数学 - JavaScript - D3.js - 解析学 - 微分法 - 関数の凹凸(増減、極値、変曲点)

解析入門〈1〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第4章(微分法)、4.3(関数の凹凸)、問題4.3-1.を取り組んでみる。

2. 
   $\begin{array}{l}f\text{'}\left(x\right)=3x^2+2ax+b\\ f\text{'}\text{'}\left(x\right)=6x+2a\\ 6x+2a=0\\ x=\frac{1}{3}a\end{array}$
3. 
   $\begin{array}{l}f\text{'}\left(x\right)=\frac{2\left(1+x^2\right)-2x·2x}{{\left(1+x^2\right)}^2}=\frac{2\left(1-x^2\right)}{{\left(1+x^2\right)}^2}\\ f\text{'}\text{'}\left(x\right)=\frac{-4x{\left(1+x^2\right)}^2-2\left(1-x^2\right)·2\left(1+x^2\right)·2x}{{\left(1+x^2\right)}^4}\\ =\frac{4x\left(1+x^2\right)\left(-1-x^2-2+2x^2\right)}{{\left(1+x^2\right)}^4}\\ =\frac{4x\left(1+x^2\right)\left(x^2-3\right)}{{\left(1+x^2\right)}^4}\\ =\frac{4x\left(x^2-3\right)}{{\left(1+x^2\right)}^3}\\ f\text{'}\left(x\right)=0,x=\pm 1\\ f\text{'}\text{'}\left(x\right)=0,x=0,\pm \sqrt{3}\\ f\left(-\sqrt{3}\right)=-\frac{\sqrt{3}}{2},f\left(-1\right)=-1,f\left(0\right)=0,f\left(1\right)=1,f\left(\sqrt{3}\right)=\frac{\sqrt{3}}{2}\end{array}$
   
   

   x		-√3		-1		0		1		√3
   f'	-	-	-	0	+	+	+	0	-	-	-
   f''	-	0	+	+	+	0	-	-	-	0	+
   f	-	-√3/2	-	-1	+	0	+	1	-	√3/2	-

   極小値: -1(x=-1)、極大値: 1(x=1)、変曲点: (-√3, -√3/2)、(0, 0)、(0, √3/2)

   コード(Emacs)

   HTML5

   <div id="graph0"></div>
   
   <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>
   <script src="sample3.js"></script>    
   

   JavaScript

   let width = 500,
       height = 500,
       padding = 20,
       f = (x) => 2 * x / (1 + Math.pow(x, 2));
   
   let points = []
   
   for (let x = -5; x <= 5; x += 0.01) {
       points.push([x, f(x)]);
   }
   
   let xscale = d3.scaleLinear()
       .domain([-5, 5])
       .range([padding, width - padding]);
   let yscale = d3.scaleLinear()
       .domain([-5, 5])
       .range([height - padding, padding]);
   let xaxis = d3.axisBottom().scale(xscale);
   let yaxis = d3.axisLeft().scale(yscale);
   let svg = d3.select('#graph0')
       .append('svg')
       .attr('width', width)
       .attr('height', height);
   
   svg.selectAll('circle')
       .data(points)
       .enter()
       .append('circle')
       .attr('cx', (d) => xscale(d[0]))
       .attr('cy', (d) => yscale(d[1]))
       .attr('r', 1)
       .attr('fill', 'green');
   
   svg.append('g')
       .attr('transform', `translate(0, ${height / 2})`)
       .call(xaxis);
   
   svg.append('g')
       .attr('transform', `translate(${width / 2}, 0)`)
       .call(yaxis);