Explanation Given: r ( t ) = 〈 t , t , e t 〉 Point = ( 0 , 0 , 1 ) 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 , e t 〉 Differentiate r ( t ) = w . r . t t r ' ( t ) = 〈 1 , 1 , e t 〉 Now, magnitude of r ' ( t ) r ' ( t ) = 1 2 + 1 2 + ( e t ) 2 = 1 + 1 + e 2 t The unit tangent vector can be calculated as T ( t ) = r ' ( t ) | r ' ( t ) | = 〈 1 , 1 , e t 〉 1 + 1 + e 2 t = 〈 1 1 + 1 + e 2 t , 1 2 + e 2 t , e t 2 + e 2 t 〉 T ( 0 ) = 〈 1 1 + 1 + 1 , 1 2 + 1 , 1 2 + 1 〉 = 〈 1 3 , 1 3 , 1 3 〉 Differentiate T ( t ) w . r . t t T ' ( t ) = 〈 − 1 2 × ( 2 + e 2 t ) − 3 2 × e 2 t × 2 , − 1 2 × ( 2 + e 2 t ) − 3 2 × e 2 t × 2 , e t 2 + e 2 t − e t × 1 2 ( 2 + e 2 t ) − 1 2 × e 2 t × 2 ( 2 + e 2 t ) 〉 = 〈 − e 2 t ( 2 + e

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 14.4, Problem 44E

To determine

To find:

$T, N and B$ For the curve $r (t) = 〈 t, t, e^{t} 〉$ at $(0, 0, 1)$

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Chapter 14.4, Problem 43E

Chapter 14.4, Problem 45E

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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 , e t 〉 at ( 0 , 0 , 1 )

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

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To find:

$T, N and B$ For the curve $r (t) = 〈 t, t, e^{t} 〉$ at $(0, 0, 1)$

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Chapter 14 Solutions

T , N and B For the curve r ( t ) = 〈 t , t , e t 〉 at ( 0 , 0 , 1 )

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

Videos

To find: T,N and B For the curve r(t)=〈t,t,et〉 at (0,0,1)

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Chapter 14 Solutions

To find:

$T, N and B$ For the curve $r (t) = 〈 t, t, e^{t} 〉$ at $(0, 0, 1)$