This is the question:
The solution says:
a) The surface is given by $z=f(x,y)$
If we see in graph as we move towards $\vec{u}=<5,0>$, $z$ increases, thus $D_{\vec{u}}f(2,-4)$ is positive.
I'm not sure what it means to 'move toward <5,0>'. Does that just mean move from the point (2,-4) towards the point $(5,0)$ in the xy plane? And how do we know that z increases? In a contour plot we can normally see numbers but in this case I don't see anything.
Similarly, for c) the solution says:
z increases rapidly, thus $D_{\vec{u}}f(2,-4)$ is positive.
Again, how do I know it's increasing 'rapidly'?
Also the solution for d) is:
At $f(2,-4)$ z is neither minima nor maxima as we can see in the graph thus the answer is NEI.
How do I know that it's not a minima or maxima? And what does that mean in terms if $f_{xx}$? I.e. how do I solve this versus finding the answer for $f_{yy}$. The only thing the solution says for e) is: Same reason as d)
Lastly,
Are these the correct answers:
a) $D_{\vec{u}}f(2,-4)$ is positive
b) $D_{\vec{u}}f(2,-4)$ is positive
c) $D_{\vec{u}}f(2,-4)$ is positive
d) NEI
e) NEI
Best Answer
There are two ways to interpret an ordered pair $(x,y)$ in the plane:
Such distinctions are used in many calculus textbooks, e.g., Stewart's Calculus.
In your context, the first interpretation is written as $(x,y)$. For instance, $(-2,4)$ is a point in the plane.
The second interpretation is written as $\langle x,y\rangle$. Any nonzero Euclidean vector can be visualized as an "arrow" that has length and direction. Arrows that are parallel, equivalently, that have the same direction and same length, are representing the same (Euclidean) vector. One such arrow representing $\langle x,y\rangle$ is one that is pointing from the origin $(0,0)$ to the point $(x,y)$.
So $\vec{u}=\langle 5,0\rangle$ is not a point, but a vector. "Moving toward $\vec{u}$" in your exercise means "moving (in the $xy$-plane) from the point $(2,-4)$ in the direction of $\vec{u}$".
What you see is the graph of a function (of two variables). Each point on the graph, has the 3D coordinate $(x,y,f(x,y))$.
The graph in the $z$-direction tells you when the function increases/decreases. The figure below is another example.
It is increasing "rapidly" if the surface is very steep.
(It does not make sense to say "a minima/maxima". These are plurals. One may say "a minimum/maximum". ) In the picture, you have a (local) min/max when you have a lowest/highest point.