Find the Work Done by F Over the Curve in the Direction

question 201

Multiple Choice

Find the work done by F over the curve in the direction of increasing t.
- $\mathbf { F } = x \mathrm { e } ^ { 3 \mathrm { x } ^ { 2 } } \mathbf { i } + \mathrm { e } ^ { 7 \mathrm { y } _ { j } } + \mathrm { e } ^ { 3 \mathrm { z } } \mathbf { k }$ ; the path is $\mathrm { C } _ { 1 } \cup \mathrm { C } _ { 2 }$ where $\mathrm { C } _ { 1 }$ is the straight line from $( 0,0,0 )$ to $( 1,1,0 )$ and $\mathrm { C } _ { 2 }$ is the straight line from $( 1,1,0 )$ to $( 1,1,1 )$

• A) \(W = \frac { e ^ { 3 } } { 6 } + \frac { e ^ { 7 } } { 7 } + \frac { e ^ { 3 } } { 3 }\)
• B) \(W = \frac { e ^ { 3 } - 1 } { 6 } + \frac { e ^ { 7 } - 1 } { 7 }\)
• C) \(W = \frac { e ^ { 3 } - 1 } { 6 } + \frac { e ^ { 7 } - 1 } { 7 } + \frac { e ^ { 3 } - 1 } { 3 }\)
• D) \(W = \frac { e ^ { 3 } - 1 } { 3 } + \frac { 2 e ^ { 7 } - 2 } { 7 } + \frac { e ^ { 3 } - 1 } { 3 }\)

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Definitions:

Production Possibilities Frontier

A curve that depicts all maximum output possibilities for two or more goods given a set of inputs (resources, labor, etc.), assuming all are used efficiently.

Labor Force

The total number of people employed and unemployed, actively seeking employment in an economy.

Production Possibilities Frontier

A curve depicting the maximum output of two goods that can be produced with given resources and technology, assuming all resources are fully utilized.

Improvement In Technology

Enhancements or advancements in technology that increase productivity or efficiency, often leading to economic growth.