Consider the paraboloid z=f(x,y)= 4+x^2+3y^2 . Beginning at (3,4,61) on the surface, find the path in the xy-plane that points in the direction of steepest descent on the surface.
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Take the gradient?
Gradient = <2x, 6y>
Your vector: 6i + 24j
@AccessDenied help! I thought I remembered all of this :-(
When they say "path" are they looking for the equation of the line or do they want the vector?
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i was imagining that the path on xy-plane was something like this with respect to an actual function:
that's just from intuition tho, i havent actually gotten into this part of calc -- i was just curious to see the answer. D:
Here's what I have so far:
1. Take the gradient (i component is partial derivative with respect to x, j component is partial derivative with respect to y.)
2. Plug your point in to get your vector:
<2(3), 6(4)> = <6, 24> = 6i +24j
3. Now think of your vector as a slope (rise/run) 24j is the rise. 6i is the run.
Slope = 24/6 = 4
4. You have a point in the xy-plane: (3,4). You have a slope: m=4
Now you can find the eqn of your line. First using y=mx+b, solve for b.
That's what I'm thinking, but I'm curious to see what other have to say.
the explanations online for similar problems are just beyond me, so I can't really tell you anything helpful. I've done a brief study on gradient, but nothing like this! :P
Path of "greatest change" is just the gradient of a function with with two independent variables. It will be a vector in the xy-plane.
that's kinda what it looks like
thank you for all for your help!
This is the PERFECT example for you :-D
I'm probably more trouble than help on this problem. Duf is the directional derivative. In R3, you end up computing the actual slope (just a number) at that point. Basically, when you look at a contour map of your function, that slope will be perpendicular to the contour at that point. Hence, direction of greatest change.
As far as "path of greatest descent," I'm still thinking that they want the equation of a line. In this case, it doesn't matter if you have -grad f, or +grad f. You'll still end up with the same line equation.