Explanation Given info. r ( t ) = 〈 t , t 2 , 2 3 t 3 〉 Point: 〈 1 , 1 , − 2 3 〉 Formula used: r ( t ) = 〈 x , y , z 〉 = 〈 x ( t ) , y ( t ) , z ( t ) 〉 r ' ( t ) = d d t 〈 x ( t ) , y ( t ) , z ( t ) 〉 The unit tangent vector is T ( t ) = r ' ( t ) | r ' ( t ) | The value of unit normal vector N ( t ) is N ( t ) = T ' ( t ) | T ' ( t ) | The value of binomial vector B ( 1 ) is calculated as B ( t ) = T ( t ) × N ( t ) Calculation: r ( t ) = 〈 t , t 2 , 2 3 t 3 〉 Differentiate r ( t ) w.r.t t r ' ( t ) = 〈 1 , 2 t , 2 t 2 〉 Now, magnitude of r ' ( t ) | r ' ( t ) | = 1 2 + ( 2 t 2 ) + ( 2 t 2 ) 2 = 1 + 4 t 2 + 4 t 4 T ( t ) = r ' ( t ) | r ' ( t ) | = 〈 1 , 2 t , 2 t 2 〉 1 + 4 t 2 + 4 t 4 T ( t ) = 〈 1 1 + 4 t 2 + 4 t 4 , 2 t 1 + 4 t 2 + 4 t 4 , 2 t 2 1 + 4 t 2 + 4 t 4 〉 T ( 1 ) = 〈 1 1 + 4 + 4 , 2 × 1 1 + 4 + 4 , 2 × 1 1 + 4 + 4 〉 = 〈 1 3 , 2 3 , 2 3 〉 T ' ( t ) = 〈 − 1 2 × ( 1 + 4 t 2 + 4 t 4 ) − 3 2 × ( 8 t + 16 t 3 ) , 2 1 + 4 t 2 + 4 t 4 − 2 t × 1 2 ( 1 + 4 t 2 + 4 t 4 ) − 1 2 ( 8 t + 16 t 3 ) ( 1 + 4 t 2 + 4 t 4 ) , 1 + 4 t 2 + 4 t 4 × 4 t − 2 t 2 × 1 2 ( 1 + 4 t 2 + 4 t 4 ) − 1 2 × ( 8 t + 16 t 3 ) ( 1 + 4 t 2 + 4 t 4 ) 〉 T ' ( 1 ) = 〈 − 1 2 ( 1 + 4 + 4 ) − 3 2 × ( 8 + 16 ) , 2 1 + 4 + 4 − 1 × ( 1 +

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 14.4, Problem 43E

To determine

To find:

T,N, and B for the curve $r (t) = 〈 t, t^{2}, \frac{2}{3} t^{3} 〉$ at $〈 1, 1, \frac{- 2}{3} 〉$

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Chapter 14.4, Problem 42E

Chapter 14.4, Problem 44E

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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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T,N, and B for the curve r ( t ) = 〈 t , t 2 , 2 3 t 3 〉 at 〈 1 , 1 , − 2 3 〉

Solution Summary: The author explains how to find T,N, and B for the curve r (t) = t, 2, 3.

Concept explainers

To find:

T,N, and B for the curve $r (t) = 〈 t, t^{2}, \frac{2}{3} t^{3} 〉$ at $〈 1, 1, \frac{- 2}{3} 〉$

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Chapter 14 Solutions

T,N, and B for the curve r ( t ) = 〈 t , t 2 , 2 3 t 3 〉 at 〈 1 , 1 , − 2 3 〉

Solution Summary: The author explains how to find T,N, and B for the curve r (t) = t, 2, 3.

Concept explainers

To find: T,N, and B for the curve r(t)=〈t,t2,23t3〉 at 〈1,1,−23〉

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Chapter 14 Solutions

To find:

T,N, and B for the curve $r (t) = 〈 t, t^{2}, \frac{2}{3} t^{3} 〉$ at $〈 1, 1, \frac{- 2}{3} 〉$