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:

Diminishing Returns

An economic principle stating that as investment in a particular area increases, the rate of profit from that investment, after a certain point, begins to decrease.

Marginal Cost

The change in total production cost that arises when the quantity produced is incremented by one unit.

Fixed Cost

Expenses that do not change in proportion to the activity of a business, such as rent, salaries, and loan payments.

Variable

A variable is an element, feature, or factor that is liable to vary or change; in mathematics and statistics, it's a quantity that can assume any of a set of values.