math trivia

School is about to begin. The janitor has all the lockers closed. All 1000 of them. Student #1 comes along and opens ALL of the lockers. Student #2 comes along and closes doors 2, 4, 6, 8, 10, etc.... Student #3 comes along and changes the state of every 3rd locker ( 3, 6, 9, 12, 15). Student #4 comes along and changes the state of every 4th locker (4, 8, 12, 16). Student #5 comes along and changes the state of every 5th locker (5, 10, 15, 20) Student #6 comes along and changes the state of every 6th locker (6, 12,18. 24) It keeps going until all 1000 students go through. What lockers are closed?

show all working for this review problem

(See attached file for full problem description and equations) --- 1. Let g : R → R+ be such a function that g ∈ C1(R) and for all x ∈ R,−1 < g'(x) < 0. Show that the sequence xn+1 := g(xn) converges to the unique fixed point of the function g, regardless of choice of x0 ∈ R. [Note: Observe that the domain of function g is not a compact interval. ] 2. Write a MATLAB program (Newton-Raphson method) for finding the root of the function f(x) = x5 − 2x3 + x + 2. Be as much accurate as you can. To proceed, store this function in an m-file, say, f.m, and its derivative in df.m. Label the whole procedure newt.m. ---

Newton-Raphson

1. let g: R→R+ be such a function that g∈ C^1(R) and for all x ∈ R, -1

Numerical Stability

(See attached file for full problem description with equations) --- There is a function f of the form for which and . Determine and , and assess the sensitivity of these parameters to slight changes in the values of f at the two indicated points. ---

Solve Finite Difference Equation

See attached file for full problem description with equation. --- Find analytically the solution of this difference equation with the given initial values: Without computing the solution recursively, predict whether such a computation would be stable. (Note: A numerical process is unstable if small errors made at one stage of the process are magnified in subsequent stages and seriously degrade the accuracy of the overall calculation.) ---