The curvature of the curve given by the vector function $r$ is
$\mathrm { k } ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ { t } ( t ) \right| ^ { 3 } }$
Use the formula to find the curvature of $\mathbf { r } ( t ) = \left( \sqrt { 19 } t , e ^ { t } , e ^ { - t } \right\}$ at the point $( 0,1,1 )$.
A) $\sqrt { 21 }$
B) $\frac { \sqrt { 2 } } { 21 }$
C) $21 \sqrt { 2 }$
D) $\frac { 21 } { \sqrt { 2 } }$
E) $\frac { \sqrt { 21 } } { 21 }$

Question

Quizplus · Accepted Answer

The curvature formula requires the first and second derivatives of the vector function. For $r(t) = (\sqrt{19}t, e^t, e^{-t})$, the first derivative $r'(t) = (\sqrt{19}, e^t, -e^{-t})$ and the second derivative $r''(t) = (0, e^t, e^{-t})$. The cross product $r'(t) \times r''(t)$ and its magnitude, along with the magnitude of $r'(t)$ cubed, are used in the curvature formula. After calculations, the curvature at the given point is found to be $\frac{\sqrt{2}}{21}$, making choice B correct.

The Curvature of the Curve Given by the Vector Function rrr

The Curvature of the Curve Given by the Vector Function $r$