Up to now all our double integrals have been volumes. Today we'll use double integrals to compute some new things, including the mass, and the first and second moments of a mass distribution that lies in the $xy$plane. Polar coordinates will be used.
math 2210
what one fool can do, another can.
Friday, March 27, 2015
Wednesday, March 25, 2015
lecture 23, polar coordinates
If the region of integration is a sector of a circle it's possible that the region is neither type 1 nor type 2. One option is to subdivide the region into two type 1 (or type 2) regions; a better option is to make an inverse substitution using the polar coordinate system (15.4). The Jacobian determinant will make an appearance.
Super hires WyoCast video.
Super hires WyoCast video.
Sunday, March 22, 2015
lecture 22, nonrectangular regions
We'll set up limits of integration for triangular and circular regions in $\mathbb{R}^2$ (15.3) and explore the mysteries of type I and type II regions. We'll do an example where reversing the order of integration converts the integral from a form where it has to be approximated to a form that can be solved by substitution.
discussion, week 8
Exams will be handed back, so no quiz this week. The discussion problems concern properties of double integrals (pages 979 and 981), identifying regions of integration (15.3), and reversing the order of integration (15.3) in order to evaluate a double integral.
Wednesday, March 18, 2015
curves, surfaces, and derivatives
In chapters 12 and 13 we described curves in $\mathbb{R}^3$ with vector functions of the form $\vec{r}(t)$. These vector functions take a single independent variable because curves are one dimensional. The derivative of the vector function is tangent to, or parallel to, the space curve.
In chapters 12 and 14 we described surfaces in $\mathbb{R}^3$ with single Cartesian equations. These equations depend on some combination of $x$, $y$, and $z$. If the equation of the surface is expressed as $g(x,y,z)=0$ (or any constant) then the gradient of the function $g(x,y,z)$ is normal to, or perpendicular to, the surface.
Tangent and normal vectors will reappear in the final weeks of the semester when we integrate the tangential component of some vector field along a curve or integrate the normal component of another vector field over a surface.
In chapters 12 and 14 we described surfaces in $\mathbb{R}^3$ with single Cartesian equations. These equations depend on some combination of $x$, $y$, and $z$. If the equation of the surface is expressed as $g(x,y,z)=0$ (or any constant) then the gradient of the function $g(x,y,z)$ is normal to, or perpendicular to, the surface.
Tangent and normal vectors will reappear in the final weeks of the semester when we integrate the tangential component of some vector field along a curve or integrate the normal component of another vector field over a surface.
Tuesday, March 17, 2015
exam two results
We graded 178 exams this time. The average is 76.6 74.6 and the quartiles are 66, 77.5 and 85. So slightly more than 50% of you scored 77.5 or higher and slightly more than 25% scored 85 or higher. Last semester the average score was 68.3 and the quartiles were 60.5, 70, and 78.5, but that exam included several optimization problems.
There is a big increase, from the first exam, in the number of people scoring 67 or below. If your score is not as high as you would like, give some thought to these lists of good and bad study habits: ten things to do and ten things not to do.
In problem 5 the median score would have been higher if more people had checked to see if their parametrization was consistent with the surface equations. And on problem 11, too many people were unable to make the connection between the tangent plane equation and the differential.
There is a big increase, from the first exam, in the number of people scoring 67 or below. If your score is not as high as you would like, give some thought to these lists of good and bad study habits: ten things to do and ten things not to do.
The median scores on the individual problems are:
14/20

4/5

5/5

5/5

3/6

5/6

6/6

8/10

3.5/5

14/16

7/8

4/8

In problem 5 the median score would have been higher if more people had checked to see if their parametrization was consistent with the surface equations. And on problem 11, too many people were unable to make the connection between the tangent plane equation and the differential.
Each full size rectangle represents 5 calcunauts. 
Each full size rectangle represents 4 calcunauts. 
Friday, March 13, 2015
lecture 21, double integrals
We'll take a stab at approximating the volume located beneath a surface and above a rectangle in the $xy$plane (15.1). After having way too much fun with the approximation, we'll compute the volume exactly using an iterated integral (15.2). I'll show a second example where the iterated integral melts into a product of two integrals.
Highdef video from WyoCast.
Highdef video from WyoCast.
Thursday, March 12, 2015
exam two solutions
(9:42 PM) Instead of kicking back after the exam, David Tobin, Brendan Taedter and Mark Fenn decided to fix my lousy solutions to problems 4, 6, and 10. I'll update the solutions as they find more mistakes.
Take a look at my solutions while the exam is still fresh in your mind. As usual, let me know if you find any errors or unclear bits. I'll make you famous.
I fixed two minor typos and added some explanation to problem 11 at 8:40 pm.
Take a look at my solutions while the exam is still fresh in your mind. As usual, let me know if you find any errors or unclear bits. I'll make you famous.
I fixed two minor typos and added some explanation to problem 11 at 8:40 pm.
exam two is history!
Exam two starts this afternoon, Thursday, March 12, at 5:15 pm. The exam covers sections 13.1, 13.2, 14.1, and 14.314.7. No electronic devices are 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. Some facts are not on the equation sheet. Among other things, you need to know the definitions of the dot and cross products.
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
We will provide you with an equation sheet. You can find the equation sheet on the last page of each mock exam. Some facts are not on the equation sheet. Among other things, you need to know the definitions of the dot and cross products.
Your exam room is a function of your discussion section. If possible, leave an empty seat between you and the next calcunaut.
To practice for the exam, use your WebAssign, discussion, and mock exam problems. Solutions are available for these problems. The mock exams include questions from section 14.8 but section 14.8 is not covered on this exam.
If you don't understand something ask questions at your discussion section and during our office hours. Spaced practice and selfexamination may produce better results than cramming.
If you don't understand something ask questions at your discussion section and during our office hours. Spaced practice and selfexamination may produce better results than cramming.
Wednesday, March 11, 2015
lecture 20, this and that
I'll work another Lagrange multiplier problem (14.8) and then estimate the volume that lies under a surface in $\mathbb{R}^3$ (15.1). On Friday we'll compute the volume exactly with an iterated integral.
More lowres video from WyoCast.
More lowres video from WyoCast.
Monday, March 9, 2015
discussion, week 7
No quiz this week due to the exam on Thursday. The problems cover material from sections 14.6 and 14.7.
Sunday, March 8, 2015
lecture 19, lagrange multiplier method
We'll determine absolute minimum and maximum values of functions of two and three variables (14.8). You might remember that a continuous function $f(x)$ is guaranteed to have both a maximum and a minimum value if we restrict the function to a closed interval (the extreme value theorem). Think about what the equivalent of a closed interval in $\mathbb{R}$ might look like in $\mathbb{R}^2$, and $\mathbb{R}^3$.
The audio worked today!
The audio worked today!
Friday, March 6, 2015
lecture 18, critical points
I'll review the directional derivative and enumerate the properties of the gradient vector. Then we'll explore for critical points for functions of two variables (14.7). We'll use the gradient vector as a kind of first derivative test to classify critical points as local maximums, local minimums, or saddle points. We'll also talk about how to use the second derivative test to classify critical points.
Wednesday, March 4, 2015
lecture 17, the directional derivative
We'll use differentials and the chain rule to assemble the directional derivative (14.6). Then we'll calculate some rates of change and investigate the properties of the gradient vector.
Tuesday, March 3, 2015
discussion, week 6
This week's discussion problems involve partial derivatives, tangent planes, linear approximations, differentials and the chain rule.
The quiz has questions about space curves and the surface they lie on (13.1), the geometric relationship between $\vec{r}(t)$ and its derivative (13.2), and about the domains of functions of two variables (14.1).
The quiz has questions about space curves and the surface they lie on (13.1), the geometric relationship between $\vec{r}(t)$ and its derivative (13.2), and about the domains of functions of two variables (14.1).
Monday, March 2, 2015
20 dollars!
Dr. Meredith Minnear (mminear2@uwyo.edu), from the Psychology Department, has a 20 dollar bill (tax free!) with your name on it. To get the money you'll have to spend an hour answering questions that test your spatial reasoning ability. The minimum wage in Wyoming is 7.25 dollars/hour.
Sunday, March 1, 2015
lecture 16, the chain rule
We'll start by looking at the similarity of tangent planes, linear approximations and differentials. Then we'll turn to the multivariable version of the chain rule (14.5).
This version of the chain rule is more complicated and more powerful than the single variable version. We'll work examples that involving a space curve, a change of variables from Cartesian to polar coordinates, and a peculiar case where the variables obey a constraint equation.
A team of engineering students could, as a senior project, vastly improve the WyoCast system. Start with an infrared light source that could be placed on the board, that the camera would pan toward and focus on.
A team of engineering students could, as a senior project, vastly improve the WyoCast system. Start with an infrared light source that could be placed on the board, that the camera would pan toward and focus on.
Friday, February 27, 2015
no office hours this afternoon
I've been fighting an infection all week and the antibiotics are not working as fast as I hoped. So I'll be snoozing on the couch this afternoon and watching my email in case you have questions.
lecture 15, differentials
We'll practice partial differentiation today by working on tangent plane, linear approximation, and differential examples.
Wednesday, February 25, 2015
lecture 14, partial derivatives and tangent planes
I'll do one more tangent line problem and then show that our slope measurements can be made more easily using partial differentiation (14.3). If there is time, we'll use the tangent lines to create a tangent plane to the surface.
More lowres video.
More lowres video.
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