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20章(面積、体積、長さ - 積分法の応用)、20.3(曲線の長さ)、点の運動する距離、問30.を取り組んでみる。
-
1秒後。
4秒後。
速度が0になる秒。
求める物体が達する最高の高さ。
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, solve
print('30.')
print('(1)')
t = symbols('t')
v = 30 - 9.8 * t
for t0 in [1, 4]:
print(f'{t0}秒後')
I = 10 + Integral(v, (t, 0, t0))
for s in [I, I.doit()]:
pprint(s)
print()
print()
print('(2)')
t0 = solve(v, t)[0]
I = 20 + Integral(v, (t, 0, t0))
for s in [I, I.doit()]:
pprint(s)
print()
print('(3)')
I = Integral(abs(v), (t, 0, 5))
for s in [I, I.doit()]:
pprint(s)
print()
I = Integral(v, (t, 0, t0)) + Integral(-v, (t, t0, 5))
for s in [I, I.doit()]:
pprint(s)
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample30.py
30.
(1)
1秒後
1
⌠
10 + ⎮ (-9.8⋅t + 30) dt
⌡
0
35.1000000000000
4秒後
4
⌠
10 + ⎮ (-9.8⋅t + 30) dt
⌡
0
51.6000000000000
(2)
3.06122448979592
⌠
20 + ⎮ (-9.8⋅t + 30) dt
⌡
0
65.9183673469388
(3)
5
⌠
⎮ │9.8⋅t - 30│ dt
⌡
0
5
⌠
⎮ │9.8⋅t - 30│ dt
⌡
0
5 3.06122448979592
⌠ ⌠
⎮ (9.8⋅t - 30) dt + ⎮ (-9.8⋅t + 30) dt
⌡ ⌡
3.06122448979592 0
64.3367346938776
$
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="0.005"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="0"> <label for="x2">x2 = </label> <input id="x2" type="number" value="100"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="0"> <label for="y2">y2 = </label> <input id="y2" type="number" value="100"> <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="sample30.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) => 30 - 9.8 * x,
g = (x) => 10 + 30 * x - 4.9 * x ** 2;
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, y1, 1, y2, 'red'],
[x1, 35.1, x2, 35.1, 'red'],
[4, y1, 4, y2, 'red'],
[x1, 51.6, x2, 51.6, 'red'],
[30 / 9.8, y1, 30 / 9.8, y2, 'red'],
[x1, 55.92, x2, 55.92, 'red'],
[5, y1, 5, y2, 'red']],
fns = [[f, 'green'],
[g, '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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