Monthly Archives: November 2013

Bonus Assignment

This optional assignment is worth some extra points, the number of which will be determined by the quality of what you turn in.

There is no penalty for not doing this assignment.

All write ups for this assignment must be typed or written very neatly by hand. It might help you to use a graphing calculator or computer algebra system such as Mathematica, Wolfram Alpha, Sage, or Maple.

Recall \(V = C([-1,1])\) is the vector space of continuous real valued functions with domain \([-1,1]\). Define an inner product on \(V\) by \(\langle f(x),g(x) \rangle = \int_{-1}^1 f(t)g(t)\,dt\) for all \(f(x),g(x)\in V\).

Here are some possible topics to explore (you may find other interesting things to include in your write up):

  1. Starting with \(\{1,x,x^2,x^3\}\), use Graham-Schmidt to find an orthonormal basis for the subspace \(P_3(\mathbb{R})\) of \(V\).
  2. Find the element of \(P_3(\mathbb{R})\) nearest to \(f(x) = e^x\) and to \(g(x) = \cos(x)\). Nearest in the sense of norm/length induced by the given inner product.
  3. Recall that the Taylor polynomial of degree 3 (which is an element of \(P_3(\mathbb{R})\)) is a cubic polynomial that you learned about in calculus. Taylor polynomials approximate functions. Find the Taylor polynomials of degree 3 for \(f(x) = e^x\) and for \(g(x) = \cos(x)\), centered at the origin.
  4. Compare and contrast the approximations from (2) to the approximations from (3). For example, is the Taylor polynomial a better approximation outside of the interval \([-1,1]\)? If \(T\) denotes the Taylor polynomial and \(Q\) denotes the approximation from (2) (for either \(f\) or \(g\)), how does \(||T – f|| \) compare to \(||Q-f||\)? and \(||T-g||\) to \(||Q-g||\)?
  5. Which type of approximation do you think is better?