三个微分中值定理的关系

1from manim import *2import numpy as np3 4class ParametricCauchyTheorem(Scene):5    def construct(self):6        # 创建标题7        title = Text("柯西中值定理的参数曲线解释", font="SimSun").to_edge(UP)8        self.play(Write(title))9        10        # 参数方程说明11        parametric_eq = MathTex(12            r"\gamma(t) = \begin{cases} x = g(t) \\ y = f(t) \end{cases}, t \in [a,b]"13        ).next_to(title, DOWN)14        self.play(Write(parametric_eq))15        16        # 参数方程解释17        param_explanation = Text(18            "用参数方程表示曲线，柯西中值定理有更直观的几何意义", 19            font="SimSun"20        ).scale(0.6).next_to(parametric_eq, DOWN)21        self.play(Write(param_explanation))22        self.wait()23        24        # 添加淡出标题的动画25        self.play(FadeOut(title), FadeOut(parametric_eq))26        27        # 创建参数曲线的坐标系28        axes = Axes(29            x_range=[-1, 3, 0.5],30            y_range=[-1, 3, 0.5],31            axis_config={"color": BLUE},32        )33        axes_labels = axes.get_axis_labels(x_label="g(t)", y_label="f(t)")34        35        # 定义函数 f(t) 和 g(t)36        def f(t):37            return t**3 - 3*t**2 + 2*t38            39        def g(t):40            return t41        42        # 计算导数43        def df(t):44            return 3*t**2 - 6*t + 245            46        def dg(t):47            return 148        49        # 区间端点50        a, b = 0, 251        52        # 满足柯西定理的c值53        c_cauchy = self.find_cauchy_c(f, df, g, dg, a, b)54        55        # 在坐标系创建前进行淡出56        self.play(57            FadeOut(param_explanation),58            Create(axes), 59            Write(axes_labels)60        )61        62        # 绘制参数曲线 (g(t), f(t))63        parametric_curve = ParametricFunction(64            lambda t: axes.c2p(g(t), f(t)),65            t_range=[0, 2.5],66            color=BLUE67        )68        69        curve_eq = MathTex(70            r"\gamma(t) = \begin{cases} x = g(t) \\ y = f(t) \end{cases}"71        ).scale(0.6).to_corner(UL)72        self.play(Create(parametric_curve), Write(curve_eq))73        74        # 参数化图示 - 添加箭头从t到曲线点75        t_line = NumberLine(76            x_range=[0, 2, 0.5],77            length=3,78            include_numbers=True,79        ).to_edge(DOWN, buff=1)80        t_label = Text("参数 t", font="SimSun").scale(0.5).next_to(t_line, LEFT)81        82        self.play(Create(t_line), Write(t_label))83        84        # 展示参数t如何映射到曲线上的点85        t_points = [0, 0.5, 1, 1.5, 2]86        animations = []87        88        for t in t_points:89            dot_t = Dot(t_line.n2p(t), color=YELLOW, radius=0.05)90            dot_curve = Dot(axes.c2p(g(t), f(t)), color=YELLOW, radius=0.05)91            arrow = Arrow(dot_t.get_center(), dot_curve.get_center(), buff=0.1, color=YELLOW_A)92            93            animations.append(AnimationGroup(94                Create(dot_t),95                Create(arrow),96                Create(dot_curve),97                lag_ratio=0.398            ))99        100        for anim in animations:101            self.play(anim, run_time=0.8)102        103        self.wait()104        self.play(*[FadeOut(obj) for obj in animations])105        106        # 标记曲线上的端点107        point_a = Dot(axes.c2p(g(a), f(a)), color=RED)108        point_b = Dot(axes.c2p(g(b), f(b)), color=RED)109        110        label_a = MathTex(r"\gamma(a)").next_to(point_a, DOWN)111        label_b = MathTex(r"\gamma(b)").next_to(point_b, DOWN)112        113        self.play(Create(point_a), Create(point_b))114        self.play(Write(label_a), Write(label_b))115        116        # 绘制从A到B的割线（参数视角下的弦）117        secant_line = Line(118            axes.c2p(g(a), f(a)),119            axes.c2p(g(b), f(b)),120            color=RED121        )122        self.play(Create(secant_line))123        124        # 创建沿曲线移动的点125        moving_point = Dot(axes.c2p(g(a), f(a)), color=YELLOW)126        moving_label = MathTex(r"\gamma(t)").next_to(moving_point, UP, buff=0.1)127        128        # 切线 - 参数曲线视角下的切向量129        tangent = always_redraw(130            lambda: Line(131                axes.c2p(132                    g(moving_point.get_center()[0]/axes.c2p(1,0)[0]) - 0.5, 133                    f(moving_point.get_center()[0]/axes.c2p(1,0)[0]) - 134                    0.5 * (df(moving_point.get_center()[0]/axes.c2p(1,0)[0])/dg(moving_point.get_center()[0]/axes.c2p(1,0)[0]))135                ),136                axes.c2p(137                    g(moving_point.get_center()[0]/axes.c2p(1,0)[0]) + 0.5, 138                    f(moving_point.get_center()[0]/axes.c2p(1,0)[0]) + 139                    0.5 * (df(moving_point.get_center()[0]/axes.c2p(1,0)[0])/dg(moving_point.get_center()[0]/axes.c2p(1,0)[0]))140                ),141                color=GREEN142            )143        )144        145        tangent_label = VGroup(146            Text("切线:", font="SimSun").scale(0.5),147            MathTex(r"\gamma'(t)").scale(0.6)148        ).arrange(RIGHT, buff=0.2).set_color(GREEN)149        tangent_label.to_corner(UR)150        151        # 在移动点的动画部分，添加参数值显示152        t_tracker = DecimalNumber(153            a,154            num_decimal_places=2,155            include_sign=False,156        ).scale(0.6).next_to(moving_point, RIGHT, buff=0.1)157 158        t_label = MathTex(r"t = ").scale(0.6).next_to(t_tracker, LEFT, buff=0.05)159 160        t_group = VGroup(t_label, t_tracker)161 162        self.play(163            Create(moving_point),164            Write(moving_label),165            Write(t_group),166            Create(tangent),167            Write(tangent_label)168        )169        170        # 修改点移动动画，同时更新t值171        def update_t_value(mob):172            # 获取当前点的x坐标，对应参数t173            x = moving_point.get_center()[0]174            # 转换回参数t值175            t_value = x / axes.c2p(1,0)[0]176            # 更新数值177            mob.set_value(t_value)178 179        t_tracker.add_updater(update_t_value)180        t_group.add_updater(lambda m: m.next_to(moving_point, RIGHT, buff=0.1))181 182        # 在点移动动画结束后记得移除updater183        t_tracker.remove_updater(update_t_value)184        t_group.remove_updater(lambda m: m.next_to(moving_point, RIGHT, buff=0.1))185        186        # 沿曲线移动点187        curve_points = []188        num_points = 30189        for i in range(num_points + 1):190            t = a + i * (c_cauchy - a) / num_points191            curve_points.append(axes.c2p(g(t), f(t)))192        193        self.play(194            MoveAlongPath(moving_point, VMobject().set_points_as_corners(curve_points)),195            MaintainPositionRelativeTo(moving_label, moving_point),196            run_time=3197        )198        199        # 高亮显示c点，此时切线平行于割线200        self.play(Flash(moving_point, color=YELLOW, flash_radius=0.3))201        202        # 标记c点203        point_c = Dot(axes.c2p(g(c_cauchy), f(c_cauchy)), color=YELLOW)204        label_c = MathTex(r"\gamma(\xi)").next_to(point_c, UP)205        206        self.play(207            ReplacementTransform(moving_point, point_c),208            ReplacementTransform(moving_label, label_c),209            FadeOut(t_group)210        )211        212        # 显示切向量和弦向量以强调平行关系213        secant_slope_display = VGroup(214            Text("弦斜率:", font="SimSun").scale(0.4),215            MathTex(r"\frac{f(b) - f(a)}{g(b) - g(a)}").scale(0.5)216        ).arrange(RIGHT, buff=0.1).set_color(RED)217        secant_slope_display.next_to(secant_line.get_center(), UP, buff=0.2)218 219        tangent_slope_display = VGroup(220            Text("切线斜率:", font="SimSun").scale(0.4),221            MathTex(r"\frac{f'(\xi)}{g'(\xi)}").scale(0.5)222        ).arrange(RIGHT, buff=0.1).set_color(GREEN).next_to(point_c, UP+RIGHT, buff=0.2)223 224        self.play(Create(point_c), Write(label_c))225        self.play(Create(secant_line))226        self.play(Write(secant_slope_display), Write(tangent_slope_display))227        228        # 等显示完斜率后，再显示柯西定理内容229        self.wait(1)  # 给观众一点时间理解斜率230 231        # 显示参数形式的柯西中值定理232        theorem_text = MathTex(233            r"\exists \xi \in (a,b) : \frac{f'(\xi)}{g'(\xi)} = \frac{f(b) - f(a)}{g(b) - g(a)}"234        ).scale(0.7)235 236        # 使用参数形式重写237        parametric_theorem = MathTex(238            r"\exists \xi \in (a,b) : \gamma'(\xi) \parallel (\gamma(b) - \gamma(a))"239        ).scale(0.7)240 241        theorem_group = VGroup(theorem_text, parametric_theorem).arrange(DOWN, buff=0.3)242        theorem_group.to_edge(DOWN)243 244        self.play(Write(theorem_text))245        self.play(Write(parametric_theorem))246        247        # 显示切向量和弦向量以强调平行关系248        secant_slope_display = VGroup(249            Text("弦斜率:", font="SimSun").scale(0.4),250            MathTex(r"\frac{f(b) - f(a)}{g(b) - g(a)}").scale(0.5)251        ).arrange(RIGHT, buff=0.1).set_color(RED)252        secant_slope_display.next_to(secant_line.get_center(), UP, buff=0.2)253 254        tangent_slope_display = VGroup(255            Text("切线斜率:", font="SimSun").scale(0.4),256            MathTex(r"\frac{f'(\xi)}{g'(\xi)}").scale(0.5)257        ).arrange(RIGHT, buff=0.1).set_color(GREEN).next_to(point_c, UP+RIGHT, buff=0.2)258 259        self.play(Create(point_c), Write(label_c))260        self.play(Create(secant_line))261        self.play(Write(secant_slope_display), Write(tangent_slope_display))262        263        # 强调平行关系264        parallel_text = Text(265            "柯西中值定理：存在点ξ使得切线斜率等于弦斜率", 266            font="SimSun"267        ).scale(0.6).next_to(parametric_theorem, UP, buff=0.3)268        269        self.play(Write(parallel_text))270        271        # 高亮显示平行关系272        self.play(273            Indicate(secant_line, color=RED, scale_factor=1.1),274            Indicate(secant_slope_display, color=RED, scale_factor=1.1),275            Indicate(tangent_slope_display, color=GREEN, scale_factor=1.1),276            run_time=2277        )278        279        self.wait(2)280 281        # 在显示参数形式的柯西中值定理部分（约在第160行），添加更直观的参数表达282        # 在parametric_theorem之后添加:283        param_explanation = Text(284            "其中 γ'(t) = (g'(t), f'(t)) 是切向量", 285            font="SimSun"286        ).scale(0.6).next_to(parametric_theorem, DOWN, buff=0.2)287        self.play(Write(param_explanation))288 289 290class UnifiedMeanValueTheorems(Scene):291    def construct(self):292        # 标题保持不变293        title = Text("从柯西到拉格朗日到罗尔：三个中值定理的联系", font="SimSun").scale(0.8).to_edge(UP)294        self.play(Write(title))295        296        # 添加初始说明297        intro_text = Text(298            "三个中值定理体现了数学中的一般到特殊的演化", 299            font="SimSun"300        ).scale(0.6).next_to(title, DOWN)301        self.play(Write(intro_text))302        self.wait()303        304        # 添加淡出标题的动画305        self.play(FadeOut(title), FadeOut(intro_text))306        307        # 创建坐标系308        axes = Axes(309            x_range=[-0.5, 3.5, 0.5],310            y_range=[-2, 4, 0.5],311            axis_config={"color": BLUE},312        )313        axes_labels = axes.get_axis_labels(x_label="g(t)", y_label="f(t)")  # 改为参数形式的标记314        315        # 选用一个函数能同时展示三个定理316        def f(t):317            return t**3 - 3*t**2 + 2*t318        319        def df(t):320            return 3*t**2 - 6*t + 2321            322        def g(t):323            return t  # 改回与ParametricCauchyTheorem类相同的定义324        325        def dg(t):326            return 1327            328        # 绘制坐标系和曲线329        self.play(330            Create(axes), 331            Write(axes_labels)332        )333        334        # 绘制参数曲线 (g(t), f(t))335        parametric_curve = ParametricFunction(336            lambda t: axes.c2p(g(t), f(t)),337            t_range=[0, 3],338            color=BLUE339        )340        341        curve_label = MathTex(342            r"\gamma(t) = \begin{cases} x = g(t) \\ y = f(t) \end{cases}"343        ).scale(0.8).to_edge(UP, buff=0.5)344        self.play(Create(parametric_curve), Write(curve_label))345        346        # 1. 柯西中值定理 - 参数曲线视角347        cauchy_title = Text("柯西中值定理（参数曲线视角）", font="SimSun").scale(0.7)348        cauchy_title.to_edge(DOWN, buff=0.1)349        350        a, b = 0.5, 2.5  # 选择参数端点351        c_cauchy = self.find_cauchy_c(f, df, g, dg, a, b)352        353        # 标记端点354        point_a = Dot(axes.c2p(g(a), f(a)), color=RED)355        point_b = Dot(axes.c2p(g(b), f(b)), color=RED)356        label_a = MathTex(r"\gamma(a)").next_to(point_a, DOWN)357        label_b = MathTex(r"\gamma(b)").next_to(point_b, DOWN)358        359        self.play(Write(cauchy_title))360        self.play(Create(point_a), Create(point_b), Write(label_a), Write(label_b))361        362        # 绘制从A到B的割线（参数视角下的弦）363        secant_line = Line(364            axes.c2p(g(a), f(a)),365            axes.c2p(g(b), f(b)),366            color=RED367        )368        self.play(Create(secant_line))369        370        # 找到满足平行条件的c点371        c_point = Dot(axes.c2p(g(c_cauchy), f(c_cauchy)), color=YELLOW)372        c_label = MathTex(r"\gamma(\xi)").next_to(c_point, UP)373 374        # 切线 - 参数曲线在c点的导数375        c_tangent_line = Line(376            axes.c2p(g(c_cauchy) - 0.5, f(c_cauchy) - 0.5 * (df(c_cauchy)/dg(c_cauchy))),377            axes.c2p(g(c_cauchy) + 0.5, f(c_cauchy) + 0.5 * (df(c_cauchy)/dg(c_cauchy))),378            color=GREEN379        )380 381        secant_slope_display = VGroup(382            Text("弦斜率:", font="SimSun").scale(0.4),383            MathTex(r"\frac{f(b) - f(a)}{g(b) - g(a)}").scale(0.5)384        ).arrange(RIGHT, buff=0.1).set_color(RED)385        secant_slope_display.next_to(secant_line.get_center(), UP, buff=0.2)386 387        tangent_slope_display = VGroup(388            Text("切线斜率:", font="SimSun").scale(0.4),389            MathTex(r"\frac{f'(\xi)}{g'(\xi)}").scale(0.5)390        ).arrange(RIGHT, buff=0.1).set_color(GREEN).next_to(c_point, UP+RIGHT, buff=0.2)391 392        self.play(Create(c_point), Write(c_label))393        self.play(Create(c_tangent_line))394        self.play(Write(secant_slope_display), Write(tangent_slope_display))395        396        # 先等待观察斜率397        self.wait(1)  398 399        # 然后再显示柯西定理公式400        cauchy_formula = MathTex(401            r"\exists \xi \in (a,b) : \frac{f'(\xi)}{g'(\xi)} = \frac{f(b) - f(a)}{g(b) - g(a)}"402        ).scale(0.7).next_to(cauchy_title, UP)403 404        # 参数化形式的解释405        cauchy_explanation = Text(406            "柯西定理表明：存在一点ξ，使得该点切线方向与弦平行", 407            font="SimSun"408        ).scale(0.5).next_to(cauchy_formula, UP)409 410        self.play(Write(cauchy_formula))411        self.play(Write(cauchy_explanation))412        413        # 突出显示切向量与弦向量平行414        self.play(415            Indicate(c_tangent_line, scale_factor=1.2, color=GREEN),416            Indicate(secant_slope_display, scale_factor=1.1, color=RED),417            Indicate(tangent_slope_display, scale_factor=1.1, color=GREEN),418            Indicate(secant_line, color=RED, scale_factor=1.1),419            run_time=2420        )421        422        self.wait(2)423        424        # 2. 拉格朗日中值定理 (g(t)=t时的特例)425        self.play(426            FadeOut(cauchy_explanation),427            FadeOut(cauchy_formula),428            FadeOut(curve_label),429            FadeOut(secant_slope_display),430            FadeOut(tangent_slope_display),431            FadeOut(c_tangent_line)432        )433        434        lagrange_title = Text("拉格朗日中值定理 (g(t)=t时的特例)", font="SimSun").scale(0.7)435        436        self.play(ReplacementTransform(cauchy_title, lagrange_title))437        438        # 为拉格朗日定理选择新的端点439        a_lag, b_lag = 0.8, 2.3  # 新的非坐标轴上点440        441        # 更新端点 - 确保在曲线上442        self.play(FadeOut(point_a), FadeOut(point_b), FadeOut(label_a), FadeOut(label_b))443 444        point_a = Dot(axes.c2p(a_lag, f(a_lag)), color=RED)  # 重新创建点445        point_b = Dot(axes.c2p(b_lag, f(b_lag)), color=RED)446        label_a = MathTex("a").next_to(point_a, DOWN)447        label_b = MathTex("b").next_to(point_b, DOWN)448 449        self.play(450            Create(point_a),451            Create(point_b),452            Write(label_a),453            Write(label_b)454        )455        456        # 更新割线457        new_secant = Line(458            axes.c2p(a_lag, f(a_lag)),459            axes.c2p(b_lag, f(b_lag)),460            color=RED461        )462        self.play(ReplacementTransform(secant_line, new_secant))463        secant_line = new_secant464        465        # 计算拉格朗日定理的c值466        c_lag = self.find_lagrange_c(f, df, a_lag, b_lag)467 468        # 先显示拉格朗日定理的公式469        lag_formula = MathTex(470            r"\exists \xi \in (a,b) : f'(\xi) = \frac{f(b) - f(a)}{b - a}"471        ).scale(0.7).next_to(lagrange_title, UP, buff=0.2)472 473        self.play(Write(lag_formula))474 475        # 更新c点位置和标签476        new_c_label = MathTex(r"\xi").next_to(axes.c2p(c_lag, f(c_lag)), UP, buff=0.1)477 478        self.play(479            c_point.animate.move_to(axes.c2p(c_lag, f(c_lag))),480            FadeOut(c_label),481            FadeIn(new_c_label)482        )483        c_label = new_c_label  # 更新标签引用484 485        # 添加c点处的切线486        lag_tangent_line = Line(487            axes.c2p(c_lag - 0.5, f(c_lag) - 0.5 * df(c_lag)),488            axes.c2p(c_lag + 0.5, f(c_lag) + 0.5 * df(c_lag)),489            color=GREEN490        )491        self.play(Create(lag_tangent_line))492 493        # 强调切线斜率等于割线斜率494        self.play(495            Indicate(lag_tangent_line, scale_factor=1.2, color=GREEN),496            Indicate(secant_line, scale_factor=1.2, color=RED),497            run_time=2498        )499 500        self.wait(1)501 502        # 3. 罗尔中值定理 (当f(a)=f(b)时的特例)503        self.play(504            FadeOut(lag_formula),505            FadeOut(c_label),506            FadeOut(c_point),507            FadeOut(lag_tangent_line),508            FadeOut(point_a),509            FadeOut(point_b),510            FadeOut(label_a),511            FadeOut(label_b),512            FadeOut(secant_line)513        )514        515        rolle_title = Text("罗尔中值定理 (f(a)=f(b)时的特例)", font="SimSun").scale(0.7)516        517        self.play(ReplacementTransform(lagrange_title, rolle_title))518        519        # 为罗尔定理选择函数值相等的两点520        # 对于f(t) = t^3 - 3t^2 + 2t = t(t-1)(t-2)，函数在t=0,1,2处为0521        rolle_a, rolle_b = 1, 2  # 选择t=1和t=2，使得f(1)=f(2)=0522 523        # 标记罗尔定理的端点 - 两个函数值相同的点524        rolle_point_a = Dot(axes.c2p(rolle_a, f(rolle_a)), color=PURPLE)525        rolle_point_b = Dot(axes.c2p(rolle_b, f(rolle_b)), color=PURPLE)526        rolle_label_a = MathTex("a").next_to(rolle_point_a, DOWN)527        rolle_label_b = MathTex("b").next_to(rolle_point_b, DOWN)528        529        self.play(530            Create(rolle_point_a),531            Create(rolle_point_b),532            Write(rolle_label_a),533            Write(rolle_label_b)534        )535 536        # 绘制罗尔区间的割线（水平线，因为f(a)=f(b)=0）537        rolle_secant = Line(538            axes.c2p(rolle_a, f(rolle_a)),539            axes.c2p(rolle_b, f(rolle_b)),540            color=PURPLE541        )542        self.play(Create(rolle_secant))543 544        # 修改f(a)=f(b)的显示方式（大约在第523行）545        # 将:546        equal_values = MathTex(r"f(a) = f(b) = 0").scale(0.7).next_to(rolle_title, UP)547 548        # 修改为:549        equal_values = MathTex(r"f(a) = f(b)").scale(0.7).next_to(rolle_title, UP)550        self.play(Write(equal_values))551 552        # 找到f'(c)=0的点553        # 解方程f'(c) = 0，即 3c^2 - 6c + 2 = 0554        # 这个方程有两个解：c ≈ 0.423 和 c ≈ 1.577555        # 对于区间[1,2]，我们需要使用1.577556        c_rolle = (6 + np.sqrt(12))/6  # 约等于1.577，在[1,2]区间内的极值点557 558        # 标记c点559        c_rolle_point = Dot(axes.c2p(c_rolle, f(c_rolle)), color=YELLOW)560        c_rolle_label = MathTex(r"\xi").next_to(c_rolle_point, DOWN)561 562        # 水平切线563        horizontal_tangent = Line(564            axes.c2p(c_rolle - 0.8, f(c_rolle)),565            axes.c2p(c_rolle + 0.8, f(c_rolle)),566            color=GREEN567        )568 569        self.play(570            Create(c_rolle_point),571            Write(c_rolle_label),572            Create(horizontal_tangent)573        )574 575        # 添加等待时间，让观众有时间观察水平切线576        self.wait(2)577 578        # 显示罗尔定理公式579        rolle_formula = MathTex(580            r"\exists \xi \in (a,b) : f'(\xi) = 0"581        ).scale(0.7).next_to(equal_values, UP, buff=0.2)582        self.play(Write(rolle_formula))583 584        # 再次等待，强调水平切线与公式的关系585        self.play(586            Indicate(horizontal_tangent, scale_factor=1.2, color=GREEN),587            Indicate(c_rolle_point, scale_factor=1.2, color=YELLOW),588            run_time=1.5589        )590        self.wait(1)591 592        # 清除所有内容，展示定理关系图593        self.play(594            *[FadeOut(mob) for mob in self.mobjects if mob != title]595        )596        597        # 创建定理关系图598        relation_diagram = VGroup()599        600        # 创建三个框和文本601        cauchy_box = Rectangle(height=1, width=4, color=BLUE)602        cauchy_text = Text("柯西中值定理", font="SimSun").scale(0.6)603        cauchy_group = VGroup(cauchy_box, cauchy_text).arrange(ORIGIN)604        605        lagrange_box = Rectangle(height=1, width=4, color=GREEN)606        lagrange_text = Text("拉格朗日中值定理", font="SimSun").scale(0.6)607        lagrange_group = VGroup(lagrange_box, lagrange_text).arrange(ORIGIN)608        609        rolle_box = Rectangle(height=1, width=4, color=RED)610        rolle_text = Text("罗尔中值定理", font="SimSun").scale(0.6)611        rolle_group = VGroup(rolle_box, rolle_text).arrange(ORIGIN)612        613        # 排列定理框614        relation_diagram.add(cauchy_group, lagrange_group, rolle_group)615        relation_diagram.arrange(DOWN, buff=1.5)616        617        # 添加箭头和说明618        arrow1 = Arrow(cauchy_group.get_bottom(), lagrange_group.get_top(), color=BLUE)619        arrow_text1 = Text("g(t)=t, g'(t)=1", font="SimSun").scale(0.4).next_to(arrow1, RIGHT)620        621        arrow2 = Arrow(lagrange_group.get_bottom(), rolle_group.get_top(), color=GREEN)622        arrow_text2 = Text("f(a)=f(b)", font="SimSun").scale(0.4).next_to(arrow2, RIGHT)623        624        # 创建公式组625        formulas = VGroup(626            MathTex(r"\frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(a)}").scale(0.5).next_to(cauchy_group, LEFT, buff=0.5),627            MathTex(r"f'(\xi) = \frac{f(b)-f(a)}{b-a}").scale(0.5).next_to(lagrange_group, LEFT, buff=0.5),628            MathTex(r"f'(\xi) = 0").scale(0.5).next_to(rolle_group, LEFT, buff=0.5)629        )630        631        # 组合所有元素632        final_diagram = VGroup(633            relation_diagram, 634            arrow1, arrow_text1, 635            arrow2, arrow_text2,636            formulas637        )638        639        final_diagram.move_to(ORIGIN)640        641        # 显示关系图642        self.play(Create(relation_diagram))643        self.play(Create(arrow1), Write(arrow_text1), Create(arrow2), Write(arrow_text2))644        self.play(Write(formulas))645        646        # 总结文字647        summary = Text(648            "三个中值定理展示了数学中特例与一般化的美妙关系", 649            font="SimSun"650        ).scale(0.6).to_edge(DOWN)651        652        self.play(Write(summary))653        654        self.wait(3)655 656    def find_cauchy_c(self, f, df, g, dg, a, b):657        """计算满足柯西中值定理的c值"""658        # 计算割线斜率659        secant_slope = (f(b) - f(a)) / (g(b) - g(a))660        661        # 定义方程 f'(\xi)/g'(\xi) = secant_slope662        def equation(x):663            return df(x) / dg(x) - secant_slope664        665        # 使用二分法求解666        left, right = a, b667        epsilon = 1e-10668        while right - left > epsilon:669            mid = (left + right) / 2670            if equation(mid) * equation(left) <= 0:671                right = mid672            else:673                left = mid674        675        return (left + right) / 2676 677    def find_lagrange_c(self, f, df, a, b):678        """计算满足拉格朗日中值定理的c值"""679        # 计算割线斜率680        secant_slope = (f(b) - f(a)) / (b - a)681        682        # 定义方程 f'(\xi) = secant_slope683        def equation(x):684            return df(x) - secant_slope685        686        # 使用二分法求解687        left, right = a, b688        epsilon = 1e-10689        while right - left > epsilon:690            mid = (left + right) / 2691            if equation(mid) * equation(left) <= 0:692                right = mid693            else:694                left = mid695        696        return (left + right) / 2