Explanation Given: r ( t ) = u ( t ) × v ( t ) , u ( 2 ) = 〈 1 , 2 , − 1 〉 , u ′ ( 2 ) = 〈 3 , 0 , 4 〉 , and v ( t ) = 〈 t , t 2 , t 3 〉 . 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) The vector r ′ ( 2 ) is the derivative of the vector function r ( t ) = u ( t ) × v ( t ) at t = 2 . Write the vector function r ( t ) as follows. r ( t ) = u ( t ) × v ( t ) Differentiate on both sides of the expression with respect to t . d d t [ r ( t ) ] = d d t [ u ( t ) × v ( t ) ] Rewrite the expression as follows. r ′ ( t ) = d d t [ u ( t ) × v ( t ) ] Substitute 2 for t , r ′ ( 2 ) = d d t [ u ( 2 ) × v ( 2 ) ] (2) Use the formula in equation (1) and rewrite the expression in equation (2) as follows. r ′ ( 2 ) = u ′ ( 2 ) × v ( 2 ) + u ( 2 ) × v ′ ( 2 ) (3) Calculation of vector v ( 2 ) : Substitute 2 for t in the expression v ( t ) = 〈 t , t 2 , t 3 〉 , v ( 2 ) = 〈 2 , 2 2 , 2 3 〉 = 〈 2 , 4 , 8 〉 Calculation of vector v ′ ( t ) : The vector v ′ ( t ) is the derivative of the vector function v ( t ) = 〈 t , t 2 , t 3 〉 . Differentiate each component of the vector function v ( t ) = 〈 t , t 2 , t 3 〉 to obtain the vector v ′ ( t ) as follows. d d t [ v ( t ) ] = d d t 〈 t , t 2 , t 3 〉 v ′ ( t ) = 〈 d d t ( t ) , d d t ( t 2 ) , d d t ( t 3 ) 〉 Use the following formula and compute the expression. d d t t n = n t n − 1 Compute the expression v ′ ( t ) = 〈 d d t ( t ) , d d t ( t 2 ) , d d t ( t 3 ) 〉 as follows. v ′ ( t ) = 〈 1 , 2 t , 3 t 2 〉 Calculation of vector v ′ ( 2 ) : Substitute 2 for t in the expression, v ′ ( 2 ) = 〈 1 , 2 ( 2 ) , 3 ( 2 2 ) 〉 = 〈 1 , 4 , 12 〉 Calculation u ′ ( 2 ) × v ( 2 ) : Substitute 〈 3 , 0 , 4 〉 for u ′ ( 2 ) and 〈 2 , 4 , 8 〉 for v ( 2 ) in the expression u ′ ( 2 ) × v ( 2 ) , u ′ ( 2 ) × v ( 2 ) = 〈 3 , 0 , 4 〉 × 〈 2 , 4 , 8 〉 Rewrite and compute the expression as follows

The vector r ′ ( 2 ) .

Question

Chapter 13.2, Problem 50E

To determine

To find: The vector r′(2) .

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Chapter 13.2, Problem 49E

Chapter 13.2, Problem 51E

Multivariable Calculus

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

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Multivariable Calculus

Vector Functions. 50E

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Chapter 13 Solutions

Multivariable Calculus

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