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Unformatted text preview: Math 53 Homework 7 Due Wednesday 10/13/10 in section This homework assignment covers various topics related to Chapter 14. With the exception of Problem 1 (same as Problem 5 of HW 6), these problems are only marginally related to the material on Midterm 1; so I recommend that you wait until after the midterm to attempt them. Work: Problems 14 below. Bonus problem (extra credit, hard): Problem 5 below. Problem 1. (Identical to Problem 5 of Homework 6.) Consider a triangle in the plane, with angles ,, . Assume that the radius of its incircle is equal to 1. a) By decomposing the triangle into six right triangles having the incenter as a common vertex, express the area A of the triangle in terms of ,, (your answer should be a symmetric expression). Then use your result to show that A can be expressed as a function of the two variables and by the formula A = cot 2 + cot 2 + tan + 2 . b) What is the set of possible values for and ? Find all the critical points of the function A in this region. c) By computing the values of A at the critical points and its behavior on the boundary of the region where it is defined, find the maximum and the minimum of A (justify your answer). Describe the shapes of the triangles corresponding to these two situations. Problem 2. Leastsquares interpolation. In experimental sciences, statistics, and many other fields, one often wishes to es tablish a linear relationship between two quantities (say x and y ). Repeated experi ments (or sampling of x and y for various individuals among the general population) give a set of experimental data ( x 1 ,y 1 ), . . . , ( x n ,y n ) (each pertaining to a different experiment or to a different individual). One then attempts to find the straight line y = mx + b that best fits the given experimental data. Leastsquares interpolation is the most common method for doing so. (Note: the goal is to find m and b , which describe the relation between x and y we are not trying to solve for x and y !). We will first work out the general formula (following a problem in the book), then apply it in an example....
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 Fall '08
 GUREVITCH
 Math, Linear Algebra, Algebra

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