Constructadirect(butunmotivated)proofofExample1asfollows,without using any knowledge of eigenvalues. Let w(x) be aC2 function such that w(0)=w(1)=0. (a) Expand 1 0 [w(x)w(x)cot(x)]2 dx and integrate the cross-term by parts. (b) Show that w2(x)cotx 0 asx 0 or 1. (c) Deduce that 1 0 [w(x)]2 dx2 1 0 [w(x)]2 dx = 1 0 [w(x)w(x)cot x]2 dx0. (d) Showthatifw(x)=sinx,thenpart(c)isanequalityandtherefore the minimum of Example 1 is 2.

Math 1550 - 004 Spring 2017 Exam I Review We'll do our first midterm test on Friday, February 17 . Sections that will be tested on this exam: 0.1: Linear Equations a. Be able to solve Applied Linear Expression Problems (Mixture, Distance, Geometry) 0.2: Quadratic Equations a. Know the Quadratic Formula (I will not ask you to find complex roots on this exam) b. Know the Principle of Zero Product 0.3: Other Types of Equations a. Know Absolute Value Function b. Be able to find the roots of quadratic expressions (I don't care what method you use) c. Be able to solve Radical and Rational Expressions 0.4: Inequalities