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(ホース、仰角、到達距離)

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36、37.を取り組んでみる。

$\begin{array}{l} 0 \leq θ \leq \frac{π}{2} \\ \frac{d}{d θ} f (θ) = \frac{2 v^{2}}{g} (\cos^{2} θ - \sin^{2} θ) \\ = \frac{2 v^{2}}{g} (\cos^{2} θ - \sin^{2} θ) \\ = \frac{2 v^{2}}{g} (1 - 2 \sin^{2} θ) \\ 1 - 2 \sin^{2} θ = 0 \\ \sin^{2} θ = \frac{1}{2} \\ \sin θ = \frac{1}{\sqrt{2}} \\ θ = \frac{1}{4} π \\ f' (\frac{π}{4}) = 0 \\ θ < \frac{π}{4} \\ f' (θ) > 0 \\ θ > \frac{π}{4} \\ f' (θ) < 0 \\ f (\frac{π}{4}) = \frac{2 v^{2}}{g} \sin \frac{π}{4} \cos \frac{π}{4} \\ = \frac{2 v^{2}}{g} \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} \\ = \frac{v^{2}}{g} \end{array}$
$\begin{array}{l} f' (t) = 1 - \frac{1}{t^{2}} \\ f' (1) = 0 \\ 0 < t < 1 \\ f' (t) < 0 \\ 1 < t \\ f' (t) > 0 \\ f (1) = 1 + \frac{1}{1} = 2 \\ f (t) \geq 2 \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, sin, cos, solve

v, g, x = symbols('v g x')
fs = [2 * v ** 2 / g * sin(x) * cos(x),
      x + 1 / x]

for i, f in enumerate(fs, 36):
    print(f'{i}.')
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    xs = solve(f1, x)
    pprint(xs)
    for x0 in xs:
        pprint(f.subs({x: x0}))
    print()

入出力結果(Terminal, IPython)

$ ./sample36.py
36.
  ⎛   2              ⎞
∂ ⎜2⋅v ⋅sin(x)⋅cos(x)⎟
──⎜──────────────────⎟
∂x⎝        g         ⎠
     2    2         2    2   
  2⋅v ⋅sin (x)   2⋅v ⋅cos (x)
- ──────────── + ────────────
       g              g      
⎡-3⋅π   -π   π  3⋅π⎤
⎢─────, ───, ─, ───⎥
⎣  4     4   4   4 ⎦
 2
v 
──
g 
  2 
-v  
────
 g  
 2
v 
──
g 
  2 
-v  
────
 g  

37.
d ⎛    1⎞
──⎜x + ─⎟
dx⎝    x⎠
    1 
1 - ──
     2
    x 
[-1, 1]
-2
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="0.001">
<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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.01" value="0.1">

<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="sample36.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_dx0 = document.querySelector('#dx0'),
    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) => x + 1 / x,
    f1 = (x) => 1 - 1 / x ** 2,
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);

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),
        dx0 = parseFloat(input_dx0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        x3 = 16 ** (1 / 3) + 1,
        lines = [[1, y1, 1, y2, 'red']],
        fns = [[f, 'green']],
        fns1 = [[]],
        fns2 = [[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]);
                }
            }
        });                 

    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), 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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
dx0 =

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2017年7月11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(ホース、仰角、到達距離)

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