2017年5月17日水曜日

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6、7.を取り組んでみる。


  1. x0 1sin 1 x 1 x>0 xxsin 1 x x x<0 xxsin 1 x x lim x0 ±x=0 lim x0 xsin 1 x =0 f0 lim h0 (0+h)sin 1 0+h 0 h = lim h0 hsin 1 h h = lim h0 sin 1 h = lim x sinx f0

  2. x0 f'( x )=2xsin 1 x + x 2 1 x 2 cos 1 x =2xsin 1 x cos 1 x x=0 f'( x )=0 x0 1sin 1 x 1 x 2 x 2 sin 1 x x 2 lim x0 ± x 2 =0 lim x0 x 2 sin 1 x =0 f( x )=0 f lim h0 f'( 0+h )f'( 0 ) h = lim h0 ( 2( 0+h )sin 1 0+h cos 1 0+h )0 h = lim h0 2hsin 1 h cos 1 h h = lim h0 ( 2sin 1 h cos 1 h ) = lim h0 ( 5 ( 2 5 sin 1 h 1 5 cos 1 h ) ) cosθ= 2 5 ,sinθ= 1 5 = lim h0 ( 5 ( sin 1 h cosθcos 1 h sinθ ) ) = lim h0 ( 5 ( sin( 1 h θ ) ) ) = 5 lim x sinx f'0

コード(Emacs)

Python 3

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

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

print('6')
x = Symbol('x', nonzero=True)
expr = x * sin(1 / x)

pprint(expr)
l = Limit(expr, x, 0, dir='+')
pprint(l)
pprint(l.doit())
l = Limit(expr, x, 0, dir='-')
pprint(l)
pprint(l.doit())

h = Symbol('h', positive=True)
expr = ((0 + h) * sin(1 / (0 + h)) - 0) / h
pprint(expr)
l = Limit(expr, h, 0)
pprint(l)
pprint(l.doit())

print('7.')
x = Symbol('x', nonzero=True)
expr = x ** 2 * sin(1 / x)
pprint(expr)
l = Limit(expr, x, 0, dir='+')
pprint(l)
pprint(l.doit())
l = Limit(expr, x, 0, dir='-')
pprint(l)
pprint(l.doit())

d = Derivative(expr, x)
pprint(d)
f1 = d.doit()
pprint(f1)

l = Limit(f1, x, 0)
pprint(l)
pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample6.py
6
     ⎛1⎞
x⋅sin⎜─⎟
     ⎝x⎠
          ⎛1⎞
 lim x⋅sin⎜─⎟
x─→0⁺     ⎝x⎠
0
          ⎛1⎞
 lim x⋅sin⎜─⎟
x─→0⁻     ⎝x⎠
0
   ⎛1⎞
sin⎜─⎟
   ⎝h⎠
        ⎛1⎞
 lim sin⎜─⎟
h─→0⁺   ⎝h⎠
<-1, 1>
7.
 2    ⎛1⎞
x ⋅sin⎜─⎟
      ⎝x⎠
      2    ⎛1⎞
 lim x ⋅sin⎜─⎟
x─→0⁺      ⎝x⎠
0
      2    ⎛1⎞
 lim x ⋅sin⎜─⎟
x─→0⁻      ⎝x⎠
0
d ⎛ 2    ⎛1⎞⎞
──⎜x ⋅sin⎜─⎟⎟
dx⎝      ⎝x⎠⎠
       ⎛1⎞      ⎛1⎞
2⋅x⋅sin⎜─⎟ - cos⎜─⎟
       ⎝x⎠      ⎝x⎠
     ⎛       ⎛1⎞      ⎛1⎞⎞
 lim ⎜2⋅x⋅sin⎜─⎟ - cos⎜─⎟⎟
x─→0⁺⎝       ⎝x⎠      ⎝x⎠⎠
<-1, 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="0.0001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1.4">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.4">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.4">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.4">

<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="sample6.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';

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

    let points = [];
    
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f6(x)]);
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f71(x)]);
    }
    let t2 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        points.push([x, f72(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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