Explanation Given data: u ( t ) = 〈 sin t , cos t , t 〉 and v ( t ) = 〈 t , cos t , sin t 〉 . Formula used: Write the expression to find the derivative of cross product of the two vectors u ( t ) and v ( t ) . d d t [ u ( t ) × v ( t ) ] = u ′ ( t ) × v ( t ) + u ( t ) × v ′ ( t ) (1) Consider the given vector u ( t ) as follows. u ( t ) = 〈 sin t , cos t , t 〉 Rewrite the expression as follows. u ( t ) = sin t i + cos t j + t k Calculation of vector u ′ ( t ) : The vector u ′ ( t ) is the derivative of the vector function u ( t ) = sin t i + cos t j + t k . Differentiate each component of the vector function u ( t ) = sin t i + cos t j + t k to obtain the vector u ′ ( t ) as follows. d d t [ u ( t ) ] = d d t [ sin t i + cos t j + t k ] u ′ ( t ) = d d t ( sin t ) i + d d t ( cos t ) j + d d t ( t ) k Use the following formulae to compute the expression. d d t sin t = cos t d d t cos t = − sin t d d t t n = n t n − 1 Compute the expression u ′ ( t ) = d d t ( sin t ) i + d d t ( cos t ) j + d d t ( t ) k by using the formulae as follows. u ′ ( t ) = ( cos t ) i + ( − sin t ) j + ( 1 ) k = ( cos t ) i − ( sin t ) j + k Consider the given vector v ( t ) as follows. v ( t ) = 〈 t , cos t , sin t 〉 Rewrite the expression as follows. v ( t ) = t i + cos t j + sin t k Calculation of vector v ′ ( t ) : The vector v ′ ( t ) is the derivative of the vector function v ( t ) = t i + cos t j + sin t k . Differentiate each component of the vector function v ( t ) = t i + cos t j + sin t k to obtain the vector v ′ ( t ) as follows. d d t [ v ( t ) ] = d d t [ t i + cos t j + sin t k ] Rewrite and compute the expression as follows. v ′ ( t ) = d d t ( t ) i + d d t ( cos t ) j + d d t ( sin t ) k = ( 1 ) i + ( − sin t ) j + ( cos t ) k = i − ( sin t ) j + ( cos t ) k Calculation of vector u ′ ( t ) × v ( t ) : Substitute ( t ) i + ( cos t ) j + ( sin t ) k for v ( t ) , and ( cos t ) i − ( sin t ) j + k for u ′ ( t ) , and compute the expression u ′ ( t ) × v ( t ) as follows. u ′ ( t ) × v ( t ) = [ ( cos t ) i − ( sin t ) j + k ] × [ ( t ) i + ( cos t ) j + ( sin t ) k ] Rewrite and compute the expression as follows. u ′ ( t ) × v ( t ) = | i j k cos t − sin t 1 t cos t sin t | = { i [ ( − sin t ) ( sin t ) − ( cos t ) ( 1 ) ] − j [ ( cos t ) ( sin t ) − ( t ) ( 1 ) ] + k [ ( cos t ) ( cos t ) − ( t ) ( − sin t ) ] } = ( − sin 2 t − cos t ) i + ( t − cos t sin t ) j + ( cos 2 t + t sin t ) k = 〈 ( − sin 2 t − cos t ) , ( t − cos t sin t ) , ( cos 2 t + t sin t ) 〉 Calculation of vector u ( t ) × v ′ ( t ) : Substitute ( sin t ) i + ( cos t ) j + ( t ) k for u ( t ) , and i − ( sin t ) j + ( cos t ) k for v ′ ( t ) and compute the expression u ( t ) × v ′ ( t ) as follows

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

8th Edition

ISBN: 9781305779198

Author: James Stewart

Publisher: Cengage Learning

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Chapter 13.2, Problem 48E

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Chapter 13.2, Problem 47E

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Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Find the limit. 3. limt0(e3ti+t2sin2tj+cos2tk)Ch. 13.1 - Find the limit. 4. limt1(t2-tt-1i+t+8j+sintlntk)Ch. 13.1 - Find the limit. 5. limt1+t21t2,tan-1t,1e2tt Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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The expression for d d t [ u ( t ) × v ( t ) ] .

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