Exam 5: Final Examination

You may use a calculator for numerical calculations, but not for symbolic computation.

You may use your note cards from Exams 1-4 and one 8.5"-by-11" page of notes. This page could be the table of discrete distributions and/or the p. 161 table of generating functions with other notes written on it.

The final exam will be comprehensive, but with special emphasis on recent material.

RECENT TOPICS

• Divide-and-conquer algorithms and recurrence relations
  • Unwinding f(n) = a f(n/b) + g(n)
  • Master theorem
  • Case study: merge sort
• Generating functions
  • "THE MOST POWERFUL WAY to deal with sequences of numbers, as far as anybody knows, is to manipulate infinite series that 'generate' those sequences." --Graham, Knuth & Patashnik
  • Ordinary generating functions (ogf)
    Two perspectives:
    • Formal power series
    • Calculus-defined power series
  • Basic manipulations
    They're like polynomials of unbounded degree.
  • Dictionary of common ogfs: see esp. p. 161.
  • Solving counting problems via generating functions
    Computations can be turned over to Maple.
  • Solving recurrence relations via generating functions

Here's an old MCS 256 exam on generating functions. (Sorry, I don't have the answers stored electronically.)

PREVIOUS TOPICS

See the summaries and documents provided on the course web site and throughout the semester
• Recurrences and mathematical induction
• Finite difference calculus
• Summation calculus
• Asymptotics
• Combinatorics
  • How to count
  • Intermediate counting techniques
• Discrete probability
  • Probability spaces, conditional probability
  • Random variables and distributions
    Special discrete distributions
  • Moments: mean, variance, etc.; probability generating function
  • Joint distributions
  • Inequalities
  • Law of Large Numbers, Central Limit Theorem
• Recurrence relations
  • Linear recurrence relations with constant coefficients
  • Solution via characteristic polynomials
    • Homogeneous case
    • Special nonhomogeneous cases
  • (Optional) Operator calculus