Freshman college student so pardon if the question is dumb. I’m currently taking an intermediate microeconomics class and we just introduced Lagrange multipliers. However, it was in a slightly different fashion to the method in my multivariable calc class, so I wanted to ask if there was a reason for this difference. In Economics they introduced the lagrangian = U(X,Y) - lamba * (I - PxX - PxY) and to optimize they set partial derivatives with respect to X, Y and lamba equal to 0. In my math class, we would optimize this by setting gradient of U equal to lamba * gradient of the constraint. Both of these methods lead to the same result: MUx/Px = MUy/Py, but I wanted to ask if there was any difference in the approaches and simply why economics would present it in a different way.
https://youtu.be/8mjcnxGMwFo&t=1m24s
At 1min 24secin the video ,he describes a circle by the equation ,
g(x,y)=x²+y²-1=0 ,this implies that the circle must lie on a plane (some z=c ,(or) in this case z=0),but how can the curve of a circle be that squishy and lie on some surface like that ,shouldn't the equation of the circle be different ,for it to lie on that weird surface and have a squishy figure ?
I understand that you have to find the grad f = lambda grad g etcetera.
But I notice that for it to be a solution if you would slightly change the constraint constant the amount of intersections will immediately change. e.g. for ; f = xy. and constraint x+y=10 the maximum point is the one where there is exactly one intersection. If you would slightly change the curve you immediately lose or gain another intersection. Sounds like demanding that there is exactly one real root to the polynomial.
Is there some way to use this?
So there are my 2 questions and i tried doing one and recieved an error for this code:
syms x y z k
f=400*x*y*z^2;
g=x^2+y^2+z^2-1;
F=f+k*g;
Fx=diff(F,x);
Fy=diff(F,y);
Fk=diff(F,k);
[xc yc,k]=solve([Fx,Fy,Fk],[x,y,k])
fval=subs(f,{x,y},{xc,yc});
[maxv i]=max(fval);
[minv j]=min(fval);
fprintf('The Highest Temperature point:(%f,%f)\n',xc(i),yc(i))
fprintf('The Minimum Temperature point:(%f,%f)\n',xc(j),yc(j))
fcontour(g)
hold on
fcountour(f)
rotate3d on
I have to submit this in a few hours so any help would really be appreciated
I have equations for two spheres and two planes and i need to find the minimum distance between the intersections of the first sphere and plane and the second sphere with the second plane.
i managed to find that the two intersections are both circles and found their center and radius but i dont know how to go on from there. any help?
the equations are as follows:
C1 is the intersection of x^2 + y^2 + z^2 = 1 and x + y + z = 1.5
the center for c1 is (0.5,0.5,0.5) and raduis r = 0.5
C2 is the intersection of x^2 + y^2 + z^2 = 9 and z = 0
center for C2 is (0,0,0) and radius r = 3.
I am an undergrad math student, I am currently taking Diff Eq, and I took Calc 3 last semester. We just got done with systems of differential equations and we talked about eigenvalues/vectors and there seems to be a solid connection between these objects and Lagrange multipliers from multivariate calculus. What is the nature of this connection? Are there any comprehensive resources to look into this? I found one site but it was explaining it like a proof instead of an explanation so it made it harder to follow. Thanks in advance for the help!
For obtaining the Lagrange Multiplier Statistics I follow this steps (Wooldridge 2019, Introductory Econometrics):
- Regress 𝑦 on the restricted set of independent variables and consider the residuals 𝑢̃;
- Regress 𝑢̃ on all of the independent variables and obtain the R-squared 𝑅^(2)_𝑢
- Compute 𝐿𝑀=𝑛⋅𝑅^(2)𝑢
- Compare 𝐿𝑀 to the appropriate critical value, 𝑐, in a 𝜒^(2)_𝑞 distribution: if 𝐿𝑀>𝑐 the null hypothesis is rejected and all the coefficients.
My doubt is about point 2., I don't understand if I should include an intercept in regressing the 𝑢̃ on the independent variables or not. If I do include the intercept the resulting 𝑅^(2)_𝑢 is the same as the one from the unrestricted regression.
I'm in high school so my use of math terminology isn't great, but I'm trying to understand Lagrange multipliers and have created a rough outline of my current understanding of how they work. I would appreciate some guidance as to whether my intuition is correct. (It is not formal or rigorous in the slightest)
Thanks for reading!
So i have a doubt about it. After doing the partial differentiation to find lambda.
I got x=1/ λ , y=3/ λ , z = 5/ λ
and
after substituting those into the constraint, at the end i got λ = ± 1
so do i do when (λ =1 and i have x=1,y=3,z=5 and λ =-1 and i have x=-1,y=-3,z=-5 ) or λ = ± 1 and i have x=± 1,y=± 3, z=± 5.
the 1st option i will have 2 points, while the 2nd one, do i have 8 possible points? ie (± 1, ± 3 ±5)= (1,3,5) , (1,3,-5).....(-1,-3,-5)
which of these two are correct?
The base of an aquarium with given volume V is made of slate and the sides are made of glass. If the slate costs nine times as much (per unit area) as glass, use Lagrange multipliers to find the dimensions of the aquarium that minimize the cost of the materials.
Here are the equations I came up with: (a=lamda)
f(x,y,z) = 9xy+2xz+2yz
9y+2z = ayz
9x+2z = axz
2x+2y = axy
xyz = v
The hardest part, of course, is solving for this system of equation. After some trying I got x = y = v/9 and z = 81/v. Apparently, this is not the correct answer but I could be inputting it incorrectly.
Any help would be much appreciated thanks!
I'm kinda new to these kinds of questions, really need your help with this one (and I also hope the LaTeX is displayed properly).
I have this function:
[;\sum_{i=1}^m \sum_{j=1}^{100} q_{ij}\log(\phi_j);]
and I also have this constraint:
[;\sum_{j=1}^{100} \phi_j=1;]
and I need to find the MLE (Maximum Likelihood Estimator) of [;\phi_j;].
So I realized that the way to solve it is by using what's called Lagrange Multipliers, but I have no idea how to use them. I only know the final answer is:
[;\frac{\sum_{i=1}^m q_{ij}}{m};]
Any help?
Im just an EE student who isnt quite satisfied with memorising the formulas in Multivariable Calc :p. As far as I understand, the lagrange multiplier λ is a proportionality constant (with a special property but its not relevant here), which between the gradient of a function f and the gradient of a constraint g, so that ∇f = λ * ∇g. Then you also need the Hessian matrix to determine if the point is a min or max or saddle.. Some folks in this reddit post have suggested that its also an eigenvalue of a matrix A so that ∇f = A * ∇g = ∇f = λ * ∇g. I dont quite understand this, what is this matrix and how can I find it? Thanks for any help!
Hi, any idea if Lagrange multipliers are a good topic for an HL maths IA? I was thinking of either tying it into economics of physics but I'm not sure how to go about doing so. Do you need to discover something 'new' in an IA? Or can you just go through your thought process of solving an interesting question? Thank you
Freshman college student so pardon if the question is dumb. I’m currently taking an intermediate microeconomics class and we just introduced Lagrange multipliers. However, it was in a slightly different fashion to the method in my multivariable calc class, so I wanted to ask if there was a reason for this difference. In Economics they introduced the lagrangian = U(X,Y) - lamba * (I - PxX - PxY) and to optimize they set partial derivatives with respect to X, Y and lamba equal to 0. In my math class, we would optimize this by setting gradient of U equal to lamba * gradient of the constraint. Both of these methods lead to the same result: MUx/Px = MUy/Py, but I wanted to ask if there was any difference in the approaches and simply why economics would present it in a different way.
https://youtu.be/8mjcnxGMwFo&t=1m24s
At 1min 24secin the video ,he describes a circle by the equation ,
g(x,y)=x²+y²-1=0 ,this implies that the circle must lie on a plane (some z=c ,(or) in this case z=0),but how can the curve of a circle be that squishy and lie on some surface like that ,shouldn't the equation of the circle be different ,for it to lie on that weird surface and have a squishy figure ?
https://youtu.be/8mjcnxGMwFo&t=1m24s
At 1min 24secin the video ,he describes a circle by the equation ,
g(x,y)=x²+y²-1=0 ,this implies that the circle must lie on a plane (some z=c ,(or) in this case z=0),but how can the curve of a circle be that squishy and lie on some surface like that ,shouldn't the equation of the circle be different ,for it to lie on that weird surface and have a squishy figure ?
