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 - 平均変化率と微分係数(微分可能)

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)、平均変化率と微分係数、問22.を取り組んでみる。

1. $\begin{array}{l} \lim_{h \to 0} \frac{- 3 (- 2 + h) + 2 - 8}{h} \\ = \lim_{h \to 0} - 3 \\ = - 3 \end{array}$
2. $\begin{array}{l} \lim_{h \to 0} \frac{\sqrt{4 + h} - 2}{h} \\ = \lim_{h \to 0} \frac{h}{h (\sqrt{4 + h} + 2)} \\ = \lim_{h \to 0} \frac{1}{\sqrt{4 + h} + 2} \\ = \frac{1}{4} \end{array}$
3. $\begin{array}{l} \lim_{h \to 0} \frac{\frac{1}{{(1 + h)}^{2}} - 1}{h} \\ = \lim_{h \to 0} \frac{1 - {(1 + h)}^{2}}{h {(1 + h)}^{2}} \\ = \lim_{h \to 0} \frac{h^{2} - 2 h}{h {(1 + h)}^{2}} \\ = \lim_{h \to 0} \frac{h - 2}{{(1 + h)}^{2}} \\ = - 2 \end{array}$

コード(Emacs)

Python 3

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

from sympy import symbols, sqrt, Limit, pprint

x, h = symbols('x h')

exprs = [(-3 * x + 2, -2),
         (sqrt(x), 4),
         (1 / x ** 2, 1)]

for i, (expr, x0) in enumerate(exprs, 1):
    print('({0})'.format(i))
    expr = (expr.subs({x: x0 + h}) - expr.subs({x: x0})) / h
    pprint(expr)
    l = Limit(expr, h, 0)
    pprint(l)
    pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample22.py
(1)
-3
 lim -3
h─→0⁺  
-3
(2)
  _______    
╲╱ h + 4  - 2
─────────────
      h      
       _______    
     ╲╱ h + 4  - 2
 lim ─────────────
h─→0⁺      h      
1/4
(3)
        1    
-1 + ────────
            2
     (h + 1) 
─────────────
      h      
             1    
     -1 + ────────
                 2
          (h + 1) 
 lim ─────────────
h─→0⁺      h      
-2
$

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">
<br>
<label for="p0">p = </label>
<input id="p0" type="number" value="2">
<label for="q0">q = </label>
<input id="q0" type="number" value="3">
<label for="p0">a = </label>
<input id="a0" type="number" min="0.001" value="4">
<label for="b0">b = </label>
<input id="b0" type="number" min="0.001" 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="sample22.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_h = document.querySelector('#h0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_h],
    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 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),
        h = parseFloat(input_h.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || h === 0) {
        return;
    }

    let points = [],
        f = (fn, x) => (fn(x + h) - fn(x)) / h,
        f1 = (x) => -3 * x + 1,
        f2 = (x) => Math.sqrt(x),
        f3 = (x) => 1 / x ** 2;
    
    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 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' : '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);

    p(
    [[f1, -2], [f2, 4], [f3, 1]]
            .map((a, i) => `(${i + 1}) ${f(...a)}`)
            .join('\n')
    );
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
h =

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2017年5月21日日曜日

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 - 平均変化率と微分係数(微分可能)

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