____ Show That $\mathbf { F }$ Is Conservative and Find a Function

question 35

Multiple Choice

____ Show that $\mathbf { F }$ is conservative and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ .
$\mathbf { F } ( x , y ) = \left( 15 x ^ { 2 } y ^ { 2 } - 14 x y ^ { 4 } \right) \mathbf { i } + \left( 10 x ^ { 3 } y - 28 x ^ { 2 } y ^ { 3 } \right) \mathbf { j } ; A ( 1 , - 2 )$ and $B ( 1 , - 1 )$

• A) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 34\)
• B) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 90\)
• C) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 144\)
• D) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 118\)

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Definitions:

Cognitive Development

The process by which a person learns to reason, solve problems, and understand the world around them through stages, as proposed by theorists like Piaget.

Vygotsky

A seminal theorist in developmental psychology, known for his emphasis on the sociocultural influences on cognitive development.

Zone Of Proximal Development

A concept developed by Lev Vygotsky that represents the difference between what a learner can do without help and what they can achieve with guidance and encouragement from a skilled partner.

Scaffolding

Vygotsky’s term for temporary cognitive structures or methods of solving problems that help the child as he or she learns to function independently.