Explanation Given: The vector curve r ( t ) = 〈 cos t , sin t , ln ( cos t ) 〉 and the point ( 1 , 0 , 0 ) . Calculation: For the given vector function r ( t ) = 〈 cos t , sin t , ln ( cos t ) 〉 and the point ( 1 , 0 , 0 ) , we have, t = 0 . Now, ‖ r ′ ( t ) ‖ = sin 2 t + cos 2 t + tan 2 t = sec t , r ″ ( t ) = 〈 − cos t , − sin t , − sec 2 t 〉 , And r ′ ( t ) × r ″ ( t ) = 〈 − sin t , cos t , − tan t 〉 × 〈 − cos t , − sin t , − sec 2 t 〉 , = 〈 − sec t − tan t sin t , sin t − sec 2 t sin t , 1 〉 ‖ r ′ ( t ) × r ″ ( t ) ‖ = ( − sec t − tan t sin t ) 2 + ( sin t − sec 2 t sin t ) 2 + 1 2 Then, T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ = 〈 − sin t , cos t , − tan t 〉 sec t = 〈 − sin t cos t , cos 2 t , sin t 〉 And, B ( t ) = r ′ ( t ) × r ″ ( t ) ‖ r ′ ( t ) × r ″ ( t ) ‖ = To determine (b) The equation of osculating plane to the given curve at the point ( 1 , 0 , 0 ) .

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ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 13.4, Problem 54E

To determine

(a)

The vectors T, N and B for r(t)=〈cost,sint,ln(cost)〉 at (1,0,0).

To determine

(b)

The equation of osculating plane to the given curve at the point (1,0,0) .

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Chapter 13.4, Problem 53E

Chapter 13.4, Problem 55E

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Chapter 13 Solutions

Calculus: Early Transcendentals

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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(a) The vectors T , N and B for r ( t ) = 〈 cos t , sin t , ln ( cos t ) 〉 at ( 1 , 0 , 0 ) .

Solution Summary: The author calculates the vector function r(t)=langle mathrmcost,

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(a)

The vectors T, N and B for r(t)=〈cost,sint,ln(cost)〉 at (1,0,0).

(b)

The equation of osculating plane to the given curve at the point (1,0,0) .

Want to see the full answer?

Chapter 13 Solutions

(a) The vectors T , N and B for r ( t ) = 〈 cos t , sin t , ln ( cos t ) 〉 at ( 1 , 0 , 0 ) .

Solution Summary: The author calculates the vector function r(t)=langle mathrmcost,

Videos

(a) The vectors T, N and B for r(t)=〈cost,sint,ln(cost)〉 at (1,0,0).

(b) The equation of osculating plane to the given curve at the point (1,0,0) .

Want to see the full answer?

Chapter 13 Solutions

(a)

The vectors T, N and B for r(t)=〈cost,sint,ln(cost)〉 at (1,0,0).

(b)

The equation of osculating plane to the given curve at the point (1,0,0) .