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();
explanation: The provided text extract is a detailed mathematical derivation and proof, followed by a Python script and a JavaScript code snippet. The main content is a proof involving a function \( f(x) \) and its derivative, leading to an inequality. The Python script and JavaScript code are used to visualize the function and its properties.
  
  The text does not contain a specific question or a question-and-answer pair. It is primarily a presentation of a mathematical proof and related code.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ## 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();
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text extract is a detailed mathematical derivation and proof, followed by a Python script and a JavaScript code snippet. The main content is a proof involving a function \( f(x) \) and its derivative, leading to an inequality. The Python script and JavaScript code are used to visualize the function and its properties.
  
  The text does not contain a specific question or a question-and-answer pair. It is primarily a presentation of a mathematical proof and related code.
  
  Label: no-question