Suppose you are climbing a hill whose shape is given by the equation z=1000−0.005x2−0.01y2, where x,y and z are measured in meters, and you are standing at a point with coordinates . The positive x-axis points east and the positive y-axis points north. If you walk due south, will you start to ascend or descend? At what rate?

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Aamina Herring · Accepted Answer

Step 1 We are given f(x,y)=1000−0.005x2−0.01y2; P=(60,40); We got to find the directional derivative of the function f at point P along υ. Through equation 9, we know that Duf(x,y)=▽f(x,y)⋅u where u is a unit vector in the direction of v Now since we are moving due south, the direction of the vector shall be υ=0i−j. Hence u=v→|v|=11[0,−1] The gradient of f is given by: ▽f(x,y)=[fx(x,y),fy(x,y)] Hence in this case fx(x,y)=−0.01x;fy(x,y)=−0.02y ▽(x,y)=[−0.01x,−0.02y] After substituting the values of Point P, ▽(60,40)=[−0.6,−0.8] Hence from the equation 9, we can write Duf(60,40)=[−0.6,−0.8]⋅d11[0,−1] Duf(60,40)=0+0.8=0.8 Now since Duf&amp;gt;0, we shall move uphill as we move south. Answer A walk due south from (60,40,966) is uphill at the rate of 0.8 m per meter of travel.

Suppose you are climbing a hill whose shape is given by the equation z=1000-0

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