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18章(曲線の性質、最大・最小 - 微分法の応用)、18.5(関数の近似、テイラーの定理)、関数の整級数展開(2)、問61-1、2、3、4、5.を取り組んでみる。
-
ロビタルの定理。
ロピタルの定理。
ロピタルの定理。
前問(4)より。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import pprint, symbols, cos, sin, exp, Limit, S, log, plot
x = symbols('x')
a, b = symbols('a b', positive=True)
fs = [((1 - cos(x)) / x ** 2, 0),
((x - sin(x)) / x ** 3, 0),
(x * (exp(1 / x) - 1), S.Infinity),
((a ** x - 1) / x, 0),
((a ** x - b ** x) / x, 0)]
print('61.')
for i, (f, x0) in enumerate(fs, 1):
print(f'({i})')
l = Limit(f, x, x0)
pprint(l)
pprint(l.doit())
try:
p = plot(f, show=False, legend=True)
p.save(f'sample1_{i}.svg')
except Exception as err:
print(type(err), err)
print()
入出力結果(Terminal, IPython)
$ ./sample61.py
61.
(1)
⎛-cos(x) + 1⎞
lim ⎜───────────⎟
x─→0⁺⎜ 2 ⎟
⎝ x ⎠
1/2
(2)
⎛x - sin(x)⎞
lim ⎜──────────⎟
x─→0⁺⎜ 3 ⎟
⎝ x ⎠
1/6
(3)
⎛ ⎛ 1 ⎞⎞
⎜ ⎜ ─ ⎟⎟
⎜ ⎜ x ⎟⎟
lim ⎝x⋅⎝ℯ - 1⎠⎠
x─→∞
1
/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/experimental_lambdify.py:232: UserWarning: The evaluation of the expression is problematic. We are trying a failback method that may still work. Please report this as a bug.
warnings.warn('The evaluation of the expression is'
(4)
⎛ x ⎞
⎜a - 1⎟
lim ⎜──────⎟
x─→0⁺⎝ x ⎠
log(a)
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.
(5)
⎛ x x⎞
⎜a - b ⎟
lim ⎜───────⎟
x─→0⁺⎝ x ⎠
log(a) - log(b)
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.
$
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" min="0" value="2"> <label for="b0">b = </label> <input id="b0" type="number" min="0" value="3"> <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="sample61.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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0],
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 f3 = (x) => x * (Math.exp(1 / 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),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
y3 = Math.log(a0),
y4 = Math.log(a0) - Math.log(b0),
lines = [[x1, 1, x2, 1, 'orange'],
[x1, y3, x2, y3, 'brown'],
[x1, y4, x2, y4, 'purple']],
f4 = (x) => (a0 ** x - 1) / x,
f5 = (x) => (a0 ** x - b0 ** x) / x,
fns = [[f3, 'red'],
[f4, 'green'],
[f5, 'blue']],
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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