Suppose you are climbing a hill whose shape is given by the...

Suppose you are climbing a hill whose shape is given by the equation $ z = 1000 - 0.005x^2- 0.01y^2 $, where $ x $, $ y $, and $ z $ are measured in meters, and you are standing at a point with coordinates $ (60, 40, 966) $. The positive $ x $-axis points east and the positive $ y $-axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

Suppose you are climbing a hill whose shape is given by the equation $ z = 1000 - 0.005x^2- 0.01y^2 $, where $ x $, $ y $, and $ z $ are measured in meters, and you are standing at a point with coordinates $ (60, 40, 966) $. The positive $ x $-axis points east and the positive $ y $-axis points north.
(a) If you walk due south, will you start to ascend or descend? At what rate?
(b) If you walk northwest, will you start to ascend or descend? At what rate?
(c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

Key Concepts

Angle of Inclination

The angle of inclination refers to the angle that the path of steepest ascent makes with the horizontal plane. It can be found by using trigonometric relationships, typically by taking the arctangent of the ratio between the vertical rate of change (given by the gradient magnitude) and the horizontal distance covered. This concept connects the calculus-based rate of change with geometric interpretations in the real world.

Gradient Vector

The gradient vector of a multivariable function is a vector composed of its partial derivatives. It points in the direction of the steepest increase (ascent) of the function, and its magnitude represents the rate of maximum change at that point. This concept is fundamental in understanding how a scalar field (such as a hill's height) changes in different directions.

Directional Derivative

The directional derivative of a function in a specific direction is the rate at which the function changes as one moves in that direction. It is obtained by computing the dot product of the gradient vector with a unit vector pointing in the specified direction. This concept is essential for determining whether one is ascending or descending on a surface while moving along any given path.

Partial Derivatives

Partial derivatives measure the rate of change of a multivariable function with respect to one variable while keeping the other variables constant. They are the building blocks used to construct the gradient vector and are vital for analyzing how each independent variable individually contributes to the behavior of the function at a given point.

Steepest Ascent

The steepest ascent is the direction in which a function increases most rapidly, and it is given by the gradient vector. The rate of ascent in this direction is equal to the magnitude of the gradient. This concept is central to optimization and in understanding how to navigate a landscape with maximum gain in elevation.

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