$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$

question 8

Multiple Choice

$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. $$\begin{array} { l } \mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } \\\mathbf { u } ( t ) = 2 \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 3 t ^ { 3 } \mathbf { k }\end{array}$$

• A) \(4 \mathbf { i } + \left( 36 t ^ { 3 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }\)
• B) \(4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t ^ { 2 } - 45 t \right) \mathbf { k }\)
• C) \(4 t \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } - \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }\)
• D) \(4 \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } + \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }\)
• E) \(4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }\)

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Definitions:

Operating Income

Earnings before interest and taxes (EBIT), a measure of a company's profitability from its core business operations.

Fixed Costs

Operating expenses of a business that remain constant regardless of the volume of production or sales.

Unit Contribution Margin

The amount each unit sold contributes towards covering fixed costs and generating profit, calculated as the selling price per unit minus the variable cost per unit.

Operating Income

Earnings before interest and taxes (EBIT), measuring a company's profit from its operational and regular business activities.