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〈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18章(曲線の性質、最大・最小 - 微分法の応用)、18.2(関数の増減の判定およびその応用)、導関数の符号と関数の増減、問2、3、4、5.を取り組んでみる。
y'=2ax+b2ax+b=0x=−b2ax -b/2a y' - 0 + y ↘︎ ↗︎
y'=3x2+2ax+b3x2+2ax+b=0D4=a2−3b-
y'=12−3x212−3x2=0x=±2−24+8=−1624−8=16x -2 2 y' - 0 + 0 - y ↘︎ -16 ↗︎ 16 ↘︎
y'=x2+2xx2+2x=0x(x+2)=0x=0,−20−83+4=43x -2 0 y' + 0 - 0 + y ↗︎ 4/3 ↘︎ 0 ↗︎
y'=3x2+1≥1増加関数で、極値はない。
y'=−2+2x−3x23x2−2x+2=0D4=1−6=−5<0y'<0減少関数で、極値はない。
y'=4x3−4x=4x(x2−1)=4x(x+1)(x−1)1−2+1=011−2+1=0x -1 0 1 y' - 0 + 0 - 0 + y ↘︎ 0 ↗︎ 1 ↘︎ 0 ↗︎
y'=4x3+2x=2x(x2+1)0x 0 y' - 0 + y ↘︎ 0 ↗︎
y'=4x3−12x2=4x2(x−3)0−27x 0 3 y' - 0 - 0 + y ↘︎ 0 ↘︎ -27 ↗︎
y=|x2(x−3)|x<3y=−x3+3x2y'=−3x2+6x=−3x(x−2)x≥3y=x3−3x2y'=3x2−6x=3x(x−2)0−8+3·4=427−27=0x 0 2 3 y' - 0 + 0 - + y ↘︎ 0 ↗︎ 4 ↘︎ 0 ↗︎
y'=1−1x21−1x2=0x2−1=0x=±1−1−1=−21+1=2x -1 0 1 y' + 0 - - 0 + y ↗︎ -2 ↘︎ ↘︎ 2 ↗︎
y=ex+xexex+xex=0x=−1e0+0e0=1x -1 y' - 0 + y ↘︎ 1 ↗︎
y'=1+2cosx1+2cosx=0cosx=−12x=23π,43π23π+2√32=23π+√343π+2·(−√32)=43π−√3x 2π/3 4π/3 y' + 0 - 0 + y ↗︎ 2π/3 + √3 ↘︎ 4π/3 - √3 ↗︎
-
y=ax3+bx2+cx+dy'=3ax2+2bx+c3a−2b+c=0−a+b−c+d=1012a+4b+c=08a+4b+2c+d=−17c=−3a+2b−a+b+3a−2b+d=1012a+4b−3a+2b=08a+4b−6a+4b+d=−172a−b+d=103a+2b=02a+8b+d=−17d=10−2a+b3a+2b=02a+8b+10−2a+b=−179b=−27b=−3a=2d=10−4−3=3c=−6−6=−12y=2x3−3x2−12x+3
c=0d=−812a+4b+c=08a+4b+2c+d=012a+4b=08a+4b−8=03a+b=02a+b−2=0b=−3a2a−3a−2=0a=−2b=6y=−2x3+6x2−8
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, sin, plot, pi, sqrt print('4 (11).') x = symbols('x') f = x + 2 * sin(x) g = 2 * pi / 3 + sqrt(3) h = 4 * pi / 3 - sqrt(3) p = plot(f, g, h, (x, 0, 2 * pi), show=False, legend=True) for i, color in enumerate(['red', 'green', 'blue']): p[i].line_color = color p.save('sample2.svg')
入出力結果(Terminal, IPython)
$ ./sample2.py 4 (11). $
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="-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="a0">a = </label> <input id="a0" type="number" value="-2"> <label for="b0">b = </label> <input id="b0" type="number" value="6"> <label for="c0">c = </label> <input id="c0" type="number" value="0"> <label for="d0">d = </label> <input id="d0" type="number" value="-8"> <br> <label for="x0">x0 = </label> <input id="x0" type="number" value="0"> <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="sample2.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_a0 = document.querySelector('#a0'), input_b0 = document.querySelector('#b0'), input_c0 = document.querySelector('#c0'), input_d0 = document.querySelector('#d0'), input_x0 = document.querySelector('#x0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_a0, input_b0, input_c0, input_d0, input_x0], 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), a = parseFloat(input_a0.value), b = parseFloat(input_b0.value), c = parseFloat(input_c0.value), d = parseFloat(input_d0.value), x0 = parseFloat(input_x0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [], f = (x) => a * x ** 3 + b * x ** 2 + c * x + d, f1 = (x) => 3 * a * x ** 2 + 2 * b * x + c, g = (x) => f1(x0) * (x - x0) + f(x0), fns = [[f, 'green']], fns1 = [[g, 'blue']]; 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]); }); 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); p(fns); }; inputs.forEach((input) => input.onchange = draw); btn0.onclick = draw; btn1.onclick = () => pre0.textContent = ''; draw();
(x) => a * x ** 3 + b * x ** 2 + c * x + d,green
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