Friday, March 24, 2017

lecture 23: double integrals on non-rectangular regions

I'll tie up any loose ends from section 13.1 and then think about integration over non-rectangular regions in the plane (13.2). We'll work an example where the order of integration becomes important.

Thursday, March 23, 2017

exam 2 solutions

While the exam is still fresh in your mind, take a look at my solutions. Please let me know if you find any bogus math or fuzzy explanations. As Bang says, "My head hurts when I look at your answers."

effective learning techniques

An engineering professor answers the question "Which is the most effective learning technique you have experienced so far?"

... I didn’t just do a math homework problem and turn it in. Instead, particularly if it was an important homework problem, I would work it and rework it fresh, spacing the practice out over several days. I wouldn’t peek at the answer unless I absolutely had to. That ensured I really could solve the problem myself—that I wasn’t just fooling myself that I knew it. After I was comfortable that I could really solve the problem by myself on paper, I then “went mental,” practicing the steps in my mind until the solution could flow like a sort of mental song. I could perform this kind of mental practice at times people often don’t think to use for studying—like in the shower, or when I was walking to class. I found that this attention to chunking eventually gave me sort of magic powers—I could glance at many problems, even ones I’d never seen before, and know virtually instantly how to solve them.

exam two approaches!

Exam two begins this afternoon (Thursday, March 22) at 5:15 pm. This exam covers section 11.6, and all of chapter 12 (except for 12.3)No electronic devices are allowed at the exam.

We will provide you with this equation sheet. Be careful, some facts are not on the equation sheet. You need to know how to compute the dot and cross products, how to integrate and differentiate, and understand the properties of the gradient vector. You should also know the cosine and sine of common angles like $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, and $2\pi$ radians.

This exam is your opportunity to demonstrate to us that you understand the material. Be sure to read each question carefully, and draw sketches where appropriate. We expect complete solutions and correct notation. Be careful with the T/F questions; think, don't react.

Your exam room is a function of the first four letters of your last name.

  • Aaaa through Hanc, go to CR 302
  • Hans through Pont, go to CR 306
  • Post through Zzzz, go to CR 310

  • We are sharing the rooms with Calculus I and II students. Make sure you are not sitting next to another Calculus III student.

    To practice for the exam, use the problems from MyMathLab, discussion, and the mock exams and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

    Wednesday, March 22, 2017

    uncertain office hours for march 22

    I have an appointment this afternoon that interferes with my office hours. I'll try to get back as soon as possible.

    lecture 22: volumes by double integrals

    We'll use double integrals to calculate the volume that lies above a rectangle (13.1) in the $xy$-plane and beneath a surface $z=f(x,y)$. If there is time, we'll also create a double integral that gives the volume above a triangle (13.2) and below the surface.

    Monday, March 20, 2017

    lecture 21: lagrange multiplier method refresher

    I'll work two or three examples where we look for the extreme values of functions on curves or surfaces.

    Friday, March 10, 2017

    lecture 20: lagrange multiplier method

    In calc I a function that is continuous on a closed interval is guaranteed to have an absolute maximum and an absolute minimum value on the interval. We'll chat about analogs to closed intervals in $\mathbb{R}^2$ and $\mathbb{R}^3$. And, we'll solve some problems (12.9) using gradient vectors to find the extreme value(s) of various functions on sets of points in both $\mathbb{R}^2$ and $\mathbb{R}^3$. A tiny amount calculus will occur.

    Wednesday, March 8, 2017

    lecture 19: critical points and the second derivatives test

    We'll use the gradient vector to identify and classify critical points for a function of two variables: $f(x,y)=x^3+y^3-3xy$. Once we know how the function works we'll use algebra to make sure we located all the critical points, and the second derivatives test to check our interpretations of those critical points (12.8).

    Tuesday, March 7, 2017

    the big picture

    In chapter 11 we learned to describe lines and curves in $\mathbb{R}^3$ using vector functions of the sort $\vec{r}(t)$. These vector functions have one independent variable, $t$, because curves are one dimensional. The derivative of the vector function is tangent to, or parallel to, the space curve.

    In chapter 12 we describe surfaces in $\mathbb{R}^3$ with single Cartesian equations that depended 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.

    Both tangent and normal vectors are used in the final weeks of the semester, when we integrate the tangential component of some vector field along a curve, or the normal component of another vector field over a surface.

    Monday, March 6, 2017

    lecture 18: tangent planes and differentials

    We'll use the gradient vector to attach tangent planes to surfaces that are described by implicit or explicit equations. The tangent plane lies very close to the surface at points near the point of attachment, so the tangent plane equation can be rearranged to give a linear approximation and a total differential. We'll work three examples.

    Friday, March 3, 2017

    lecture 17: properties of the gradient vector

    We'll finish the example problem from Wednesday and then look at the properties of the gradient vector (12.6). They are:
    1. The range of the directional derivative is $-| \, \vec{\nabla}f \,| \le D_{\hat{u}}f \le | \, \vec{\nabla}f \,|$.
    2.  $\vec{\nabla}f$ is the direction in which $f$ increases most rapidly, aka the direction of maximum increase.
    3. $-\vec{\nabla}f$ is the direction in which $f$ decreases most rapidly, aka the direction of maximum decrease.
    4. $\vec{\nabla}f$ is perpendicular to level curves of $f(x,y)$ in $\mathbb{R}^2$ or level surfaces of $f(x,y,z)$ in $\mathbb{R}^3$.
    The fourth property gives us a spiffy way to create tangent planes to surfaces. And, tangent planes are a gateway to linear approximations.

    Wednesday, March 1, 2017

    lecture 16: chain rule and directional derivatives

    I'll work two more chain rule examples. In one case, we'll rid the world of the scourge of implicit differentiation (12.5).

    Then we'll find the rate of change of a function in an arbitrary direction in the function's domain. Dot products will appear as will an amazing vector, the gradient vector, that is constructed from the first derivatives of the function (12.6).

    Monday, February 27, 2017

    lecture 15: partial derivatives, and the chain rule

    We'll talk more about partial derivatives (12.4) today; we'll look at implicit differentiation and we'll compute some second and third derivatives. If there is time, we'll look at the multivariable chain rule (12.5). It's a nice excuse to compute partial derivatives and, maybe, look at a total derivative.

    Friday, February 24, 2017

    lecture 14: partial derivatives

    We'll talk more about traces and level curves and compute slopes on surfaces in various directions. And then we'll talk about partial derivatives (12.4).

    Wednesday, February 22, 2017

    lecture 13: functions of two variables

    We'll use planes and quadric surfaces to create functions of two variables (12.2). We'll examine the domains of these functions, construct level curve maps, and compute slopes on the surfaces. We may even compute some partial derivatives (12.4).

    Monday, February 20, 2017

    exam one results (updated, again)

    Last Thursday, 153 brave calcunauts took the first exam; four others took a make-up exam on March 2. The average is now 77.7 and the quartile scores are 70, 79, and 87. So, slightly more than 50% of you have scores of 79 or higher.

    If you are unhappy with your score, talk to your discussion leader, or me. Make an appointment if you can't attend our office hours.

    lecture 12: quadric surfaces

    We'll review planes (12.1) by working several examples. Then we'll turn to the more interesting quadric surfaces. We've played with several of these, including spheres, circular cylinders and cones.

    As you read through section 12.1 look at the images of the surfaces. 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 those traces as lines, circles, parabolas or hyperbolas. This is how we match a particular quadric equation with its graph. This is how we visualize surfaces.

    Friday, February 17, 2017

    lecture 11: planes and surfaces

    We'll review the main points about differentiating and integrating vector functions by working a few examples (11.6, 11.7).

    But then we have to jump into chapter 12. Before long you will be finding partial derivatives and working with the multivariable version of the chain rule. We'll start by talking about planes (12.1). I'll show how to assemble a plane equation from a point and a normal vector. Some of our examples will involve cross products and lines.