Find Max Gradient at P of z=2x^2+3y^2

[savva]

Homework Statement

For the surface z=2x^2+3y^2, find
(i) the gradient at the point P (2,1,11) in the direction making an angle a with the x-axis;
(ii) the maximum gradient at P and the value of a for which it occurs.

Homework Equations

ma=\partialz/\partialx(cos(a))+\partialz/\partialy(sin(a))

If dma/da = 0 and d^2ma/da^2<0 the ma, is a maximum for that value of a

The Attempt at a Solution

\partial

(i) Firstly I calculated:
\partialz/\partialx = 4x
\partialz/\partialy = 6y

Therefore applying equation: ma=\partialz/\partialx(cos(a))+\partialz/\partialy(sin(a)) at P(2,1,11)

We get:

ma=8cos(a)+6sin(a)
(I am quite sure this is correct, I don't have answers so if I am doing something wrong can someone please inform me.

(ii) This is where I am stuck, I can't quite get the correct working out, can anyone please help out, below is the calculations I attempted:

If dma/da = 0 and d^2ma/da^2<0 the ma, is a maximum for that value of a.

dma/da = -8sin(a)+6cos(a)

dma/da=0 Therefore: -8sin(a) + 6cos(a) = 0 Therefore: 8sin(a)=6cos(a)

This is what I was up to, got stuck here, not sure if what I done is correct, if it is, how do I proceed from here?

Thanks in advance
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I have now calculated part (ii), I realized I have a problem in part (i), I need to find a to get the gradient ma, can anybody help me out with this?

 

[savva]

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[ehild]

Therefore applying equation: ma=\partialz/\partialx(cos(a))+\partialz/\partialy(sin(a)) at P(2,1,11)

We get:

ma=8cos(a)+6sin(a)
(I am quite sure this is correct, I don't have answers so if I am doing something wrong can someone please inform me.

It is correct for the derivative at the given point along the direction defined by the angle α.

(ii) This is where I am stuck, I can't quite get the correct working out, can anyone please help out, below is the calculations I attempted:

If dma/da = 0 and d^2ma/da^2<0 the ma, is a maximum for that value of a.

dma/da = -8sin(a)+6cos(a)

dma/da=0 Therefore: -8sin(a) + 6cos(a) = 0 Therefore: 8sin(a)=6cos(a)

This is what I was up to, got stuck here, not sure if what I done is correct, if it is, how do I proceed from here?

Isolate α: sin(α)/cos(α)=6/8 =tan(α)=0.75 α=?
Getting α, substitute it back to mα.

ehild