Fall 2020

Emily Proctor

Please read the syllabus to learn all the details about the course.

We'll use our Canvas site for turning in homework, and here is a chronological list of the videos and notes for the semester.

- Week beginning September 7
- Due Thursday, September 10:
- For Thursday, September 10: make a 90-second audio recording introducing yourself to the class. Start with your name and your class year, and then tell us a little something about yourself, perhaps a happy math memory. See our Canvas site for more details. When you are done, upload your audio file via Canvas.

- Due Friday, September 11:
- Read Sections 1.1 and 1.2. (Note: you won't be responsible for knowing about the symmetric form of a line in R^n.)
- Watch Sections 1.1 and 1.2 video: Part 1/notes.
- From Section 1.1 do problems: 6, 7, 10, 15, 17, 19 (property 2 only), 21a, 22, 23.
- From Section 1.2 do problems: 9, 10, 11, 17, 19, 28, 30, 35, 43.
- Bring questions from the homework to class with you on Friday, and we will talk more about it then. Your homework (pdf format) will be due via Canvas by 5pm on Friday.
- Note: if you would like to do your homework directly on a tablet and create a pdf that way, rather than writing it and scanning it, that is a perfectly great alternative as well.
- Here is an expanded version of the notes that I presented in today's video, if you would like to take a look.

- Week beginning September 14
- Due Monday, September 14:
- Due Wednesday, September 16:
- Read Section 1.5 and Section 1.6 (p.48-51).
- Watch Sections 1.5 and 1.6 videos Part 1, Part 2, Part 3, and notes.
- In part of these videos, I to talk a bit about the parametric equations of a plane. With this in mind, it might help you to go back before watching and take a look again at Problem 1.1.22.
- The notes that are posted here are a slightly expanded version of the notes I used in the videos. These notes include proofs of the Cauchy-Schwarz inequality and the triangle inequality, in case you would like to see them.
- From Section 1.5 do problems: 1, 4, 8, 11, 12, 14, 16, 19, 22, 23, 25. (Problem 25 is a distance problem. Even though we didn't cover the method in class, I put this problem on the assignment to prompt you to read Examples 7, 8, and 9 in the book. The goal is to help you understand material about projections and dot products better, so reading these examples is as important as doing the problem.)
- From Section 1.6 do problems: 6, 9, 10, 11, 12.

- Due Friday, September 18:
- Read Section 1.7.
- Watch Section 1.7 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 1.7 do problems: 1, 9, 11, 15, 18, 19, 20a, 23, 26, 29, 32, 33, 35 (see hint below), 38, 42ab (see note below).
- Hint for 1.7.35: There are two inequalities given in Problem 35: 2cos(phi)\leq rho and rho\leq 3. Consider each separately. For the first, try multiplying through by rho.
- For 1.7.42, make note of which method for describing the given region is simpler. Later on, when we are integrating in three dimensions, this type of thinking will help you to set up integrals so that they are as simple to compute as possible.

- Week beginning September 21
- Due Monday, September 21:
- Due Wednesday, September 23:
- Read Section 2.1, p.92-95.
- Watch Section 2.1 videos Part 4/notes and Part 5/notes.
- From Section 2.1, do problems: 32*, 33, 37, 38**, 39, 40, 41, 42, 46.
- *Problems 2.1.32, 33, and 37 are just asking for some level surfaces. You do not need to put them together into a graph (it would be impossible!).
- **Problem 2.1.38 is highlighting an important concept. Pay attention to this one, in conjunction with today's video and the italicized comment in the middle of p.93.
- Here is the chart of quadric surfaces mentioned in today's video.
- If you looking to practice some more with cylindrical and spherical coordinates, problems 1.7.23-35 would be good extra problems to play with. These are completely optional, to work on on your own, as much or as little as you like.

- Due Friday, September 25:
- Read Section 2.2. This is a relatively long section, but the author does a good job of describing things, so it's worth it to take some time to read it through. You do not need to read the addendum at the end unless you are curious.
- Watch Section 2.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 2.2, do problems: 3, 7, 8*, 11, 12, 13, 19, 23, 33, 35, 39**, 45**. 47.
- *For problem 2.2.8, it may help to know that |y| can be defined as a piecewise function: |y| = (y if y\geq 0) and (-y if y\leq 0).
- **Briefly justify your answers for problems 2.2.39 and 45.

- Week beginning September 28
- Due Monday, September 28:
- Due Wednesday, September 30:
- Read Section 2.3, p.118-123
- Watch Section 2.3 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 2.3, do problems: 35, 37, 40, 42, 45.
- In case it is helpful, here is an example of how to compute the equation for a tangent plane.
- Make note of problem 45; it is a bridge between what we are covering here and what you will be learning about for Friday.
- Start preparing for our exam on Thursday (October 8) of next week. It will cover from the beginning of the semester through Monday's assignment next week, which will be on Section 2.5.

- Due Friday, October 2:
- Read Section 2.3, p.123-128. Read Section 2.4, p.133-135.
- Watch Section 2.3 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 2.3 do problems: 26, 27, 33, 44, 59.
- From Section 2.4 do problems: 2, 5.
- The material from the end of the videos about the interpretation of the derivative goes into a bit more depth than the book does. We'll be thinking this way later on in the semester. Since there are fewer problems due than usual, it would be worth it to take another look or two at the reading/video/notes now to help that material sink in.

- Week beginning October 5:
- Due Monday, October 5:
- Read Section 2.5.
- Watch Section 2.5 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 2.5 do problems: 2, 3, 4, 5, 8, 11, 13, 24, 25, 28, 29.
- Continue to prepare for the exam this Thursday, October 8. The exam will cover from the beginning of the semester through Section 2.5. I will pass it out to you by 8am on Thursday morning and it will be due back on Friday, October 9 by the end of class time (11:10am Eastern).
- I will hold an optional review session on Wednesday evening, 6:30-8pm Eastern. The zoom link for this is the same as the link for office hours, which you can find on our Canvas site. I won't come with an agenda, so please bring any questions you have!
- Here, again, is our reflective assignment for you to contemplate in the background over the next two weeks. This assignment is due at class time on Friday, October 16.

- Due Wednesday, October 7:
- Due Friday, October 9:
- I will pass out the exam at 8am Eastern on Thursday; keep an eye on your email for instructions on where to find it. Please submit your completed exam via Canvas by the end of class time, 11:10am Eastern. Note that this is a firm deadline; see the exam for details.
- We will not have class on Friday. I hope you enjoy some time away from math!

- Week beginning October 12
- Due Monday, October 12:
- Read Section 3.1.
- Watch Section 3.1 videos Part 1/notes and Part 2/notes.
- From Section 3.1 do problems: 3, 5, 9, 11b*, 12b*, 17 (give a vector equation), 19, 25**, 27, 29***, 30, 33.
- * For Problems 11b and 12b, you do not need the pictures from parts a in order to do parts b.
- ** If you would like it, here is a description about how to think about Problem 25.
- *** In Problem 29, note that if ||x(t)|| is constant, then so is ||x(t)||^2. Problem 27 might be of help here.
- Continue to keep in mind our reflective assignment, due at class time on Friday, October 16.

- Due Wednesday, October 14:
- Read Section 2.6 p. 164-168.
- Watch Section 2.6 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- In order to make today's video more meaningful, it'd be worth it to review the notion that every surface in R^3 can be thought of as the level surface of some function.
- From Section 2.6 do problems: 18, 20, 23, 25, 26, 29 (just do method b), 31, 34, 36.

- Due Friday, October 16:
- Read Section 3.2 p.203-205.
- Watch Section 3.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.
- From Section 3.2 do problems: 3, 5, 6, 10, 11, 12ab, 14*, 15**.
- Turn in your response to the reflective assignment by class time on Friday. There is a link on Canvas for this.
- *For Problem 14, to compute an indefinite integral of f(t) from a to infinity, compute the definite integral from a to b, then take the limit as b goes to infinity.
- **In Problem 3.2.15, you'll need to use the polar conversion formulas x=rcos(theta), y=rsin(theta). Theta is the defining parameter here. This means that as theta increases, the distance from the point on the curve to the origin (i.e. r) changes, depending on theta. Thus, the curve goes counterclockwise around the origin, but moves closer or farther away from the origin as it goes.

- Week beginning October 19
- Due Monday, October 19:
- Read Section 3.3 p.221-224 and Section 3.4 p.227-232.
- Watch Sections 3.3 video Part 1/notes.
- Watch Section 3.4 videos Part 1/notes and Part 2/notes.
- From Section 3.3 do problems: 3, 4, 24a.
- From Section 3.4 do problems: 4, 10, 13, 14, 15, 16*, 20, 23, 28ab*, 31**.
- *The phrase "f and g are functions of class C^2" in Problems 3.4.16 and 3.4.28b means that f and g can both be differentiated two times and their second derivatives are continuous. It's a technical requirement that ensures that any appropriate theorems can be applied here.
- **For Problem 3.4.31, it might be helpful to go back and look in your notes at the place where we derived the formula D_uf=(grad f) dot u.

- Due Wednesday, October 21:
- Read Section 4.1 p.244-256 and Section 4.2 p.263-267.
- Watch Section 4.1 videos Part 1/notes and Part 2/notes.
- Watch Section 4.2 video Part 1/notes.
- From Section 4.1 do problems: 4, 5, 11, 19, 20, 22*, 23*, 25.
- *For Problems 22 and 23, multiply out the matrices so that you arrive at an actual polynomial.

- Due Friday, October 23:
- Read Section 4.2 p.267-274.
- Watch Section 4.2 videos Part 2/notes and Part 3/notes.
- From Section 4.2 do problems: 1*, 3, 8, 11, 12, 22a, 29**, 31***, 42.
- *For Problem 1b, see Example 3 p.265, as well as p.249.
- **For Problem 29, your work will be much easier if you minimize the *square* of the distance rather than the distance itself.
- ***For Problem 31, confirm that your answer is indeed a maximum.
- The version of the second derivative test given on p.268 of the book is a more general version (for functions R^n to R) than the version I did in the video. For the specific version I did in the video (for functions R^2 to R), see Example 5 on p.269.
- I did not give the full proof of the second derivative test for functions R^2 to R in the video, but if you are curious about it, here are the notes I wrote up that give the proof. It's a really beautiful application of diagonalizability, which you learned about in linear algebra. So, it is not required but I still encourage you to take a look at the notes, both to see why the second derivative test works, and for more practice with reading proofs.
- Here is a write-up of an example that I did during today's video, in case you want to look more closely at the details.
- I talked about the Extreme Value Theorem at the end of the video, but only talked through the strategy of an example. I didn't assign you any problems based on the Extreme Value Theorem, but take a look at the following details of the example, along with Examples 8 and 9 in the book to get an idea of how the theorem can and can't be used in determining the extreme values of a function. We will use this theorem a bit in the next section, when we work on Lagrange multipliers.

- Week beginning October 26
- Due Monday, October 26:
- Read Section 4.3 p.278-284
- Watch Section 4.3 videos Part 1/notes and Part 2/notes.
- In order to get more out of the video for today, it might be helpful to review how gradients and level sets are related (specifically Theorem 6.4 p.164) before watching.
- From Section 4.3 do problems: 1*, 6, 13a, 21, 22, 23**.
- For Problem 1, once again, remember that minimizing the square of the distance will make your work simpler!
- For Problem 23, you are trying to find the maximum and minimum value of f over the entire closed disk (not just the boundary circle). To find critical points on the interior, you can use the method of finding critical points that we learned in Section 4.2. Since the boundary of the disk is a level set, you can find critical points on the boundary by using Lagrange multipliers. Once you have found all of the critical points, make an argument about which critical point(s) give(s) the absolute maximum and which critical point(s) give(s) the absolute minimum.

- Due Wednesday, October 28: