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)
- Nebo(Windows アプリ)
- iPad Pro + Apple Pencil
- MyScript Nebo(iPad アプリ)
- 参考書籍
解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.2(高次偏導関数、テイラーの定理)、問題10-(c).を取り組んでみる。
-
よって、 求める3次のテイラー多項式は、
macOS High Sierraの標準搭載されているグラフ作成ソフト、Grapher で作成。
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, Derivative, factorial
a, b = symbols('a, b', nonzero=True)
x, y = symbols('x, y')
f = 1 / sqrt((1 + x) * (1 + y))
d = {x: 0, y: 0}
Dx = Derivative(f, x, 1)
Dy = Derivative(f, y, 1)
Dxx = Derivative(f, x, 2)
Dyy = Derivative(f, y, 2)
Dxy = Derivative(Dx, y, 1)
Dxxx = Derivative(f, x, 3)
Dyyy = Derivative(f, y, 3)
Dxxy = Derivative(Dxx, y, 1)
Dxyy = Derivative(Dyy, x, 1)
expr = f.subs(d) + (Dx.subs(d) * x + Dy.subs(d) * y) + 1 / factorial(2) * (Dxx.subs(d) * x ** 2 + 2 * Dxy.subs(d) * x * y + Dyy.subs(d) * y ** 2) + \
1 / factorial(3) * (Dxxx.subs(d) * x ** 3 + 3 * Dxxy.subs(d) * x ** 2 *
y + 3 * Dxyy.subs(d) * x * y ** 2 + Dyyy.subs(d) * y ** 3)
for t in [f, expr, expr.doit()]:
pprint(t)
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample10.py
1
───────────────────
_________________
╲╱ (x + 1)⋅(y + 1)
⎛ 3 ⎞│ ⎛ 2 ⎞│
3 ⎜ d ⎛ 1 ⎞⎟│ 2 ⎜ d ⎛ 1 ⎞⎟│
x ⋅⎜───⎜─────────⎟⎟│ x ⋅⎜───⎜─────────⎟⎟│
⎜ 3⎜ _______⎟⎟│ 2 ⎜ 2⎜ _______⎟⎟│ 2
⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0 3⋅x ⋅y ⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0 3⋅x⋅y x⋅y
─────────────────────── - ────── + ─────────────────────── - ────── + ─── + x⋅
6 16 2 16 4
⎛ 3 ⎞│ ⎛ 2 ⎞│
3 ⎜ d ⎛ 1 ⎞⎟│ 2 ⎜ d ⎛ 1 ⎞⎟│
y ⋅⎜───⎜─────────⎟⎟│ y ⋅⎜───⎜─────────⎟⎟│
⎜ 3⎜ _______⎟⎟│ ⎜ 2⎜ _______⎟⎟│
⎛d ⎛ 1 ⎞⎞│ ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0 ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0 ⎛d
⎜──⎜─────────⎟⎟│ + ─────────────────────── + ─────────────────────── + y⋅⎜─
⎜dx⎜ _______⎟⎟│ 6 2 ⎜d
⎝ ⎝╲╱ x + 1 ⎠⎠│x=0 ⎝
⎛ 1 ⎞⎞│
─⎜─────────⎟⎟│ + 1
y⎜ _______⎟⎟│
⎝╲╱ y + 1 ⎠⎠│y=0
3 2 2 2 3 2
5⋅x 3⋅x ⋅y 3⋅x 3⋅x⋅y x⋅y x 5⋅y 3⋅y y
- ──── - ────── + ──── - ────── + ─── - ─ - ──── + ──── - ─ + 1
16 16 8 16 4 2 16 8 2
$
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="-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="y0">y0 = </label> <input id="y0" type="number" value="1"> <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="sample10.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_y0 = document.querySelector('#y0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_y0],
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, y) => 1 / Math.sqrt((1 + x) * (1 + y)),
g = (x, y) => 1 - (x + y) / 2 + (3 * x ** 2 + 2 * x * y + 3 * y ** 2) / 8 -
(5 * x ** 3 + 3 * x ** 2 * y + 3 * x * y ** 2 + 5 * y ** 3) / 16;
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),
y0 = parseFloat(input_y0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
fns = [[(x) => f(x, y0), 'red'],
[(x) => g(x, y0), 'green']];
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]);
}
}
});
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) => d[2] || 'green');
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.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
p(fns.join('\n'));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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