Consider the function f(x,y)=−3x^2+2y^2 .
Find the the directional derivative of f at the point (−4,−1) in the direction given by the angle =pi.
Find the unit vector which describes the direction in which f is increasing most rapidly at (−4,−1) .

Question

Consider the function f(x,y)=−3x^2+2y^2 . 
Find the the directional derivative of f at the point  (−4,−1)  in the direction given by the angle  =pi.
Find the unit vector which describes the direction in which f is increasing most rapidly at (−4,−1) .

amistre64 · Answer

gradf, dot, u

amistre64 · Answer

and a function increase the most in the direction of the gradient

anonymous · Answer

I got the gradient vector to be <-6x, 4y> Then I got the gradient vecotr of (-4,-1)=<24, -4>. I got the unit vector as (1/sqrt(10))<3,1> but am still not getting the right answer.

amistre64 · Answer

$$\nabla f\cdot \vec u=|\nabla f|~|\vec u|~cos(\alpha)$$
this is at a maximum when cos(a) = 1, so the direction of grad f and u are in the same direction, right?

amistre64 · Answer

as such,  i agree with 24,-4    divide by sqrt(24^2 + 4^2) to make it a unit

amistre64 · Answer

or:  (6,-1) is in the same direction,  and we get (6/sqrt(37),-1/sqrt(37))

amistre64 · Answer

do you ahve the right answer?  or is this just tellin gyou you are in error?

anonymous · Answer

No i still dont have the right answer. Where did the (6, -1) come from?

anonymous · Answer

Because by using pi you get the (-1, 0) and then use that to find (3,1) then get the unit vector from that.

amistre64 · Answer

part 1 or part 2,

anonymous · Answer

I got part 2. but still don't know how to Find the the directional derivative of f at the point  ( -4, -1) in the direction given by the angle =pi

amistre64 · Answer

so the direction is (-1,0) correct?

anonymous · Answer

Yes

amistre64 · Answer

Find the the directional derivative of f 
          at the point  (−4,−1)  
            in the direction given by the angle  =pi.

(-24,4).(-1,0) = 24

therefore, in the direction of (-1,0) at the point (-4,-1) we have a slope of 24 for a tangent line to the surface

anonymous · Answer

Okay so isnt the directional derivative the point (24, -4) multiplied by the unit vector?

anonymous · Answer

Because when I type that in I still am not getting the correct response.

amistre64 · Answer

there is no multiplication of vectors.  there is dot product and cross product

anonymous · Answer

Sorry I meant the dot product. I thought it's be<24, -4> dot product

amistre64 · Answer

the dotproduct is not dependant on a unit vector.

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