2017年6月29日木曜日

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.2(関数の増減の判定およびその応用)、最大・最小問題、問30.を取り組んでみる。


  1. f( x )=x+ 2 x1 +2+ x 2 + ( 2 x1 +2 ) 2 f'( x )=1+ 2 ( x1 ) 2 + 1 2 ( x 2 + ( 2 x1 +2 ) 2 ) 1 2 ( 2x+2( 2 x1 +2 )· 2 ( x1 ) 2 ) =1+ 2 ( x1 ) 2 + ( x 2 + ( 2 x1 +2 ) 2 ) 1 2 ( x+( 2 x1 +2 )· 2 ( x1 ) 2 ) = ( x1 ) 2 2 ( x1 ) 2 + ( ( x1 ) 2 +4 ) 1 2 ( ( x1 ) 3 4 ( x1 ) 2 ) ( x1 ) 2 2 ( x1 ) 2 + ( ( x1 ) 2 +4 ) 1 2 ( ( x1 ) 3 4 ( x1 ) 2 )=0 ( x1 ) 2 2+ ( ( x1 ) 2 +4 ) 1 2 ( ( x1 ) 3 4 )=0 ( ( x1 ) 2 2 ) ( ( x1 ) 2 +4 ) 1 2 + ( x1 ) 3 4=0 ( ( x1 ) 2 2 ) 2 ( ( x1 ) 2 +4 ) ( ( x1 ) 3 4 ) 2 =0 ( ( x1 ) 4 4 ( x1 ) 2 +4 )( ( x1 ) 2 +4 )( ( x1 ) 6 8 ( x1 ) 3 +16 )=0 8 ( x1 ) 3 12 ( x1 ) 2 =0 ( x1 ) 2 ( 2( x1 )3 )=0 2x5=0 x= 5 2 f( 5 2 )= 5 2 + 2 5 2 1 +2+ 25 4 + ( 2 5 2 1 +2 ) 2 = 5 2 + 4 3 +2+ 25 4 + ( 4 3 +2 ) 2 = 35 6 + 25 4 + 100 9 = 35 6 +5 1 4 + 4 9 = 35 6 +5 25 36 = 35 6 + 25 6 =10

コード(Emacs)

Python 3

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

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

x = symbols('x')

fs = [(x + 2 / (x - 1) + 2 + sqrt(x ** 2 + (2 / (x - 1) + 2) ** 2), (2, 10))]

for i, (f, (x1, x2)) in enumerate(fs, 30):
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, x1, x2), show=False, legend=True)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample30.py
  ⎛         ___________________            ⎞
  ⎜        ╱                 2             ⎟
d ⎜       ╱   2   ⎛      2  ⎞           2  ⎟
──⎜x +   ╱   x  + ⎜2 + ─────⎟   + 2 + ─────⎟
dx⎝    ╲╱         ⎝    x - 1⎠         x - 1⎠
         ⎛      2  ⎞                   
       2⋅⎜2 + ─────⎟                   
         ⎝    x - 1⎠                   
   x - ─────────────                   
                 2                     
          (x - 1)                 2    
──────────────────────── + 1 - ────────
     ___________________              2
    ╱                 2        (x - 1) 
   ╱   2   ⎛      2  ⎞                 
  ╱   x  + ⎜2 + ─────⎟                 
╲╱         ⎝    x - 1⎠                 
[5/2]
$

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="1.1">
<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="15">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.1">

<label for="x0">x = </label>
<input id="x0" type="number" min="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="sample30.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0],
    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 / (x - 1) + 2 + Math.sqrt(x ** 2 + (2 / (x - 1) + 2) ** 2),
    f1 = (x) => 1 - 2 / (x - 1) ** 2 + 1 / 2 * (x ** 2 + (2 / (x - 1) + 2) ** 2) ** (-1 / 2) * (2 * x + 2 * (2 / (x - 1) + 2) * (-2) / (x - 1) ** 2),
    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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],        
        lines = [[5 / 2, y1, 5 / 2, y2, 'red'],
                 [x1, 10, x2, 10, 'red']],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'blue']];

    fns.forEach((o) => {
        let [fn, color] = o;
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;

        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;
        
        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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'));
    p(fns1.join('\n'));
};

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








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