Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.4 Taylor Polynomials - Exercises - Page 493: 31


$$ 0.183$$

Work Step by Step

Since the error bound is given by $$\left|T_{n}(x)-f(x)\right| \leq K \frac{|x-a|^{n+1}}{(n+1) !}$$ where $ \left|f^{(n+1)}(u)\right| \leq K$, then for $$f(x)=e^x\ \to\ f^{(n)} (x)= e^x$$ For $x\in [0, 1.1] $ choose $K = e^{1.1}$ and \begin{align*} \left|e^{1.1}-T_{3}(1.1)\right| &\leq K \frac{(1.1-0)^{4}}{4 !}\\ &=\frac{e^{1.1} 1.1^{4}}{24}\\ & \approx 0.183 \end{align*}
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