2017年7月7日金曜日

学習環境

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34.を取り組んでみる。


  1. 円形の池の中心をO、PRの長さをx、∠ROQのなす角をΘとする。

    0x1 0θπ RQ= 1 2 x 2 = 1 x 2 RQ= 1 2 ( sin θ 2 )·2=sin θ 2 1 x 2 =sin θ 2 arcsin 1 x 2 = θ 2 θ=2arcsin 1 x 2 f( x )= x 2 + π· θ 2π 4 = x 2 + θ 8 = x 2 + arcsin 1 x 2 4 f'( x )= 1 2 + 1 4 1 1( 1 x 2 ) · 1 2 ( 1 x 2 ) 1 2 ( 2x ) = 1 2 1 4 1 x 2 1 2 1 4 1 x 2 =0 2 1 x 2 1=0 1 x 2 = 1 2 1 x 2 = 1 4 x 2 = 3 4 x= 3 2 f'( 3 2 )=0 0x< 3 2 f'( x )>0 3 2 <x1 f'( x )<0

    よって最小の時間、最大の時間はそれぞれ以下のようになる。

    f( 0 )= arcsin1 4 = π 2 4 = π 8 f( 1 )= 1 2 > π 8 π 8 f( 3 2 )= 3 2 2 + arcsin 1 ( 3 2 ) 2 4 = 3 +arcsin 1 2 4 = 3 + π 6 4 = 6 3 +π 24

コード(Emacs)

Python 3

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

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

print('34.')
x = symbols('x', positive=True)
f = x / 2 + asin(sqrt(1 - x ** 2)) / 4
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
s = solve(f1)
pprint(s)

pprint(f.subs({x: s[0]}))
print(pi / 8 < 1 / 2)
p = plot(f, (x, 0, 1), show=False, legend=True)
p.save('sample34.svg')

入出力結果(Terminal, IPython)

$ ./sample34.py
34.
  ⎛        ⎛   __________⎞⎞
  ⎜        ⎜  ╱    2     ⎟⎟
d ⎜x   asin⎝╲╱  - x  + 1 ⎠⎟
──⎜─ + ───────────────────⎟
dx⎝2            4         ⎠
1          1       
─ - ───────────────
2        __________
        ╱    2     
    4⋅╲╱  - x  + 1 
⎡√3⎤
⎢──⎥
⎣2 ⎦
π    √3
── + ──
24   4 
True
$

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="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" min="0" max="1" step="0.01" value="0.5">

<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="sample34.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) => x / 2 + Math.asin(Math.sqrt(1 - x ** 2)) / 4,
    g = (x) => Math.sqrt(1 / 4 - (x - 1 / 2) ** 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 = [],
        x3 = Math.sqrt(3) / 2,
        y3 = g(x0),
        lines = [[0, y1, 0, y2, 'red'],
                 [x3, y1, x3, y2, 'red'],
                 [x0, y1, x0, y2, 'brown'],
                 [0, 0, x0, y3, 'blue'],
                 [x0, y3, 1, 0, 'blue']],
        fns = [[f, 'orange'],
               [g, 'green']],
        fns1 = [],
        fns2 = [];

    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();








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