2017年6月18日日曜日

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)、不等式・方程式への応用、問6、7、8.を取り組んでみる。


  1. f( x )=x1logx f'( x )=1 1 x f( 1 )=11log1=0 f( x )0 x1logx0 x1logx
    x 01
    f'-0+
    f ↘︎0↗︎

  2. x1<xlogx f( x )=xlogxx+1 f'( x )=logx+11 =logx >log1 =0 1 x < logx x1 logx<x1 f( x )=x1logx f'( x )=1 1 x >0 logx x1 <1 1 x < logx x1 <1

  3. f( 0 )=log10+0=0 f(x)=log(1+x)x+ x 2 2 f'(x)= 1 1+x 1+x = 11x+x+ x 2 1+x = x 2 1+x >0 f( x )>0 x x 2 2 <log(1+x) f(x)=x x 2 2 + x 3 3 log(1+x) f'(x)=1x+ x 2 1 x+1 = x+1 x 2 x+ x 3 + x 2 1 x+1 = x 3 x+1 >0 f( x )>0 log(1+x)<x x 2 2 + x 3 3 x x 2 2 <log(1+x)<x x 2 2 + x 3 3

    一般化。

    偶数と奇数により場合分け。

    g n ( 0 )log( 1+x )=0 d dx ( g 2k (x)log(1+x)) = d dx (( g 2k1 (x)logx) x 2k 2k ) = x 2k1 x+1 x 2k1 = x 2k1 x 2k x 2k1 x+1 = x 2k x+1 <0 g 2k (x)log(1+x)<0 g 2k ( x )<log( 1+x ) d dx ( g 2k+1 (x)log(1+x)) == d dx (( g 2k (x)logx)+ x 2k+1 2k+1 ) = x 2k x+1 + x 2k = x 2k + x 2k+1 + x 2k x+1 = x 2k+1 x+1 >0 g 2k+1 (x)log(1+x)>0 g 2k+1 ( x )>log( 1+x )

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, plot

print('6.')
x = symbols('x', positive=True)
p = plot(x - 1, log(x), (x, 0.00001, 10), show=False, legend=True)
for i, color in enumerate(['red', 'green']):
    p[i].line_color = color
p.save('sample6.svg')

print('7.')
p = plot(1 / x, log(x) / (x - 1), 1, (x, 1.00001, 10), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample7.svg')

入出力結果(Terminal, IPython)

$ ./sample6.py
6.
7.
$

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="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="2">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="1">



<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'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    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 term = (n) => (x) => (-1) ** (n - 1) * x ** n / n,
    f = (x) => Math.log(1 + x);

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),
        n0 = parseFloat(input_n0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        gn = (x) => range(1, n0 + 1).reduce((prev, i) => prev + term(i)(x), 0),
        fns = [[f, 'green'], [gn, 'blue']];

    console.log(gn(10));
    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]);
            }
        }
    });
    
    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);
};

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








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