2017年6月28日水曜日

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15、16、17、18.を取り組んでみる。


  1. F'( x )=Q ( x 2 + b 2 ) 3 2 3 2 Qx ( x 2 + b 2 ) 5 2 ·2x =Q ( x 2 + b 2 ) 3 2 ( 13 x 2 ( x 2 + b 2 ) 1 ) 13 x 2 ( x 2 + b 2 ) 1 =0 x 2 + b 2 3 x 2 =0 x 2 = b 2 2 x=± b 2 F( b 2 )=Q b 2 ( b 2 2 + b 2 ) 3 2 =Q b 2 ( 3 2 b 2 ) 3 2 = 2Q 3 3 b 2

  2. f( y )=y ( hy ) 1 2 f'( y )= ( hy ) 1 2 y 2 ( hy ) 1 2 ( hy ) 1 2 y 2 ( hy ) 1 2 =0 hy y 2 =0 y= 2 3 h

  3. f( x )= ( x2 ) 2 + x 2 +9 =2 x 2 4x+13 f'( x )=4x4 =4( x1 ) ( 1,0 )

  4. 正三角形の一辺の長さをa、円の半径をrとする。

    L=3a+2πr r= L3a 2π 0a L 3 f( a )= 1 2 a 2 3 2 +π r 2 = 3 4 a 2 +π L 2 +9 a 2 6aL 4 π 2 = 3 π a 2 +9 a 2 6La+ L 2 4π g( a )=( 3 π+9 ) a 2 6La+ L 2 g'( a )=2( 3 π+9 )a6L =2( ( 3 π+9 )a3L ) a= 3L 3 π+9

    正三角形と縁で囲まれる面積の和を最小とするには、長さLのワイヤーを 3L 3 π+9 L 3L 3 π+9 という2つの部分に切ればいい。

    g( 0 )= L 2 g( L 3 )=( 3 π+9 ) L 2 9 6L L 3 + L 2 =( 3 π+9 9 2+1 ) L 2 =( π 3 3 +12+1 ) L 2 = π 3 3 L 2 < L 2 a=0

    最大にするには、長さLのワイヤーで円を作ればいい。

コード(Emacs)

Python 3

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

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

x = symbols('x')

Q = 1
h = 1
b = 2
L = 5
fs = [(Q * x * (x ** 2 + b ** 2) ** (-3 / 2), (0, 5)),
      (x * (h - x) ** (1 / 2), (0, h)),
      ((x - 2) ** 2 + x ** 2 + 9, (0, 2)),
      (1 / 2 * x ** 2 * sqrt(3) / 2 + pi * ((L - 3 * x) / (2 * pi)) ** 2, (0, L / 3))]

for i, (f, (x1, x2)) in enumerate(fs, 15):
    print('{0}.'.format(i))
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, x1, x2), show=False, legend=True)
    p.save('sample{0}.svg'.format(i))
    print()

入出力結果(Terminal, IPython)

$ ./sample15.py
15.
  ⎛          -1.5⎞
d ⎜  ⎛ 2    ⎞    ⎟
──⎝x⋅⎝x  + 4⎠    ⎠
dx                
                 -2.5           -1.5
       2 ⎛ 2    ⎞       ⎛ 2    ⎞    
- 3.0⋅x ⋅⎝x  + 4⎠     + ⎝x  + 4⎠    
[-1.4142135623731, 1.4142135623731]

16.
d ⎛          0.5⎞
──⎝x⋅(-x + 1)   ⎠
dx               
                -0.5           0.5
- 0.5⋅x⋅(-x + 1)     + (-x + 1)   
[0.666666666666667]

17.
d ⎛ 2          2    ⎞
──⎝x  + (x - 2)  + 9⎠
dx                   
4⋅x - 4
[1]

18.
  ⎛                       2⎞
d ⎜         2   (-3⋅x + 5) ⎟
──⎜0.25⋅√3⋅x  + ───────────⎟
dx⎝                 4⋅π    ⎠
           18⋅x - 30
0.5⋅√3⋅x + ─────────
              4⋅π   
[1.03868059752328]

$

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="-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="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">
<br>
<label for="l0">L = </label>
<input id="l0" type="number" min="0" value="20">
<label for="x0">x = </label>
<input id="x0" type="number" min="0" step="0.1" 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="sample15.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_l0 = document.querySelector('#l0'),
    input_x0 = document.querySelector('#x0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_l0, 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 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),
        l0 = parseFloat(input_l0.value),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || l0 < 3 * x0) {
        return;
    }    

    let points = [[9, 0, 'red']],
        x3 = -x0 * 1 / 2,
        y3 = x0 * Math.sqrt(3) / 2,
        r0 = (l0 - 3 * x0) / (2 * Math.PI),
        lines = [[0, 0, -x0, 0, 'blue'], [-x0, 0, x3, y3, 'blue'],
                 [x3, y3, 0, 0, 'blue']],
        f = (x) => Math.sqrt(r0 ** 2 - (x - r0) ** 2) + r0,
        g = (x) => -Math.sqrt(r0 ** 2 - (x - r0) ** 2) + r0,
        fns = [[f, 'green'], [g, 'green']];

    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();








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