On a certain mountain, the elevation z above a point (x, y)...

On a certain mountain, the elevation $z$ above a point $(x, y)$ in an $x y$ -plane at sea level is $z=2000-0.02 x^{2}-0.04 y^{2}$, where $x, y$, and $z$ are in meters. The positive $x$ -axis points east, and the positive $y$ -axis north. A climber is at the point $(-20,5,1991)$ (a) If the climber uses a compass reading to walk due west, will she begin to ascend or descend? (b) If the climber uses a compass reading to walk northeast, will she ascend or descend? At what rate? (c) In what compass direction should the climber begin walking to travel a level path (two answers)?

On a certain mountain, the elevation $z$ above a point $(x, y)$ in an $x y$ -plane at sea level is $z=2000-0.02 x^{2}-0.04 y^{2}$, where $x, y$, and $z$ are in meters. The positive $x$ -axis points east, and the positive $y$ -axis north. A climber is at the point $(-20,5,1991)$
(a) If the climber uses a compass reading to walk due west, will she begin to ascend or descend?
(b) If the climber uses a compass reading to walk northeast, will she ascend or descend? At what rate?
(c) In what compass direction should the climber begin walking to travel a level path (two answers)?

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Key Concepts

Gradient Vector

The gradient vector of a function is composed of its partial derivatives with respect to each variable. It points in the direction of greatest rate of increase of the function and its magnitude represents the maximum rate of increase at that point. This concept is fundamental in multivariable calculus and optimization and is used to determine the steepness of a slope or surface in various directions.

Directional Derivative

The directional derivative measures the rate of change of a function in any specified direction. It is calculated as the dot product of the gradient vector with a unit vector representing the direction of interest. This concept is essential in understanding how a function varies as one moves in a particular direction, making it invaluable in fields ranging from physics to engineering.

Level Curve (or Contour Line)

A level curve of a function of two variables is a set of points where the function takes on a constant value. These curves provide a way to visualize the topography of a surface, such as hills or valleys, and help in understanding regions where there is no change in elevation or function value. They are especially important in applications like topographic mapping and meteorology.

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