Calculus 3 Suppose you are climbing a hill whose shape is given by the equation

Question

Calculus 3

Suppose you are climbing a hill whose shape is given by the equation below, where x, y, and z are measured in meters, and yo are standing at a point with coordinates 140, 240, 926). The positive x-axis points east and the positive y-axis points north. z = 1600-0.(05a,2-0.011/2 (a) If you walk due south, will you start to ascend or descend? ascend O descend At what rate? (b) If you walk northwest, will you start to ascend or descend? O ascend descend At what rate? (Round the answer to two decimal places.)

Explanation / Answer

a) If you walk South:
The question is asking for the value of the directional derivative in the direction of -j. This is the negative of partial y. Since zy=0.02y, the value is equal to 0.02(240) = 4.8. Being positive, this means that the hill slopes upward in this direction.

b) If you walk NorthWest:
As above, except u=1/sqrt(2)<-1;1> is the direction: Duz(140,240) =<-0.01(140), -0.02(240)> . u
=-3.4/sqrt(2) =-2.40

c) The slope is largest in the direction of the gradient<-0.01x,-0.02y>= (-1.4,-4.8).
The rate in that direction is |grad f |= sqrt(1.4^2+4.8^2)=25
angle above horizontal= 87.70

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