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:

Annual Rainfall

The total amount of precipitation, measured in units of length, that falls in a specific area over the course of one year.

Shoreline Erosion

The process by which coastal land is worn away by natural forces such as wave action, currents, or tidal movements.

Least Squares Regression

A statistical method used to fit a regression line through a set of points in such a way that minimizes the sum of the squares of the vertical distances of the points from the line.

Coefficient Of Determination

A statistical measure represented by \(R^2\) that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s).