2017年5月28日日曜日

学習環境

数学読本〈4〉数列の極限,順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.3(導関数とその計算)、曲線の接線の方程式、問34.を取り組んでみる。


    1. f'( x )= x 2 +2x3 x 2 +2x3=5 x 2 +2x8=0 ( x+4 )( x2 )=0 x=4,2

    2. x 2 +2x3=0 ( x+3 )( x1 )=0 x=3,1

    3. y 0 = 1 3 x 0 3 + x 0 2 3 x 0 y 0 +27=( x 0 2 +2 x 0 3 ) x 0 y 0 = x 0 3 +2 x 0 2 3 x 0 27 1 3 x 0 3 + x 0 2 3 x 0 = x 0 3 +2 x 0 2 3 x 0 27 x 0 3 +3 x 0 2 9 x 0 =3 x 0 3 +6 x 0 2 9 x 0 81 2 x 0 3 +3 x 0 2 81=0 ( x 0 3 )( 2 x 0 2 +9 x 0 +27 )=0 x=3

コード(Emacs)

Python 3

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

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

x = symbols('x')
f = Rational(1, 3) * x ** 3 + x ** 2 - 3 * x
f1 = Derivative(f, x)
pprint(f)
pprint(f1)
f1 = f1.doit()
pprint(f1)

print('(1)')
pprint(solve(f1 - 5, x, dict=True))

print('(2)')
pprint(solve(f1.doit(), x, dict=True))

print('(3)')
x0 = symbols('x0', real=True)
pprint(solve(f.subs({x: x0}) - (f1.subs({x: x0})
                                * x - 27).subs({x: x0}), x0, dict=True))

入出力結果(Terminal, IPython)

$ ./sample34.py
 3           
x     2      
── + x  - 3⋅x
3            
  ⎛ 3           ⎞
d ⎜x     2      ⎟
──⎜── + x  - 3⋅x⎟
dx⎝3            ⎠
 2          
x  + 2⋅x - 3
(1)
[{x: -4}, {x: 2}]
(2)
[{x: -3}, {x: 1}]
(3)
[{x₀: 3}]
$

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">

<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="sample34.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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) => 1 / 3 * x ** 3 + x ** 2 - 3 * x,
    f1 = (x) => x ** 2 + 2 * x - 3,
    g = (x, x0, y0) => f1(x0) * (x - x0) + y0;

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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    let lines = [];
    for (let x = x1; x <= x2; x += 100 * dx) {
        let y = f(x),
            y1 = g(x1, x, y),
            y2 = g(x2, x, y);

        if (Math.abs(y1) < Infinity && Math.abs(y2) < Infinity) {
            lines.push([x1, y1, x2, y2]);
        }
    }
    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, i) => i <= 1 ? 'black' : 'blue');
    
    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', 'red');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
};

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







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