You CAN Ace Calculus
external links you may find helpful |
---|
Single Variable Calculus |
---|
Multi-Variable Calculus |
---|
Acceleration Vector |
Arc Length (Vector Functions) |
Arc Length Function |
Arc Length Parameter |
Conservative Vector Fields |
Cross Product |
Curl |
Curvature |
Cylindrical Coordinates |
Lagrange Multipliers |
Line Integrals |
Partial Derivatives |
Partial Integrals |
Path Integrals |
Potential Functions |
Principal Unit Normal Vector |
Differential Equations |
---|
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
| |
free ideas to save on bags & supplies |
---|
Help Keep 17Calculus Free |
---|
The Method of Lagrange Multipliers is used to find maximums and minimums of a function subject to one or more constraints. We could also say that we want to optimize the function or find the extreme values of the function.
We highly recommend that you download the notes for this topic from
Dr Chris Tisdell. Look for the pdf link entitled Extreme values + Lagrange multipliers.
In basic calculus, we learned that finding the critical points gives us information about maximums, minimums (and saddlepoints). We use the same idea here, i.e. locations where the derivative is zero gives us possible locations of maxs/mins. We use this method to integrate the constraints into the equation.
Basic Technique |
---|
If we are given a function \(f(x,y,z)\) that we want to optimize (find maximums, minimums or both) subject to a constraint \(g(x,y,z) = 0\), we set up the gradient equation \( \nabla f = \lambda \nabla g \). [ For a version of the equations that do not use the gradient, see below. ]
The variable \( \lambda \) is just a number (independent of \(x\), \(y\) and \(z\)) called a Lagrange multiplier. We introduce \( \lambda \) during the course of solving this problem but it will not appear in our answer.
Remember that the gradient is a vector. So, the above equation gives us three equations ( one for each variable ) and four unknowns ( \(x\), \(y\), \(z\) and \( \lambda \) ). Using also the constraint equation \(g(x,y,z) = 0\), we can now (theoretically) solve for all four unknowns.
This next video clip explains this technique in more detail.
video by Dr Chris Tisdell
So, why does this work? It seems kind of strange that introducing another variable would enable us to optimize a function. Here is a great video clip that explains this. His use of graphs is very good to visualize what is going on.
video by Dr Chris Tisdell
Two Constraints |
---|
Okay, so now you know how to handle one constraint, if you are given two constraints, you just add another lagrange multiplier. We usually use the Greek letter mu, \( \mu \). The equation then looks like \( \nabla f = \lambda \nabla g + \mu \nabla h \) where the function we want to optimize is \(f(x,y,z)\) and the constraint equations are \(g(x,y,z) = 0\) and \(h(x,y,z) = 0\). Here is a video explaining this in more detail, including an example.
video by Dr Chris Tisdell
Equations That Do Not Use The Gradient |
---|
An alternate version of the equations using the Lagrange Method that do not use the gradient is given below. We present equations with three variables and two constraints. As you would expect, you can use the same ideas with two variables (drop \(z\)) and one constraint (drop \( h(x,y,z) \)).
Equations And Set Up |
---|
Optimize \(f(x,y,z)\) |
Constraints \(g(x,y,z)=0\) and \(h(x,y,z) = 0\) |
\( L(x,y,z,\lambda, \mu) = f(x,y,z) - \lambda g(x,y,z) - \mu h(x,y,z) \) |
Equations To Solve |
\( \partial L / \partial x = 0 ~~~ \to ~~~ \partial f / \partial x - \lambda ~ \partial g / \partial x - \mu ~ \partial h / \partial x = 0 \) |
\( \partial L / \partial y = 0 ~~~ \to ~~~ \partial f / \partial y - \lambda ~ \partial g / \partial y - \mu ~ \partial h / \partial y = 0 \) |
\( \partial L / \partial z = 0 ~~~ \to ~~~ \partial f / \partial z - \lambda ~ \partial g / \partial z - \mu ~ \partial h / \partial z = 0 \) |
\( g(x,y,z) = 0 \) |
\( h(x,y,z) = 0 \) |
5 equations and 5 unknowns |
Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems |
---|
Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on. |
Instructions - - Unless otherwise instructed, follow these guidelines.
1. Optimize the functions subject to the given constraints.
2. Show whether they are maximums or minimums.
3. If you are not told whether to find maximums or minimums, find all of them.
4. Give all answers in exact form.
Basic Problems |
---|
optimize: \( f(x,y) = x^2+y^2 \) | constraint: \( 2x+6y=2000 \) |
Problem Statement |
---|
optimize: \( f(x,y) = x^2+y^2 \) | constraint: \( 2x+6y=2000 \) |
Solution |
---|
This solution is in two consecutive videos.
video by Krista King Math
video by Krista King Math
close solution |
optimize: \( f(x,y) = x^2+y^2 \) | constraint: \( xy=1 \) |
Problem Statement |
---|
optimize: \( f(x,y) = x^2+y^2 \) | constraint: \( xy=1 \) |
Solution |
---|
This solution is in two consecutive videos.
video by Krista King Math
video by Krista King Math
close solution |
minimize: \( C(x,y) = 6x^2 + 12y^2 \) | constraint: \( x + y = 90 \) |
Problem Statement |
---|
minimize: \( C(x,y) = 6x^2 + 12y^2 \) | constraint: \( x + y = 90 \) |
Solution |
---|
video by PatrickJMT
close solution |
optimize: \( f(x,y) = xy \) | constraint: \( x^2 + 2y^2 = 1 \) |
Problem Statement |
---|
optimize: \( f(x,y) = xy \) | constraint: \( x^2 + 2y^2 = 1 \) |
Solution |
---|
video by Dr Chris Tisdell
close solution |
minimize: \( f(x,y,z) = x^2 + y^2 + z^2 \) | constraint: \( 2x + y - z = 1 \) |
Problem Statement |
---|
minimize: \( f(x,y,z) = x^2 + y^2 + z^2 \) | constraint: \( 2x + y - z = 1 \) |
Solution |
---|
video by Dr Chris Tisdell
close solution |
maximize: \( f(x,y,z) = x^2+2y-z^2 \) | constraint: \( 2x=y\) and \( y+z=0 \) |
Problem Statement |
---|
maximize: \( f(x,y,z) = x^2+2y-z^2 \) | constraint: \( 2x=y\) and \( y+z=0 \) |
Solution |
---|
video by Dr Chris Tisdell
close solution |
optimize: \( f(x,y) = 3x+4y \) | constraint: \( x^2+y^2 = 1 \) |
Problem Statement |
---|
optimize: \( f(x,y) = 3x+4y \) | constraint: \( x^2+y^2 = 1 \) |
Solution |
---|
video by Dr Chris Tisdell
close solution |
Determine the point on the surface of \( xyz = 1 \) that is closest to the origin and satifies \( x > 0 \), \( y > 0 \) and \( z > 0 \).
Problem Statement |
---|
Determine the point on the surface of \( xyz = 1 \) that is closest to the origin and satifies \( x > 0 \), \( y > 0 \) and \( z > 0 \).
Solution |
---|
video by Dr Chris Tisdell
close solution |
maximize: \( T(x,y) = 6xy \) | constraint: \( x^2 + y^2 = 8 \) |
Problem Statement |
---|
maximize: \( T(x,y) = 6xy \) | constraint: \( x^2 + y^2 = 8 \) |
Solution |
---|
video by Dr Chris Tisdell
close solution |
We have a thin metal plate that occupies the region in the xy-plane \( x^2 + y^2 \leq 25 \). If \( f(x,y) = 4x^2 - 4xy + y^2 \) denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.
Problem Statement |
---|
We have a thin metal plate that occupies the region in the xy-plane \( x^2 + y^2 \leq 25 \). If \( f(x,y) = 4x^2 - 4xy + y^2 \) denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.
Solution |
---|
video by Dr Chris Tisdell
close solution |
optimize: \( f(x,y,z) = xyz \) | constraint: \( x^2 + y^2 + z^2 = 3 \) |
Problem Statement |
---|
optimize: \( f(x,y,z) = xyz \) | constraint: \( x^2 + y^2 + z^2 = 3 \) |
Solution |
---|
video by Krista King Math
close solution |
Find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to \( x^3 + y^3 = 16 \).
Problem Statement |
---|
Find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to \( x^3 + y^3 = 16 \).
Solution |
---|
video by MIP4U
close solution |
Find the rectangle with maximum perimeter that can inscribed in the ellipse \( x^2 + 4y^2 = 4 \).
Problem Statement |
---|
Find the rectangle with maximum perimeter that can inscribed in the ellipse \( x^2 + 4y^2 = 4 \).
Solution |
---|
video by MIT OCW
close solution |
Intermediate Problems |
---|
optimize: \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \) | constraint: \( 3x + 5y = 47 \) |
Problem Statement |
---|
optimize: \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \) | constraint: \( 3x + 5y = 47 \) |
Solution |
---|
This solution is in three consecutive videos.
video by Krista King Math
video by Krista King Math
video by Krista King Math
close solution |
optimize: \( f(x,y,z) = x^2+y^2+z^2 \) | constraints: \( x+y+z = 1 \); \( x+2y+3z = 6 \) |
Problem Statement |
---|
optimize: \( f(x,y,z) = x^2+y^2+z^2 \) | constraints: \( x+y+z = 1 \); \( x+2y+3z = 6 \) |
Solution |
---|
video by Krista King Math
close solution |
optimize: \( f(x,y,z) = 3x-y-3z \) | constraints: \( x+y-z=0 \); \( x^2+2z^2 = 1 \) |
Problem Statement |
---|
optimize: \( f(x,y,z) = 3x-y-3z \) | constraints: \( x+y-z=0 \); \( x^2+2z^2 = 1 \) |
Solution |
---|
This solution is presented in two consecutive videos.
video by PatrickJMT
video by PatrickJMT
close solution |
optimize: \( f(x,y,z) = x^2+x+2y^2+3z^2 \) | constraint: \( x^2+y^2+z^2 = 1 \) |
Problem Statement |
---|
optimize: \( f(x,y,z) = x^2+x+2y^2+3z^2 \) | constraint: \( x^2+y^2+z^2 = 1 \) |
Solution |
---|
video by MIT OCW
close solution |