. Differential and integral calculus, an introductory course for colleges and engineering schools. root lies between 1 and 2(the polynomial is obviously single-valued,and no polynomial has a discontinuity).That is, the graph cuts OX between thepoint (1, - 5) and (2, 5). We next for x and find that P() = , andhence by theorem 1 a root lies between 2. We continue to substitute values for x asfollows: P() = ; hence the root lies between and 2P() = ; hence the root lies between and () = ; hence the root lies between and (
. Differential and integral calculus, an introductory course for colleges and engineering schools. root lies between 1 and 2(the polynomial is obviously single-valued,and no polynomial has a discontinuity).That is, the graph cuts OX between thepoint (1, - 5) and (2, 5). We next for x and find that P() = , andhence by theorem 1 a root lies between 2. We continue to substitute values for x asfollows: P() = ; hence the root lies between and 2P() = ; hence the root lies between and () = ; hence the root lies between and () = ; hence the root lies between and () = ; hence the root lies between and We have now found one root correct to the second place of decimals,viz., By continuing the process we can find as many decimal places of the root as desired.* * In textbooks of algebra may be found a very expeditious method of makingthese substitutions, — a method known as Horners Method for Approximatingto the Roots of Equations. Horners Method applies only to bytheorem 1. 52 SOME GENERAL PROPERTIES OF FUNCTIONS 67 52. Zeros of Functions. A value of the argument which makesa function zero is termed a zero of the function. The terms rootand zero are exactly equivalent, except that the term root is usu-ally confined to polynomials, while the term zero is applicable toany function. For example, 1 is a root of the polynomial x2 — 1, and is also a zero of it, while on the other hand - is a zero of cos x, but is not usually called a root of it. Theorems 1 and 1 applyof course not only to polynomials, but to all real, single-valued,continuous functions, and consequently the method of the preced-ing article may be applied to finding the zeros of functions that arenot polynomials. Example. Let us find the zeros of f(x) = x — 2 sin x. To find the zeros of this function is equivalent to finding values of xthat will make x = 2 sin x, and this is equiv
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