## 2017年11月28日火曜日

### 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 - 大きさの程度(直線、導関数、二階微分、極値、変曲点、関数の凹凸、グラフの描画)

1. $\begin{array}{}f\text{'}\left(x\right)={e}^{2x}+x{e}^{2x}·2\\ ={e}^{2x}\left(1+2x\right)\\ f\text{'}\text{'}\left(x\right)={e}^{2x}2\left(1+2x\right)+{e}^{2x}·2\\ =2{e}^{2x}\left(1+2x+1\right)\\ =2{e}^{2x}\left(2+2x\right)\\ =4{e}^{2x}\left(x+1\right)\end{array}$
$\begin{array}{}f\text{'}\left(x\right)=0\\ x=-\frac{1}{2}\\ x<-\frac{1}{2}\\ f\text{'}\left(x\right)<0\\ x>-\frac{1}{2}\\ f\text{'}\left(x\right)>0\end{array}$
$\begin{array}{}f\text{'}\text{'}\left(x\right)=0\\ x=-1\\ x<-1\\ f\text{'}\left(x\right)<0\\ x>-1\\ f\text{'}\text{'}\left(x\right)>0\end{array}$
$\begin{array}{}f\left(-\frac{1}{2}\right)=-\frac{1}{2}{e}^{-1}\\ f\left(-1\right)=-{e}^{-2}\\ f\left(0\right)=0\\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=\infty \\ \underset{x\to \infty }{\mathrm{lim}}f\left(x\right)=0\end{array}$

問題の関数のグラフ。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, Derivative, plot

x = symbols('x')
f = x * exp(2 * x)
for n in range(1, 3):
D = Derivative(f, x, n)
for t in [D, D.doit()]:
pprint(t)
print()
print()

p = plot(f, (x, -2, 2), ylim=(-2, 2), show=False, legend=True)

p.save('sample1.svg')


$./sample1.py d ⎛ 2⋅x⎞ ──⎝x⋅ℯ ⎠ dx 2⋅x 2⋅x 2⋅x⋅ℯ + ℯ 2 d ⎛ 2⋅x⎞ ───⎝x⋅ℯ ⎠ 2 dx 2⋅x 4⋅(x + 1)⋅ℯ  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="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-2"> <label for="x2">x2 = </label> <input id="x2" type="number" value="2"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-2"> <label for="y2">y2 = </label> <input id="y2" type="number" 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="sample1.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) => x * Math.exp(2 * x), f1 = (x) => Math.exp(2 * x) * (1 + 2 * x), f2 = (x) => 4 * Math.exp(2 * x) * (x + 1); 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 = [], lines = [[-1/2, y1, -1/2, y2, 'red'], [-1, y1, -1, y2, 'brown']], fns = [[f, 'green'], [f1, 'blue'], [f2, 'orange']], 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]) .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();