2017年6月11日日曜日

学習環境

数学読本〈4〉数列の極限,順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.5(いろいろな関数の導関数)、高次導関数、問62.を取り組んでみる。


    1. f'( x )= a x sin( logx )+ b x cos( logx ) f''( x )= a x 2 sin( logx ) a x 2 cos( logx ) b x 2 cos( logx ) b x 2 sin( logx ) x 2 f''( x )+xf'( x )+f( x ) =asin( logx )acos( logx )bcos( logx )bsin( logx ) asin( logx )+bcos( logx )+acos( logx )+bsin( logx ) =0

    2. x n f ( n ) = x n d dx f ( n1 ) ( x ) = x n d dx ( a n1 x n1 cos( logx )+ b n1 x n1 sin( logx ) ) = x n ( a n1 ( n1 ) x n2 x 2n2 cos( logx ) a n1 x n sin( logx ) b n1 ( n1 ) x n2 x 2n2 sin( logx )+ b n1 x n cos( logx ) ) = a m1 ( n1 )cos( logx ) a n1 sin( logx ) b n1 ( n1 )sin( logx )+ b n1 cos( logx ) =( a n1 ( n1 )+ b n1 )cos( logx )+( a n1 b n1 ( n1 ) )sin( logx )

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, log, Derivative

print('62.')
print('(1)')
x, a, b = symbols('x a b')

f = a * cos(log(x)) + b * sin(log(x))

f1 = Derivative(f, x, 1)
f2 = Derivative(f, x, 2)

for func in [f, f1, f2]:
    pprint(func)
    fn = func.doit()
    pprint(fn.factor())
    print()

eq = f + x * f1.doit() + x ** 2 * f2.doit()
pprint(eq)
pprint(eq.expand())

入出力結果(Terminal, IPython)

$ ./sample62.py
62.
(1)
a⋅cos(log(x)) + b⋅sin(log(x))
a⋅cos(log(x)) + b⋅sin(log(x))

∂                                
──(a⋅cos(log(x)) + b⋅sin(log(x)))
∂x                               
-(a⋅sin(log(x)) - b⋅cos(log(x))) 
─────────────────────────────────
                x                

  2                               
 ∂                                
───(a⋅cos(log(x)) + b⋅sin(log(x)))
  2                               
∂x                                
-(-a⋅sin(log(x)) + a⋅cos(log(x)) + b⋅sin(log(x)) + b⋅cos(log(x))) 
──────────────────────────────────────────────────────────────────
                                 2                                
                                x                                 

                                  ⎛  a⋅sin(log(x))   b⋅cos(log(x))⎞
a⋅sin(log(x)) - b⋅cos(log(x)) + x⋅⎜- ───────────── + ─────────────⎟
                                  ⎝        x               x      ⎠
0
$

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="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="21">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">

<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="sample62.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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 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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => a0 * Math.cos(Math.log(x)) + b0 * Math.sin(Math.log(x)),
        f1 = (x) => 1 / x *
        (-a0 * Math.sin(Math.log(x)) + b0 * Math.cos(Math.log(x))),
        f2 = (x) => 1 / x ** 2 *
        ((a0 - b0) * Math.sin(Math.log(x)) - (a0 + b0) * Math.cos(Math.log(x)));
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'red']);
        }
    }
    let t1 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'green']);
        }
    }
    let t2 = points.length;    
    for (let x = x1; x <= x2; x += dx) {
        let y = f2(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'blue']);
        }
    }
    let t3 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = x * f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'brown']);
        }
    }
    let t4 = points.length;    
    for (let x = x1; x <= x2; x += dx) {
        let y = x ** 2 * f2(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'purple']);
        }
    }
    let lines = [];

    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();








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