2017年5月23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)、17.3(導関数とその計算)、定数倍の微分、和や差の微分、問24、25、26.を取り組んでみる。


    1. 60 x 5

    2. 2x4

    3. 3 x 2 4

    4. f'( x )=3 x 4 2 x 2 8 =12 x 3 4x

  1. ds dt s=9.6t

  2. dS dr =2πr dV dr =4π r 2

コード(Emacs)

Python 3

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

from sympy import symbols, sqrt, Derivative, pprint, pi, Rational

x = symbols('x')

print('問24.')
exprs = [
    10 * x ** 6,
    x ** 2 - 4 * x + 5,
    x ** 3 - 4 * x,
    (x ** 2 - 2) * (3 * x ** 2 + 4)
]

for i, expr in enumerate(exprs, 1):
    print('({0})'.format(i))
    pprint(expr)
    d = Derivative(expr, x)
    pprint(d)
    pprint(d.doit().expand())

print('問25')
t = symbols('t')
s = 4.9 * t ** 2
pprint(s)
d = Derivative(s, t)
pprint(d)
pprint(d.doit())

print('問26')
r = symbols('r')
s = pi * r ** 2
pprint(s)
d = Derivative(s, r)
pprint(d)
pprint(d.doit())

v = Rational(4, 3) * pi * r ** 3
pprint(v)
d = Derivative(v, r)
pprint(d)
pprint(d.doit())

入出力結果(Terminal, IPython)

$ ./sample24.py
問24.
(1)
    6
10⋅x 
d ⎛    6⎞
──⎝10⋅x ⎠
dx       
    5
60⋅x 
(2)
 2          
x  - 4⋅x + 5
d ⎛ 2          ⎞
──⎝x  - 4⋅x + 5⎠
dx              
2⋅x - 4
(3)
 3      
x  - 4⋅x
d ⎛ 3      ⎞
──⎝x  - 4⋅x⎠
dx          
   2    
3⋅x  - 4
(4)
⎛ 2    ⎞ ⎛   2    ⎞
⎝x  - 2⎠⋅⎝3⋅x  + 4⎠
d ⎛⎛ 2    ⎞ ⎛   2    ⎞⎞
──⎝⎝x  - 2⎠⋅⎝3⋅x  + 4⎠⎠
dx                     
    3      
12⋅x  - 4⋅x
問25
     2
4.9⋅t 
d ⎛     2⎞
──⎝4.9⋅t ⎠
dt        
9.8⋅t
問26
   2
π⋅r 
d ⎛   2⎞
──⎝π⋅r ⎠
dr      
2⋅π⋅r
     3
4⋅π⋅r 
──────
  3   
  ⎛     3⎞
d ⎜4⋅π⋅r ⎟
──⎜──────⎟
dr⎝  3   ⎠
     2
4⋅π⋅r
$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<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">

<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="sample24.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 f1 = (x) => 10 * x ** 6,
    f2 = (x) => x ** 2 - 4 * x + 5,
    f3 = (x) => x ** 3 - 4 * x,
    f4 = (x) => (x ** 2 - 2) * (3 * x ** 2 + 4);

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 = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f2(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f3(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t3 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f4(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    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('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d, i) =>
              i < t1 ? 'red' :
              i < t2 ? 'green' :
              i < t3 ? 'blue' : 'orange');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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', 'black');
    
    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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