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The question as presented, this is from Calculus Vol. II Section 8.14 #4

A differentiable scalar field $f$ has, at the point $(1,2)$, directional derivatives $+2$ in the direction toward $(2,2)$ and $−2$ in the direction toward $(1,1)$. Determine the gradient vector at $(1,2)$ and compute the directional derivative in the direction toward $(4,6)$.

Right off the bat, I think there must be a typo in the question because the directional derivatives in the direction of $(2,2)$ should be equal to the directional derivatives in the direction of $(1,1)$. So instead, I thought I would look at the answer for $(4,6)$ and try to find out what the typo was in the original problem. In the back of the book we have the answers as:

The gradient vector at $(1,2)$ is $(2,2)$, and the directional derivative in the direction toward $(4,6)$ is $\frac {14} 5$.

These don't seem to be consistent either, with each other or with the original question. Is this a series of typos, or am I misinterpreting something?

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With the word "toward" there, I think they mean the displacement vectors between (1,2) and (2,2) resp. (1,1) for the two given directional vectors. – anon Aug 17 '12 at 15:08
@anon Yes, all is well with that interpretation. Care to make that into an answer? – process91 Aug 17 '12 at 15:10
up vote 1 down vote accepted

By "in the direction toward," I believe what is meant is that the displacement vector between the base point and the second point is the direction in the directional derivatives.

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If you were standing at the origin, then the directional derivatives in the direction of $(1,1)$ and $(2,2)$ would be the same. But you are standing at $(1,2)$, so the point $(2,2)$ lies towards the east, and the point $(1,1)$ towards the south.

Since the function is sloping down towards the south, it must be rising with the same slope towards the north. Hence the information about the directional derivatives tells you that ${\partial f\over \partial x}(1,2)=2$ and that ${\partial f\over\partial y}(1,2)=2$. The rest should now follow easily.

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