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

question 36

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 $\mathbf { r } ( t ) = \left\langle \sqrt { 13 } t , e ^ { t } , e ^ { - t } \right)$ at the point $( 0,1,1 )$ .
Select the correct answer.

• A) \(\sqrt { 15 }\)
• B) \(\frac { \sqrt { 2 } } { 15 }\)
• C) \(15 \sqrt { 2 }\)
• D) \(\frac { 15 } { \sqrt { 2 } }\)
• E) \(\frac { \sqrt { 15 } } { 15 }\)

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Definitions:

Sample Size

The number of observations or data points collected in a study, used to make inferences about a population.

Standard Error

A statistical measure that quantifies the amount of variation or dispersion of a sample statistic compared to the true population parameter.

Standard Deviation

A statistic that measures the dispersion or variability of a dataset relative to its mean.

Sample Size

The number of observations or data points collected in a study, which impacts the study’s ability to detect meaningful effects or differences.