Метод касательных. Метод Ньютона.

[@OnTheWayToTheDirection]

Code Snippet: from manim import * from math import log2 class MovingAndZoomingCamera(MovingCameraScene): def construct(self): self.camera.frame.save_state() # 1. Построение графика и осей: axes = Axes( x_range=[-11, 11, 1], y_range=[-11, 11, 1], x_length = 7, y_length = 7, axis_config={"color": BLUE}, ).move_to(LEFT*3) labels = axes.get_axis_labels(x_label="x", y_label="y") graph = axes.plot(lambda x: (x/2)**3, x_range=[-4, 4], color=GREEN) self.play(Create(axes, run_time = 4)) self.play(Write(labels)) self.play(Create(graph)) # показываем что мы ищем: reshenie_dot = Dot(axes.coords_to_point(0,0), color = PINK).scale(0.8) podpis = MathTex("x^*", color = PINK).move_to(reshenie_dot.get_center() + LEFT/2.5 + UP/2.5).scale(0.8) self.play( Create(reshenie_dot), Write(podpis), run_time = 2 ) self.wait() # 2. Построение крайних значений функции: # -------------------------------------------------------- Rpoint = Dot().move_to(axes.coords_to_point(4, 8)) self.play(Create(Rpoint)) # * self.wait() R_point_proecsia = Dot().move_to(axes.coords_to_point(4, 0)) # точка проекц на ось Х. R_point_proecsia.scale(0.5) R_Dline = DashedLine( # пунктирная линия Rpoint.get_center(), R_point_proecsia.get_center() ) # -------------------------------------------------------- Lpoint = Dot().move_to(axes.coords_to_point(-4, -8)) self.play(Create(Lpoint)) # * self.wait() L_point_proecsia = Dot().move_to(axes.coords_to_point(-4, 0)) # точка проекц на ось Х. L_point_proecsia.scale(0.5) L_Dline = DashedLine( # пунктирная линия Lpoint.get_center(), L_point_proecsia.get_center() ) self.play(Create(L_Dline), Create(L_point_proecsia), Create(R_Dline), Create(R_point_proecsia)) self.wait() # -------------------------------------------------------- # 3. Подписываем крайние точки: L_label = MathTex("x_{a}").move_to(L_point_proecsia.get_center() + UP/3).scale(0.6) R_label = MathTex("x_{b}").move_to(R_point_proecsia.get_center() + DOWN/3).scale(0.6) self.play(Create(L_label), Create(R_label)) self.wait() # 4. Проводим касаетльную #- R_label_new = MathTex("x_{0}", color = RED).move_to(R_point_proecsia.get_center() + DOWN/3).scale(0.6) self.play(Transform(R_label, R_label_new), Flash(R_label_new, flash_radius=0.25), run_time = 3) #- target_point = Dot().move_to(axes.coords_to_point(2.667, 0)).scale(0.5) casatelnaya = Line( axes.coords_to_point(4, 8), target_point.get_center(), color = BLUE ) x1_text = MathTex("x_{1}", color = RED).move_to(target_point.get_center() + DOWN/3).scale(0.5) # self.play(self.camera.frame.animate.move_to(casatelnaya).set(width=10)) self.play(Create(casatelnaya), Create(target_point)) self.play(Create(x1_text)) self.wait() # 5. Начинаем писать формулу: angle = Angle( axes.x_axis, # направление 1 Line(axes.coords_to_point(2.667, 0), axes.coords_to_point(4, 8)), # направление 2 radius=0.2, color = ORANGE ) value = MathTex(r"\alpha", color = ORANGE).move_to(angle.get_center() + UP/5 + RIGHT/7).scale(0.7) self.play(Create(angle), Create(value)) L1 = MathTex( "x_1 - x_0", " = ", "-", r"AB", r"\cdot", r"\cos(", r"\alpha", ")" ).move_to([4,3,0]) L1[0].color = RED L1[2].color = RED L1[3].color = BLUE L1[6].color = ORANGE # 6. Обозначения для касательно A = Text("A", color=BLUE).move_to(x1_text.get_center()).scale(0.6) B = Text("B", color=BLUE).move_to(Rpoint.get_center()+UP/3+RIGHT/6).scale(0.6) self.play(Write(L1)) self.play( ReplacementTransform(x1_text, A), # не используй Transform!!!! Flash(x1_text, flash_radius=0.25), run_time = 3 ) self.play(Create(B)) self.wait() self.play(Wiggle( VGroup( A, B, casatelnaya, L1[3] ) ), run_time = 2) self.wait()

[@OnTheWayToTheDirection]

# 7. отмечаю f(x_0) на оси Y: R_point_proecsia_y = Dot().move_to(axes.coords_to_point(0, 8)).scale(0.5) F_ot_X_dashLIne = DashedLine( # пунктирная линия Rpoint.get_center(), R_point_proecsia_y.get_center() ) self.play(Create(R_point_proecsia_y), Create(F_ot_X_dashLIne), run_time = 2) self.wait() # 8. Подписываем значение. f_ot_x_Label = MathTex("f(x_0)", color = GREEN).move_to(R_point_proecsia_y.get_center()+LEFT/1.8).scale(0.7) self.play(Create(f_ot_x_Label)) self.wait() # 9. Меняем формулу L2 = MathTex( "x_1 - x_0", " = ", "-", r"\frac{f(x_0)}{\sin(\alpha)}", r"\cdot", r"\cos(", r"\alpha", ")" ).move_to(L1.get_center() + DOWN).scale(0.7) L2[0].color = RED L2[3].color = BLUE L2[6].color = ORANGE self.play(Create(L2), run_time = 3) self.wait() L3 = MathTex( "x_1 - x_0", " = ", "-", r"f(x_0) \cdot", r"ctg(", r"\alpha", ")" ).move_to(L2.get_center() + DOWN).scale(0.7) L3[0].color = RED L3[-2].color = ORANGE self.play(Create(L3), run_time = 3) self.wait() L4 = MathTex( "x_1 - x_0", " = ", "-", r"\frac{f(x_0)}{tg(\alpha)}", ).move_to(L3.get_center() + DOWN).scale(0.7) L4[0].color = RED L4[2].color = RED L4[3][-2:-1].color = ORANGE self.play(Create(L4), run_time = 3) self.wait() L5 = MathTex( "x_1 - x_0", " = ", "-", r"\frac{f(x_0)}{f'(x_0)}", ).move_to(L4.get_center() + DOWN).scale(0.7) L5[0].color = RED L5[2].color = RED self.play(Create(L5), run_time = 3) self.wait() L6 = MathTex( "x_1", " = ", "x_0" "-", r"\frac{f(x_0)}{f'(x_0)}", ).move_to(L5.get_center() + DOWN).scale(0.7) L6[0].color = RED L6[2].color = RED self.play(Create(L6), run_time = 3) self.wait() self.play(FadeOut(VGroup(L1,L2,L3,L4,L5))) self.play(L6.animate.move_to(L1.get_center())) self.play(L6.animate.scale(1.4)) self.wait() # 10. Подготовка к построению 2ой касательной: self.play(self.camera.frame.animate.move_to(axes.coords_to_point(2,0)).set(width=10)) self.wait() # 11. Убираем лишние обозначения: self.play(FadeOut(casatelnaya, B, angle, value), run_time = 2) x1_text = MathTex("x_{1}", color = RED).move_to(target_point.get_center() + DOWN/3).scale(0.5) self.play( ReplacementTransform(A, x1_text), # не используй Transform!!!! Flash(x1_text, flash_radius=0.25), run_time = 3 ) self.wait() # 12. Снова делаем необходимые построения: Rpoint = Dot().move_to(axes.coords_to_point(2.667, 2.45)).scale(0.5) R_point_proecsia_y = Dot().move_to(axes.coords_to_point(0, 2.45)).scale(0.5) target_point = Dot().move_to(axes.coords_to_point(1.777, 0)).scale(0.5) F_ot_X_dashLIne = DashedLine( # пунктирная линия Rpoint.get_center(), R_point_proecsia_y.get_center() ) F_to_X_dashLIne = DashedLine( # пунктирная линия Rpoint.get_center(), axes.coords_to_point(2.667, 0) ) casatelnaya = Line( Rpoint.get_center(), target_point.get_center(), color = BLUE ) f_ot_x_Label = MathTex("f(x_1)", color = GREEN).move_to(R_point_proecsia_y.get_center()+LEFT/1.8).scale(0.7) x2_text = MathTex("x_{2}", color = RED).move_to(target_point.get_center() + DOWN/3).scale(0.4) self.play( Create(f_ot_x_Label), Create(F_to_X_dashLIne), Create(F_ot_X_dashLIne), Create(casatelnaya), Create(x2_text), Create(R_point_proecsia_y), Create(target_point), Create(Rpoint), run_time = 3 ) self.wait()

[@OnTheWayToTheDirection]

# 13. Снова записываем формулу: self.play(Restore(self.camera.frame)) # перемещение камеры на исходную позицию L7 = MathTex( "x_2", " = ", "x_1" "-", r"\frac{f(x_1)}{f'(x_1)}", ).move_to(L6.get_center() + DOWN*1.3) L7[0].color = RED L7[2].color = RED self.play(Create(L7)) self.wait() # 14. Продолжаем её до n: L8 = MathTex( "..." ).move_to(L7.get_center() + DOWN) self.play(Create(L8)) self.wait() L9 = MathTex( "x_n", " = ", "x_{n-1}" "-", r"\frac{f(x_{n-1})}{f'(x_{n-1})}", ).move_to(L8.get_center() + DOWN) L9[0].color = RED L9[2].color = RED self.play(Create(L9)) self.wait(10)