Topics Covered:

      The elementary functions as solutions of ODE's:
         The natural logarithm and exponential functions.
      Integration:
         Definitions & criteria.  The first FTC.  Convergence of Darboux & 
         Riemann sums.  Linearity, monotonicity, & additivity over intervals.
         The second FTC.  The existence of solutions of ODE's.  Two classical 
         integration methods.  Approximation of integrals.
      Approximation by Taylor polynomials:         
         Definitions.  Lagrange's remainder theorem.  Convergence.  A power 
         series for logarithm.  The Cauchy integral remainder formula and the 
         binomial expansion.  Example:  An infinitely differentiable function 
         that is not analytic.  The Weierstrass Approximation Theorem. 
      Convergence of sequences & series of functions:
         Sequences & series of numbers.  Pointwise & uniform convergence of 
         sequences of functions.  Uniform limits of functions.  Power series.  
         A veeeeeeeery special example.
      Euclidean n-space (Rn):
         The linear structure and inner product.  Convergence of sequences.  
         Subsets:  Interiors, exteriors, and boundaries.
         

WRITING UP HOMEWORK SOLUTIONS: You are expected to show all essential steps in solving a problem. Draw appropriate figures. If in doubt, PUT IT IN! Make your solution flow. Ask yourself the question: "If I were the reader, how would I score this problem?'' Numerical answers alone, single words "yes" or "no", etc., are NEVER, EVER acceptable as complete answers. Writing up homework exercises is your opportunity to improve your mathematical communication skills.  But, of course, you have already been "honed to a fine edge" by Professor Wiggins concerning these matters.  So I am sure that all this is second nature...

Homework Sets: 
   
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