2017年7月30日日曜日

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)、関数の整級数展開(2)、問61-6、7、8、9、10.を取り組んでみる。


    1. f( x )= ( 1+ a x ) x logf( x )=xlog( 1+ a x ) logf( x )= log( 1+ a x ) 1 x g( x )=log( 1+ a x ) h( x )= 1 x lim x g( x )=0 lim x h( x )=0 g'( x )= 1 1+ a x · a x 2 h'( x )= 1 x 2

      ロピタルの定理より。

      lim x log( 1+ a x ) 1 x = lim x g'( x ) h'( x ) = lim x a 1+ a x =a lim x logf( x )=a lim x f( x )= e a

    2. f( x )= x 1 x logf( x )= logx x g( x )=logx h( x )=x lim x g( x )= lim x h( x )= g'( x )= 1 x h'( x )=1

      ロピタルの定理より。

      lim x logx x = lim x g'( x ) h'( x ) = lim x 1 x =0 lim x logf( x )=0 lim x f( x )=1

    3. f( x )= 1+x ( 1+ 1 2 x ) g( x )= x 2 lim x0 f( x )=0 lim x0 g( x )=0 f'( x )= 1 2 ( 1+x ) 1 2 1 2 = 1 2 ( ( 1+x ) 1 2 1 ) g'( x )=2x lim x0 f'( x )=0 lim x0 g'( x )=0 f''( x )= 1 2 · 1 2 ( 1+x ) 3 2 = 1 4 ( 1+x ) 3 2 g''( x )=2

      ロピタルの定理より。

      lim x0 f( x ) g( x ) = lim x0 f'( x ) g'( x ) = lim x0 f''( x ) g''( x ) = lim x 1 4 ( 1+x ) 3 2 2 = 1 8

    4. x 2 sin 2 x x 2 sin 2 x = x 2 sin 2 x · x 2 sin 2 x x 4 lim x0 x 2 sin 2 x =1 f( x )= x 2 sin 2 x g( x )= x 4 lim x0 f( x )=0 lim x0 g( x )=0 f'( x )=2x2sinxcosx g'( x )=4 x 3 lim x0 f'( x )=0 lim x0 g'( x )=0 f''( x )=22( cos 2 x sin 2 x ) =2( 1( 12 sin 2 x ) ) =4 sin 2 x g''( x )=12 x 2

      ロピタルの定理より。

      lim x0 f( x ) g( x ) = lim x0 f'( x ) g'( x ) = lim x0 f''( x ) g''( x ) = lim x0 4 sin 2 x 12 x 2 = 1 3 lim x0 x 2 sin 2 x x 2 sin 2 x = 1 3

    5. f( x )= ( a x + b x 2 ) 1 x logf( x )= log a x + b x 2 x g( x )=log a x + b x 2 =log e xloga + e xlogb 2 h( x )=x lim x0 g( x )=log1=0 lim x0 h( x )=0 g'( x )= 1 2 ( e xloga loga+ e xlogb logb ) e xloga + e xlogb 2 = e xloga loga+ e xlogb logb e xloga + e xlogb h'( x )=1

      ロピタルの定理より。

      lim x0 g( x ) h( x ) = lim x0 g'( x ) h'( x ) = lim x0 e xloga loga+ e xlogb logb e xloga + e xlogb = loga+logb 2 =log ( ab ) 1 2 lim x0 logf( x )=log ( ab ) 1 2 lim x0 f( x )= ( ab ) 1 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, sqrt, Rational, Limit, S, plot

x = symbols('x')
a, b = symbols('a b', positive=True)
fs = [((1 + a / x) ** x, S.Infinity),
      (x ** (1 / x), S.Infinity),
      ((sqrt(1 + x) - (1 + Rational(1, 2) * x)) / x ** 2, 0),
      (1 / sin(x) ** 2 - 1 / x ** 2, 0),
      (((a ** x + b ** x) / 2) ** (1 / x), 0)]

print('61.')
for i, (f, x0) in enumerate(fs, 6):
    print(f'({i})')
    l = Limit(f, x, x0)
    pprint(l)
    pprint(l.doit())
    try:
        p = plot(f, show=False, legend=True)
        p.save(f'sample1_{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample61_6.py
61.
(6)
           x
    ⎛a    ⎞ 
lim ⎜─ + 1⎟ 
x─→∞⎝x    ⎠ 
 a
ℯ 
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.

(7)
    x ___
lim ╲╱ x 
x─→∞     
1
<class 'OverflowError'> complex exponentiation

(8)
     ⎛  x     _______    ⎞
     ⎜- ─ + ╲╱ x + 1  - 1⎟
     ⎜  2                ⎟
 lim ⎜───────────────────⎟
x─→0⁺⎜          2        ⎟
     ⎝         x         ⎠
-1/8

(9)
     ⎛   1      1 ⎞
 lim ⎜─────── - ──⎟
x─→0⁺⎜   2       2⎟
     ⎝sin (x)   x ⎠
1/3

(10)
          _________
         ╱  x    x 
        ╱  a    b  
 lim x ╱   ── + ── 
x─→0⁺╲╱    2    2  
√a⋅√b
<class 'ValueError'> The same variable should be used in all univariate expressions being plotted.

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" min="0" value="3">

<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="sample61_6.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'),
    input_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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 f7 = (x) => x ** (1 / x),
    f9 = (x) => 1 / Math.sin(x) ** 2 - 1 / 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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        y3 = Math.exp(a0),
        y4 = 1,
        y5 = 1 / 3,
        y6 = Math.sqrt(a0 * b0),
        lines = [[x1, y3, x2, y3, 'yellow'],
                 [x1, y4, x2, y4, 'brown'],
                 [x1, y5, x2, y5, 'skyblue'],
                 [x1, y6, x2, y6, 'purple']],
        f6 = (x) => (1 + a0 / x) ** x,
        f10 = (x) => ((a0 ** x + b0 ** x) / 2) ** (1 / x),
        fns = [[f6, 'red'],
               [f7, 'green'],
               [f9, 'blue'],
               [f10, 'orange']],
        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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