Find the Gradient Field of the Function $$f ( x , y , z ) = \left( x ^ { 8 } y ^ { 2 } - y ^ { 9 } z ^ { 3 } \right) e ^ { - z ^ { 2 } }$$

question 190

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \left( x ^ { 8 } y ^ { 2 } - y ^ { 9 } z ^ { 3 } \right) e ^ { - z ^ { 2 } }$$

• A)\({\nabla} f= 8 x ^ { 7 } y ^ { 2 } e ^ { - z ^ { 2 } } \mathbf { i } + \left( 2 x ^ { 8 } y - 9 y ^ { 8 } z ^ { 3 } \right) e ^ { - z ^ { 2 } } \mathbf { j } + y ^ { 2 } z e ^ { - z ^ { 2 } } \left[ y ^ { 7 } z \left( 2 z ^ { 2 } - 3 \right) - 2 x ^ { 8 } \right] \mathbf { k }\)
• B) \({\nabla} f = 8 \mathrm { x } ^ { 7 } \mathbf { i } + \left( 9 \mathrm { y } ^ { 8 } - 2 \mathrm { y } \right) \mathbf { j } + 2 \mathrm { z } ^ { 4 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathbf { k }\)
• C) \({\nabla} f = 8 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 2 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathrm { i } + \left( 2 \mathrm { x } ^ { 8 } \mathrm { y } - 9 \mathrm { y } ^ { 8 } \mathrm { z } ^ { 3 } \right) \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathrm { j } + \left( 3 - 2 \mathrm { z } ^ { 2 } \right) \mathrm { y } ^ { 9 } \mathrm { z } ^ { 2 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathbf { k }\)
• D) \(\nabla f = 8 x ^ { 7 } y ^ { 2 } \mathbf { i } + \left( 9 y ^ { 8 } z ^ { 3 } - 2 x ^ { 8 } y \right) \mathbf { j } + \left( 2 z ^ { 2 } - 3 \right) y { } ^ { 9 } { } ^ { 2 } \mathbf { k }\)

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Definitions:

Epigenetic Inheritance

Inheritance that involves changes in how a gene is expressed without any change in that gene’s nucleotide sequence.

Chromosome 15

A specific chromosome in humans that carries genetic information, including genes associated with various disorders and physical traits.

Repressible Genes

Genes that can be turned off by the products of other genes in response to certain conditions.

TATA Box

A component of a eukaryotic promoter region; consists of a sequence of bases located about 30 base pairs upstream from the transcription initiation site.