Friday, October 24, 2014

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 simply 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 20, Business Auditorium
  • Section 21, CR 129
  • Section 22, CR 133
  • Section 23, Business Auditorium
  • 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

    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.

    constrained optimization

    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.

    Tuesday, October 7, 2014

    lecture 16, directional derivatives

    Please breeze through section 14.6 before class on Wednesday. The theorem in box 15, and the final section, about the significance of the gradient vector, will take your breath away.

    We'll develop the directional derivative and work several examples. We'll also investigate the magical properties of the magical gradient vector.

    Monday, October 6, 2014

    discussion, week 6

    This week's quiz has questions from sections 13.1, 13.2 and 14.1.

    Like last week, the discussion problems are all over the place. They cover topics from sections 13.2, 14.3, 14.4, and 14.5. 

    Sunday, October 5, 2014

    lecture 15, chain rule

    Please skim section 14.5 before class. Pay attention to the examples and the content of the red-lined boxes. Look at a few problems at the end of the section.

    The multivariable version of the chain rule is more complicated and more powerful than the single variable version. We'll do three examples that involve a space curve, a change of variables from Cartesian to polar coordinates, and a peculiar case where the variables obey a constraint equation.

    Saturday, October 4, 2014

    the equality of mixed partials

    Clairaut's theorem (because I stated it wrong in class): Suppose $f(x,y)$ is defined on an open subset $D$ of $\mathbb{R}^2$. If the second-order mixed partial derivatives $f_{xy}$ and $f_{yx}$ exist and are continuous on $D$ then $f_{xy}=f_{yx}$ at every point in $D$.

    Thursday, October 2, 2014

    lecture 14, tangent planes

    We'll go long in tangent planes, linear approximations, and differentials (section 14.4). Examples will be worked.

    discussion problem solutions

    Here are my solutions to the discussion problems for week 5 and week 4. Let me know if you find mistakes or parts that are unclear.