Explanation Given info: r ( t ) = < cos t , sin t , t 2 > Concept used: The curvature at the point t can be calculated as k ( t ) = ‖ r ' ( t ) × r ' ' ( t ) ‖ ‖ r ' ( t ) ‖ 3 Calculation: r ( t ) = < cos t , sin t , t 2 > Do single differentiation, r ' ( t ) = < − sin t , cos t , 2 t > Again do differentiation w . r . t t r ' ' ( t ) = < − cos t , − sin t , 2 > Now, find the value of r ' ( t ) and r ' ( t ) × r ' ' ( t ) at t = π 2 r ' ( t ) = < − sin t , cos t , 2 t > ‖ r ' ( π 2 ) ‖ = ( − 1 ) 2 + 0 2 + π 2 = 1 + π 2 r ' ( t ) × r ' ' ( t ) = < − sin t , cos t , 2 t > × < − cos t , − sin t , 2 > r ' ( π 2 ) × r ' ' ( π 2 ) = < − 1 , 0 , π > × < 0 , − 1 , 2 > ( π 2 ) × r ' ' ( π 2 ) = |

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ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 14.4, Problem 13E

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Evaluate the curvature at the given point $r (t) = < \cos t, \sin t, t^{2} > at t = \frac{π}{4}$

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Chapter 14.4, Problem 12E

Chapter 14.4, Problem 14E

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Chapter 14 Solutions

EBK CALCULUS

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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Evaluate the curvature at the given point r ( t ) = < cos t , sin t , t 2 > at t = π 4

Evaluate the curvature at the given point $r (t) = < \cos t, \sin t, t^{2} > at t = \frac{π}{4}$