Monday, April 20, 2015

exam three results

We graded only 166 exams this time but five brave calcunauts will take a makeup exam next week.

The average score is 75 and the quartiles are 68, 78, and 85.25; slightly more than 82 people have scores of 78 or higher and slightly more than 41 people have scores of 85.25 or higher. For comparison, last semester the average was 77.4 and the quartiles were 68.75, 78, and 88.

The median scores on the individual problems are:

16/20
10/10
7/10
10/10
3/6
3/8
5/8
12/12
6/8
7/8

Scores would have been higher if more people had sketched the regions of integration. The hardest problems for most people (5, 6, and 7) required some amount of algebra. If you had problems with integration or partial differentiation, practice both before the next exam. And, if you had problems writing limits of integration in polar or spherical coordinates, practice both before the next exam.

Each full size rectangle represents three brave calcunauts.
Each full size rectangle represents five brave calcunauts.
Each full size rectangle represents four brave calcunauts.

discussion

Exams will be handed back so there is no quiz. The discussion problems cover material from sections 16.1 through 16.3.

Sunday, April 19, 2015

lecture 33, fundamental theorem

In single variable calculus the fundamental theorem of calculus makes computing integrals almost as easy as differentiation. Today we'll construct a fundamental theorem for line integrals and take it for a test drive. This version of the theorem works only for conservative force fields, fields that are derived from potential functions: $\vec{F}=\vec{\nabla}f$. We'll figure out how to tell if a force field is conservative and how to compute it's potential function if it is conservative.

There is a version of the fundamental theorem for every region of integration in the table.

Friday, April 17, 2015

lecture 32, line integrals and vector fields

We'll see that the two types of line integrals are, in fact, the same (16.2). Even so, for convenience, we will continue to compute them in slightly different ways. We'll talk also about how to identify the graphs of vector functions of two variables (16.1) using two simple examples from the book.  I'll introduce a third field and a simple path and show you how to determine qualitatively if the line integral is positive, zero, or negative. We'll then compute the line integral, getting a value that agrees with the qualitative result.

All afternoon classes are canceled. Be safe; stay off the roads.

This video from the MIT series shows more examples of how to compute line integrals using geometric reasoning.

Thursday, April 16, 2015

a proposal

Exam four, less than four weeks away now, covers all of chapter 16. The extensions of the fundamental theorem, which we will start on next week, are the heart of the course. Because I want you to concentrate on this material (and the line and surface integrals of chapter 16) as much as you can, I have a proposal for you. If your score on exam four is higher than at least one of your first three exams scores then I'll drop the lowest of your first three exam scores and double the weight of your four exam score.

I tried this last semester and it had a great effect; a number of people were able to raise their grades, the pass rate increased, and people who might have skipped the exam in other semesters stayed with the course to the end. 

exam three solutions

Brent Pearce has eyes like an eagle; he noticed the coordinates of the critical points in problem 5 were in the wrong order. Fixed at 9:30 am, April 17.

Here are my solutions to the exam. Please tell me if you find a mistake or an unclear explanation.

exam 3 is history!

Exam three started at 5:15 pm this afternoon. The exam covers sections 14.8, 15.1-15.5 and 15.7-15.10.

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 integration and partial differentiation.

Section 14.8 covers the Lagrange multiplier method; example problems are in the mock exams for exam 2.

Your exam room is a function of your discussion section. If possible, leave an empty seat between you and the next calcunaut.

  • Section 20, Berry Center 138
  • Section 21, Berry Center 138
  • Section 22, Classroom Bldg 214
  • Section 23, Geology (old part) 216
  • Section 24, Classroom Bldg 214
  • Section 25, Geology (old part) 216

  • 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.

    The Union of Concerned Cognitive Scientists want you to remember that spaced practice and self-examination may produce better results than cramming.

    Wednesday, April 15, 2015

    lecture 31, line integrals

    On Monday we used the arc length element $ds=\sqrt{(dx)^2+(dy)^2+(dz)^2}$ and three parametric equations to compute physical properties of wires (16.2, one dimensional mass distributions). Today we will use the differential of position, $d\vec{r}=\langle dx,dy,dz \rangle$, to compute the work that a force does on a mass as the mass moves through space (16.2).

    Monday, April 13, 2015

    discussion, week 11

    There is no quiz this week due to the exam on Thursday afternoon. The discussion problems cover material from sections 15.8 to 15.10. Don't look at the solutions until you've tried the problems.

    lecture 30, line integrals

    After two semesters of single variable calculus and four weeks of double and triple integrals you are an expert at integrating over one, two, and three dimensional regions like the ones shown in the table. These regions are flat in the sense that they are just walled off portions of $\mathbb{R}$, $\mathbb{R}^2$, or $\mathbb{R}^3$.

    This week we will fill out the top row of the table by integrating over one-dimensional curves that live in $\mathbb{R}^2$ or $\mathbb{R}^3$. We'll be going back and forth between sections 16.2 and 16.1.
    In each region of integration the red points are inside the region and the black points are on the boundary. If we peeled the sphere away we would see a red ball underneath. 

    Thursday, April 9, 2015

    lecture 29, inverse substitution

    We'll solve several double integrals with complex regions of integration by applying inverse substitutions (15.10). If you are reading along, the early part of section 15.10 is needlessly scary; a safe place to start is the redlined box on page 1044. This video shows another example of an inverse substitution.

    Wednesday, April 8, 2015

    not everything is complicated

    Mechanical engineers and mathematicians join forces to study wrinkles (Quanta Magazine).

    lecture 28, triple integrals

    This is day five of triple integrals! I'll review the spherical coordinate system and explain how we can find limits for $\rho$ and $\phi$ by plotting cross sections in the $rz$ half plane. Along the way we'll compute the center of mass for a hemisphere of uniform density and the moment of inertia for a ball of uniform density that is tangent to the origin.

    I was in a rush to finish, so here is a solution of the latter problem.

    Tuesday, April 7, 2015

    constrained optimization

    Here are four ways to find the extreme values of $f(x,y)=4x^2+y^2$ on the unit circle $x^2+y^2=1$. The unit circle is a closed and bounded set of points in $\mathbb{R}^2$. The closed aspect should be clear; saying that the set is bounded means every point on the circle is a finite distance from the origin.

    This video from the MIT series shows how to solve a geometry problem by the Lagrange multiplier method.


    Monday, April 6, 2015

    lecture 27, triple integrals

    We'll work several examples where we write the integrals in cylindrical coordinates (15.8). Then we'll move on to spherical coordinates (15.9).

    Sunday, April 5, 2015

    discussion, week 10

    The quiz includes two questions about partial differentiation and one about regions of integration in the $xy$-plane.

    This week's problems are varied, but most concern triple integrals in cylindrical coordinates.

    Thursday, April 2, 2015

    lecture 26, triple integrals

    We'll tie up some loose ends in the last example from Wednesday (15.7).

    The solid regions you'll integrate over in engineering and physics courses are often symmetric with respect to one (cylinders and cones) or more (spheres) of the coordinate axes. In these cases it may make sense to make a change of variables to cylindrical (15.8) or spherical coordinates (15.9). We'll explore cylindrical coordinates first.

    Wednesday, April 1, 2015

    lecture 25, triple integrals

    We'll look at several, slightly, complicated regions of integration today, ones where we can use our projection and parametrization skills. We are sticking with Cartesian coordinates (15.7).

    Monday, March 30, 2015

    lecture 24, triple integrals

    We are skipping section 15.6 but we will compute a few surface areas in the second half of chapter 16 after figuring out how to parametrize surfaces.

    Today we'll start on triple integrals in Cartesian or rectangular coordinates (15.7). The integrals we set up will be compute the volume or moment of inertia of various solid regions.