数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 平均値の定理 - 増加・減少関数(断面、台形、貯水槽、傾斜角、極小値、極大値の存在、光の照度、光源の強さ、距離、比例、反比例、半円形と長方形、扇形の面積)

解析入門 原書第3版
原著

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
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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5章(平均値の定理)、3(増加・減少関数)、補充問題29、30、31、32、33.を取り組んでみる。

29. 
    

    傾斜角をΘとする。

    $\begin{array}{l}f\left(\theta \right)=\frac{1}{2}\left(10+2·10\cos \theta +10\right)10\sin \theta \\ =\left(20+20\cos \theta \right)\sin \theta \\ =20\left(1+\cos \theta \right)\sin \theta \\ \\ f\text{'}\left(\theta \right)=20\left(-\sin \theta \sin \theta +\left(1+\cos \theta \right)\cos \theta \right)\\ =20\left(-{\sin }^2\theta +\cos \theta +{\cos }^2\theta \right)\\ =20\left(2{\cos }^2\theta +\cos \theta -1\right)\\ 2{\cos }^2\theta +\cos \theta -1=0\\ \left(2\cos \theta -1\right)\left(\cos \theta +1\right)=0\\ \cos \theta =\frac{1}{2}\\ \theta =\frac{\pi }{3}\end{array}$
30. 
    $\begin{array}{l}f\text{'}\left(x\right)=2x-\frac{a}{x^2}\\ =\frac{1}{x^2}\left(2x^3-a\right)\end{array}$
    1. 
       $\begin{array}{l}f\text{'}\left(2\right)=16-a\\ a=16\end{array}$
    2. 
       $\begin{array}{l}f\text{'}\left(-3\right)=-54-a\\ a=-54\end{array}$
    3. 
       $\begin{array}{l}2x^3-a=0\\ x^3=\frac{a}{2}\\ f\text{'}\left({\left(\frac{a}{2}\right)}^{\frac{1}{3}}\right)=0\\ \\ a\ge 0\\ x<0,0<x<{\left(\frac{a}{2}\right)}^{\frac{1}{3}}\\ f\text{'}\left(x\right)<0\\ x>{\left(\frac{a}{2}\right)}^{\frac{1}{3}}\\ f\text{'}\left(x\right)>0\\ \\ a<0\\ x<{\left(\frac{a}{2}\right)}^{\frac{1}{3}}\\ f\text{'}\left(x\right)<0\\ {\left(\frac{a}{2}\right)}^{\frac{1}{3}}<x<0\\ f\text{'}\left(x\right)>0\\ \\ 0<x\\ f\text{'}\left(x\right)>0\end{array}$
31. 
    

    強さaの光源からの距離をxとする。

    $\begin{array}{l}f\left(x\right)=a\frac{1}{x^2}+b\frac{1}{{\left(c-x\right)}^2}\\ =\frac{a}{x^2}+\frac{b}{{\left(x-c\right)}^2}\\ \\ f\text{'}\left(x\right)=\frac{-2ax}{x^4}+\frac{-b2\left(x-c\right)}{{\left(x-c\right)}^4}\\ =\frac{-2a}{x^3}+\frac{-b2}{{\left(x-c\right)}^3}\\ =\frac{-2\left(a{\left(x-c\right)}^3+b{x}^3\right)}{x{\left(x-c\right)}^3}\\ a{\left(x-c\right)}^3+b{x}^3=0\\ -\frac{a}{b}={\left(\frac{x}{x-c}\right)}^3\\ {\left(\frac{a}{b}\right)}^{\frac{1}{3}}=-\frac{x}{x-c}\\ x=\frac{c}{{\left(\frac{a}{b}\right)}^{\frac{1}{3}}+1}\end{array}$
32. 
    

    長方形の上の辺の長さをa、横の辺の長さをb、半円形の半径をrとする。

    $\begin{array}{l}a+2b+2\pi r\frac{1}{2}=6\hfill \\ a+b+\pi r=6\hfill \\ r=\frac{1}{2}a\hfill \\ a+b+\frac{a\pi }{2}=6\hfill \\ b=6-(1+\frac{\pi }{2})a\hfill \\ f(a)=ab+\frac{1}{2}\pi r^2\hfill \\ =a(6-(1+\frac{\pi }{2})a)+\frac{1}{2}\pi \frac{a^2}{4}\hfill \\ =(\frac{1}{8}\pi -(1+\frac{\pi }{2}))a^2+6a\hfill \\ \hfill \\ f\text{'}(a)=2(\frac{1}{8}\pi -(1+\frac{\pi }{2}))a+6\hfill \\ 2(\frac{1}{8}\pi -(1+\frac{\pi }{2}))a+6=0\hfill \\ (-1-\frac{3}{8}\pi )a+3=0\hfill \\ a=\frac{3}{\frac{3}{8}\pi +1}=\frac{24}{3\pi +8}\hfill \\ b=6-(1+\frac{\pi }{2})\frac{24}{3\pi +8}\hfill \\ =\frac{18\pi +24-24-12}{3\pi +8}\hfill \\ =\frac{18\pi -12}{3\pi +8}\hfill \end{array}$
33. 
    

    半径をr、角をΘとする。

    $\begin{array}{l}2r+2\pi r\frac{\theta }{2\pi }=16\\ \theta =\frac{2\pi \left(16-2r\right)}{2\pi r}\\ =\frac{2\left(8-r\right)}{r}\\ \\ f\left(r\right)=\pi r^2\frac{\theta }{2\pi }\\ =\frac{r^2}{2}·\frac{2\left(8-r\right)}{r}\\ =r\left(8-r\right)\\ =8r-r^2\\ \\ f\text{'}\left(r\right)=8-2r\\ r=4\\ \theta =\frac{2\left(8-4\right)}{4}\\ =2\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve, Rational, sin, cos, pi

print('29.')
x = symbols('x', positive=True)
f = Rational(1, 2) * (10 + 2 * 10 * cos(x) + 10) * sin(x)

d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
pprint(solve(f1))

print('30')
a = symbols('a')
f = x ** 2 + a / x
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
pprint(solve(f1))

for x0 in [2, -3]:
    pprint(solve(f1.subs({x: x0}), a))

print('32')
f = x * (6 - (1 + pi / 2) * x) + Rational(1, 2) * pi * x ** 2 / 4
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
pprint(solve(f1))

print('33.')
f = x ** 2 / 2 * 2 * (8 - x) / x
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)
pprint(solve(f1))

入出力結果(Terminal, IPython)

$ ./sample29.py
29.
d                          
──((10⋅cos(x) + 10)⋅sin(x))
dx                         
                                2   
(10⋅cos(x) + 10)⋅cos(x) - 10⋅sin (x)
⎡π⎤
⎢─⎥
⎣3⎦
30
∂ ⎛a    2⎞
──⎜─ + x ⎟
∂x⎝x     ⎠
  a       
- ── + 2⋅x
   2      
  x       
⎡⎧      3⎫⎤
⎢⎨a: 2⋅x ⎬⎥
⎣⎩       ⎭⎦
[16]
[-54]
32
  ⎛   2                      ⎞
d ⎜π⋅x      ⎛    ⎛    π⎞    ⎞⎟
──⎜──── + x⋅⎜- x⋅⎜1 + ─⎟ + 6⎟⎟
dx⎝ 8       ⎝    ⎝    2⎠    ⎠⎠
  ⎛  π    ⎞     ⎛    π⎞   π⋅x    
x⋅⎜- ─ - 1⎟ - x⋅⎜1 + ─⎟ + ─── + 6
  ⎝  2    ⎠     ⎝    2⎠    4     
⎡   24  ⎤
⎢───────⎥
⎣8 + 3⋅π⎦
33.
d             
──(x⋅(-x + 8))
dx            
-2⋅x + 8
[4]
$

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" value="0.1">

<label for="a0">a = </label>
<input id="a0" type="number" 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="sample29.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'),
    input_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0, input_a0],
    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),
        dx0 = parseFloat(input_dx0.value),
        a0 = parseFloat(input_a0.value);

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

    let points = [],
        x3 = (a0 / 2) ** (1 / 3),
        lines = [],
        f = (x) => x ** 2 + a0 / x,
        f1 = (x) => 2 * x - a0 / x ** 2,
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[g, 'orange']];

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

                if (Math.abs(y) < Infinity) {
                    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();

(x) => x ** 2 + a0 / x,green

(x0) => (x) => f1(x0) * (x - x0) + f(x0),orange

r = dx =
x1 = x2 =
y1 = y2 =
dx0 = a =