直接编写 Manim CE 0.19 场景代码生成动画的进阶…

# 梯度下降动画（Manim CE 0.19）

一、核心思路

• 函数：f(x) = x⁴ − 3x + 1，f'(x) = 4x³ − 3
• 起点x=4，学习率需较小（建议0.01~0.02）避免振荡，迭代20步
• 每步画出：当前点、该点切线、x值文本标签

二、完整代码

```python

from manim import *

import numpy as np

def f(x): return x**4 - 3*x + 1

def df(x): return 4*x**3 - 3

class GradientDescent(Scene):

def construct(self):

axes = Axes(

x_range=[-2.2, 4.5, 1],

y_range=[-4, 200, 50],

x_length=11, y_length=6,

axis_config={"include_numbers": True},

).to_edge(DOWN)

curve = axes.plot(lambda x: f(x), color=BLUE)

labels = axes.get_axis_labels(x_label="x", y_label="f(x)")

self.play(Create(axes), Create(curve), Write(labels))

x = 4.0

lr = 0.01

path = VMobject(color=YELLOW)

for i in range(20):

dot = Dot(axes.c2p(x, f(x)), color=RED, radius=0.07)

x_val = axes.c2p(x, 0)[0]

# 切线（局部短段）

slope = df(x)

p = axes.c2p(x, f(x))

dx, dy = 0.8, 0.8 * slope * axes.y_length / axes.x_length

tangent = Line(p - np.array([dx, dy, 0]),

p + np.array([dx, dy, 0]),

color=GREEN, stroke_width=2)

tag = MathTex(f"x_{{{i}}}={x:.3f}", font_size=24).next_to(dot, UR, buff=0.1)

if i == 0:

self.play(Create(dot), Create(tangent), Write(tag))

else:

self.play(Create(dot), Create(tangent), Write(tag), run_time=0.4)

# 更新x并留轨迹

x_new = x - lr * df(x)

seg = Line(axes.c2p(x,