Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 9 - Infinite Series - 9.7 Maclaurin And Taylor Polynomials - Exercises Set 9.7 - Page 657: 1

Answer

$f(0) + f '(0)x + \frac{f''(0)x^{2}}{2!}+\frac{f'''(0)x^{3}}{3!}$

Work Step by Step

As $f$ can be differentiated $3$ times at $0$, we use the definition of the Maclaurin series $p_n(x) = \displaystyle\sum_{n=0}^N \frac{f^{n}(0)x^{n}}{n!}$ for $N=3$ to find the Mclaurin polynomial: $p_3(x)=\displaystyle\sum_{n=0}^3 \frac{f^{n}(0)x^{n}}{n!}=f(0)+f '(0)x+\frac{f''(0)x^{2}}{2!}+\frac{f'''(0)x^{3}}{3!}$
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