%
%	gbb60166@gmail.com	https://unilab.gbb60166.jp/prekou/prekou.htm
%
% aspectratio= は 1610, 169, 149, 54, 32 の中から選べる（省略時は 43）
% C:\texlive\2023\texmf-dist\tex\latex\beamer\beamer.cls
%\documentclass[20pt,dvipdfmx,usepdftitle=false,hyperref={pdfpagemode=FullScreen},aspectratio=169]{beamer}
%\documentclass[20pt,dvipdfmx,usepdftitle=false,hyperref={pdfstartview={XYZ null null 8.00}},aspectratio=169]{beamer}
\documentclass[20pt,dvipdfmx,usepdftitle=false,aspectratio=169]{beamer}
% pdfの栞の字化けを防ぐ
%\AtBeginDvi{\special{pdf:tounicode EUC-UCS2}}
% テーマ
\usetheme{Copenhagen}
% navi. symbolsは目立たないが，dvipdfmxを使うと機能しないので非表示に
\setbeamertemplate{navigation symbols}{} 
\usepackage{bxdpx-beamer,pxjahyper,minijs}
%\usepackage{graphicx}
%\usepackage{tkokugo,furikana,tsayusen,shiika,sfkanbun,jdkintou,plext}
\usepackage{type1cm}
\usepackage[deluxe]{otf}
%\usepackage{tikzsymbols}
%\usepackage[dvipdfmx]{graphicx}%
\def\pgfsysdriver{pgfsys-dvipdfmx.def}%（graphicxパッケージを使用しない場合はこの行を有効に）
%\def\pgfsysdriver{pgfsys-dvips.def}%デフォルト
%\usepackage[e]{esvect}
\usepackage{tikz}%（これで、pgfとpgfforが読み込まれます。）
%\usepackage{tikz-3dplot}
%\usetikzlibrary{calc}
%\usetikzlibrary{arrows.meta,patterns,angles,intersections,backgrounds}
%\usepackage{animate}
%\usepackage{pgfplots}
%\pgfplotsset{compat=newest}
%\pgfplotsset{compat=newest, every axis/.append style={line width=1pt}}
%\PassOptionsToPackage{dvipdfmx}{graphicx}
\nofiles

\hypersetup{% Jul 18 2023
	pdftitle={プレ高数学科（LaTeX Beamer で汎用PDFプレゼンを作る）},
	pdfauthor={gbb60166@gmail.com},
	pdfsubject={https://unilab.gbb60166.jp/prekou/prekou.htm}
	}

% フォントはお好みで
%\usepackage{txfonts}

\mathversion{bold}
\renewcommand{\familydefault}{\sfdefault}
\renewcommand{\seriesdefault}{bx}
\renewcommand{\kanjifamilydefault}{\gtdefault}
\renewcommand{\kanjiseriesdefault}{bx}
\setbeamerfont{title}{size=\normalsize,series=\bfseries}
\setbeamerfont{frametitle}{size=\normalsize,series=\bfseries}
\setbeamertemplate{frametitle}[default][center]
\usefonttheme{professionalfonts}

% 参考にしたURL
% http://windom.phys.hirosaki-u.ac.jp/fswiki/wiki.cgi?page=LaTeX+Beamer%A4%C7%A5%D7%A5%EC%A5%BC%A5%F3%A5%C6%A1%BC%A5%B7%A5%E7%A5%F3

\newcommand{\Slash}[1]{\ooalign{\hfil\kern-3pt／\hfil\crcr$#1$}}
\everymath{\displaystyle}

\def\maruwaku#1{\begin{tikzpicture}[scale=0.7, baseline={([yshift=-22pt] current bounding box.north)}]
\filldraw[color=CUDBlue, line width=1pt, rounded corners=2pt] (-0.1,0)--(2.1,0)--(2.1,1.1)--(-0.1,1.1)--cycle;
\draw(1,0.5) node[white]{#1};
\end{tikzpicture}
}

\setbeamersize{text margin left=5mm,text margin right=5mm}
\fboxrule=0.8pt

\makeatletter
\def\hooklen#1#2{\settowidth{\@tempdima}{\(#1\)}
%\advance\@tempdima by.3ex
% ↑ 数式モードで式の前後に入るスペースを制御したかったが、
%    難しいのでやめた。段々難解なコードになっているのでやめた方がよい？
\hbox to\@tempdima{\hfil \(#2\)\hfil}}
%■等号付き不等号 by 美文書作成入門
\newcommand{\myleqq}{\mathrel{\mathpalette\gl@align<}}
\newcommand{\mygeqq}{\mathrel{\mathpalette\gl@align>}}
\newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@
	\ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
\makeatother

% カラーユニバーサルデザインを調べたつもりだがあまり自信がありません
% http://www.fukushihoken.metro.tokyo.jp/kiban/machizukuri/kanren/color.files/colorudguideline.pdf
% http://jfly.iam.u-tokyo.ac.jp/colorset/

% ■ アクセントカラー小面積を目立たせる高彩度色
\definecolor{CUDRed}{RGB}{255,75,0}
\definecolor{CUDGreen}{RGB}{3,175,122}
\definecolor{CUDBlue}{RGB}{0,90,255}
\definecolor{CUDCyan}{RGB}{77,196,255}
\definecolor{CUDMagenta}{RGB}{153,0,153}
\definecolor{CUDYellow}{RGB}{255,241,0}
\definecolor{CUDBrown}{RGB}{128,64,0}

\definecolor{CUDOrange}{RGB}{246,170,0}

% ■ ベースカラー広い面積の塗り分けに用いる低・中彩度色
\definecolor{CUDPink}{RGB}{255,202,191}
\definecolor{CUDBrightGreen}{RGB}{119,217,168}
\definecolor{CUDLime}{RGB}{216,242,85}
\definecolor{CUDCream}{RGB}{255,255,128}
\definecolor{CUDBrightCyan}{RGB}{191,228,255}
%\definecolor{CUD}{RGB}{}%

\setbeamercolor{CUDBrightGreen}{fg=black,bg=CUDBrightGreen!50}
\setbeamercolor{CUDBrightCyan}{fg=black,bg=CUDBrightCyan}
\setbeamercolor{CUDCream}{fg=black,bg=CUDCream!75}

%\setbeamercolor{frametitle}{use=structure,fg=white,bg=structure.fg}
%\setbeamercolor{palette primary}{use=structure,fg=white,bg=structure.fg}
%\setbeamercolor{frametitle}{bg=CUDOrange}
%\setbeamercolor{palette primary}{bg=CUDOrange}
%\setbeamercolor{frametitle}{bg=CUDCyan!25,fg=black}
%\setbeamercolor{palette primary}{bg=CUDCyan!25,fg=black}
%\setbeamercolor{frametitle}{bg=CUDCream!75,fg=black!80}
%\setbeamercolor{palette primary}{bg=CUDCream!75,fg=black!80}


\begin{document}

\title{プレ高数学科}\author{gbb60166}

%■ 本文と数式モードの間に挿入される空白スペース
%\mathsurround=0pt
%■ \( \log\log \)でlogとlogの間に挿入される細スペース（省略形は \,）
%\thinmuskip=3mu%■default
%■ \( a+b \)で+の前後に挿入される中スペース（省略形は \>）
%\medmuskip=4mu plus 2mu minus 4mu%■default
\medmuskip=1mu plus 1mu minus 1mu
%■ \( x=5 \)で=の前後に挿入される太スペース（省略形は \;）
%\thickmuskip=5mu plus 5mu%■default
\thickmuskip=3mu plus 2mu minus 2mu
% mu は math units の省略で長さの単位

\newcommand{\myfrac}[2]{\mathchoice%■ 修正 Oct 8,2023
	{%\fontsize{20}{20}\selectfont%	\displaystyle
		\frac{\raisebox{-.6ex}{\(#1\)}}{\raisebox{.4ex}{\(#2\)}}}%
	{\textstyle{\frac{#1}{#2}}}%	\textstyle
	{%\fontsize{13}{13}\selectfont%	\scriptstyle
		\frac{\raisebox{0ex}{\(#1\)}}{\raisebox{-.4ex}{\(#2\)}}}%
	{%\fontsize{11}{11}\selectfont%	\scriptscriptstyle
		\frac{\raisebox{0ex}{\(#1\)}}{\raisebox{-.4ex}{\(#2\)}}}%
	}

\newcommand{\ittan}{\begin{tikzpicture}[scale=0.7,
	baseline={([yshift=-21pt] current bounding box.north)}]
\filldraw[very thick,draw=CUDBlue,fill=CUDYellow,rounded corners=1pt]
	(0,0)--++(0.8,0)--++(0,0.8)--++(-0.8,0)--cycle;
\draw(0.4,0.4) node[CUDBlue]
	{\fontsize{7pt}{7pt}\selectfont
		\shortstack{一旦\\[-2pt]停止}};
\end{tikzpicture}
	}



%■■■■■■■■■■■　実験エリア　■■■■■■■■■■■■■■■■




%\end{document}
%■■■■■■■■■■■　ここから本文　■■■■■■■■■■■■■■■■

\begin{frame}[t]
\frametitle{関数の長さ（積分）}
\maruwaku{問題}\hskip1ex
\begin{minipage}[t]{.7\textwidth}
\(y=x^\frac{\,3\,}{2}\)の、区間\(0\myleqq x \myleqq\myfrac{\,4\,}{3}\)に
おける長さを求めなさい。
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{公式はこれです}
\maruwaku{公式}\hskip1ex
\begin{minipage}[t]{.8\textwidth}
\(y=f(x)\)の区間\(a\myleqq x\myleqq b\)における\\[1ex]
長さ$s$は

\bigskip
\centerline{%
\begin{beamercolorbox}[wd=0.9\textwidth,ht=4ex,dp=2.4ex,shadow=true,
	rounded=true, center]{CUDCream}
\( s=\int_a^b \sqrt{1+\{f'(x)\}^2}\quad dx \)
\end{beamercolorbox}
}
\end{minipage}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{まず\(f'(x)\)を求める}
\(f(x)=x^\frac{\,3\,}{2}\)を微分すると
\(f'(x)=\myfrac{\,3\,}{2}\, x^\frac{\,1\,}{2}\)となる。\ittan

\bigskip
積分する区間は\(0\myleqq x \myleqq \myfrac{\,4\,}{3}\)なので、
公式に代入して
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{公式の代入して整理する}
\vspace*{-3.6ex}
\begin{eqnarray*}
& & \int_a^b \sqrt{1+\{f'(x)\}^2}\quad dx \\
&=& \int_0^\frac{\,4\,}{3}
	\sqrt{1+\Bigl(\myfrac{\,3\,}{2}\, x^\frac{\,1\,}{2}\Bigr)^2}\ dx \\
&=& \int_0^\frac{\,4\,}{3} \sqrt{1+\myfrac{\,9\,}{4}\, x}\quad dx
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{置換積分を使う}
\vspace*{-4ex}
\begin{eqnarray*}
&=& \int_0^\frac{\,4\,}{3} \sqrt{1+\myfrac{\,9\,}{4}\, x}\quad dx \\
&=& \int_0^\frac{\,4\,}{3}
	\Bigl(1+\myfrac{\,9\,}{4}\, x\Bigr)^\frac{\,1\,}{2}\ dx
	\quad\ittan
\end{eqnarray*}

\(u=1+\myfrac{\,9\,}{4}\, x\)と置いて\textcolor{CUDRed}{置換積分}をする
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setbeamercolor{frametitle}{bg=CUDCream!75,fg=black!80}
\setbeamercolor{palette primary}{bg=CUDCream!75,fg=black!80}

\begin{frame}[t]
%\frametitle{\( dx \)を\( du \)で表すと}
\frametitle{\( u=1+\tfrac{\,9\,}{4}\, x\)の置換積分は…}
\(u=1+\myfrac{\,9\,}{4}\, x\)の両辺を\(x\)で微分すると

\vspace*{-2ex}
\begin{eqnarray*}
\myfrac{du}{dx} &=& \myfrac{\,9\,}{4}\\[1.5ex]
9\, dx &=& 4\, du\\[1.5ex]
dx &=& \myfrac{\,4\,}{9}\, du \quad\ittan
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{積分の区間を求める}
\(u=1+\myfrac{\,9\,}{4}\, x\)なので
\(\textcolor{CUDRed}{0}\myleqq x \myleqq
	\textcolor{CUDBlue}{\myfrac{\,4\,}{3}}\)のとき

\qquad
\(x=\hooklen{\myfrac{\,4\,}{3}}{\textcolor{CUDRed}{0}}\)では\quad
\(1+\myfrac{\,9\,}{4}\times\hooklen{\myfrac{\,4\,}{3}}{\textcolor{CUDRed}{0}}
	= \colorbox{CUDRed}{\textcolor{white}{$1$}} \)\quad\ittan

\bigskip
\qquad
\(x=\textcolor{CUDBlue}{\myfrac{\,4\,}{3}}\)では\quad
\(1+\myfrac{\,9\,}{4}\times\textcolor{CUDBlue}{\myfrac{\,4\,}{3}}
	= \colorbox{CUDBlue}{\textcolor{white}{$4$}} \)\quad\ittan

\bigskip
なので\( \colorbox{CUDRed}{\textcolor{white}{$1$}}\myleqq u
	\myleqq \colorbox{CUDBlue}{\textcolor{white}{$4$}} \)となる。
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setbeamercolor{frametitle}{use=structure,fg=white,bg=structure.fg}
\setbeamercolor{palette primary}{use=structure,fg=white,bg=structure.fg}

\begin{frame}[t]
\frametitle{あとは、これを計算するだけ}
元に戻って、すべてをまとめると
\begin{eqnarray*}
\int_{\textcolor{CUDRed}{0}}^{\textcolor{CUDBlue}{\frac{\,4\,}{3}}}
	\Bigl( \textcolor{CUDGreen}{1+\myfrac{\,9\,}{4}\, x}\Bigr)^\frac{\,1\,}{2}
		\textcolor{CUDMagenta}{dx}
	&=& \int_{\text{\colorbox{CUDRed}{\textcolor{white}{$1$}}}}
		^{\text{\colorbox{CUDBlue}{\textcolor{white}{$4$}}}}
		{\textcolor{CUDGreen}{u}}^\frac{\,1\,}{2}\, 
		\textcolor{CUDMagenta}{\myfrac{\,4\,}{9}\, du} \\[1ex]
&=& \myfrac{\,4\,}{9} \int_1^4 u^\frac{\,1\,}{2}\, du
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{ひたすら計算}
\vspace*{-3.6ex}
\begin{eqnarray*}
&=& \myfrac{\,4\,}{9} \int_1^4 u^\frac{\,1\,}{2}\, du \\[1ex]
&=& \myfrac{\,4\,}{9} \left[\myfrac{\,2\,}{3}u^\frac{\,3\,}{2}\right]_1^4
	\quad =\myfrac{\,4\,}{9}\times\myfrac{\,2\,}{3}
		\left[u^\frac{\,3\,}{2}\right]_1^4 \\[1ex]
&=& \myfrac{\,8\,}{27} \bigl(4^\frac{\,3\,}{2}-1^\frac{\,3\,}{2}\bigr)
	\quad = \myfrac{\,8\,}{27} \bigl(4\sqrt{4}-1\bigr)
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\frametitle{やっと答えにたどり着いた}
\vspace*{-3ex}
\begin{eqnarray*}
&=& \myfrac{\,8\,}{27} \bigl(4\sqrt{4}-1\bigr)
	\quad = \myfrac{\,8\,}{27} \left(8-1\right) \\[1ex]
&=& \myfrac{\,8\,}{27} \times 7 \quad = \myfrac{\,56\,}{27}\quad\text{\□答}
\end{eqnarray*}
\vskip2ex
\centerline{頭がついて来ますか？}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\end{document}
