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

question 27

Multiple Choice

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) \(\frac { \sqrt { 21 } } { 21 }\)

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Definitions:

General Adaptation Syndrome

A three-stage model (alarm, resistance, exhaustion) that describes the physiological changes the body undergoes when under stress.

Resistance Stage

The second stage of the General Adaptation Syndrome, where the body adapts to and resists stress through continued hormone and energy production.

Alarm Stage

The initial reaction of the body to stress, activating the fight or flight response as part of the general adaptation syndrome.

Exhaustion Stage

The exhaustion stage is the final phase of the stress response, where the body's ability to resist stressors diminishes significantly after prolonged exposure, leading to depletion of energy and possible health deterioration.