C-star-algebras. Banach spaces by Unknown Author

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Hint: First calculate the area of the rectangle enclosing ^ABC, and then subtract the areas of the three right triangles. (b) Calculate the area of the triangle with vertices (1, 3), (4, 1), and (10, 4). Hint: Work with an enclosing rectangle and three right triangles, as in part (a). (c) Using the same technique that you used in parts (a) and (b), show that the area of the triangle in the following figure is given by A ϭ 12 1x1y2 Ϫ x2y1 ϩ x2y3 Ϫ x3y2 ϩ x3y1 Ϫ x1y3 2 Remark: If we use absolute value signs instead of the parentheses, then the formula will hold regardless of the relative positions or quadrants of the three vertices.

A) Solve the given equation for F in terms of C. Then set up a table and graph the equation in a C-F coordinate system (C on the horizontal axis, F on the vertical). (b) Use a graphing utility to obtain a graph of 5F Ϫ 9C ϭ 160. SOLUTION (a) We have 5F Ϫ 9C ϭ 160 5F ϭ 9C ϩ 160 9 F ϭ C ϩ 32 5 adding 9C to both sides dividing both sides by 5 (1) Using equation (1) now, we pick values for C and compute the corresponding values for F. To avoid working with fractions, we take values for C that 34 Chapter 1 Fundamentals TABLE 2 are multiples of 5, as indicated in Table 2.

1, we’ll derive this formula and look at some of its implications. ) The Quadratic Formula The solutions of the quadratic equation ax 2 ϩ bx ϩ c ϭ 0, where a xϭ EXAMPLE 5 0, are given by Ϫb Ϯ 2b2 Ϫ 4ac 2a Using the quadratic formula to solve a quadratic equation Use the quadratic formula to solve the equation 2x 2 ϭ 3 Ϫ 4x. SOLUTION We first rewrite the given equation as 2x 2 ϩ 4x Ϫ 3 ϭ 0, so that it has the form ax 2 ϩ bx ϩ c ϭ 0. By comparing these last two equations, we see that a ϭ 2, b ϭ 4, and c ϭ Ϫ3.

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