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Suppose that a mountain climber is at the point $(500, 300, 3390)$ on the surface of a mountain described by the function $h(x,y)=4000-.001x^2-.004y^2$. In what direction would the climber have the steepest up slope? What is that slope?

$\nabla h(x,y) = h_x(x,y)i+h_y(x,y)j$

$h_x = -.002xi$

$h_y = -.008yj$

$\langle -.002x,-.008y\rangle$

$\nabla h(500,300) = \langle -.002(500), -.008(300)\rangle = \langle -1,-2.4\rangle$

Since this is a negative vector, I assume this would be the steepest down slope, so the steepest up slope would be the opposite?

$\langle 1,2.4 \rangle$

Does this solution answer both what the steepest up slope is and the the what the slope is?

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The gradient will give you the direction of steepest ascent. So in this case moving in the direction $(-1,-2.4)$ will increase the height you are at the most you do not need to take the negative that would be wrong. This may seem a little counter intuitive at first but think about if the peak of a mountain was in both negative $x$ and $y$ values then you would still have to move towards this negative direction to increase your height the most. The slope is simply the magnitude of the gradient vector so just square both components, add then up and square root and you will have your slope.

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