Math 110.109, Calculus II (Physical Science and Engineering)

Spring 2011 Course Lecture Synopses

 

Week 1:  January 31, 2011

 

http://www.mathematics.jhu.edu/brown/courses/s11/109.htm

 

 

 

 

·       Monday, January 31:  Welcome to the course!  After a not-so-brief introduction to the purpose and focus of this course, as well as where it sits in your mathematical present and future, and how I will run it and such, I started on the content.  Starting in Chapter 7, Techniques of Integration, I used the idea that integration is the opposite of differentiation to establish the central focus of this chapter;  that integration, or anti-differentiation, in general is much more difficult, if even possible at times, to do.  But the patterns found in the differentiation rules we already have studied can lead to techniques of integration that are remarkably powerful in those cases where the pattern can be recognized.  Undoing the Power Rule (Section 3.1) , which works for all real numbers , is solely a matter of recognizing the pattern:  .  Sometimes this rule is referred to as the Anti-Power Rule, and works for almost all real numbers  (can you see where it doesn’t?  Do you know why it doesn’t?)  I started the discussion by bringing this up and then rehashing the Substitution Method, which I call the Anti-chain Rule.  The reason is that the pattern one looks for in an integrand to use this method, is ;  the integrand is a product of functions, one factor of which is a composition of two functions and the other factor is the derivative of the “inside” function.   This pattern of the integrand is precisely the derivative of a composition of two functions , where  is the anti-derivative of .  Thus, we get , where  and .  I used the example of  to illustrate.  This leads to the next step.  Another integration technique derived from a rule of differentiation is the technique called “Integration by Parts”.  Recall the Product Rulein differentiation:   (Note that this implies that the derivative of a product is NOT the product of the derivatives!)  Since each side of the equality is a function of , the respective anti-derivatives will also be equal (up to a constant!)  Thus we get a new relationship by integrating:  .  We can use the Fundamental Theorem of Calculus on the left hand side, and solve for one of the right hand side elements to get what is commonly called Integration by Parts rule, or what I call the Anti-product Rule:  .  Why this may create an advantage can almost be immediately seen.  Suppose you had a product of functions as an integrand.  Since it is obvious that the anti-derivative of a product will not in general be a product of the anti-derivatives.  But if the two factors are such that 1) the antiderivative of  is easy to calculate, while the derivative of  is easier to work with in an integral, then the right-hand side would be easier to calculate than the original problem.  I ended the lecture with the example , and a quick idea of just how this technique would work well here.

·       Wednesday, February 2: