Multivariable calculus yummy in my tummy

[mathwiz123]

[Question]
Hi, my teacher gave us this problem, and he couldn't figure out why
method was incorrect and why I got the answer I did.

Given we know the gradient slope = <-56,1.886> at the point (2,0) on a
surface f(x,y), in what direction, expressed as a unit vector, is f
increasing most rapidly?





[Difficulty]
I solved the problem like this:

Max slope = magnitude of gradient slope
(gradient slope) dot (unit vector) = 56.03
-56.03Ux + 1.866Uy = 56.03
sqt(Ux^2 + Uy^2) = 1

Solving the system Ux= -.9977 Uy=.0672

The only problem is, by definition, I should be able to get the unit
vector by taking the gradient slope vector and dividing by the
magnitude of the gradient vector. or,

<-56/56.03, 1.886/56.03> = <Ux,Uy> = <-.9994,.0336>

The weird thing is, the correct Uy value is almost exactly half of
mine...what's going on?

[Thoughts]

I tried a similar technique for finding where the ∇f = 0

<-56, 1.866> dot (unit vector) = 0

-56.03Ux + 1.886Uy = 0
sqt(Ux^2 + Uy^2) = 1

Solving the system this time I got the correct answer, why here and
not there?

 

[HallsofIvy]

[Question]
Hi, my teacher gave us this problem, and he couldn't figure out why
method was incorrect and why I got the answer I did.

Given we know the gradient slope = <-56,1.886> at the point (2,0) on a
surface f(x,y), in what direction, expressed as a unit vector, is f
increasing most rapidly?





[Difficulty]
I solved the problem like this:

Max slope = magnitude of gradient slope
(gradient slope) dot (unit vector) = 56.03
-56.03Ux + 1.866Uy = 56.03
sqt(Ux^2 + Uy^2) = 1

Solving the system Ux= -.9977 Uy=.0672

I understand that the 56.03 is the magnitude of the given vector but the coefficient of Ux should be -56, not -56.03.

The only problem is, by definition, I should be able to get the unit
vector by taking the gradient slope vector and dividing by the
magnitude of the gradient vector. or,

<-56/56.03, 1.886/56.03> = <Ux,Uy> = <-.9994,.0336>

The weird thing is, the correct Uy value is almost exactly half of
mine...what's going on?

[Thoughts]

I tried a similar technique for finding where the ∇f = 0

<-56, 1.866> dot (unit vector) = 0

-56.03Ux + 1.886Uy = 0
sqt(Ux^2 + Uy^2) = 1

Solving the system this time I got the correct answer, why here and
not there?

 

[tacman]

Hi there! It looks like you have the right idea in solving this problem. Your method for finding the unit vector in the direction of maximum increase is correct, as you correctly determined the unit vector to be <-.9994,.0336>.

As for the discrepancy in your values for Uy, it could be due to rounding errors or a slight mistake in your calculations. It's always a good idea to double check your work and make sure your calculations are accurate.

The reason your second method for finding where ∇f = 0 worked is because you are essentially finding the critical points of the function, which is a different concept from finding the direction of maximum increase. In this case, the critical point is (2,0), which is where the gradient slope is equal to 0.

Overall, it seems like you have a good understanding of multivariable calculus and are on the right track. Keep up the good work!