2017年9月23日土曜日

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20章(面積、体積、長さ - 積分法の応用)、20.1(面積)、面積の公式、問8.を取り組んでみる。


    1. M(a+b2,a2+b22)ddxx2=2x

      Aにおける放物線の接線の方程式。

      y=2a(xa)+a2=2axa2

      Bにおける放物線の接線の方程式。

      y=2b(xb)+b2=2bxb2

      点Qを求める。

      2axa2=2bxb2x=a2b22(ab)=a+b2Q(a+b2,2a·a+b2a2)Q(a+b2,ab)

      よって、点MとQのx座標は等しいので、MとQを結ぶ直線MQは放物線の軸と平行である。


    2. P(a+b2,(a+b2)2)P(a+b2,a2+2ab+b24)

      線分MQの中点。

      (a+b2,a2+b22+ab2)=(a+b2,a2+2ab+b24)

      よって、直線MQと放物線の交点Pは線分MQの中点である。


    3. 点A、Bを通る直線の方程式。

      y=b2a2ba(xa)+a2=(a+b)(xa)+a2=(a+b)xab

      放物線と弦ABで囲まれる図形の面積。

      S1=ba(((a+b)xab)x2)dx=[a+b2x2abx13x3]ba=(a+b2b2ab213b3)(a+b2a2a2b13a3)=3ab2+3b36ab22b33a33a2b+6a2b+2a36=b33ab2+3a2ba36=16(ba)3

    4. 直線AQの方程式。

      S2=a+b2a(x2(2axa2))dx+ba+b2(x2(2bxb2))dx=a+b2a(x22ax+a2)dx+ba+b2(x22bx+b2)dx=a+b2a(xa)2dx+ba+b2(xb)2dx=[13(xa)3]a+b2a+[13(xb)3]ba+b2=13([(xa)3]a+b2a+[(xb)3]ba+b2)=13((a+b2a)3(a+b2b)3)=13((ba2)3(ab2)3)=13((ba2)3+(ba2)3)=13(2(ba2)3)=13((ba)34)=112(ba)3

    5. 点Pは線分MQの中点なので、△PABの面積は三角形QABの面積の半分。

      S3=12(S1+S2)=12(16(ba)3+112(ba)3)=12·312(ba)3=18(ba)3

    6. S1:S2:S3=16(ba)3:112(ba)3:18(ba)3=16:112:18=4:2:3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Derivative, solve

print('8.')
x, a, b = symbols('x a b')
f = x ** 2
M = ((a + b) / 2, f.subs({x: (a + b) / 2}))
f1 = Derivative(f, x, 1)
fa = f1.subs({x: a}) * (x - a) + f.subs({x: a})
fb = f1.subs({x: b}) * (x - b) + f.subs({x: b})

Qx = solve(fa - fb, x)[0]
Q = (Qx, fa.subs({x: Qx}))

for t in [M, Q]:
    pprint(t[0].doit().factor())

入出力結果(Terminal, Jupyter(IPython))

$ ./sample8.py
8.
a + b
─────
  2  
a + b
─────
  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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="-1">
<label for="b0">b = </label>
<input id="b0" 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="sample8.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'),
    input_b0 = document.querySelector('#b0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_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 f = (x) => x ** 2,
    f1 = (x) => 2 * x,
    g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);

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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
        

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        fa = g(a0),
        fb = g(b0),
        lines = [[a0, f(a0), b0, f(b0), 'blue'],
                 [a0, f(a0), 0, 0, 'blue'],
                 [0, 0, b0, f(b0), 'blue']],
        fns = [[f, 'green']],
        fns1 = [[fa, 'blue'],
                [fb, 'blue']],
        fns2 = [];

    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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
-10-8-6-4-20246810-10-8-6-4-20246810
(x) => x ** 2,green
(x) => f1(x0) * (x - x0) + f(x0),blue
(x) => f1(x0) * (x - x0) + f(x0),blue




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