2017年6月26日月曜日

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24.を取り組んでみる。


  1. 図の長さをとる場合。(微分法)


    1. 0<x<1 f( x )=2x 1 x 2 =2 x 2 x 4 f'( x )= ( x 2 x 4 ) 1 2 ( 2x4 x 3 ) = x 1 ( 1 x 2 ) 1 2 2x( 12 x 2 ) = 2( 12 x 2 ) 1 x 2 12 x 2 =0 x= 1 2 f'( 1 2 )=0 x< 1 2 f'( x )>0 x> 1 2 f'( x )<0 f( 1 2 )=2 1 2 1 4 =1

    2. 0x1 f( x )=2·2x+2 1 x 2 =4x+2 1 x 2 =2( 2x+ 1 x 2 ) g( x )=2x+ 1 x 2 g'( x )=2+ ( 1 x 2 ) 1 2 ( 2x ) 2 =2x ( 1 x 2 ) 1 2 2x ( 1 x 2 ) 1 2 =0 2=x ( 1 x 2 ) 1 2 4= x 2 ( 1 x 2 ) 1 4( 1 x 2 )= x 2 x 2 = 4 5 x= 2 5 f( 2 5 )= 8 5 +2 1 4 5 = 8 5 + 2 5 = 10 5 =2 5

    図の角をとる場合。(三角関数の公式)


    1. 0θ π 2 f( θ )=2cosθsinθ =sin2θ 02θπ 2θ= π 2 θ= π 4 f( π 4 )=sin π 2 =1

    2. 0θ π 2 f( θ )=2·2cosθ+2sinθ =2( 2cosθ+sinθ ) g( θ )=2cosθ+sinθ = 5 ( 2 5 cosθ+ 1 5 sinθ ) sinα= 2 5 ,cosα= 1 5 g( θ )= 5 ( sinαcosθ+cosαsinθ ) = 5 sin( α+θ ) 0α+θ π 2 +α α+θ= π 2 f( π 2 α )=2 5

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sqrt, Derivative, solve, Pow, cos, sin, plot, pi

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

f = 2 * x * sqrt(1 - Pow(x, 2))
g = 2 * (2 * x + sqrt(1 - Pow(x, 2)))
h = 2 * cos(x) * sin(x)
l = 2 * (2 * cos(x) + sin(x))
fs1 = [f, g]
fs2 = [h, l]
fss = [(fs1, (0, 1)), (fs2, (0, pi / 2))]
colors = ['red', 'green', 'blue', 'orange']

for i, (fs, (x1, x2)) in enumerate(fss, 1):
    p = plot(*fs, 1, 2 * sqrt(5), (x, x1, x2), show=False, legend=True)
    for func in fs:
        d = Derivative(func, x, 1)
        pprint(d)
        f1 = d.doit()
        pprint(f1)
        pprint(solve(f1, x))

    for j, color in enumerate(colors):
        p[j].line_color = color

    p.save('sample24_{0}.svg'.format(i))

入出力結果(Terminal, IPython)

$ ./sample24.py
  ⎛       __________⎞
d ⎜      ╱    2     ⎟
──⎝2⋅x⋅╲╱  - x  + 1 ⎠
dx                   
          2            __________
       2⋅x            ╱    2     
- ───────────── + 2⋅╲╱  - x  + 1 
     __________                  
    ╱    2                       
  ╲╱  - x  + 1                   
⎡-√2   √2⎤
⎢────, ──⎥
⎣ 2    2 ⎦
  ⎛           __________⎞
d ⎜          ╱    2     ⎟
──⎝4⋅x + 2⋅╲╱  - x  + 1 ⎠
dx                       
       2⋅x         
- ───────────── + 4
     __________    
    ╱    2         
  ╲╱  - x  + 1     
⎡2⋅√5⎤
⎢────⎥
⎣ 5  ⎦
d                  
──(2⋅sin(x)⋅cos(x))
dx                 
       2           2   
- 2⋅sin (x) + 2⋅cos (x)
⎡-3⋅π   -π   π  3⋅π⎤
⎢─────, ───, ─, ───⎥
⎣  4     4   4   4 ⎦
d                      
──(2⋅sin(x) + 4⋅cos(x))
dx                     
-4⋅sin(x) + 2⋅cos(x)
[-2⋅atan(2 + √5), -2⋅atan(-√5 + 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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="2">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" min="0" max="1" step="0.1" value="0.1">

<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="sample24.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 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),
        y0 = Math.sqrt(1 - x0 ** 2);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        x3 = 1 / Math.sqrt(2),
        x4 = 2 / Math.sqrt(5),
        lines = [[-x0, 0, x0, 0, 'green'], [x0, 0, x0, y0, 'green'],
                 [x0, y0, -x0, y0, 'green'], [-x0, y0, -x0, 0, 'green'],
                 [x3, 0, x3, y2, 'red'], [x4, 0, x4, y2, 'orange']],
        f = (x) => Math.sqrt(1 - x ** 2),
        fns = [[f, 'blue']],
        fns1 = [];

    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();








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