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 - 指数関数(指数関数、対数関数、累乗、接線の方程式)

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)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
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)、2(指数関数)、練習問題9、10、11、12、13、14、15、16、17、18、19、20.を取り組んでみる。

$\begin{array}{l} y' = \frac{1}{x} \\ g (x) = \frac{1}{e} (x - e) + \log e \\ = \frac{1}{e} x - 1 + 1 \\ = \frac{x}{e} \end{array}$
$\begin{array}{l} y' = \log x + x \cdot \frac{1}{x} = \log x + 1 \\ g (x) = (\log e + 1) (x - e) + e \log e \\ = 2 x - 2 e + e \\ = 2 x - e \end{array}$
$\begin{array}{l} g (x) = (\log 2 + 1) (x - 2) + 2 \log 2 \\ = (\log 2 + 1) x - 2 \log 2 - 2 + 2 \log 2 \\ = (\log 2 + 1) x - 2 \end{array}$
$\begin{array}{l} y' = \frac{1}{x^{3}} \cdot 3 x^{2} = \frac{3}{x} \\ g (x) = \frac{3}{e} (x - e) + \log e^{3} \\ = \frac{3}{e} x - 3 + 3 \\ = \frac{3}{e} x \end{array}$
$\begin{array}{l} y' = \frac{- \frac{1}{x}}{{(\log x)}^{2}} \\ = - \frac{1}{x {(\log x)}^{2}} \\ g (x) = - \frac{1}{e {(\log e)}^{2}} (x - e) + \frac{1}{\log e} \\ = - \frac{1}{e} (x - e) + 1 \\ = - \frac{1}{e} x + 2 \end{array}$
$\begin{array}{l} g (x) = - \frac{1}{2 {(\log 2)}^{2}} (x - 2) + \frac{1}{\log 2} \\ = - \frac{1}{2 {(\log 2)}^{2}} x + \frac{1}{{(\log 2)}^{2}} + \frac{1}{\log 2} \end{array}$
$\begin{array}{l} y' = e^{2 x} 2 = 2 e^{2 x} \\ g (x) = 2 e^{2} (x - 1) + e^{2} \\ = 2 e^{2} x - e^{2} \end{array}$
$\begin{array}{l} y' = e^{x} + x e^{x} = e^{x} (1 + x) \\ g (x) = e^{2} (1 + 2) (x - 2) + 2 e^{2} \\ = 3 e^{2} x - 6 e^{2} + 2 e^{2} \\ = 3 e^{2} x - 4 e^{2} \end{array}$
$\begin{array}{l} g (x) = e^{5} (1 + 5) (x - 5) + 5 e^{5} \\ = 6 e^{5} - 30 e^{5} + 5 e^{5} \\ = 6 e^{5} - 25 e^{5} \end{array}$
$\begin{array}{l} y' = e^{- x} + x e^{- x} (- 1) \\ = e^{- x} (1 - x) \\ g (x) = e^{- 0} (1 - 0) (x - 0) + 0 \\ = x \end{array}$
$\begin{array}{l} y' = e^{- x} (- 1) = - e^{- x} \\ g (x) = - e^{0} (x - 0) + e^{0} \\ = - x + 1 \end{array}$
$\begin{array}{l} y' = 2 x e^{- x} + x^{2} e^{- x} (- 1) \\ = 2 x e^{- x} - x^{2} e^{- x} \\ = e^{- x} x (2 - x) \\ g (x) = e^{- 1} (2 - 1) (x - 1) + e^{- 1} \\ = e^{- 1} x - e^{- 1} + e^{- 1} \\ = e^{- 1} x \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, log, Derivative, E, exp, plot

x = symbols('x')
fs = [(log(x), E),
      (x * log(x), E),
      (x * log(x), 2),
      (log(x ** 3), E),
      (1 / log(x), E),
      (1 / log(x), 2),
      (exp(2 * x), 1),
      (x * exp(x), 2),
      (x * exp(x), 5),
      (x * exp(-x), 0),
      (exp(-x), 0),
      (x ** 2 * exp(-x), 1)]

for i, (f, x0) in enumerate(fs, 9):
    print(f'{i}.')
    D = Derivative(f, x, 1)
    f1 = D.doit()
    g = f1.subs({x: x0}) * (x - x0) + f.subs({x: x0})
    for t in [D, f1, g.expand()]:
        pprint(t)
        print()
    print()

    p = plot(f, g, (x, x0 - 0.5, x0 + 0.5), show=False, legend=True)
    for j, color in enumerate(['green', 'blue']):
        p[j].line_color = color
    p.save(f'sample{i}.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample9.py
9.
d         
──(log(x))
dx        

1
─
x

   -1
x⋅ℯ  


10.
d           
──(x⋅log(x))
dx          

log(x) + 1

2⋅x - ℯ


11.
d           
──(x⋅log(x))
dx          

log(x) + 1

x⋅log(2) + x - 2


12.
d ⎛   ⎛ 3⎞⎞
──⎝log⎝x ⎠⎠
dx         

3
─
x

     -1
3⋅x⋅ℯ  


13.
d ⎛  1   ⎞
──⎜──────⎟
dx⎝log(x)⎠

   -1    
─────────
     2   
x⋅log (x)

     -1    
- x⋅ℯ   + 2


14.
d ⎛  1   ⎞
──⎜──────⎟
dx⎝log(x)⎠

   -1    
─────────
     2   
x⋅log (x)

      x         1         1   
- ───────── + ────── + ───────
       2      log(2)      2   
  2⋅log (2)            log (2)


15.
d ⎛ 2⋅x⎞
──⎝ℯ   ⎠
dx      

   2⋅x
2⋅ℯ   

     2    2
2⋅x⋅ℯ  - ℯ 


16.
d ⎛   x⎞
──⎝x⋅ℯ ⎠
dx      

   x    x
x⋅ℯ  + ℯ 

     2      2
3⋅x⋅ℯ  - 4⋅ℯ 


17.
d ⎛   x⎞
──⎝x⋅ℯ ⎠
dx      

   x    x
x⋅ℯ  + ℯ 

     5       5
6⋅x⋅ℯ  - 25⋅ℯ 


18.
d ⎛   -x⎞
──⎝x⋅ℯ  ⎠
dx       

     -x    -x
- x⋅ℯ   + ℯ  

x


19.
d ⎛ -x⎞
──⎝ℯ  ⎠
dx     

  -x
-ℯ  

-x + 1


20.
d ⎛ 2  -x⎞
──⎝x ⋅ℯ  ⎠
dx        

   2  -x        -x
- x ⋅ℯ   + 2⋅x⋅ℯ  

   -1
x⋅ℯ  


$

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="sample9.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) => Math.log(x),
    f1 = (x) => 1 / x,
    g = (x) => f1(Math.E) * (x - Math.E) + f(Math.E);

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 = [[Math.E, y1, Math.E, y2, 'red']],
        fns = [[f, 'green']],
        fns1 = [[g, 'blue']],
        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11月3日金曜日

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 - 指数関数(指数関数、対数関数、累乗、接線の方程式)

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