Math410 (Bayly) Homework 4 (due Monday 8 March, with Exam 2)

Section 2.4: 13, (19)

Section 2.5: (6), 7, 8, (9)

Section 3.4: 2, 3, (6), 13, (14), (16)

Section 4.2: (1), 3, (10), 15

Section 4.3: (5)

AND use techniques of section 2.5 to rank UA, Stanford, Washington, and Oregon basketball teams, based on games played amongst each other so far.  You will get a linear system with no solution (probably), but you can use least squares to get a “closest” solution.  Feel free to use calculator or Prof Rychlik’s applet at http://www.math.arizona.edu/~rychlik (you have to scroll down to find the Gaussian Elimination link).

Extra Credit/ Honors Credit: FUNCTION SPACES (see second half of section 3.4). A function f(x) defined on an interval of values can be thought of as an infinite-dimensional vector , with the “entries” being the values taken by f(x). We can, if we wish, define infinite dimensional matrices as functions of two variables A(x,y), with multiplication defined by integration instead of summation: . (Though in fact the subject of functional analysis deals with “matrices” or linear operators that can’t always be represented by functions of two variables.) The concepts of dot product and length can be easily generalized: , , and the functions f and g are orthogonal if .

(1) Function approximation: suppose we want to approximate the function f(x)=sin(x) by a cubic polynomial g(x)=, for x in the range from 0 to . We know that the Taylor polynomialdoes a very good job for x close to zero, but gets less accurate for larger values of x. Find the values of a and b for which g(x) is the best approximation to sin(x), and see how close they are to the Taylor values 1 and -1/6 .

(2) Differential equations: suppose we want to solve the initial-value problem withand, for x in the range from 0 to 2.

(a) Find the exact solution in terms of exponentials.

(b) If we did not know about exponentials, we could seek an approximate solution in the form , which satisfies the initial conditions for any value of k. Find the value of k that minimizes (using 0 and 2 as limits of integration).

(c) Graph the exact and approximate solutions and see how close they are.