数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 導関数とその計算 - 積および商の微分(ニュートン商の極限、平方根)

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)、積および商の微分、問31、32.を取り組んでみる。

$\begin{array}{l} \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \\ = \lim_{h \to 0} \frac{x + h - x}{h (\sqrt{x + h} + \sqrt{x})} \\ = \lim_{h \to 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} \\ = \frac{1}{2 \sqrt{x}} \\ = \frac{1}{2} x^{- \frac{1}{2}} \end{array}$
1. $- 20 x^{- 21}$
2. $- \frac{1}{2} x^{- \frac{3}{2}}$
3. $\frac{3}{2} x^{\frac{1}{2}}$
4. $\frac{1}{3} x^{- \frac{2}{3}}$
5. $\frac{6}{5} x^{\frac{1}{5}}$
6. $(x^{\frac{5}{4}} - x^{\frac{6}{5}}) + (x + 1) (\frac{5}{4} x^{\frac{1}{4}} - \frac{6}{5} x^{\frac{1}{5}})$
7. $\frac{\frac{1}{2} x^{- \frac{1}{2}} (x^{2} + 1) - x^{\frac{1}{2}} \cdot 2 x}{{(x^{2} + 1)}^{2}}$
8. $\frac{(2 x - 1) x^{\frac{1}{2}} - (x^{2} - x + 4) \frac{1}{2} x^{- \frac{1}{2}}}{x}$

コード(Emacs)

Python 3

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

from sympy import symbols, Limit, sqrt, pprint, Derivative

x, h = symbols('x, h')

print('31.')
l = Limit((sqrt(x + h) - sqrt(x)) / h, h, 0)
pprint(l)
pprint(l.doit())

d = Derivative(sqrt(x), x)
pprint(d)
pprint(d.doit())

入出力結果(Terminal, IPython)

$ ./sample31.py
31.
             _______
     -√x + ╲╱ h + x 
 lim ───────────────
h─→0⁺       h       
 1  
────
2⋅√x
d     
──(√x)
dx    
 1  
────
2⋅√x
$

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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<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="sample31.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) => Math.sqrt(x),
    f1 = (x, x0, y0) => 1 / 2 * x0 ** (-1 / 2) * (x - x0) + y0;

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 = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    let lines = [];
    for (let x = x1; x <= x2; x += 10 * dx) {
        let y = f(x),
            y1 = f1(x1, x, y),
            y2 = f1(x2, x, y);

        if (Math.abs(y1) < Infinity && Math.abs(y2) < Infinity) {
            lines.push([x1, y1, x2, y2]);
        }
    }
    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, i) => i === 0 ? 'black' : 'blue');
    
    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', 'red');
    
    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 =

Kamimura's blog

2017年5月26日金曜日

数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 導関数とその計算 - 積および商の微分(ニュートン商の極限、平方根)

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