Gradient of a two term equation

[slain4ever]

Homework Statement

1. for the surface z=9 - x^2 - y^2 find,
i) the gradient at (1,1,7) in the direction making an angle alpha with the x-axis
ii) the max gradient at the point (1,1,7) and the value of alpha for which it occurs


2. find the stationary point of z=x^2 +2x +3y^2 -3xy + 5 and determine the nature of this point

Homework Equations

this is what i need, I am not sure of the equation you use to differentiate an equation like this.

The Attempt at a Solution

I know you have to partially diff it with respect to x and then with respect to y,
but I don't know how to combine the 2 together so you can just put the numbers in and find the gradient.

 

[vela]

Look up the directional derivative in your textbook. That's what you need for (i).

 

[slain4ever]

ah ok I read that and looked through the example but in the example the alpha was given

so i ended up with du f(x,y) = -2(1)cos(alpha) + -2(1)sin(alpha)

I'm not sure what to do next, if i put 7 on the left hand side the equation is false.

Or is this the answer? gradient m = -2cos(alpha) + -2sin(alpha) and i simply leave it at that and go onto part ii?EDIT: ok I am going to assume that the answer is: m = -2cos(alpha) + -2sin(alpha)

so now using that i got 2 * SQRT(2) = max gradient
and alpha = -135 degrees and alpha = 225 degrees
is this correct?

 

[slain4ever]

also for the 2nd question I got x=-4 and y=-2 but I don't know what it is asking for when it says determine the nature. does it just mean if it's a minimum or maximum?

 

[vela]

ah ok I read that and looked through the example but in the example the alpha was given

so i ended up with du f(x,y) = -2(1)cos(alpha) + -2(1)sin(alpha)

I'm not sure what to do next, if i put 7 on the left hand side the equation is false.

Or is this the answer? gradient m = -2cos(alpha) + -2sin(alpha) and i simply leave it at that and go onto part ii?

Yes, that's right.

EDIT: ok I am going to assume that the answer is: m = -2cos(alpha) + -2sin(alpha)

so now using that i got 2 * SQRT(2) = max gradient
and alpha = -135 degrees and alpha = 225 degrees
is this correct?

How did you find these angles? One is correct; the other isn't.

also for the 2nd question I got x=-4 and y=-2 but I don't know what it is asking for when it says determine the nature. does it just mean if it's a minimum or maximum?

I think so. Remember it doesn't have to be a minimum or a maximum. It could also be a saddle point, for example.

 

[slain4ever]

i found the angles by solving for alpha 2*2^.5 = -2cos(alpha) - 2sin(alpha)
this gave alpha=45(8n-3)

then i just solved for n=0 and n=1
I'm not sure how many solutions this should have a bit more explanation of how to do the final step would be appreciated.from graphing i believe the point to be a minimum right?

 

[vela]

Oh, sorry, I completely missed the sign on -135 degrees. That's the same angle as 225 degrees, so obviously, both answers are correct.

What was your reasoning that the maximum gradient was 2\sqrt{2}? It's correct, but I'm wondering how you figured it out.

In the second problem, the point is a minimum. You should be able to show it analytically by calculating derivatives.

 

[slain4ever]

2 * root(2) was found like this
m=-2xi -2yj
substitute 1,1
m(1,1)=-2i-2j
then take sqrt of the squares added together
SQRT((-2)^2 + (-2)^2)
= 2*sqrt(2)

but with the angles, when you solve the equation how many answers should I give since obviously the solution to anthing=sin(alpha) has infinite solutions, should I give the 2 angles already stated? more? less? leave the answer as alpha=45(8n-3)?

 

[vela]

OK, so you've implicitly used the fact that m=∇f points along the direction of maximal change. That's what I was trying to figure out if you knew.

You do realize that the angles -135 degrees and 225 degrees specify the same angle and direction, right?

 

[slain4ever]

oh wow i just saw that, i honestly thought they were different directions and there were 2* n number of solutions. Herp derp
ok so i guess i just submit 225 as the angle and I am done, thank you for your time and assistance