Unit 8: Topics covered in videos

  • Patterns. (Count, Cases, Position. OddQ, EvenQ, PrimeQ. Counting all entries in a list which match a certain pattern, such as strictly greater than 10.)
  • Piecewise functions using patterns. (Review of the old way of making piecewise functions, then a new way using patterns. The old way with the keyword Piecewise is our preferred way of making them.)
  • Dice rolls. (Making `or' in Mathematica using two vertical lines. Investigating the probability that the maximum among 3 rolls of a 6-sided die is either 5 or 6. Having a variable n to stand for the number of trials.)
  • Newton's method with NestList. (The idea of Newton's method for finding a root of a function f(x). The motivation for the formula in terms of the tangent line to the graph. Implementing Newton's method for the function f(x) = x^2 - 2 using NestList.)
  • Newton's method analysis. (Writing a function that takes as input an x-value and as output returns 5 steps of Newton's method. Counting how many inputs get us close to the actual zero, using patterns. What's the input for which we are the furthest from a zero after 5 iterations of Newton's method? What does that input correspond to in terms of the graph?)
  • The Collatz conjecture. (Studying the Collatz conjecture using Piecewise and NestList or NestWhileList. Making `not equals' using exclamation point and then equals.)
  • Collatz conjecture analysis. (Making a list of lists representing the Collatz conjecture values for various inputs. For every starting value from 1000 to 2000, what's the most number of steps it takes until we get to 1? Using Extract in conjunction with Position.)
  • Random walk analysis. (Making a variant of our usual random walks, where we stop only when we reach a specific value, rather than a specific distance from the origin. Making a list of lists, where each inside list corresponds to one of these new random walks. Investigating the average length of one of these random walks. Counting how many of them are longer than a specific length using a pattern.)
  • Plotting two-dimensional random walks. (Grouping commands using parentheses. Choosing a random point on a circle. Plotting the points corresponding to the random walk. Changing the colors of the points. Norm, Hue, Style.)