2017年6月6日火曜日

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)、三角関数の微分、問44、45、46.を取り組んでみる。


  1. ( cotx )' =( 1 tanx )' =( cosx sinx )' = sin 2 x cos 2 x sin 2 x = 1 sin 2 x

    1. 3cos3x

    2. 2cos2x

    3. 4 cos 2 x

    4. 2sinxcosx

    5. 5 cos 4 xsinx

    6. 2tanx· 1 cos 2 x

    7. 4xcos( 2 x 2 3 )

    8. 4 x 3 3 x 2 cos 2 ( x 4 x 3 )

    9. cos 2 x sin 2 x

    10. 3 sin 2 ( 3x4 )

    11. 6 sin 2 2xcos2x

    12. 1 2 ( 1+cosx ) 1 2 sinx

    13. sinx( 1sinx )+ cos 2 x ( 1sinx ) 2

    14. 1 cos 2 x ( 1+tanx )( 1tanx ) 1 cos 2 x ( 1+tanx ) 2

    1. y'=cosx cos π 4 = 1 2

    2. y'=sinx sin π 6 = 1 2

    3. y'= cos 2 x sin 2 x=1 cos 2 3π 2 sin 2 3π 2 = 1 2 1 2 =1

    4. y'= 1 cos 2 x 1 cos 2 ( π 4 ) =2

    5. y'= cosx sin 2 x cos( π 3 ) sin 2 ( π 3 ) = 1 2 3 4 = 4 2·3 = 2 3

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, tan, pi, Derivative

print('46.')
x = symbols('x')

funcs = [(sin(x), pi / 4),
         (cos(x), pi / 6),
         (sin(x) * cos(x), 3 * pi / 2),
         (tan(x), - pi / 4),
         (1 / sin(x), -pi / 3)]

for i, (f, x0) in enumerate(funcs, 1):
    print('({})'.format(i))
    pprint(f)
    d = Derivative(f, x)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(f1.subs({x: x0}))

入出力結果(Terminal, IPython)

$ ./sample44.py
46.
(1)
sin(x)
d         
──(sin(x))
dx        
cos(x)
√2
──
2 
(2)
cos(x)
d         
──(cos(x))
dx        
-sin(x)
-1/2
(3)
sin(x)⋅cos(x)
d                
──(sin(x)⋅cos(x))
dx               
     2         2   
- sin (x) + cos (x)
-1
(4)
tan(x)
d         
──(tan(x))
dx        
   2       
tan (x) + 1
2
(5)
  1   
──────
sin(x)
d ⎛  1   ⎞
──⎜──────⎟
dx⎝sin(x)⎠
-cos(x) 
────────
   2    
sin (x) 
-2/3
$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<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="x0">x0 = </label>
<input id="x0" type="number" step="0.1" value="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="sample44.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_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_x0],
    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 / Math.tan(x),
    f1 = (x) => -1 / Math.sin(x) ** 2;

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),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        g = (x) => f1(x0) * (x - x0) + f(x0);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [[x1, g(x1), x2, g(x2)]];

    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, i) => i <= 1 ? 'black' : 'blue');
    
    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', 'red');
    
    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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