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: 15

Answer

$$ \kappa(t)=\frac{e^t}{\left(1+e^{2t}\right)^{3 / 2}}, $$ $$ \kappa(3)= \frac{e^3}{\left(1+e^{6}\right)^{3 / 2}}, $$

Work Step by Step

We have $$ \kappa(t)=\frac{\left|f^{\prime \prime}(t)\right|}{\left(1+f^{\prime}(t)^{2}\right)^{3 / 2}}=\frac{e^t}{\left(1+e^{2t}\right)^{3 / 2}}, $$ and hence $$ \kappa(3)= \frac{e^3}{\left(1+e^{6}\right)^{3 / 2}}, $$
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