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)