2017年10月11日水曜日


数学読本〈5〉

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.3(曲線の長さ)、曲線 y = f(x) (a ≤ x ≤ b)の長さ、問26.を取り組んでみる。


    1. ∫ 0 a 1+ ( e x − e −x 2 ) 2 dx = ∫ 0 a 1+ e 2x −2+ e −2x 2 2 dx = 1 2 ∫ 0 a 2 2 + e 2x −2+ e −2x dx = 1 2 ∫ 0 a 2+ e 2x + e −2x dx = 1 2 ∫ 0 a ( e x + e −x ) 2 dx = 1 2 ∫ 0 a ( e x + e −x )dx = 1 2 [ e x − e −x ] 0 a = 1 2 ( ( e a − e −a )−( 1−1 ) ) = e a − e −a 2

    2. ∫ 0 1 1+ ( 2x ) 2 dx u=2x du dx =2 x=0 u=0 x=1 u=2 1 2 ∫ 0 2 1+ u 2 du u= e t − e −t 2 du dt = e t + e −t 2 u=0 t=0 u=2 e t − e −t 2 =2 e t − e −t =4 ( e t ) 2 −1=4 e t ( e t ) 2 −4 e t −1=0 e t >0 e t =2+ 4+1 =2+ 5 log e t =log( 2+ 5 ) t=log( 2+ 5 ) 1 2 ∫ 0 2 1+ u 2 du = 1 2 ∫ 0 log( 2+ 5 ) 1+ ( e t − e −t 2 ) 2 e t + e −t 2 dt = 1 2 ∫ 0 log( 2+ 5 ) 4+ e 2t + e −2t −2 2 2 e t + e −t 2 dt = 1 2 3 ∫ 0 log( 2+ 5 ) e 2t + e −2t +2 ( e t + e −t )dt = 1 2 3 ∫ 0 log( 2+ 5 ) ( e t + e −t ) 2 ( e t + e −t )dt = 1 2 3 ∫ 0 log( 2+ 5 ) ( e t + e −t )( e t + e −t )dt = 1 2 3 ∫ 0 log( 2+ 5 ) ( e t + e −t ) 2 dt = 1 2 3 ∫ 0 log( 2+ 5 ) ( e 2t +2+ e −2t )dt = 1 2 3 [ 1 2 e 2t +2t− 1 2 e −2t ] 0 log( 2+ 5 ) = 1 2 3 ( ( 1 2 e 2log( 2+ 5 ) +2log( 2+ 5 )− 1 2 e −2log( 2+ 5 ) )−( 1 2 − 1 2 ) ) = 1 2 3 ( 1 2 ( 2+ 5 ) 2 +2log( 2+ 5 )− 1 2 ( 2+ 5 ) −2 ) = 1 2 3 ( 1 2 ( 4+5+4 5 )+2log( 2+ 5 )− 1 2( 4+5+4 5 ) ) = 1 2 3 ( 1 2 ( 9+4 5 )+2log( 2+ 5 )− 1 2( 9+4 5 ) ) = 1 2 3 ( 1 2 ( 9+4 5 )+2log( 2+ 5 )− 9−4 5 2( 81−16·5 ) ) = 1 2 3 ( 1 2 ( 9+4 5 )+2log( 2+ 5 )− 9−4 5 2( 81−80 ) ) = 1 2 3 ( 1 2 ( 9+4 5 )+2log( 2+ 5 )− 9−4 5 2 ) = 1 2 3 ( 8 5 2 +2log( 2+ 5 ) ) = 1 4 ( 2 5 +log( 2+ 5 ) )

    3. ∫ 1 3 1+ ( 1 x ) 2 dx = ∫ 1 3 x 2 +1 x dx t= x 2 +1 dt dx = 1 2 ( x 2 +1 ) − 1 2 2x = x x 2 +1 = x t x=1 t= 2 x=3 t= 10 ∫ 2 10 t x t x dt = ∫ 2 10 t 2 x 2 dt t= x 2 +1 t 2 = x 2 +1 x 2 = t 2 −1 ∫ 2 10 t 2 x 2 dt = ∫ 2 10 t 2 t 2 −1 dt = ∫ 2 10 t 2 −1+1 t 2 −1 dt = ∫ 2 10 ( 1+ 1 t 2 −1 )dt = ∫ 2 10 1dt + ∫ 2 10 1 t 2 −1 dt = [ t ] 2 10 + ∫ 2 10 1 ( t+1 )( t−1 ) dt 1 ( t+1 )( t−1 ) = A t+1 + B t−1 1 ( t+1 )( t−1 ) = At−A+Bt+B ( t+1 )( t−1 ) 1 ( t+1 )( t−1 ) = ( A+B )t+( B−A ) ( t+1 )( t−1 ) A+B=0 B−A=1 2B=1 B= 1 2 A=− 1 2 [ t ] 2 10 + ∫ 2 10 1 ( t+1 )( t−1 ) dt = 10 − 2 + 1 2 ∫ 2 10 ( −1 t+1 + 1 t−1 )dt = 10 − 2 + 1 2 [ −log( t+1 )+log( t−1 ) ] 2 10 = 10 − 2 + 1 2 ( ( −log( 10 +1 )+log( 10 −1 ) )−( −log( 2 +1 )+log( 2 −1 ) ) ) = 10 − 2 + 1 2 log ( 10 −1 )( 2 +1 ) ( 10 +1 )( 2 −1 ) = 10 − 2 + 1 2 log ( 10 −1 ) 2 ( 2 +1 ) 2 ( 10−1 )( 2−1 ) = 10 − 2 + 1 2 log ( 10 −1 ) 2 ( 2 +1 ) 2 9 = 10 − 2 + 1 2 log ( ( 10 −1 )( 2 +1 ) 3 ) 2 = 10 − 2 +log ( 10 −1 )( 2 +1 ) 3

コード(Emacs)

Python 3

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

print('26.')

x, a, b = symbols('x a b')
L = lambda f: Integral(sqrt(1 + Derivative(f, x) ** 2), (x, a, b))

fs = [((exp(x) + exp(-x)) / 2, (0, a)),
      (x ** 2, (0, 1)),
      (log(x), (1, 3))]

for i, (f0, (a0, b0)) in enumerate(fs, 1):
    print(f'({i})')
    I = L(f0).subs({a:a0, b:b0})
    for t in [I, I.doit()]:
        pprint(t)
        print()
    p = plot(f0, show=False, legend=True)
    p.save(f'sample26_{i}.svg')
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample26.py
26.
(1)
a                               
⌠                               
⎮       _____________________   
⎮      ╱               2        
⎮     ╱  ⎛  ⎛ x    -x⎞⎞         
⎮    ╱   ⎜d ⎜ℯ    ℯ  ⎟⎟         
⎮   ╱    ⎜──⎜── + ───⎟⎟  + 1  dx
⎮ ╲╱     ⎝dx⎝2     2 ⎠⎠         
⌡                               
0                               

a                         
⌠                         
⎮    __________________   
⎮   ╱  2⋅x        -2⋅x    
⎮ ╲╱  ℯ    + 2 + ℯ      dx
⌡                         
0                         
──────────────────────────
            2             


(2)
1                        
⌠                        
⎮      _______________   
⎮     ╱         2        
⎮    ╱  ⎛d ⎛ 2⎞⎞         
⎮   ╱   ⎜──⎝x ⎠⎟  + 1  dx
⎮ ╲╱    ⎝dx    ⎠         
⌡                        
0                        

asinh(2)   √5
──────── + ──
   4       2 


(3)
3                            
⌠                            
⎮      ___________________   
⎮     ╱             2        
⎮    ╱  ⎛d         ⎞         
⎮   ╱   ⎜──(log(x))⎟  + 1  dx
⎮ ╲╱    ⎝dx        ⎠         
⌡                            
1                            

-√2 - asinh(1/3) + log(1 + √2) + √10


$

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">

<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="sample26.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) => 1 / 2 * (Math.exp(x) + Math.exp(-x)),
    g = (x) => x ** 2,
    h = (x) => Math.log(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 = [],
        fns = [[f, 'red'],
               [g, 'green'],
               [h, 'blue']],
        fns1 = [],
        fns2 = [];

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

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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();




































						












時刻:


11:00




ラベル:


                                              ,
                                            


                                              ,
                                            


                                              ,
                                            


                                              ,
                                            


                                              ,
                                            
























0
コメント:















コメントを投稿


















                                    

                                  










コメントの投稿(Atom)