数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(曲線と点との距離、最小点、最小値)

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

解析入門原書第3版 (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第5章(平均値の定理)、3(増加・減少関数)、補充問題11、12、13、14.を取り組んでみる。

$\begin{array}{l} y = \pm \sqrt{x^{2} - 16} \\ f (x) = x^{2} + {(\sqrt{x^{2} - 16} - 6)}^{2} \\ = x^{2} + x^{2} - 16 + 36 - 12 \sqrt{x^{2} - 16} \\ = 2 x^{2} - 12 \sqrt{x^{2} - 16} + 20 \\ = 2 (x^{2} - 6 \sqrt{x^{2} - 16} + 10) \\ f' (x) = 2 (2 x - 3 {(x^{2} - 16)}^{- \frac{1}{2}} 2 x) \\ = 4 x (1 - 3 {(x^{2} - 16)}^{- \frac{1}{2}}) \\ 1 - 3 {(x^{2} - 16)}^{- \frac{1}{2}} = 0 \\ 1 - \frac{x^{2} - 16}{9} = 0 \\ x^{2} - 25 = 0 \\ x = \pm 5 \\ (\pm 5, 3) \end{array}$
$\begin{array}{l} x = y^{2} - 1 \\ f (y) = y^{2} + {(y^{2} - 1)}^{2} \\ f' (y) = 2 y + 2 (y^{2} - 1) 2 y \\ = 2 y (1 + 2 y^{2} - 2) \\ = 2 y (2 y^{2} - 1) \\ y = 0, \pm \frac{1}{\sqrt{2}} \\ (- 1, 0), (- \frac{1}{2}, \pm \frac{1}{\sqrt{2}}) \end{array}$
$\begin{array}{l} x = \frac{2}{5} y^{2} - 1 \\ f (y) = y^{2} + {(\frac{2}{5} y^{2} - 1)}^{2} \\ f' (y) = 2 y + 2 (\frac{2}{5} y^{2} - 1) \frac{4}{5} y \\ = 2 y (1 + \frac{8}{25} y^{2} - \frac{4}{5}) \\ = \frac{2}{25} y (25 + 8 y^{2} - 20) \\ = \frac{2}{25} y (8 y^{2} + 5) \\ y = 0 \\ (- 1, 0) \end{array}$
$\begin{array}{l} f (x) = {(x - 9)}^{2} + {(2 x^{2})}^{2} \\ f' (x) = 2 (x - 9) + 2 (2 x^{2}) 4 x \\ = 2 x - 18 + 16 x^{3} \\ = 2 (8 x^{3} + x - 9) \\ = 2 (x - 1) (8 x^{2} + 8 x + 9) \\ (1, 2) \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve, plot, sqrt

x = symbols('x')

fss = [[-sqrt(x ** 2 - 16), sqrt(x ** 2 - 16)],
       [-sqrt(x + 1), sqrt(x + 1)],
       [-sqrt(5 / 2 * (x + 1)), sqrt(5 / 2 * (x + 1))],
       [2 * x ** 2]]


for i, fs in enumerate(fss, 11):
    print('{0}.'.format(i))
    d = Derivative(fs[0], x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(*fs, show=False, legend=True)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample11.py
11.
  ⎛    _________⎞
d ⎜   ╱  2      ⎟
──⎝-╲╱  x  - 16 ⎠
dx               
    -x      
────────────
   _________
  ╱  2      
╲╱  x  - 16 
[0]
12.
d ⎛   _______⎞
──⎝-╲╱ x + 1 ⎠
dx            
    -1     
───────────
    _______
2⋅╲╱ x + 1 
[]
13.
d ⎛   _____________⎞
──⎝-╲╱ 2.5⋅x + 2.5 ⎠
dx                  
     -1.25     
───────────────
  _____________
╲╱ 2.5⋅x + 2.5 
[]
14.
d ⎛   2⎞
──⎝2⋅x ⎠
dx      
4⋅x
[0]
$

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="-15">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="15">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-15">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="15">
<br>
<label for="x0">x = </label>
<input id="x0" type="number" step="0.1" 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="sample11.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0],
    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) => 2 * x ** 2,
    g = (x) => Math.sqrt((x - 9) ** 2 + f(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),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [[9, 0, 'red']],
        lines = [[x0, f(x0), 11, 1, 'blue'], [x0, y1, x0, y2, 'brown']],
        fns = [[f, 'green'], [g, 'orange']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(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('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);

    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
x =

Kamimura's blog

2017年6月27日火曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(曲線と点との距離、最小点、最小値)

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