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, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第19章(細分による加法 - 積分法)、19.2(不定積分の計算)、置換積分法、問13.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, Integral, sqrt, log, plot print('13.') x = symbols('x') n = symbols('n', positive=True, integer=True) fs = [x * sqrt(3 - x), x * sqrt(3 - x), log(x) ** 2 / x, x / sqrt(x ** 2 + 1), x / (x ** 2 + 1) ** n] for i, f in enumerate(fs, 1): print(f'({i})') I = Integral(f, x) for o in [I, I.doit()]: pprint(o.factor()) print() try: p = plot(f, show=False, legend=True) p.save(f'sample13_{i}.svg') except Exception as err: print(type(err), err) print()
入出力結果(Terminal, IPython)
$ ./sample13.py 13. (1) ⌠ ⎮ ________ ⎮ x⋅╲╱ -x + 3 dx ⌡ ⎧ 2 _______ _______ _______ ⎪2⋅ⅈ⋅x ⋅╲╱ x - 3 2⋅ⅈ⋅x⋅╲╱ x - 3 12⋅ⅈ⋅╲╱ x - 3 │x│ ⎪──────────────── - ─────────────── - ────────────── for ─── > 1 ⎪ 5 5 5 3 ⎨ ⎪ 2 ________ ________ ________ ⎪ 2⋅x ⋅╲╱ -x + 3 2⋅x⋅╲╱ -x + 3 12⋅╲╱ -x + 3 ⎪ ─────────────── - ────────────── - ───────────── otherwise ⎩ 5 5 5 (2) ⌠ ⎮ ________ ⎮ x⋅╲╱ -x + 3 dx ⌡ ⎧ 2 _______ _______ _______ ⎪2⋅ⅈ⋅x ⋅╲╱ x - 3 2⋅ⅈ⋅x⋅╲╱ x - 3 12⋅ⅈ⋅╲╱ x - 3 │x│ ⎪──────────────── - ─────────────── - ────────────── for ─── > 1 ⎪ 5 5 5 3 ⎨ ⎪ 2 ________ ________ ________ ⎪ 2⋅x ⋅╲╱ -x + 3 2⋅x⋅╲╱ -x + 3 12⋅╲╱ -x + 3 ⎪ ─────────────── - ────────────── - ───────────── otherwise ⎩ 5 5 5 (3) ⌠ ⎮ 2 ⎮ log (x) ⎮ ─────── dx ⎮ x ⌡ 3 log (x) ─────── 3 (4) ⌠ ⎮ x ⎮ ─────────── dx ⎮ ________ ⎮ ╱ 2 ⎮ ╲╱ x + 1 ⌡ ________ ╱ 2 ╲╱ x + 1 (5) ⌠ ⎮ -n ⎮ ⎛ 2 ⎞ ⎮ x⋅⎝x + 1⎠ dx ⌡ ⎧ ⎛ 2 ⎞ ⎪ log⎝x + 1⎠ ⎪ ─────────── for n = 1 ⎪ 2 ⎪ ⎨ 2 ⎪ x 1 ⎪- ─────────────────────────── - ─────────────────────────── otherwise ⎪ n n n n ⎪ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎩ 2⋅n⋅⎝x + 1⎠ - 2⋅⎝x + 1⎠ 2⋅n⋅⎝x + 1⎠ - 2⋅⎝x + 1⎠ <class 'ValueError'> The same variable should be used in all univariate expressions being plotted. $
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="0.0001" value="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-5"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-5"> <label for="y2">y2 = </label> <input id="y2" type="number" value="5"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="2" step="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="sample13.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'), pre0 = document.querySelector('#output0'), width = 600, height = 600, padding = 50, 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_n0 = document.querySelector('#n0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_n0], 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 f3 = (x) => Math.log(x) ** 2 / x; 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), n0 = parseInt(input_n0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [], f5 = (x) => x / (x ** 2 + 1) ** n0, fns = [[f3, 'green'], [f5, 'orange']], fns1 = [], fns2 = []; fns.forEach((o) => { let [fn, color] = o; for (let x = x1; x <= x2; x += dx) { let y = fn(x); if (Math.abs(y) < Infinity) { points.push([x, y, color]); } } }); fns1.forEach((o) => { let [fn, color] = o; lines.push([x1, fn(x1), x2, fn(x2), color]); }); fns2.forEach((o) => { let [fn, color] = o; for (let x = x1; x <= x2; x += dx0) { let g = fn(x); lines.push([x1, g(x1), x2, g(x2), color]); } }); let xscale = d3.scaleLinear() .domain([x1, x2]) .range([padding, width - padding]); let yscale = d3.scaleLinear() .domain([y1, y2]) .range([height - padding, padding]); 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();
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