Find T, N, and B for the Given Space Curve $$r ( t ) = \frac { 4 } { 3 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 4 } { 3 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + 1 \mathrm { tk }$$

question 97

Multiple Choice

Find T, N, and B for the given space curve.
- $$r ( t ) = \frac { 4 } { 3 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 4 } { 3 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + 1 \mathrm { tk }$$

• A) \(T = \frac { 2 } { 3 } \sqrt { 1 + t } \mathbf { i } + \frac { 2 } { 3 } \sqrt { 1 - t } j + \frac { 1 } { 3 } k ; N = \frac { 1 } { 2 } \sqrt { 2 - 2 } \mathrm { ti } + \frac { 1 } { 2 } \sqrt { 2 + 2 t j } ; B = \frac { 1 } { 6 } \sqrt { 2 + 2 t i } + \frac { 1 } { 6 } \sqrt { 2 - 2 } \mathbf { j } + \frac { 2 \sqrt { 2 } } { 3 } \mathbf { k }\)
• B) \(\mathrm { T } = \frac { 2 } { 3 } \sqrt { 1 + \mathrm { t } } \mathrm { i } - \frac { 2 } { 3 } \sqrt { 1 - \mathrm { t } } \mathrm { j } + \frac { 1 } { 3 } \mathrm { k } ; \mathrm { N } = \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + \frac { 1 } { 2 } \sqrt { 2 + 2 \mathrm { t } } ; \mathbf { B } = - \frac { 1 } { 6 } \sqrt { 2 + 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 6 } \sqrt { 2 - 2 \mathrm { t } } \mathbf { j } + \frac { 2 \sqrt { 2 } } { 3 } \mathrm { k }\)
• C) \(\mathrm { T } = 2 \sqrt { 1 + \mathrm { t } } - 2 \sqrt { 1 - t } \mathbf { j } + 1 \mathbf { k } ; \mathbf { N } = \frac { 3 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + \frac { 3 } { 2 } \sqrt { 2 + 2 t } \mathbf { j } ; \mathbf { B } = - \frac { 1 } { 2 } \sqrt { 2 + 2 t \mathrm { i } } + \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } + 2 \sqrt { 2 } \mathbf { k }\)
• D) \(\mathrm { T } = \frac { 1 } { 3 } \sqrt { 1 + \mathrm { t } } - \frac { 1 } { 3 } \sqrt { 1 - \mathrm { t } } \mathrm { j } + \frac { 1 } { 3 } \mathrm { k } ; \mathrm { N } = \frac { 1 } { 2 } \sqrt { 2 - 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 2 } \sqrt { 2 + 2 \mathrm { t } } ; \mathbf { B } = - \frac { 1 } { 6 } \sqrt { 2 + 2 \mathrm { t } } \mathrm { i } + \frac { 1 } { 6 } \sqrt { 2 - 2 \mathrm { t } } + \frac { \sqrt { 2 } } { 3 } \mathbf { k }\)

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Definitions:

Punishment

A consequence designed to reduce or eliminate undesirable behavior by presenting an adverse outcome following that behavior.

Organizational Behaviour

The study of how individuals and groups interact within organizational settings, focusing on aspects like leadership, culture, performance, and motivation.

Learned Behaviour

Actions or reactions developed as a result of experiences or training, rather than being innate.

Extinction

In behavioral psychology, the gradual weakening and disappearance of a conditioned response when it is no longer reinforced.