The Torsion of a Curve Defined By $\mathbf { r } ( t )$

question 56

Multiple Choice

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by
$$\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right) \cdot \mathbf { r } ^ { ttt } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right| ^ { 2 } }$$
Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 5 t \mathbf { i } + \sin 5 t \mathbf { j } + 4 t \mathbf { k }$ .

• A) \(\frac { 21 } { 41 }\)
• B) \(\frac { 9 } { 41 }\)
• C) \(\frac { 20 } { 41 }\)
• D) \(\frac { 80 } { 41 }\)

Recognize the significance of urban centers and their growth in the American West by 1900.

Understand the role of the federal government and legislation in shaping the West.

Comprehend the cultural myths and realities of the West as propagated by figures like William "Buffalo Bill" Cody.

Acknowledge the challenges and dangers of life in the American West, including conflict, disease, and natural disasters.

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Definitions:

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A term describing individuals who have a strong desire to accomplish significant tasks, achieve mastery, and reach high standards of excellence.

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