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: 1


$$ r'(t) = \lt 8t,9\gt, \quad T(t)= \frac{ \lt8t,9\gt}{\sqrt{64t^2+81}}, \quad T(1)=\frac{ \lt 8,9\gt}{\sqrt{145}}.$$

Work Step by Step

Since $ r(t) = \lt 4t^2, 9t\gt$, then $ r'(t) = \lt 8t,9\gt$ and $\|r'(t)\|=\sqrt{64t^2+81}=\sqrt{59}$. Hence, we have $$T(t)=\frac{r'(t)}{\|r'(t)\|}=\frac{ \lt8t,9\gt}{\sqrt{64t^2+81}}, \quad T(1)=\frac{ \lt 8,9\gt}{\sqrt{145}}.$$
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