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
- 参考書籍
数学読本〈4〉数列の極限,順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.5(いろいろな関数の導関数)、逆三角関数とその微分、問55、56、57.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, asin, atan, sqrt, exp, Derivative x = symbols('x') print('57.') funcs = [asin(x / 2), atan(- x / 3), asin((2 * x + 1) / sqrt(5)), exp(atan(x))] for i, func in enumerate(funcs, 1): print('({})'.format(i)) pprint(func) d = Derivative(func, x) f1 = d.doit() pprint(f1) pprint(factor(f1)) pprint(expand(f1)) pprint(f1.factor())
入出力結果(Terminal, IPython)
$ ./sample55.py 57. (1) ⎛x⎞ asin⎜─⎟ ⎝2⎠ 1 ───────────────── __________ ╱ 2 ╱ x 2⋅ ╱ - ── + 1 ╲╱ 4 1 ──────────────────── __________________ ╲╱ -(x - 2)⋅(x + 2) 1 ───────────────── __________ ╱ 2 ╱ x 2⋅ ╱ - ── + 1 ╲╱ 4 1 ──────────────────── __________________ ╲╱ -(x - 2)⋅(x + 2) (2) ⎛x⎞ -atan⎜─⎟ ⎝3⎠ -1 ────────── ⎛ 2 ⎞ ⎜x ⎟ 3⋅⎜── + 1⎟ ⎝9 ⎠ -3 ────── 2 x + 9 -1 ────────── ⎛ 2 ⎞ ⎜x ⎟ 3⋅⎜── + 1⎟ ⎝9 ⎠ -3 ────── 2 x + 9 (3) ⎛√5⋅(2⋅x + 1)⎞ asin⎜────────────⎟ ⎝ 5 ⎠ 2⋅√5 ───────────────────────── __________________ ╱ 2 ╱ (2⋅x + 1) 5⋅ ╱ - ────────── + 1 ╲╱ 5 2 ───────────────────── __________________ ╱ 2 ╲╱ - 4⋅x - 4⋅x + 4 2⋅√5 ───────────────────────── __________________ ╱ 2 ╱ 4⋅x 4⋅x 4 5⋅ ╱ - ──── - ─── + ─ ╲╱ 5 5 5 2 ───────────────────── __________________ ╱ 2 ╲╱ - 4⋅x - 4⋅x + 4 (4) atan(x) ℯ atan(x) ℯ ──────── 2 x + 1 atan(x) ℯ ──────── 2 x + 1 atan(x) ℯ ──────── 2 x + 1 atan(x) ℯ ──────── 2 x + 1 $
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"> <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="sample55.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'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2], 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 f = (x) => Math.exp(Math.atan(x)), f1 = (x) => Math.exp(Math.atan(x)) / (1 + x ** 2), g = (x0) => (x) => f1(x0) * (x - x0) + f(x0); 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); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = []; for (let x = x1; x <= x2; x += dx) { let y = f(x); if (Math.abs(y) < Infinity) { points.push([x, y]); } } let lines = []; for (let x = x1; x <= x2; x += 100 * dx) { let g0 = g(x); lines.push([x1, g0(x1), x2, g0(x2), 'blue']); } 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', 'green'); svg.append('g') .attr('transform', `translate(0, ${height - padding})`) .call(xaxis); svg.append('g') .attr('transform', `translate(${padding}, 0)`) .call(yaxis); }; inputs.forEach((input) => input.onchange = draw); btn0.onclick = draw; btn1.onclick = () => pre0.textContent = ''; draw();
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