2017年7月3日月曜日

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22.を取り組んでみる。


  1. f( x )= x 2 + p 2 + ( ax ) 2 + q 2 f'( x )= 1 2 ( x 2 + p 2 ) 1 2 2x 1 2 ( ( ax ) 2 + q 2 ) 1 2 2( ax ) = ( x 2 + p 2 ) 1 2 x ( ( ax ) 2 + q 2 ) 1 2 ( ax ) ( x 2 + p 2 ) 1 2 x ( ( ax ) 2 + q 2 ) 1 2 ( ax )=0 x x 2 + p 2 ax ( ax ) 2 + q 2 =0 cos θ 1 cos θ 2 =0 cos θ 1 =cos θ 2

コード(Emacs)

Python 3

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

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

a, p, q, x = symbols('a p q x', nonnegative=True)

f = sqrt(x ** 2 + p ** 2) + sqrt((a - x) ** 2 + q ** 2)
d = Derivative(f, x, 1)
pprint(d)
f1 = d.doit()
pprint(f1)
s = solve(f1, x)
pprint(s)
for x0 in s:
    pprint(x0)
    print()
    y1 = (p / x0).factor()
    y2 = (q / (a - x0)).factor()
    pprint(y1)
    print()
    pprint(y2)
    print(y1 == y2)

入出力結果(Terminal, IPython)

$ ./sample22.py 
  ⎛   _________      _______________⎞
∂ ⎜  ╱  2    2      ╱  2          2 ⎟
──⎝╲╱  p  + x   + ╲╱  q  + (a - x)  ⎠
∂x                                   
     x               -a + x      
──────────── + ──────────────────
   _________      _______________
  ╱  2    2      ╱  2          2 
╲╱  p  + x     ╲╱  q  + (a - x)  
⎡ a⋅p    a⋅p ⎤
⎢─────, ─────⎥
⎣p - q  p + q⎦
 a⋅p 
─────
p - q

p - q
─────
  a  

-(p - q) 
─────────
    a    
False
 a⋅p 
─────
p + q

p + q
─────
  a  

p + q
─────
  a  
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="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="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.5">
<label for="p0">p = </label>
<input id="p0" type="number" min="0"  value="5">
<label for="q0">q = </label>
<input id="q0" type="number" min="0"  value="10">
<label for="a0">a = </label>
<input id="a0" type="number" min="0"  value="15">
<label for="x0">x = </label>
<input id="x0" type="number" min="0"  value="2">


<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="sample22.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'),
    input_p0 = document.querySelector('#p0'),
    input_q0 = document.querySelector('#q0'),
    input_a0 = document.querySelector('#a0'),
    input_x0 = document.querySelector('#x0'),
    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_p0, input_q0, input_a0, 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),
        dx0 = parseFloat(input_dx0.value),
        p0 = parseFloat(input_p0.value),
        q0 = parseFloat(input_q0.value),
        a0 = parseFloat(input_a0.value),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2 || x0 > a0) {
        return;
    }    

    let points = [],
        x3 = a0 * p0 / (p0 + q0),
        f =
        (x) => Math.sqrt(x ** 2 + p0 ** 2) + Math.sqrt((a0 - x) ** 2 + q0 ** 2),
        f1 =
        (x) => x / Math.sqrt(x ** 2  + p0 ** 2) -
        (a0 - x) / Math.sqrt((a0 - x) ** 2 + q0 ** 2),
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        lines = [[0, 0, 0, p0, 'green'],
                 [a0, 0, a0, q0, 'green'],
                 [0, p0, x0, 0, 'green'],
                 [x0, 0, a0, q0, 'green'],
                 [x3, y1, x3, y2, 'red']],
        fns = [[f, 'blue']],
        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();








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