2017年6月15日木曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第5章(平均値の定理)、1(最大点と最小点の定理)、練習問題1-13.を取り組んでみる。


  1. 2x2=0 x=1

  2. 4x3=0 x= 3 4

  3. 6x1=0 x= 1 6

  4. 2x+2=0 x=1

  5. 4x+3=0 x= 3 4

  6. 3 x 2 =0 x=0

  7. 3 x 2 3=0 x 2 =1 x=±1

  8. cosxsinx=0 tanx=1 x= π 4 +nπ

  9. sinx=0 sinx=0 x=nπ

  10. cosx=0 x= π 2 +nπ

  11. 正方形の1辺の長さを x 、高さを y 、箱の体積をVとする。

    C= x 2 +4xy V= x 2 y y= x 4 + C 4x V= x 3 4 + C 4 x= x 4 ( x 2 C ) V'= 3 4 x 2 + C 4 3 4 x 2 + C 4 =0 3 x 2 +C=0 x= C 3 y= 1 4 C 3 + C 4 C 3 = C 4 3 + C 3 4 C = C+3C 4 3C = 2C 4 3C = C 2 3

  12. 底面の半径を r 、高さを h とする。

    C=h·2rπ+π r 2 V=π r 2 h h= Cπ r 2 2πr V=π r 2 Cπ r 2 2πr = r 2 ( Cπ r 2 )= Crπ r 3 2 V'= C3π r 2 2 C3π r 2 2 =0 r= C 3π h= Cπ r 2 2πr = Cπ C 3π 2π C 3π = C C 3 2 πC 3 = 2 C 6 π 3 = C 3π

    1. C=2 x 2 +4xy V= x 2 y y= C2 x 2 4x V= x( C2 x 2 ) 4 V'= 1 4 ( C6 x 2 ) C6 x 2 =0 x= C 6 y= C2 C 6 4 C 6 = 2 3 C· 1 4 6 C = C 6

    2. C=h·2rπ+2π r 2 V=π r 2 h h= C2π r 2 2πr V=π r 2 C2π r 2 2πr = Cr2π r 3 2 V'= C6π r 2 2 C6π r 2 =0 r= C 6π h= C2π C 6π 2π C 6π = C C 3 2 Cπ 6 = 2C 6 3·2 Cπ = 2C 3π

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, solve, Derivative, plot

x = symbols('x')
fs = [x ** 2 - 2 * x + 5, sin(x) + cos(x)]
for i, f in enumerate(fs):
    d = Derivative(f, x, 1)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(solve(f1, x, dict=True))
    p = plot(f, f1, show=False, legend=True)
    for i, color in enumerate(['green', 'blue']):
        p[i].line_color = color
    p.save('sample1_{}.svg'.format(i))
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
d ⎛ 2          ⎞
──⎝x  - 2⋅x + 5⎠
dx              
2⋅x - 2
[{x: 1}]

d                  
──(sin(x) + cos(x))
dx                 
-sin(x) + cos(x)
⎡⎧   -3⋅π ⎫  ⎧   π⎫⎤
⎢⎨x: ─────⎬, ⎨x: ─⎬⎥
⎣⎩     4  ⎭  ⎩   4⎭⎦

$

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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" step="0.01" value="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="sample1.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 f = (x) => Math.sin(x) + Math.cos(x),
    f1 = (x) => Math.cos(x) - Math.sin(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),
        x0 = parseFloat(input_x0.value);
        
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [];

    [[f, 'red'], [f1, 'green']]
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    [[g(x0), 'blue']]
        .forEach((o) => {
            let [f, color] = o;

            lines.push([x1, f(x1), x2, f(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(${padding}, 0)`)
        .call(yaxis);
    p(`f1(${x0}) = ${f1(x0)}`);
};

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








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