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:

Enzyme Function

The role of an enzyme in catalyzing specific chemical reactions, speeding up the reaction without being consumed in the process.

pH

A measure of the acidity or alkalinity of a solution, with a range from 0 to 14.

Denaturation

A structural change in proteins or nucleic acids, often irreversible, caused by external factors like pH, temperature, or chemicals.

Polar Covalent

A type of chemical bond where a pair of electrons is unequally shared between two atoms, causing a distribution of electrical charge.