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 - 最大・最小問題(曲線と点の距離の最小値、指数関数、対数関数)

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)、最大・最小問題、問21、22.を取り組んでみる。

$\begin{array}{l} f (x) = {(x - 4)}^{2} + {(x^{3})}^{2} \\ = {(x - 4)}^{2} + x^{6} \\ f' (x) = 2 (x - 4) + 6 x^{5} \\ = 6 x^{5} + 2 x - 8 \\ f' (x) = 0 \\ x = 1 \\ x < 1 \\ f' (x) < 0 \\ x > 1 \\ f' (x) > 0 \\ (1, 1^{3}) = (1, 1) \end{array}$
1. $\begin{array}{l} (x, e^{x}) \\ (x_{0}, x_{0}) \\ x - x_{0} = - (e^{x} - x_{0}) \\ x_{0} = \frac{x + e^{x}}{2} \\ f (x) = {(x - \frac{x + e^{x}}{2})}^{2} + {(e^{x} - \frac{x + e^{x}}{2})}^{2} \\ = x^{2} - x (x + e^{x}) + e^{2 x} - e^{x} (x + e^{x}) + \frac{{(x + e^{x})}^{2}}{2} \\ = x^{2} - x^{2} - x e^{x} + e^{2 x} - x e^{x} - e^{2 x} + \frac{x^{2} + 2 x e^{x} + e^{2 x}}{2} \\ = - x e^{x} + \frac{x^{2} + e^{2 x}}{2} \\ f' (x) = - e^{x} - x e^{x} + x + e^{2 x} \\ = x (1 - e^{x}) + e^{x} (e^{x} - 1) \\ f' (0) = 1 \\ x < 0 \\ f' (x) < 0 \\ x > 0 \\ f' (x) > 0 \\ (0, e^{0}) = (0, 1) \end{array}$
2. $\begin{array}{l} f (x) = {(x - \frac{x + e^{x}}{2})}^{2} + {(e^{x} - \frac{x + e^{x}}{2})}^{2} \\ 2 \sqrt{f (0)} = 2 \sqrt{\frac{1}{4} + \frac{1}{4}} \\ = \frac{2}{\sqrt{2}} \\ = \sqrt{2} \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Pow, plot, exp, Derivative, solve

x = symbols('x', real=True)

p = plot(Pow(x, 3), show=False)
p.save('sample21.svg')

f = (x - (x + exp(x)) / 2) ** 2 + (exp(x) - (x + exp(x)) / 2) ** 2
d = Derivative(f, x)
pprint(d)
pprint(d.expand())

f1 = d.doit()
pprint(f1)
# f1 = f1.expand()
pprint(f1.expand())
s = solve(f1, x)

pprint(s)
for x0 in s:
    for n in range(-1, 2):
        pprint(x0 + n)
        pprint(f1)
        pprint(f1.expand())
        result = f1.subs({x: x0 + n})
        pprint(result)
        pprint(result.is_positive)
        pprint(result.is_zero)
        pprint(result.is_negative)
        pprint(exp(x0 + n))
        print()

入出力結果(Terminal, IPython)

$ ./sample21.py
  ⎛          2           2⎞
  ⎜⎛       x⎞    ⎛     x⎞ ⎟
d ⎜⎜  x   ℯ ⎟    ⎜x   ℯ ⎟ ⎟
──⎜⎜- ─ + ──⎟  + ⎜─ - ──⎟ ⎟
dx⎝⎝  2   2 ⎠    ⎝2   2 ⎠ ⎠
  ⎛ 2           2⋅x⎞
d ⎜x       x   ℯ   ⎟
──⎜── - x⋅ℯ  + ────⎟
dx⎝2            2  ⎠
⎛       x⎞            ⎛     x⎞           
⎜  x   ℯ ⎟ ⎛ x    ⎞   ⎜x   ℯ ⎟ ⎛   x    ⎞
⎜- ─ + ──⎟⋅⎝ℯ  - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ  + 1⎠
⎝  2   2 ⎠            ⎝2   2 ⎠           
     x        2⋅x    x
- x⋅ℯ  + x + ℯ    - ℯ 
[0]
-1
⎛       x⎞            ⎛     x⎞           
⎜  x   ℯ ⎟ ⎛ x    ⎞   ⎜x   ℯ ⎟ ⎛   x    ⎞
⎜- ─ + ──⎟⋅⎝ℯ  - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ  + 1⎠
⎝  2   2 ⎠            ⎝2   2 ⎠           
     x        2⋅x    x
- x⋅ℯ  + x + ℯ    - ℯ 
⎛       -1⎞                          ⎛ -1    ⎞
⎜  1   ℯ  ⎟ ⎛   -1    ⎞   ⎛      -1⎞ ⎜ℯ     1⎟
⎜- ─ - ───⎟⋅⎝- ℯ   + 1⎠ + ⎝-1 + ℯ  ⎠⋅⎜─── + ─⎟
⎝  2    2 ⎠                          ⎝ 2    2⎠
False
False
True
 -1
ℯ  

0
⎛       x⎞            ⎛     x⎞           
⎜  x   ℯ ⎟ ⎛ x    ⎞   ⎜x   ℯ ⎟ ⎛   x    ⎞
⎜- ─ + ──⎟⋅⎝ℯ  - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ  + 1⎠
⎝  2   2 ⎠            ⎝2   2 ⎠           
     x        2⋅x    x
- x⋅ℯ  + x + ℯ    - ℯ 
0
False
True
False
1

1
⎛       x⎞            ⎛     x⎞           
⎜  x   ℯ ⎟ ⎛ x    ⎞   ⎜x   ℯ ⎟ ⎛   x    ⎞
⎜- ─ + ──⎟⋅⎝ℯ  - 1⎠ + ⎜─ - ──⎟⋅⎝- ℯ  + 1⎠
⎝  2   2 ⎠            ⎝2   2 ⎠           
     x        2⋅x    x
- x⋅ℯ  + x + ℯ    - ℯ 
         ⎛  ℯ   1⎞            ⎛  1   ℯ⎞
(-ℯ + 1)⋅⎜- ─ + ─⎟ + (-1 + ℯ)⋅⎜- ─ + ─⎟
         ⎝  2   2⎠            ⎝  2   2⎠
True
False
False
ℯ

$

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="x0">x = </label>
<input id="x0" type="number" step="0.1" 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="sample21.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_x0],
    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.exp(x),
    g = (x) => x,
    h = (x) => Math.log(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),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        a = (x0 + Math.exp(x0)) / 2,
        lines = [[x0, f(x0), a, a, 'brown']],
        fns = [[f, 'red'], [h, 'green']],
        fns1 = [[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]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;

        lines.push([x1, fn(x1), x2, fn(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);
    
    p(fns.join('\n'));
    p(fns1.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
x =

Kamimura's blog

ほしい物リスト

2017年6月24日土曜日

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 - 最大・最小問題(曲線と点の距離の最小値、指数関数、対数関数)

0 コメント:

コメントを投稿