Preview Activity 11.7.1.
Below 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 \(x\)-coordinate (location in the east-west or horizontal direction) and \(y\)-coordinates (location in the north-south or vertical direction).
(a)
Using the contour plot 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.
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\(\displaystyle h_x(A)\)
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\(\displaystyle h_y(A)\)
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\(\displaystyle h_x(B)\)
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\(\displaystyle h_y(B)\)
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\(\displaystyle h_x(C)\)
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\(\displaystyle h_y(C)\)
(b)
Suppose you are standing at point \(B\text{.}\) Do you expect the elevation to increase, decrease, or remain constant if you take a step in the northeast direction? Write a sentence to explain your reasoning.
(c)
Suppose you are standing at point \(B\text{.}\) Do you expect the elevation to increase, decrease, or remain constant if you take a step in the southeast direction? Write a sentence to explain your reasoning.
(d)
Suppose you are standing at point \(A\text{.}\) Do you expect the elevation to increase, decrease, or remain constant if you take a step in the southwestdirection? Write a sentence to explain your reasoning.
(e)
Suppose you are standing at point \(C\text{.}\) In what direction would you take a step to move in the steepest downhill direction? Write a sentence to explain your reasoning.
(f)
Suppose you are standing at point \(A\text{.}\) In what direction would you take a step to move in the steepest uphill direction? Write a sentence to explain your reasoning.
(g)
Suppose you are standing at point \(B\text{.}\) Rank the following directions in order of steepness (from most steep and uphill to level to most steep downhill): northeast, north, east, west, south
