Explanation Given: The curve is: r ( t ) = 〈 2 cos t , 2 sin t , 4 〉 . And the point P is ( 2 , 2 , 4 ) . Formula Used: Let C be a smooth curve represented by r on an open interval I . The unit tangent vector at t is defined as: T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ , r ′ ( t ) ≠ 0 A line L parallel to v = 〈 a , b , c 〉 and passing through the point P ( x 1 , y 1 , z 1 ) is represented by the parametric equation: x = x 1 + a t y = y 1 + b t z = z 1 + c t Calculation: Consider the provided curve, r ( t ) = 〈 2 cos t , 2 sin t , 4 〉 The derivative of r ( t ) is: r ′ ( t ) = d d t 〈 2 cos t , 2 sin t , 4 〉 = 〈 − 2 sin t , 2 cos t , 0 〉 Then, the unit tangent vector is: T ( t ) = r ′ ( t ) ‖ r ′ ( t ) ‖ = 1 ( − 2 sin t ) 2 + ( 2 cos t ) 2 + ( 0 ) 2 〈 − 2 sin t , 2 cos t , 0 〉 = 〈 − sin t , cos t , 0 〉 Since the provided curve is at the point P ( 2 , 2 , 4 )

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Question

Chapter 12.4, Problem 11E

To determine

To calculate: The unit tangent vector and a set of parametric equations for the tangent line to the space curve r(t)=〈2cost,2sint,4〉 at point P (2,2,4).

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Chapter 12.4, Problem 10E

Chapter 12.4, Problem 12E

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Chapter 12 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Prob. 2E Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Finding the Domain In Exercises 3-10, find the...Ch. 12.1 - Finding the Domain In Exercises 3-10, find the...Ch. 12.1 - Prob. 7E Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Evaluating a function In Exercises 11 and 12...

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The unit tangent vector and a set of parametric equations for the tangent line to the space curve r ( t ) = 〈 2 cos t , 2 sin t , 4 〉 at point P ( 2 , 2 , 4 ) .

Solution Summary: The author explains how to calculate the unit tangent vector and parametric equations for a smooth curve represented by r on an open interval

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To calculate: The unit tangent vector and a set of parametric equations for the tangent line to the space curve r(t)=〈2cost,2sint,4〉 at point P (2,2,4).

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Chapter 12 Solutions