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

question 120

Multiple Choice

The curvature of the curve given by the vector function $r$ is
$$k ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ {t } ( t ) \right| ^ { 3 } }$$ 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:

Sulfides

A group of minerals containing sulfur (S) bonded with a metal.

Pyrite

A common, pale bronze to brass yellow, iron sulfide mineral, commonly called “fool’s gold.”

Element S

Refers to Sulfur, a chemical element with the symbol S and atomic number 16, known for its bright yellow crystals.

Rock-forming Minerals

Rock-forming minerals are minerals that make up the majority of Earth's crust and are the basic building blocks of rocks, including quartz, feldspar, and mica among others.