数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 大きさの程度(直線と指数関数の和と差、曲線の描画、導関数、二階微分、極限、極値点、変曲点、漸近線)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
Nebo(Windows アプリ)
iPad Pro + Apple Pencil
MyScript Nebo(iPad アプリ)
参考書籍

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8章(指数関数と対数関数)、4(大きさの程度)、練習問題21.を取り組んでみる。

$\begin{array}{l} f' (x) = 1 - e^{x} \\ f'' (x) = - e^{x} \\ f' (x) = 0 \\ e^{x} = 1 \\ x = 0 \\ f'' (x) < 0 \\ \lim_{x \to \infty} f (x) = - \infty \\ \lim_{x \to - \infty} f (x) = - \infty \end{array}$

関数の曲線のグラフの描画。
$\begin{array}{l} f' (x) = 1 + e^{x} > 0 \\ f'' (x) = e^{x} > 0 \\ \lim_{x \to \infty} f (x) = \infty \\ \lim_{x \to \infty} f (x) = - \infty \end{array}$

関数の曲線のグラフの描画。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, Derivative, Limit, oo, solve

x = symbols('x', real=True)
fs = [x - exp(x), x + exp(x)]

for i, f in enumerate(fs):
    print(f'({chr(ord("a") + i)})')
    for n in range(1, 3):
        Dn = Derivative(f, x, n)
        fn = Dn.doit()
        for t in [Dn, fn, solve(fn, x)]:
            pprint(t)
            print()
        print()
    for x0 in [-oo, oo]:
        l = Limit(f, x, x0)
        for t in [l, l.doit()]:
            pprint(t)
            print()
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample21.py
(a)
d ⎛     x⎞
──⎝x - ℯ ⎠
dx        

   x    
- ℯ  + 1

[0]


  2        
 d ⎛     x⎞
───⎝x - ℯ ⎠
  2        
dx         

  x
-ℯ 

[zoo]


     ⎛     x⎞
 lim ⎝x - ℯ ⎠
x─→-∞        

-∞


    ⎛     x⎞
lim ⎝x - ℯ ⎠
x─→∞        

-∞



(b)
d ⎛     x⎞
──⎝x + ℯ ⎠
dx        

 x    
ℯ  + 1

[]


  2        
 d ⎛     x⎞
───⎝x + ℯ ⎠
  2        
dx         

 x
ℯ 

[]


     ⎛     x⎞
 lim ⎝x + ℯ ⎠
x─→-∞        

-∞


    ⎛     x⎞
lim ⎝x + ℯ ⎠
x─→∞        

∞



$

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="0.001" 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">

<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="sample21.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) => x - Math.exp(x),
    g = (x) => x + Math.exp(x),
    h = (x) => x;

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 = [],
        lines = [[x1, y1, x2, y2, 'red']],
        fns = [[f, 'green'],
               [g, 'blue']],
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .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]);
            }
        });
    
    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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2017年12月18日月曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 大きさの程度(直線と指数関数の和と差、曲線の描画、導関数、二階微分、極限、極値点、変曲点、漸近線)

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