2017年7月15日土曜日

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)、18.5(関数の近似、テイラーの定理)、1次の近似式、問53、54.を取り組んでみる。


    1. 10.02 3 = ( 10+0.02 ) 3 = 10 3 ( 1+ 2 1000 ) 3 1000·( 1+ 3·2 1000 ) =1006

    2. ( 9.012 ) 1 2 = ( 9( 1+ 12 9000 ) ) 1 2 =3 ( 1+ 1 750 ) 1 2 3( 1+ 1 2·750 ) =3+ 1 2·250 =3.002

    3. 1.984 1 ( 20.016 ) 1 = 2 1 ( 10.008 ) 1 1+0.008 2 =0.504

    4. 7.9904 1 3 = ( 80.0096 ) 1 3 = 8 1 3 ( 10.0012 ) 1 3 1+0.0004 2 =0.5002

    1. f( x )= e x f'( x )= e x f( 0 )=1 f'( 0 )=1 e x 1+x

    2. f( x )=tanx f'( x )= 1 cos 2 x f( 0 )=0 f'( 0 )=1 tanx0+1x=x

    3. f( h )=sin( a+h ) f'( h )=cos( a+h ) f( 0 )=sina f'( 0 )=cosa sin( a+h )sina+hcosa

    4. f( h )=log( a+h ) f'( h )= 1 a+h f( 0 )=loga f'( 0 )= 1 a log( a+h )loga+ h a

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, log, tan, exp, plot

print('54.')

x = symbols('x')
fs = [(exp(x), 1 + x),
      (tan(x), x),
      (sin(2 + x), sin(2) + x * cos(2)),
      (log(2 + x), log(2) + x / 2)]

for i, (f, g) in enumerate(fs, 1):
    print(f'({i})')
    p = plot(f, g, (x, -0.1, 0.1), show=False, legend=True)
    for j, color in enumerate(['red', 'green']):
        p[j].line_color = color
    p.save(f'sample54_{i}.svg')

入出力結果(Terminal, IPython)

$ ./sample53.py
54.
(1)
(2)
(3)
(4)
$

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="sample53.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 f1 = (x) => Math.exp(x),
    f2 = (x) => 1 + x,
    g1 = (x) => Math.tan(x),
    g2 = (x) => x,
    h1 = (x) => Math.sin(2 + x),
    h2 = (x) => Math.sin(2) + x * Math.cos(2),
    l1 = (x) => Math.log(2 + x),
    l2 = (x) => Math.log(2) + x / 2;

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 = [[f1, 'red'], [f2, 'green'],
               [g1, 'blue'], [g2, 'orange'],
               [h1, 'purple'], [h2, 'brown'],
               [l1, 'skyblue'], [l2, 'yellow']],
        fns1 = [],
        fns2 = [];

    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);

    [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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