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(三角関数(正弦、余弦、正接、逆正弦、逆余弦、逆正接)、累乗(べき乗))

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5.を取り組んでみる。

1. $\frac{1}{1 + {(\log x)}^{2}} \cdot \frac{1}{x}$
2. $\begin{array}{l} \frac{1}{\cos (3 x + 5)} (- \sin (3 x + 5)) 3 \\ = - 3 \tan (3 x + 5) \end{array}$
3. $e^{\sin (2 x)} \cos (2 x) 2 = 2 e^{\sin (2 x)} \cos (2 x)$
4. $e^{\arccos x} \frac{- 1}{\sqrt{1 - x^{2}}} = - \frac{e^{\arccos x}}{\sqrt{1 - x^{2}}}$
5. $\frac{1}{e^{x}} e^{x} = 1$
6. $\frac{e^{x} - x e^{x}}{e^{2 x}} = \frac{1 - x}{e^{x}}$
7. $e^{e^{x}} \cdot e^{x} = e^{e^{x} + x}$
8. $\begin{array}{l} e^{- \arcsin x} \frac{- 1}{\sqrt{1 - x^{2}}} \\ = - \frac{e^{- \arcsin x}}{\sqrt{1 - x^{2}}} \end{array}$
9. $\frac{e^{x}}{\cos^{2} (e^{x})}$
10. $\begin{array}{l} y = x^{\sqrt{x}} \\ \log y = \log x^{\sqrt{x}} \\ \log y = \sqrt{x} \log x \\ \frac{d}{d x} \log y = \frac{1}{2} x^{- \frac{1}{2}} \log x + \sqrt{x} \cdot \frac{1}{x} \\ \frac{d}{d y} (\log y) \cdot \frac{d y}{d x} = \frac{1}{2} x^{- \frac{1}{2}} \log x + \sqrt{x} \cdot \frac{1}{x} \\ \frac{1}{y} \frac{d y}{d x} = \frac{1}{2} x^{- \frac{1}{2}} \log x + x^{- \frac{1}{2}} \\ \frac{d y}{d x} = (\frac{1}{2} \log x + 1) x^{- \frac{1}{2}} y \\ \frac{d y}{d x} = (\frac{1}{2} \log x + 1) x^{- \frac{1}{2}} x^{\sqrt{x}} \\ \frac{d y}{d x} = (\frac{1}{2} \log x + 1) x^{\sqrt{x} - \frac{1}{2}} \end{array}$
11. $\begin{array}{l} y = x^{x^{\frac{1}{3}}} \\ \log y = \log x^{x^{\frac{1}{3}}} \\ \log y = x^{\frac{1}{3}} \log x \\ \frac{d}{d x} \log y = \frac{1}{3} x^{- \frac{2}{3}} \log x + x^{\frac{1}{3}} \frac{1}{x} \\ \frac{d}{d y} \log y \cdot \frac{d y}{d x} = \frac{1}{3} x^{- \frac{2}{3}} \log x + x^{- \frac{2}{3}} \\ \frac{1}{y} \frac{d y}{d x} = (\frac{1}{3} \log x + 1) x^{- \frac{2}{3}} \\ \frac{d y}{d x} = (\frac{1}{3} \log x + 1) x^{- \frac{2}{3}} y \\ \frac{d y}{d x} = (\frac{1}{3} \log x + 1) x^{- \frac{2}{3}} x^{x^{\frac{1}{3}}} \\ \frac{d y}{d x} = (\frac{1}{3} \log x + 1) x^{x^{\frac{1}{3}} - \frac{2}{3}} \end{array}$
12. $\frac{e^{x} + 1}{\sqrt{1 - {(e^{x} + x)}^{2}}}$
13. $e^{\tan x} \cdot \frac{1}{\cos^{2} x} = \frac{e^{\tan x}}{\cos^{2} x}$
14. $\frac{e^{x}}{1 + e^{2 x}}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, log, sin, cos, tan, asin, acos, atan, Rational, sqrt, Derivative, plot

print('5.')
x = symbols('x')
fs = [atan(log(x)),
      log(cos(3 * x + 5)),
      exp(sin(2 * x)),
      exp(acos(x)),
      log(exp(x)),
      x / exp(x),
      exp(exp(x)),
      exp(-asin(x)),
      tan(exp(x)),
      x ** sqrt(x),
      x ** (x ** Rational(1 / 3)),
      asin(exp(x) + x),
      exp(tan(x)),
      atan(exp(x))]


for i, f in enumerate(fs):
    c = chr(ord("a") + i)
    print(f'({c})')
    try:
        D = Derivative(f, x, 1)
        f1 = D.doit()
        for t in [D, f1]:
            pprint(t)
            print()
        print()
        p = plot(f, f1, show=False, legend=True)
        for j, color in enumerate(['red', 'green']):
            p[j].line_color = color
        p.save(f'sample5_{c}.png')
    except Exception as err:
        print(type(err), err)

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
5.
(a)
d               
──(atan(log(x)))
dx              

       1       
───────────────
  ⎛   2       ⎞
x⋅⎝log (x) + 1⎠


(b)
d                    
──(log(cos(3⋅x + 5)))
dx                   

-3⋅sin(3⋅x + 5) 
────────────────
  cos(3⋅x + 5)  


(c)
d ⎛ sin(2⋅x)⎞
──⎝ℯ        ⎠
dx           

   sin(2⋅x)         
2⋅ℯ        ⋅cos(2⋅x)


(d)
d ⎛ acos(x)⎞
──⎝ℯ       ⎠
dx          

    acos(x)  
  -ℯ         
─────────────
   __________
  ╱    2     
╲╱  - x  + 1 


(e)
d ⎛   ⎛ x⎞⎞
──⎝log⎝ℯ ⎠⎠
dx         

1


(f)
d ⎛   -x⎞
──⎝x⋅ℯ  ⎠
dx       

     -x    -x
- x⋅ℯ   + ℯ  


(g)
  ⎛ ⎛ x⎞⎞
d ⎜ ⎝ℯ ⎠⎟
──⎝ℯ    ⎠
dx       

    ⎛ x⎞
 x  ⎝ℯ ⎠
ℯ ⋅ℯ    


/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/experimental_lambdify.py:232: UserWarning: The evaluation of the expression is problematic. We are trying a failback method that may still work. Please report this as a bug.
  warnings.warn('The evaluation of the expression is'
/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1109: RuntimeWarning: invalid value encountered in double_scalars
  cos_theta = dot_product / (vector_a_norm * vector_b_norm)
/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1105: RuntimeWarning: invalid value encountered in subtract
  vector_b = (z - y).astype(np.float)
/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1104: RuntimeWarning: invalid value encountered in subtract
  vector_a = (x - y).astype(np.float)
/opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1007: RuntimeWarning: overflow encountered in double_scalars
  pos_bottom = ('data', 0) if yl*yh <= 0 else 'center'
(h)
d ⎛ -asin(x)⎞
──⎝ℯ        ⎠
dx           

   -asin(x)  
 -ℯ          
─────────────
   __________
  ╱    2     
╲╱  - x  + 1 


(i)
d ⎛   ⎛ x⎞⎞
──⎝tan⎝ℯ ⎠⎠
dx         

⎛   2⎛ x⎞    ⎞  x
⎝tan ⎝ℯ ⎠ + 1⎠⋅ℯ 


(j)
d ⎛ √x⎞
──⎝x  ⎠
dx     

 √x ⎛log(x)   1 ⎞
x  ⋅⎜────── + ──⎟
    ⎝ 2⋅√x    √x⎠


(k)
  ⎛ ⎛  6004799503160661⎞⎞
  ⎜ ⎜ ─────────────────⎟⎟
  ⎜ ⎜ 18014398509481984⎟⎟
d ⎜ ⎝x                 ⎠⎟
──⎝x                    ⎠
dx                       

 ⎛  6004799503160661⎞                                                         
 ⎜ ─────────────────⎟                                                         
 ⎜ 18014398509481984⎟                                                         
 ⎝x                 ⎠ ⎛      6004799503160661⋅log(x)                  1       
x                    ⋅⎜──────────────────────────────────── + ────────────────
                      ⎜                   12009599006321323    120095990063213
                      ⎜                   ─────────────────    ───────────────
                      ⎜                   18014398509481984    180143985094819
                      ⎝18014398509481984⋅x                    x               

   
   
   
  ⎞
──⎟
23⎟
──⎟
84⎟
  ⎠


(l)
d ⎛    ⎛     x⎞⎞
──⎝asin⎝x + ℯ ⎠⎠
dx              

         x           
        ℯ  + 1       
─────────────────────
    _________________
   ╱           2     
  ╱    ⎛     x⎞      
╲╱   - ⎝x + ℯ ⎠  + 1 


(m)
d ⎛ tan(x)⎞
──⎝ℯ      ⎠
dx         

⎛   2       ⎞  tan(x)
⎝tan (x) + 1⎠⋅ℯ      


(n)
d ⎛    ⎛ x⎞⎞
──⎝atan⎝ℯ ⎠⎠
dx          

    x   
   ℯ    
────────
 2⋅x    
ℯ    + 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="0.001" 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">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="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="sample5.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) => Math.log(Math.cos(3 * x + 5)),
    f1 = (x) => -3 * Math.tan(3 * x + 5)
    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;
    }    

    let points = [],
        lines = [],
        fns = [[f, 'red']],
        fns1 = [],
        fns2 = [[g, 'green']];

    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 =
dx0 =

Kamimura's blog

2017年10月30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 - 指数関数(三角関数(正弦、余弦、正接、逆正弦、逆余弦、逆正接)、累乗(べき乗))

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