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 2 years ago
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 xyplane that points in the direction of steepest descent on the surface.
 2 years ago
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 xyplane that points in the direction of steepest descent on the surface.

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brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3Gradient = <2x, 6y> Your vector: 6i + 24j @AccessDenied help! I thought I remembered all of this :(

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3When they say "path" are they looking for the equation of the line or do they want the vector?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0i was imagining that the path on xyplane was something like this with respect to an actual function: dw:1333233857061:dw that's just from intuition tho, i havent actually gotten into this part of calc  i was just curious to see the answer. D:

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3Here's what I have so far: 1. Take the gradient <fx, fy> (i component is partial derivative with respect to x, j component is partial derivative with respect to y.) Gradient: <2x,6y> 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 xyplane: (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. 4=4(3) +b b=8 y=4x8 That's what I'm thinking, but I'm curious to see what other have to say.

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0the 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

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3Path of "greatest change" is just the gradient of a function with with two independent variables. It will be a vector in the xyplane.

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3http://mathforum.org/mathimages/index.php/Gradients_and_Directional_Derivatives

unmathguy
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1333234900760:dw that's kinda what it looks like

unmathguy
 2 years ago
Best ResponseYou've already chosen the best response.0thank you for all for your help!

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3AWESOME!!!! http://mathhelpforum.com/calculus/77210directionalderivativedirectionsteepestdescent.html This is the PERFECT example for you :D

brinethery
 2 years ago
Best ResponseYou've already chosen the best response.3I'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.
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