18.02SC | Fall 2010 | Undergraduate

Multivariable Calculus

2. Partial Derivatives

Part C: Lagrange Multipliers and Constrained Differentials

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In this part we will study a new type of optimization problem: that of finding the maximum (or minimum) value of a function w = f(x, y, z) when we are only allowed to consider points (x, y, z) which are constrained to lie on a surface. The technique we will use to solve these problems is called Lagrange multipliers.

» Session 39: Statement of Lagrange Multipliers and Example
» Session 40: Proof of Lagrange Multipliers
» Session 41: Advanced Example
» Session 42: Constrained Differentials
» Session 43: Clearer Notation
» Session 44: Example
» Problem Set 6

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Overview

In this session you will:

  • Watch a lecture video clip and read board notes
  • Read course notes and examples
  • Watch a recitation video

Lecture Video

Video Excerpts

Clip: Lagrange Multipliers by Example

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.


Reading and Examples

Lagrange Multipliers (PDF)

Recitation Video

Lagrange Multipliers

Problems and Solutions

Problems: Lagrange Multipliers (PDF)

Solutions (PDF)

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Overview

In this session you will:

  • Watch a lecture video clip and read board notes
  • Watch a recitation video
  • Do problems and use solutions to check your work

Lecture Video

Video Excerpts

Clip: Advanced Example

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.


Recitation Video

Lagrange Multipliers (3 Variables)

Problems and Solutions

Problems: Lagrange Multipliers (PDF)

Solutions (PDF)

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Overview

In this session you will:

  • Watch a lecture video clip and read board notes
  • Read course notes and examples
  • Watch a recitation video
  • Review another example

Lecture Video

Video Excerpts

Clip: Clearer Notation

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Reading and Examples

Non-Independent Variables (PDF)

Recitation Video

The Chain Rule with Constraints

Reading and Examples

Non-Independent Variables (PDF)

Problems and Solutions

Problems: The Chain Rule with Constraints (PDF)

Solutions (PDF)

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« Previous | Next »

Overview

In this session you will:

  • Watch a lecture video clip and read board notes
  • Watch a recitation video
  • Do problems and use solutions to check your work

Lecture Video

Video Excerpts

Clip: Example

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.



Recitation Video

Gradients - Composition

Problems and Solutions

Problems: Chain Rule with Constraint (PDF)

Solutions (PDF)

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Course Info

Instructor
Departments
As Taught In
Fall 2010
Learning Resource Types
Simulations
Exams with Solutions
Problem Sets with Solutions
Lecture Videos
Lecture Notes