Find the Gradient Vector Field Of $f$ $$f ( x , y , z ) = \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 4 z ^ { 2 } }$$

question 40

Multiple Choice

Find the gradient vector field of $f$ .
$$f ( x , y , z ) = \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 4 z ^ { 2 } }$$

• A)
• \(\frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• B)
• \(\frac { x } { \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 4 z ^ { 2 } } } \mathbf { i } + \frac { 2 y } { \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 4 z ^ { 2 } } } \mathbf { j } + \frac { 4 z } { \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 4 z ^ { 2 } } } \mathbf { k }\)
• C) \(\frac { 2 x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 2 y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { 2 z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• D) \(\frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• E) None of these

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Definitions:

Generalized Expectancies

Beliefs about how often actions lead to rewards versus punishment.

Reinforcement Value

How enticing a particular reward is.

External Locus

The belief that one's life events are largely controlled by external forces or fate rather than one's own efforts.

Unconditioned Response

In classical conditioning, it is the automatic, natural reaction elicited by an unconditioned stimulus without prior learning or conditioning.