2017年6月23日金曜日

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18、19、20.を取り組んでみる。


  1. f'( x )=3 x 2 12 =3( x 2 4 ) =3( x+2 )( x2 ) f( 2 )=8+24+10=26 f( 2 )=824+10=6 f( 3 )=27+36+10=19 f( 5 )=12560+10=75

    最大値 75、最小値 -6。


  2. y=1x 0x1 f( x )= x 3 +2 ( 1x ) 3 f'( x )=3 x 2 +6 ( 1x ) 2 ( 1 ) =3 x 2 6( x 2 2x+1 ) =3( x 2 +2( x 2 2x+1 ) ) =3( x 2 4x+2 ) x 2 4x+2=0 x=2± 42 =2± 2 x=2 2 f( 0 )=2 f( 1 )=1 f( 2 2 )= ( 2 2 ) 3 +2 ( 1( 2 2 ) ) 3 =8+3·2·2+3·4·( 2 )2 2 +2 ( 2 1 ) 3 =8+1212 2 2 2 +2( 2 2 +3 2 +3·2( 1 )1 ) =2014 2 +4 2 +6 2 122 =64 2

    最大値 2、最小値 64 2


  3. S=h·2πr+2π r 2 h= S2π r 2 2πr V=hπ r 2 = S2π r 2 2πr ·π r 2 = 1 2 r( S2π r 2 ) d dr V= 1 2 ( S6π r 2 ) 6π r 2 =S r= S 6π h= S2π· S 6π 2π S 6π = 2S 3 1S 3 =2 r:h=1:2

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, plot

x = symbols('x')
fs = [(x ** 3 - 12 * x + 10, (-3, 5)),
      (x ** 3 + 2 * (1 - x) ** 3, (0, 1))]

for i, (f, (x1, x2)) in enumerate(fs, 18):
    print('{0}.'.format(i))
    pprint(f)
    p = plot(f, (x, x1, x2), show=False)
    p.save('sample{0}.svg'.format(i))
    print()

入出力結果(Terminal, IPython)

$ ./sample18.py
18.
 3            
x  - 12⋅x + 10

19.
 3             3
x  + 2⋅(-x + 1) 

$

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="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>

<label for="s0">S = </label>
<input id="s0" type="number" min="0" 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="sample18.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_s0 = document.querySelector('#s0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_s0],
    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),
        s0 = parseFloat(input_s0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],        
        lines = [],
        h = (x) => (s0 - 2 * Math.PI * x ** 2) / (2 * Math.PI * r),
        v = (x) => h(x) * Math.PI * x ** 2,
        f = (x) => h(x) / x,
        fns = [[(x) => x, 'red'], [h, 'green'], [v, 'blue'], [f, 'brown']];

    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]);
            }
        }
    });
    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'));
};

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








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