**Unit 2: Partial Differentiation**
*For single-variable functions, the derivative of the graph at a
specified point is the slope of the tangent line at that point. This is
because the dependent variable is only changing with regards to a single
independent variable. In this course, we will consider functions of
multiple variables. Because these functions have outputs that depend on
multiple variables, each variable contributes to the rate of change of
the function independently. Thus, we refer to the function’s rate of
change with respect to a single specified variable as a* partial
derivative*, as it does not describe the entire rate of change of the
function. Moreover, for functions of multiple variables, there will no
longer be a single tangent line at a specified point. For example, in
the case of a function of two variables, we consider a tangent plane as
opposed to a tangent line. The concept of a partial derivative is
important to study concepts such as tangent spaces, extrema, etc. in the
context of functions of 2 or more variables. *

*As in Single-Variable Calculus, Multivariable Calculus studies the
maximum and minimum values for given functions due to their numerous
applications. Lagrange Multipliers is a method of finding maximum and
minimum values for functions subject to constraints that uses partial
derivatives; however, the idea of optimization with constraints has no
one-variable analogue.*

*Again, you should focus on the geometric meaning of the concepts
introduced in this unit.*

**Unit 2 Time Advisory**

This unit will take you approximately 25.5 hours to complete.

☐ Subunit 2.1: 9.25 hours

☐ Subunit 2.1.1: 1.75 hour

☐ Subunit 2.1.2: 1.25 hour

☐ Subunit 2.1.3: 1.25 hour

☐ Subunit 2.1.4: 5 hours

☐ Subunit 2.2: 16.25 hours

☐ Subunit 2.2.1: 4.5 hours

☐ Subunit 2.2.2: 3.25 hours

☐ Subunit 2.2.3: 4.75 hours

☐ Subunit 2.2.4: 3.75 hours

**Unit2 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:

- Determine the domain of a function.
- Determine whether the domain of a function is open or closed, bounded or unbounded, connected or not connected.
- Sketch the graph of the domain of a function.
- Sketch the graph of a given function.
- Identify the characteristics of a function from a graph of its level curves.
- Evaluate limits.
- Show that limits do not exist by evaluating the limit along two different paths.
- Determine whether a given function is continuous.
- Evaluate partial derivatives.
- Evaluate higher order partial derivatives.
- Describe the geometrical significance of a directional derivative.
- Find the directional derivative.
- Describe the relation between existence of partial derivatives, continuity, and differentiability.
- Evaluate derivatives of functions using the chain rule.
- Use the method of Lagrange Multipliers to find the extrema of functions subject to constraints.

**2.1 Partial Derivatives**
**2.1.1 Level Curves and Partial Derivatives**
- **Activity: The Saylor Foundation: Math Insight's “Level Sets”**
Link: The Saylor Foundation: Math
Insight’s “Level
Sets” (PDF)

```
Also Available in:
[HTML and Java](http://mathinsight.org/level_sets)</span>
NOTE: In order to view the Java applets within this resource you
must click on the [HTML and
Java](http://mathinsight.org/level_sets) link, as the PDF version
does not support the Java applets.
<span style="background-color: transparent;"> Instructions: Please
click on the webpage linked above and work through the notes and the
applets. Feel free to go through more examples by clicking on the
link at the bottom of the page. </span>
This activity should take approximately 1 hour to complete.
Terms of Use: The linked resource above is released under
a [Creative Commons Attribution-NonCommercial-ShareAlike 3.0
Unported
License](http://creativecommons.org/licenses/by-nc-sa/3.0/), it is
attributed to Duane Q. Nykamp and the original version can be
found [here](http://mathinsight.org/level_sets).
```

**Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video Lecture 8: Partial Derivatives”**Link: MIT: Denis Auroux’s*Multivariable Calculus:*“Video Lecture 8: Partial Derivatives” (Adobe Flash, iTunes or Mp4)

Also available in: Adobe Flash, iTunes or Mp4

Instructions: Please view the entire video lecture (46:13). You may also click on the “Transcript” tab on the page to read the lecture.This reading should take approximately 15 minutes to read and review.

Terms of Use: The video above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.**Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives:”**Link: MIT: Denis Auroux’s*Multivariable Calculus:*“II. Partial Derivatives” (PDF)

Instructions: Read the entire PDF document (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.This activity should take approximately 15 minutes to complete.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.

**2.1.2 Max-Min Problems and Least Squares**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 9: Max-Min and Least Squares”**
Link: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
9: Max-Min and Least
Squares” (Youtube)

Also available in: Adobe Flash, iTunes, or
Mp4

Instructions: Please view the entire lecture (49:44). You may also
click on the “Transcript” tab on the page to read the lecture.

Terms of Use: The video above is released under Creative Commons
Attribution-NonCommercial-ShareAlike
3.0. It is
attributed to Denis Auroux and the original version can be found
here.

**2.1.3 Second Derivative Test**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 10: Second Derivative Test”**
Link: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
10: Second Derivative
Test” (YouTube)

Also available in: Adobe Flash, iTunes, or
Mp4

Instructions: Please view the entire lecture (52:18). You may also
click on the “Transcript” tab on the page to read the lecture.

```
This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-10-second-derivative-test/).
```

**2.1.4 Limits, Continuity, and Partial Differentiation**
- **Reading: East Tennessee State University: Jeff Knisley’s
Multivariable Calculus Online: “Chapter 2: Partial Differentiation:”
“2.1 Functions of 2 Variables,” “2.2 Limits and Continuity,” and
“2.3 Partial Derivatives”**
Links: East Tennessee State University: Jeff Knisley’s
*Multivariable Calculus Online:* “Chapter 2: Partial
Differentiation:” “2.1 Functions of 2
Variables,” (HTML
and Java) “2.2 Limits and
Continuity,” (HTML)
and “2.3 Partial
Derivatives” (HTML
and Java)

Instructions: Please click on each link above to sections “2.1
Functions of 2 Variables,” “2.2 Limits and Continuity,” and “2.3
Partial Derivatives.” Read all four parts for each individual
section. Begin by reading the first webpage for each section linked
above, and then select the other Parts at the top of each webpage to
continue reading (After finishing Part 4 of "2.3 Partial
Derivatives", you will have read 12 Parts total).

```
This reading should take approximately 1 hour and 30 minutes to
complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Functions of 2 Variables Exercises,” “Limits and Continuity Exercises,” and “Partial Derivatives Exercises”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“Functions of 2 Variables Exercises,” (HTML and Java) “Limits and Continuity Exercises,” (HTML) and “Partial Derivatives Exercises” (HTML and Java)

Instructions: Please click on the “Functions of 2 Variables Exercises” link above, and at the top of this webpage, click on the “Exercises” link to access the question sets. Work through exercises 7, 9, 15, 19, and 25. Similarly, click on the “Limits and Continuity Exercises” link above, and once on the webpage, select the “Exercises” link at the top of the webpage to access the question sets. Complete exercises 5, 13, 17, and 25. Finally, click on the “Partial Derivatives Exercises” link above, and at the top of this webpage, click on the “Exercises” link to be redirected to the question sets. Try to solve exercises 5, 11, 19, 25, and 29. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and then finding sections 2.1-2.3.These exercises should take approximately 3 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Activity: The Saylor Foundation: Math Insight's "Introduction to Partial Derivatives"**Link: The Saylor Foundation: Math Insight’s “Introduction to Partial Derivatives” (PDF)Also Available in:

HTML and Java*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.

Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.

This activity should take approximately 30 minutes to complete.

Terms of Use: The linked resource above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.

**2.2 Differentiation and Chain Rule**
**2.2.1 Chain Rule**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 11: Chain Rule”**
Link: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
11: Chain
Rule” (YouTube)

```
Also available in: [Adobe Flash, iTunes, or
Mp4](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/)
Instructions: Please view the entire lecture (50:09). You may also
click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/).
```

**Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives”**Link: MIT: Denis Auroux’s*Multivariable Calculus:*“II. Partial Derivatives” (PDF)

Instructions: Read the entire PDF file (4 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.**Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.5 Linearization and the Hessian” and “2.6 The Chain Rule”**Links: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“Chapter 2: Partial Differentiation:” “2.5 Linearization and the Hessian” (HTML and Java) and “2.6 The Chain Rule” (HTML)

Instructions: Please click the links titled “Section 2.5 Linearization and the Hessian” and “Section 2.6 The Chain Rule” above. For each section, read Parts 1-4 in their entirety. You may access each part by clicking on the links at the top of each webpage.These exercises should take approximately 3 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.5 Linearization and the Hessian” and “2.6 The Chain Rule”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“2.5 Linearization and the Hessian” (HTML) and “2.6 The Chain Rule” (HTML)

Instructions: Please click the link titled “2.5 Linearization and the Hessian” posted above, and then select the “Exercises” link at the top of the webpage to access the questions. Try to solve exercises 1, 9, 21, and 27. Then, click on the “2.6 The Chain Rule” link posted above, and select the “Exercises” link at the upper right corner of the webpage to access the questions. Complete exercises 5, 13, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and then scrolling down to find sections 2.5 and 2.6.These exercises should take approximately 2 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**2.2.2 Gradient**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 12: Gradient”**
Link: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
12:
Gradient” (YouTube)

Also available in: Adobe Flash, iTunes, or
Mp4

```
Instructions: Please view entire video lecture (50:10). You may
also click on the “Transcript” tab on the page to read the
lecture.
This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-12-gradient/).
```

**Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.7 Properties of the Gradient”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“Chapter 2: Partial Differentiation:” “2.7 Properties of the Gradients” (HTML and Java)

Instructions: Please read all four parts of “2.7 Properties of the Gradients.” Begin by reading “Part 1: Gradients and Level Curves” linked above, and then select the links for Parts 2-4 at the top of the webpage to continue reading.This reading should take approximately 30 minutes to read and review.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.7 Properties of the Gradients Exercises”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“2.7 Properties of the Gradients Exercises” (HTML)

Instructions: Please click on the link above titled “2.7 Properties of the Gradients Exercises.” On this webpage, click on the link titled “Exercises” at the upper right corner of the webpage to access the questions. Try to complete exercises 3, 13, and 15. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.7.

These exercises should take approximately 1 hour to complete.Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Activity: The Saylor Foundation: Math Insight's “An Introduction to the Directional Derivative and the Gradient”**Link: The Saylor Foundation: Math Insight’s “An Introduction to the Directional Derivative and the Gradient” (PDF)Also Available in:

HTML and Java*NOTE: In order to view the Java applets within this resource you must click on the HTML and Java link, as the PDF version does not support the Java applets.

Instructions: Click on the webpage linked above and work through the notes and the applets. Feel free to work on more examples or read more sections by clicking the relevant links on the bottom of the page.

This activity should take approximately 30 minutes to complete.

Terms of Use: The linked resource above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License, it is attributed to Duane Q. Nykamp and the original version can be found here.

**2.2.3 Optimization and Lagrange Multipliers**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 13: Lagrange Multipliers”**
Link: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
13: Lagrange
Multipliers” (YouTube)

Also available in: Adobe Flash, iTunes, or
Mp4

Instructions: Please view the entire video lecture (50:10). You
may also click on the “Transcript” tab on the page to read the
lecture.

```
This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-13-lagrange-multipliers/).
```

**Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.8 Optimization” and “2.9 Lagrange Multipliers”**Links: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“Chapter 2: Partial Differentiation:” “2.8 Optimization” (HTML and Java) and “2.9 Lagrange Multipliers” (HTML and Java)Instructions: Please click the links titled “Section 2.8 Optimization” and “Section 2.9 Lagrange Multipliers” found above. Read Parts 1-4 of each section. Make sure to click on the links to each part at the top of each webpage to complete this reading assignment.

This reading should take approximately 1 hour to read and review.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.8 Optimization Exercises” and “2.9 Lagrange Multipliers Exercises”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“2.8 Optimization Exercises” (HTML and Java) and “2.9 Lagrange Multipliers Exercises” (HTML and Java)

Instructions: Please click on the link above titled “2.8 Optimization Exercises”, and then select the “Exercises” link at the top right corner of the webpage to access the question sets. Solve exercises 1, 17, and 25. Similarly, click on the link titled “2.9 Lagrange Multipliers Exercises” above, and then click on the link titled “Exercises” at the upper right corner of the webpage to access the questions. Work on exercises 5, 7, and 15. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to sections 2.8 and 2.9.These exercises should take approximately 2 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Activity: University of Minnesota: Jonathan Rogness’ Multivariable Calculus and Vector Analysis: “Lagrange Multipliers”**Link: University of Minnesota: Jonathan Rogness’*Multivariable Calculus and Vector Analysis*: “Lagrange Multipliers” (HTML and Java)Instructions: Click on the webpage linked above and work through the notes and the applets.

This activity should take approximately 30 minutes to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**2.2.4 Partial Differential Equations**
- **Lecture: MIT: Denis Auroux’s Multivariable Calculus: “Video
Lecture 14: Non-Independent Variables” and “Video Lecture 15:
Partial Differential Equations”**
Links: MIT: Denis Auroux’s *Multivariable Calculus:* “Video Lecture
14: Non-Independent
Variables” (YouTube)
and “Video Lecture 15: Partial Differential
Equations” (YouTube)

Also available in:

Adobe Flash, iTunes, or
Mp4 (Lecture
14)

Adobe Flash, iTunes, or
Mp4 (Lecture
15)

```
Instructions: Please view the video lectures in their entirety
(49:11 and 45:23, respectively). You may also click on the
“Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-14-non-independent-variables/)
(Lecture 14) and
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-15-partial-differential-equations/)
(Lecture 15).
```

**Reading: MIT: Denis Auroux’s Multivariable Calculus: “II. Partial Derivatives:”**Link: MIT: Denis Auroux’s*Multivariable Calculus:*“II. Partial Derivatives” (PDF)

Instructions: Open the link above and read the entire PDF document (3 pages). Please note that this reading is paired with the video lecture above so please read it after watching the video.This reading should take approximately 15 minutes to study.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Denis Auroux and the original version can be found here.**Reading: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “Chapter 2: Partial Differentiation:” “2.4: Partial Differential Equations”**Link: East Tennessee State University: Jeff Knisley’s*Multivariable Calculus Online:*“Chapter 2: Partial Differentiation:” “2.4: Partial Differential Equations” (HTML)

Instructions: Please read Parts 1-4 of “Section 2.4: Partial Differential Equations” from the link posted above. Begin by reading “Part 1: Partial Differential Equations” linked above, and then click on the links to Parts 2-4 at the top of the webpage.This reading should take approximately 30 minutes to study.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: East Tennessee State University: Jeff Knisley’s Multivariable Calculus Online: “2.4: Partial Differential Equations Exercises”**Link: East Tennessee State University’s Jeff Knisley’s*Multivariable Calculus Online:*“2.4: Partial Differential Equations Exercises” (HTML)

Instructions: Please click on the “2.4: Partial Differential Equations Exercises” link above, and once on the webpage, click on the “Exercises” link at the top right corner of the webpage to redirect to the exercise sets. Complete exercises 11, 21, and 23. Check the solutions by clicking on the link titled “Answers to Selected Odd Exercises” (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.4.These exercises should take approximately 1 hour to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.