Minimum distance to the origin Find the points on the surface z^2=x y+4 closest to the origin.

Name: Problem 22, Chapter 14: Thomas Calculus
Uploaded: 2019-07-02T19:59:49-08:00
Duration: 5 min 16 s
Channel: Brittany Knowlton
Description: Minimum distance to the origin Find the points on the surface z^2=x y+4 closest to the origin.

Minimum distance to the origin Find the points on the...

Key Concepts

Surface Constraint Analysis

Surface constraint analysis involves studying the geometric and algebraic properties of a surface defined by an equation. Understanding the characteristics of the surface is crucial in optimization problems because the nature of the constraint influences the potential locations of the optimal solution, and it helps in setting up the necessary equations for methods like Lagrange multipliers.

Distance Function Minimization

Distance function minimization is the process of finding the point or points that minimize the distance, often measured using Euclidean distance, from a given reference point to a set or surface. By typically minimizing the squared distance, one can simplify the calculations and avoid dealing with the square root while preserving the location of the minimum.

Constrained Optimization

Constrained optimization involves finding the maximum or minimum of a function subject to one or more constraints. The constraints, usually given as equations or inequalities, restrict the domain of the function over which the optimization is performed, requiring that the solution not only optimizes the function but also satisfies these conditions.

Lagrange Multipliers

Lagrange multipliers is a method used to solve constrained optimization problems by introducing additional variables (the multipliers) that link the objective function with the constraints. This technique converts the constrained problem into a system of equations whose solutions yield the points where the objective function may achieve extreme values under the given restrictions.

Transcript

00:01 Okay, so for this question, we're given the surface x, y, z is equal to one, and we want to find the points on this that are origin.

00:12 So we are minimizing, and so that means the function that are minimizing is the distance function.

00:23 We want to minimize it at the origins of zero, zero, so that means our function here is just going to be x squared plus y squared plus z squared.

00:35 And so that means our constraint is going to be that function that was given to us.

00:42 So we're wanting that to be our g.

00:44 And i want to set equal zero.

00:45 So we're going to move everything to one side.

00:47 So i'll get x, y, z minus 1, is equal to zero.

00:52 So we want to find the points on the surface that are closest to the origin.

00:56 So we want to minimize this distance.

00:58 So i'm going to be using equation 1 here about the gradient of f is equal to the lambda times gradient of g.

01:06 And i want to find those values of x, y, z that satisfy that.

01:11 So when i take the gradient of f of my distance function here, i get 2xi plus 2y plus 2y, k.

01:22 When i take the gradient of g, when i take the derivative with respect to x here first, i get y -z -i, and then for when i take it with y, i get x, j, and then x, y, and then x -y, okay.

01:42 So i'm going to solve that the gradient of f is equal to lambda times the gradient of g.

01:47 So that's going to mean that 2xi plus 2y plus 2y plus 2 zk is equal to lambda times g.

01:59 So lambda y zi plus lambda x zj plus lambda x y k.

02:11 Okay, so now i can make three different equations here by taking the coefficients of the same variables and setting them equal to one another.

02:19 So i get 2x is equal to lambda y z.

02:29 2x is equal to lambda y z...

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