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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. Least-squares 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. Least-squares 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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