Wednesday, October 7, 2015

charlie gone!

Just a reminder, I'll be away for a few days, so no class Friday and no office hours Wednesday or Friday. Bang has office hours Friday morning from 9-10 AM in Ross 244.

the differential is useful, sections 14.5 and 14.6

We'll do two more chain rule problems (14.5) using the differential. One involves a change of variables from Cartesian to polar coordinates. In the other problem we will rid the planet of implicit differentiation. You are welcome World!

Next we'll use the differential to find the rate of change of a function in an arbitrary direction (14.6) in the domain of the function. Dot products will appear.

Monday, October 5, 2015

discussion, week 6

This week's discussion problems involve partial derivatives, tangent planes, linear approximations, differentials.

This week's quiz concerns level curves for functions of two variables and partial derivatives.

tangent planes continued, section 14.4

We'll finish our work on 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.

Friday, October 2, 2015

linear approximations and differentials, section 14.4

We'll practice partial differentiation today by computing some first and second derivatives.

In single variable calculus tangent line are a big deal. Anyone with a straight edge can see that there is a unique tangent line for each point on a smooth curve. The tangent line equation is wildly useful; it is used in linear approximations, in defining differentials, and as an explanation for l'Hospital's rule.

Now we are dealing with smooth surfaces in $\mathbb{R}^3$ and we've seen that tangent lines are not unique on surfaces. But there will be a unique tangent plane at each point on a smooth surface. We'll figure out how to write equations for tangent planes and use those equations to create linear approximations and differentials.

Wednesday, September 30, 2015

partial differentiation, sections 14.1 and 14.3

We'll play with the hyperbolic paraboloid described by $z=x^2+3xy+y^2$. You can plot the level curves for this surface by typing

contour plot of z=x^2+3x*y+y^2 for x=-4 to 4 and y=-4 to 4

into the search box at The level curves are hyperbolas and intersecting lines. This map shows a critical point (a saddle point) at $(0,0)$ and suggests the surface is symmetric about the plane $y=x$. 

We'll look at two traces that pass through the point $(1,2,11)$ on the surface. We'll compute the slopes on those traces, using chapter 13 methods, and see that this smooth surface in $\mathbb{R}^3$ has different slopes in different directions at $(1,2,11)$  Remember, in single variable calculus smooth curves in $\mathbb{R}^2$ had unique slopes at a given point.

We'll finish up by talking about how to compute partial derivatives of some other functions.

Tuesday, September 29, 2015

discussion, week 5

Exams will be passed back, so no quiz this week. The discussion problems cover material from 13.1, 13.2 and 14.1. The problems from 13.1 involve parametrizing closed curves with trig functions.

exam one results

If you took this exam you may view your score and current total score in WebAssign by clicking on the Grades link. Don't click on the Grades link if you will be taking the makeup exam - the view will terrify you.

We graded 223 exams over the weekend. There are four perfect scores (our long drought has ended)! The quartile scores are 69, 80, and 90. So slightly more than 50% of the group scored 80 or higher and slightly more than 25% scored 90 or higher. Last semester the quartiles were similar: 70, 80, and 90.

The median scores on the nine problems are 16/20, 15/20, 8/8, 4/8, 8/8, 8/8, 12/12, 8/8, and 8/8.

Problem 4 has the lowest score. In part a, many people failed to see that the line's tangent vector is normal to the plane. In part b, way too many people jumped to the conclusion that the equation for the y-axis is y=0. It is not; y=0 is the equation for the xz-plane. The Cartesian equations for the y-axis are x=0 and z=0; in $\mathbb{R}^3$ it takes two equations to describe a one dimensional object.

If you scored 66 or below (20.1% of the group), or you want an A in the course, give some thought to these lists of good and bad study habits: 
ten things to do and ten things not to do.

Each full size rectangle represents five brave calcunauts.

Many people wrote beautiful solutions for this exam. Here are just two examples.

Click to embiggen.

Monday, September 28, 2015

functions of several variables, 14.1

I'll start with a quick summary of sections 13.1 and 13.2 and then we'll meet some charming functions of two variables. We'll use quadric surfaces and planes to create the functions. We'll figure out what the domains of these functions look like, and we'll create contour maps (topographic maps) to help visualize the functions. Also, we'll see that, unlike curves, these functions may have a multiple slopes at a particular point in their domains. This observation leads us to partial derivatives and directional derivatives.

Friday, September 25, 2015

vector calculus, section 13.2

We'll differentiate and integrate vector functions. We'll speak forbidden words like speed, velocity and acceleration. Prepare to be shocked.

exam one solutions

Take a look at my solutions while the exam is still fresh in your mind. I corrected my typo in problem 8c, but let me know if you find any errors in the solutions. I'll make you famous.

Victor Anthony found a mistake in my solution to 4a. Of course, that one also caused an error in 4b. Both are fixed. Bravo Victor!

I fixed a typo in my solution to problem 6.

Tiantian Zheng asked a most excellent question about the angle between two planes. As the image shows, there are two angles that one could measure. One is acute, the other is obtuse, and the two are supplementary (they sum to $180^\circ$).

The angle between the red and blue normal vectors equals the acute angle between the planes. And, the angle between the red and green normal vectors equals the obtuse angle between the planes.

You might convince yourself that it works this way by rotating the tilted plane so that it becomes either perpendicular or parallel to the horizontal plane. When the planes are perpendicular the red vector is perpendicular to the blue and green vectors. When the planes are parallel the red vector is parallel to the blue and green vectors.

Thursday, September 24, 2015

exam one is history

Exam one begins this afternoon at 5:15 pm. The exam covers only chapter 12. 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. You need to know how to measure distance between points in $\mathbb{R}^3$, the sphere equation, and 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, Business Auditorium
  • Section 21, Business Auditorium
  • Section 22, History 57
  • Section 23, Geology 216
  • Section 24, Geology 216
  • Section 25, History 57

  • 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. Spaced practice and self-examination may produce better results than cramming.

    mother of all nerdfests

    I'll be in the MAC from 2:30-4:30 today. Is it true they have AC?

    Wednesday, September 23, 2015

    wolframalpha pro access

    Caleb Wilkins sent me this link to the WolframAlpha website. 

    calculus of single variable vector functions, section 13.2

    We'll parametrize (13.1) another space curve or two and then talk about how to differentiate vector functions (13.2) constructed from the parametric equations that describe those space curves. One of our goals is to compute the slope of a space curve.

    Tuesday, September 22, 2015

    discussion, week 4

    There is no quiz this week due to the exam on Thursday.

    This week's problems cover material from all of chapter 12; use the discussion to review for the exam. You are welcome to attend as many discussion sections as you like.

    Remember, Discussion section 21, which meets Tuesday from 11 - 12:15 PM, has moved to a new room: Physical Sciences 239.

    Monday, September 21, 2015

    traces and space curves, sections 12.6 and 13.1

    We'll work one more example of a quadric surface, the hyperbolic paraboloid, from 12.6. Except for spheres, please don't memorize the equations of specific surfaces.

    I do want you to know how to construct traces for a given surface in horizontal and vertical planes, and be able to identify the traces as lines, parabolas, ellipses and hyperbolas. The mock exams are a good example of the questions you might be asked on the exam.

    After working the example we'll talk about how to parametrize space curves (13.1).

    Friday, September 18, 2015

    room change!

    Discussion section 21, which meets Tuesday from 11 - 12:15 PM, is moving to a new room: Physical Sciences 239. The rumor that a significant biohazard was discovered in AG 2018 is completely false.

    quadric surfaces, section 12.6

    As you look at the images of the surfaces in section 12.6, pay attention to the curves drawn on those surfaces. These space curves are called traces, and they represent intersections between planes and the surfaces themselves. The traces help our brains interpret the images as curvy 2-dim objects living in $\mathbb{R}^3$.

    Our main objective is to describe the traces with Cartesian equations and then interpret them as lines, circles, parabolas or hyperbolas. This is how we match a particular quadric equation with its graph. This is how we visualize surfaces.

    Thursday, September 17, 2015

    do you have complete notes?

    UDSS is looking for a volunteer to take notes for this course. Students who volunteer will be eligible to earn community service hours and will be entered in several gift card drawings throughout the semester. Please contact Les Brown at 766-6189 or lbrown31 AT