Any Calculus III wiz?

[Smooth Operater]

Hello Guys!

I am currently stucked on a HW question. I am sure there are some intelligent math students here. If you are bored, hope you guys can help me out?


The temperature T(x,y) at points of the xy-plane is given by
T(x,y) = x^2-2y^2.

Along what curve through (2,-1) of the plane should an ant move to continue experiencing "maximum rate of cooling"?



I think you need to use direction derivatives to solve this question.
Direction derivatives=gradientT * unit vector.

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[BubbleBobble]

Consider: What direction must the unit vector be to maximize the directional derivative?

 

[patzan]

Just do your own homework, dude.

 

[Smooth Operater]

thanx Bubble, for helping me out, i tried the question, and this is my first attempt after wondering for 30 min.

First

I compute the gradient.

Fx = 2x
Fy = -4y

which give me (4,4)

the unit vector: (x-1,y- -1) <--not sure if i did it right.

then do the dot product,

(4,4)(x-1,y - -1) = 0

which gives me a two variable equation, should the equation of the curve have only one variable?




I know I need to find the lowest negative slope. But, I am not sure how.

And from bubble's suggestion
i find the vector that will give me the maximize the directional derivative. therefore, the unit vector is -1/(x^(1/3)) * (4,4) .
it's negative cuz i wanna to the max. cooling which means max of decreasing.



but, what's next? I am kinda lost.

big
thanx for those trying to help me out!

 

[Smooth Operater]

the answer should be y= 4/x^2 is all worked out correctly, but I am not sure how? 🙄

 

[DMO]

I do agree with the above poster that you should do your own homework, but it isn't everyday that students post all their homework in which others can pick up the slack. This is just one problem that is proving difficult. After all, isn't a forum all about the exchange of ideas, problems and solutions? That's how the scientific community fuctions. Just my 2 cents. Please don't send mail bombs to my house, I just bought a new 25 dollar mailbox.

😀