Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.4 Curvature - Exercises - Page 734: 14



Work Step by Step

The curvature $\kappa$ for a plane curve system is: $\kappa (t)= \dfrac{||r'(t) \times r''(t)||}{||r'(t)||^3}$ We have: $r'(t) =\lt \sinh t, \cosh t, 1 \gt$ and $r''(t) =\lt \cosh t, \sinh t, 0 \gt$ Thus, $\kappa(t) = \dfrac{||-\sinh ti+\cosh t j-k||}{||\lt \sinh t, \cosh t, 1 \gt||^3} \\=\dfrac{\sqrt {\sinh^2 t+\cosh^2 t+1}}{[\sinh^2t+\cosh^2 t+1]^{3/2}}=\dfrac{1}{\cosh (2t)+1}$ Now, we will compute $\kappa (t)$ at $t=0$ Therefore, $\kappa(0)= \dfrac{1}{\cosh 2(0)+1}=\dfrac{1}{2}$
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