Explanation Given: The vector-valued function r ( t ) = 〈 8 , 3 sin 2 t , 3 cos 2 t 〉 , at 0 ≤ t ≤ π , t = π 4 . Definition used: Tangent Vector: “Let r ( t ) = 〈 f ( t ) , g ( t ) , h ( t ) 〉 where f , g and h are differentiable functions on ( a , b ) . Then r has a derivative on ( a , b ) and r ′ ( t ) = 〈 f ′ ( t ) , g ′ ( t ) , h ′ ( t ) 〉 . Provided r ′ ( t ) ≠ 0 , r ′ ( t ) is a tangent vector at the point corresponding to r ( t ) .” The unit tangent vector can be given by, T ( t ) = r ′ ( t ) | r ′ ( t ) | . Derivative rule: (1) Constant rule : d d t ( c t ) = c d d t ( t ) (2) d d t ( sin a t ) = a cos a t [ a is constant ] (3) d d t ( cos a t ) = − a sin a t [ a is constant ] Calculation: Obtain the derivative of the vector-valued function r ( t ) = 〈 8 , 3 sin 2 t , 3 cos 2 t 〉 . r ′ ( t ) = d d t ( r ( t ) ) = d d t 〈 8 , 3 sin 2 t , 3 cos 2 t 〉 r ′ ( t ) = 〈 d d t ( 8 ) , d d t ( 3 sin 2 t ) , d d t ( 3 cos 2 t ) 〉 Apply the constant rule and derivative rules (2) and (3), r ′ ( t ) = 〈 8 d d t ( 1 ) , 3 d d t ( sin 2 t ) , 3 d d t ( cos 2 t ) 〉 = 〈 0 , 3 ( 2 ) cos 2 t , 3 ( 2 ) ( − sin 2 t ) 〉 = 〈 0 , 6 cos 2 t , − 6 sin 2 t 〉 Thus, the tangent vector is r ′ ( t ) = 〈 0 , 6 cos 2 t , − 6 sin 2 t 〉

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 14, Problem 28RE

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To find: The unit tangent vector for the parametrized curves.

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Chapter 14 Solutions

Calculus: Early Transcendentals (3rd Edition)

Ch. 14.1 - Restrict the domain o f the vector function in...Ch. 14.1 - Explain why the curve in Example 5 lies on the...Ch. 14.1 - How many independent variables does the function...Ch. 14.1 - How many dependent scalar variables does the...Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 14.1 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 14.1 - Find a function r(t) for the line passing through...Ch. 14.1 - Find a function r(t) whose graph is a circle of...

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The unit tangent vector for the parametrized curves.

Solution Summary: The author explains the unit tangent vector for the parametrized curves: r(t)=langle 8,3mathrmsin2t

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