ANNOUNCEMENTS
MTH 254H — Fall 2016

12/7/16

The final has been graded, and course grades have been assigned. I believe they should show up online tomorrow.

You can get your exam back if you stop by my office next term.
If you'd like to know your exam score before then, send me an email request using a campus address.

12/5/16

Below are the answers to the final.

Full solutions can be seen in my office—but probably not until next term, when you can also pick up your graded exam.

1. $8/3$

2. $78\pi$ g

3. (a) positive (b) not enough information (c) negative

4. saddle points at $(+1,-1)$ and $(-1,+1)$; local min ($-4$) at $(+1,+1)$; local max ($+4$) at $(-1,-1)$

5. min: $-10$, which occurs at $(-\frac65,-\frac85)$; max: $+10$, which occurs at $(+\frac65,+\frac85)$;

6. $40$

7. (a) $0$ ft/mi (b) $-5$ ft/mi (c) $\frac{5}{\sqrt2}$ ft/mi

8. (a) $4$ (b) $0$ (c) $\vec{0}$ (d) $2\,\vec{k}$ (out of page)

9. $8$ m/s in direction $-\hat\imath$

10. (a) [many answers possible] (b) $-10x\cos(2\pi y)\hat\imath+10\pi x^2\sin(2\pi y)\hat\jmath$ (c) $-18$ ft/mi (d) $-90$ ft/mi

EC. $k/5$, or equivalently $fg$

12/1/16

Extra office hours: I will be in my office starting at 1 PM on Sunday, 12/4.

• I will stay as long as there are people waiting to talk with me.
• I will not leave before 3 PM, but will most likely do so then if there's nobody left.
• If you plan to arrive after 3 PM please let me know in advance.
• Bring your phone! The building may be locked; if so, call me at (541) 737-5159.
• I may also be able to hear you if you stand below my window (on the North side)...

11/30/16

We showed in class today that $dA = \left| \frac{\partial\vec{r}}{\partial u} \times \frac{\partial\vec{r}}{\partial v} \right| du\,dv$ for any variables $u$, $v$, generalizing the expressions for $dA$ in rectangular and polar coordinates.

• Further discussion and a general formula involving a determinant known as a Jacobian appear in the online text.
• The Jacobian expression $dx\,dy = \left| \frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv$ generalizes the notion of subsitution, in which $dx=\frac{dx}{du}\,du$.

Pictures of the board from today's class can be found here, here, and here.

11/28/16

Further information about curvature can be found in my online book on differential forms.

Here's a review problem based on today's discussion of arclength:

Find the length of the curve given by $\vec{r}=2\cos^2\theta\,\hat\imath+2\sin^2\theta\,\hat\jmath$ as $\theta$ goes from $0$ to $\pi/2$.

11/27/16

Here are some suggested problems from Briggs/Cochran that may help you review.
(See also the suggested problems listed below for the midterm.)

• §11.1: 1–43;
• §11.3: 13–18, 38–42;
• §11.4: 7–28;
• §11.6: 13–20;
• §11.7: 7–14;
• §12.6: 9–26 (and then perhaps 27–48);
• §12.8: 9–28, 37–44, 54–57;
• §12.9: 5–10 (and then perhaps 17–26, 38–43).
• Chapter 12 review: 30, 39–46, 50–55, 59, 66, 67, 84–96
• Chapter 13 review: 2–10, 17, 18, 42, 43, 46, 47

(Only do as many of the problems in a given group as you feel you need — doing all of them is too many!)

11/26/16

The final will be Monday 12/5/16 from 9:30–11:20 AM in Gilm 234.

• The final will be somewhat less than twice as long as the midterm
• It will cover material from the entire course, but with an emphasis on material since the midterm.
• The main new topics (roughly 55–65% of the exam) are:
  • vectors & vector functions;
  • gradient;
  • optimization;
• The old material (roughly 35–45%) is described below in the midterm announcement.
• Together, these topics correspond roughly to §11, §12, & §13 in Briggs/Cochran.
• You may bring two 3″×5″ index cards (both sides) of handwritten notes, or the equivalent.
• Other rules are as announced below for the midterm.
• Friday's lecture will be devoted to review. Come prepared to ask questions!

11/16/16

Lab writeup for Friday:

Write up your answers to today's "roller coaster" activity.

Some of the questions have more than one answer, so make sure that you clearly state your assumptions.

11/7/16

As I mentioned in class today, further information about the second derivative test can be found in §5.2.

11/6/16

Two mathematicians are talking on the telephone. Both are in the continental United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?

Give up? Some hints can be found here.

11/4/16

Further discussion of the hill activity can be found in this article (by a former MTH 255 TA who is now a math professor), as well as in this followup article.

11/1/16

Lab writeup for Wednesday (with apologies for the delay in posting this task):

Write up your answers to Monday's "hillside" activity, together with a brief discussion of your results.

Make a note of your measurements; you'll need them for Wednesday's activity.

10/29/16

Below are the answers to the midterm. Full solutions can be seen in my office.

1. (a) FALSE (b) FALSE

2. $18$

3. $0$

4. $15\pi/8$

5. $16/3$

6. (a) $3y^2~$ (b) $-3y\,\sin(3xy)~$ (c) $-3\sin(3xy)-9xy\,\cos(3xy)~$

7. (a) $\sim5^\circ/\hbox{m}$, $\sim-1^\circ/\hbox{m}$ (b) [many answers possible] (c) [many answers possible]

8. $-9\pi$

EC. $16k\pi$ grams

10/23/16

As mentioned in class, if you were confused by the wording of problem 1a on the midterm, send me a short email message to that effect.

10/23/16

Here is a selection of problems from Briggs/Cochran that may help you review. Do only as many as you feel you need to.

• §12.4: 7–39; §12.5: 7–14 & 60;
• §13.2: 9–34 & 43–50; §13.3: 19 & 23; §13.4: 25–34 (optional challenge: 35–38); §13.5: 15–22 & 30–31.

Here are two somewhat challenging integration problems:

• Determine the center of mass of a solid hemisphere of uniform (that is, constant) density.
• Evaluate $\int_0^3\int_y^3 3\,\sqrt{x^2+16} \>\>dx\,dy$.    (Answer: 61)

10/19/16

Lab writeup for Friday:

Write up a short (perhaps half a page) summary of today's chain rule activity.

Include a brief discussion of how well the two derivations of $\frac{\partial f}{\partial r}$ agree.

10/16/16

Here are some further resources that may be helpful while preparing for the midterm:

• Good practice questions can be found in the sections on multiple integrals in any (multivariable) calculus textbook.
• See the entry below (dated 9/22/16) regarding textbooks on reserve at the library.
• A rough match between sections in Briggs/Cochran and the material we've covered in class is on the schedule page.
• The review questions in the Hughes Hallett text at the end of each chapter are particularly good.

A good strategy when integrating is to always ask yourself:

• What are you adding up? (Volume? Chocolate?)
• How are you chopping? (Draw a line in the region!)
• What are the limits? (Inner limits: from one end of your line to the other; Outer limits: from the first such line to the last.)

A good problem-solving strategy is to always start by writing down what you know and what you want.

10/15/16

The midterm will be Wednesday 10/26/16 in class.

• The topics to be covered on the midterm are
  • multiple integration;
  • partial differentiation.
• The exam is closed book, and calculators may not be used.
• You may bring one 3″×5″ index card (both sides) of handwritten notes.
• Please write your exams in pencil or black ink (blue ink is OK).
• Please turn off all electronic devices, such as cell phones and alarms; this also includes personal music players.
• Monday's class (10/24/16) will be devoted to review. Come prepared to ask questions!

10/14/16

Lab writeup for Monday:

Write up your answers to the "Go" question (only).

You should explain one of your answers in some detail (a few sentences), but do not need to do so for the rest.

There is an unfortunate typo on the homework assignment for Monday: $H$ should be replaced by $T$ everywhere.

A corrected version has been uploaded to the homework page.

10/12/16

With apologies, I did not succeed when I tried to upload this week's HW assignment on Monday. It should now be visible.

10/10/16

Lab writeup for Wednesday: (followup to Friday's cone activity)

Write up one way to find the volume of a right circular cone using multiple integrals.

• You may submit more than one solution, but at least one of them must involve a double or triple integral.
• Your writeup should explain in words what you are integrating, where you are integrating, what coordinates you are using, and how you are chopping (and in what order).

Copies of the pages I showed in class today are available online:

• A summary of integration in 1, 2, and 3 dimensions;
• The pages that I passed around showing two ways of answering the last homework question using spherical coordinates;
• Full details of the "subtraction" method for solving the same problem, as outlined in class.

If you did not get the correct answer ($28\pi/15$) to the last homework question you are strongly encouraged to attempt it again in cylindrical coordinates.

Feel free to submit your work to me for confirmation.

10/7/16

Lab followup:

You do not need to write anything up from today's activity, but do be prepared on Monday to quickly set up (but not evaluate) a multiple integral for the volume of the cone.

(Sigh; the announcement below didn't get uploaded in time...)

Room change: For today only (F 10/7), we will meet in LInC 303.

(There is a College of Science event just outside our classroom, and setup will be taking place during class.)

10/6/16

You can find out more about the reasons we will use the "physics" convention for the names of the spherical coordinates in our paper:

Spherical Coordinates, Tevian Dray and Corinne A. Manogue, College Math. J. 34, 168–169 (2003)

The short answer is that most students will need to switch conventions at some point during their education, so this might as well be done sooner rather than later.

10/5/16

You can find more information about the computation of $dV$ in spherical coordinates in §1.12, of the online text.

Expressions for $dV$ in both cylindrical and spherical coordinates can be found in §1.16.

At the end of class today, I encouraged you to work out the volume of a sphere (of radius $a$, say) using a triple integral in spherical coordinates. (You should of course recognize the answer.)

As a bonus problem, can you use the methods from today's class in order to determine the surface area of a sphere?

Yes, you should know how to use trig substitutions to evaluate integrals.

The most common substitutions are $x=a\sin\theta$ to simplify $\sqrt{a^2-x^2}$, and $x=a\tan\theta$ to simplify $\sqrt{a^2+x^2}$.

You should also be comfortable with the basic trig identities:

$\sin^2\theta+\cos^2\theta=1$
$\sin2\theta=2\sin\theta\cos\theta$
$\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$

10/4/16

At the end of class yesterday, I suggested a method for evaluating $\int\limits_0^1\int\limits_y^1 e^{x^2} \,dx\,dy$.

You are encouraged to complete the computation and evaluate this integral.

10/3/16

My office hours have changed:

• I have canceled my after-class office hours on MW (for the simple reason that I don't seem to make it back to my office in time). I will however remain after class as long as it takes to answer any remaining questions.
• I have slightly shortened my before-class office hours on MW (to ensure that I get to class on time).
• I have added an hour on M morning, from 10:30–11:30 AM.
• In addition, I am usually available MW mornings from 10 AM on, but it's safest to email me beforehand.

9/28/16

Lab writeup for Friday:

Write up a short description of your group's work on today's activity.

A reasonable length for your complete writeup is a full page.
You may email me a photo of your in-class drawings, then refer to the photo in your writeup.

9/26/16

You are encouraged to work through at least one, and preferably two, ways of chopping the cylinder in today's activity.

Make sure that you obtain the corect answer for the volume!
You do not need to turn this in, but feel free to do so if you would like feedback.

I have started updating the schedule...

9/23/16

Lab writeup for Monday:

Write up a short description of your group's work on today's activity.

• See the guidelines on the homework page.
• Lab writeups are a relatively small part of your grade; don't stress out now trying to get it just right.

A reasonable length for your complete writeup is a full page.

Below are some suggestions for improving the presentation of your written work.
(It is not necessary to follow all of these suggestions all of the time.)

• Restate the problem in your own words.
  (This is not necessary if the answer includes a description of the problem, e.g. "the thermostat is located at...".)
• Use (mostly) complete sentences (with the math included as grammatically correct parts).
• Don't write a book — keep it short and sweet.
• Don't use scratch paper; use blue or black ink (or pencil).
• Don't use a multicolumn format.

The goal of your writeups should be to be able to understand them 5 years from now without any additional information.

The criteria I will use to evaluate written work can be found here.

9/22/16

Below are some resources you may find helpful.

• Online materials suitable for reviewing precalculus concepts can be found here.
• Several standard calculus textbooks are on reserve in the Valley Library, including Briggs/Cochran (the current text in MTH 254) and Hughes Hallett (the previous text).
  (If you are not logged in, you may need to return to this page and follow the link a second time...)
  You are strongly encouraged to use one or both of these books regularly as a source of practice problems. The Hughes Hallett text in particular has "Exercises", which are more-or-less routine, "Problems", which are more conceptual, and "Check Your Understanding" questions at the end of each chapter, which are True/False questions that can be surprisingly difficult. See me if you are having difficulty choosing appropriate problems.
• Both Maple and Mathematica are available in the math computer lab in the back of Kidder 108.
  Wolfram Alpha is of course also available online.

9/20/16

My official office hours have been posted on the course home page.

I am usually in my office on Monday and Wednesday mornings, typically from 9 AM until noon.
Tuesdays and Thursdays are usually not good days to find me, although there are exceptions.
Clicking on the calendar icon on the home will bring up my full weekly schedule, which is also available here.

5/19/16

Make sure you read the note about textbooks, and take a look at the grading policy.

I reserve the right to make small changes to these rules.