泰勒级数逼近

1# -*- coding: utf-8 -*-2from manim import *3import numpy as np4 5class TaylorSeriesApproximation(Scene):6    def construct(self):7        # 设置默认字体8        Text.set_default(font="SimSun")9        10        # 标题11        title = VGroup(12            Text("泰勒级数逼近", font="SimSun"),13            MathTex("e^x")14        ).arrange(RIGHT, buff=0.3).scale(0.6)15        title.to_edge(UP, buff=0.3)16        self.play(Write(title))17        self.wait(1)18 19        # 创建坐标系20        axes = Axes(21            x_range=[-4, 4, 1],22            y_range=[-1, 4, 1],23            axis_config={"include_tip": True},24            x_length=10,25            y_length=626        ).scale(0.8)27        axes.shift(DOWN * 1.2)28        29        # 添加坐标轴标签30        x_label = Text("x").scale(0.6)31        y_label = Text("y").scale(0.6)32        x_label.next_to(axes.x_axis, RIGHT)33        y_label.next_to(axes.y_axis, UP)34        35        axes_group = VGroup(axes, x_label, y_label)36        self.play(Create(axes_group))37 38        # 添加局部放大坐标系39        magnified_axes = Axes(40            x_range=[-0.1, 0.1, 0.02],41            y_range=[0.9, 1.1, 0.05],42            axis_config={43                "include_tip": True,44                "numbers_to_exclude": ["all"],45                "tick_size": 0.05,46            },47            x_length=3,48            y_length=3,49            tips=True50        ).scale(0.8)51        magnified_axes.to_corner(UR, buff=0.5)52        53        # 添加放大区域标题54        magnified_title = Text("零点附近放大图", font="SimSun").scale(0.4)55        magnified_title.next_to(magnified_axes, UP, buff=0.2)56        57        # 添加放大区域的边框58        magnified_box = SurroundingRectangle(59            magnified_axes,60            buff=0.2,61            color=YELLOW,62            stroke_width=263        )64        65        self.play(66            Create(magnified_axes),67            Write(magnified_title),68            Create(magnified_box)69        )70 71        # 在主坐标系中标记放大区域72        zoom_region = Rectangle(73            width=0.3,74            height=0.3,75            color=YELLOW,76            stroke_width=277        ).move_to(axes.c2p(0, 1))78        79        # 连接线80        connection_lines = VGroup(81            Line(82                zoom_region.get_corner(UR),83                magnified_box.get_corner(DL),84                color=YELLOW,85                stroke_width=186            ),87            Line(88                zoom_region.get_corner(DR),89                magnified_box.get_corner(DL),90                color=YELLOW,91                stroke_width=192            )93        )94        95        self.play(96            Create(zoom_region),97            Create(connection_lines)98        )99 100        # 绘制原始的 e^x 函数101        exp_curve = axes.plot(102            lambda x: np.exp(x),103            color=BLUE,104            x_range=[-4, 4, 0.01]105        )106        magnified_exp = magnified_axes.plot(107            lambda x: np.exp(x),108            color=BLUE,109            x_range=[-0.1, 0.1, 0.01]110        )111        112        exp_label = Text("e^x", color=BLUE).scale(0.5)113        exp_label.next_to(exp_curve.point_from_proportion(0.8), UP)114        115        self.play(116            Create(exp_curve),117            Create(magnified_exp),118            Write(exp_label)119        )120 121        # 展示泰勒级数公式122        taylor_formula = MathTex(123            r"e^x = ", r"1", r" + ", r"x", r" + ", r"\frac{x^2}{2!}", r" + ", 124            r"\frac{x^3}{3!}", r" + ", r"\frac{x^4}{4!}", r" + \cdots"125        ).scale(0.6)126        taylor_formula.to_edge(UP, buff=1.0)127        self.play(Write(taylor_formula))128 129        # 显示每一项的函数形式130        terms_explanation = VGroup(131            MathTex(r"S_0(x) = 1", color=RED),132            MathTex(r"S_1(x) = 1 + x", color=GREEN),133            MathTex(r"S_2(x) = 1 + x + \frac{x^2}{2!}", color=YELLOW),134            MathTex(r"S_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}", color=PURPLE),135            MathTex(r"S_4(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}", color=ORANGE)136        ).arrange(DOWN, aligned_edge=LEFT, buff=0.2).scale(0.5)137        138        # 将函数形式放在右边放大图下方139        terms_explanation.next_to(magnified_box, DOWN, buff=0.5)140        141        # 定义颜色142        colors = [RED, GREEN, YELLOW, PURPLE, ORANGE]143        144        # 初始化累加曲线变量145        current_sum = None146        magnified_sum = None147        current_boxes = None148        149        # 定义每一步要高亮的项的范围150        term_ranges = [151            [1],           # f₀: 只高亮 1152            [1, 2, 3],     # f₁: 高亮 1 + x153            [1, 2, 3, 4, 5],  # f₂: 高亮 1 + x + x²/2!154            [1, 2, 3, 4, 5, 6, 7],  # f₃: 高亮到 x³/3!155            [1, 2, 3, 4, 5, 6, 7, 8, 9]  # f₄: 高亮到 x⁴/4!156        ]157        158        for i in range(5):159            # 创建当前步骤的高亮框160            box = SurroundingRectangle(161                VGroup(*[taylor_formula[j] for j in term_ranges[i]]),162                color=colors[i],163                buff=0.1,164                stroke_width=1.5,165                fill_opacity=0166            )167            168            # 显示当前项的函数形式和高亮框169            if current_boxes:170                self.play(171                    Write(terms_explanation[i]),172                    ReplacementTransform(current_boxes, box)  # 替换高亮框173                )174            else:175                self.play(176                    Write(terms_explanation[i]),177                    Create(box)178                )179            180            current_boxes = box181            182            # 修改累加函数，n=0时只显示常数项1183            def get_current_approximation(x, current_i):184                if current_i == 0:185                    return np.ones_like(x)  # 返回常数1186                return sum(x**n / np.math.factorial(n) for n in range(current_i + 1))187            188            # 在主坐标系中绘制累加曲线189            new_sum = axes.plot(190                lambda x: get_current_approximation(x, i),191                color=colors[i],192                x_range=[-4, 4, 0.01]193            )194            195            # 在放大坐标系中绘制累加曲线196            new_magnified_sum = magnified_axes.plot(197                lambda x: get_current_approximation(x, i),198                color=colors[i],199                x_range=[-0.1, 0.1, 0.01]200            )201            202            # 显示累加曲线203            if current_sum:204                self.play(205                    ReplacementTransform(current_sum, new_sum),206                    ReplacementTransform(magnified_sum, new_magnified_sum)207                )208            else:209                self.play(210                    Create(new_sum),211                    Create(new_magnified_sum)212                )213            214            current_sum = new_sum215            magnified_sum = new_magnified_sum216            self.wait(1)217 218        # 添加误差说明219        error_text = Text(220            "随着项数增加，近似误差逐渐减小",221            color=YELLOW222        ).scale(0.4)223        error_text.to_edge(DOWN, buff=0.5)224        self.play(Write(error_text))225        self.wait(2)226 227def main():228    scene = TaylorSeriesApproximation()229    scene.render()230 231if __name__ == "__main__":232    main()