2017年7月18日火曜日

学習環境

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、2章(方程式と関数)、2.6(関数の最大値・最小値)、2.1.6(接線の傾きと極値)、問題3.を取り組んでみる。


    1. y0 ax+b0 x b a mx+n= ax+b m 2 x 2 +2mnx+ n 2 =ax+b m 2 x 2 +( 2mna )x+ n 2 b=0 ( 2mna ) 2 4 m 2 ( n 2 b )=0 4 m 2 n 2 + a 2 4mna4 m 2 n 2 +4 m 2 b=0 a 2 4mna+4 m 2 b=0 n= a 2 +4 m 2 b 4ma y=mx+ a 2 +4 m 2 b 4ma 4may=4 m 2 ax+ a 2 +4 m 2 b 4( ax+b ) m 2 4aym+ a 2 =0 4( ax+b ) m 2 4a ax+b m+ a 2 =0 ( 2 ax+b ma ) 2 =0 m= a 2 ax+b

    2. ax+b0 x b a mx+n= 1 ax+b am x 2 +( an+bm )x+bn1=0 ( an+bm ) 2 4am( bn1 )=0 a 2 n 2 + b 2 m 2 +2abmn4abmn+4am=0 a 2 n 2 + b 2 m 2 2abmn+4am=0 ( anbm ) 2 +4am=0 ( anbm ) 2 =4am am0 anbm=±2 am n= ±2 am +bm a y=mx+ bm±2 am a ay=amx+bm±2 am ayamxbm=±2 am ay( ax+b )m=± am ( ax+b ) 2 m 2 2ay( ax+b )m+ a 2 y 2 =4am ( ax+b ) 2 m 2 +( 2a 1 ax+b ( ax+b )+4a )m+ a 2 ( 1 ax+b ) 2 =0 ( ax+b ) 2 m 2 +2am+ a 2 ( 1 ax+b ) 2 =0 ( ( ax+b )m+ a ax+b ) 2 =0 ( ax+b )m+ a ax+b =0 m= a ( ax+b ) 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, sqrt

x, a, b, m, n = symbols('x a b m n')

f = m * x + n
fs = [sqrt(a * x + b),
      1 / (a * x + b)]

for i, f0 in enumerate(fs, 1):
    print(f'[{i}]')
    pprint(f0)
    ms = solve(f - f0, m)
    for m0 in ms:
        pprint(m0)
        pprint(m0.expand())
        pprint(m0.factor())
    print()

入出力結果(Terminal, IPython)

$ ./sample3.py
[1]
  _________
╲╱ a⋅x + b 
       _________
-n + ╲╱ a⋅x + b 
────────────────
       x        
        _________
  n   ╲╱ a⋅x + b 
- ─ + ───────────
  x        x     
 ⎛      _________⎞ 
-⎝n - ╲╱ a⋅x + b ⎠ 
───────────────────
         x         

[2]
   1   
───────
a⋅x + b
-a⋅n⋅x - b⋅n + 1
────────────────
  x⋅(a⋅x + b)   
    a⋅n⋅x         b⋅n           1     
- ────────── - ────────── + ──────────
     2            2            2      
  a⋅x  + b⋅x   a⋅x  + b⋅x   a⋅x  + b⋅x
-(a⋅n⋅x + b⋅n - 1) 
───────────────────
    x⋅(a⋅x + b)    

$

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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="dx0">a = </label>
<input id="dx0" type="number" min="0" value="0.1">

<label for="a0">a = </label>
<input id="a0" type="number" min="0" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" min="0" value="3">

<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="sample3.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_a0, input_b0],
    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 / (x ** 2 + 25 ** 2) ** (3 / 2),
    f1 = (x) =>
    ((x ** 2 + 25 ** 2) ** (3 / 2) -
     x * 3 / 2 * (x ** 2 + 25 ** 2) ** (1 / 2) * 2 * x) /
    (x ** 2 + 25 ** 2) ** 3,
    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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);

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

    let points = [],
        lines = [],
        f = (x) => Math.sqrt(a0 * x + b0),
        mf = (x) => a0 / (2 * Math.sqrt(a0 * x + b0)),
        g = (x) => 1 / (a0 * x + b0),
        mg = (x) => - a0 / (a0 * x + b0) ** 2,
        hf = (x0) => (x) => mf(x0) * (x - x0) + f(x0),
        hg = (x0) => (x) => mg(x0) * (x - x0) + g(x0),
        fns = [[f, 'red'],
               [g, 'green']],
        fns1 = [],
        fns2 = [[hf, 'blue'],
                [hg, '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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