数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 関数の増減の判定およびその応用 - 不等式・方程式への応用(相加平均、相乗平均の一般化)

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)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
参考書籍

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂和夫(著)、岩波書店)の第18章(曲線の性質、最大・最小 - 微分法の応用)、18.2(関数の増減の判定およびその応用)、不等式・方程式への応用、問13、14.を取り組んでみる。

1. $\begin{array}{l} f' (t) = a - a t^{a - 1} \\ = a (1 - t^{a - 1}) \\ t < 1 \\ f' (t) > 0 \\ t = 1 \\ f' (0) = 0 \\ t > 1 \\ f' (t) > 0 \\ f (1) = a + b - 1 = 0 \\ a t + b - t^{a} \geq 0 \\ a t + b \geq t^{a} \end{array}$
2. $\begin{array}{l} a \frac{x}{y} + b \geq {(\frac{x}{y})}^{a} \\ a x + b y \\ \geq x^{a} y^{- a + 1} \\ = x^{a} y^{- (1 - b) + 1} \\ = x^{a} y^{b} \end{array}$
1. $\begin{array}{l} f' (x) = (n + 1) {(\frac{b + x}{n + 1})}^{n} \cdot \frac{1}{n + 1} - {(\frac{b}{n})}^{n} \\ = {(\frac{b + x}{n + 1})}^{n} - {(\frac{b}{n})}^{n} \\ \frac{b + x}{n + 1} = \frac{b}{n} \\ n b + n x = n b + b \\ x = \frac{b}{n} \\ x < \frac{b}{n} \\ f' (x) < 0 \\ x = \frac{b}{n} \\ f' (x) = 0 \\ x > \frac{b}{n} \\ f' (x) > 0 \\ f (\frac{b}{n}) = {(\frac{b + \frac{b}{n}}{n + 1})}^{n + 1} - {(\frac{b}{n})}^{n} \frac{b}{n} \\ = {(\frac{\frac{(n + 1) b}{n}}{n + 1})}^{n + 1} - {(\frac{b}{n})}^{n + 1} \\ = {(\frac{b}{n})}^{n + 1} - {(\frac{b}{n})}^{n + 1} \\ = 0 \\ {(\frac{b + x}{n + 1})}^{n + 1} - {(\frac{b}{n})}^{n} x \geq 0 \\ {(\frac{b + x}{n + 1})}^{n + 1} \geq {(\frac{b}{n})}^{n} x \end{array}$
2. $\begin{array}{l} a_{1} \geq a_{1} \\ {(\frac{(a_{1} + \dots + a_{n}) + a_{n + 1}}{n + 1})}^{n + 1} \\ \geq {(\frac{a_{1} + \dots + a_{n}}{n})}^{n} a_{n + 1} \\ \geq a_{1} \dots a_{n} a_{n + 1} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, exp, plot
import random

print('13.')
x = symbols('x', positive=True)
a = random.random()
b = 1 - a
y = random.random()
eqs = [a * x + b * y, x ** a * y ** b]

p = plot(*eqs, (x, 0.01, 10), show=False, legend=True)

for i, color in enumerate(['green', 'blue']):
    p[i].line_color = color

p.save('sample13.svg')

入出力結果(Terminal, IPython)

$ ./sample13.py
13.
$

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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" min="0.01" 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="sample13.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_n0 = document.querySelector('#n0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0, input_b0],
    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),
        n0 = parseFloat(input_n0.value),
        b0 = parseFloat(input_b0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],        
        lines = [],
        f = (x) => ((b0 + x) / (n0 + 1)) ** (n0 + 1),
        g = (x) => (b0 / n0) ** n0 * x,
        fns = [[f, 'green'], [g, 'blue']];

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

            if (Math.abs(y) < Infinity) {
                points.push([x, y, 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);
    
    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
n = b =

Kamimura's blog

2017年6月21日水曜日

数学 - Python - JavaScript - 曲線の性質、最大・最小 - 微分法の応用 – 関数の増減の判定およびその応用 - 不等式・方程式への応用(相加平均、相乗平均の一般化)

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