2017年6月19日月曜日

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)、3(増加・減少関数)、練習問題1-10.を取り組んでみる。


  1. f'( x )=3 x 2 0

    増加である範囲: ( , )

    減少である範囲: ϕ


  2. f'(x)=(x2)(x3)+(x1)(x3)+(x1)(x2) =3 x 2 12x+11 x= 6± 3633 3 =2± 1 3

    増加の範囲: x2 1 3 ,2+ 1 3 x

    減少の範囲: 2 1 3 x2+ 1 3


  3. f'( x )=2x1

    増加の範囲: 1 2 x

    減少の範囲: x 1 2


  4. f'( x )=cosxsinx

    増加の範囲: 2nπx π 4 +2nπ, 5 4 π+2nπx2π+2nπ

    減少の範囲: π 4 +2nπx 5 4 π+2nπ


  5. f'( x )=2cos2x

    増加の範囲: 0x π 4 , 3 4 πx 5 4 π, 7 4 πx2π

    減少の範囲: π 4 x 3 4 π, 5 4 πx 7 4 π


  6. f'( x )=4 x 3 6x=2x( 2 x 2 3 ) x=0,± 3 2

    増加の範囲: 3 2 x0, 3 2 x

    減少の範囲: x 3 2 ,0x 3 2


  7. f'( x )=3 x 2 +1>0

    増加の範囲: ( , )

    減少の範囲: ϕ


  8. f'( x )=3 x 2 +2 x=± 2 3

    増加の範囲: 2 3 x 2 3

    減少の範囲: x 2 3 , 2 3 x


  9. f'( x )=6 x 2 0

    増加の範囲: ( , )

    減少の範囲: ϕ


  10. f'( x )=10x

    増加の範囲: 0x

    減少の範囲: x0

コード(Emacs)

Python 3

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

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

x = symbols('x')
f = sin(x) + cos(x)
f1 = Derivative(f, x).doit()

pprint(solve(f1, x, dict=True))

p = plot(f, show=False)
p.save('sample1.svg')

入出力結果(Terminal, IPython)

$ ./sample1.py
⎡⎧   -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="-4">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="4">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-4">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="4">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0.1" value="0.1">

<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_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0],
    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 ** 4 - 3 * x ** 2 + 1,
    f1 = (x) => 4 * x ** 3 - 6 * 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),
        dx0 = parseFloat(input_dx0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }
    let points = [],
        lines = [],
        fns = [[f, 'red']],
        fns1 = [[g, 'blue']];

    fns1
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });

    fns
        .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]);
                }
            }
        });                 
    
    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);
    
};

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








0 コメント:

コメントを投稿