2017年8月1日火曜日

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36、37、38、39、40.を取り組んでみる。

    • x±1 f( x )= x 2 1+2 x 2 1 =1+ 2 x 2 1 f'( x )= 4x ( x 2 1 ) 2
    • 増加する範囲。

      x<1,1<x<0
    • 減少する範囲。

      0<x<1,1<x
    • 極大点。

      x=0
    • 極小点。

    • x±2 f( x )= x 2 4+3 x 2 4 =1+ 3 x 2 4 f'( x )= 6x ( x 2 4 ) 2
    • 増加する範囲。

      x<2,2<x<0
    • 減少する範囲。

      0<x<2,2<x
    • 極大点。

      x=0
    • 極小点。

    • x±1 f( x )= x+1 x1 = x1+2 x1 =1+ 2 x1 f'( x )= 2 ( x1 ) 2 <0
    • 増加する範囲。

      ϕ
    • 減少する範囲。

      { ±1 }
    • 極大点。

    • 極小点。

    • f'( x )= 2 3 x 1 3 9 x 2 + x 2 3 1 2 ( 9 x 2 ) 1 2 ( 2x ) = 2 3 x 1 3 ( 9 x 2 ) 1 2 x 5 3 ( 9 x 2 ) 1 2 = 1 3 ( 9 x 2 ) 1 2 x 1 3 ( 2( 9 x 2 )3 x 2 ) = 1 3 ( 9 x 2 ) 1 2 x 1 3 ( 185 x 2 ) 185 x 2 =0 x=± 18 5 =± 3 2 5 =± 3 10 5
    • 増加する範囲。

      3x< 3 10 5 ,0<x< 3 10 5
    • 減少する範囲。

      3 10 5 <x<0, 3 10 5 <x3
    • 極大点。

      x=± 3 10 5
    • 極小点。

      x=0
    • f'( x )= 4 3 ( x4 ) 1 3 + 4 3 ( x+4 ) 1 3 = 4 3 ( ( x4 ) 1 3 + ( x+4 ) 1 3 ) = 4 3 ( x+4 ) 1 3 ( ( x4 ) 1 3 ( x+4 ) 1 3 +1 ) = 4 3 ( x+4 ) 1 3 ( ( x 2 16 ) 1 3 +1 ) ( x 2 16 ) 1 3 +1=0 ( x 2 16 ) 1 3 =1 x 2 16=1 x 2 =15 x=± 15
    • 増加する範囲。

      4<x< 15 , 15 <x
    • 減少する範囲。

      x<4, 15 <x< 15
    • 極大点。

      x= 15
    • 極小点。

      x=4, 15

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, S, plot, sqrt, Rational

x = symbols('x')
fs = [(x ** 2 + 1) / (x ** 2 - 1),
      (x ** 2 - 1) / (x ** 2 - 4),
      (x + 1) ** 2 / (x ** 2 - 1),
      (x ** 2) ** Rational(1, 3) * sqrt(9 - x ** 2),
      ((x - 4) ** 4) ** Rational(1, 3) + 2 * ((x + 4) ** 2) ** Rational(1, 3)]

for i, f in enumerate(fs, 36):
    print(f'({i})')
    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())
    p = plot(f, show=False, legend=True)
    p.save(f'sample{i}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample36.py
(36)
  ⎛ 2    ⎞
d ⎜x  + 1⎟
──⎜──────⎟
dx⎜ 2    ⎟
  ⎝x  - 1⎠
             ⎛ 2    ⎞
 2⋅x     2⋅x⋅⎝x  + 1⎠
────── - ────────────
 2                2  
x  - 1    ⎛ 2    ⎞   
          ⎝x  - 1⎠   
[0]
    ⎛ 2    ⎞
    ⎜x  + 1⎟
lim ⎜──────⎟
x─→∞⎜ 2    ⎟
    ⎝x  - 1⎠
1
     ⎛ 2    ⎞
     ⎜x  + 1⎟
 lim ⎜──────⎟
x─→-∞⎜ 2    ⎟
     ⎝x  - 1⎠
1

(37)
  ⎛ 2    ⎞
d ⎜x  - 1⎟
──⎜──────⎟
dx⎜ 2    ⎟
  ⎝x  - 4⎠
             ⎛ 2    ⎞
 2⋅x     2⋅x⋅⎝x  - 1⎠
────── - ────────────
 2                2  
x  - 4    ⎛ 2    ⎞   
          ⎝x  - 4⎠   
[0]
    ⎛ 2    ⎞
    ⎜x  - 1⎟
lim ⎜──────⎟
x─→∞⎜ 2    ⎟
    ⎝x  - 4⎠
1
     ⎛ 2    ⎞
     ⎜x  - 1⎟
 lim ⎜──────⎟
x─→-∞⎜ 2    ⎟
     ⎝x  - 4⎠
1

(38)
  ⎛       2⎞
d ⎜(x + 1) ⎟
──⎜────────⎟
dx⎜  2     ⎟
  ⎝ x  - 1 ⎠
             2          
  2⋅x⋅(x + 1)    2⋅x + 2
- ──────────── + ───────
           2       2    
   ⎛ 2    ⎞       x  - 1
   ⎝x  - 1⎠             
[]
    ⎛       2⎞
    ⎜(x + 1) ⎟
lim ⎜────────⎟
x─→∞⎜  2     ⎟
    ⎝ x  - 1 ⎠
1
     ⎛       2⎞
     ⎜(x + 1) ⎟
 lim ⎜────────⎟
x─→-∞⎜  2     ⎟
     ⎝ x  - 1 ⎠
1

(39)
  ⎛   __________    ____⎞
d ⎜  ╱    2      3 ╱  2 ⎟
──⎝╲╱  - x  + 9 ⋅╲╱  x  ⎠
dx                       
         ____          __________    ____
      3 ╱  2          ╱    2      3 ╱  2 
    x⋅╲╱  x       2⋅╲╱  - x  + 9 ⋅╲╱  x  
- ───────────── + ───────────────────────
     __________             3⋅x          
    ╱    2                               
  ╲╱  - x  + 9                           
⎡                                            3/2                              
⎢         ⎛  3 ____  2/3    5/6  2/3 3 ___  ⎞     ⎛  3 ____  2/3    5/6  2/3 3
⎢-3⋅√10   ⎜  ╲╱ 18 ⋅5      3   ⋅5   ⋅╲╱ 6 ⋅ⅈ⎟     ⎜  ╲╱ 18 ⋅5      3   ⋅5   ⋅╲
⎢───────, ⎜- ─────────── - ─────────────────⎟   , ⎜- ─────────── + ───────────
⎣   5     ⎝       10               10       ⎠     ⎝       10               10 

       3/2⎤
 ___  ⎞   ⎥
╱ 6 ⋅ⅈ⎟   ⎥
──────⎟   ⎥
      ⎠   ⎦
    ⎛   __________    ____⎞
    ⎜  ╱    2      3 ╱  2 ⎟
lim ⎝╲╱  - x  + 9 ⋅╲╱  x  ⎠
x─→∞                       
∞⋅ⅈ
     ⎛   __________    ____⎞
     ⎜  ╱    2      3 ╱  2 ⎟
 lim ⎝╲╱  - x  + 9 ⋅╲╱  x  ⎠
x─→-∞                       
∞⋅ⅈ

(40)
  ⎛   __________        __________⎞
d ⎜3 ╱        4      3 ╱        2 ⎟
──⎝╲╱  (x - 4)   + 2⋅╲╱  (x + 4)  ⎠
dx                                 
               __________                  
  ⎛2⋅x   8⎞ 3 ╱        2         __________
2⋅⎜─── + ─⎟⋅╲╱  (x + 4)       3 ╱        4 
  ⎝ 3    3⎠                 4⋅╲╱  (x - 4)  
───────────────────────── + ───────────────
                2              3⋅(x - 4)   
         (x + 4)                           
[-√15, √15]
    ⎛   __________        __________⎞
    ⎜3 ╱        4      3 ╱        2 ⎟
lim ⎝╲╱  (x - 4)   + 2⋅╲╱  (x + 4)  ⎠
x─→∞                                 
∞
     ⎛   __________        __________⎞
     ⎜3 ╱        4      3 ╱        2 ⎟
 lim ⎝╲╱  (x - 4)   + 2⋅╲╱  (x + 4)  ⎠
x─→-∞                                 
∞

$

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">

<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="sample36.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 f1 = (x) => (x ** 2 + 1) / (x ** 2 - 1),
    f2 = (x) => (x ** 2 - 1) / (x ** 2 - 4),
    f3 = (x) => (x + 1) ** 2 / (x ** 2 - 1);
    
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 = [],
        fns = [[f1, 'orange'],
               [f2, 'green'],
               [f3, 'blue']],
        fns1 = [],
        fns2 = [];

    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();







0 コメント:

コメントを投稿

Comments on Google+: