Day 30 in MAT227

Today:

• Announcements:
  • Final exam: 6:45 - 8:45 p.m., Wednesday, December 15, 2010.

  • It's time for course evaluations: those are to be done on-line. Please visit the site http://eval.nku.edu/ to do so.

  • The Calc Lab will be open Monday, Tuesday and Wednesday of exam week during the regularly scheduled hours.

  • Your (late) test two corrections are returned. You may still turn in a test one correction.

  • Assignment 12.1 returned

  • RE: Assignment 10.1 (returned last time: 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....

  • If you'd like, I'll come in for a review session the afternoon before the final (Is there interest?)

• 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 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:
    

    We can sometimes use that to our advantage, however; for example, if we want to imagine a point travelling around a circle, then is a parameter, and by letting it run over multiples of 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

• Review: we've gone through a lot of material this semester. Here are the highlights, as I see them:

  • Labs!
    • I'm going to ask questions about the labs. You should be familiar with each lab.
    • This is as good a time as any to mention that units are important!

  • Chapter 5:
    • Approximating areas with rectangles
    • LRR, RRR, Midpoint; Trapezoidal (when you have two estimates, you can get a third by taking a suitable average).
    • The Midpoint method is the best of the lot in general, but usually makes an error half the size of the trapezoidal method, and in the opposite direction. What does that suggest?
    • Are integrals all about areas?
    • Riemann sums (partitions, subintervals, limits, ...)
    • Definite and indefinite integrals
    • The Fundamental Theorem of Calculus, Part I
    • The Fundamental Theorem of Calculus, Part II
    • Special summation formulas
    • Substitution (the chain rule backwards)

  • Chapter 6:
    • This is what it's all about:
    • Areas between curves
    • Volumes
    • Volumes of revolution
    • Average value of a function

  • Chapter 7:
    • Why e?
    • Exponential functions -- easy to invert, and extremely important for modeling purposes.
    • Inverse functions, especially logs
    • We only need base e exponentials and logs (ln)
    • Easy to invert a function given by a graph
    • inverse trigonometric functions
    • L'Hopital's rule

  • Chapter 8:
    • Integration by parts (the product rule backwards)

  • Chapter 9:
    • Arc length and surface area
    • This example illustrated how easy it is to create an integral for a quantity of interest -- arc length -- using only elementary pieces of mathematics (e.g. Pythagorean theorem)

  • Chapter 10:
    • Differential equations -- easily created
    • F=ma -- mother of many physical differential equations
    • Newton's law of cooling

  • Chapter 12:
    • Parametric curves
    • Introduction to the cycloid
    • Motions of planets follow ellipses
    • Graphing ellipses and other parametric curves
    • Speed of a particle versus slope of the tangent line
    • Polar coordinates