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Lecture 7, 9/7/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A
- Error estimates: more examples.
e.g., Approximate $\sin x $ by $x$ for $|x|<\frac12$.
Sol. $\fcolorbox{white}{yellow}{The series is alternating, the error is estimated by $\textit{ the next term:}$ }$
$$ \text{error} \le \frac{|x|^3}{3!} \le \frac{ \Big(\frac12\Big)^3}{3!} = \frac1{2^3 \; 3!} = 0.021. $$
- Applications of Power Series.
- Pendulum.
- High order derivatives. e.g.,
$$ \frac{d^5}{d x^5} \Bigg( \frac1x \sin x^2 \Bigg)_{x=0} =
\frac{d^5}{d x^5} \Bigg(\frac1x \Bigg[ x^2 - \frac{ (x^2 )^3}{3!} + \cdots\Bigg]\Bigg)_{x=0}
= \fcolorbox{white}{yellow}{$5! \Big( \text{ the coefficient of } x^5 \Big)$} = 5!\Big( -\frac1 {3!} \Big)= -20.
$$
- Limit. e.g.,
$$ \lim_{x\to 0} \frac{1-\cos(2x)}{x^2} = \lim_{x\to 0} \frac1{x^2} \Bigg( 1 - \Bigg[ 1 - \frac{(2x)^2}{2!} + \cdots \Bigg] \Bigg)
= = \lim_{x\to 0} \frac1{x^2} \Bigg( \frac{(2x)^2}{2!} - \cdots \Bigg)
= \frac{2^2}{2!} = 2.$$