数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 一般の指数関数(微分、導関数、累乗(べき乗)、不等号)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
Nebo(Windows アプリ)
iPad Pro + Apple Pencil
MyScript Nebo(iPad アプリ)
参考書籍

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8章(指数関数と対数関数)、3(指数関数)、練習問題12.を取り組んでみる。

$\begin{array}{l} f (x) = \frac{x}{p} + \frac{1}{q} - x^{\frac{1}{p}} \\ f' (x) \\ = \frac{1}{p} - \frac{1}{p} x^{\frac{1}{p} - 1} \\ = \frac{1}{p} (1 - x^{\frac{1}{p} - 1}) \\ f' (1) = \frac{1}{p} (1 - 1) = 0 \\ p \geq 1 \\ \frac{1}{p} \leq 1 \\ \frac{1}{p} - 1 \leq 0 \\ 0 < x < 1 \\ f' (x) < 0 \\ 1 < x \\ f' (x) > 0 \\ f (1) = \frac{1}{p} + \frac{1}{q} - 1 = 0 \end{array}$

よって、

$\begin{array}{l} f (x) \geq 0 \\ \frac{x}{p} + \frac{1}{q} - x^{\frac{1}{p} - 1} \geq 0 \\ x^{\frac{1}{p} - 1} \leq \frac{x}{p} + \frac{1}{q} \end{array}$

（証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Derivative, solve

x, p, q = symbols('x, p, q')
f = x / p + 1 / q - x ** (1 / p)
D = Derivative(f, x, 1)
f1 = D.doit()
for t in [D, f1, f1.factor()]:
    pprint(t)
    print()

pprint(solve(1 / p + 1 / q - 1, q))

入出力結果(Terminal, Jupyter(IPython))

$ ./sample12.py
∂ ⎛  p ___   1   x⎞
──⎜- ╲╱ x  + ─ + ─⎟
∂x⎝          q   p⎠

    p ___
1   ╲╱ x 
─ - ─────
p    p⋅x 

    p ___
x - ╲╱ x 
─────────
   p⋅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="0.001" value="0.005">
<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">
<br>
<label for="p0">p0 = </label>
<input id="p0" type="number" min="1" value="2">

<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="sample12.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_p0 = document.querySelector('#p0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_p0],
    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 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),
        p0 = parseFloat(input_p0.value),
        q0 = p0 / (p0 - 1);

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

    let points = [],
        lines = [[1, y1, 1, y2, 'red']],
        f = (x) => x / p0 + 1 / q0,
        g = (x) => x ** (1 / p0)
        fns = [[f, 'green'],
               [g, 'blue']],
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), 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();

r = dx =
x1 = x2 =
y1 = y2 =
p0 =

Kamimura's blog

2017年11月24日金曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 一般の指数関数(微分、導関数、累乗(べき乗)、不等号)

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