2017年5月11日木曜日

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 lim_{x→+∞} f(x), 極限 lim_{x→-∞} f(x)、問6、7、8.を取り組んでみる。




    1. lim x+ x 3 ( 1+ 2 x 2 )=

    2. lim x x 3 ( 1+ 2 x 2 )=

    3. lim x+ x 4 ( 1 6 x )=

    4. lim x x 4 ( 1 6 x )=

    5. lim x+ 1 x( x+1 ) =0

    6. lim x 2+ 1 x 2 6+ 5 x = 1 3

    7. lim t+ 2 t +3 t 3 10 t 2 1 =

    8. lim t 2 t +3 t 3 10 t 2 1 =

    9. 1

    10. lim t x+ 2 11 x 10 1 3 10 x 10 =

  1. lim x+ f( x )=+ n=2k lim x f( x )=+ n=2k+1 lim x f( x )=

    1. lim x x+1x+1 x+1 + x1 =0

    2. lim x 1 1+ 1 x 2 =1

    3. lim x 1 1+ 1 x 2 =1

    4. lim x x x 2 +x +x = lim x 1 1+ 1 x +1 = 1 2

    5. lim x x x 2 +x +x = lim x 1 1+ 1 x +1 =

コード(Emacs)

Python 3

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

from sympy import Symbol, Limit, sqrt, S, pprint

x = Symbol('x')

exprs = [(sqrt(x + 1) - sqrt(x - 1), x, S.Infinity),
         (x / sqrt(x ** 2 + 1), x, S.Infinity),
         (x / sqrt(x ** 2 + 1), x, S.NegativeInfinity),
         (sqrt(x ** 2 + x) - x, x, S.Infinity),
         (sqrt(x ** 2 + x) - x, x, S.NegativeInfinity)]

for t in exprs:
    pprint(Limit(*t).doit())

入出力結果(Terminal, IPython)

$ ./sample6.py
0
1
-1
1/2
∞
$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" 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="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_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    inputs = [input_x1, input_x2],
    p = (x) => pre0.textContent += x + '\n';

let f1 = (x) => Math.sqrt(x + 1) - Math.sqrt(x - 1),
    f2 = (x) => x / Math.sqrt(x ** 2 + 1),
    f3 = (x) => Math.sqrt(x ** 2 + x) - x;

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) {
        points.push([x, f2(x)]);
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += 0.001) {
        if (x <= -1 || 0 <= x) {
            points.push([x, f3(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' : 'blue');

    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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