Explanation Given data: The vector function r ( t ) = 〈 t 2 , ln t , t ln t 〉 and point ( 1 , 0 , 0 ) . Formula used: Consider the three three-dimensional vectors such as a = 〈 a 1 , a 2 , a 3 〉 and b = 〈 b 1 , b 2 , b 3 〉 . Cross product of vectors: Write the expression for cross product of vectors a and b ( a × b ) . a × b = | i j k a 1 a 2 a 3 b 1 b 2 b 3 | a × b = ( a 2 b 3 − b 2 a 3 ) i − ( a 1 b 3 − b 1 a 3 ) j + ( a 1 b 2 − b 1 a 2 ) k (1) Write the expression for curvature of vector function r ( t ) . k ( t ) = | r ′ ( t ) × r ″ ( t ) | | r ′ ( t ) | 3 (2) Here, r ′ ( t ) is first derivative of vector function r ( t ) , and r ″ ( t ) is second derivative of vector function r ( t ) . Write the expression for magnitude of vector a ( | a | ) . | a | = a 1 2 + a 2 2 + a 3 2 (3) Here, a 1 , a 2 and a 3 are the x , y , and z -coordinates of vector respectively. The point on vector function r ( t ) = 〈 t 2 , ln t , t ln t 〉 is ( t 2 , ln t , t ln t ) . The curvature is located at point ( 1 , 0 , 0 ) . Hence equate the components of points ( t 2 , ln t , t ln t ) and ( 1 , 0 , 0 ) . t 2 = 1 t = 1 t = ± 1 ln t = 0 t = e 0 t = 1 t ln t = 0 ln t = 0 t = e 0 t = 1 Therefore, the value of t is 1 from conclusions. Write the vector function. r ( t ) = 〈 t 2 , ln t , t ln t 〉 Apply differentiation with respect to t on both sides of equation. r ′ ( t ) = d d t 〈 t 2 , ln t , t ln t 〉 = 〈 d d t ( t 2 ) , d d t ( ln t ) , d d t ( t ln t ) 〉 = 〈 2 t , 1 t , ( t ( d d t ( ln t ) ) + ( d d t ( t ) ) ( ln t ) ) 〉 { ∵ d d x ( x ) = 1 d d x ( x n ) = n x n − 1 d d x ( u v ) = u v ′ + u ′ v } = 〈 2 t , 1 t , ( t ( d d t ( ln t ) ) + ( d d t ( t ) ) ( ln t ) ) 〉 Simplify the equation. r ′ ( t ) = 〈 2 t , 1 t , ( t ( 1 t ) + ( 1 ) ( ln t ) ) 〉 r ′ ( t ) = 〈 2 t , 1 t , ( 1 + ln t ) 〉 (4) Substitute 1 for t , r ′ ( 1 ) = 〈 2 ( 1 ) , 1 1 , ( 1 + ln ( 1 ) ) 〉 = 〈 2 , 1 , ( 1 + 0 ) 〉 { ∵ ln ( 1 ) = 0 } = 〈 2 , 1 , 1 〉 Find the value of | r ′ ( 1 ) | by using equation (3)

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ISBN: 9781337772020

Author: Stewart

Publisher: VST

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Question

Chapter 10.8, Problem 20E

To determine

To find: The curvature of r(t)=〈t2,lnt,tlnt〉 at point (1,0,0) .

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Chapter 10.8, Problem 19E

Chapter 10.8, Problem 21E

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The curvature of r ( t ) = 〈 t 2 , ln t , t ln t 〉 at point ( 1 , 0 , 0 ) .

Solution Summary: The author explains how to find the curvature of r(t)=langle t

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To find: The curvature of r(t)=〈t2,lnt,tlnt〉 at point (1,0,0) .

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Chapter 10 Solutions