Give an upper bound for the difference between the function $f(x) = e^x$ and its second degree Taylor polynomial centered at $a = 0$ for $0 \le x \le 0.18$. The maximum value of the absolute third derivative on the interval $[0, 0.18]$ is $M_3 = \text{________}$ The upper bound is $|R_2(x)| \le \text{________}$
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$f(x) = e^x$ $f'(x) = e^x$ $f''(x) = e^x$ $f'''(x) = e^x$ The absolute value of the third derivative is $|f'''(x)| = e^x$. Since $e^x$ is an increasing function, the maximum value of $|f'''(x)|$ on the interval $[0, 0.18]$ occurs at $x = 0.18$. $M_3 = |f'''(0.18)| Show more…
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