2017年7月1日土曜日

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20.を取り組んでみる。


  1. 直円柱の高さをh、両端の円の半径をrとする。

    V=π r 2 h+ 4 3 π r 3 h= V 4 3 π r 3 π r 2 = 3V4π r 3 3π r 2 f( r )=2πrh+2·4π r 2 =2πr· 3V4π r 3 3π r 2 +2·4π r 2 =2· 3V4π r 3 +12π r 3 3r =2· 3V+8π r 3 3r f'( r )=2 24π r 2 ·3r( 3V+8π r 3 )3 9 r 2 =2· 24π r 3 3V8π r 3 3 r 2 =2· 16π r 3 3V 3 r 2 16π r 3 3V=0 r 3 = 3V 16π r= ( 3V 16π ) 1 3 = 1 2 ( 3V 2π ) 1 3 h= V 4 3 π 3V 16π πr ( 3V 16π ) 2 3 = V V 4 πr ( 3V 16π ) 2 3 = 3V 4πr ( 3V 16π ) 2 3 = 1 r · ( 3V·4 π ) 1 3 = 1 r · ( 12V π ) 1 3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, pi, Pow, Derivative, solve

V, r = symbols('V r', positive=True)

h = (3 * V - 4 * pi * Pow(r, 3)) / (3 * pi * Pow(r, 2))
f = 2 * pi * r * h + 2 * 4 * pi * Pow(r, 2)
d = Derivative(f, r, 1)
f1 = d.doit()
rs = solve(f1, r)
pprint(rs)
for r0 in rs:
    pprint(r0)
    pprint(h.subs({r: r0}))

入出力結果(Terminal, IPython)

$ ./sample20.py
⎡ 2/3 3 ___ 3 ___⎤
⎢2   ⋅╲╱ 3 ⋅╲╱ V ⎥
⎢────────────────⎥
⎢      3 ___     ⎥
⎣    4⋅╲╱ π      ⎦
 2/3 3 ___ 3 ___
2   ⋅╲╱ 3 ⋅╲╱ V 
────────────────
      3 ___     
    4⋅╲╱ π      
 2/3 3 ___ 3 ___
2   ⋅╲╱ 3 ⋅╲╱ V 
────────────────
     3 ___      
     ╲╱ π
$     

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="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="25">
<br>

<label for="v0">v0 = </label>
<input id="v0" type="number" min="0" step="1" value="1000">

<label for="r1">r1 = </label>
<input id="r1" 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="sample20.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_v0 = document.querySelector('#v0'),
    input_r1 = document.querySelector('#r1'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_v0, input_r1],
    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) => Math.sqrt(x ** 2 + 64 / 9 * x ** 2 / (2 * x - 9) ** 2),
    f1 = (x) =>
    1 / 2 * (x ** 2 + 64 / 9 * x ** 2 / (2 * x - 9 ) ** 2) ** (-1 / 2) *
    (2 * x + 64 / 9 *
     (2 * x * (2 * x - 9) ** 2 - x ** 2 * 2 * (2 * x - 9) * 2) /
     (2 * x - 9) ** 4),
    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),
        v0 = parseFloat(input_v0.value),
        r1 = parseFloat(input_r1.value);

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

    let points = [],
        h0 = (v0 - 4 / 3 * Math.PI * r1 ** 3) / (Math.PI * r1 ** 2),
        x3 = 2 * r1,
        x4 = 1 / 2 * (3 * v0 / (2 * Math.PI)) ** (1 / 3),
        y3 = r1 + h0,
        lines = [[0, r1, 0, y3, 'green'],
                 [x3, r1, x3, y3, 'green'],
                 [r1, y1, r1, y2, 'red'],
                 [x4, y1, x4, y2, 'blue']],
        f = (x) => Math.sqrt(r1 ** 2 - (x - r1) ** 2) + y3,
        g = (x) => -Math.sqrt(r1 ** 2 - (x - r1) ** 2) + r1,
        fns = [[f, 'green'],
               [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);

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

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








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