2017年7月18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3、4、5.を取り組んでみる。


    1. x軸との交点(-1, 0)。

      y軸との交点(1, 0)。


    2. 臨界点。

      f'( x )= x 2 +1( x+1 )2x ( x 2 +1 ) 2 = x 2 2x+1 ( x 2 +1 ) x 2 +2x1=0 x=1± 1+1 =1± 2

    3. 増加する範囲。

      1 2 x1+ 2

    4. 減少する範囲。

      x1 2 ,1+ 2 x

    5. 極大点。

      1+ 2

      極小点。

      1 2

    6. lim x f( x )=0

      lim x f( x )=0


    7. 未定義な区間は無い。


    1. ( nπ,0 )

      ( 0,0 )


    2. f'( x )=2sinxcosx=sin2x sin2x=0 2x=nπ x= nπ 2

    3. 2nπ2x(2n+1)π nπx( n+ 1 2 )π

    4. π+2nπ2x2π+2nπ π 2 +nπxπ+nπ

    5. 極大点。

      ( n+ 1 2 )π

      極小点。

      π+nπ

    6. lim x± f( x ) は循環。


    7. 未定義な区間はない。


    1. ( π 2 +nπ,0 ) ( 0,1 )

    2. f'( x )=2cosxsinx=sin2x sin2x=0 2x=nπ x= nπ 2

    3. π+2nπ2x2π+2nπ π 2 +nπxπ+nπ

    4. 2nπ2xπ+2nπ nπx π 2 +nπ

    5. 極大点。

      π+2nπ

      極小点。

      π 2 +nπ

    6. lim x± f( x ) は収束しない。


    7. 未定義な区間はない。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, solve, Derivative, Limit, S, sin, cos, plot

x = symbols('x')
fs = [(x + 1) / (x ** 2 + 1),
      sin(x) ** 2,
      cos(x) ** 2]

for i, f in enumerate(fs, 3):
    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('sample3.svg')

入出力結果(Terminal, IPython)

$ ./sample3.py
3.
x + 1 
──────
 2    
x  + 1
[-1]
1
d ⎛x + 1 ⎞
──⎜──────⎟
dx⎜ 2    ⎟
  ⎝x  + 1⎠
  2⋅x⋅(x + 1)     1   
- ─────────── + ──────
           2     2    
   ⎛ 2    ⎞     x  + 1
   ⎝x  + 1⎠           
[-1 + √2, -√2 - 1]
    x + 1 
lim ──────
x─→∞ 2    
    x  + 1
0
     x + 1 
 lim ──────
x─→-∞ 2    
     x  + 1
0

4.
   2   
sin (x)
[0, π]
0
d ⎛   2   ⎞
──⎝sin (x)⎠
dx         
2⋅sin(x)⋅cos(x)
⎡   π     3⋅π⎤
⎢0, ─, π, ───⎥
⎣   2      2 ⎦
       2   
lim sin (x)
x─→∞       
<0, 1>
        2   
 lim sin (x)
x─→-∞       
<0, 1>

5.
   2   
cos (x)
⎡π  3⋅π⎤
⎢─, ───⎥
⎣2   2 ⎦
1
d ⎛   2   ⎞
──⎝cos (x)⎠
dx         
-2⋅sin(x)⋅cos(x)
⎡   π     3⋅π⎤
⎢0, ─, π, ───⎥
⎣   2      2 ⎦
       2   
lim cos (x)
x─→∞       
<0, 1>
        2   
 lim cos (x)
x─→-∞       
<0, 1>

$

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="0.005">
<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="-0.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">
<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="sample3.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) => Math.sin(x) ** 2,
    f1 = (x) => Math.sin(2 * x),
    g = (x) => Math.cos(x) ** 2,
    g1 = (x) => -Math.sin(2 * x),
    hf = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
    hg = (x0) => (x) => g1(x0) * (x - x0) + g(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 = [],
        fns = [[f, 'blue'],
               [g, 'green'],],
        fns1 = [],
        fns2 = [[hf, 'brown'],
                [hg, '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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