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20章(面積、体積、長さ - 積分法の応用)、20.4(簡単な微分方程式)、二三の応用、問44.を取り組んでみる。
-
P(x, y)を平面上の任意の点とする。
点Pを通る曲線を考える。
この曲線の点Pにおける接線の傾きを考える。
求める曲線は、点Pにおいて問題の曲線の傾きと直交する接線をもつ曲線。
求める曲線をy = f(x)とおく。
x = 0のとき(y軸、直線)、原点において、問題の曲線の傾きは0なので、すべての問題の曲線と直交する。
よって、求める曲線。
コード(Emacs)
Python 3
#!/usr/bin/env python3 from sympy import pprint, symbols, sqrt, plot, Rational, Derivative, solve print('44.') k, x, C = symbols('k x C', real=True) f = k * x ** 3 g1 = sqrt(C - x ** 2 / 3) g2 = -g1 gs = [g1.subs({C: C0}) for C0 in range(1, 6)] + \ [g2.subs({C: C0}) for C0 in range(1, 6)] for t in [f, g1, g2]: pprint(t) print() k0 = Rational(1, 20) C0 = 1 Df = Derivative(f, x, 1).doit().subs({k: k0}) Dg = Derivative(g1, x, 1).doit().subs({C: C0}) x0 = solve((f - g1).subs({k: k0, C: C0}), x)[0] pprint(x0) h1 = Df.subs({k: k0, x: x0, C: C0}) * (x - x0) + f.subs({k: k0, x: x0, C: C0}) h2 = Dg.subs({k: k0, x: x0, C: C0}) * (x - x0) + g1.subs({k: k0, x: x0, C: C0}) p = plot(f.subs({k: k0}), h1, h2, *gs, (x, -5, 5), ylim=(-5, 5), show=False) p[0].line_color = 'red' p[1].line_color = 'green' p[2].line_color = 'orange' p.save('sample44.svg')
入出力結果(Terminal, Jupyter(IPython))
$ ./sample44.py 44. 3 k⋅x ________ ╱ 2 ╱ x ╱ C - ── ╲╱ 3 ________ ╱ 2 ╱ x - ╱ C - ── ╲╱ 3 $
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="k0">k = </label> <input id="k0" type="number" value="1"> <label for="c0">C = </label> <input id="c0" type="number" value="1"> <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="sample44.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_k0 = document.querySelector('#k0'), input_c0 = document.querySelector('#c0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_k0, input_c0], 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), k0 = parseFloat(input_k0.value), c0 = parseFloat(input_c0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [], f = (x) => k0 * x ** 3, g1 = (x) => Math.sqrt(c0 - x ** 2 / 3), g2 = (x) => -g1(x), fns = [[f, 'red'], [g1, 'green'], [g2, 'green']], 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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