$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} $

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

  1. 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 $\textit{ the next term:}$ }$ $$ \text{error} \le \frac{|x|^3}{3!} \le \frac{ \Big(\frac12\Big)^3}{3!} = \frac1{2^3 \; 3!} = 0.021. $$
  2. Applications of Power Series.
    1. Pendulum.
    2. High order derivatives. e.g., $$ \frac{d^5}{d x^5} \Bigg( \frac1x \sin x^2 \Bigg)_{x=0} = \frac{d^5}{d x^5} \Bigg(\frac1x \Bigg[ x^2 - \frac{ (x^2 )^3}{3!} + \cdots\Bigg]\Bigg)_{x=0} = \fcolorbox{white}{yellow}{$5! \Big( \text{ the coefficient of } x^5 \Big)$} = 5!\Big( -\frac1 {3!} \Big)= -20. $$
    3. Limit. e.g., $$ \lim_{x\to 0} \frac{1-\cos(2x)}{x^2} = \lim_{x\to 0} \frac1{x^2} \Bigg( 1 - \Bigg[ 1 - \frac{(2x)^2}{2!} + \cdots \Bigg] \Bigg) = = \lim_{x\to 0} \frac1{x^2} \Bigg( \frac{(2x)^2}{2!} - \cdots \Bigg) = \frac{2^2}{2!} = 2.$$