Discrete Calculus & Probability
MCS 256: Discrete Calculus & Probability
Course Outline
- Introduction to discrete mathematics
- Thinking recursively
- The principle of mathematical induction
- Sequences
- Triangular numbers
- Unwinding/unfolding/backtracking a recurrence
- Arithmetic sequences
- Geometric sequences
- Sums
- Financial mathematics
- Simple interest
- Compound interest
- Nominal rates
- Continuous compounding
- Present values of cash flows
- Pascal's triangle and the binomial theorem
- More proofs by mathematical induction
- Difference calculus
- Summation calculus
- Summation/sigma notation
- Delimited form
- General sigma notation
- Product/pi notation
- Summation methods
- Technique 0: Look it up. (More later)
- Technique 1: Guess the answer; prove it by induction.
- Technique 2: Use divided differences and the
interpolating polynomial
- Technique 3: Discrete calculus
- Summation calculus
- Basic manipulations
- Fundamental Theorem of Calculus
- Fundamental Theorem of Discrete Calculus:
the telescoping sum formula
- Antidifferences/indefinite sums
(analagous to antiderivatives/indefinite integrals)
- A short table of antidifferences
- Summation by parts
- Ordinary and factorial powers
- Synthetic division
- Stirling numbers of the first and second kinds
- Solving first-order recurrence relations
- Exam 1
- Asymptotics
- Calculus review: behavior of functions f(x) as x -> infinity
- Limits of rational functions at infinity
- L'Hospital's rule
- Asymptotics definitions
- Relations comparing functions with respect to growth rate
- Little oh, Big Oh, Big Omega, Big Theta
- Stirling's approximation