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)、高次導関数、問62.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, sin, cos, log, Derivative
print('62.')
print('(1)')
x, a, b = symbols('x a b')
f = a * cos(log(x)) + b * sin(log(x))
f1 = Derivative(f, x, 1)
f2 = Derivative(f, x, 2)
for func in [f, f1, f2]:
pprint(func)
fn = func.doit()
pprint(fn.factor())
print()
eq = f + x * f1.doit() + x ** 2 * f2.doit()
pprint(eq)
pprint(eq.expand())
入出力結果(Terminal, IPython)
$ ./sample62.py
62.
(1)
a⋅cos(log(x)) + b⋅sin(log(x))
a⋅cos(log(x)) + b⋅sin(log(x))
∂
──(a⋅cos(log(x)) + b⋅sin(log(x)))
∂x
-(a⋅sin(log(x)) - b⋅cos(log(x)))
─────────────────────────────────
x
2
∂
───(a⋅cos(log(x)) + b⋅sin(log(x)))
2
∂x
-(-a⋅sin(log(x)) + a⋅cos(log(x)) + b⋅sin(log(x)) + b⋅cos(log(x)))
──────────────────────────────────────────────────────────────────
2
x
⎛ a⋅sin(log(x)) b⋅cos(log(x))⎞
a⋅sin(log(x)) - b⋅cos(log(x)) + x⋅⎜- ───────────── + ─────────────⎟
⎝ x x ⎠
0
$
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="-1"> <label for="x2">x2 = </label> <input id="x2" type="number" value="21"> <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="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="sample62.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 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 = [],
f = (x) => a0 * Math.cos(Math.log(x)) + b0 * Math.sin(Math.log(x)),
f1 = (x) => 1 / x *
(-a0 * Math.sin(Math.log(x)) + b0 * Math.cos(Math.log(x))),
f2 = (x) => 1 / x ** 2 *
((a0 - b0) * Math.sin(Math.log(x)) - (a0 + b0) * Math.cos(Math.log(x)));
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 'red']);
}
}
let t1 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f1(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 'green']);
}
}
let t2 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f2(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 'blue']);
}
}
let t3 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = x * f1(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 'brown']);
}
}
let t4 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = x ** 2 * f2(x);
if (Math.abs(y) < Infinity) {
points.push([x, y, 'purple']);
}
}
let lines = [];
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);
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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