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)、最大・最小問題、問21、22.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, Pow, plot, exp, Derivative, solve
x = symbols('x', real=True)
p = plot(Pow(x, 3), show=False)
p.save('sample21.svg')
f = (x - (x + exp(x)) / 2) ** 2 + (exp(x) - (x + exp(x)) / 2) ** 2
d = Derivative(f, x)
pprint(d)
pprint(d.expand())
f1 = d.doit()
pprint(f1)
# f1 = f1.expand()
pprint(f1.expand())
s = solve(f1, x)
pprint(s)
for x0 in s:
for n in range(-1, 2):
pprint(x0 + n)
pprint(f1)
pprint(f1.expand())
result = f1.subs({x: x0 + n})
pprint(result)
pprint(result.is_positive)
pprint(result.is_zero)
pprint(result.is_negative)
pprint(exp(x0 + n))
print()
入出力結果(Terminal, IPython)
$ ./sample21.py
⎛ 2 2⎞
⎜⎛ x⎞ ⎛ x⎞ ⎟
d ⎜⎜ x ℯ ⎟ ⎜x ℯ ⎟ ⎟
──⎜⎜- ─ + ──⎟ + ⎜─ - ──⎟ ⎟
dx⎝⎝ 2 2 ⎠ ⎝2 2 ⎠ ⎠
⎛ 2 2⋅x⎞
d ⎜x x ℯ ⎟
──⎜── - x⋅ℯ + ────⎟
dx⎝2 2 ⎠
⎛ x⎞ ⎛ x⎞
⎜ x ℯ ⎟ ⎛ x ⎞ ⎜x ℯ ⎟ ⎛ x ⎞
⎜- ─ + ──⎟⋅⎝ℯ - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ + 1⎠
⎝ 2 2 ⎠ ⎝2 2 ⎠
x 2⋅x x
- x⋅ℯ + x + ℯ - ℯ
[0]
-1
⎛ x⎞ ⎛ x⎞
⎜ x ℯ ⎟ ⎛ x ⎞ ⎜x ℯ ⎟ ⎛ x ⎞
⎜- ─ + ──⎟⋅⎝ℯ - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ + 1⎠
⎝ 2 2 ⎠ ⎝2 2 ⎠
x 2⋅x x
- x⋅ℯ + x + ℯ - ℯ
⎛ -1⎞ ⎛ -1 ⎞
⎜ 1 ℯ ⎟ ⎛ -1 ⎞ ⎛ -1⎞ ⎜ℯ 1⎟
⎜- ─ - ───⎟⋅⎝- ℯ + 1⎠ + ⎝-1 + ℯ ⎠⋅⎜─── + ─⎟
⎝ 2 2 ⎠ ⎝ 2 2⎠
False
False
True
-1
ℯ
0
⎛ x⎞ ⎛ x⎞
⎜ x ℯ ⎟ ⎛ x ⎞ ⎜x ℯ ⎟ ⎛ x ⎞
⎜- ─ + ──⎟⋅⎝ℯ - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ + 1⎠
⎝ 2 2 ⎠ ⎝2 2 ⎠
x 2⋅x x
- x⋅ℯ + x + ℯ - ℯ
0
False
True
False
1
1
⎛ x⎞ ⎛ x⎞
⎜ x ℯ ⎟ ⎛ x ⎞ ⎜x ℯ ⎟ ⎛ x ⎞
⎜- ─ + ──⎟⋅⎝ℯ - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ + 1⎠
⎝ 2 2 ⎠ ⎝2 2 ⎠
x 2⋅x x
- x⋅ℯ + x + ℯ - ℯ
⎛ ℯ 1⎞ ⎛ 1 ℯ⎞
(-ℯ + 1)⋅⎜- ─ + ─⎟ + (-1 + ℯ)⋅⎜- ─ + ─⎟
⎝ 2 2⎠ ⎝ 2 2⎠
True
False
False
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="0.0001" 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="x0">x = </label> <input id="x0" type="number" step="0.1" 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="sample21.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_x0 = document.querySelector('#x0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
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 f = (x) => Math.exp(x),
g = (x) => x,
h = (x) => Math.log(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),
x0 = parseFloat(input_x0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
a = (x0 + Math.exp(x0)) / 2,
lines = [[x0, f(x0), a, a, 'brown']],
fns = [[f, 'red'], [h, '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.join('\n'));
p(fns1.join('\n'));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
0 コメント:
コメントを投稿