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)、17.3(導関数とその計算)、定数倍の微分、和や差の微分、問24、25、26.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from sympy import symbols, sqrt, Derivative, pprint, pi, Rational
x = symbols('x')
print('問24.')
exprs = [
10 * x ** 6,
x ** 2 - 4 * x + 5,
x ** 3 - 4 * x,
(x ** 2 - 2) * (3 * x ** 2 + 4)
]
for i, expr in enumerate(exprs, 1):
print('({0})'.format(i))
pprint(expr)
d = Derivative(expr, x)
pprint(d)
pprint(d.doit().expand())
print('問25')
t = symbols('t')
s = 4.9 * t ** 2
pprint(s)
d = Derivative(s, t)
pprint(d)
pprint(d.doit())
print('問26')
r = symbols('r')
s = pi * r ** 2
pprint(s)
d = Derivative(s, r)
pprint(d)
pprint(d.doit())
v = Rational(4, 3) * pi * r ** 3
pprint(v)
d = Derivative(v, r)
pprint(d)
pprint(d.doit())
入出力結果(Terminal, IPython)
$ ./sample24.py
問24.
(1)
6
10⋅x
d ⎛ 6⎞
──⎝10⋅x ⎠
dx
5
60⋅x
(2)
2
x - 4⋅x + 5
d ⎛ 2 ⎞
──⎝x - 4⋅x + 5⎠
dx
2⋅x - 4
(3)
3
x - 4⋅x
d ⎛ 3 ⎞
──⎝x - 4⋅x⎠
dx
2
3⋅x - 4
(4)
⎛ 2 ⎞ ⎛ 2 ⎞
⎝x - 2⎠⋅⎝3⋅x + 4⎠
d ⎛⎛ 2 ⎞ ⎛ 2 ⎞⎞
──⎝⎝x - 2⎠⋅⎝3⋅x + 4⎠⎠
dx
3
12⋅x - 4⋅x
問25
2
4.9⋅t
d ⎛ 2⎞
──⎝4.9⋅t ⎠
dt
9.8⋅t
問26
2
π⋅r
d ⎛ 2⎞
──⎝π⋅r ⎠
dr
2⋅π⋅r
3
4⋅π⋅r
──────
3
⎛ 3⎞
d ⎜4⋅π⋅r ⎟
──⎜──────⎟
dr⎝ 3 ⎠
2
4⋅π⋅r
$
コード(Emacs)
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="1"> <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"> <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="sample24.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 f1 = (x) => 10 * x ** 6,
f2 = (x) => x ** 2 - 4 * x + 5,
f3 = (x) => x ** 3 - 4 * x,
f4 = (x) => (x ** 2 - 2) * (3 * x ** 2 + 4);
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 = [];
for (let x = x1; x <= x2; x += dx) {
let y = f1(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t1 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f2(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t2 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f3(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
let t3 = points.length;
for (let x = x1; x <= x2; x += dx) {
let y = f4(x);
if (Math.abs(y) < Infinity) {
points.push([x, y]);
}
}
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('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d, i) =>
i < t1 ? 'red' :
i < t2 ? 'green' :
i < t3 ? 'blue' : 'orange');
svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]])
.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', 'black');
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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