If f ′ ( x 0 ) ≠ 0, f ′ ( x 0 ) ≠ 0, this tangent line intersects the x x-axis at some point ( x 1, 0 ). We then draw the tangent line to f f at x 0. By sketching a graph of f, f, we can estimate a root of f ( x ) = 0. Newton’s method makes use of the following idea to approximate the solutions of f ( x ) = 0. In cases such as these, we can use Newton’s method to approximate the roots. No simple formula exists for the solutions of this equation. For example, consider the task of finding solutions of tan ( x ) − x = 0. Similar difficulties exist for nonpolynomial functions. No formula exists that allows us to find the solutions of f ( x ) = 0. For example, consider the functionį ( x ) = x 5 + 8 x 4 + 4 x 3 − 2 x − 7. Also, if f f is a polynomial of degree 5 5 or greater, it is known that no such formulas exist. Although formulas exist for third- and fourth-degree polynomials, they are quite complicated. However, for polynomials of degree 3 3 or more, finding roots of f f becomes more complicated. If f f is the second-degree polynomial f ( x ) = a x 2 + b x + c, f ( x ) = a x 2 + b x + c, the solutions of f ( x ) = 0 f ( x ) = 0 can be found by using the quadratic formula. If f f is the first-degree polynomial f ( x ) = a x + b, f ( x ) = a x + b, then the solution of f ( x ) = 0 f ( x ) = 0 is given by the formula x = − b a. Describing Newton’s MethodĬonsider the task of finding the solutions of f ( x ) = 0. This technique makes use of tangent line approximations and is behind the method used often by calculators and computers to find zeroes. In this section, we take a look at a technique that provides a very efficient way of approximating the zeroes of functions. For most functions, however, it is difficult-if not impossible-to calculate their zeroes explicitly. In many areas of pure and applied mathematics, we are interested in finding solutions to an equation of the form f ( x ) = 0. 4.9.4 Apply iterative processes to various situations.4.9.3 Recognize when Newton’s method does not work.4.9.2 Explain what an iterative process means.4.9.1 Describe the steps of Newton’s method.
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