2017年6月25日日曜日

学習環境

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5、6、7.を取り組んでみる。


  1. ab=64 b= 64 a f( a )= b 2 + ( a 2 ) 2 = 64 2 a 2 + a 2 4 f'( a )=2· 64 2 a 3 + a 2 2· 64 2 a 3 + a 2 =0 4· 64 2 + a 4 =0 a 4 = 4 7 a 4 = 2 14 a= 2 7 2 a=8 2 b= 64 8 2 = 8 2 =4 2

  2. a+b=4 b=4a f( a )= a 2 + b 2 = a 2 +168a+ a 2 =2( a 2 4a+8 ) f'( a )=2( 2a4 ) =4( a2 ) a=2 b=42=2

  3. x,24x 0x24 2πr=x r= x 2π 4a=24x a=6 x 4 f( x )= r 2 π+ a 2 = x 2 4 +36 x 2 16 3x = 3 16 x 2 3x+36 f'( x )= 3 8 x3 3 8 x3=0 x=8 f( 0 )=36 f( 8 )=1224+36=24 f( 24 )=72

    最小値: 8 cm で円、16 cm で正方形を作る。

    最大値: 24 cm で円を作る。

コード(Emacs)

Python 3

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

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

x = symbols('x', negative=False)

fs = [65 ** 2 / x ** 2 + x ** 2 / 4,
      2 * (x ** 2 - 4 * x + 8)]

for i, f in enumerate(fs, 5):
    print('{0}.'.format(i))
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, 0, 5), show=False)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample5.py
5.
  ⎛ 2       ⎞
d ⎜x    4225⎟
──⎜── + ────⎟
dx⎜4      2 ⎟
  ⎝      x  ⎠
x   8450
─ - ────
2     3 
     x  
[√130, -√130⋅ⅈ, √130⋅ⅈ]
6.
d ⎛   2           ⎞
──⎝2⋅x  - 8⋅x + 16⎠
dx                 
4⋅x - 8
[2]
$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-25">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="50">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" max="24" value="10">
b = <span id="b0"></span>

<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="sample5.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_a0 = document.querySelector('#a0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0],
    span_b0 = document.querySelector('#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),
        a = parseFloat(input_a0.value),
        b = 24 - a,
        r0 = a / (2 * Math.PI);

    span_b0.innerText = b;
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }    
    
    let points = [[x2 / 2, y2 / 2, 'green']],
        lines = [[0, 0, 0, b, 'green'],
                 [0, b, -b, b, 'green'],
                 [-b, b, -b, 0, 'green'],
                 [-b, 0, 0, 0, 'green']],
        fns = [];

    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]);
                }
            }
        });                 
    
    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');

    r = xscale(r0) - xscale(0);
    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();








b =

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