Use the Properties of the Derivative to Find $D _ { t } ( \mathbf { r } ( t ) \times \mathbf { u } ( t ) ) \text { given the following }$

question 10

Multiple Choice

Use the properties of the derivative to find
$D _ { t } ( \mathbf { r } ( t ) \times \mathbf { u } ( t ) ) \text { given the following }$ vector-valued functions. $\begin{array} { l } \mathbf { r } ( t ) = 4 \cos 4 t \mathbf { j } + 4 \sin 4 t \mathbf { k } \\\mathbf { u } ( t ) = \frac { 5 } { t } \mathbf { i } + 4 \cos 4 t \mathbf { j } + 4 \sin 4 t \mathbf { k }\end{array}$

• A) \(\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }\)
• B) \(\left( \frac { 80 \cos ( 4 t ) } { t } + \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }\)
• C) \(\left( \frac { 80 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 20 \sin ( 4 t ) } { t } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }\)
• D) \(\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { i } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } - \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }\)
• E) \(\left( \frac { 80 \cos ( 4 t ) } { t } - \frac { 20 \sin ( 4 t ) } { t ^ { 2 } } \right) \mathbf { j } + \left( \frac { 20 \cos ( 4 t ) } { t ^ { 2 } } + \frac { 80 \sin ( 4 t ) } { t } \right) \mathbf { k }\)

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Definitions:

Hypothetical Thinking

The ability to think about situations, propositions, or problems that are not based on immediate reality or current experiences.

Conservation of Mass

A fundamental principle in physics stating that mass is neither created nor destroyed in a closed system through any chemical or physical transformation.

Conservation of Volume

A cognitive ability in child development where the child understands that the volume of a substance remains constant even if its appearance changes.

Class Inclusion

A concept in cognitive development referring to the ability to understand that a category can include smaller subcategories.