If $\mathbf { u } ( t ) = \left( - \sqrt { t } \sin t , t , t ^ { \frac { 2 } { 3 } } \right)$

question 35

Essay

If $\mathbf { u } ( t ) = \left( - \sqrt { t } \sin t , t , t ^ { \frac { 2 } { 3 } } \right)$ and $\mathbf { v } ( t ) = \left( - \sqrt { t } \sin t , \cos ^ { 2 } t , - t ^ { \frac { 1 } { 3 } } \right)$ , compute $\frac { d } { d t } ( \mathbf { u } ( t ) \cdot \mathbf { v } ( t ) )$ and $\frac { d } { d t } ( \mathbf { u } ( t ) \cdot \mathbf { u } ( t ) )$ .

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Definitions:

Atlantic Slave Trade

The transatlantic trading patterns which were established from the 16th through the 19th centuries, which transported millions of Africans to the Americas as slaves.

Southern Ohio

A region in the United States characterized by its geographical location within the state of Ohio, often noting its distinct cultural and historical features.

New England Settlers

Refers to the early colonists and immigrants who established the first English colonies in the northeastern part of what is now the United States.

Cotton Gin

A machine invented by Eli Whitney in 1793 that quickly and efficiently removed seeds from cotton fibers, significantly impacting the agricultural economy of the United States.