2017年6月7日水曜日

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)、指数関数・対数関数の微分、問47、48、49、50、51、52、53.を取り組んでみる。


  1. 1 x ·( 1 )= 1 x

    1. 3 e 3x

    2. 2x e x 2

    3. 4 4x = 1 x

    4. 2 x log2

    5. 2x e x x 2 e x

    6. 2x ( x 2 +1 )log10

    7. e x sinx+ e x cosx

    8. 1 x ( logx ) 2

    9. 1logx x 2

    10. e x logx+ e x 1 x

    11. e x cos e x

    12. 3 e sin3x cos3x

    13. sinx cosx =tanx

    14. 1 x sin( logx )

    1. y'= e x y1=x

    2. y'= 1 x y0=x1

    3. y'=2 e 2x y e 2 =2 e 2 ( x1 )

    4. y'= 1 x y1= 1 e ( xe )

    5. y'= e x +x e x y=x

    6. y'= 1logx x 2 y=x1

  2. y'= e x y e a = e a ( xa ) 0 e a = e a ( 0a ) e a =a e a a=1 ye=e( x1 ) y=ex

  3. y'= 1 x yloga= 1 a ( xa ) 0loga= 1 a ( 0a ) loga=1 loga=1 a=e yloge= 1 e ( xe ) y1= x e 1 y= x e

  4. f( x )= a x f'( x )= a x loga lim h0 a 0+h a 0 h = lim h0 a 0+h a 0 h =f'( 0 ) =loga

  5. h= 1 n nh0 lim n ( 1+ 1 n ) n = lim h0 ( 1+h ) h =e

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, Derivative, Limit, S

print('47.')
x = symbols('x', negative=True)

f = log(-x)
d = Derivative(f, x, 1)
f1 = d.doit()
pprint(d)
pprint(f1)

print('52.')
h = symbols('h')
a = symbols('a', positive=True)
f = (a ** h - 1) / h
l = Limit(f, h, 0)
pprint(l)
pprint(l.doit())

print('53.')
n = symbols('n', positive=True, integer=True)
l = Limit((1 + 1 / n) ** n, n, S.Infinity)
pprint(l)
pprint(l.doit())

入出力結果(Terminal, IPython)

$ ./sample47.py
47.
d          
──(log(-x))
dx         
1
─
x
52.
      h    
     a  - 1
 lim ──────
h─→0⁺  h   
log(a)
53.
           n
    ⎛    1⎞ 
lim ⎜1 + ─⎟ 
n─→∞⎝    n⎠ 
ℯ
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="3">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="100">

<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="sample47.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_y1, input_y2,
             input_n0],
    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 = (n) => (1 + 1 / n) ** n;

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseInt(input_n0.value);
        
    
    if (r === 0 || y1 > y2) {
        return;
    }

    let points = [];
    
    for (let x = 1; x <= n0; x += 1) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [[0, Math.E, n0, Math.E]];

    let xscale = d3.scaleLinear()
        .domain([0, n0])
        .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([[0, 0, n0, 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', '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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