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(指数関数、対数関数、級数)

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10、11、12.を取り組んでみる。

$\begin{array}{l} f (x) = e^{x} - x - 1 \\ f' (x) = e^{x} - 1 \\ e^{x} - 1 = 0 \\ x = 0 \\ f (0) = 1 - 0 - 1 = 0 \\ 1 + x < e^{x} \\ 1 - x < e^{- x} \\ \frac{1}{1 - x} > e^{x} \\ 1 + x < e^{x} < \frac{1}{1 - x} \end{array}$
$\begin{array}{l} f_{0} (x) = e^{x} - 1 \\ f_{0}' (x) = e^{x} > 0 \\ f_{0} (0) = 0 \\ x < 0 \\ f_{0} (x) < 0 \\ x > 0 \\ f_{0} (x) > 0 \\ f_{1} (x) = e^{x} - (1 + \frac{x}{1!}) = e^{x} - 1 - x \\ f_{1}' (x) = e^{x} - 1 \\ f_{1}' (0) = 1 \\ x < 0 \\ f'_{1} (x) < 0 \\ x > 0 \\ f'_{1} (x) > 0 \\ f_{1} (0) = 1 - 1 - 0 = 0 \\ f_{1} (x) > 0 \\ f_{n}' (x) = f_{n - 1} (x) \\ f_{2 k}' (x) = f_{2 k - 1} (x) \\ f_{2 k}' (x) > 0 \\ f_{2 k} (0) = 0 \\ x < 0 \\ f_{2 k} (x) < 0 \\ x > 0 \\ f_{2 k} (x) > 0 \\ f_{2 k + 1}' (x) = f_{2 k} (x) \\ x < 0 \\ f_{2 k + 1}' (x) < 0 \\ x > 0 \\ f_{2 k + 1}' (x) > 0 \\ f_{2 k + 1} (0) = 0 \\ f_{2 k + 1} (x) > 0 \end{array}$
$\begin{array}{l} g (x) = f (x) e^{- x} \\ g' (x) = f' (x) e^{- x} - f (x) e^{- x} \\ = e^{- x} (f' (x) - f (x)) \\ g (0) = f (0) e^{- 0} = 1 \\ x > 0 \\ g' (x) > 0 \\ g (x) > 1 \\ f (x) e^{- x} > 1 \\ f (x) > e^{x} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, exp, plot

print('10.')
x = symbols('x')
p = plot(1 + x, exp(x), 1 / (1 - x),
         (x, 0.1, 0.9), show=False, legend=True)

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

p.save('sample10.svg')

入出力結果(Terminal, IPython)

$ ./sample10.py
10.
$

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="-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="n0">n = </label>
<input id="n0" type="number" min="0" step="1" value="0">



<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="sample10.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    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 factorial = (n) => range(1, n + 1).reduce((x, y) => x * y, 1),
    term = (k) => (x) => x ** k / factorial(k),
    gn = (n) => (x) =>
    range(0, n + 1).reduce((prev, next) => prev + term(next)(x), 0);

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

    let points = [],        
        lines = [],
        g = gn(n0),
        f = (x) => Math.exp(x) - g(x),
        fns = [[f, 'green']];

    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);
};

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

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

Kamimura's blog

2017年6月20日火曜日

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(指数関数、対数関数、級数)

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