Directional derivative

[Lionking21]

Hi again guys, I have found the directional derivative to 0, and was a bit unsure about the following with maximal? I Got a result with 0,1, budt NOT sure about it.

 

[Dr.Peterson]

You're saying that your answer to the first question is 0, and to the second is <1, 0>? That looks right to me.

What are you unsure of? It may help to show your work along with your answers, so we can see if perhaps your work is wrong though your answer happens to be right.

[Edit: I mistyped 1 where I meant 0.]

 

[Lionking21]

No, my answer is 0 to the directional derivative. Taking the partial derivative x’=ycos(xy) and y’=xcos(xy), then inputing the values 0 and 1. It gives x’=1 and y’=0. The last Would be to multiply with unit vector. So 1*0+0*1=0. I am just unsure about the second one

 

[Dr.Peterson]

No, my answer is 0 to the directional derivative.

I fixed that above! Sorry.

Taking the partial derivative x’=ycos(xy) and y’=xcos(xy), then inputing the values 0 and 1. It gives x’=1 and y’=0. The last Would be to multiply with unit vector. So 1*0+0*1=0. I am just unsure about the second one

You don't really mean x' and y', do you? What notation were you taught for this? I think you mean f_x and f_y.

Your work is correct, otherwise.

So, what is your work for finding the direction of fastest growth? What do you call the vector that gives this direction?

The relevant theorem is toward the bottom of this page: http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx

 

[Lionking21]

Yeah that is the exact notation. I have a formula for the next question. The gradient divided with the length of gradient

 

[Dr.Peterson]

Any vector in the direction of the gradient will be an answer to the second question, since they just ask which is in the right direction; the gradient divided by its length is specifically the unit vector in that direction. It happens that all the options shown are unit vectors; so that formula gives the answer. If that were not true, you could just check the ratio of components of each vector.

I presume you found that the gradient is <1, 0>, which happens to be a unit vector, so that is the answer.

 

[Lionking21]

Yes, exactly thanks!