Let f(x)polynomial T1o(x) centered at a = 2.(b)3+#; defined on[0, +0). Find the 10th Taylor

Answered: Let f(x) = , defined on [0, +). Find…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 3E

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Question

Let f(x)
polynomial T1o(x) centered at a = 2.
(b)
3+#; defined on
[0, +0). Find the 10th Taylor

Expert Solution

Step 1

Taylor series is nothing but expressing the infinitely differentiable function in terms of its derivatives. Let us first see the Taylor series expansion of f(x) around a is defined as following.

$k^{t h}$ derivative of function at a is denoted as $f^{k} (a)$

$\begin{array}{rcl} f (x) & = & f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} + . . . \\ = & \sum_{k = 1}^{\infty} \frac{f^{(k)} (a)}{k!} {(x - a)}^{k} \end{array}$ .

This is the Taylor series expansion.

Let us define the Taylor polynomial.

$\begin{array}{rcl} T_{0} (x) & = & f (a) \\ T_{1} (x) & = & f (a) + \frac{f' (a)}{1!} (x - a) \\ T_{2} (x) & = & f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} . . \\ . \\ T_{n} (x) & = & \sum_{k = 1}^{n} \frac{f^{(k)} (a)}{k!} {(x - a)}^{k} \end{array}$

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