e-book Differential Equations and Polynomials Volume 2

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Editorial Reviews. About the Author. Dr. Mehran Basti is the author of the volume 1 of the same Differential Equations and Polynomials Volume 2 Kindle Edition. by Dr. Mehran Basti (Author).
Table of contents

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Given a Taylor series for f f at a , the n th partial sum is given by the n th Taylor polynomial p n.

Therefore, to determine if the Taylor series converges to f , f , we need to determine whether. Suppose that f f has derivatives of all orders on an interval I containing a. Then the Taylor series. For each of the following functions, find the Maclaurin series and its interval of convergence. Show that the Maclaurin series converges to cos x cos x for all real numbers x. In this project, we use the Maclaurin polynomials for e x to prove that e is irrational.

The proof relies on supposing that e is rational and arriving at a contradiction. In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. Find the value of the Taylor polynomial p n of f f at the indicated point. In the following exercises, find the Taylor series of the given function centered at the indicated point.

Compare the maximum difference with the square of the Taylor remainder estimate for sin x. Compare the maximum difference with the square of the Taylor remainder estimate for cos x.

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Compare the maximum error with the Taylor remainder estimate. Taylor approximations and root finding. Want to cite, share, or modify this book?

Skip to Content. Calculus Volume 2 6. Table of contents. Answer Key. Learning Objectives 6.

Describe the procedure for finding a Taylor polynomial of a given order for a function. Estimate the remainder for a Taylor series approximation of a given function. Theorem 6. Uniqueness of Taylor Series If a function f f has a power series at a that converges to f f on some open interval containing a , then that power series is the Taylor series for f f at a.

Media Visit the MacTutor History of Mathematics archive to read brief biographies of Brook Taylor and Colin Maclaurin and how they developed the concepts named after them. Example 6. Figure 6. Checkpoint 6.

Use these two polynomials to estimate 11 3. Convergence of Taylor Series Suppose that f f has derivatives of all orders on an interval I containing a. Solution Using the n th Maclaurin polynomial for e x found in Example 6. Student Project Proving that e is Irrational In this project, we use the Maclaurin polynomials for e x to prove that e is irrational. Let R n x R n x denote the remainder when using p n x p n x to estimate e x. Write down the formula for the n th Maclaurin polynomial p n x p n x for e x and the corresponding remainder R n x.

Show that s n! R n 1 is an integer.

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Kauers, M. Leykin, A. In: Mathematical Software, Beijing, pp. Spivak M. Calculus on manifolds. New-York, Amsterdam, W. Benjamin, Inc. Kurs visshey algebri [Course of advanced algebra]. Moscow, Nauka Publ. Moscow, Mir Publ. Mastering Simulink 4. Sinergetika i problemy teorii upravleniya [Synergetics and problems of control theory.

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