Day 29 in MAT227

Last time: More Section 12.2

Next time: Review

Today:

Announcements:
- It's time for course evaluations: those are to be done on-line. Please visit the site http://eval.nku.edu/ to do so.
- Your (late) test two corrections are due today. You can hand in your test one corrections up to the final -- I didn't put a deadline on those.
- Assignment 10.1 returned (Law of Cooling).
  - For your consideration, Part I
    1. Works fine!
    2. It is a rate constant: k tells how fast the exponential grows or dies.
    3. 1/time. Just do the "unit math"
    4. The metal pot will equilibrate faster -- hence, it has the larger k
  - For your consideration, Part II
    1. Say "asymptotes to the ambient temperature, 5 degrees C"
    2. Say "asymptotes to the 0", steadily decreasing.
    3. Plug and chug.
    4. half life: yes, half way to the goal post (5 degrees C). "half life" is characteristic of every exponential problem.
    5. If not well stirred, you have a non-homogenous fluid, and you have to use PDEs (partial differential equations)
    6. The ambient temperature would also be changing (in fact, you'd have (at least) three different temperatures to consider: that of the room, that of the water, that of the soup....
- Assignment 12.1 due (remember, revised due dates....)
  This is the last homework assignment you are to turn in. For 12.2 and 12.3, I've assigned a few of the simpler problems. I'll expect you to do something similar to those for the exam: pp. 620-621, #2, 3, 12, 17, 23; pp. 627-630, #4, 6, 8, 9, 10, 13, 14, 22.
  You might want to take a look at those before next time, so that you can ask any questions. These types of problems are likely targets for final exam questions!
- Next time we'll be reviewing for the final. If you'd like, I'll come in for a review session the afternoon before the final (Is there interest?)
Let's spend a little more time on the lab from last time: Planetary Orbits (ellipses and more!).
Section 12.3: Polar Coordinates
- We're going to be using a different coordinate system to describe points in the plane -- no big deal, except that you're most likely unfamiliar with the new system (called polar coordinates):
  
  Coordinate r is called the radial coordinate and $\theta$ is called the angular coordinate.
- If horizontal and vertical lines are very easily expressed in the usual Cartesian coordinates system, it's circles and lines centered at the origin that are easily expressed in polar coordinates. Here are the "grid lines" in polar coordinates:
- One of the problems of polar coordinates is their non-uniqueness:
  
  ${(r,\theta)=(r,\theta+2n\pi)}$
  We can sometimes use that to our advantage, however; for example, if we want to imagine a point travelling around a circle, then $\theta$ is a parameter, and by letting it run over multiples of $2\pi$ the point makes multiple trips around the circle....
- One of the advantages of polar coordinates is that they allow us to create curves that can't be represented as functions of Cartesian coordinates y(x).
- Examples:
  - Understanding polar coordinates:
    - #1, p. 627
    - #2, p. 627
  - Converting between coordinate systems:
    - #3, p. 627
    - #5, p. 628
    - #20, p. 628
  - Converting equations:
    - #16, p. 628
    - #20, p. 628
    - #33, p. 629

Website maintained by Andy Long. Comments appreciated.