2017年6月12日月曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第6章(関数の近似、極限の計算)、6.2(極限の計算)、問題1、2.を取り組んでみる。


    1. x= 1 y xy0 lim y0 e y 1 y = lim y0 e y =1

    2. f( x )= x 1 x logf( x )= logx x lim x logf( x ) = lim x logx x = lim x 1 x 1 =0 lim x x 1 x =1

    3. lim x0 e xloga e xlogb x = lim x0 ( e xloga loga e xlogb logb ) =logalogb

    4. lim x0 1 1 1 x 2 3 x 2 = lim x0 1 2 ( 1 x 2 ) 1 2 ( 2x ) 1 x 2 6x = lim x0 ( 1 x 2 ) 1 2 6 = 1 6

    5. lim x π 2 sinx1 cosx = lim x π 2 cosx sinx =0

    6. lim x π 2 sinx+xcosx sinx =1

    1. lim x0 1 1 1+x 2x = lim x0 1 ( 1+x ) 2 2 = 1 2

    2. lim x0 x 2 sin 2 x x 2 sin 2 x = lim x0 x 2 sin 2 x x 4 · x 2 sin 2 x = lim x0 x 2 sin 2 x x 4 = lim x0 2x2sinxcosx 4 x 3 = lim x0 2xsin2x 4 x 3 = lim x0 22cos2x 12 x 2 = lim x0 1cos2x 6 x 2 = lim x0 2sin2x 12x = lim x0 4cos2x 12 = 1 3

    3. lim x0 1cosx 1 cos 2 x 1 = lim x0 cos 2 x cos 3 x 1 cos 2 x = lim x0 2cosxsinx+3cosxsinx 2cosxsinx = lim x0 2+3 2 = 1 2

    4. x logx = 1 1 x logx = 1 log x 1 x y= x 1 x logy= logx x x0y1 lim y1 y1 logy = lim y1 1 1 y =1

    5. y= 1 sinx x0+y xy= x sinx f( x )= y x logf( x )=xlogy logf( x )=xy logy y x0+xy1 y logy y 0 lim x0+ f( x )=1 lim x0+ ( 1 sinx ) x =1

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Limit, pi, E, plot

x = symbols('x')
expr = (E - (1 + x) ** (1 / x)) / x
for dir in ['+', '-']:
    l = Limit(expr, x, 0, dir=dir)
    pprint(l)
    pprint(l.doit())

p = plot(expr, show=False)
p.save('sample1.svg')

入出力結果(Terminal, IPython)

$ ./sample1.py
       x _______    
     - ╲╱ x + 1  + ℯ
 lim ───────────────
x─→0⁺       x       
ℯ
─
2
       x _______    
     - ╲╱ x + 1  + ℯ
 lim ───────────────
x─→0⁻       x       
ℯ
─
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) => (1 + x) ** (1 / x),
    g = (x) => Math.E - f(x),
    h = (x) => x,
    l = (x) => g(x) / h(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;
    }

    let points = [],
        lines = [];

    [[f, 'red'], [g, 'green'], [h, 'blue'], [l, 'brown']].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('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.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.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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