Day 25 in MAT227

Today:

• Announcements:
  • Assignment 8.2 due
  • Exam is not yet graded - how did it go?

• Last time we glanced at section 9.1: Arc Length and Surface Area
  • What kinds of problems are we looking at in this section? Arc length and surface area calculations for areas of revolution:
    

  • Let's start with arc length:
    

    The usual trick is to discretize something continuous, and then pass to a limit:

    

    Now hopefully we can figure out how the formula will look:

    

    So what's the general formula?

    

  • For surface area (for areas of revolution)
    

    We discretize bits of the surface, and then make a series of little bands; take the limit, and get the general formula:

    

  • Examples:
    • #5, p. 475
    • #12, p. 475
    • #39, p. 476
    • #51, p. 477

• Today I want to talk about differential equations, albeit briefly.
  • You've already solved a differential equation:
    

    The solution is -- so there's an unknown constant k (in this case, it corresponds to the "initial value"). If we tell what y equals at t=0, then we'll get the particular solution, (and that's often very important, that we get the exact solution and unique solution given the data).

    We know that exponential functions have the property that the derivative is proportional to its function value (and is the special function for which the constant of proportionality is exactly 1).

  • As indicated on page 505, differential equations can be a little different:

    

    or

    

    The first doesn't involve y, and the second involves the second derivative of y.

  • The general method for solving differential equation is to stare at them until the solution comes to you!;): what are solutions to the two differential equations above?

  • Today we'll do a lab in which we consider "Newton's Law of Cooling". Before we start, however, let's make up a few differential equations of our own: consider the following functions, and figure out differential equations that they solve:
    • 
    • 
    • 
    We could play this game forever, and get lots of interesting differential equations. But where do differential equations appear in real life?

  • Here's one place: Newton's Law of Cooling (Lab)
  • and

Links:

• Derivation of Newton's Law of Cooling