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

question 215

Multiple Choice

Find the work done by F over the curve in the direction of increasing t.
- $$\mathbf { F } = - \frac { 1 } { 16 \mathrm { x } ^ { 2 } } \mathbf { i } + \frac { 3 \mathrm { z } } { 4 \mathrm { x } } \mathbf { j } + \frac { 1 } { 48 \mathrm { x } ^ { 2 } } \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \frac { \cos 4 \mathrm { t } } { 4 } \mathbf { i } + \frac { \sin 4 \mathrm { t } } { 4 } \mathbf { j } + 3 \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq \frac { \pi } { 16 }$$

• A) \(W = \frac { \sqrt { 2 } } { 4 } + \frac { 9 } { 256 } \pi ^ { 2 }\)
• B) \(W = \frac { \sqrt { 2 } } { 4 } + \frac { 9 } { 512 } \pi ^ { 2 }\)
• C) \(W = \frac { \sqrt { 2 } } { 8 } + \frac { 9 } { 512 } \pi ^ { 2 }\)
• D) \(W = \frac { \sqrt { 2 } + 2 } { 4 } + \frac { 9 } { 512 } \pi ^ { 2 }\)

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Definitions:

Sensory Receptor

Cells or structures specialized to detect changes in the environment (such as light, heat, or pressure) and trigger appropriate responses.

Deep Pressure

A form of tactile sensory input characterized by firm, steady pressure applied to the body, often used therapeutically to reduce stress or anxiety.

Light Touch

A sensory perception triggered by a light pressure that activates mechanoreceptors in the skin.

Memory Engram

A hypothetical unit or physical substrate within the brain theorized to store memories, encompassing the changes in neural networks that represent a single memory.