Exam 5: Final Exam

Exam 5: Final Examination

Tuesday, May 27, 8:00-10:00 a.m.

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

You may use your note cards from Exams 1-3 and your one-sided 8.5"-by-11" page of notes for Exam 4 and one new 3"-by-5" note card. The p. 161 table of generating functions will be provided.

The final exam will focus on recent material, but a few points will relate to earlier topics.

RECENT TOPICS

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)
  - 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
Recurrence relations
- Linear recurrence relations
  - ... with constant coefficients: Solution via characteristic polynomials
    - Homogeneous case
    - Special nonhomogeneous cases
  - ... first-degree with variable coefficients: summation factor/backtracking techniques
- Divide-and-conquer algorithms and recurrence relations
  - f(n) = a f(n/b) + g(n)
  - Master Theorem
Here's an old MCS 256 exam on generating functions. (Sorry, I don't have the answers stored electronically.)
PREVIOUS TOPICS
- 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
- Generating functions
- Recurrence relations