Wednesday, October 29, 2014

lecture 25, triple integrals

We are skipping over 15.6, but don't worry. We'll compute surface areas in chapter 16. Over the next day or two skim through section 15.7. Pay attention to examples 2, 4, and 5. They are the bestest.

I'll finish that scintillating problem about the centroid of a pizza slice and then think about about how to compute the volume of a tetrahedron using a triple integral.

Tuesday, October 28, 2014

Midterm grades are due by noon Wednesday. I computed midterm scores from a weighted average of your homework and your best exam score. The quartiles are 73.5, 81.6, and 91.2 and I put the C/D grade boundary at 69. Nearly 87% of you have midterm scores above the cutoff.

Why did I use your best exam score, rather than your average score? Because I plan to replace your lowest exam score with your final exam score, assuming it's higher. I am betting your score on the final exam will exceed your lowest score to date. I am betting those of you with one low exam score will try to avoid a second low score.

Be careful. There is difficult material to come, and planning to do well on an exam is not the same as having a plan to do well on an exam

exam two results

We graded 169 exams over the weekend. The average score is 68.3 and the quartiles are 60.5, 70, 78.5. So slightly more than 50% of you (plural) have scores of 70 or higher. Problems 1 (T/F) and 5 (lines and curves) did the most damage. Most of you did well on the other problems. The median scores for the individual problems are:

 12/20 5/5 5/5 4/5 9/18 7/10 4/7 14/14 9/10 0/4 4/4

For comparison purposes, in the past two semesters exam 2 means were 67 (Fall 2013) and 71 (Spring 2014) so this result is not unusual. We think exam 2 scores tend to be lower because people have more exams to deal with at mid semester.

Each full-size rectangle in this histogram represents 10 brave calcunauts.

discussion, week 9

We will be returning exams in discussion sections this week, so no quiz this week. If you have questions about grading talk to Andrew, Curtis or me. As you look at my solutions remember that there is usually more than one way to solve a problem.

Several of this week's problems show that reversing the order of integration may simplify the integration. In problem 3 you get to identify the region of integration by looking at the limits of integration. On pages 3 and 4 you'll use differentials and cross products to compute the area of a region in the $xy$-plane.

Monday, October 27, 2014

lecture 24, applications of double integrals

We'll compute areas, masses, centroids and (maybe) moments of inertia by converting double integrals in Cartesian coordinates to double integrals in polar coordinates. The applications are mainly from section 15.5.

Friday, October 24, 2014

exam 2 solutions

The person (or group) finding the most missteakes in my solutions to exam 2 will win a valuable prize!

Aziz Alattar believes the differentiation in problem 3 should be performed in the indicated order. He is correct. (fixed 10/24 8:23 pm).

lecture 23, iterated integrals, change of variables

Over the weekend, browse section 15.4 which looks at double integrals in polar coordinates. There are three redlined boxes to look at.

I'll talk about the differences between type 1 and type 2 regions and the double integrals associated with those regions. Then we'll look at how trigonometric substitution can greatly simplify certain double integrals.

Thursday, October 23, 2014

exam 2 is history!

Exam two starts at 5:15 pm this afternoon. The exam covers sections 13.1, 13.2, 14.1, and 14.3-14.8.

Andrew has office hours from 10-11 in Ross 207; Curtis will be in the math lab at 2:45. My office hours are from 2:40-3:30.

Calculators and computers are not allowed at the exam. We will provide you with an equation sheet. You can find the equation sheet on the last page of each mock exam. Practice your partial differentiation.

To practice for the exam, use your WebAssign, discussion, and mock exam problems. Solutions are available for these problems. If you don't understand something ask questions at your discussion section and during our office hours.

If you bombed exam one, consider changing your strategy. Spaced practice and self-examination may produce better results than cramming.

Your exam room = f(your discussion section). You will share the room with Calculus II students. Be sure to leave an empty seat, or a Calc II student, between you and the next Calc III student.

• Section 21, CR 129
• Section 22, CR 133
• Section 24, CR 129
• Section 25, CR 133
• Wednesday, October 22, 2014

lecture 22, iterated integrals

Remembering there is an exam tomorrow, I'll start by quizzing everyone about the magical properties of the gradient vector.

I'll review the last problem from Monday. (People in the noon section were outraged that I did not provide a drawing of that volume. I spent most of yesterday taking calls from their lawyers.) Then we'll look at a new problem where the region of integration is of type 2.

For the record, type 1 regions have descriptions like $D= [a,b] \times [g_1(x),g_2(x)]$. In type 2 regions the descriptions look like $D= [h_1(y),h_2(y)] \times [a,b]$. In both cases $a$ and $b$ are numbers. In an attempt to make your heads explode, we'll look at a region that is both type 1 and type 2.

Tuesday, October 21, 2014

practice quizzes

If you have clicked on the Personal Study Plan in WebAssign you've seen the practice quizzes for each section of the text. Many of the problems are similar (or identical) to homework problems but a few cover topics we have skipped over. Ignore those problems.

Many of your homework problems have a Practice Another Version button that appears after the due date has passed.

discussion, week 8

This week's discussion problems cover critical points and maximums and minimums. Don't look at the solutions until you have tried solving the problems on your own.

Monday, October 20, 2014

lecture 21, iterated integrals

I'll remind you about the iterated integrals (15.2) we discussed on Friday and then work another example. Then we'll move on to look at iterated integrals over non rectangular regions (15.3).

Please scan the figures and examples in section 15.3.  Pay attention to the section on properties of double integrals (pages 993 and 994).

Friday, October 17, 2014

lecture 20, double integrals

We'll finish approximating our first double integral (15.1) and then compute the integral exactly (15.2).

Thursday, October 16, 2014

discussion problem solutions

Let me know if you find errors or unclear parts in these solutions.

Tuesday, October 14, 2014

lecture 19, double integrals

We rushed through the applications of partial derivatives, but you'll have a chance to work optimization problems with Curtis and Andrew next week. Now it's time to move on to chapter 15.

Please cast your gaze upon section 15.1 before class. Pay attention to examples 1 and 2 and the redlined box at the bottom of page 976.

I'll talk about how to approximate volumes in $\mathbb{R}^3$. Examples will be worked.

Monday, October 13, 2014

discussion, week 7

This week's quiz has questions about partial derivatives.

The discussion problems are centered around the magical properties of the gradient vector. They cover topics from sections 14.1, 14.6, and 14.7.

lecture 18, maximums and minimums

Skim through section 14.8 today. It discusses a very cool method for finding the maximum and minimum value of a function on a constraint curve or surface. The method involves gradient vectors.

In single variable calculus the extreme value theorem says that continuous function will have an absolute minimum and an absolute maximum on a closed interval. The max and min may occur at critical points inside the interval or at the endpoints. We'll see how the extreme value theorem plays out with functions of two variables. We'll also see why the Lagrange multiplier method (14.8) works. Examples will be worked.

Thursday, October 9, 2014

lecture 17, maximum and minimum values

Please begin skimming section 14.7. Focus on the part that deals with locating and classifying critical points.

I'll review the properties of the magical gradient vector by looking at a graph of a nonlinear function; the graph combines level curves and gradient vectors. We'll ooh and ah at the gradient vectors that are perpendicular to the level curves. Then we'll move on to locating and classifying the critical points of the function.

discussion problem solutions

Let's hope a few of these solutions are correct. Let me know if you find errors or bits that are unclear.