Find a Function F Such That $\mathbf { F } = \nabla f$

question 77

Essay

Find a function f such that $\mathbf { F } = \nabla f$ and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the curve C. $\mathbf { F } ( x , y ) = e ^ { 2 y } \mathbf { i } + \left( 1 + 2 x e ^ { 2 y } \right) \mathbf { j } ; C : \mathbf { r } ( t ) = t e ^ { t } \mathbf { i } + ( 1 + t ) \mathbf { j } ; 0 \leq t \leq 1$

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Definitions:

Foreseeable Future

A period of time over which events can be predicted or expected to occur based on current knowledge or analysis.

Intragenerational Equity

The ability to meet the needs of current generations.

Eco-Justice

Refers to the fair treatment and meaningful involvement of all people regardless of race, color, national origin, or income with respect to the development, implementation, and enforcement of environmental laws, regulations, and policies.

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A measure that evaluates the general well-being of individuals and societies, encompassing not only wealth and employment but also the built environment, physical and mental health, education, recreation and leisure time, and social belonging.