ) = d d t v → ( t ) Calculation: r → ( t ) = 〈 t − 1 , t , t 2 〉 Differentiate w . r . t t r → ' ( t ) = d d t r → ( t ) = d d t 〈 t − 1 , t , t 2 〉 = 〈 d d t t − 1 , d d t t , d d t t 2 〉 = 〈 − t 2 , 1 , 2 t 〉 ⋯ ⋯ ( 1 ) v ( t ) = ‖ r ' ( t ) ‖ = ‖ 〈 − t 2 , 1 , 2 t 〉 ‖ = ( − t 2 ) + 1 2 + ( 2 t 2 ) = t 4 + 1 + 4 t 2 = 1 + 4 t 2 + t 4 ⋯ ⋯ ( 2 ) r → ' ' ( t ) = d d t r → ' ( t ) = d d t 〈 − t − 2 , 1 , 2 t 〉 = 〈 d d t − t − 2 , d d t 1 , d d t 2 t 〉 = 〈 − ( − 2 ) t − 3 , 0 , 2 〉 = 〈 2 t − 3 , 0 , 2 〉 ⋯ ⋯ ( 3 ) v ' ( t ) = 1 2 1 + t 4 + 4 t 2 × ( 4 t 3 + 8 t ) { Apply chain rule of differentiation } = 2 t 3 + 4 t 1 + t 4 + 4 t 2 ⋯ ⋯ ( 4 ) Substitute t = − 1 in equation ( 1 ) , ( 2 ) , ( 3 ) and ( 4 ) to get the value r → ' ( − 1 ) , v → ( − 1 ) , r → ' ' ( − 1 ) and v ' ( − 1 ) r → ' ( − 1 ) = 〈 { − ( − 1 ) } − 2 , 1 , 2 × ( − 1 ) 〉 〈 1 , 1 , − 2 〉 v → ( −

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 14.4, Problem 50E

To determine

To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 t^{- 1}, t, t^{2} 〉 at t = - 1$

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Chapter 14.4, Problem 49E

Chapter 14.4, Problem 51E

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Chapter 14 Solutions

CALCULUS (CLOTH)

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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Normal vector ( N → ) at the indicated point. 〈 t − 1 , t , t 2 〉 at t = − 1

Solution Summary: The author calculates a normal vector ( N ) at the indicated point.

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To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 t^{- 1}, t, t^{2} 〉 at t = - 1$

Want to see the full answer?

Chapter 14 Solutions

Normal vector ( N → ) at the indicated point. 〈 t − 1 , t , t 2 〉 at t = − 1

Solution Summary: The author calculates a normal vector ( N ) at the indicated point.

Concept explainers

Videos

To calculate: Normal vector (N→) at the indicated point. 〈t−1,t,t2〉 at t=−1

Want to see the full answer?

Chapter 14 Solutions

To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 t^{- 1}, t, t^{2} 〉 at t = - 1$