2017年7月20日木曜日

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6章(曲線をえがくこと)、2(曲線をえがくこと)、練習問題9、10.を取り組んでみる。


    1. x軸との交点。

      x 4 2 x 3 +1=0 ( x1 )( x 3 x 2 x1 )=0 ( 1,0 )

      y軸との交点は(0, 1)。


    2. 臨界点。

      f'( x )=4 x 3 6 x 2 =2 x 2 ( 2x3 ) x=0, 3 2

    3. 増加する範囲。

      3 2 x

    4. 減少する範囲。

      x 3 2

    5. 極大点は無い。

      極小点。

      3 2

    6. lim x f( x )=

      lim x f( x )=


    7. 未定義の区間は無い。


    1. x軸との交点。

      x± 2 2 x 2 1=0 x 2 = 1 2 ( ± 1 2 ,0 )

      y軸との交点は ( 1 2 ,0 )


    2. f( x )= 2 x 2 4+3 x 2 2 =2+ 3 x 2 2 f'( x )= 3·2x ( x 2 2 ) 2 = 6x ( x 2 2 ) 2 x=0

    3. x< 2 , 2 <x0

    4. 0x< 2 , 2 <x

    5. 極大点 x=0

      極小点は無い。


    6. lim x f( x )=2

      lim x f( x )=2


    7. x=± 2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, S, tan, plot

x = symbols('x')
fs = [x ** 4 - 2 * x ** 3 + 1,
      (2 * x ** 2 - 1) / (x ** 2 - 2)]

for i, f in enumerate(fs, 9):
    print(f'{i}.')
    pprint(f)
    pprint(solve(f))
    pprint(f.subs({x: 0}))
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1))
    for x0 in [S.Infinity, -S.Infinity]:
        l = Limit(f, x, x0)
        pprint(l)
        pprint(l.doit())
    print()

p = plot(fs[0], show=False, legend=True)
p.save('sample9.svg')

入出力結果(Terminal, IPython)

$ ./sample9.py
9.
 4      3    
x  - 2⋅x  + 1
⎡                        __________                                           
⎢   1   ⎛  1   √3⋅ⅈ⎞    ╱ √33   19                  4                1        
⎢1, ─ + ⎜- ─ - ────⎟⋅3 ╱  ─── + ──  + ─────────────────────────────, ─ + ─────
⎢   3   ⎝  2    2  ⎠ ╲╱    9    27                       __________  3        
⎢                                       ⎛  1   √3⋅ⅈ⎞    ╱ √33   19         ⎛  
⎢                                     9⋅⎜- ─ - ────⎟⋅3 ╱  ─── + ──       9⋅⎜- 
⎣                                       ⎝  2    2  ⎠ ╲╱    9    27         ⎝  

                                            __________                        
         4                 ⎛  1   √3⋅ⅈ⎞    ╱ √33   19   1          4          
──────────────────────── + ⎜- ─ + ────⎟⋅3 ╱  ─── + ── , ─ + ──────────────── +
              __________   ⎝  2    2  ⎠ ╲╱    9    27   3         __________  
1   √3⋅ⅈ⎞    ╱ √33   19                                          ╱ √33   19   
─ + ────⎟⋅3 ╱  ─── + ──                                     9⋅3 ╱  ─── + ──   
2    2  ⎠ ╲╱    9    27                                       ╲╱    9    27   

     __________⎤
    ╱ √33   19 ⎥
 3 ╱  ─── + ── ⎥
 ╲╱    9    27 ⎥
               ⎥
               ⎥
               ⎦
1
d ⎛ 4      3    ⎞
──⎝x  - 2⋅x  + 1⎠
dx               
   3      2
4⋅x  - 6⋅x 
[0, 3/2]
    ⎛ 4      3    ⎞
lim ⎝x  - 2⋅x  + 1⎠
x─→∞               
∞
     ⎛ 4      3    ⎞
 lim ⎝x  - 2⋅x  + 1⎠
x─→-∞               
∞

10.
   2    
2⋅x  - 1
────────
  2     
 x  - 2 
⎡-√2   √2⎤
⎢────, ──⎥
⎣ 2    2 ⎦
1/2
  ⎛   2    ⎞
d ⎜2⋅x  - 1⎟
──⎜────────⎟
dx⎜  2     ⎟
  ⎝ x  - 2 ⎠
             ⎛   2    ⎞
 4⋅x     2⋅x⋅⎝2⋅x  - 1⎠
────── - ──────────────
 2                 2   
x  - 2     ⎛ 2    ⎞    
           ⎝x  - 2⎠    
[0]
       2    
    2⋅x  - 1
lim ────────
x─→∞  2     
     x  - 2 
2
        2    
     2⋅x  - 1
 lim ────────
x─→-∞  2     
      x  - 2 
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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" value="0.1">

<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="sample9.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_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_dx0],
    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) => (2 * x ** 2 - 1) / (x ** 2 - 2),
    f1 = (x) => -3 * 2 * x / (x ** 2 - 2) ** 2,
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);
    
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),
        dx0 = parseFloat(input_dx0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[-Math.sqrt(2), y1, -Math.sqrt(2), y2, 'red'],
                 [Math.sqrt(2), y1, Math.sqrt(2), y2, 'red']],
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'orange']];

    fns
        .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]);
                }
            }
        });                 

    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(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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