Day 9 in MAT229

Today:

• Announcements:
  • Your assignment over 6.3 will be due tomorrow, Wednesday.
  • I'm going to consolidate two sections from chapter 9 today.
  • You have a quiz opportunity over sections 9.3 and 9.4 today.
  • Note: the old syllabus had "6.7" listed next, rather than 6.6 (inverse trig functions). So I've changed the syllabus.

• quiz opportunity

• We spent a good part of yesterday talking about one particular property of logarithms:
  

  So if the derivative of the log of the function is simple to calculate, then we're in business when we want f'(x)....

• Today we'll start by looking over this lab: Logs and Exponentials as Models

  We'll be looking at these uses of logs and exponentials first, then turning to differential equations -- the subject of chapter 9.

• Mathematical interlude: differential equations, especially those utilizing exponentials and logs in different ways (as well as a reminder about integration).

  Mathematical notions:

  • Section 9.3 is called separable equations: it illustrates a type of differential equation that we can actually solve by straightforward integration. Separable equations are a special type of differential equation, which are equations that link functions and their derivatives in an equation (or system of equation).

  • Note the emphasis on "differentials", however, the "dt" and "dx" that folks like to drop from their integrals. [Don't be a dx-dropper!]

  • Many physical phenomena have dynamics modeled by interactions between functions and their derivatives. The simplest differential equations are linear, like the following:
    • An example of a differential equation is this:
      

      The solution is something we've been talking about: what function has a derivative which is proportional to the function itself? This suggests the solution, which 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).

      An alternate way of thinking about this is as follows: we can "separate" the variables (thinking of the operation of differentiation as a quotient):

      

      Now integrate both sides:

      

      and solve:

      

      or

      

      Since the right-hand side is positive, so is the left. we can just write

      

    • Differential equations can be a little different (not necessarily more difficult, but different):

      

      or

      

      The first doesn't involve y, and the second involves the second derivative of y. Can you think of a function that "solves" it?

    • 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?

    • The applications are endless. Here are a few:
      • Population (interest, half-life, etc.):
        

      • Mixing problem (p. 623):
        

      • The Logistic Model (an "improvement" on the exponential population model, because it includes the idea of a carrying capacity):
        

      Each of these differential equations is separable, and can be solved by straight integration.

    • Here's another application Newton's Law of Cooling (Lab)

  Links:

  • Derivation of Newton's Law of Cooling