2017年6月14日水曜日

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4章(正弦と余弦)、5(2つの基本的な極限)、練習問題1-18.を取り組んでみる。


  1. lim h0 sin2h h = lim k0 2 sink k =2

  2. lim h0 sin3h h = lim k0 3 sink k =3

  3. lim h0 sinh 3h = 1 3

  4. lim h0 tan2h sinh = lim h0 2 tan2h 2h · h sinh =2

  5. lim h0 cos2h 1+sinh =1

  6. lim h0 sin h 2 h = lim h0 h· ( sinh h ) 2 =0

  7. lim h0 sin2h 3h = lim h0 2h 3 sin2 h 2 2 h 2 =0

  8. lim h0 sin h 3 h 3 =1

  9. lim h0 sin2 h 3 h 3 = lim h0 2 sin2 h 3 2 h 3 =2

  10. lim h0 hsinh sin2 h 2 = lim h0 1 2 sinh h 2 h 2 sin2 h 2 = 1 2

  11. lim h0 2 3 sinh h · sin2h 2h · 3h sin3h = 2 3

  12. lim h0 tanh hcosh = lim h0 sinh h · 1 cos 2 h =1

  13. lim h0 tan 2 h hsinh = lim h0 sinh h cos 2 h =1

  14. lim h0 sin 2 h htanh = lim h0 sinh h ·cosh=1

  15. lim h0 tan 3 h h 2 sinhcosh = lim h0 sin 2 h h 2 cos 4 h =1

  16. lim h0 sin 2 h cosh h 2 = lim h0 1 cosh · sin 2 h h 2 =1

  17. lim h0 1cosh sinh = lim h0 1 cos 2 h sinh( 1+cosh ) = lim h0 sin 2 h sinh( 1+cosh ) = lim h0 sinh 1+cosh =0

  18. lim h0 1 cos 2 h sin 2 h cos 2 h ( 1+ cos 2 h ) = lim h0 cos 2 h sin 2 h sin 2 h( 1+ cos 2 h ) = 1 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Limit, cos, tan

h = symbols('h')

for dir in ['+', '-']:
    l = Limit((1 - cos(h)) / tan(h) ** 2, h, 0, dir=dir)
    pprint(l)
    pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample1.py
     -cos(h) + 1
 lim ───────────
h─→0⁺     2     
       tan (h)  
1/2
     -cos(h) + 1
 lim ───────────
h─→0⁻     2     
       tan (h)  
1/2
$

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="-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="sample1.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 f = (x) => Math.sin(x) / x,
    g = (x) => Math.tan(x) / 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);
        
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [];

    [[f, 'red'], [g, 'green'], [(x) => Math.sin(x), 'blue'],
     [(x) => Math.tan(x), 'orange'], [(x) => x, 'brown']]
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(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);
};

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







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