数学 - Python - 代数学 - 1次関数、2次関数 - 2次関数の最大と最小 - 定義域

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第6章(1次方程式、2次方程式)、5(2次関数の最大と最小)の問15の解答を求めてみる。

1. $\begin{array}{l} - 5 \leq x \leq 0 \\ y = x^{2} + 4 x \\ = {(x + 2)}^{2} - 4 \end{array}$
  
  x が-5 のとき最大値
  
  $25 - 20 = 5$
  
  x が-2のとき最小値-4。
2. $\begin{array}{l} y = {(x - 2)}^{2} + 1 \\ x = - 2, y = 17 \\ x = 1, y = 2 \end{array}$
3. $\begin{array}{l} y = - 2 (x^{2} - 3 x) - \frac{9}{2} \\ = - 2 {(x - \frac{3}{2})}^{2} \\ x = \frac{3}{2}, y = 0 \\ x = - 1, y = - \frac{25}{2} \end{array}$
4. $\begin{array}{l} y = - \frac{1}{2} (x^{2} + \frac{5}{2} x) + 3 \\ = - \frac{1}{2} {(x + \frac{5}{4})}^{2} + \frac{25}{32} + 3 \\ = - \frac{1}{2} {(x + \frac{5}{4})}^{2} + \frac{121}{32} \\ x = - \frac{5}{4}, y = \frac{121}{32} \\ x = 2, y = - 2 - \frac{5}{2} + 3 = - \frac{3}{2} \end{array}$

#!/usr/bin/env python3 from sympy import symbols, plot, Rational print('15.') x = symbols('x') ys = [x ** 2 + 4 * x, x ** 2 - 4 * x + 5, -2 * x ** 2 + 6 * x - Rational(9, 2), -x ** 2 / 2 - 5 * x / 4 + 3] maxs = [5, 17, 0, Rational(121, 32)] mins = [-4, 2, -Rational(25, 2), -Rational(3, 2)] intervals = [(x, a, b) for a, b in [(-5, 0), (-2, 1), (-1, Rational(3, 2)), (-Rational(5, 2), 2)]] for i, (y, mi, ma, interval) in enumerate(zip(ys, mins, maxs, intervals), 1): p = plot(y, mi, ma, interval, ylim=(min(mins), max(maxs)), legend=True, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color p.save(f'sample15_{i}.png') p.show()

% ./sample15.py 15. /opt/local/Library/Frameworks/Python.framework/Versions/3.8/lib/python3.8/site-packages/sympy/plotting/plot.py:1065: MatplotlibDeprecationWarning: The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later. self.ax[i].spines['left'].set_smart_bounds(True) /opt/local/Library/Frameworks/Python.framework/Versions/3.8/lib/python3.8/site-packages/sympy/plotting/plot.py:1066: MatplotlibDeprecationWarning: The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later. self.ax[i].spines['bottom'].set_smart_bounds(False) %

Kamimura's blog

2020年6月2日火曜日

数学 - Python - 代数学 - 1次関数、2次関数 - 2次関数の最大と最小 - 定義域

0 コメント:

コメントを投稿