text: ## 2017年11月24日金曜日
  
  ### 数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 一般の指数関数(微分、導関数、累乗(べき乗)、不等号)
  
  1. $\begin{array}{}f\left(x\right)=\frac{x}{p}+\frac{1}{q}-{x}^{\frac{1}{p}}\\ f\text{'}\left(x\right)\\ =\frac{1}{p}-\frac{1}{p}{x}^{\frac{1}{p}-1}\\ =\frac{1}{p}\left(1-{x}^{\frac{1}{p}-1}\right)\\ f\text{'}\left(1\right)=\frac{1}{p}\left(1-1\right)=0\\ p\ge 1\\ \frac{1}{p}\le 1\\ \frac{1}{p}-1\le 0\\ 00\\ f\left(1\right)=\frac{1}{p}+\frac{1}{q}-1=0\end{array}$
  
  よって、
  
  $\begin{array}{}f\left(x\right)\ge 0\\ \frac{x}{p}+\frac{1}{q}-{x}^{\frac{1}{p}-1}\ge 0\\ {x}^{\frac{1}{p}-1}\le \frac{x}{p}+\frac{1}{q}\end{array}$
  
  （証明終）
  
  コード(Emacs)
  
  Python 3
  
  #!/usr/bin/env python3
  from sympy import pprint, symbols, Derivative, solve
  
  x, p, q = symbols('x, p, q')
  f = x / p + 1 / q - x ** (1 / p)
  D = Derivative(f, x, 1)
  f1 = D.doit()
  for t in [D, f1, f1.factor()]:
  pprint(t)
  print()
  
  pprint(solve(1 / p + 1 / q - 1, q))
  
  
  $./sample12.py ∂ ⎛ p ___ 1 x⎞ ──⎜- ╲╱ x + ─ + ─⎟ ∂x⎝ q p⎠ p ___ 1 ╲╱ x ─ - ───── p p⋅x p ___ x - ╲╱ x ───────── p⋅x$
  
  
  HTML5
  
  <div id="graph0"></div>
  <pre id="output0"></pre>
  <label for="r0">r = </label>
  <input id="r0" type="number" min="0" value="0.5">
  <label for="dx">dx = </label>
  <input id="dx" type="number" min="0" step="0.001" value="0.005">
  <br>
  <label for="x1">x1 = </label>
  <input id="x1" type="number" value="-10">
  <label for="x2">x2 = </label>
  <input id="x2" type="number" value="10">
  <br>
  <label for="y1">y1 = </label>
  <input id="y1" type="number" value="-10">
  <label for="y2">y2 = </label>
  <input id="y2" type="number" value="10">
  <br>
  <label for="p0">p0 = </label>
  <input id="p0" type="number" min="1" value="2">
  
  <button id="draw0">draw</button>
  <button id="clear0">clear</button>
  
  <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="sample12.js"></script>
  
  
  JavaScript
  
  let div0 = document.querySelector('#graph0'),
  pre0 = document.querySelector('#output0'),
  width = 600,
  height = 600,
  btn0 = document.querySelector('#draw0'),
  btn1 = document.querySelector('#clear0'),
  input_r = document.querySelector('#r0'),
  input_dx = document.querySelector('#dx'),
  input_x1 = document.querySelector('#x1'),
  input_x2 = document.querySelector('#x2'),
  input_y1 = document.querySelector('#y1'),
  input_y2 = document.querySelector('#y2'),
  input_p0 = document.querySelector('#p0'),
  inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
  input_p0],
  p = (x) => pre0.textContent += x + '\n',
  range = (start, end, step=1) => {
  let res = [];
  for (let i = start; i < end; i += step) {
  res.push(i);
  }
  return res;
  };
  
  let draw = () => {
  pre0.textContent = '';
  
  let r = parseFloat(input_r.value),
  dx = parseFloat(input_dx.value),
  x1 = parseFloat(input_x1.value),
  x2 = parseFloat(input_x2.value),
  y1 = parseFloat(input_y1.value),
  y2 = parseFloat(input_y2.value),
  p0 = parseFloat(input_p0.value),
  q0 = p0 / (p0 - 1);
  
  if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
  return;
  }
  
  let points = [],
  lines = [[1, y1, 1, y2, 'red']],
  f = (x) => x / p0 + 1 / q0,
  g = (x) => x ** (1 / p0)
  fns = [[f, 'green'],
  [g, 'blue']],
  fns1 = [],
  fns2 = [];
  
  fns
  .forEach((o) => {
  let [f, color] = o;
  for (let x = x1; x <= x2; x += dx) {
  let y = f(x);
  
  points.push([x, y, color]);
  }
  });
  
  fns1
  .forEach((o) => {
  let [f, color] = o;
  
  lines.push([x1, f(x1), x2, f(x2), color]);
  });
  
  fns2
  .forEach((o) => {
  let [f, color] = o;
  
  for (let x = x1; x <= x2; x += dx0) {
  let g = f(x);
  lines.push([x1, g(x1), x2, g(x2), color]);
  }
  });
  
  let xscale = d3.scaleLinear()
  .domain([x1, x2])
  let yscale = d3.scaleLinear()
  .domain([y1, y2])
  
  let xaxis = d3.axisBottom().scale(xscale);
  let yaxis = d3.axisLeft().scale(yscale);
  div0.innerHTML = '';
  let svg = d3.select('#graph0')
  .append('svg')
  .attr('width', width)
  .attr('height', height);
  
  svg.selectAll('line')
  .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
  .enter()
  .append('line')
  .attr('x1', (d) => xscale(d[0]))
  .attr('y1', (d) => yscale(d[1]))
  .attr('x2', (d) => xscale(d[2]))
  .attr('y2', (d) => yscale(d[3]))
  .attr('stroke', (d) => d[4] || 'black');
  
  svg.selectAll('circle')
  .data(points)
  .enter()
  .append('circle')
  .attr('cx', (d) => xscale(d[0]))
  .attr('cy', (d) => yscale(d[1]))
  .attr('r', r)
  .attr('fill', (d) => d[2] || 'green');
  
  svg.append('g')
  .attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
  .call(yaxis);
  
  [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
  };
  
  inputs.forEach((input) => input.onchange = draw);
  btn0.onclick = draw;
  btn1.onclick = () => pre0.textContent = '';
  draw();