A Bug Is Crawling on the Surface $z = x ^ { 2 } + x y + 2 y ^ { 2 }$

question 233

Essay

A bug is crawling on the surface $z = x ^ { 2 } + x y + 2 y ^ { 2 }$ . When he reaches the point $( 2,1,8 )$ he wants to avoid vertical change. In which direction should he head? (He wants the directional derivative in the $Z$ -direction to be zero.)

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Definitions:

Central Nervous

Pertains to the central nervous system, which consists of the brain and spinal cord, responsible for processing and transmitting information throughout the body.

Limbic

Refers to the limbic system, a complex system of nerves and networks in the brain, involved with instinct and mood. It controls the basic emotions and drives.

Autonomic Nervous System

The part of the nervous system that controls involuntary actions of the muscles and glands.

Internal Organs

Organs located inside the body that perform essential functions, such as the heart, liver, kidneys, and lungs.