2017年5月18日木曜日

学習環境

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題8.を取り組んでみる。


    1. 1sin 1 x n 1 | x m | x m sin 1 x n | x m | lim x0 ±| x m |=0 lim x0 x m sin 1 x n =0 f0

    2. lim h0 f( h )f( 0 ) h = lim h0 h m1 sin 1 h n m=1 lim h0 f( h )f( 0 ) h =sin10 f0 m>1 | h m1 sin 1 h n || h m1 | lim h0 | h m1 |=0 lim h0 f( h )f( 0 ) h =0 f0

    3. f'( x )=m x m1 sin 1 x n + x m · n x n1 x 2n cos 1 x n =m x m1 sin 1 x n n x mn1 cos 1 x n cosθ= m x n1 m 2 x 2( m1 ) + n 2 x 2( mn1 ) sinθ= n x mn1 m 2 x 2( m1 ) + n 2 x 2( mn1 ) f'( x )= m 2 x 2( m1 ) + n 2 x 2( mn1 ) sin( 1 x n θ ) mn1>0 m>n+1

    4. lim h0 f'( h )f'( 0 ) h = lim h0 m h m1 sin 1 h n n h mn1 cos 1 h n h = lim h0 ( m h m2 sin 1 h n n h mn2 cos 1 h n ) mn2>0 m>n+2

    5. f''( x ) =m( m1 ) x m2 sin 1 x n +m x m1 n x n1 x 2n cos 1 x n n( mn1 ) x mn2 cos 1 x n n x mn1 ( sin 1 x n ) n x n1 x 2n =m( m1 ) x m2 sin 1 x n mn x mn2 cos 1 x n n( mn1 ) x mn2 cos 1 x n n 2 x m2n2 sin 1 x n m2n2>0 m>2n+2

コード(Emacs)

Python 3

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

from sympy import Symbol, symbols, sin, Limit, pprint, Derivative

print('8.')
x = Symbol('x', nonzero=True)
m, n = symbols('m n', integer=True, positive=True)
f = x ** m * sin(1 / x ** n)

for i in range(3):
    fi = Derivative(f, x, i)
    pprint(fi)
    pprint(fi.doit())

入出力結果(Terminal, IPython)

$ ./sample8.py
8.
 m    ⎛ -n⎞
x ⋅sin⎝x  ⎠
 m    ⎛ -n⎞
x ⋅sin⎝x  ⎠
d ⎛ m    ⎛ -n⎞⎞
──⎝x ⋅sin⎝x  ⎠⎠
dx             
   m    ⎛ -n⎞      m  -n    ⎛ -n⎞
m⋅x ⋅sin⎝x  ⎠   n⋅x ⋅x  ⋅cos⎝x  ⎠
───────────── - ─────────────────
      x                 x        
  2             
 d ⎛ m    ⎛ -n⎞⎞
───⎝x ⋅sin⎝x  ⎠⎠
  2             
dx              
 m ⎛ 2    ⎛ -n⎞          -n    ⎛ -n⎞        ⎛ -n⎞    2  -n    ⎛ -n⎞    2  -2⋅n 
x ⋅⎝m ⋅sin⎝x  ⎠ - 2⋅m⋅n⋅x  ⋅cos⎝x  ⎠ - m⋅sin⎝x  ⎠ + n ⋅x  ⋅cos⎝x  ⎠ - n ⋅x    ⋅
───────────────────────────────────────────────────────────────────────────────
                                                     2                         
                                                    x                          

   ⎛ -n⎞      -n    ⎛ -n⎞⎞
sin⎝x  ⎠ + n⋅x  ⋅cos⎝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="0.0001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">
<br>
<label for="m0">m = </label>
<input id="m0" type="number" min="1" step="1" value="1">
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="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="sample8.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_m = document.querySelector('#m0'),
    input_n = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_m, input_n],
    p = (x) => pre0.textContent += x + '\n';

let f6 = (x) => x === 0 ? 0 : x * Math.sin(1 / x),
    f71 = (x) => x === 0 ? 0 : x ** 2 * Math.sin(1 / x),
    f72 = (x) => x === 0 ? 0 : 2 * x * Math.sin(1 / x) - Math.cos(1 / x);

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),
        m = parseInt(input_m.value, 10),
        n = parseInt(input_n.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => x ** m * Math.sin(x ** -n),
        f1 = (x) => (m * x ** m * Math.sin(x ** -n) - n * x ** (m - n) * Math.cos(x ** -n)) / x,
        f2 = (x) =>
        x ** m *
        (m ** 2 * Math.sin(x ** -n) -
         2 * m * n * x ** -n * Math.cos(x ** -n) -
         m * Math.sin(x ** - n) +
         n ** 2 * x ** -n * Math.cos(x ** -n) -
         n ** 2 * x ** (-2 * n) * Math.sin(x ** -n) +
         n * x ** -n * Math.cos(x ** -n)) / x ** 2;
    
    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f(x)]);
        }
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f1(x)]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        if (x !== 0) {
            points.push([x, f2(x)]);
        }
    }    
    
    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('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .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);
    p(f6(0));
    p(f71(0));
    p(f72(0));
}

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();








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