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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Preview Activity 11.7.1 .
Figure 11.7.2 is a contour plot showing the elevations for a region of a nearby park. We will be referring to
\(h(x,y)\) as the function of two variables that gives the elevation as a function of x-coordinates (location in East-West, horizontal direction) and y-coordinates (location North-South, vertical direction).
Figure 11.7.2. A contour plot representing the elevation of locations in a nearby park with three locations labeled
(a)
Using
Figure 11.7.2 and treating the elevation as a multivariable function
\(h(x,y)\text{,}\) state whether each of the following is positive, negative, or zero. Write a sentence to justify your reasoning.
\(\displaystyle h_x(A)\)
\(\displaystyle h_y(A)\)
\(\displaystyle h_x(B)\)
\(\displaystyle h_y(B)\)
\(\displaystyle h_x(C)\)
\(\displaystyle h_y(C)\)
(b) Suppose we are at point B. Would you expect the elevation to be increasing, decreasing, or constant if you took a step in the Northeast direction?
(c) Suppose we are at point B. Would you expect the elevation to be increasing, decreasing, or constant if you took a step in the Southeast direction?
(d) Suppose we are at point A. Would you expect the elevation to be increasing, decreasing, or constant if you took a step in the Southwest direction?
(e) Suppose we are at point C. In what direction would you take a step to move in the steepest downhill direction?
(f) Suppose we are at point A. In what direction would you take a step to move in the steepest uphill direction?
(g) Suppose we are at point B. Rank the following directions in order of steepness (from most steep and uphil to level to most steep downhill): Northeast, North, East, West, South
