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)

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

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。

傾斜角をΘとする。
$\begin{array}{l} f (θ) = \frac{1}{2} (10 + 2 \cdot 10 \cos θ + 10) 10 \sin θ \\ = (20 + 20 \cos θ) \sin θ \\ = 20 (1 + \cos θ) \sin θ \\ f' (θ) = 20 (- \sin θ \sin θ + (1 + \cos θ) \cos θ) \\ = 20 (- \sin^{2} θ + \cos θ + \cos^{2} θ) \\ = 20 (2 \cos^{2} θ + \cos θ - 1) \\ 2 \cos^{2} θ + \cos θ - 1 = 0 \\ (2 \cos θ - 1) (\cos θ + 1) = 0 \\ \cos θ = \frac{1}{2} \\ θ = \frac{π}{3} \end{array}$
f'( x )=2x− a x 2 = 1 x 2 ( 2 x 3 −a )
1. $\begin{array}{l} f' (2) = 16 - a \\ a = 16 \end{array}$
2. $\begin{array}{l} f' (- 3) = - 54 - a \\ a = - 54 \end{array}$
3. $\begin{array}{l} 2 x^{3} - a = 0 \\ x^{3} = \frac{a}{2} \\ f' ({(\frac{a}{2})}^{\frac{1}{3}}) = 0 \\ a \geq 0 \\ x < 0, 0 < x < {(\frac{a}{2})}^{\frac{1}{3}} \\ f' (x) < 0 \\ x > {(\frac{a}{2})}^{\frac{1}{3}} \\ f' (x) > 0 \\ a < 0 \\ x < {(\frac{a}{2})}^{\frac{1}{3}} \\ f' (x) < 0 \\ {(\frac{a}{2})}^{\frac{1}{3}} < x < 0 \\ f' (x) > 0 \\ 0 < x \\ f' (x) > 0 \end{array}$
強さaの光源からの距離をxとする。
$\begin{array}{l} f (x) = a \frac{1}{x^{2}} + b \frac{1}{{(c - x)}^{2}} \\ = \frac{a}{x^{2}} + \frac{b}{{(x - c)}^{2}} \\ f' (x) = \frac{- 2 a x}{x^{4}} + \frac{- b 2 (x - c)}{{(x - c)}^{4}} \\ = \frac{- 2 a}{x^{3}} + \frac{- b 2}{{(x - c)}^{3}} \\ = \frac{- 2 (a {(x - c)}^{3} + b x^{3})}{x {(x - c)}^{3}} \\ a {(x - c)}^{3} + b x^{3} = 0 \\ - \frac{a}{b} = {(\frac{x}{x - c})}^{3} \\ {(\frac{a}{b})}^{\frac{1}{3}} = - \frac{x}{x - c} \\ x = \frac{c}{{(\frac{a}{b})}^{\frac{1}{3}} + 1} \end{array}$
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。
$\begin{array}{l} a + 2 b + 2 π r \frac{1}{2} = 6 \\ a + b + π r = 6 \\ r = \frac{1}{2} a \\ a + b + \frac{a π}{2} = 6 \\ b = 6 - (1 + \frac{π}{2}) a \\ f (a) = a b + \frac{1}{2} π r^{2} \\ = a (6 - (1 + \frac{π}{2}) a) + \frac{1}{2} π \frac{a^{2}}{4} \\ = (\frac{1}{8} π - (1 + \frac{π}{2})) a^{2} + 6 a \\ f' (a) = 2 (\frac{1}{8} π - (1 + \frac{π}{2})) a + 6 \\ 2 (\frac{1}{8} π - (1 + \frac{π}{2})) a + 6 = 0 \\ (- 1 - \frac{3}{8} π) a + 3 = 0 \\ a = \frac{3}{\frac{3}{8} π + 1} = \frac{24}{3 π + 8} \\ b = 6 - (1 + \frac{π}{2}) \frac{24}{3 π + 8} \\ = \frac{18 π + 24 - 24 - 12}{3 π + 8} \\ = \frac{18 π - 12}{3 π + 8} \end{array}$
半径をr、角をΘとする。
$\begin{array}{l} 2 r + 2 π r \frac{θ}{2 π} = 16 \\ θ = \frac{2 π (16 - 2 r)}{2 π r} \\ = \frac{2 (8 - r)}{r} \\ f (r) = π r^{2} \frac{θ}{2 π} \\ = \frac{r^{2}}{2} \cdot \frac{2 (8 - r)}{r} \\ = r (8 - r) \\ = 8 r - r^{2} \\ f' (r) = 8 - 2 r \\ r = 4 \\ θ = \frac{2 (8 - 4)}{4} \\ = 2 \end{array}$

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();

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

Kamimura's blog

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2017年7月6日木曜日

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)

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