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## Math 10C Spring 2016 Calculus III

### Weeks 9-Finals

• 6/8/16 Practice final solutions are available here.
• 6/5/16 Just as a reminder, as I said in class, you may bring TWO 8.5x11, double-sided "cheat sheets" to the final. I highly recommend re-writing them from scratch (as opposed to simply re-using the ones you wrote for the midterms), because the process of actually making cheat sheets is the best studying you can do for this test.

• 6/5/16 We are at 80% on CAPEs. Therefore a third grading option is available: 50% final, 30% better midterm, 20% homework (in addition to the original two options.)
• 6/5/16 One more day to do CAPES. We are at 78%.
• 6/3/16 Just to remind you all, there is a practice final, an actual test from 2014. The two integral problems are more difficult than what will be on the exam and are defined on funny domains we didn't cover. So therefore, for a problem more representative of what you'll actually find on our test, replace problems 7 and 8 with:

"Evaluate the double integral $\int \int_R (x^2+xy)\,dA$, where $R$ is the square in the plane having vertices $0,0),(1,1)$, $(1,0)$, and ${\color{red}{(0,1)}}$" and "Evaluate the following integral in both orders: $\int_0^1 \int_{\color{red}0}^1 4 y \,{\color{red}{\sin(x)}}\,dx\,dy.$" [the bottom limit has been replaced by $0$, and there is no square inside the $\sin$]. Solutions will be posted on Tuesday (and also, these alternate versions).

• 5/31/16 Though I announced this in class and Piazza (please go there, especially because we know that not everyone can make office hours or even lecture/section at times), my Monday office hour this week is changed to Wednesday, 6/1 (at the same time: 3:00PM-5:00PM).
• 5/22/16 It's CAPE season!! (http://cape.ucsd.edu). Please do CAPE reviews for the class. I can see your participation level (but of course, not your actual reviews, until after the final). Since CAPE reviews are very informative and help improve my teaching, as well as affect which classes get assigned to whom, we want as many of you to do it as possible. Therefore, if 80% or more of the class participates in CAPE reviews, I'll do the following to help out:
• Your lowest homework score will be dropped.
• The last day of lecture (and quite possibly the last two days) will be a review of the most important concepts (instead of trying to wedge in more multiple integrals)... and...
• The grading option that is "the best of two" will give only 50% of the weight to the final, not 60%.
• ### Weeks 5-8

• 5/18/16 Midterm 2 seats are assigned. Same procedure as last time. If you are left-handed, let me know. But to recap: In order look up which seat to sit in, check your grades in TED. One of the "grades" will be called "MT1 Seat". This is the seat number (in Center Hall 101, where your midterm will be held—it's different from Lecture!) you will be sitting in. Check this chart to locate your seat ahead of time.
• 5/17/16  Solutions to the second practice are available here.
• 5/16/16 I will have extra office hours to help out with your midterms, on Wednesday, 3PM-5PM (right after class). Also, my normal Thursday 12-2 office hours are still on.
• 5/14/16  As you all know, Midterm 2 is on Thursday, 5/19. Same procedures as last time: it will be in Center Hall 101, 7:00PM-8:00PM. Seats will be assigned; check TED on Tuesday night. If you are a left-hander, let me know (but if you already let me know last time, your preference has been saved). Also make sure to work on Homework 8, even though its due date is after the midterm. Also, don't forget the practice midterms. Now for the topics: basically, Chapters 14 and 15 (except Sections 14.5, parts of 14.7, 14.8, and 15.3).

• Partial Derivative Basics (Sections 14.1-14.2). Basically, know how to compute partial derivatives: just the usual derivatives where you treat the other variable as a constant. In particular, much confusion in partial derivatives stems from not being able to see what is constant, and wondering why is zero sometimes and not zero at other times. To clarify, consider these two rules:
• $\frac{\partial}{\partial x} (x+c) = 1$ [the constant goes to zero and drops out, because it's added to $x$]
• but
• $\frac{\partial}{\partial x} (cx) = c$ [the constant stays around, because it's multiplying $x$].
• The Differential and Gradient (Sections 14.3-14.4). The differential $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ and gradient $\nabla f = \frac{\partial f }{\partial x} \vec{i} +\frac{\partial f }{\partial x} \vec{j}$ both carry exactly the same information (namely, bundling the two partials together to form a new entity with superpowers), but are used in different contexts. The differential is for calculating small changes in $z$ given small changes in $x$ and $y$.

• Find directional derivatives: partials in any direction you want, not just in $\vec i$ and $\vec j$. This is why it has superpowers: from just derivatives in two directions, it is enough to find them in all directions.
• Find the direction of the maximum rate of change, as well as the maximum rate of change itself: the max rate of change is the magnitude $\|\nabla f\|$, and the direction is the unit vector $\nabla f/\|\nabla f\|$.
• The Chain Rule. (Section 14.6) In particular, remember those diagrams: multiply along each pathway to the variable you want to find the partial with respect to, and add up the results of all the paths.
• Second Partial Derivatives (Section 14.7). Partials are so fun that we can do them over and over again. Remember that the mixed $f_{xy}$ is the same as $f_{yx}$. [Skip the thing about Taylor polynomials]
• Classification of Critical Points (Section 15.1). This is undoubtedly a trickier section, because some of those tests are hard to remember offhand. Make sure to write this on your cheat sheet:
• Critical point is where $f_x = f_y = 0$ or $\nabla f = \vec 0$.
• Take $D = f_{xx}f_{yy} -f_{xy}^2$ at each critical point, called the discriminant.
• If $f_{xx} > 0$ and $D >0$, it is a minimum
• If $f_{xx} < 0$ and $D>0$, it is a maximum
• If $D < 0$, it is a saddle point (and guaranteed to neither be min nor max).
• If $D = 0$, scream bloody murder; it could be anything. Barring a major boneheaded mistake on my part, I will not put an example with $D=0$ on the test.
• Optimization problems (Section 15.2). This is basically the word-problem and applications version of Section 15.1. Be able to translate whatever "cost" function into a formula. I will try to minimize nasty algebra with this one, and will give a better example on Monday.

The following topics will not be covered:

• Anything explicitly from Midterm 1. By explicit, I mean, there will be no question directly asking you "Find the dot product of ...". However, if you need to use the dot product (say, the directional derivative of something, $\nabla f \cdot \vec u$), that's fair game.
• Gradients and directional derivatives for functions of 3 or more variables (Section 14.5).
• The Physical Chemistry example in 14.6 (the relevant topic is partial derivatives with respect to whole different sets of variables).
• Taylor polynomials (part of Section 14.7)
• General theory of differentiability (Section 14.8)
• Any critical points for which $D=0$.
• Minimization/Maximization with constraints/Lagrange Multipliers (Section 15.3, even though we'll start talking about that on Monday and Wednesday).
• 4/27/16  Midterm 1 has been graded; please go to your sections or office hours to pick them up. The statistics were:  Mean 19.90 (79.60%) Median 20 (80%) Standard Deviation 3.61 (14.44%) Min 8 (32%) Max 25 (100%)
That average is right on the mark, so there does not need to be a curve. This is the distribution of scores:
• 4/25/16  Homework Assignments 6-8 have been released. In addition, the due dates of HW4-HW8 have been changed to Thursdays at 11:59pm instead of Wednesdays at 11:59 pm.

• ### Weeks 1-4

• 4/20/16  Midterm 1 Seats have been assigned. In order look up which seat to sit in, check your grades in TED. One of the "grades" will be called "MT1 Seat". This is the seat number (in Center Hall 101, where your midterm will be held—it's different from Lecture!) you will be sitting in (see the screenshot below; the seat assignment has been underlined):

The left-handed seats have been left unassigned, so if you are in fact left-handed, let me know (your preference, of course, will be recorded for next time).

Check the Center Hall seating chart to see in advance where to sit.

You must sit in your assigned seat (unless there is a problem, in which case, of course, we will reseat you) and we will check. We also reserve the right to reseat anyone as well.

Good luck; and remember I have office hours tomorrow at 12; come if you have last-minute questions.

• 4/19/16  Solutions to the first practice are available here.
• 4/18/16  As you all know, there is a midterm on Thursday 4/21 at 7:00-8:00pm in Center Hall 101. (Note, critically, it will NOT be in lecture!!) Here's what you will need to bring:
• A blue book
• An 8.5" x 11" handwritten cheat sheet (double-sided). Don't photocopy; writing your cheat sheet is the best damn studying you will ever do; I guarantee it.

Other notes:

• No calculators are allowed (or needed)
• No other electronic devices--turn those phones off!
• Seats will be assigned. We will check. Look on TED the night before (or email me, if you are taking the course through Extension and cannot access TED). If you are left-handed, please let me know; it will affect your seat assignment.

Topics (Reading: Sections 12.1-12.4 and 13.1-13.3 and the first part of 13.4)

• Basic geometry of 3D space
• Basic graphs of functions z=f(x,y) [NO drawing, only matching!]
• Contour plots [NO drawing, only matching and/or finding their equations]
• Slicing (know what they are and contour plots as special case)
• Linear functions: how to recognize when something is linear.
• Vector basics, displacements
• Basic vector operations (addition, subtraction, and scalar multiplication)
• Magnitude and direction of vectors
• Resolving vectors into components
• Velocity vectors
• Dot products
• Very basics of cross products: their definition as directed area and the formula defining it.

These topics will NOT be covered:

• Actual drawing of any 3D graphs
• Actual drawing of contour plots
• Finding equations of surfaces for anything more complicated than planes
• Dealing with angles between vectors that are anything other than standard unit circle special angles from trig (ONLY: 30, 45, 60, 90, 120, 135, 150, or 180-degree angles can show up. HINT: write sines and cosines of this on your cheat sheet) [Without a calculator, you can't compute sines and cosines or find arc cosine of anything more complicated than this]
• More complex applications of cross products
• Anything from Chapter 14
• 4/18/16  An-Vy's Review Session is tonight in Ledden Auditorium, 6:00pm-7:30pm. Jay's review session is tomorrow in the same place, same time.
• 4/17/16  Just as a reminder, Practice Midterm Exams are available. Solutions to the first practice are available here.
• 4/16/16  One of your TAs, Jay Bonthius, is holding a Review Session for Midterm 1 on Tuesday at 6pm in Ledden Auditorium. You do not have to be enrolled in Jay's sections to attend. More details about the Midterm will be posted this weekend. In addition, one of the TAs (An-Vy) for the other 10C Lecture will be holding one at the same time 6pm but on Monday, in HSS 1330 (capacity there is more limited, so go to Jay's if you can) (Updated; see new announcement above).
• 4/2/16  The due date for the Syllabus Quiz has been extended to Wednesday, April 6 at 11:59 PM to accommodate late enrollers.
• 3/30/16  In case anyone started Homework 1 (NOT the Syllabus Quiz), note that has been changed to a multiple choice version that is more consistent with how you will be tested on graphing problems (which we'll talk about in class today) so you will have to start it over.
• 3/30/16  Not that it matters much, since you haven't even had any discussion sections yet, but Section B08 has a new TA, Hanyi Wang. Anyone who has written down the contact information for the previous TA should update their info accordingly.
• 3/30/16  Homework Assigments 1-3 (up to the Midterm 1) have been released on WileyPLUS. The due dates are as indicated on the calendar at 11:59 PM on that day.
• 3/29/16  The Course Calendar is up, as a rough guide to your study planning. It may vary as the class progresses, so bookmark it and check often.
• 3/29/16  No discussion sections today (the first Tuesday of the quarter).
• 3/25/16  Initial webpage is up! Not everything is here yet!

• Course Syllabus (printable version here)  Updated 3/26/16  You are responsible for knowing the information and policies in the syllabus.
• Course Calendar  Updated 5/1/16  Important dates for the course in a convenient calendar format [not up yet]
• Homework (textbook assignments - these are additional practice and are not graded.)  Updated 3/25/16
• WileyPLUS (online assignments - graded)  Updated 3/27/16
• Piazza  Online discussion forum for Math 10C where you can ask anything (even anonymously) and get lecture notes and clarification.