数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 関数の連続性 - 中間値の定理(一般化)

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)、17.2(関数の連続性)、中間値の定理、問17、18、19、20.を取り組んでみる。

$\begin{array}{l} f (2) = 3 > 0 \\ f (0) = - 5 < 0 \\ - 5 < c < 3 \\ f (c) = 0 \end{array}$
1. $\begin{array}{l} \sin π - π \cos π = π > 0 \\ \sin \frac{3}{2} π - \frac{3}{2} π \cos \frac{3}{2} π = - 1 < 0 \end{array}$
2. $\begin{array}{l} 9 - 12 + 2 = - 1 < 0 \\ 27 - 18 + 2 = 11 > 0 \end{array}$
3. $\begin{array}{l} \log_{10} 10 - \frac{10}{20} = 1 - \frac{1}{2} > 0 \\ \log_{10} 100 - \frac{100}{20} = 2 - 5 < 0 \end{array}$
$\begin{array}{l} x \neq 0 \\ \sum_{i = 0}^{n} a_{i} x^{i} = x^{n} \sum_{i = 0}^{n} \frac{a_{i}}{x^{n - i}} \\ \lim_{x \to \infty} x^{n} \sum_{i = 0}^{n} \frac{a_{i}}{x^{n - i}} \\ = a_{n} \lim_{x \to \infty} x^{n} \\ n = 2 k + 1 \\ a_{n} > 0 \\ a_{n} \lim_{x \to \infty} x^{n} = \infty > 0 \\ a_{n} \lim_{x \to - \infty} x^{n} = - \infty < 0 \\ a_{n} < 0 \\ a_{n} \lim_{x \to \infty} x^{n} = \infty < 0 \\ a_{n} \lim_{x \to - \infty} x^{n} = - \infty > 0 \end{array}$
$\begin{array}{l} f (a) < m < f (b) \\ f (a) - m < 0 \\ f (b) - m > 0 \\ g (x) = f (x) - m \\ g (a) < 0 \\ g (b) > 0 \\ a < c < b \\ g (c) = 0 \\ g (c) = f (c) - m \\ f (c) = m \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import Symbol, sin, cos, log, pi, pprint

x = Symbol('x', real=True)

exprs = [(sin(x) - x * cos(x), pi, 3 / 2 * pi),
         (3 ** x - 6 * x + 2, 2, 3),
         (log(x, 10) - x / 20, 10, 100)]
for i, (expr, x1, x2) in enumerate(exprs, 1):
    print('({0})'.format(i))
    pprint(expr)
    print('x = {0}'.format(x1))
    y1 = expr.subs({x: x1})
    pprint(y1)
    print('x = {0}'.format(x2))
    y2 = expr.subs({x: x2})
    pprint(y2)
    pprint(y1 * y2 < 0)

入出力結果(Terminal, IPython)

$ ./sample17.py
(1)
-x⋅cos(x) + sin(x)
x = pi
π
x = 1.5*pi
-1
True
(2)
 x          
3  - 6⋅x + 2
x = 2
-1
x = 3
11
True
(3)
  x     log(x)
- ── + ───────
  20   log(10)
x = 10
1/2
x = 100
     log(100)
-5 + ────────
     log(10) 
True
$

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="0.01">
<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">

<label for="n0">n = </label>
<input id="n0" type="number"  min="1" step="2" value="5">

<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="sample17.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_n = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_n],
    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 f17 = (x) => x ** 3 + x ** 2 - 2 * x - 5;

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),
        n = parseInt(input_n.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        ans = range(0, n)
        .map((i) => (Math.random() < 0.5 ? -1 : 1) * Math.random() * 2)
        .concat([(Math.random() < 0.5 ? -1 : 1) * Math.random() + 1]),
        f19 = (x) => ans.map((a, i) => a * x ** i).reduce((x, y) => x + y);
    
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f17(x)]);
    }
    let t = points.length;
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f19(x)]);
    }
    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 < t ? 'green' : 'blue');


    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();

r = dx =
x1 = x2 =
y1 = y2 = n =

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2017年5月18日木曜日

数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 関数の連続性 - 中間値の定理(一般化)

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