18.02SC | Fall 2010 | Undergraduate

# Multivariable Calculus

2. Partial Derivatives

## Part A: Functions of Two Variables, Tangent Approximation and Optimization

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We start this unit by learning to visualize functions of several variables using graphs and level curves. Following this we will study partial derivatives. These will be used in the tangent approximation formula, which is one of the keys to multivariable calculus. It ties together the geometric and algebraic sides of the subject and is the higher dimensional analog of the equation for the tangent line found in single variable calculus. We will use it in part B to develop the chain rule.

We will apply our understanding of partial derivatives to solving unconstrained optimization problems. (In part C we will solve constrained optimization problems.)

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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
• Read another set of course notes
• Work with a Mathlet to reinforce lecture concepts
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Functions of Two Variables: Graphs

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

Graphing Functions of Two Variables (PDF)

### Problems and Solutions

Problems: Graphing a Function (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 two recitation videos
• Work with a Mathlet to reinforce lecture concepts
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Level Curves and Contour Plots

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

Level Curves and Contour Plots (PDF)

### Problems and Solutions

Problems: Contour Plots (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: Partial Derivatives

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

Partial Derivatives (PDF)

### Problems and Solutions

Problems: Partial Derivatives (PDF)

Solutions (PDF)

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« Previous | Next »

### 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
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Approximation Formula

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

The Tangent Approximation (PDF)

Non-Geometric Argument for the Approximation Formula (PDF)

### Examples

Tangent Approximation (PDF)

### Problems and Solutions

Problems: Tangent Approximation (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
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Optimization Problem

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

Critical Points (PDF)

### Problems and Solutions

Problems: Critical Points (PDF)

Solutions (PDF)

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« Previous | Next »

### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Review another example
• Watch a recitation video
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Least Squares

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

Least Squares Interpolation (PDF)

### Examples

Least Squares (PDF)

### Problems and Solutions

Problems: Least Squares (PDF)

Solutions (PDF)

« Previous | Next »

« Previous | Next »

### Overview

In this session you will:

• Watch two lecture video clips and read board notes
• Read course notes and examples
• Review an example
• Work with a Mathlet to reinforce lecture concepts
• Watch a recitation video
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip 2: Second Derivative Test

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

Second Derivative Test (PDF)

### Examples

Second Derivative Test (PDF)

### Problems and Solutions

Problems: Second Derivative Test (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

### Lecture Video

#### Video Excerpts

Clip: Example

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

### Recitation Video

#### Max/Min

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## Course Info

Fall 2010
##### Learning Resource Types
Simulations
Exams with Solutions
Problem Sets with Solutions
Lecture Videos
Lecture Notes