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Calculus / Calculus: Early Transcendentals 1 / Chapter 9 / Problem 27RE

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Taylor series Write out the first three terms of the

Chapter 1, Problem 27RE

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QUESTION:

Write out the first three terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

\(f(x)=e^{3x},\ a=0\)

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QUESTION:

Write out the first three terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

\(f(x)=e^{3x},\ a=0\)

ANSWER:

Solution 27RE

Step 1:

In this problem we have to find the first three terms of the Taylor polynomial for function $f(x)={e}^{3x}$ with the given center point a=0.

Taylor series is given by $f(x)=\sum\limits_{k\ =\ 0}^{\infty{}}\frac{{f}^{(k)}(a)}{k!}{x}^{k}$

In our case, $f(x)\approx{}\sum\limits_{k\ =\ 0}^{\infty{}}\frac{{f}^{(k)}(a)}{k!}{x}^{k}\ =\sum\limits_{k\ =\ 0}^{2}\frac{{f}^{(k)}(a)}{k!}{x}^{k}$

So, what we need to calculate derivatives, evaluate them at the given point.

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