2017年6月17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)、導関数の符号と関数の増減、問2、3、4、5.を取り組んでみる。


  1. y'=2ax+b 2ax+b=0 x= b 2a
    x-b/2a
    y'-0+
    y↘︎↗︎

  2. y'=3 x 2 +2ax+b 3 x 2 +2ax+b=0 D 4 = a 2 3b

    1. y'=123 x 2 123 x 2 =0 x=±2 24+8=16 248=16
      x -22
      y'-0+0-
      y ↘︎-16↗︎16↘︎

    2. y'= x 2 +2x x 2 +2x=0 x( x+2 )=0 x=0,2 0 8 3 +4= 4 3
      x-20
      y'+0-0+
      y↗︎4/3↘︎0↗︎

    3. y'=3 x 2 +11

      増加関数で、極値はない。


    4. y'=2+2x3 x 2 3 x 2 2x+2=0 D 4 =16=5<0 y'<0

      減少関数で、極値はない。


    5. y'=4 x 3 4x =4x( x 2 1 ) =4x( x+1 )( x1 ) 12+1=0 1 12+1=0
      x-101
      y'-0+0-0+
      y↘︎0↗︎1↘︎0↗︎

    6. y'=4 x 3 +2x =2x( x 2 +1 ) 0
      x0
      y'-0+
      y↘︎0↗︎

    7. y'=4 x 3 12 x 2 =4 x 2 ( x3 ) 0 27
      x03
      y'-0-0+
      y↘︎0↘︎-27↗︎

    8. y=| x 2 ( x3 ) | x<3 y= x 3 +3 x 2 y'=3 x 2 +6x =3x( x2 ) x3 y= x 3 3 x 2 y'=3 x 2 6x =3x( x2 ) 0 8+3·4=4 2727=0
      x023
      y'-0+0-+
      y↘︎0↗︎4↘︎0↗︎

    9. y'=1 1 x 2 1 1 x 2 =0 x 2 1=0 x=±1 11=2 1+1=2
      x-101
      y'+0--0+
      y↗︎-2↘︎↘︎2↗︎

    10. y= e x +x e x e x +x e x =0 x=1 e 0 +0 e 0 =1
      x-1
      y'-0+
      y↘︎1↗︎

    11. y'=1+2cosx 1+2cosx=0 cosx= 1 2 x= 2 3 π, 4 3 π 2 3 π+2 3 2 = 2 3 π+ 3 4 3 π+2·( 3 2 )= 4 3 π 3
      x2π/34π/3
      y'+0-0+
      y↗︎2π/3 + √3↘︎4π/3 - √3↗︎

    1. y=a x 3 +b x 2 +cx+d y'=3a x 2 +2bx+c 3a2b+c=0 a+bc+d=10 12a+4b+c=0 8a+4b+2c+d=17 c=3a+2b a+b+3a2b+d=10 12a+4b3a+2b=0 8a+4b6a+4b+d=17 2ab+d=10 3a+2b=0 2a+8b+d=17 d=102a+b 3a+2b=0 2a+8b+102a+b=17 9b=27 b=3 a=2 d=1043=3 c=66=12 y=2 x 3 3 x 2 12x+3

    2. c=0 d=8 12a+4b+c=0 8a+4b+2c+d=0 12a+4b=0 8a+4b8=0 3a+b=0 2a+b2=0 b=3a 2a3a2=0 a=2 b=6 y=2 x 3 +6 x 2 8

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, plot, pi, sqrt

print('4 (11).')
x = symbols('x')
f = x + 2 * sin(x)
g = 2 * pi / 3 + sqrt(3)
h = 4 * pi / 3 - sqrt(3)

p = plot(f, g, h, (x, 0, 2 * pi), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample2.svg')

入出力結果(Terminal, IPython)

$ ./sample2.py
4 (11).
$

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="-2">
<label for="b0">b = </label>
<input id="b0" type="number" value="6">
<label for="c0">c = </label>
<input id="c0" type="number" value="0">
<label for="d0">d = </label>
<input id="d0" type="number" value="-8">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" value="0">


<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="sample2.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'),
    input_c0 = document.querySelector('#c0'),
    input_d0 = document.querySelector('#d0'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0, input_c0, input_d0, 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),
        a = parseFloat(input_a0.value),
        b = parseFloat(input_b0.value),
        c = parseFloat(input_c0.value),
        d = parseFloat(input_d0.value),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        f = (x) => a * x ** 3 + b * x ** 2 + c * x + d,
        f1 = (x) => 3 * a * x ** 2 + 2 * b * x + c,
        g = (x) => f1(x0) * (x - x0) + f(x0),
        fns = [[f, 'green']],
        fns1 = [[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]);
    });
    
    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);
};

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









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