2017年6月27日火曜日

学習環境

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25、26、27、28.を取り組んでみる。


  1. OPA=α tanα= 9 x tan( α+θ )= 16 x tan( α+θ )= tanα+tanθ 1tanαtanθ 16 x = 9 x +tanθ 1 9 x tanθ 16 x 144 x 2 tanθ= 9 x +tanθ 7 x =( 144 x 2 +1 )tanθ tanθ= 7 x( 144 x 2 +1 ) = 7x 144+ x 2 d dx tanθ=7 144+ x 2 x·2x ( 144+ x 2 ) 2 =7 x 2 144 ( 144+ x 2 ) 2 x=12

  2. 半径の長さをr、中心角の大きさをθ、面積をSとする。

    2r+2πr θ 2π =12 2r+rθ=12 θ= 122r r S=π r 2 θ 2π = θ r 2 2 =( 6r )r = r 2 +6r d dr S=2r+6 =2( r3 ) r=3( m ) θ=2( rad ) S=9( m 2 )

  3. 光源の高さをhとする。

    f( h )= h h 2 + a 2 ( h 2 + a 2 ) = h ( h 2 + a 2 ) 3 2 f'( h )= ( h 2 + a 2 ) 3 2 h 3 2 ( h 2 + a 2 ) 1 2 2h ( h 2 + a 2 ) 3 = ( h 2 + a 2 ) 1 2 ( h 2 + a 2 3 h 2 ) ( h 2 + a 2 ) 3 2 h 2 + a 2 =0 h= a 2

  4. S=π( h 2 + r 2 ) 2πr 2π h 2 + r 2 =πr ( h 2 + r 2 ) 1 2 S 2 = π 2 r 2 ( h 2 + r 2 ) h 2 = S 2 π 2 r 4 π r 2 V= 1 3 π r 2 h V 2 = 1 9 π 2 r 4 h 2 = 1 9 π r 4 S 2 π 2 r 4 π r 2 = 1 9 r 2 ( S 2 π 2 r 4 ) = 1 9 ( r 2 S 2 π 2 r 6 ) d dr ( r 2 S 2 π 2 r 6 )=2 S 2 r6 π 2 r 5 =2r( S 2 3 π 2 r 4 ) S 2 =3 π 2 r 4 h 2 = S 2 π 2 r 4 π r 2 = 3 π 2 r 4 π 2 r 4 π r 2 =2 r 2 r:h=1: 2

コード(Emacs)

Python 3

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

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

x = symbols('x')

fs = [(25, 7 * x / (144 + x ** 2), (0, 20)),
      (27, x / (x ** 2 + 10 ** 2) ** (3 / 2), (0, 10))]

for i, f, (x1, x2) in fs:
    d = Derivative(f, x, 1)
    pprint(d)
    f1 = d.doit()
    pprint(f1)
    pprint(solve(f1, x))
    p = plot(f, (x, x1, x2), show=False, legend=True)
    p.save('sample{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample25.py
d ⎛  7⋅x   ⎞
──⎜────────⎟
dx⎜ 2      ⎟
  ⎝x  + 144⎠
         2              
     14⋅x          7    
- ─────────── + ────────
            2    2      
  ⎛ 2      ⎞    x  + 144
  ⎝x  + 144⎠            
[-12, 12]
  ⎛            -1.5⎞
d ⎜  ⎛ 2      ⎞    ⎟
──⎝x⋅⎝x  + 100⎠    ⎠
dx                  
                   -2.5             -1.5
       2 ⎛ 2      ⎞       ⎛ 2      ⎞    
- 3.0⋅x ⋅⎝x  + 100⎠     + ⎝x  + 100⎠    
[-7.07106781186548, 7.07106781186548]
$

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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="0.02">
<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" min="0" value="5">

<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="sample25.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 = [],        
        lines = [],
        f = (x) => x / (x ** 2 + a0 ** 2) ** (3 / 2),
        f1 = (x) =>
        (x ** 2 + a0 ** 2) ** (1 / 2) * (x ** 2 + a0 ** 2 - 3 * x ** 2) /
        (x ** 2 + a0 ** 2) ** 3,
        g = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [],
        fns2 = [[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]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;
        
        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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);
    
    p(fns.join('\n'));
    p(fns1.join('\n'));
};

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








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