Explanation Given: r → ( t ) = 〈 t 2 , t 3 〉 at t = 1 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 2 , t 3 〉 r → ' ( t ) = d d t r → ( t ) = d d t 〈 t 2 , t 3 〉 = 〈 d d t t 2 , d d t t 3 〉 = 〈 2 t , 3 t 2 〉 r → ' ( t ) = ( 2 t , 3 t 2 ) ⋯ ⋯ ( 1 ) v ( t ) = ‖ r → ' ( t ) ‖ = ( 2 t ) 2 + ( 3 t 2 ) 2 = 4 t 2 + 9 t 4 ⋯ ⋯ ( 2 ) r → ' ' ( t ) = d d t r → ' ( t ) = d d t 〈 2 t , 3 t 2 〉 = 〈 d d t 2 t , d d t 3 t 2 〉 = 〈 2 , 6 t 〉 = ( 2 , 6 t ) ⋯ ⋯ ( 3 ) v ' ( t ) = 1 2 4 t 2 + 9 t 4 × ( 8 t + 36 t 3 ) = 4 t + 18 t 3 4 t 2 + 9 t 4 ⋯ ⋯ ( 4 ) Now, Substitute t = 1 in equation ( 1 ) , ( 2 ) , ( 3 ) and ( 4 ) to find r → ' ( 1 ) , v → ( 1 ) , r → ' ' ( 1 ) , v → ' ( 1 ) r ' ( 1 ) = 〈 2 , 3 〉 v → ' ( 1 ) = 4 + 9 = 13 r ' ' ( 1 ) = 〈 2 , 6 〉 v → ' ( 1 ) = 4 + 18 4 + 9 = 22 13 Since, N ( t ) = v ( t ) r → ' ' ( t ) − v ' ( t ) r → ' ( t ) ‖ v ( t ) r → ' ' ( t ) − v ' ( t ) ...

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Question

Chapter 14.4, Problem 47E

To determine

To calculate:

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

$〈 t^{2}, t^{3} 〉, t = 1$

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Chapter 14.4, Problem 46E

Chapter 14.4, Problem 48E

Chapter 14 Solutions

EBK CALCULUS

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 2 , t 3 〉 , t = 1

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

Concept explainers

Videos

To calculate:

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

$〈 t^{2}, t^{3} 〉, t = 1$

Want to see the full answer?

Chapter 14 Solutions

Normal vector ( N → ) at the indicated point. 〈 t 2 , t 3 〉 , t = 1

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

Concept explainers

Videos

To calculate: Normal vector (N→) at the indicated point. 〈t2,t3〉,t=1

Want to see the full answer?

Chapter 14 Solutions

To calculate:

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

$〈 t^{2}, t^{3} 〉, t = 1$