Explanation Given: The vector-valued function is: r ( t ) = e 2 t i + cos t j − sin 3 t k at point P ( 1 , 1 , 0 ) . Formula used: Unit tangent vector T ( t ) = r ( t ) ‖ r ′ ( t ) ‖ Calculation: Consider the provided vector-valued function, r ( t ) = e 2 t i + cos t j − sin 3 t k Differentiate with respect to t. r ′ ( t ) = 2 e 2 t i − sin t j − 3 cos 3 t k And, the modulus will be: ‖ r ′ ( t ) ‖ = 4 e 4 t + sin 2 t + 9 cos 2 3 t Now, use tangent vector formula

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Chapter 12, Problem 45RE

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To calculate: The unit tangent vector T(t) for r(t)=e2ti+costj−sin3tk and a set of parametric equations for the tangent line to the space curve at point P(1,1,0).

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Chapter 12, Problem 44RE

Chapter 12, Problem 46RE

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Chapter 12 Solutions

CALCULUS EARLY TRANSCENDENTAL FUNCTIONS

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The unit tangent vector T ( t ) for r ( t ) = e 2 t i + cos t j − sin 3 t k and a set of parametric equations for the tangent line to the space curve at point P ( 1 , 1 , 0 ) .

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To calculate: The unit tangent vector T(t) for r(t)=e2ti+costj−sin3tk and a set of parametric equations for the tangent line to the space curve at point P(1,1,0).

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Chapter 12 Solutions