Suppose That the Temperature at the Point $( x , y , z )$

question 11

Multiple Choice

Suppose that the temperature at the point $( x , y , z )$ is given by $T ( x , y , z ) = e ^ { - z ^ { 2 } } \sin ( x y )$ .If you are at the point $( 1,2 \pi , 0 )$ , in which direction should you go to decrease your temperature the fastest?

• A) \(2 \pi \vec { i } + \vec { j } + 0 \vec { k }\)
• B) \(2 \pi \vec { i } - \vec { j } + 0 \vec { k }\)
• C) \(2 \pi \vec { i } + \vec { j } - \vec { k }\)
• D) \(- 2 \pi \vec { i } - \vec { j } + 0 \vec { k }\)
• E)None of the above; the gradient of T is zero at \(( 1,2 \pi , 0 )\) .

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Definitions:

Conjugated Alkyne

A hydrocarbon compound that contains a conjugated system of alternating double and triple bonds within its carbon chain.

Intramolecular Aldol

A chemical reaction where an aldol condensation occurs within a single molecule, forming a cyclic compound.

2,6-heptanedione

An organic compound characterized by a seven-carbon chain with ketone groups located at the second and sixth carbons.

Crossed Aldol Reaction

An aldol reaction where two different aldehydes or ketones are combined in the presence of a base or acid to produce a β-hydroxy aldehyde or ketone.