2017年5月9日火曜日

学習環境

数学読本〈4〉数列の極限,順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.1(関数と極限)、極限値の計算、問4.を取り組んでみる。


    1. lim x1 ( x 2 +ax+b ) = lim x1 ( x1 ) x 2 +ax+b x1 =0·5 =0 1+a+b=0 b=1a lim x1 x 2 +ax1a x1 = lim x1 ( x1 )( x+a+1 ) x1 = lim x1 ( x+a+1 ) =2+a =5 a=3 b=13=4

    2. lim x3 ( x 2 +ax+6 ) = lim x3 ( x 2 ( 3+b )x+3b ) x 2 +ax+6 x 2 ( 3+b )x+3b = lim x3 ( 993b+3b ) 2 5 = lim x3 0· 2 5 =0 9+3a+6=0 a=5 = lim x3 x 2 5x+6 x 2 ( 3+b )x+3b = lim x3 ( x2 )( x3 ) ( x3 )( xb ) = 1 3b = 2 5 2( 3b )=5 b= 1 2

    3. lim x2 ( x 3 +ax+b ) = lim x2 x 3 +ax+b x 2 x2 ( x 2 x2 ) =3·0 =0 8+2a+b=0 b=2a8 lim x2 x 2 x2 x 3 +ax2a8 = lim x2 ( x2 )( x+1 ) ( x2 )( x 2 +2x+a+4 ) = lim x2 x+1 x 2 +2x+a+4 = 3 4+4+a+4 = 3 12+a = 1 3 12+a=9 a=3 b=2

    4. lim x1 ( a x+1 b ) = lim x1 ( x1 ) a x+1 b x1 =0· 2 =0 a 2 b=0 b=a 2 lim x1 a x+1 b x1 =a lim x1 x+1 2 x1 =a lim x1 x+12 ( x1 )( x+1 + 2 ) =a lim x1 1 x+1 + 2 = a 2 2 = 2 a=4 b=4 2

コード(Emacs)

Python 3

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

from sympy import Symbol, Limit, sqrt, pprint

x = Symbol('x')

for i, t in enumerate([
        ((x ** 2 + 3 * x - 4) / (x - 1), x, 1),
        ((x ** 2 + -5 * x + 6) / (x ** 2 - (3 + 1/2) * x + 3 * 1/2), x, 3),
        ((x ** 2 - x - 2) / (x ** 3 - 3 * x - 2), x, 2),
        ((4 * sqrt(x + 1) - 4 * sqrt(2)) / (x - 1), x, 1)
                      ]):
    print('({})'.format(i + 1))
    pprint(Limit(*t).doit())

入出力結果(Terminal, IPython)

$ ./sample4.py
(1)
5
(2)
0.400000000000000
(3)
1/3
(4)
√2
$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">

<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="sample4.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_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    inputs = [input_x1, input_x2],
    p = (x) => pre0.textContent += x + '\n';

let f1 = (x) => (x ** 2 + 3 * x -4) / (x - 1),
    f2 = (x) => (x ** 2 - 5 * x + 6) / (x ** 2 - 7 / 2 * x + 1.5),
    f3 = (x) => (x ** 2 - x - 2) / (x ** 3 - 3 * x - 2),
    f4 = (x) => (4 * Math.sqrt(x + 1) - 4 * Math.sqrt(2)) / (x - 1);

let draw = () => {
    pre0.textContent = '';

    let x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value);
    
    let points = [];

    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 1) {
            points.push([x, f1(x)]);
        }
    }
    let t1 = points.length;

    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 3) {
            points.push([x, f2(x)]);
        }
    }
    let t2 = points.length;
    
    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 2) {
            points.push([x, f3(x)]);
        }
    }
    let t3 = points.length;
    for (let x = x1; x <= x2; x += 0.001) {
        if (x !== 1) {
            points.push([x, f4(x)]);
        }
    }
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]);
    let yscale = d3.scaleLinear()
        .domain([x1, x2])
        .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', 1)
        .attr('fill', (d, i) =>
              i < t1 ? 'red' :
              i < t2 ? 'green' :
              i < t3 ? 'blue' : 'gray');

    svg.append('g')
        .attr('transform', `translate(0, ${yscale(0)})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${xscale(0)}, 0)`)
        .call(yaxis);
}

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






















						

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