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

---

Definitions:

Service Businesses

Companies that primarily sell intangible goods or services rather than physical objects, such as consultancy, repair, and maintenance firms.

Not As Critical

Referring to elements or issues that are less urgent or important in the context of a given discussion or situation.

Gentrification

Gentrification is the process of neighborhood transformation characterized by the renovation of housing, inflow of affluent residents, and displacement of lower-income families and businesses.

Comparison Shop

The practice of comparing prices, features, and quality of products or services from different vendors to make an informed purchasing decision.