2017年10月21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8章(指数関数と対数関数)、1(対数関数)、練習問題6.を取り組んでみる。


    1. f(x) = x - log x とおく。

      f'( x )=1 1 x f''( x )= 1 x 2 = 1 x 2

      f(1) = 1 - log 1 = 1 - 0 = 1となる。

      また、f''(x) > 0 より、fは強増加なので、f(x) > 0となる。

      以上より、全ての x > 1 に対し、x - log x > 0、log x < x が成り立つ。(証明終)


    2. f(x) = x - log(1 + x) とおく。

      f(0) = 0 - log(1 + 0) = 0 - log 1 = 0 - 0 = 0。

      f'( x )=1 1 1+x f''( x )= 1 ( 1+x ) 2 = 1 ( 1+x ) 2

      ここで、f''(x) > 0 なので、関数fは強増加。

      よって、すべての x > 0 に対し、f(x) > 0、すなわち、x - log(1 + x) > 0、log(1 + x) < x が成り立つ。(証明終)

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, plot, Derivative

print('6.')
x = symbols('x')
fs = [log(x), log(1 + x)]

for i, f in enumerate(fs):
    c = chr(ord('a') + i)
    print(f'({c})')
    f0 = x - f
    for s in range(1, 3):
        D = Derivative(f0, x, s)
        for t in [D, D.doit()]:
            pprint(t)
            print()
        print()

    p = plot(x, f, show=False, legend=True)
    for k, color in enumerate(['red', 'green']):
        p[k].line_color = color
    p.save(f'sample6_{c}.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
(a)
d             
──(x - log(x))
dx            

    1
1 - ─
    x


  2            
 d             
───(x - log(x))
  2            
dx             

1 
──
 2
x 


(b)
d                 
──(x - log(x + 1))
dx                

      1  
1 - ─────
    x + 1


  2                
 d                 
───(x - log(x + 1))
  2                
dx                 

   1    
────────
       2
(x + 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="0.001" 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="sample6.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 fa = (x) => Math.log(x) + 1,
    fb = (x) => Math.log(x + 1);

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 = [[fa, 'red'],
               [fb, 'blue']],
        fns1 = [[(x) => x]],
        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]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), 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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