概説微分積分の問題3.16から3.18までの解答。
ソフト使って解く問題だったのでSymPyを使ってみた。機能を探すのにかなり手間取ったけど、数式を代数的に扱えて面白い。∞はooだったりする。SymPyとnumpyとscipyとの関連性はまだよくわからず。
ただしPlotはpygletがまともに動かなくて駄目だった。この辺はMatplotlibに任せるべきだったかもしれないけど、力尽きた。
import sympy as sym from sympy.abc import x, y from sympy.geometry import Point, Line from sympy.utilities.autowrap import autowrap print "[3.16 (1)]" f1 = (x ** 2 - 1) * (x + 2) * (x - 3) * sym.sin(2 * sym.pi * x) df1 = sym.diff(f1, x) print df1.subs(x, 2.) print Line(Point(2., f1.subs(x, 2)), slope=df1.subs(x, 2.)).equation(), "= 0" print "[3.16 (2)]" f2 = (sym.sqrt(1 + x ** 2) + sym.sqrt(1 - x ** 2)) / \ (sym.sqrt(1 + x ** 2) - sym.sqrt(1 - x ** 2)) df2 = sym.diff(f2, x) print df2 print "[3.17 (1)]" f3 = sym.sin(x) - 3 * sym.sin(2 * x) + 2 * sym.cos(3 * x) # sym.Plot(f3, [x, 0, 2 * sym.pi, 100]) df3 = sym.diff(f3, x) df3_callable = autowrap(df3) print sym.mpmath.findroot(df3_callable, (0, 1.5), solver='anderson') print sym.mpmath.findroot(df3_callable, (1.5, 3), solver='anderson') print sym.mpmath.findroot(df3_callable, (3, 4), solver='anderson') print sym.mpmath.findroot(df3_callable, (4, 4.7), solver='anderson') print sym.mpmath.findroot(df3_callable, (4.7, 5), solver='anderson') print sym.mpmath.findroot(df3_callable, (5, 6.5), solver='anderson') print "[3.17 (2)]" f4 = 3 * sym.exp(- x ** 2) \ - 2 * sym.exp(- 2 * (x - 1) ** 2) - 2 * sym.exp(- 2 * (x + 1) ** 2) \ + sym.exp(- 3 * (x - 2) ** 2) + sym.exp(- 3 * (x + 2) ** 2) # sym.Plot(f4, [x, -3, 3]) df4 = sym.diff(f4, x) df4_callable = autowrap(df4) print sym.mpmath.findroot(df4_callable, (-3, -2), solver='anderson') print sym.mpmath.findroot(df4_callable, (-2, -1), solver='anderson') print sym.mpmath.findroot(df4_callable, (-1, 1), solver='anderson') print sym.mpmath.findroot(df4_callable, (1, 2), solver='anderson') print sym.mpmath.findroot(df4_callable, (2, 3), solver='anderson') print "[3.18 (1)]" print sym.limit(x ** (1. / x), x, sym.oo) print "[3.18 (2)]" print sym.limit((x - sym.sin(x)) / (sym.atan(x) - x) , x, 0)
[3.16 (1)] -24.0*pi 24.0*pi*x + 1.0*y - 48.0*pi = 0 [3.16 (2)] (-x/(x**2 + 1)**(1/2) - x/(-x**2 + 1)**(1/2))*((-x**2 + 1)**(1/2) + (x**2 + 1)**(1/2))/(-(-x**2 + 1)**(1/2) + (x**2 + 1)**(1/2))**2 + (x/(x**2 + 1)**(1/2) - x/(-x**2 + 1)**(1/2))/(-(-x**2 + 1)**(1/2) + (x**2 + 1)**(1/2)) [3.17 (1)] 0.921193032136374 2.17913969064355 3.51092140472367 4.645447617455 4.71238898038469 6.02205784978526 [3.17 (2)] -2.10620408177109 -1.15895914580887 0.0 1.15895914580887 2.10620408177109 [3.18 (1)] 1 [3.18 (2)] -1/2
