Greenwood press
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book-20600
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- Linear Functions
74
POLAR COORDINATES POLYNOMIAL FUNCTIONS 75 POLYNOMIAL FUNCTIONS A polynomial function f (x) has a general equation f (x) = a 1 x n + a 2 x n−1 +a 3 x n−2 + . . . + a z , where coefficients and constants are associated with a and exponents associated with n are positive integers. Linear functions, such as y = 3x − 5, and quadratic functions, such as r = 3w 2 − 5w + 7, are polynomial functions that have numerous applications discussed elsewhere in this book (see Linear Functions and Quadratic Functions). Polynomial functions with degree three or greater are found in applications associated with volume and financial planning. Empty open-faced containers such as crates are put together by attaching a net of five rectangles. A rectangular piece of plastic can be cut so that it can turn into an open-faced rectangular prism when folded at its seams. If a square piece is cut out of each corner of a rectangle, then four folds will form a net with five rectangles that can be formed to develop the prism, as shown below. A manufacturer is probably interested in finding the location to cut the square from the corners so that the consumers will be able to fill the crate with the most amount of material. In essence, the goal is to maximize the volume based on a fixed amount of material. Suppose that square corners are removed from a rec- tangular sheet of plastic with dimensions of 6 feet by 4 feet. Each side of the prism can be represented in terms of the edge length, x, of the square that was removed from the corners, as shown below. The volume of the crate, v, is the product of its dimensions, so it can be rep- resented by the equation v = x(6 − 2x)(4 − 2x). This equation is a polynomial function, because it is the factored form of v = 4x 3 − 20x 2 + 24x. A relative maximum of this function on a graph, as shown on the following page, within a domain between 0 and 2 feet occurs when x ≈ 0.78 feet, or about 9.4 inches. This means that the crate with the largest possible volume will occur when squares with an edge length of 9.4 inches are cut from the corners. Open-faced prism with dimensions x by l − 2x by w − 2x formed by cutting squares with side length x out of the corners of rectangular sheet with dimensions l by w. Open-faced prism formed by cutting squares with edge length, x, out of the corners of a rectangular sheet with dimensions of 6 feet by 4 feet. Long-term investing uses a polynomial function to account for money that is invested each year. Suppose an account was set up so that you contributed money each year towards your retirement based on a fixed percentage of interest, assum- ing that you continued to add a minimum amount of money to the account each year and did not withdraw money at any time. The total amount of money, m, in the bank after n years based on an annual interest rate of r percent can be repre- sented by the function m = a 1 (1 + r 100 ) n +a 2 (1 + r 100 ) n−1 +a 3 (1 + r 100 ) n−2 + . . . + a z , where the coefficients, a, are the individual amounts of money deposited into the account after each year. For example, if $500 is deposited at the end of the first year, $700 at the end of the second year, $800 at the end of the third year, and $400 at the end of the fourth year, then the total amount of money in the account at the end of the fourth year is determined by the equation m = 500(1 + r 100 ) 3 + 700(1 + r 100 ) 2 + 800(1 + r 100 ) + 400. This means that the initial deposit of $500 will compound three times, the second deposit of $700 will compound two times, and so on. If an employee uses this retirement plan for only four years and wants to know the value of the account 21 years after the first investment, then the equation would be rewritten to m = 500(1 + r 100 ) 20 + 700(1 + r 100 ) 19 + 800(1 + r 100 ) 19 + 400(1 + r 100 ) 18 . This information is useful for people in their financial planning so that they can learn how to save money for their children’s education and their own retirement. online sources for further exploration Antenna pattern correction Application of polynomial functions Download 1.81 Mb. Do'stlaringiz bilan baham: |
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