2017年9月29日金曜日

学習環境

逆転インバース(前編)|数学ガールの秘密ノートの割り算の微分、分数関数の微分を、《暗記テスト》でも《思考テスト》でもなく、《定義テスト》で考えてみる。

定義テストで考えてみた理由は、関数の微分、導関数を求めるのに累乗(べき乗)、三角関数、指数関数・対数関数等のいろいろ簡単な公式を使用して計算を多くしているうちに、元々の定義を忘れがちになる場合があるかなぁと思ったから。

ということで、早速定義に立ち返って分数関数の微分を求めてみる。

lim h g( x+h ) f( x+h ) g( x ) f( x ) h = lim h g( x+h )f( x )g( x )f( x+h ) f( x )f( x+h ) h = lim h g( x+h )f( x )g( x )f( x+h ) f( x )f( x+h )h = 1 f( x ) lim h g( x+h )f( x )g( x )f( x+h )+g( x )f( x )g( x )f( x ) f( x+h )h = 1 f( x ) lim h 1 f( x+h ) · ( g( x+h )f( x )g( x )f( x ) )( g( x )f( x+h )g( x )f( x ) ) h = 1 f( x ) lim h 1 f( x+h ) ·( g( x+h )g( x ) h f( x )g( x ) f( x+h )f( x ) h · ) = 1 f( x ) · 1 f( x ) ( g'( x )f( x )g( x )f'( x ) )

ということで、次のことが成り立つ。

d dx ( g( x ) f( x ) )= g'( x )f( x )g( x )f'( x ) ( f( x ) ) 2

一応 Python(SymPy)で確認。

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Function, Derivative, Limit

x, h = symbols('x, h')
f = Function('f')
g = Function('g')
F = g(x) / f(x)

for t in [f, g, F]:
    pprint(t)
    print()

D = Derivative(F, x, 1)
for t in [D, D.doit()]:
    pprint(t.factor())
    print()


for d in ["-", "+"]:
    l = Limit((F.subs({x: x + h}) - F) / h, h, 0)
    for t in [l, l.doit()]:
        pprint(t)
        print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample25.py
f

g

g(x)
────
f(x)

d ⎛g(x)⎞
──⎜────⎟
dx⎝f(x)⎠

 ⎛       d               d       ⎞ 
-⎜- f(x)⋅──(g(x)) + g(x)⋅──(f(x))⎟ 
 ⎝       dx              dx      ⎠ 
───────────────────────────────────
                2                  
               f (x)               

     ⎛g(h + x)   g(x)⎞
     ⎜──────── - ────⎟
     ⎜f(h + x)   f(x)⎟
 lim ⎜───────────────⎟
h─→0⁺⎝       h       ⎠

 ⎛       ⎛ d        ⎞│            ⎛ d        ⎞│    ⎞ 
-⎜- f(x)⋅⎜───(g(ξ₁))⎟│     + g(x)⋅⎜───(f(ξ₁))⎟│    ⎟ 
 ⎝       ⎝dξ₁       ⎠│ξ₁=x        ⎝dξ₁       ⎠│ξ₁=x⎠ 
─────────────────────────────────────────────────────
                         2                           
                        f (x)                        

     ⎛g(h + x)   g(x)⎞
     ⎜──────── - ────⎟
     ⎜f(h + x)   f(x)⎟
 lim ⎜───────────────⎟
h─→0⁺⎝       h       ⎠

 ⎛       ⎛ d        ⎞│            ⎛ d        ⎞│    ⎞ 
-⎜- f(x)⋅⎜───(g(ξ₁))⎟│     + g(x)⋅⎜───(f(ξ₁))⎟│    ⎟ 
 ⎝       ⎝dξ₁       ⎠│ξ₁=x        ⎝dξ₁       ⎠│ξ₁=x⎠ 
─────────────────────────────────────────────────────
                         2                           
                        f (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="-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" step="0.0001" value="0.05">

<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="sample.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),
    g = (x) => x ** 2,
    f1 = (x) => Math.cos(x),
    g1 = (x) => 2 * x,
    fg = (x) => f(x) / g(x),
    fg1 = (x) => (f1(x) * g(x) - f(x) * g1(x)) / g(x) ** 2,
    h = (x0) => (x) => fg1(x0) * (x - x0) + fg(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, 'red'],
               [g, 'green'],
               [fg, 'blue']],
        fns1 = [],
        fns2 = [[h, 'orange']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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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