2017年9月15日金曜日

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7章(逆関数)、1(逆関数の定義)、練習問題7、8.を取り組んでみる。


  1. f'( x )= 2x x 4 = 2 x 3 <0( 0<x1 )

    よって、0 < x ≤ 1 で 関数fは強減少なので、逆関数gが存在する。定義される範囲は 1 ≤ y。

    y= 1 x 2 x 2 = 1 y x= 1 y g( y )= 1 y

  2. f( x )= x 2 +11 x 2 +1 =1 1 x 2 +1 f'( x )= 2x ( x 2 +1 ) 2 = 2x ( x 2 +1 ) 2 >0( 0<x5 )

    よって、逆関数gが存在する。定義される範囲は 0 ≤ y ≤ 25/26。

    y=1 1 x 2 +1 ( y1 )( x 2 +1 )=1 x 2 +1= 1 1y x 2 = 1 1y 1 x 2 = y 1y x= y 1y g( y )= y 1y

コード(Emacs)

Python 3

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

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

x, y = symbols('x y', real=True)
fs = [1 / x ** 2,
      x ** 2 / (x ** 2 + 1)]

for i, f in enumerate(fs, 7):
    print(f'({i})')
    s = solve(y - f, x)
    pprint(s)
    p = plot(f, show=False, legend=True)
    p.save(f'sample{i}_0.svg')
    p = plot(*s, show=False, legend=True)
    p.save(f'sample{i}_1.svg')
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample7.py
(7)
⎡     ___      ___⎤
⎢    ╱ 1      ╱ 1 ⎥
⎢-  ╱  ─ ,   ╱  ─ ⎥
⎣ ╲╱   y   ╲╱   y ⎦

(8)
⎡     _______      _______⎤
⎢    ╱  -y        ╱  -y   ⎥
⎢-  ╱  ───── ,   ╱  ───── ⎥
⎣ ╲╱   y - 1   ╲╱   y - 1 ⎦

$

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">

<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="sample7.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 fx7 = (x) => 1 / x ** 2,
    fy7 = (y) => 1 / Math.sqrt(y),
    fx8 = (x) => x ** 2 / (x ** 2 + 1),
    fy8 = (y) => Math.sqrt(y / (1 - y));

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);

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

    let points = [],
        lines = [],
        fns = [[(x) => x, 'red'],
               [fx7, 'green'],        
               [fy7, 'blue'],
               [fx8, 'orange'],
               [fy8, 'purple']]
        fns1 = [],
        fns2 = [];

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

                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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