Math 110.107, Calculus II (Biological and Social Sciences)

Fall 2010 Course Lecture Synopses

 

Week 11:  November 8 through November 12

 

http://www.mathematics.jhu.edu/brown/courses/f10/107.htm

 

 

 

 

·       Monday, November 8: 

·       Wednesday, November 10: Today, I talked about another type of equilibrium solution at the origin for the  system .  Suppose  has complex eigenvalues (remember this means that the quadratic formula used to solve the characteristic equation has negative discriminant.  In this case, like in the case we did on Monday, the two eigenvalues are distinct, but are not real (they are complex conjugates  ).  This presents a problem because the eigenvectors (the vector solutions to  ) will also not be real.  It turns out, though that the eigenvectors can be solved for in the usual way, and they are also complex conjugates of each other .  This is a problem because the solutions to a system of real differential equations must also be real.  They are, but must be constructed in a different way.  The book does not construct the solutions.  Instead, they simply go back into the slope field to “see” the solutions. Then they discuss the oscillatory nature of the solutions by graphing each solution separately.  I do not like this, so I constructed the solutions explicitly.  Using the data above, I wrote out the solutions as .  I then analyzed this solution for clues as to what is going on in the slope field where the solutions live.  First, only the  -value really determines whether solution collapse into the origin, or move away from it.  Second, the  -value ensures that the sines and cosines make the solutions oscillate.  They also make sure that there is NO straight line motion along eigendirections like in the case we developed in the last class.  Using many examples in JODE on the projector, we studied the cases for  (the stable spiral or the spiral sink),  (the unstable spiral or the spiral source), and  (the center, also stable but solutions do not collapse to the origin.  Then for the last part of the class, I introduced Chapter 12 with a little bit of lighthearted probability, counting, and why one would need calculus to do probability. 

·      Friday, November 12: Today, we learned how to count, with three counting principles of counting from Section 12.1.  The first was the idea that for an experiment consisting of many tasks performed in order, the total number of possible outcomes is the product of the total number of outcomes for each task.  With examples like, flipping coins, tossing dice, playing cards and arranging pictures on a wall, we went through many examples.  The second is the idea of a permutation, a choice and arrangement of a subset of objects from another set.  The key to a permutation is that it is an ordered subset of  objects out of  total objects.  When the arrangement or order of the choosing matters, the counting is called a permutation, and the total number of permutations of the  objects out of  is given by .  We also went over the definition of the factorial used here.  The last principle we talk about was how to choose  objects out of  when the order of the choosing does not matter.  One can count by again using the total number of permutations, but one must then discount the total by the number of ways one can arrange the subset of  objects.  There are exactly  ways to arrange  objects. These are called combinations and the total number of ways one can choose  objects out of  when order does not matter is .  Again, we went through many examples.