数学 - Python - 解析学 - 微分と基本的な関数 - 微分係数、導関数 - 変化率(運動、質点、移動距離、加速度、半球、体積、円錐形)

解析入門 原書第3版
原著

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• Pythonからはじめる数学入門(参考書籍)

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第3章(微分係数、導関数)、補充問題(変化率) 7、8、9、10、11.を取り組んでみる。

7. 
   $\begin{array}{l}s\text{'}\left(t\right)=4t-1\\ s\text{'}\text{'}\left(t\right)=4\\ 4t-1=0\\ t=\frac{1}{4}\\ 加速度4\end{array}$
8. 
   $\begin{array}{l}y={\left(x\left(t\right)\right)}^2-6x\left(t\right)\\ \frac{dy}{dt}=2x\left(t\right)\frac{dx}{dt}-6\frac{dx}{dt}\\ \frac{2x\left(t\right)\frac{dx}{dt}-6\frac{dx}{dt}}{\frac{dx}{dt}}=4\\ 2x\left(t\right)=4\\ x\left(t\right)=2\\ \left(2,-8\right)\end{array}$
9. 
   $\begin{array}{l}\pi r^2·\frac{dh}{dt}=4\\ \frac{dh}{dt}=\frac{4}{\pi r^2}\\ h^2+{\left(10-h\right)}^2+2r^2=2·{10}^2\\ h=5\\ 25+25+2r^2=200\\ r^2=75\\ \frac{dh}{dt}=\frac{4}{75\pi }\left(フィート/分\right)\end{array}$
10. 
    $\begin{array}{l}d\left(t\right)=\sqrt{{\left(100+50t\right)}^2+{\left(60t\right)}^2}\\ d\text{'}\left(t\right)=\frac{1}{2}{\left({\left(100+50t\right)}^2+{\left(60t\right)}^2\right)}^{-\frac{1}{2}}\left(2\left(100+50t\right)50+2·60t·60\right)\\ d\text{'}\left(t\right)={\left({\left(100+50t\right)}^2+{\left(60t\right)}^2\right)}^{-\frac{1}{2}}\left(\left(100+50t\right)50+60t·60\right)\\ ={10}^{-1}{\left({\left(10+5t\right)}^2+{\left(6t\right)}^2\right)}^{-\frac{1}{2}}\left(\left(100+50t\right)50+60t·60\right)\\ ={\left({\left(10+5t\right)}^2+{\left(6t\right)}^2\right)}^{-\frac{1}{2}}\left(\left(100+50t\right)5+60t·6\right)\\ d\text{'}\left(\frac{3}{2}\right)={\left({\left(10+\frac{15}{2}\right)}^2+9^2\right)}^{-\frac{1}{2}}\left(175·5+90·6\right)\\ ={\left({\left(\frac{35}{2}\right)}^2+9^2\right)}^{-\frac{1}{2}}\left(175·5+90·6\right)\\ =\frac{175·5+90·6}{\sqrt{\frac{{35}^2+9^2·2^2}{2^2}}}\\ =\frac{2\left(175·5+90·6\right)}{\sqrt{{35}^2+9^2·2^2}}\end{array}$
11. 
    $\begin{array}{l}\frac{\pi {\left(3h\left(t\right)\right)}^{2}h\left(t\right)}{3}=3t\\ h\left(t\right)={\left(\frac{t}{\pi }\right)}^{\frac{1}{3}}\\ h\text{'}\left(t\right)=\frac{1}{3}{\left(\frac{t}{\pi }\right)}^{-\frac{2}{3}}·\frac{1}{\pi }\\ {\left(\frac{t}{\pi }\right)}^{\frac{1}{3}}=4\\ t=4^3\pi \\ h\text{'}\left(4^3\pi \right)=\frac{1}{3}{\left(\frac{4^3\pi }{\pi }\right)}^{-\frac{2}{3}}\frac{1}{\pi }\\ =\frac{1}{48\pi }\left(m/分\right)\end{array}$

コード(Emacs)

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

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

print('10.')
t = symbols('t', positive=True)
l = sqrt((100 + 50 * t) ** 2 + (60 * t) ** 2)
pprint(l)

l1 = Derivative(l, t)
pprint(l1)

lt = l1.subs({t: 1.5})
pprint(lt)

lt0 = lt.doit()
pprint(lt0)

pprint(lt0 - 2 * (175 * 5 + 90 * 6) /
       sqrt(35 ** 2 + 9 ** 2 * 2 ** 2) < 0.00001)

入出力結果(Terminal, IPython)

$ ./sample7.py
10.
   _________________________
  ╱       2               2 
╲╱  3600⋅t  + (50⋅t + 100)  
  ⎛   _________________________⎞
d ⎜  ╱       2               2 ⎟
──⎝╲╱  3600⋅t  + (50⋅t + 100)  ⎠
dt                              
⎛  ⎛   _________________________⎞⎞│     
⎜d ⎜  ╱       2               2 ⎟⎟│     
⎜──⎝╲╱  3600⋅t  + (50⋅t + 100)  ⎠⎟│     
⎝dt                              ⎠│t=1.5
71.9052708731530
True
$